Encyclopedia of solid earth geophysics [2 ed.] 9783030586300, 3030586308

Table of contents :
Contents
About the Editor
Contributors
Preface to the Second Edition
Preface to the First Edition
Acknowledgments
A
Absolute Age Determinations: Radiometric
Definition
Introduction
Radioactivity and the Systematics of Its Use as a Chronometer
The Application of Radiometric Dating
Summary and Conclusions
Cross-References
Bibliography
Anthropogenic Seismicity Related to Exploitation of Georesources
Definition
Overview
Mechanisms of Interaction Between Technological Activity and the Rock Mass
Induced, Triggered, or Natural
Specific Features
Collaborative Undertakings for Research into Anthropogenic Seismicity
Summary
Bibliography
Archaeomagnetism
Synonyms
Definition
Basic Features
Archaeological Site Sampling
Directional Analyses
Paleo-intensity Analyses
Magnetic Dating Based on the Direction of Remanence
Relative Dating
Chronological Dating
Magnetic Polarity and Polarity Excursion Dating
Magnetic Dating Based on the Intensity of Remanence
Errors
General Errors
Errors Specific to Directional Dating
Errors Specific to Intensity Dating
Overall Error Assessment
Specialized Applications
Murals and Plaster
Object Reconstruction
Archaeological Sediments
Magnetic ``Sourcing´´
Summary
Cross-References
Bibliography
Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India
Definition and Introduction
Triggered vis-à-vis Induced Earthquakes
Artificial Water Reservoir-Triggered Earthquakes
Reservoir-Triggered Seismicity (RTS): Global Distribution
Important Factors for RTS
Common Characteristics of RTS Sequences
Mechanism of Triggered Earthquakes
Koyna, India
How Long Triggered Earthquakes Will Continue at Koyna?
Short-Term Earthquake Forecast at Koyna
Near-Source Studies at Koyna
Pilot Borehole Location
Summary
Cross-References
Bibliography
B
Biogeophysics
Definition
Introduction
Geophysical Detection of Cells and Biofilms
Geophysical Detection of Metabolic By-Products and Mineral Weathering
Geophysical Detection of Microbially Mediated Redox Processes
Geophysical Detection of Microbe-Mineral Transformations
Summary
Cross-References
Bibliography
Body Waves
Synonyms
Definition
Types of Body Waves
Wave Propagation
Waves on a Seismogram
Refraction
Reflection
Velocity
Velocity of P- and S-waves
P-waves
S-waves
Attenuation
Summary: Results of Regional and Global Studies
Body-Wave Modeling
Cross-References
Bibliography
Borehole Seismic Networks and Arrays
Synonyms
Definition
Introduction
Early Developments
Current Instrumentation and Installation
Noise-Suppression and Signal-Detection Advantages
Networks Versus Arrays
Illustrative Examples
Conclusions
Cross-References
Bibliography
C
Characteristic Earthquakes and Seismic Gaps
Definition
Introduction
History
Assumptions
Small Characteristic Earthquakes
Modified Seismic Gap Hypothesis
Challenges to the Seismic Gap Model
Conclusions
Cross-References
Bibliography
Continental Crustal Structure
Definition
Introduction
Methods
Cratons
Orogens
Extensional Areas
Constraints on Crustal Petrology
Summary
Cross-References
Acknowledgments
Bibliography
Continental Drift
Definition
Introduction
Alfred Wegener
Conclusions
Cross-References
Bibliography
Continental Rifts
Synonyms
Definition
Introduction
Rift, Graben, and Taphrogen
General Properties of Rifts
Structure of Rifts
Sedimentation in Rifts
Magmatism in Rifts
Metamorphism in Rifts
Kinds of Rifts
Geometric Classification of Rifts (See Figs. 7 and 8)
Kinematic Classification of Rifts (See Fig. 8)
Dynamic (Genetic) Classification of Rifts (See Fig. 6)
Concluding Note
Bibliography
Core Dynamo
Synonyms
Definition
Introduction
Dynamo Model Concept and Equations
Model Setup
Dynamo Equations
Parameters
Energetics of the Geodynamo
Flow Pattern and Dynamo Mechanism
Flow Outside Tangent Cylinder
Flow Inside Tangent Cylinder
Field Properties
Field Strength Inside the Core
Scaling Laws
Magnetic Field Geometry
Secular Variation
Time Scales
Advection vs. Diffusion of Field
Magnetic Field Reversals
Summary
Cross-References
Bibliography
Core-Mantle Coupling
Definition, Scope, and Aims
The Four Coupling Processes
The Viscous Torque
The Topographic Torque
The Gravitational Torque
The Magnetic Torque
Synthesis
Cross-References
Acknowledgments
Bibliography
Crustal Reflectivity (Oceanic) and Magma Chamber
Synonyms
Definition
Introduction
Methods
Crustal Structure at Fast Spreading Centers
Plot Triplication
Axial Magma Chamber (Melt Lens)
Lower Crust
Oceanic Moho
Crustal Thickness
Upper Mantle
Faults in the Oceanic Crust and Upper Mantle
Summary
Cross-References
Bibliography
Curie Temperature
Synonyms
Definition
Magnetic Minerals and Curie Temperature
Geomagnetic Implications
Cross-References
Bibliography
D
Deep Scientific Drilling
Definition
Introduction
Goals of Scientific Drilling
Drilling, Sampling, and Monitoring Techniques
Dynamics of Continental Crust
Volcanic Systems
Mid-ocean Ridge Magmatism
Active Volcano Drilling and Subduction-Induced Volcanism
Mantle Plume Volcanism
Impact Craters
Active Fault Zones
Nojima Fault Projects, Japan
SAFOD, California
Taiwan Chelungpu Drilling Project, Taiwan
Wenchuan Earthquake Fault Scientific Drilling Project (WFSD), Sichuan, China
Geophysical Observatory in the Eastern Sea of Marmara, GONAF
Deep Scientific Drilling to Study Reservoir-Triggered Earthquakes in Koyna, India
Subduction Zone Fault Drilling at Sea
Japan Trench Fast Drilling Project (JFAST)
Unconventional Resources: Geoenergy
The Iceland Deep Drilling Project, IDDP
Summary
Cross-References
Bibliography
Deep Seismic Reflection and Refraction Profiling
Synonyms
Definition
Deep Seismic Reflection and Refraction Profiling
Notational Notes
Introduction
Basic Principles
Data Acquisition
Field Layouts
Special Processing and Interpretation Tools
Main Results
Some Recent Developments
Research Problems
Summary
Cross-References
Bibliography
Differential Rotation of the Earth´s Inner Core
Definition
Background
Seismological Observations
Concluding Remarks
Cross-References
Bibliography
E
Earth Rotation
Definition
Introduction
Mathematical Formulation of Earth Rotation
Definition and Observation of Earth Orientation Parameters
IERS International Earth Rotation and Reference Systems Service
Summary
Cross-References
Bibliography
Earth Tides
Definition
The Tidal Force
The Earth´s Response
Summary
Cross-References
Bibliography
Earth, Density Distribution
Definition
The Mean Density
The Moment of Inertia Constraint
Self-Compression
Free Oscillation Data
A Density Gradient Anomaly
Extrapolation to Zero Pressure and Inferences About Composition
Lateral Density Variations
Summary
Cross-References
Bibliography
Earth´s Structure, Core
Definition
Overall Structure
Radial and Lateral Variations
Irregular Surface
Mushy Zone at the Top
Temporal Changes
Possible Physical Mechanisms
Cross-References
Bibliography
Earth´s Structure, Global
Definition
Introduction
1D Structure of the Earth
Characterizing the Interior of the Earth with Earthquake Waves
From 1D Models to 3D Tomographic Models
Mantle Convection Jigsaw
Seismic 3D Tomography: The Earth Scanner
An Anisotropic and Anelastic Earth
Outlook: New Challenges
Cross-References
Bibliography
Further Reading
Earth´s Structure, Lower Mantle
Synonyms
Definition
Introduction
Tools for Studying the Lower Mantle
Three-Dimensional Structure and Dynamics
Future Progress
Summary
Cross-References
Bibliography
Earth´s Structure, Upper Mantle
Definition
Tomographic Techniques
Global Models
Regional Models
Outlook
Cross-References
Bibliography
Earthquake Lights
Synonyms
Definition
Classification of Earthquake Lights
Examples of EQL
Example of Type 3 EQL
Source of EQL and Associated Electromagnetic (EM) Phenomena
Conclusion
Cross-References
Bibliography
Earthquake Precursors and Prediction
Definitions and Introduction
History
Nonmechanical Short-Term Precursors
Difficulties Inherent to Short-Term Precursors
Macro-anomalies
Geochemical/Hydrological Precursors
Satellite IR Images as Possible Surface Temperature
Electric, Magnetic, and EM Precursors
Telluric Current Precursors: The VAN Method
Ultralow Frequency (ULF) Precursors
Higher-Frequency EM Emission
Lithosphere-Atmosphere-Ionosphere Coupling
Possible Mechanism of Pre-seismic EM Emissions
Critical Phenomena
Summary and Future Outlook
Cross-References
Bibliography
Earthquake Prediction, M8 Algorithm
Definition
Introduction
The M8 Algorithm
The MSc Algorithm
Targeting Mega-earthquakes
Conclusion
Cross-References
Bibliography
Earthquake Rupture: The Inverse Problem
Definition
Introduction
Definition of Some Common Terms Used in Inverse Theory
The Method
Constraints from Laboratory Experiments
Discussion of Results of Inversion
Super-Shear Wave Rupture Speeds
Earthquake Rupture Complexity
Large Earthquakes on Transform Faults
Summary
Cross-References
Bibliography
Earthquake Sounds
Definition
Introduction
Records of Sounds
How Are Sounds Produced?
Applications of Sounds
Infrasound
Sonification, Education, and Art
Summary
Cross-References
Acknowledgments
Bibliography
Earthquake Source Theory
Definition
Introduction
Point Source Models
Finite Source Models
Kinematic Models
Dynamical Fault Models
Future Work
Bibliography
Earthquake, Aftershocks
Definition
Introduction
Scaling Laws for Aftershocks
Causes of Aftershocks
Aftershocks and Earthquake Hazard
Summary
Cross-References
Bibliography
Earthquake, Archaeoseismology
Synonyms
Definition
Introduction
Archaeoseismic Observations
Methods
Summary
Cross-References
Bibliography
Earthquake, Focal Mechanism
Definition
Representation by a Double-Couple
Focal Mechanisms from P-Wave First Motions
Stress Release
Fundamental Results
Modern Focal Solutions: Inversion of the Moment Tensor
Evaluation Tools for Double-Couple Solutions
Non-double-Couple Solutions
Conclusion
Cross-References
Bibliography
Earthquake, Foreshocks
Synonyms
Definition
Introduction
Characteristics of Foreshocks
Causes of Foreshocks
Can Foreshocks Be Used to Predict Earthquakes?
Summary
Cross-References
Bibliography
Earthquake, Magnitude
Synonyms
Definition
Original Definition of Earthquake Magnitude
Local Magnitude Scales
Teleseismic Magnitude Scales
Surface-Wave Magnitudes
Body-Wave Magnitudes
Relationships Between Magnitude Scales and Released Seismic Energy Es
Magnitude ``Saturation´´
Non-Saturating Magnitude Scales
Relationship Between Mw, Me, and Classical Magnitudes
Summary and Conclusions
Cross-References
Bibliography
Earthquakes and Crustal Deformation
Definition
Types of Earthquakes and Related Phenomena Defined
Elastic Rebound Theory
Pre-seismic Crustal Deformation
Co-seismic Crustal Deformation
Post-seismic Crustal Deformation
Observational Advances and New Discoveries
Perspectives
Cross-References
Bibliography
Earthquakes in the Himalaya
Definition
Introduction
Geodynamic Models of Himalaya-Tibetan Orogenic System
Crustal Deformation and Strain Buildup Along Himalayan Arc
Mid-Crustal Seismicity and Associated Seismic Swarm Along Main Himalayan Thrust
Factors Influencing Himalayan Earthquakes
Role of Subducted Ridge
Stress Triggering Process
Non-tectonic Deformation
Human-Induced Deformation
Future Directions: Key Questions to Be Answered
Cross-References
Bibliography
Earthquakes on Oceanic Transform Faults, Unexpectedly Large
Definition
Introduction
Rupture on Conjugate Faults in an Oceanic Setting
Rupture on a Fault Normal to the Preexisting Zones of Weakness
Simultaneous Earthquake Rupture on Conjugate Faults in an Oceanic Setting
A Great Earthquake on a Fossil Fracture Zone
Summary
Cross-References
Bibliography
Earthquakes, Did You Feel It?
Definition and Introduction
Historical Context
The DYFI System
Citizen Science Seismology
Outlook
Cross-References
Bibliography
Earthquakes, Early and Strong Motion Warning
Synopsis
Introduction
Uses of Early Warning
Approaches to Early Warning
Implementations of Early Warning
Summary
Cross-References
Bibliography
Earthquakes, Energy
Definition and Calculation
Summary and Outlook
Cross-References
Bibliography
Earthquakes, Intensity
Synonyms
Definition
Macroseismic Scales and Intensity Assignment
Nature of Sensors
History of Intensity Scales
Conversion of Intensity Scales
Macroseismic Surveys
Intensity Attenuation with Distance
Conversion of Intensities to Magnitudes
Relation of Intensities to Recorded Strong Ground Motion
Conclusions
Cross-References
Bibliography
Earthquakes, Intraplate
Synonyms
Definition
Mechanism of Intraplate Earthquakes
Summary and Conclusions
Cross-References
Bibliography
Earthquakes, Location Techniques
Synonyms
Definition
Introduction
Early History
Single-Event Location
Location Uncertainty
Multiple-Event Location
Waveform Cross-Correlation
Relative Location
Time Reversal and Migration Location
Summary
Cross-References
Bibliography
Earthquakes, PAGER
Definition and Introduction
The PAGER Process
The PAGER Earthquake Impact Scale
Ongoing PAGER Developments
Summary
Cross-References
Bibliography
Earthquakes, ShakeCast
Definition
Background
ShakeCast System Overview
Outlook
Cross-References
Bibliography
Earthquakes, ShakeMap
Definition and Introduction
Historical Context
Related Systems and Uses
Summary
Cross-References
Bibliography
Earthquakes, Strong-Ground Motion
Definition
Introduction
Description of Strong-Ground Motion
Earthquake Source Effects on Strong-Ground Motion: Near-Field Effects
Wave-Propagation Effects
Strong-Ground Motion Modeling
Strong-Motion Simulation: An Example
Conclusions
Cross-References
Bibliography
Earthquakes, Volcanogenic
Definition
Volcano Seismology
Introduction
Volcano-Tectonic Earthquakes
Rock-Fall and Lahar Events
Low-Frequency Earthquakes
Volcanic Tremor
Very-Long-Period Earthquakes
Volcano Seismology in a Wider Volcanological Context
Estimation of Magma Ascent Rate
Combining Seismicity with Infrasonic Signals
Detection of Lahars and Pyroclastic Flows
Detecting Stress Changes in Volcanic Settings
Summary and Conclusions
Cross-References
Bibliography
Electrical Properties of Rocks
Definition
Introduction
Definition of Electrical Conductivity (Resistivity)
Electrical Conduction
Electrical Conductivity of Liquid-Bearing Rocks
Electrical Conductivity of Crust
Electrical Conductivity of the Earth´s Mantle
Origin of High Conductive Anomalies in the Mantle
Oceanic Asthenosphere
Mantle Transition Zone
The Base of the Lower Mantle
Conclusions
Cross-References
Bibliography
Electrical Resistivity Surveys and Data Interpretation
Synonyms
Definition
Introduction
Traditional Profiling and Sounding Surveys
Two-Dimensional Resistivity Imaging Surveys
Three-Dimensional Resistivity Imaging Surveys
4-D Time-Lapse Surveys
Summary
Cross-References
Acknowledgments
Bibliography
Electromagnetic Methods, Imaging Magma Bodies
Definition: Essential Concepts
Observations
Interpretation
Conclusions
Cross-References
Bibliography
Electromagnetic Pulsations and Magnetic Storms
Definition
Electromagnetic Pulsations
Classification of Magnetic Pulsations
Generation Mechanisms of Magnetic Pulsations
Substorms and Magnetic Storms
Relationship Between Magnetic Pulsations and Substorms and Magnetic Storms
Magnetic Storms and Society
Future Outlook
Cross-References
Acknowledgments
Bibliography
Electronic Geophysical Year
Definition
Introduction and Concept
History
Scope
Operations
Achievements
Summary
Cross-References
Bibliography
Energy Budget of the Earth
Definition
Total Heat Loss of the Earth
Continental Heat Flow
Oceanic Heat Flow
Young Sea Floor
Old Sea Floor
Bathymetry
Age Distribution of the Sea Floor
Hot Spots
Energy Budget of the Mantle
Radiogenic Heat Production
Heat Flux From the Core
Secular Cooling
Present and Long Term Cooling Rates
Perspectives
Summary
Cross-References
Bibliography
Energy Partitioning of Seismic Waves
Synonyms
Definition
Summary
Acknowledgments
Bibliography
Equatorial Electrojet
Definition
Cause
Characteristics of the EEJ
Uses of EEJ Observations
Cross-References
Bibliography
F
Fault Zone Guided Waves
Synonyms
Definition
A Short and Recent History of Fault Zone Guided Wave Seismology and Controversies
Types of Fault Zone Guided Waves and Their Particle Motions
The Effects of Fault Geometry, Velocity Structure, Anisotropy, and Heterogeneity
Conclusions
Cross-References
Bibliography
Fiber Optic Distributed Strain Sensing for Seismic Applications
Definition
Introduction
Optical Fibers
Cable Coupling
Determination of Strain Changes Along Optical Fiber
DAS Versus Conventional Seismic Sensing
Localization of the Sensor Position
Selected Case Studies
Summary and Conclusions
Acknowledgments
Bibliography
Fiber-Optic Sensing in Geophysics, Temperature Measurements
Definition
Introduction
Optical Fibers
Sensor Configurations, Sensing Methods
Point Sensors, Sensor Arrays
Distributed Sensing
Interrogation Techniques
DTS Based on Raman Scattering
Implementation
Temperature Accuracy and Resolution
Hydrogen Darkening
Sensor Cables, Deployment Methods
Permanent Installation
Temporary/Retrievable Installation
Applications, Methods of Evaluation
Passive Monitoring
Formation Temperatures, Heat Flow Studies
Flow Profiling
Well Cementing and Well Treatments
High-Temperature Applications
Active Methods
Fluid Logging, Thermal Tracer/Dilution Techniques
Heat-Pulse Method
Other Applications in Related Disciplines
Summary
Cross-References
Bibliography
Floating Seismographs (MERMAIDS)
Definition
History
Technical Description
Signal Discrimination
A Typical Mission
Other Sensors
Multi-MERMAID
Recent Results
Summary
Cross-References
Bibliography
Fractal Scaling of Earthquakes
Synonyms
Definition
Introduction
Scaling of Earthquake Sources
Scaling of Distribution in Slip Roughness
Unified Scaling Law
Statistical Treatment of Fractal Heterogeneities in Earth
Seismic Attenuation in Fractal Media
Summary and Outlook
Cross-References
Acknowledgments
Bibilography
Fractals and Chaos
Synonyms
Definition
Introduction
Basic Concepts
Power Law (Scaling Behavior)
Fractional Brownian Motion (fBm)
Fractional Gaussian Noise (fGn)
Fractal Signal Analysis
Analyzing Geophysical Time Series
Fractal Behavior of Various Physical Properties
Fractals in Seismology
Fractal Porous Media
Chaos
Summary
Bibliography
Free Oscillations of the Earth
Definition
Introduction
Toroidal Oscillations of a Uniform Elastic Sphere
Spheroidal Oscillations of a Uniform Elastic Sphere
Oscillations of an SNREI Earth Model
Effect of the Rotation of the Earth
Jeans´ Formula
Continuous Free Oscillations of the Earth
Conclusions
Cross-References
Acknowledgments
Bibliography
G
Geodesy, Figure of the Earth
Definition
Background
Mathematical Figures of the Earth
Reference Coordinate System
Datum
Summary
Cross-References
Bibliography
Geodesy, Ground Positioning, and Levelling
Definition
Introduction
Ground Positioning
Levelling
Summary
Cross-References
Bibliography
Geodesy, Networks, and Reference Systems
Definition
Introduction
Geodetic Reference Systems
Theory of Reference Systems
Metrology
Global Geodetic Reference Systems
International Terrestrial Reference System (ITRS)
World Geodetic System 1984 (WGS84)
Continental Reference Systems
National Geodetic Reference Systems
Historical Reference Systems
Global Geodesy
International Services
Fundamental Station for Geodesy
Instrumentation of a Fundamental Station
Summary
Cross-References
Bibliography
Geodesy, Physical
Definition
Historical Development of Geodesy
Gravity Field of the Earth
Properties of the Gravitational Potential
Disturbing Potential
Geoid and Reference Ellipsoid
Methods of In Situ Measurements
Satellite Geodesy
Determining the Geoid
Gravity Reduction
Gravity Models
Conclusions
Cross-References
Bibliography
Geodetic Pendulums, Horizontal Ultra Broad Band
Synonyms
Definition
Instrument Description
Grotta Gigante Horizontal Pendulums
Summary
Cross-References
Bibliography
Links to Geodynamic Observatories with Active Tiltmeters
Geodynamics
Definition
Cross-References
Bibliography
Geoelectromagnetism
Definitions and Scope
Introduction
Magnetovariational Sounding (MVS) and Magnetovariational Profiling (MVP)
Deep Mantle MVS Studies
Magnetovariational Profiling
Array Transfer Function
Magnetotelluric Soundings
Basic Formulas
Interpretation
Application to Geosciences and Future Directions
Cross-References
Bibliography
Geoid
Synonyms
Definition
Introduction
Geoid and Heights
Geoid Modeling from Gravity Data
Measuring Height of the Everest
Geoid Anomaly and Tectonics
GRACE and Coseismic Deformation
Tectonic Forces
Bibliography
Geoid Determination, Computational Methods
Definition
Introduction
The Remove-Compute-Restore Technique
Computation of the GM-Contributions
Computation of the Δg-Contribution
Computation of the Terrain Contributions
Evaluation of Satellite Altimetry Contributions
Summary
Cross-References
Bibliography
Geoid Determination, Theory and Principles
Definition
Basic Principles: the Gravity, Normal, and Disturbing Potentials
The Stokes and Molodensky Geodetic BVPs
Analytical Solutions to the Geodetic BVPs
The Operational Solution to the Geodetic BVPs
The Treatment of the Topography
The Oceanic Geoid and Gravity from Satellite Altimetry
Summary
Cross-References
Bibliography
Geoid Undulation, Interpretation
Synonyms
Definition
Introduction
Direct Problem
Inverse Problem
Truncated Geoid
Some Other Applications
Summary
Cross-References
Bibliography
Geomagnetic Excursions
Synonyms
Definition
Geographical and Temporal Extent
Field Configuration During Excursions
Origin
Summary
Cross-References
Bibliography
Geomagnetic Field, Global Pattern
Definition
Introduction
Observations
Spherical Harmonic Analysis
Maps of the Geomagnetic Field
Recent Changes in the Geomagnetic Field
Summary
Cross-References
Bibliography
Geomagnetic Field, IGRF
Definition
Datasets
Basic Mathematical Expression
IGRF Evolving
Summary or Conclusions
Cross-References
Bibliography
Geomagnetic Field, Measurement Techniques
Introduction
Earth´s Surface Magnetic Field Measurements
General Observatory Setting
Absolute Calibration of a Vector Magnetometers
Near-Earth Magnetic Field Measurements
Summary
Cross-References
Bibliography
Geomagnetic Field, Polarity Reversals
Introduction: The Discovery of Geomagnetic Reversals
The Geomagnetic Polarity Timescale
The Changing Frequency over Time and the Duration of Geomagnetic Reversals
The Morphology of Geomagnetic Reversals
Dynamo Mechanisms and Reversals
Outlook: The Future of Geomagnetic Reversals
Acknowledgments
Bibliography
Geomagnetic Field, Secular Variation
Definition
Cross-References
Bibliography
Geomagnetic Field, Theory
Definition
Introduction
Kinematic Dynamo Theory
Energetics of the Geodynamo
Magnetohydrodynamic Theory of the Geodynamo
Time Dependence of the Geodynamo
Outlook
Cross-References
Bibliography
Geomagnetically Induced Currents
Definition
Introduction
Space Weather Events and Geomagnetically Induced Currents
Generation of GIC Events
Future Outlook
Cross-References
Acknowledgments
Bibliography
Geophysical Well Logging
Synonyms
Definition
Introduction
Wellbore Environment
Calibration and Standardization of Logs
Benefits and Limitations of Logging
Principles
Instrumentation
Fundamentals of Log Analysis
Cross Plots
Applications of Geophysical Well Logging
In the Oil Industry
Indirect-Imaging Technologies: Acoustic and Electrical
Hydrogeologic Applications
Mineral Exploration
Civil Engineering, Rock Mechanics, and Environmental Preservation
Summary
Cross-References
Bibliography
Geothermal Heat Pumps
Definition
The Resource
Geothermal Heat Pump (GHP) Technology
Heating and Cooling with GHPs
Design, Costs
Licensing, Environmental Benefits
Applications, Development Trends
Summary
Cross-References
Bibliography
Geothermal Record of Climate Change
Definition
Introduction
The Geothermal Method of Temperature Reconstruction
Tutorial
A Global Geothermal Climate Change Database
Temperature-Depth Profiles
Repeat Temperature Logs
Temperature Reconstruction Analysis
Summary
Cross-References
Bibliography
GPS, Data Acquisition, and Analysis
Definition
Other Satellite-Based Navigation Systems
Satellite Positioning
Point Positioning
Relative Positioning
System Description
Introduction
Satellites, Signals, and System Time
GPS Ground Instrumentation
Antenna
Receiver
Measurements
Code Pseudo-range Model
Carrier Phase Model
Remarks
Single and Double Differences
Errors and Biases
Introduction
Receiver-Dependent Errors
Measurement Noise
Receiver Clock Errors
Antenna-Specific Errors
Cycle Slips
Satellite-Dependent Biases
Orbit and Satellite Clock Errors
Antenna-Specific Errors
Signal Propagation Biases
Ionospheric Refraction
Tropospheric Refraction
Summary
Positioning Modes
Point Positioning
Relative Positioning
High-Precision Positioning
Conclusion
Cross-References
Bibliography
GPS, Tectonic Geodesy
Synonyms
Definition
Introduction
History of Tectonic Geodesy
Plate Motions
Steady ``Interseismic Deformation´´
Transient Fault Movements
Coseismic Deformation
Real-Time Coseismic Deformation
Postseismic Deformation
Volcano Deformation
Glacial Isostatic Adjustment and Sea Level Studies
Summary
Cross-References
Bibliography
Gravimeters
Synonyms
Definition
Introduction
Performance Requirements
Technology
Sensor Technology
Fused Quartz
Metal
Superconducting
Inertial-Grade Accelerometers
Design Principles
Gravity Sensor Natural Period, Sensitivity, and Seismic Noise
Displacement Sensing and Feedback
Error Sources
Intrinsic Error Sources
Scale Factor Changes
Instrument Noise
Long-Term Drift
External Noise Sources
Sensor Motion and Orientation
Temperature, Pressure, and Magnetic Field
Transportation Effects
Gravimeters
Summary
Cross-References
Bibliography
Gravity Anomalies, Interpretation
Definition
Introduction
Formulation of the Problem
Gravity Anomalies: Conditions and Enhancement
Forward Method
Inverse Method
Summary
Cross-References
Bibliography
Gravity Data, Advanced Processing
Synonyms
Definition
Introduction
Spectral Modeling
Potential Field Transformations
Euler and Werner Deconvolution
Euler Deconvolution
Extended Euler Deconvolution
Tensor Deconvolution
Wavelet Analysis, with Advantages and Limitations
Introduction
The CWT and Windowed Fourier Transform
Applications
Summary
Cross-References
Bibliography
Gravity Data, Regional-Residual Separation
Definition
Introduction
Regional-Residual Separation
Analytical Methods
Possible Drawbacks in Regional Computation
New Approach Based on Finite Element Analysis
Accuracy
Paradox Basin, Utah (Frequency-Domain)
Residual Gravity Anomalies
Klamath Mountains and Cascade Range (Model-Based Space-Domain)
Recent Studies on FEM Applications
Summary
Cross-References
Bibliography
Gravity Field of the Earth
Definition and Scope
Introduction
Historical Notes
Newton´s Law of Gravitation
Spatial and Temporal Variation Above the Surface
Spherical Harmonics
Green´s Functions
The Disturbing Potential
Spectral Contributions
Low-Degree Harmonics
Low-Degree Harmonics as Density Moments
Normal Reference Field
Earth´s Internal Gravitational Field
Summary
Bibliography
Gravity Field, Temporal Variations from Space Techniques
Definition
The GRACE Mission (2002-2017)
Modeling Temporal Geoid Variations
GRACE Data Errors
Applications of GRACE
Hydrology
Land Ice Loss from GRACE
Ocean Mass Change
GIA Effect
Coseismic and Post-seismic Deformations Due to Large Earthquakes
Summary
Cross-References
Bibliography
Gravity Field, Time Variations from Surface Measurements
Synonyms
Definition
Introduction
Concepts and Applications
Tides
Sea Level Variation
Atmospheric Pressure
Polar Motion
Free Oscillations of the Earth
Glacial Isostatic Adjustment
Volcanic Mass Changes
Crustal Deformation Associated with Earthquakes
Hydrology
Oil, Gas and Geothermal Reservoir Monitoring
Summary
Cross-References
Bibliography
Gravity Measurements, Absolute
Synonyms
Definition
Introduction
History
Principles of Modern Measurements
Macroscopic Falling Test Mass
Microscopic Falling Test Masses
Instrumentation
Applications to Geophysics
Summary
Bibliography
Gravity Method, Airborne
Synonyms
Definition
Introduction
Principle Idea of Airborne Gravity
Attitude Compensation Platforms
Scalar Gravity Meters
Vector Gravity Meters
Gravity Gradient Meters
Processing, Reductions, and Corrections
Applications and Aircraft
Summary
Cross-References
Bibliography
Gravity Method, Principles
Synonyms
Definition
Introduction
Potential Function
Newton´s Law and Newtonian Potential
Anomalies
Ambiguity and Principle of Equivalence
The Inversion Problem
Units and Gravitational Constant
Summary
Cross-References
Bibliography
Gravity Method, Satellite
Definition
Introduction
Geodesy, Gravity Field, and Scientific Applications: Why?
Classical Ways of Measuring the Earth´s Gravity Field
New Satellite Missions
Overview of Satellite Dynamics Methods
The Representation of the Gravitational Potential
The Solution of the Dynamical Problem
Reference System and Time Scale
Equations of Motion
Integration of the Equations of Motion
Variational Equations and Their Integration
The Observation Equations for the Inverse Problem
General Form of the Equations
Example: Range Measurement Generic Observation Equation
The Solution of the Inverse Problem
Progress Made: A Short Summary
Conclusion and Vision of the Future of Gravity Mapping from Satellite
Cross-References
Bibliography
Gravity Method, Surface
Definition
Introduction
Measuring Equipment
Gravity Measurements
Elevation and Position Measurements
Land Gravity Survey
Preprocessing of Field Data
Drift Correction
Base Loop Correction
Marine Gravity Survey
Moving Platform Correction
Earth Tide Correction
Drift Correction
Network or Crossover Adjustments
Design of Survey
Regional Surveys for Geodynamic Studies
Hydrocarbon Exploration
Reconnaissance Gravity Survey
Detailed/Integrated Gravity Survey
Surveys for Minerals, Groundwater, and Engineering Sites
Reconnaissance Survey
Detailed/Integrated Gravity Survey
Summary
Cross-References
Bibliography
Gravity Modeling, Theory and Computation
Definition
The Gravity Modeling from a Theoretical Point of View
The Linear Inverse Gravimetric Problem
The Nonlinear Inverse Gravimetric Problem
Spectral Approaches
Summary
Cross-References
Bibliography
Gravity, Data to Anomalies
Definition
Introduction
Key Parameters
Observed Gravity
Anomalous Gravity in Geophysics
Computing Model Gravity at the Measurement Point
A Note on Units
Normal Gravity (Fig. 2b)
Free Air Correction (Fig. 2c)
Bouguer Correction (Fig. 2d)
Terrain Correction (Fig. 2e)
Anomaly Computation
Other Factors
Atmospheric Correction
Further Isolating Anomalies of Interest
Distant Relief
The Geophysical ``Indirect Effect´´
Gravity Anomalies for Geodetic Purposes
Computation Example: The Central Andes
Summary
Bibliography
Gravity, Global Models
Synonyms
Definition
Global Gravitational Models
Introduction
Local and Regional Gravimetric Models
Signal Representation and Data Characteristics
The Recent Gravity-Mapping Satellite Missions
Beyond the Sensitivity of Satellite Data
State-of-the-Art Global Gravitational Modeling
Model Evaluation and Accuracy Assessment
Summary
Cross-References
Bibliography
Gravity, Gradiometry
Definition and Scope
Introduction
Mathematical Foundations
Gravitational Gradients
Other Coordinate Systems
Invariants
Models and Modeling
Reference Field and Power Spectral Density
Measurement Error Analysis
Gradiometry
Applications
Summary
Bibliography
Great Earthquakes
Definition
Introduction
1 November 1755: The Advance of Rationalism
18 April 1906: The Archetype of Earthquakes
26 December 2004: Cataclysm
11 March 2011: Prepared or Not Prepared?
Summary
Cross-References
Acknowledgments
Bibliography
H
Heat and Ground Water Flow
Definition
Energy Equation for Ground Water Flow
Heat Advection
Hydrothermal Systems
Heat Convection
Conclusions
Cross-References
Bibliography
Heat Flow Determinations, Continental
Definition
Introduction
Temperature
Thermal Conductivity
In Situ Thermal Conductivity Measurement in Boreholes
Sampling
Summary
Cross-References
Bibliography
Heat Flow, Continental
Synonyms
Definition
Introduction
Fourier´s Law
Geothermal Gradient
Sources of Heat
Mechanisms of Heat Transport
Global Distribution of Continental Heat Flow
Paleo-Heat Flow
Summary
Cross-References
Bibliography
Heat Flow, Seafloor: Methods and Observations
Definition
History of Observations
Methods
Shallow Measurements in Marine Sediments
Deep Borehole Measurements
Estimating Heat Flux From the Depth Limit of Gas Hydrate Stability
Example Studies
Heat Flow Signals from Hydrothermal Circulation
Dependence of Heat Flux on Age and the Global Average
The Signature of Subduction
Bottom-Water Temperature Perturbations
Summary
Cross-References
Bibliography
Height Systems, Vertical Datums and Their Unification
Definition
Introduction
Height Systems
Vertical Datum Definition and Realization
Geodetic BVP Solutions for Single and Multiple Vertical Datums
Determination of Vertical Datum Offsets
Summary and Outlook
Cross-References
Bibliography
High-Frequency Seismology
Synonyms
Definition
Introduction
Basic Wave Propagation Principle
Acoustic Wave Propagation
Elastic Wave Propagation
Elastic Wave Equation
Theory of Seismic Wave Attenuation and Its Effects on Seismic Waveform
Methods of Analysis of High-frequency Seismograms
Scattering of Seismic Waves in Heterogeneous Media
Coda Wave Analysis
Seismogram Analysis of the 1999 Chamoli Earthquake
Summary and Outlook
Cross-References
Acknowledgments
Bibliography
I
Impact Craters on Earth
Definition
The Terrestrial Record
Introduction
Distribution
Morphology
Geology of Impact Structures
Geophysics of Impact Structures
Impacts and Earth History
Concluding Remarks
Cross-References
Bibliography
Instrumentation, Electrical Resistivity
Definition
Introduction
Basic Resistivity Instruments
Electrode Types
Multielectrode and Multichannel Systems
Full-Waveform Systems
Capacitively Coupled Systems
Automated Resistivity Monitoring Systems
Summary
Cross-References
Acknowledgments
Bibliography
Instrumentation, EM
Definition
Introduction
Electric Field Measurement
Magnetic Field Measurement
EM Transmitters
Data Acquisition Systems
Navigation
Summary
Cross-References
Bibliography
International Geophysical Year
Definition
Introduction and Concept
History
Scope
Operations
Achievements
Summary
Cross-References
Bibliography
International Gravity Formula
Definition
Cross-References
Bibliography
International Polar Year 2007-2008
Concept
History
Scope
Achievements
Major Advances in Polar Knowledge and Understanding
Infrastructure Legacy
A New Generation
Excite and Stimulate the Public
Challenges Ahead
Summary
Cross-References
Bibliography
International Year of Planet Earth
Definition
Background and Activities
Legacy
Perspectives
The Earth Science Matters Foundation
GEOPARKS as a Tool
Global Geoscience Initiative
Summary
Cross-References
Bibliography
Inverse Theory, Artificial Neural Networks
Definition
Introduction
Geophysical Inversion
Artificial Neural Networks (ANNs)
Geophysical Inversion Using Artificial Neural Networks
Summary
Cross-References
Bibliography
Inverse Theory, Global Optimization
Definition
Global Optimization
Introduction
Method
Background
Optimization
Simulated Annealing
Genetic Algorithm
Neighborhood Algorithm (NA)
Particle Swarm Optimization
Trans-dimensional Inversion
Applications
Summary
Cross-References
Bibliography
Inverse Theory, Linear
Introduction
Linear Inverse Problems
Generalized Inverses
Least Square Inverse
Minimum Norm Inverse
Damped Least Square Inverse
Weighted Damped Least Square Inverse
Rao-Mitra Inverse
Backus-Gilbert Method
Continuous Inverse Problem
Discrete Inverse Problem
Trade-Off Between Resolution and Error Propagation
Summary
Cross-References
Bibliography
Inverse Theory, Monte Carlo Method
Definition
Introduction
Nonlinearity and Multimodal Fitness Functions
Bayesian Inference
Markov Chain Monte Carlo
Fixed Dimension Approach
Transdimensional Approach
A Simple Example
Summary
Cross-References
Acknowledgments
Bibliography
Inverse Theory, Singular Value Decomposition
Definition
Description
Inverse of A
Rank-Deficient Matrix and Its Inverse
Ill-Condition Matrix
Sensitivity to Errors in Data
Resolution Matrices
Example
Summary
Cross-References
Acknowledgments
Bibliography
Isostasy
Synonyms
Definition
Historical Background
Concepts and Applications
Current Investigations
Current Controversies and Gaps in Knowledge
The Long-Term Strength of the Lithosphere
Time-Scales of Isostatic Adjustment
Isostasy and Landscape Evolution
Long-Wavelength Gravity and Topography Anomalies, Isostasy, and Mantle Dynamics
Conclusions
Bibliography
Isostasy, Thermal
Definition
Introduction
Thermal Isostasy
Oceanic Thermal Isostasy
Continental Thermal Isostasy
Normalizing Compositional Elevation
Continental Thermal State
Revealing Thermal Isostasy on Continents
The North American Cordillera
Australia
Thermal Isostasy on Terrestrial Planets
Summary
Cross-References
Bibliography
K
KTB Depth Laboratory: A Window into the Upper Crust
Definition
Introduction
Scientific Background
KTB Data Infrastructure
KTB Depth Laboratory Utilization
Geoscientific Experiments
Test and Calibration of New and Existing Instruments, Methods, and Installations
Commercial Utilization of the KTB Wells
Outreach and Education
Long-Term Utilization of a Deep Borehole Site
Summary
Cross-References
Bibliography
L
Legal Continental Shelf: Geology, Geophysics, and Tectonics
Synonyms
Definition
Introduction
Defining the Outer Edge of the Continental Margin and the Limits of the Continental Shelf
Prolongation of the Land Mass
Role of Geology and Tectonics
Summary
Bibliography
Lithosphere, Continental
Definition and Introduction
Mechanical Models
Thermal Models
Seismological/Compositional Models: Tectosphere
Lithosphere-Asthenosphere Boundary
Summary and Conclusions
Cross-References
Bibliography
Lithosphere, Continental: Thermal Structure
Definition
Introduction
The Thermal Boundary Layer of Mantle Convection
Thermal Structure
Heat Flux at the Base of the Continental Crust and at the Base of the Lithosphere
Heat Flux and Heat Production
The Moho Heat Flux
Lithospheric Geotherms
Thermal Transients
Secular Transients
Summary
Cross-References
Appendix: Thermal Conductivity
Lattice Conductivity
Bibliography
Lithosphere, Mechanical Properties
Synonyms
Definition
Introduction
Mechanical Properties at Different Timescales
Sources of Information on the Mechanical Properties of the Lithosphere
Observations of Flexural Behavior and Effective Long-Term Strength of the Lithosphere
Rheological Properties of Lithosphere According to Rock Mechanics Data
Elastic Properties
Brittle-Plastic Properties
Viscous-Ductile Properties
Mechanical Properties of Oceanic Lithosphere Versus Continental Lithosphere
Mechanical Properties of the Lithosphere and Four Modes of Horizontal Deformation
Uncertainties of Data on the Mechanical Properties of the Lithosphere
Mechanical Properties of the Lithosphere and Styles of Tectonic Deformation
Links Between Different Timescales: Burger´s Rheology Model
Summary
Cross-References
Bibliography
Lithosphere, Oceanic
Definition
Introduction
The Mantle Component of the Lithosphere
Models Constrained by the Cooling of the Lithosphere as It Ages
Models Constrained by Rheological Observations
Models Constrained by Seismic and Magnetotelluric Observations
Surface Waves Studies
Studies Using Seismic Body Wave Phases
Other Geophysical Studies
The Crustal Component of the Oceanic Lithosphere
Samples of Crustal Rocks
Ophiolites, Ocean Drilling, and the Ocean Crust
Seismic Studies of the Ocean Crust
The Midocean Ridges
Hydrothermal Activity
Along-Axis Variations in Ridge Processes and Slow-Spreading Ridges
Summary
Cross-References
Bibliography
Lithosphere, Oceanic: Thermal Structure
Definition
Introduction
Key Observations
Heat Flux
Seafloor Depth
Depth of the Lithosphere-Asthenosphere Boundary
Other Observational Constraints
Ways of Understanding Lithospheric Thermal Structure
Boundary-Layer Cooling
Plate Cooling Models
Influence of Plumes
Driving Forces for Lithospheric Motion
Summary
Cross-References
Bibliography
Lithospheric Magnetic Anomalies from Satellite Data
Synonyms
Definition
Introduction
The Lithospheric Field from Space
Mathematical Background
Global Models of Lithospheric Magnetic Anomalies
Summary
Cross-References
Bibliography
M
Magnetic Anisotropy
Definition
Introduction
Theoretical Principles
Magnetocrystalline Anisotropy
Shape Anisotropy
Magnetostriction: Stress Anisotropy
Exchange Anisotropy
Anisotropy of Magnetic Susceptibility (AMS)
AMS: Applications in Geology and Geophysics
Anisotropy of the Magnetic Remanence (AMR)
AARM: Anisotropy of Anhysteretic Remanent Magnetization
AIRM: Anisotropy of Isothermal Remanent Magnetization
HFAMS: High-Field Anisotropy of Magnetic Susceptibility
Summary
Cross-References
Bibliography
Magnetic Anomalies: Interpretation
Definition
Introduction
The Sources of Crustal and Lithospheric Fields
Measuring Magnetic Anomalies from Planes and Ships
Satellite Measurements
World Magnetic Anomalies: WDMAM
Qualitative Interpretations
Quantitative Geological Interpretations
Summary
Cross-References
Bibliography
Magnetic Anomaly Map, Global
Definition
Introduction
Early Developments
Aeromagnetic and Marine Compilations
Satellite Compilations
Preparation of the Global Magnetic Anomaly Map
Methods
Line Leveling of Data
Least Squares Collocation
Merging Satellite Models
Other Candidate Models for WDMAM
The Gamma Model
The Leeds Model
Earth Magnetic Anomaly Grid (EMAG2)
NGDC-720 Lithospheric Magnetic Model
Enhanced Magnetic Model
World Digital Magnetic Anomaly MAP Version 2.0 (WDMAM2.0)
Summary
Cross-References
Bibliography
Magnetic Data Enhancements and Depth Estimation
Synonyms
Definition
Introduction
Standard Magnetic Data Processing Corrections
Diurnal Correction
Network Adjustment (Leveling)
Microleveling
Elevation Correction
De-culturing of Magnetic Field Data
Gridding of Magnetic Field Data
IGRF Correction
Reduction to Pole (RTP) Transform
The Pseudo-Gravity Transform
Enhancements of Magnetic Data
Image Display of Magnetic Field Data
Vertical Continuation of the Field
Vertical Gradients of the Field
Bandpass and Matched Filters
Regional-Residual Separation
The Total Gradient (Analytic Signal)
The Tilt Filter
The Normalized Source Strength (NSS)
Multiscale Edge Detection (``Worms´´)
Mapping Extreme Field Variations
Analytic Derivation of Magnetic Source Parameters
The Werner Method
The Naudy Method
Euler Deconvolution
Targeted (``Sweet-Spot´´) Inversion
Summary
Cross-References
Bibliography
Magnetic Domains
Definition
Introduction
Classical Models of Magnetic Domain Structure
Exchange Energy
Magnetic Anisotropy Energy
Magnetostatic Energy
Energy and Width of the Domain Wall
Domain Width Versus Grain Size
Single-Domain/Two-Domain Transition Size do
Domains and Domain Walls Near Crystal Surfaces
Temperature Dependence of Domain Structure
Micromagnetic Models
Experimental Studies of Magnetic Domains
Methods of Imaging Domains and Domain Walls
Styles of Domains Observed in Magnetic Minerals of Paleomagnetic Significance
Number of Domains Versus Grain Size
Domain Wall Widths
Experimental Evidence for Local Energy Minimum (lEm) Domain States
LEM States and Thermomagnetic Treatments
Evolution of Magnetic Domain Structures at Elevated Temperatures
Magnetic Domain Structure in Magnetite at Low Temperatures
Cross-References
Bibliography
Magnetic Gradiometry
Definition
On the Conceptual Design of Gradiometers
Magnetometers
Advantages and Limitations
Areas of Application
Summary
Cross-References
Bibliography
Magnetic Gradiometry in Archaeo-geophysics
Definition
Advantages of Magnetic Gradiometers
Instruments
Procedure in the Field
Case Studies
Do Different Measurement Instruments Deliver Different Results?
Conclusion
Cross-References
Bibliography
Magnetic Methods, Airborne
Synonyms
Definition
Introduction
Making Magnetic Measurements from the Air
Equipment
The Importance of Source-Sensor Separation
Survey Design
Survey Specifications
Data Processing
Correction for Temporal Changes in the Geomagnetic Field
Removal of Geomagnetic Field
Tie-Line Leveling
Microleveling
Gridding
Discussion of Data Processing
Summary
Cross-References
Bibliography
Magnetic Methods, Principles
Definition
Introduction: Magnetic Field on Surface of Earth
Major Aspects of Probing the Geomagnetic Field
Core/Main Field and Its Changes
Developments in Observational Methods
Resulting Knowledge
External Field
Developments in Observational Methods
Resulting Knowledge
Lithospheric Field
Developments in Observational Methods
Resulting Knowledge
Induced Field
Methods to Study Subsurface Conductivity Structure
Methods to Study Geomagnetically Induced Currents (GIC) in the Earth´s Surface
Paleo Field
Developments in Observational Methods
Resulting Knowledge
Summary
Cross-References
Acknowledgments
Bibliography
Magnetic Methods, Surface
Definition
Magnetic Observatory
Measurements at Magnetic Observatories
Observations from this Data
Products and Uses of Observatory Data
Land and Marine Surveys
Principles of Survey Measurements
Land Surveys
Marine Magnetic Surveys
Instrumentation Technologies
Summary
Cross-References
Bibliography
Magnetic Modeling, Theory, and Computation
Definition
Introduction
Global Field Modeling
Spatial and Temporal Description
Inverse Problem and Implementation
Recent Developments in Magnetic Field Modeling
Regional Field Modeling
Harmonic Splines
Wavelet Analysis
Interpretation
Core Magnetic Field at Core-Mantle Boundary
Lithospheric Magnetic Field over the Arctic and African Regions
Conclusions
Acknowledgments
Bibliography
Magnetometers
Synonyms
Definition
Introduction
Scalar (Quantum) Magnetometers
Background Physics
Proton Magnetometers
Overhauser Magnetometer
Optically Pumped Magnetometers
Measurement of Frequency
Absolute Accuracy and Precision
Vector Magnetometers
Fluxgate Magnetometer
Absolute Measurement
Quantum Vector Magnetometers
Summary
Cross-References
Bibliography
Magnetotelluric Data Processing
Definition
Introduction
MT Transfer Functions
Least Squares Transfer Function Estimation
Robust Estimation
Leverage
Remote Reference
Error Estimates
Strike and Distortion Analysis
Summary
Cross-References
Bibliography
Magnetotelluric Interpretation
What Is Magnetotellurics?
Fundamental Concepts: Surface Impedance and Apparent Resistivity
Principles of Depth Sounding
Effects of Lateral Heterogeneities
Current Strategies for MT Interpretation
Case Studies
Summary
Cross-References
Bibliography
Magnetotellurics, Crustal Imaging
Definition
Costa Rican Subduction Zone
Ethiopian Rift and Afar
Himalayas
San Andreas Fault
Antarctica (Vestfold Hills and Rauer Group)
Dharwar Craton, India
Gawler Craton, Australia
Summary
Bibliography
Magnetovariation Studies
Definition
Interpretation
Summary
Cross-References
Bibliography
Mantle Convection
Synonyms
Definition
Introduction and History
Basics of Thermal or Free Convection
Rayleigh-Bénard Convection
Convective Onset and the Rayleigh Number
Thermal Boundary Layers and the Nusselt Number
Patterns of Convection: Upwellings, Downwellings, Plumes, and Slabs
Plumes and Slabs in the Mantle: Simple View
Energy Sources for Mantle Convection and the Earth´s Thermochemical History
Effects of Mantle Material Properties
Mantle Rheology
Compressibility, Melting, and Solid Phase Changes
Mantle Geochemistry and Mantle Mixing
Mantle Convection and the Generation of Plate Tectonics
The Plate Generation Problem
Summary
Bibliography
Mantle D Layer
Definition and Introduction
Seismic Velocity Models for D
Seismic Velocity Discontinuities in D
Post-perovskite in D
Large Low Shear Velocity Provinces in D
Ultra-Low Velocity Zones in D
Seismic Velocity Anisotropy in D
Summary
Cross-References
Bibliography
Mantle Plume: Spreading Ridge Interactions
Synonyms
Definition
Spreading Ridge System
Mantle Plume
Dynamics of Proximity of Spreading Ridge Activity and Hotspot Volcanism
Crustal Structure Beneath the Spreading Ridge, Volcanic Ridge and on-Spreading-Axis Hotspot Setting
Summary
Bibliography
Mantle Plumes
Definition
Hotspots
Global Hotspot Distribution and Hotspot Fixity
Evidence for Mantle Plumes
Geochemistry of Mantle Plumes
Genesis and Dynamics of Plumes and Superplumes
Plume Melting and Plume Strength
Continental Flood Basalts and Continental Breakup
LIP Magmatism and Environmental Effects
Plume-Lithosphere Interaction
Plume-Ridge Interaction
Summary and Conclusion
Future Directions
Cross-References
Bibliography
Mantle Viscosity
Definition
Introduction
Methods of Measurement
Non-Newtonian Rheology from Seismic Anisotropy
Natural Experiments
Mantle Viscosity from Glacial Isostatic Adjustment
Observational Data Sets
Results of Inversion of the Totality of the Data to Determine Mantle Viscosity Depth Dependence
Compatibility of the GIA-Based Viscosity Model with that Required by Models of the Mantle Convection Process
Summary
Cross-References
Bibliography
Mapping of Magnetic Variations from Space
Definition
Measurements from Space
POGO Satellite Series
MAGSAT Mission
Decade of Geopotential Field Research
Modeling of Satellite Measurements
The Future
Summary
Cross-References
Bibliography
Microcontinents
Definition
Introduction
Examples of Present-Day Microcontinents
The Jan Mayen Microcontinent in the North Atlantic Ocean
The Batavia and Gulden Draak Microcontinents in the Eastern Indian Ocean
Formation Mechanisms of Microcontinents
Summary and Conclusions
Cross-References
Bibliography
N
Numerical Methods for Flow in Fractured Porous Media
Definition
Introduction
Numerical Models for Fractured Porous Media
Continuum Fracture Models
Discrete Fracture Models
Discretization Schemes for DFM
An Example of Computational Workflow
Conclusion
Cross-References
Acknowledgments
Bibliography
Numerical Methods, Boundary Element
Definition
Introduction
How Does BEM Work?
What Can BEM Do?
Applications of the Boundary Element Method
Cross-References
Bibliography
Numerical Methods, Domain Decomposition
Definition
Introduction
Algorithms Without Overlap
Methods with Overlap
Application to Geophysical and Geological Problems
Cross-References
Bibliography
Numerical Methods, Finite Difference
Definition
Introduction
Theory of Wave Propagation and Fundamental Concepts
Finite-Difference Approximations
Accuracy and Numerical Dispersion
Lax-Wendroff Corrections, Optimally Accurate FD Schemes and Time Dispersion Correction
Time Stepping and Stability
Boundary Conditions
Topography and Conformal Mapping of Grids
Source Implementations
Conclusions
Cross-References
Bibliography
Numerical Methods, Finite Element
Definition
General Introduction
Finite Element Model of 2-D Poisson Equation
Finite Element Models of the 2-D Navier-Stokes Equations
Velocity-Pressure (Mixed) Model
Penalty-Finite Element Model
Summary
Cross-References
Acknowledgments
Bibliography
Numerical Methods, Multigrid
Definition
Introduction
History
Two-Grid Scheme
Multigrid
Nonlinear Multigrid
Generalizations
Beyond Partial Differential Equations
Geophysical Applications
Summary
Cross-References
Bibliography
O
Ocean Bottom Seismics
Synonyms
Introduction
Instrumentation
Ocean Bottom Seismometers
Ocean Bottom Cables
Ocean Bottom Seismic Nodes
Ocean Bottom Seismic Sources
Applications
P-Waves
P-to-S Converted Waves
Wide-Aperture (Wide-Azimuth) Studies
Passive-Source Seismology
Future Developments
Conclusions
Cross-References
Acknowledgments
Bibliography
Ocean, Spreading Center
Definition
Introduction
Morphology and Spreading Rates of Mid-Oceanic Ridge
Internal Structure of the Mid-Oceanic Ridge
Relationships Between Spreading Rate, Seismic Structure, and Ridge-Axis Morphology
Summary
Cross-References
Bibliography
Orogenic Belts
Synonyms
Partial Synonyms
Definition
Introduction
The Orogenic Architecture
Subduction-Controlled Orogens
Collision-Controlled Orogens
Species of Orogens
Orogenic Collages
Magmatism in Orogens
Subduction Magmatism
Collision Magmatism
Metamorphism in Orogens
Subduction-Related Metamorphism
Collision-Related Metamorphism
Geomorphology of Orogenic Belts
Economic Deposits in Orogenic Belts
Hydrocarbons
Coal
Metals
Building Materials
Summary
Bibliography
Further Reading
P
Paleomagnetic Field Intensity
Synonyms
Definition
Introduction
Absolute Palaeointensity Measurements
Relative Palaeointensity Measurements
Cosmogenic Isotope Palaeointensity Measurements
Geophysical Applications
Summary
Cross-References
Bibliography
Paleomagnetism, Magnetostratigraphy
Synonyms
Definition
Magnetic Stratigraphy
Introduction
Detrital Remanent Magnetization
Field and Laboratory Techniques
Calibration and Correlation
Strengths and Weaknesses of Magnetic Stratigraphy
Some Applications and Implications
Summary
Cross-References
Bibliography
Paleomagnetism, Measurement Techniques and Instrumentation
Synonyms
Definition
Introduction
Techniques of Sampling, Measurement and Analysis
Field Sampling
Measurement and Treatment
Analysis
Statistics
Determination of Palaeomagnetic Poles
Instrumentation
Spinner Magnetometers
Cryogenic Magnetometers
Alternating Field Demagnetizers
Thermal Demagnetization
A Perspective on Seventy Years of Palaeomagnetism
Summary
Cross-References
Bibliography
Paleomagnetism, Polar Wander
Definition and Introduction
Measure of True Polar Wander
Is a Reference Frame Based on Hotspots Valid?
Modeling of True Polar Wander
An Integrated Explanation for Polar Wander?
Conclusions
Cross-References
Bibliography
Paleomagnetism, Principles
Definition
Introduction
The Geocentric Axial Dipole
Rock Magnetism
Remanent Magnetizations
Statistical Testing of Paleomagnetic Data
Virtual Geomagnetic Poles and Apparent Polar Wander Paths
Magnetic Polarity Reversals
Environmental Magnetism
Summary
Cross-References
Bibliography
Paleoseismology
Synonyms
Definition
Introduction
Paleoseismic Records
Fault Identification and Slip Rate
Evidence for Paleoearthquake
Time Constraints of Paleoearthquakes
Magnitude and Frequency of Paleoearthquakes
Off-Fault Paleoseismology
Seismic Hazard Assessment Based on Paleoseismology
Issues in Paleoseismology and Perspectives
Summary
Cross-References
Bibliography
Plate Motions in Time: Inferences on Driving and Resisting Forces
Definition
Introduction
Plate Tectonics and the Rheology of the Lithosphere
Computer Models of the Faulted Lithosphere Coupled with Global Mantle Circulation Models
Recent Plate Motion Changes
Kinematics of the Southern Pacific and Atlantic, and Its Relations to Climate Variations and Topographic Growth in the Andes
Plate Motion Changes in the Indian Ocean
Conclusions
Cross-References
Bibliography
Plate Tectonics, Precambrian
Definition
Introduction
Modern-Style Plate Tectonics and Criteria for Recognizing Plate Tectonics in the Precambrian Record
Evolution of Earth´s Geodynamic Regime and the Onset of Plate Tectonics
The Hadean Eon
The Archean-Proterozoic Eons
Summary
Cross-References
Bibliography
Plate-Driving Forces
Definition
Introduction
Balance Between Buoyancy Forces and Viscous Dissipation
Plate-Driving Forces: Lithospheric Models
Plate-Driving Forces: Whole-Mantle Models
Summary
Cross-References
Bibliography
Plates and Paleoreconstructions
Definition
Earthquakes and Plates
Plate Compositions, Sizes, and Shapes
Plate Tectonics
Plate Motions and Euler´s Theorem
Sliding Motions
Divergent Motions
Convergent Motions
Reference Frames
Plates as Reference Frames
Paleomagnetism as a Reference Frame
Hot-Spots as a Reference Frame
Paleoreconstructions
Conservation of Continental Shapes
Record of Continental Separation in the Ocean-Floor
Paleozoic and Precambrian Composites
Paleo-Plate Reconstructions
Conclusions
Cross-References
Bibliography
Poroelasticity
Definition
Basic Concepts
Drained and Undrained Deformation
Linear Stress-Strain Formulation for Poroelastic Media
Gassmann´s Equation and Fluid Substitution
Fluid Substitution
Skempton Coefficient
Three-Dimensional Consolidation
Dynamic Poroelasticity and Wave Propagation in Saturated Porous Medium
Bibliography
Propagation of Elastic Waves: Fundamentals
Definition
Summary
Introduction
Newton´s Second Law
Cauchy´s Stress and Strain
Hooke´s Law
Navier´s Equation
D´Alembert Solution of Wave Equation
Fourier Transforms and Plane Waves
Green´s Function for Homogeneous Space
Green´s Function Retrieval from Correlations
Energy Densities at Given Points and Directions
The Horizontal-to-Vertical Spectral Ratio
Concluding Remarks
Cross-References
Acknowledgments
Bibliography
R
Radioactivity in Earth´s Core
Definition
Introduction
Potassium Radioactivity in the Earth´s Core
U (and th) Radioactivity in the Earth´s Core
Conclusions
Cross-References
Bibliography
Radiogenic Heat Production in the Continental Crust
Definition
Crustal Thickness and Heat Production
Crustal Differentiation
Crustal Temperature and Strength
Summary: Crustal Stabilization
Cross-References
Bibliography
Radiogenic Heat Production of Rocks
Synonyms
Definition
Radiogenic Heat Generation
Tabulated Data
Calculated Heat Generation Rate
Measuring Techniques
Heat Generation and Geoneutrinos
Summary
Cross-References
Bibliography
Recovery of Source Magnetization Direction from Magnetic Field Data
Definition
Introduction
The Expression of Magnetization Direction in Magnetic Field Data
Magnetization Contrast and Anomaly Separation
Magnetization Direction and Source Location
Magnetization Direction and Source Shape
Complex Distributions of Magnetization
Homogeneity and Inhomogeneity of Magnetization
Magnetic Field Studies and Paleomagnetism
Conclusions
Cross-References
Bibliography
Remanent Magnetism
Definition
Introduction
Source of RM
Natural Remanent Magnetization (NRM)
Thermoremanent Magnetization (TRM)
Detrital Remanent Magnetization (DRM
Chemical Remanent Magnetization (CRM)
Viscous Remanent Magnetization (VRM)
Isothermal Remanent Magnetization (IRM)
Applications of Remanent Magnetization
Summary
Cross-References
Bibliography
Remote Sensing and GIS Techniques for Tectonic Studies
Synonyms
Definition
Introduction
Data and Method
Remote Sensing
Tectonic Studies
Geographic Information System (GIS)
Summary
Cross-References
Bibliography
Remote Sensing, Applications to Geophysics
Definition
Introduction
Application of Remote Sensing to Geophysics
Summary
Cross-References
Bibliography
S
SAR Interferometry
Synonyms
Definition
Introduction
Fundamentals of SAR Imaging and SAR Data
Fundamental Principles of InSAR
InSAR Processing
Outlook for InSAR Geodesy
Cross-References
Bibliography
Satellite Altimetry
Synonyms
Definition
Satellite Altimetry
Measurement Principles in Altimetry
Oceanography by Altimetry
Hydrology by Altimetry
Ice and Cryosphere Monitoring
Summary
Acknowledgments
Bibliography
Seafloor Spreading
Definition
Introduction
The Revolution
Ridge Axis Geometry, Morphology, and Crustal Formation
Hydrothermal Vents
Summary
Cross-References
Bibliography
Sedimentary Basins
Definition
Basin Types
Basin-Forming Mechanisms
Conceptual Models of Basin Formation
The Configuration of the Sediment Fill
The Configuration of the Crust Beneath Sedimentary Basins
The Configuration of the Mantle Lithosphere
Heat Flow in Sedimentary Basins
Summary
Cross-References
Bibliography
Seismic Data Acquisition and Processing
Definition
Introduction
Seismic Data Acquisition
Seismic Sources
Seismic Receivers
Seismic Recorder
Acquisition Geometry
Seismic Data Processing
Introduction
Signal vs. Noise
Kinematics of the Seismic Signal (Primaries)
NMO
Dipping Bed
Many Reflectors: Layer-Cake
Velocities in Seismics
NMO Stretch
Semblance: A Measure of Signal Alignment
Velocity: Processing Point of View
Amplitude Changes along the Propagation Path
Geometrical Spreading
Absorption
Energy Partitioning at Interfaces
Waveforms: Convolution, Deconvolution
Convolutional Model of the Seismic Trace
Deconvolution as Inverse Filtering
Wavelet Processing
Deterministic Deconvolution
Statistical Deconvolution
The Processing Flow: Putting it all Together
Preprocessing
Prestack Processing
Poststack Processing: Positioning Properly
No-Stack Processing: Imaging Complex Structures
Special Processing
Some Recent Developments
Summary
Cross-References
Bibliography
Seismic Diffraction
Definition
Introduction
Diffraction Theories
Geometrical Theory of Diffraction
Kirchhoff Approximation
Perturbation Theory: Born and Rytov Approximation
Numerical Methods Used to Compute Diffraction Wavefield
An Example of Seismic Diffraction from a Single Fracture
Diffraction Tomography
Summary
Cross-References
Bibliography
Seismic Discontinuities in the Transition Zone
Definition
Mineral Physics Data on the Phase Transitions in the TZ
Seismic Methods
Topography and Sharpness of TZ Discontinuities
The Issue of Water in the TZ
Summary
Cross-References
Bibliography
Seismic Hazard
Definition
Introduction
Cornell-McGuire PSHA Methodology
Estimation of Seismic Source Parameters
Numerical Computation of PSHA
Deaggregation of Seismic Hazard
Modifications of Cornell-McGuire PSHA Procedure and Alternative Models
Source-Free PSHA Procedures
Alternative Earthquake Recurrence Models
Ground Motion Models
Uncertainties in PSHA
Logic Tree
Controversy
Summary
Cross-References
Bibliography
Seismic Imaging, Overview
Definition
Inversion Theory
Five Types of Seismic Imaging Methods
Migration
Least Squares Migration
Full Waveform Inversion
Wave Equation Traveltime Inversion
Migration Velocity Analysis
Recent Advances in Seismic Imaging
Multiscale Waveform Inversion
Phase-Encoded Multisource Waveform Inversion
Current Status and Future of Seismic Imaging
Cross-References
Bibliography
Seismic Instrumentation
Synonyms
Definition
Introduction
Requirements for Instruments
Inertial Seismometers: Basic Principles
Mechanical Design
Electronic Design
Calibration
Installation
Displacement Seismometry Using Satellites
Deformation Seismometers
Seismic Data Recording
Summary
Cross-References
Bibliography
Seismic Microzonation
Definition
Introduction
Methodology
Example 1: Seismic Microzonation Based on Geomorphological Classification Maps
Example 2: Seismic Microzonation Based on Dense Borehole Data and GIS
Example 3: Seismic Microzonation Based on Microtremor Measurements
Summary
Cross-References
Bibliography
Seismic Monitoring of Nuclear Explosions
Definition
Introduction
Basic Properties of Earthquake and Explosion Signals
The Different Steps in Explosion Monitoring
Detection
Association
Location
Methods of Identification
Yield Estimation
Special Events
Evasion
Event Detection Capability of the International Monitoring System
Summary
Cross-References
Bibliography
Seismic Noise
Definition
Introduction
Types of Seismic Noise
Enhancing Signal Over Noise
Use of Noise as Signal
Summary
Cross-References
Bibliography
Seismic Phase Nomenclature: The IASPEI Standard
Definition
Introduction
Standard Letters, Signs, and Syntax Used for Describing Seismic Phases
Capital Letters
Exceptions
Lowercase Letters and Signs
Syntax of Generating Complex Phase Names
Examples for Creating Complex Standard Phase Names
Refracted and Converted Refracted Wave
Pure Reflected Waves
Reflected Waves with Conversion at the Reflection Point
Ray-Path Diagrams for Some of the IASPEI Standard Phases
IASPEI Standard Seismic Phase List
Crustal Phases
Mantle Phases
Core Phases
Near-Source Surface Reflections and Conversions (Depth Phases)
Surface Waves
Acoustic Phases
Amplitude Measurements
Unidentified Arrivals
Summary
Cross-References
Bibliography
Seismic Properties of Rocks
Synonyms
Definition
Introduction
Measurement Techniques
Rock Velocities
Velocity Anisotropy
Summary
Cross-References
Bibliography
Seismic Quiescence and Activation
Definition
Introduction
Main Phases of Seismic Quiescence and Activation
Physical Mechanisms Leading to Seismic Quiescence and Activation
Summary
Cross-References
Bibliography
Seismic Ray Theory
Definition
Introduction
Basic Equations of the Seismic Ray Method
Eikonal Equation. Polarization Vector
Ray Tracing and Travel-Time Computation
Transport Equation. Computation of Ray-Theory Amplitudes
Dynamic Ray Tracing. Paraxial Approximations
Coupling Ray Theory for S Waves in Anisotropic Media
Effects of Structural Interfaces
Ray-Theory Elastodynamic Green Function
Chaotic Rays. Lyapunov Exponents
Ray Perturbation Methods
Ray Perturbation Method for Weakly Dissipative Media
Concluding Remarks. Applications, Modifications, and Extensions of the Ray Method
Cross-References
Acknowledgments
Bibliography
Seismic Reservoir Characterization
Definition
Introduction
Seismic Data Processing
Seismic-Well Tie
Seismic Rock Physics
Seismic Inversion for Reservoir Properties
Applications of Machine Learning Methods in Reservoir Characterization
Summary
Cross-References
Acknowledgments
Bibliography
Seismic Seiches
Definition
Bibliography
Seismic Signals in Well Water Observations
Synonyms
Definition
Introduction
Co- and Post-seismic Signals
Transient Signals
Preseismic Signals
Mechanisms
Summary
Bibliography
Seismic Stereotomography
Synonyms
Definition
History
Data and Model in Slope Tomography
Applications of Slope Tomography
Challenges and Perspectives
Conclusions
Cross-References
Bibliography
Seismic Structure at Mid-Ocean Ridges
Definition
Introduction
Mantle Structure
Crustal Structure
Summary
Cross-References
Bibliography
Seismic Tomography
Definition
History
Onset Times
Model Parameterization and Inversion
Normal Modes and Surface Waves
Finite-Frequency Tomography
Summary
Cross-References
Bibliography
Seismic Velocity and Temperature Relationships
Definition
Acknowledgments
Bibliography
Seismic Viscoelastic Attenuation
Synonyms
Definition
Introduction
How Do Seismic Waves Attenuate?
Geometric Spreading
Intrinsic Viscoelastic Attenuation
Scattering Attenuation
Linear Viscoelasticity
Rheology
Anelastic Hysteresis
Q and Complex Velocity
The Q-1(ω) Relaxation Spectrum
Velocity Dispersion
Effects of Scattering
Equivalent Medium
Stochastic Dispersion
Effects of Anisotropy
Measurement and Modeling Attenuation
The Attenuation Operator for Body Waves
Free Oscillations and Surface Waves
Numerical Modeling
Interpretation of Attenuation Measurements in the Earth
Shear Versus Bulk Attenuation
Frequency Dependence
Thermal Activation
Regional Variations
Strain Dependence
Summary
Cross-References
Bibliography
Seismic Wave Propagation in Real Media: Numerical Modeling Approaches
Synonyms
Definition
Introduction
Direct Methods
Finite Difference Method
Anisotropy
Pseudospectral Method
Finite Element and Spectral Element Methods
Discontinuous Galerkin Method
Integral Equation Methods
Fast Multipole Method in Elastodynamics
Analytic Solutions for Nonhomogeneous Media
Equivalent Media Theories
Asymptotic Ray Tracing Methods
Poroelastic and Thermoelastic Media
Boundaries
Summary
Cross-References
Acknowledgments
Bibliography
Seismic Waves, Scattering
Definition
Introduction
Statistical Description of the Earth
The Coherent and Incoherent Fields
The Propagation Regimes
Analysis of Transmission Fluctuations
Envelope Modeling: Markov Approximation
Envelope Modeling: Radiative Transfer
Global-Scale Scattering
Interferometry with Scattered Waves
Green Function Retrieval
Monitoring Temporal Variations
Summary
Bibliography
Seismic Zonation
Definition
Introduction
Methodology
Example: Seismic Zonation Map of China (2001)
Summary
Cross-References
Bibliography
Seismic, Ambient Noise Correlation
Definition
Introduction
Theoretical Basis for the Interpretation of Noise Records at Two Stations
Applications in Seismology
Noise Source Origin and Distribution
Noise-based Seismic imaging
Noise-based Monitoring
Summary
Cross-References
Bibliography
Seismic, Artificial Intelligence to Neural Intelligence for Advanced Interpretation
Definition
Introduction: A Motivation for Automating Interpretation
Examples of Meta-attributes
Example 1: Thinned Fault Cube (TFC) Meta-attribute
Example 2: Intrusion Cube (IC) Meta-attribute
Example 3: Sill Cube (SC) Meta-attribute
Example 4: Fluid Cube (FC) Meta-attribute
Example 5: Mass Transport Deposit Cube (MTDC) Meta-attribute
Conclusion
Bibliography
Seismic, Migration
Synonyms
Definition
History
Mapping Using Pencil and Paper: Mechanical Migration
Digital Migration
Poststack Migration
Prestack Migration
2-D, 3-D, Wide-Azimuth 3-D Acquisition, and Migration
Deep Crustal Imaging
Time Migration, Depth Migration
Purposes of Migration
Structural Imaging
Velocity Estimation
Migrated Amplitude Analysis for Rock Property Determination
Imaging Conditions
Isochron Imaging
Poststack Migration: Exploding Reflector Model
Reflected Wavefield Amplitude Normalized by Source Wavefield Amplitude
Deconvolution and Cross-Correlation Imaging Conditions
Migration Techniques
Integral (Kirchhoff) Migration
Practical Considerations in Kirchhoff´s Migration
Dip Limits
Velocity Field
Beam Migration
One-Way Wave-Equation Migration (OWEM)
Finite-Difference Migration
Depth Step
Frequency-Wave Number (f-k) Migration
Stolt Stretch Factor
Two-Way Wave-Equation Migration (Reverse-Time Migration)
Migration Examples
The Evolving Role of Migration in Seismic Data Processing
Selection of Migration Algorithm
Some Aspects of Migration
Summary
Cross-References
Bibliography
Seismic, Receiver Function Technique
Synonyms
Definition
Receiver Function Technique
Separation of P and S Waves
Deconvolution
Moveout Correction and Summation
CCP Stack and Migration
Waveform Modeling
Two Data Examples
New Technical Development of the S-Receiver Function Method
Summary
Cross-References
Bibliography
Seismic, Reflectivity Method
Synonyms
Definition
The Reflectivity Method
Introduction
Plane Waves
Simple Plane Wave Synthetics
Theory
Computational Issues
Applications
Extension to Laterally Heterogeneous Media
Summary
Cross-References
Bibliography
Seismic, Super-Virtual Refraction Interferometry
Definition
Introduction
Virtual Refraction Interferometry
Super-Virtual Refraction Interferometry (SVI)
SVI in Practice
Iterative Super-Virtual Refraction Interferometry
Very Noisy Data
Summary
Bibliography
Seismic, Velocity, and Density Relationships
Definition
General Relation Between Velocity and Density
Cross-References
Acknowledgments
Bibliography
Seismic, Waveform Modeling and Tomography
Synonyms
Definition
Introduction
Waveform Modeling
Inverse Method
Shot-Encoded Waveform Tomography
The Restarted L-BFGS Algorithm
Preprocessing for Field Data Waveform Tomography
Seismic Reflection Waveform Tomography
Summary
Cross-References
Bibliography
Seismicity, Intraplate
Definition
Historical Context
Definitions and Classification
Reasons to Distinguish Intraplate Earthquakes
Do Intraplate and Interplate Earthquakes Scale Differently?
What Drives Intraplate Earthquakes?
Examples
Controversy
Summary or Conclusions
Cross-References
Bibliography
Seismicity, Subduction Zone
Definition
Introduction
Interplate Earthquakes
Trench-Outer Rise Intraplate Earthquakes
Shallow Inland Intraplate Earthquakes
Intraslab Earthquakes
Intermediate-Depth Earthquakes
Deep Earthquakes
Summary
Cross-References
Bibliography
Seismogram Interpretation
Synonyms
Definition
Introduction
Crustal Waves: Recording Distances 0-10
Body Waves Traversing the Mantle: Recording Distances 10-103
Body Waves Traversing the Core: Recording Distances 103 and Beyond
Body Waves from Intermediate-Focus and Deep-Focus Earthquakes
Surface Waves
Volcanic Earthquakes and Unusual Seismic Sources
Conclusions
Cross-References
Bibliography
Seismological Networks
Definition
Introduction
Global Seismic Networks
Worldwide Standardized Seismograph Network
International Federation of Digital Seismograph Networks
International Monitoring System
Regional Seismic Networks
Euro-Med Region
United States
China
Large-Scale Arrays
International Registry of Seismograph Stations
Summary
Cross-References
Acknowledgments
Bibliography
Seismology and Environment
Definition
Introduction
Broadband Seismic Noise
Continuous Monitoring of the Earth Environment
Spatiotemporally Localized Events
Glacial Earthquakes
Cyclones (Hurricanes-Typhoons)
Landslides
New Challenges
Cross-References
Bibliography
Seismology, Global Earthquake Model
Definition
Introduction
The GEM Initiative
The GEM Foundation
Scientific Framework
Institutional Framework
Milestones and Scientific Products
The Global Earthquake Model
Global Earthquake Hazard
Global Earthquake Risk
Global Social Vulnerability and Integrated Risk Map
Software and Analysis Tools
OpenQuake Engine and Built-In Toolkits
Interactive Global Earthquake Map Viewer Tool
OpenQuake Platform
Integrated Risk Modeling Toolkit (IRMT)
Building Taxonomy
Building Classification Survey
Inventory Data Capture Tools (IDCT)
Global Databases
Global Instrumental Earthquake Catalogue (1900-2009)
Global Historical Earthquake Catalogue and Archive (1000-1903)
Updated Active Fault Database
Updated Vulnerability Database
Global Exposure Database (GED4GEM and GED4ALL)
Current Status
Future Program
Cross-References
Acknowledgments
Bibliography
GEM Foundation Website
GEM OpenQuake Website
Github Website
Google Apps Website
UNDRR Website
ISC Website
IASPEI Website
QGIS Website
Seismology, Monitoring of CTBT
Definition
Introduction
Evaluation and Design of the Seismic Monitoring System: A Systems Engineering Perspective
Advances in Seismology, Potentially Relevant to CTBT Monitoring
``Forensic Seismology´´: Evidences and Judgments
Concluding Remarks: Interaction Between Science and CTBT Monitoring
Cross-References
Bibliography
Seismology, Rotational
Definition
Introduction
Early Attempts in Studying Rotational Motions
Measuring Rotational Motions
Large Ring Laser Gyros
G Ring Laser and Recording Teleseisms
Strong-Motion Inertial Angular Sensors
Discussions
Linear and Nonlinear Elasticity
Near-Field Seismology
Using Explosions to Study Rotational Motions
Processing Collocated Measurements of Translations and Rotations
Conclusion
Cross-References
Bibliography
Sentinel Satellites Initiate New Era in Earth Observatories
The Sentinel Satellites
Sentinels and Landsat
Free and Open Access to Data
Widening the Market and the Access to Data
Financial Challenges
Shear-Wave Splitting: New Geophysics and Earthquake Stress-Forecasting
Synonyms
Definition
Introduction
Fundamental Features of Shear-Wave Splitting in the Crust
Classes of Anisotropic Symmetry and Shear-Wave Splitting
Cusps
Shear-Wave Window
Shear-Wave Singularities
Sources of Shear-Wave Splitting
Shear-Wave Splitting in the Upper Mantle
Shear-Wave Splitting in Hydrocarbon Reservoirs
Shear-Wave Splitting Above Small Earthquakes
The New Geophysics
Stress-Forecasting Earthquakes
Summary
Cross-References
Bibliography
Single and Multichannel Seismics
Definition
Introduction
Single-Channel Seismics
Multichannel Seismics
Commonly Used High-Resolution Single- and Multichannel Seismic Sources
Processing Considerations for High-Resolution Single- and Multichannel Seismic Profiles
Summary
Cross-References
Bibliography
Slow Earthquake
Definition
Early Studies on Slow and Silent Earthquake
Long-Term Slow Slip Events
Low-Frequency Tremor and Short-Term Slow Slip
Pre-Seismic Slip
Scaling Relation
Mechanism of Slow Earthquake
Concluding Remarks
Cross-References
Acknowledgments
Bibliography
Spherical Harmonic Analysis Applied to Potential Fields
Synonyms
Definition
Introduction: Basic Formulas
Boundary Value Problems (BVP) of Potential Theory
Spherical Harmonic Analysis (SHA): Numerical Techniques
Practical Aspects
Summary
Cross-References
Bibliography
Statistical Seismology
Synonyms
Definition
Introduction
Empirical Relations
Precursors
Stochastic Models of Earthquake Occurrence
Physics-Based Models
Testing of Forecasting Methods
Operational Earthquake Forecasting
Conclusion
Cross-References
Bibliography
Subduction Zones
Definition
Introduction
Morphology and Nomenclature
Outer Rise, Trench, and Forearc
Thrust Zone
Kinematics
Thermal Structure, Plate Buoyancy, Mantle Flow
Fate of the Downgoing Plate: Devolatilization and Metamorphism
Subduction Channel
Melting and the Volcanic Arc
Seismic Imaging of the Subduction Zone
Subduction Zone Hazards
Summary
Cross-References
Bibliography
Surface Waves
Definition
Group and Phase Velocity
Surface Wave Generation and Propagation
Structure Studies Using Surface Waves
Fundamental Mode Studies (Dispersion)
Noise Cross-Correlation Methodology
Global Studies of Structure Using Surface Waves
Inversion of Surface Wave Dispersion Data
Crustal Corrections
Inversion Using Overtones
Time Domain Surface Wave Inversions
Effects of Scattering and the Born Approximation
Upper Mantle Anisotropy from Surface Wave Studies
Surface Wave Attenuation
Source Studies Using Surface Waves
The Earth´s Background Noise Spectrum: Hum and Microseisms
Summary
Cross-References
Bibliography
T
T Waves
Definition and Introduction
Sources of T Waves
Human Perception
Use in Comprehensive Nuclear-Test Ban Treaty Monitoring
Acoustic Thermometry of Ocean Climate
Quantification
A Few Geophysical Applications
Conclusion
Cross-References
Bibliography
Tectonic Geomorphology
Definition
Tectonically Generated First-Order (103-102 Km Scale) Landscapes
Transform Settings
Convergent Settings
Divergent Settings
Signatures of Morphotectonic Landscapes
Direct Evidences
Geomorphic Markers of Transform Setting
Geomorphic Markers of Divergent Setting
Geomorphic Markers of Convergent Setting
Geomorphic Markers in Multiple Settings
Indirect Evidences
Geomorphic Indices
Tectonic Geomorphology and Paleoseismological Studies of Tectonic-Landforms
Global Case Studies
Summary
Acknowledgments
Bibliography
Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat
Synonyms
Definition
Thermal Storage Properties
Heat Capacity
Isobaric and Isochoric Specific Heat Capacity
Measuring Techniques
Calculated Heat Capacity
Temperature Dependence
Volumetric Heat Capacity: Thermal Capacity
Latent Heat
Summary
Cross-References
Acknowledgments
Bibliography
Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity
Synonyms
Definition
General Remark
Thermal Conductivity
Measuring Techniques
Indirect Methods
Thermal Conductivity of Minerals and Rocks
Radiative Thermal Conductivity
Variation with Temperature
Variation with Pressure
Variation with Other Factors
Thermal Diffusivity
Measuring Techniques
Variation with Temperature
Variation with Pressure
Variation with Other Factors
Summary
Cross-References
Acknowledgments
Bibliography
Time-Dependent Seismic Hazard Assessment
Definition
Time-Dependent Seismic Hazard
Time-Dependent Seismic Hazard: Practical Tools and Case Examples of Application
Conclusions and Discussion
Cross-References
Bibliography
Time-Reversal in Seismology
Definition
History
Time-Reversal Before Modern Electronics
Modern Time-Reversal
Location of Seismic Sources
Related Seismic Methods
Time-Reversal and Complexity
Summary
Cross-References
Acknowledgments
Bibliography
TOPO-EUROPE: From the Deep Earth to the Surface of Continental Europe and Its Margins
Definition
Introduction
Deep Earth
Lithosphere Structure
Mantle-Lithosphere Interactions
Sedimentary Basins and Geo-Resources
Analogue Tectonic Modeling
Summary and Conclusions
Acknowledgments
Bibliography
Total Electron Content in Seismo-ionospheric Studies
Synonyms
Definition
Introduction
Methodology for Estimating Total Electron Content (TEC) from GPS Data
Propagation Characteristics of CIDs
Amplitudes of CID
Propagation Speed of CIDs
Directivity of CID Propagation
Dip-slip Verus Strike-slip Earthquakes
Preseismic Ionospheric Anomalies
Influence of Equatorial Ionization Anomaly on CID
Absence of CIDs
Applications of CIDs in Ionospheric Seismology
Summary
Cross-References
Acknowledgments
Bibliography
Tracking Earth´s Water in Motion from Satellite Gravity Observations
Introduction
GRACE Satellite Gravimetry
Quantifying Mass Change Using GRACE Gravity Solutions
Monitoring Land Water Cycle Using GRACE
Monitoring Ice and Ocean Mass Change Using GRACE
Summary
Bibliography
Trans-European Suture Zone
Synonyms
Definition
Introduction
Geologic-Tectonic Overview
Geophysical Signature of TESZ
Crustal Profile
Gravity and Magnetic Signature
Crustal Structure from Seismic Data
Lithospheric Mantle in TESZ
Conclusions
Bibliography
Traveltime Tomography Using Controlled-Source Seismic Data
Synonyms
Definition
Background
Forward Modeling
Traveltime Tomography and Inverison
Algorithms
Model Assessment
More Examples
Future
Cross-References
Bibliography
Tsunami
Definition
Tsunami = Killer Wave?
Tsunami Characteristics
Tsunami Period, Velocity, and Wavelength
Tsunami Eigenfunctions
Tsunami Excitation
Tsunami Excitation by Earthquakes
Tsunami Excitation from Submarine Landslides
Tsunami Excitation from Impacts
Tsunami Propagation
Tsunami Shoaling and Run-up
Tsunami Samples
Tsunami Forecasting
Summary
Cross-References
Tsunami Watch and Warning Centers
Definition
Introduction
Types of Tsunamis
Regional Tsunami Warning Systems
Key Components of a Tsunami Warning Center
Observation Network
Seismic Network
Sea Level Network
Tsunami Forecast Modeling
Pre-run Model Scenario Database
Inundation Modeling and Mapping
Real-Time Tsunami Modeling for Propagation and Inundation
Communication Infrastructure
Decision Support System
Typical Bulletins Issued by TWCs
Service Levels
Conclusion
Acknowledgments
Bibliography
Tsunami: Bay of Bengal
Definition
Introduction
Tsunami Hazard in the Bay of Bengal
Plate Motion Along the Northern Sunda Arc
Seismicity of the Northern Sunda Arc Region and Tsunami Hazard
Indo-Burmese Arc
Irrawaddy Region
Andaman Sumatra Arc Region
Summary
Cross-References
Bibliography
U
Uncertainties in Geomodelling Related to Geophysical Inversion
Synonyms
Definition
Introduction
Uncertainty
Error and Accuracy
Sensitivity
Uncertainties in the Transformations of Earth Science Problems into a Mathematical Model: Reasoning Behind Modeling and Basic ...
Reduction and Simplicity Concepts in Geosciences
Interface Representations as Reduction of Reality
Scale and Typical Representations
Considerations of Complexity
Mathematical Models of Interfaces
General Interface Description
Map-Based Interpolation Approaches
Explicit and Implicit 3-D Interpolations
Uncertainty in the Detection of Model Interfaces by Geological and Geophysical Methods
As Model Describing Initial State
Joint Inverse Frameworks
Uncertainties in the Output of Material Parameters
Relevance of Rock Physics Models
Visualization of Uncertainties as Output of Density Models: An Example from the Central Andes
Summary
Cross-References
Bibliography
Unified Scaling Law for Earthquakes that Generalizes the Fundamental Gutenberg-Richter Relationship
Definition
Introduction
Unified Scaling Law for Earthquakes
Conclusion
Cross-References
Bibliography
V
Vertical Seismic Profiling
Synonyms
Definition
Introduction
Acquisition Techniques
VSP Applications
Summary
Cross-References
Bibliography
Very Long Baseline Interferometry
Synonyms
Definition
Introduction
Methods and Techniques in Very Long Baseline Interferometry
Physical Principles
Data Calibration and Imaging
VLBI Networks
Geodetic and Geophysical Applications
Reference Systems and Tectonics
Earth Rotation
Summary
Cross-References
Bibliography
W
Wavelet Analysis
Synonyms
Definition
Introduction
Continuous Wavelet Transform
The Inversion Formula
Use of the Physical Frequency
Wavelets
Wavelet Spectrum Amplitude and Phase
Prograde and Retrograde Wavelet Spectrum
Discrete Wavelet Transform
Use of Wavelets in Geophysical Time-Series Analysis
Instantaneous Parameters
Polarization Analysis
Dispersion Analysis
Summary
Cross-References
Bibliography
Author Index
Subject Index

Citation preview

Edited by Harsh K. Gupta

of E A R T H S C I E N C E S S E R I E S

2ND EDITION

ENCYCLOPEDIA

ENCYCLOPEDIA of SOLID EARTH GEOPHYSICS

ENCYCLOPEDIA of SOLID EARTH GEOPHYSICS

Encyclopedia of Earth Sciences Series ENCYCLOPEDIA OF SOLID EARTH GEOPHYSICS Volume Editor Harsh K. Gupta National Geophysical Research Institute Council of Scientific and Industrial Research Hyderabad, India

Editorial Board Kusumita Arora National Geophysical Research Institute Council of Scientific and Industrial Research (CSIR) Hyderabad, India

Ajay Manglik National Geophysical Research Institute Council of Scientific and Industrial Research (CSIR) Hyderabad, India

Anny Cazenave Laboratoire d'Etudes en Géophysique et Océanographie Spatiales (LEGOS) CNES & Observatoire Midi-Pyrénées Toulouse, France

Sukanta Roy Ministry of Earth Sciences, Govt. of India Borehole Geophysics Research Laboratory Karad, India

Eric Robert Engdahl Denver, CO, USA

Kalachand Sain Wadia Institute of Himalayan Geology Uttarakhand, India

Rainer Kind Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences Potsdam, Germany

Seiya Uyeda Tokyo, Japan

Aims of the Series The Encyclopedia of Earth Sciences Series provides comprehensive and authoritative coverage of all the main areas in the Earth Sciences. Each volume comprises a focused and carefully chosen collection of contributions from leading names in the subject, with copious illustrations and reference lists. These books represent one of the world’s leading resources for the Earth Sciences community. Previous volumes are being updated and new works published so that the volumes will continue to be essential reading for all professional earth scientists, geologists, geophysicists, climatologists, and oceanographers as well as for teachers and students. See the back of this volume for a current list of titles in the Encyclopedia of Earth Sciences Series. Go to http://www.springerlink.com/reference-works/ to visit the “Earth Sciences Series” online.

About the Series Editor Professor Charles W. Finkl has edited and/or contributed to more than eight volumes in the Encyclopedia of Earth Sciences Series. He has been the Executive Director of the Coastal Education and Research Foundation and Editor-in-Chief of the international Journal of Coastal Research for the past 35 years. He is also the Series Editor of the Coastal Research Library (Springer). In addition to these duties, he is Distinguished University Professor Emeritus at Florida Atlantic University (FAU) (Boca Raton, Florida). He is a graduate of Oregon State University (Corvallis) and the University of Western Australia (Perth). Work experience includes the International Nickel Company of Australia (Perth), Coastal Planning & Engineering (Boca Raton, Florida), and Technos Geophysical Consulting (Miami, Florida). He has published numerous peer-reviewed technical research papers and edited or co-edited and contributed to many books. Dr. Finkl is a Certified Professional Geological Scientist (Arvada, Colorado), a Certified Professional Soil Scientist (Madison, Wisconsin), a Certified Wetland Scientist (Lawrence, Kansas), and a Chartered Marine Scientist (London). Academically, he served as a Demonstrator at the University of Western Australia, Courtesy Professor at Florida International University (Miami), Program Professor and Director of the Institute of Coastal and Marine Studies at Nova Southeastern University (Port Everglades, Florida) and Full Professor at FAU. During his career, he acquired field experience in Australia; the Bahamas; Puerto Rico, Jamaica; Brazil; Papua New Guinea and other SW Pacific islands; southern Africa; Western Europe; and the Pacific Northwest, Midwest, and Southeast USA. Dr. Finkl is a member of several professional societies including the Geological Society of America; Soil Science Society of America; Institute of Marine Engineering, Science and Technology; and the Society of Wetland Specialists. He is a recipient of the International Beach Advocacy Award (Florida Shore & Beach Preservation Association), Certificate of George V. Chilingar Medal of Honor (Russian Academy of Natural Sciences), and Lifetime Commitment to Coastal Science Award (International Coastal Symposium).

Founding Series Editor Professor Rhodes W. Fairbridge (deceased) has edited more than 24 Encyclopedias in the Earth Sciences Series. During his career he has worked as a petroleum geologist in the Middle East, been a WW II intelligence officer in the SW Pacific and led expeditions to the Sahara, Arctic Canada, Arctic Scandinavia, Brazil and New Guinea. He was Emeritus Professor of Geology at Columbia University and was affiliated with the Goddard Institute for Space Studies.

ENCYCLOPEDIA OF EARTH SCIENCES SERIES

ENCYCLOPEDIA of SOLID EARTH GEOPHYSICS Second Edition

edited by

HARSH K. GUPTA National Geophysical Research Institute Council of Scientific and Industrial Research Hyderabad, India

Editor Harsh K. Gupta National Geophysical Research Institute Council of Scientific and Industrial Research Hyderabad, India

ISBN: 978-3-030-58630-0 This publication is available also as: Electronic publication under ISBN 978-3-030-58631-7 and Print and electronic bundle under ISBN 978-3-030-58632-4

Cover illustration: C.B. 3D ILLUSTRATION & ANIMATION | illustration by Christoph Burgstedt Every effort has been made to contact the copyright holders of the figures and tables which have been reproduced from other sources. Anyone who has not been properly credited is requested to contact the publishers, so that due acknowledgement may be made in subsequent editions. All rights reserved for the contributions: Earthquake Sounds; Electrical Resistivity Surveys and Data Interpretation; Gravity, Data to Anomalies; Gravity, Global Models; Instrumentation, Electrical Resistivity; Sentinel Satellites Initiate New Era in Earth Observatories © Springer Nature Switzerland AG 2021 1st edition published by: © Springer Science þ Business Media B.V. 2011 The original Encyclopedia of Solid Earth Geophysics was compiled by David E. James, and was first published in the Encyclopedia of Earth Sciences Series in 1989. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

About the Editor

xiii

Contributors

64

Core Dynamo Ulrich R. Christensen

78

Core-Mantle Coupling Paul H. Roberts and Jonathan M. Aurnou

86

xv

Preface to the Second Edition Preface to the First Edition

Continental Rifts A. M. Celâl Şengör

xxxi xxxiii

Acknowledgments Absolute Age Determinations: Radiometric Richard W. Carlson Anthropogenic Seismicity Related to Exploitation of Georesources Stanislaw Lasocki and Beata Orlecka-Sikora

xxxv 1

Crustal Reflectivity (Oceanic) and Magma Chamber Satish C. Singh

101

Curie Temperature Vincenzo Pasquale

112

Deep Scientific Drilling Ulrich Harms and Harold Tobin

115

15

19

Deep Seismic Reflection and Refraction Profiling Kabir Roy Chowdhury

33

Differential Rotation of the Earth’s Inner Core Xiaodong Song

144

Biogeophysics Lee Slater and Estella Atekwana

37

Earth Rotation Harald Schuh and Sigrid Böhm

149

Body Waves Mahmoud Mohamed Selim Saleh

44

Earth Tides John M. Wahr

156

Borehole Seismic Networks and Arrays Marco Bohnhoff and Peter Malin

53

Earth, Density Distribution Frank D. Stacey and Paul M. Davis

160

Characteristic Earthquakes and Seismic Gaps David D. Jackson and Yan Y. Kagan

56

Earth’s Structure, Core Lianxing Wen

164

Continental Crustal Structure Rolf Meissner and Hartmut Kern

63

Earth’s Structure, Global Jean-Paul Montagner

166

Continental Drift Alan G. Smith

Archaeomagnetism Donald H. Tarling Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India Harsh K. Gupta

8

127

vi

CONTENTS

Earth’s Structure, Lower Mantle Edward J. Garnero, Allen K. McNamara and James A. Tyburczy

176

Earth’s Structure, Upper Mantle Guust Nolet

Earthquakes, Energy Domenico Di Giacomo and Peter Borman

288

Earthquakes, Intensity Gottfried Grünthal and Roger M. W. Musson

292

183 189

Earthquakes, Intraplate Prantik Mandal

299

Earthquake Lights John S. Derr, France St-Laurent, Friedemann T. Freund and Robert Thériault

Earthquakes, Location Techniques Clifford H. Thurber

301

Earthquake Precursors and Prediction Toshiyasu Nagao, Masashi Kamogawa and Seiya Uyeda

193

Earthquake Prediction, M8 Algorithm Alik Ismail-Zadeh and Vladimir Kossobokov

204

Earthquake Rupture: The Inverse Problem Shamita Das

208

Earthquake Sounds Andrew J. Michael

220

Earthquake Source Theory Raul Madariaga

224

Earthquake, Aftershocks Mian Liu and Seth Stein

Earthquakes, PAGER 308 David J. Wald, Kishor S. Jaiswal, Kristin D. Marano and Michael Hearne Earthquakes, ShakeCast Kuo-wan Lin, David J. Wald and Daniel Slosky

312

Earthquakes, ShakeMap 316 David J. Wald, C. Bruce Worden, Eric M. Thompson and Michael Hearne Earthquakes, Strong-Ground Motion Giuliano F. Panza, Cristina La Mura, Fabio Romanelli and Franco Vaccari

321

229

Earthquakes, Volcanogenic J. W. Neuberg

329

Earthquake, Archaeoseismology Klaus-G. Hinzen

232

Electrical Properties of Rocks Takashi Yoshino

339

Earthquake, Focal Mechanism Emile A. Okal

236

Earthquake, Foreshocks Mian Liu

241

Electrical Resistivity Surveys and Data Interpretation M. H. Loke, D. F. Rucker, J. E. Chambers, P. B. Wilkinson and O. Kuras

243

Electromagnetic Methods, Imaging Magma Bodies Kathryn A. Whaler

350

Earthquake, Magnitude Peter Borman

Electromagnetic Pulsations and Magnetic Storms G. S. Lakhina and B. T. Tsurutani

354

Earthquakes and Crustal Deformation Robert McCaffrey Earthquakes in the Himalaya Dibyashakti Panda, Bhaskar Kundu and Vineet K. Gahalaut Earthquakes on Oceanic Transform Faults, Unexpectedly Large Shamita Das

344

254 262

Electronic Geophysical Year 359 William K. Peterson, Daniel N. Baker, C. E. Barton, Peter Fox, Mark A. Parsons and Emily A. CoBabe-Ammann Energy Budget of the Earth Jean-Claude Mareschal and Claude Jaupart

361

Energy Partitioning of Seismic Waves Kalachand Sain

368

Equatorial Electrojet Archana Bhattacharyya

372

Fault Zone Guided Waves Peter Eric Malin

375

274

Earthquakes, Did You Feel It? David J. Wald, Vincent Quitoriano and James W. Dewey

278

Earthquakes, Early and Strong Motion Warning Richard M. Allen

282

CONTENTS

Fiber Optic Distributed Strain Sensing for Seismic Applications Thomas Reinsch, Philippe Jousset and Charlotte M. Krawczyk

vii

Geomagnetic Field, IGRF Aude Chambodut

500

Geomagnetic Field, Measurement Techniques Mioara Mandea and Anca Isac

502

384

Geomagnetic Field, Polarity Reversals Carlo Laj

507

Floating Seismographs (MERMAIDS) Yann Hello and Guust Nolet

395

Geomagnetic Field, Secular Variation Monika Korte

514

Fractal Scaling of Earthquakes Simanchal Padhy and Vijay P. Dimri

399

Geomagnetic Field, Theory Friedrich H. Busse

515

Fractals and Chaos Ravi P. Srivastava, Nimisha Vedanti and Vijay P. Dimri

405

Geomagnetically Induced Currents G. S. Lakhina, R. Hajra and B. T. Tsurutani

523

412

Geophysical Well Logging Miroslav Kobr

527

Free Oscillations of the Earth Sarva Jit Singh and Sunita Rani

423

Geothermal Heat Pumps Ladislaus Rybach

537

Geodesy, Figure of the Earth Kusumita Arora

427

Geothermal Record of Climate Change Michael G. Davis, David S. Chapman and Robert N. Harris

541

Geodesy, Ground Positioning, and Levelling Stelios P. Mertikas Geodesy, Networks, and Reference Systems Hayo Hase

434

GPS, Data Acquisition, and Analysis Carine Bruyninx, Wim Aerts and Juliette Legrand

547

Geodesy, Physical V. Chakravarthi

442

GPS, Tectonic Geodesy Jeffrey T. Freymueller

558

Geodetic Pendulums, Horizontal Ultra Broad Band Carla Braitenberg

447

Gravimeters Andrew Hugill

578

Geodynamics Alessandro M. Forte

452

Gravity Anomalies, Interpretation Mikhail K. Kaban

585

Geoelectromagnetism Antal Adam and Laszló Szarka

454

Gravity Data, Advanced Processing Christopher J. Swain and Jonathan F. Kirby

591

Geoid Paramesh Banerjee

465

Gravity Data, Regional-Residual Separation Kumarendra Mallick, Vasanthi Anthwar and Krishna Kant Sharma

596

Geoid Determination, Computational Methods Michael G. Sideris

470 Gravity Field of the Earth Christopher Jekeli

608

Geoid Determination, Theory and Principles Michael G. Sideris

476

Geoid Undulation, Interpretation Petr Vaníček

482

Geomagnetic Excursions Martha Schwartz

486

Geomagnetic Field, Global Pattern Susan Macmillan

488

Fiber-Optic Sensing in Geophysics, Temperature Measurements Jan Henninges and Ali Masoudi

379

Gravity Field, Temporal Variations from Space Techniques Anny Cazenave, J. L. Chen and G. Ramillien

621

Gravity Field, Time Variations from Surface Measurements Virendra M. Tiwari and Jacques Hinderer

626

Gravity Measurements, Absolute Mark A. Zumberge

633

viii

CONTENTS

Gravity Method, Airborne Uwe Meyer

637

International Gravity Formula Hans-Jürgen Götze

789

Gravity Method, Principles Hans-Jürgen Götze

640

International Polar Year 2007–2008 David J. Carlson

790

Gravity Method, Satellite G. Balmino

645

International Year of Planet Earth Eduardo F. J. de Mulder and Wolfgang Eder

792

Gravity Method, Surface Dinesh Chandra Mishra and Virendra M. Tiwari

656

Inverse Theory, Artificial Neural Networks William A. Sandham and David J. Hamilton

796

Gravity Modeling, Theory and Computation Jean-Pierre Barriot and Lydie Sichoix

662

Inverse Theory, Global Optimization Mrinal K. Sen and Paul L. Stoffa

807

Gravity, Data to Anomalies Ron Hackney

668

Inverse Theory, Linear Pravin K. Gupta

814

Gravity, Global Models Nikolaos K. Pavlis

677

Inverse Theory, Monte Carlo Method Malcolm Sambridge and Kerry Gallagher

821

Gravity, Gradiometry Christopher Jekeli

692

Inverse Theory, Singular Value Decomposition Ajay Manglik

827

Great Earthquakes Roger M. W. Musson

708

Isostasy A. B. Watts

831

Heat and Ground Water Flow Vincenzo Pasquale, Massimo Verdoya and Paolo Chiozzi

717

Isostasy, Thermal Derrick Hasterok and David S. Chapman

847

Heat Flow Determinations, Continental John H. Sass and Graeme Beardsmore

722

Heat Flow, Continental Paul Morgan

727

Heat Flow, Seafloor: Methods and Observations Earl E. Davis and Andrew T. Fisher

736

KTB Depth Laboratory: A Window into the Upper Crust Ulrich Harms and Jochem Kück

855

Legal Continental Shelf: Geology, Geophysics, and Tectonics Elana Geddis, Ray Wood and Vaughan Stagpoole

861

Lithosphere, Continental David E. James

866

749

Lithosphere, Continental: Thermal Structure Claude Jaupart and Jean-Claude Mareschal

872

High-Frequency Seismology Simanchal Padhy

757

Lithosphere, Mechanical Properties Evgueni Burov

884

Impact Craters on Earth Richard A. F. Grieve and Gordon R. Osinski

769

Lithosphere, Oceanic James S. McClain

893

Lithosphere, Oceanic: Thermal Structure Earl E. Davis and David S. Chapman

906

Height Systems, Vertical Datums and Their Unification Michael G. Sideris

Instrumentation, Electrical Resistivity 776 M. H. Loke, O. Kuras, J. E. Chambers, D. F. Rucker and P. B. Wilkinson Instrumentation, EM Steven Constable

782

International Geophysical Year Ralph W. Baird

786

Lithospheric Magnetic Anomalies from Satellite Data Stavros Kotsiaros Magnetic Anisotropy Leonardo Sagnotti

915 923

CONTENTS

ix

Magnetic Anomalies: Interpretation E. Thébault

935

Mapping of Magnetic Variations from Space Nandini Nagarajan

1115

Magnetic Anomaly Map, Global Kumar Hemant Singh

943

Microcontinents Carmen Gaina and Joanne Whittaker

1120

Magnetic Data Enhancements and Depth Estimation Clive Foss

957

Numerical Methods for Flow in Fractured Porous Media Luca Formaggia, Alessio Fumagalli and Anna Scotti

1125

Magnetic Domains Susan L. Halgedahl

971

Magnetic Gradiometry Harald von der Osten-Woldenburg

983

Magnetic Gradiometry in Archaeo-geophysics Harald von der Osten-Woldenburg

987

Numerical Methods, Domain Decomposition Alfio Quarteroni and Luca Formaggia

993

Numerical Methods, Finite Difference Johan O. A. Robertsson and Joakim O. Blanch

1137

Magnetic Methods, Airborne Mike Dentith

999

Numerical Methods, Finite Element J. N. Reddy

1145

Magnetic Methods, Principles Kusumita Arora

Numerical Methods, Multigrid Wim A. Mulder

1149

Ocean Bottom Seismics Ingo A. Pecher, Jörg Bialas and Ernst R. Flueh

1155

Magnetic Methods, Surface Nandini Nagarajan

Numerical Methods, Boundary Element Michele Cooke

1007

Magnetic Modeling, Theory, and Computation 1015 Mioara Mandea, Carmen Gaina and Vincent Lesur

1130

1133

1029

Ocean, Spreading Center K. S. Krishna

1162

Magnetometers Ivan Hrvoic

1036

Orogenic Belts A. M. Celâl Şengör

1166

Magnetotelluric Data Processing Gary Egbert Magnetotelluric Interpretation John F. Hermance

1042

Paleomagnetic Field Intensity 1187 Andrew Biggin, Greig A. Paterson, Neil Suttie and John Shaw

Magnetotellurics, Crustal Imaging Prasanta K. Patro

1050

Paleomagnetism, Magnetostratigraphy Donald R. Prothero

Magnetovariation Studies Nandini Nagarajan

1056

Mantle Convection David Bercovici and Elvira Mulyukova

1059

Paleomagnetism, Measurement Techniques and Instrumentation T. Radhakrishna, J. D. A. Piper and Asanulla R. Mohamed

Mantle D00 Layer Thorne Lay

Paleomagnetism, Polar Wander Jean Besse, Vincent Courtillot and Marianne Greff

1215

1079 1085

Paleomagnetism, Principles William Lowrie

1225

Mantle Plume: Spreading Ridge Interactions K. S. Krishna and M. Ismaiel

1094

Paleoseismology Shinji Toda

1235

Mantle Plumes Cinzia G. Farnetani and Albrecht W. Hofmann Mantle Viscosity W. R. Peltier

1107

Plate Motions in Time: Inferences on Driving and Resisting Forces Giampiero Iaffaldano and Hans-Peter Bunge

1193

1202

1248

x

CONTENTS

Plate Tectonics, Precambrian Y. J. Bhaskar Rao, T. Vijaya Kumar and E. V. S. S. K. Babu

1256

Plate-Driving Forces Alessandro M. Forte

Seismic Discontinuities in the Transition Zone Lev P. Vinnik

1390

Seismic Hazard Andrzej Kijko

1394

1267 1274

Seismic Imaging, Overview Gerard T. Schuster

1407

Plates and Paleoreconstructions Alan G. Smith

1280

Seismic Instrumentation Duncan Carr Agnew

1419

Poroelasticity Ran Bachrach

Seismic Microzonation Fumio Yamazaki and Yoshihisa Maruyama

1425

Propagation of Elastic Waves: Fundamentals Francisco J. Sánchez-Sesma and Ursula Iturrarán-Viveros Radioactivity in Earth’s Core V. Rama Murthy Radiogenic Heat Production in the Continental Crust Claude Jaupart and Jean-Claude Mareschal Radiogenic Heat Production of Rocks Christoph Clauser Recovery of Source Magnetization Direction from Magnetic Field Data Clive Foss Remanent Magnetism Laurie Brown and Suzanne McEnroe

1283 Seismic Monitoring of Nuclear Explosions 1429 Paul G. Richards, Zhongliang Wu, Won-Young Kim and David P. Schaff 1293 Seismic Noise Dhananjay Kumar and Imtiaz Ahmed 1298 1304

Seismic Phase Nomenclature: The IASPEI Standard Johannes Schweitzer, Dmitry A. Storchak and Peter Borman

1442

1447

Seismic Properties of Rocks Nikolas I. Christensen

1459

Seismic Quiescence and Activation Gennady Sobolev

1464

Seismic Ray Theory Vlastislav Červený and Ivan Pšenčík

1472

Seismic Reservoir Characterization Dhananjay Kumar

1487

1310 1319

Remote Sensing and GIS Techniques for Tectonic Studies Semere Solomon and Woldai Ghebreab

1325

Remote Sensing, Applications to Geophysics Hojjatollah Ranjbar

Seismic Seiches Art McGarr

1492

1330 1335

Seismic Signals in Well Water Observations R. K. Chadha

1493

SAR Interferometry Masato Furuya

1343

Seismic Stereotomography Gilles Lambaré and Thibaut Allemand

1498

Satellite Altimetry Stelios P. Mertikas and Constantine Kokolakis

1349

Seismic Structure at Mid-Ocean Ridges Donald W. Forsyth

1502

Seafloor Spreading Richard N. Hey

1353

Seismic Tomography Guust Nolet

1507

Sedimentary Basins Magdalena Scheck-Wenderoth Seismic Data Acquisition and Processing Kabir Roy Chowdhury

1365

Seismic Diffraction Enru Liu

1385

Seismic Velocity and Temperature Relationships Kalachand Sain Seismic Viscoelastic Attenuation Vernon F. Cormier

1511 1512

CONTENTS

Seismic Wave Propagation in Real Media: Numerical Modeling Approaches Ursula Iturrarán-Viveros and Francisco J. Sánchez-Sesma

xi

1525

Seismology, Rotational William H. K. Lee

1685

Seismic Waves, Scattering Ludovic Margerin

1537

Sentinel Satellites Initiate New Era in Earth Observatories Randy Showstack

Seismic Zonation Yanxiang Yu, Mengtan Gao and Guangyin Xu

1550

Shear-Wave Splitting: New Geophysics and Earthquake Stress-Forecasting Stuart Crampin

1687

Seismic, Ambient Noise Correlation Michel Campillo, Philippe Roux and Nikolai M. Shapiro

1557

Seismic, Artificial Intelligence to Neural Intelligence for Advanced Interpretation 1562 Kalachand Sain and Priyadarshi Chinmoy Kumar Seismic, Migration Samuel H. Gray and Nittala Satyavani

1567

Seismic, Receiver Function Technique Rainer Kind and Xiaohui Yuan

1580

Seismic, Reflectivity Method Mrinal K. Sen

1592

1673

Single and Multichannel Seismics Tamás Tóth

1697

Slow Earthquake Teruyuki Kato

1705

Spherical Harmonic Analysis Applied to Potential Fields Nikolaos K. Pavlis

1714

Statistical Seismology David A. Rhoades, Annemarie Christophersen and Sebastian Hainzl

1724

Subduction Zones Geoffrey A. Abers

1728

Surface Waves Barbara Romanowicz

1738

1602 1607

T Waves Emile A. Okal

1751

Seismic, Velocity, and Density Relationships Kalachand Sain

1608

Tectonic Geomorphology Sampat Kumar Tandon and Vinee Srivastava

1753

Seismic, Waveform Modeling and Tomography Yanghua Wang and Ying Rao Seismicity, Intraplate Paul Bodin

1621

Seismicity, Subduction Zone Akira Hasegawa

1625

Seismogram Interpretation Ota Kulhanek and Leif Persson

1635

Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity Christoph Clauser

1644

Time-Dependent Seismic Hazard Assessment Zhongliang Wu

1787

Seismological Networks István Bondár and Eric Robert Engdahl

1655

Time-Reversal in Seismology Carène Larmat and Clarence S. Clay

1790

Seismology and Environment Jean-Paul Montagner, Anne Mangeney and Eléonore Stutzmann Seismology, Global Earthquake Model John F. Schneider, Jephraim Oro, Anselm Smolka, Peter Suhadolc and Zhongliang Wu

1661

Seismology, Monitoring of CTBT Zhongliang Wu and Paul G. Richards

1669

Seismic, Super-Virtual Refraction Interferometry Sherif M. Hanafy

Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat Christoph Clauser

TOPO-EUROPE: From the Deep Earth to the Surface of Continental Europe and Its Margins Sierd Cloetingh and TOPO-EUROPE Working Group Total Electron Content in Seismo-ionospheric Studies J. K. Catherine and R. Rajesh

1760

1769

1794

1802

xii

Tracking Earth’s Water in Motion from Satellite Gravity Observations J. L. Chen Trans-European Suture Zone H. Thybo and I. M. Artemieva Traveltime Tomography Using Controlled-Source Seismic Data Colin A. Zelt

CONTENTS

1813

Unified Scaling Law for Earthquakes that Generalizes the Fundamental Gutenberg-Richter Relationship 1893 Vladimir Kossobokov

1819 Vertical Seismic Profiling 1897 James W. Rector III and Maria-Daphne Mangriotis 1828

Very Long Baseline Interferometry Helmut Wiesemeyer and Axel Nothnagel

1902

Tsunami Steven N. Ward

1848

Wavelet Analysis Mikhail Kulesh

1909

Tsunami Watch and Warning Centers Shailesh Nayak, Srinivasa Kumar Tummala and E. Pattabhi Rama Rao

1868

Author Index

1917

Subject Index

1921

Tsunami: Bay of Bengal Vineet K. Gahalaut

1880

Uncertainties in Geomodelling Related to Geophysical Inversion Hans-Jürgen Götze and Florian Wellmann

1885

About the Editor

Harsh K. Gupta is currently a member of the Atomic Energy Regulatory Board (India) and president of the Geological Society of India. He has previously been a member of the National Disaster Management Authority (India); secretary to the Government of India, Department of Ocean Development; director of the National Geophysical Research Institute, Hyderabad; vice chancellor of Cochin University of Science and Technology; and professor at the University of Texas at Dallas, USA. He is internationally respected for his work on artificial water reservoir–triggered earthquakes and developing criteria to distinguish them from normal earthquakes. As leader of the third Indian scientific expedition to Antarctica during 1983–1984, he succeeded in setting up India’s first permanent Antarctic research station “Dakshin Gangotri” in just one Antarctic summer, which is still a record.

Prof. Gupta chaired the Steering Committee of the Global Seismic Hazard Program. After the disastrous 2004 Sumatra earthquake, he spearheaded the establishment of the Indian Tsunami Early Warning System. He has published over 200 articles in reviewed journals, authored 5 books that have been published by Elsevier and Springer. Prof. Gupta is a recipient of the Shanti Swarup Bhatnagar Prize, Waldo E. Smith Medal of American Geophysical Union, USSR Academy of Sciences “100 years of International Geophysics Memorial Medal,” the Axford Gold Medal of the Asia Oceania Geosciences Society (AOGS), and the National Mineral Award of Excellence, among many honors. He is a fellow of the Indian National Science Academy, the World Academy of Sciences, and American Geophysical Union. Prof. Gupta has been president of the IUGG and the AOGS and is the founder president of the Asian Seismological Commission.

Contributors

Geoffrey A. Abers Lamont-Doherty Earth Observatory Columbia University Palisades, NY, USA

Kusumita Arora National Geophysical Research Institute Council of Scientific and Industrial Research (CSIR) Hyderabad, India

Antal Adam Geodetic and Geophysical Research, Institute of the Hungarian Academy of Sciences Sopron, Hungary

I. M. Artemieva Marine Geodynamics GEOMAR Helmholtz Center for Ocean Research Kiel, Germany and Department of Geophysics Stanford University Palo Alto, CA, USA

Wim Aerts KU Leuven, Electrical Engineering (ESAT) TC Diepenbeek Campus Diepenbeek, Belgium Duncan Carr Agnew Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography University of California San Diego La Jolla, CA, USA Imtiaz Ahmed BP Houston, TX, USA

Estella Atekwana College of Earth, Ocean, and Environment University of Delaware Newark, DE, USA Jonathan M. Aurnou Department of Earth and Space Sciences University of California Los Angeles, CA, USA

Thibaut Allemand CGG Massy, France

E. V. S. S. K. Babu Geochronology Division CSIR-National Geophysical Research Institute Hyderabad, India

Richard M. Allen Berkeley Seismological Laboratory, Department of Earth and Planetary Science University of California Berkeley, CA, USA

Ran Bachrach Geophysics and Planetary Sciences Department Tel Aviv University Tel Aviv, Israel

Vasanthi Anthwar National Geophysical Research Institute Hyderabad, India

Ralph W. Baird Baird Petrophysical International Houston, TX, USA

xvi

CONTRIBUTORS

Daniel N. Baker Laboratory for Atmospheric and Space Physics University of Colorado Boulder, CO, USA

Joakim O. Blanch BHP Houston, TX, USA

G. Balmino Centre National d’Etudes Spatiales Toulouse, France

Paul Bodin Pacific Northwest Seismic Network (PNSN) University of Washington Seattle, WA, USA

Paramesh Banerjee Earth Observatory of Singapore Prayukti Geomatics Pte Ltd Singapore, Singapore Jean-Pierre Barriot Observatoire Géodésique de Tahiti University of French Polynesia Faaa-Tahiti, French Polynesia C. E. Barton Australian National University Canberra, ACT, Australia Graeme Beardsmore University of Melbourne Parkville, VIC, Australia and Hot Dry Rocks Pty Ltd Windsor, VIC, Australia David Bercovici Department of Earth and Planetary Sciences Yale University New Haven, CT, USA

Jean Besse Laboratoire de Paléomagnétisme Institut de Physique du Globe de Paris Paris, France

Archana Bhattacharyya Indian Institute of Geomagnetism Navi Mumbai, India Jörg Bialas Leibniz-Institut für Meereswissenschaften University of Kiel Kiel, Germany

Andrew Biggin Geomagnetism Lab, Department of Earth, Ocean and Ecological Sciences University of Liverpool Liverpool, UK

Sigrid Böhm Research Division Higher Geodesy, Department of Geodesy and Geoinformation TU Wien Vienna, Austria Marco Bohnhoff GFZ German Research Center for Geosciences Geomechanics and Scientific Drilling Potsdam, Germany István Bondár Research Centre for Astronomy and Earth Sciences Geodetic and Geophysical Institute Kövesligethy Radó Seismological Observatory Budapest, Hungary Peter Borman (Deceased) Carla Braitenberg Department of Mathematics and Geosciences University of Trieste Trieste, Italy Laurie Brown Department of Geosciences University of Massachusetts Amherst, MA, USA Carine Bruyninx Department of Reference Systems and Planetology Royal Observatory of Belgium Brussels, Belgium Hans-Peter Bunge Geophysics Section, Department of Earth and Environmental Sciences Ludwig Maximilians University of Munich Munich, Germany Evgueni Burov (Deceased)

CONTRIBUTORS

Friedrich H. Busse Institute of Physics, University of Bayreuth Bayreuth, Germany Michel Campillo Observatoire de Grenoble Université Joseph Fourier and CNRS Grenoble, France David J. Carlson British Antarctic Survey IPY International Programme Office Cambridge, UK Richard W. Carlson Department of Terrestrial Magnetism Carnegie Institution of Washington Washington, DC, USA J. K. Catherine CSIR-NGRI Hyderabad, India Anny Cazenave LEGOS, Observatoire Midi Pyrénées Toulouse, France Vlastislav Červený Department of Geophysics, Faculty of Mathematics and Physics Charles University Praha, Czech Republic

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J. L. Chen Center for Space Research University of Texas Austin, TX, USA Paolo Chiozzi Department of Earth, Environment and Life Sciences University of Genoa Genoa, Italy Nikolas I. Christensen Department of Earth and Ocean Sciences University of British Columbia Vancouver, BC, Canada Ulrich R. Christensen Max-Planck-Institut für Sonnensystemforschung Göttingen, Germany Annemarie Christophersen GNS Science Lower Hutt, New Zealand Christoph Clauser Institute for Applied Geophysics and Geothermal Energy RWTH Aachen University Aachen, Germany Clarence S. Clay (Deceased)

R. K. Chadha CSIR:National Geophysical Research Institute Hyderabad, India

Sierd Cloetingh Tectonics Group, Department of Earth Sciences Utrecht University Utrecht, The Netherlands

V. Chakravarthi Centre for Earth, Ocean and Atmospheric Sciences University of Hyderabad Hyderabad, India

Emily A. CoBabe-Ammann Research and Innovations Office University of Colorado Boulder, CO, USA

J. E. Chambers Geophysical Tomography Team British Geological Survey Keyworth, Nottingham, UK

Steven Constable Institute of Geophysics and Planetary Physics Scripps Institution of Oceanography La Jolla, CA, USA

Aude Chambodut Institut de Physique du Globe de Strasbourg Université de Strasbourg/Ecole et Observatoire des Sciences de la Terre, CNRS Strasbourg Cédex, France

Michele Cooke Geosciences Department University of Massachusetts at Amherst Amherst, MA, USA

David S. Chapman Department of Geology and Geophysics The University of Utah Salt Lake City, UT, USA

Vernon F. Cormier Physics Department University of Connecticut Storrs, CT, USA

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Vincent Courtillot Institut de Physique du Globe de Paris Paris, France Stuart Crampin British Geological Survey Murchison House Edinburgh, UK Shamita Das Department of Earth Sciences University of Oxford Oxford, UK Earl E. Davis Pacific Geoscience Centre Geological Survey of Canada Sidney, BC, Canada Michael G. Davis Department of Physical Sciences Arkansas Tech University Russellville, AR, USA Paul M. Davis Earth and Space Sciences UCLA Los Angeles, CA, USA Eduardo F. J. de Mulder Earth Science Matters Foundation Haarlem, The Netherlands Mike Dentith School of Earth Sciences (M004) The University of Western Australia Crawley, WA, Australia John S. Derr Tijeras, NM, USA

CONTRIBUTORS

Wolfgang Eder GeoCentre, Geobiology University of Göttingen Göttingen, Germany Gary Egbert College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, OR, USA Eric Robert Engdahl Center for Imaging the Earth’s Interior, Department of Physics University of Colorado at Boulder Boulder, CO, USA Cinzia G. Farnetani Institut de Physique du Globe de Paris Paris, France Andrew T. Fisher Department of Earth and Planetary Sciences University of California at Santa Cruz Santa Cruz, CA, USA Ernst R. Flueh Leibniz-Institut für Meereswissenschaften University of Kiel Kiel, Germany Luca Formaggia MOX, Department of Mathematics Politecnico di Milano Milan, Italy Donald W. Forsyth Department of Geological Sciences Brown University Providence, RI, USA

James W. Dewey U.S. Geological Survey, Denver Federal Center Lakewood, CO, USA

Alessandro M. Forte Department of Geological Sciences University of Florida Gainesville, FL, USA

Domenico Di Giacomo Department 2: Physics of the Earth GFZ German Research Center for Geosciences Potsdam, Germany

Clive Foss Mineral Resources CSIRO Lindfield, NSW, Australia

Vijay P. Dimri CSIR-National Geophysical Research Institute Hyderabad, India

Peter Fox Rensselaer Polytechnic Institute Troy, NY, USA

CONTRIBUTORS

Friedemann T. Freund Code SCR NASA Ames Research Center/San Jose State University Moffett Field, CA, USA

Samuel H. Gray CGGVeritas Calgary, AB, Canada

Jeffrey T. Freymueller Department of Earth and Environmental Sciences Michigan State University East Lansing, MI, USA

Marianne Greff Institut de Physique du Globe de Paris Paris, France

Alessio Fumagalli Department of Mathematics Politecnico di Milano Milan, Italy Masato Furuya Department of Natural History Sciences Hokkaido University Sapporo, Japan Vineet K. Gahalaut National Geophysical Research Institute (CSIR) Hyderabad, India Carmen Gaina Department of Geosciences, CEED – Centre for Earth Evolution and Dynamics University of Oslo Oslo, Norway Kerry Gallagher UMR 6118 – Géosciences Rennes Geosciences, Université de Rennes 1 Rennes Cedex, France Mengtan Gao Institute of Geophysics China Earthquake Administration Beijing, China

Richard A. F. Grieve Department of Earth Sciences University of Western Ontario London, ON, Canada and Institute for Earth and Space Exploration University of Western Ontario London, ON, Canada Gottfried Grünthal Helmholtz Centre Potsdam German Research Centre for Geosciences GFZ Potsdam, Germany Harsh K. Gupta National Geophysical Research Institute Council of Scientific and Industrial Research Hyderabad, India Pravin K. Gupta Department of Earth Sciences Indian Institute of Technology Roorkee Roorkee, Uttarakhand, India Ron Hackney Petroleum and Marine Division Geoscience Australia Canberra, ACT, Australia

Edward J. Garnero School of Earth and Space Exploration Arizona State University Tempe, AZ, USA

Sebastian Hainzl GFZ German Research Centre for Geosciences Potsdam, Germany

Elana Geddis Harbour Chambers Wellington, New Zealand

R. Hajra Indian Institute of Technology Indore Simrol, Indore, India

Woldai Ghebreab Department of Geology and Environmental Science University of Akron Akron, OH, USA Hans-Jürgen Götze Institute of Geosciences, Working group Satellite- and Aerogeophysics Christian-Albrechts-Universität zu Kiel Kiel, Germany

Susan L. Halgedahl Department of Geology and Geophysics University of Utah Salt Lake City, UT, USA David J. Hamilton Consultraining Edinburgh, UK

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CONTRIBUTORS

Sherif M. Hanafy College of Petroleum Engineering and Geosciences (CPG) King Fahd University of Petroleum and Minerals (KFUPM) Dhahran, Saudi Arabia Ulrich Harms Scientific Drilling GFZ German Research Centre for Geosciences Potsdam, Germany

Robert N. Harris College of Earth, Ocean, and Atmospheric Sciences Oregon State University Corvallis, OR, USA Hayo Hase Geodätisches Observatorium Wettzell Bundesamt für Kartographie und Geodäsie Bad Kötzting, Germany Akira Hasegawa RCPEV, Graduate School of Science Tohoku University Sendai, Japan Derrick Hasterok Earth Sciences Department University of Adelaide Adelaide, SA, Australia Michael Hearne U.S. Geological Survey Denver Federal Center Lakewood, CO, USA Yann Hello IRD Geoazur and Université Côte d’AZUR Sophia Antipolis, France

Jacques Hinderer Institut de Physique du Globe de Strasbourg, CNRS Université de Strasbourg Strasbourg, France Klaus-G. Hinzen Department of Geosciences Universtity of Cologne Cologne, Germany Albrecht W. Hofmann Lamont-Doherty Earth Observatory Columbia University Palisades, NY, USA and Max-Planck-Institut für Chemie Mainz, Germany Ivan Hrvoic GEM Systems, Inc Markham, ON, Canada Andrew Hugill Scintrex Ltd Concord, ON, Canada Giampiero Iaffaldano Department of Geosciences and Natural Resource Management University of Copenhagen Copenhagen, Denmark Anca Isac Geological Institute of Romania Surlari National Geomagnetic Observatory Bucharest, Romania M. Ismaiel Centre for Earth, Ocean and Atmospheric Sciences University of Hyderabad Hyderabad, India

John F. Hermance Department of Geological Sciences Brown University Providence, RI, USA

Alik Ismail-Zadeh Institute of Applied Geosciences Karlsruhe Institute of Technology Karlsruhe, Germany and Institute of Earthquake Prediction Theory and Mathematical Geophysics Russian Academy of Sciences (IEPT RAS) Moscow, Russia

Richard N. Hey Hawaii Institute of Geophysics and Planetology, School of Ocean and Earth Science and Technology University of Hawaii Honolulu, HI, USA

Ursula Iturrarán-Viveros Facultad de Ciencias, Departmento de Mathemáticas Universidad Nacional Autónoma de México, Ciudad Universitaria México City, Mexico

Jan Henninges Geoenergy GFZ German Research Centre for Geosciences Potsdam, Germany

CONTRIBUTORS

David D. Jackson Department of Earth and Space Sciences University of California Los Angeles Los Angeles, CA, USA Kishor S. Jaiswal U.S. Geological Survey Denver Federal Center Lakewood, CO, USA David E. James Department of Terrestrial Magnetism Carnegie Institution of Washington Washington, DC, USA Claude Jaupart Université de Paris Institut de Physique du Globe Paris, France Christopher Jekeli Division of Geodetic Science, School of Earth Sciences Ohio State University Columbus, OH, USA Philippe Jousset GFZ German Research Centre for Geosciences Potsdam, Germany Mikhail K. Kaban Department 1: Geodesy and Remote Sensing, Sektion 1.3: Earth-System Modelling Deutsches GeoForschungsZentrum Potsdam – GFZ Potsdam, Germany Yan Y. Kagan Department of Earth and Space Sciences University of California Los Angeles Los Angeles, CA, USA Masashi Kamogawa Division for Earthquake Prediction Research Global Center for Asian and Regional Research, University of Shizuoka Shizuoka, Japan Teruyuki Kato Earthquake Research Institute The University of Tokyo Bunkyo-ku, Tokyo, Japan Hartmut Kern Kiel, Germany

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Andrzej Kijko Natural Hazard Centre, Africa University of Pretoria Pretoria, Republic of South Africa Won-Young Kim Lamont-Doherty Earth Observatory Columbia University Palisades, NY, USA Rainer Kind Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences Potsdam, Germany and Freie Universität Fachrichtung Geophysik Berlin, Germany Jonathan F. Kirby School of Earth and Planetary Sciences Curtin University Perth, WA, Australia Miroslav Kobr Institute of Hydrogeology, Engineering Geology and Applied Geophysics Charles University in Prague Prague, Czech Republic Constantine Kokolakis Geodesy and Geomatics Engineering Laboratory, School of Mineral Resources Engineering Technical University of Crete Chania, Greece Monika Korte Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences Potsdam, Germany Vladimir Kossobokov Institute of Earthquake Prediction Theory and Mathematical Geophysics Russian Academy of Sciences (IEPT RAS) Moscow, Russia Stavros Kotsiaros Planetary Magnetospheres Lab NASA Goddard Space Flight Center Greenbelt, MD, USA and University of Maryland College Park College Park, MD, USA and Technical University of Denmark (DTU) Kongens Lyngby, Denmark

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CONTRIBUTORS

Charlotte M. Krawczyk Institute for Applied Geosciences Technical University Berlin Berlin, Germany and GFZ German Research Centre for Geosciences Potsdam, Germany K. S. Krishna Centre for Earth, Ocean and Atmospheric Sciences University of Hyderabad Hyderabad, India Jochem Kück Scientific Drilling GFZ German Research Centre for Geosciences Potsdam, Germany

Mikhail Kulesh Information Technology DIS AG Mahlow, Germany Ota Kulhanek Department of Earth Sciences, Section of Seismology Uppsala University Uppsala, Sweden Dhananjay Kumar Chevron Houston, TX, USA and BP Houston, TX, USA Priyadarshi Chinmoy Kumar Wadia Institute of Himalayan Geology Dehradun, India T. Vijaya Kumar Geochronology Division CSIR-National Geophysical Research Institute Hyderabad, India

Cristina La Mura Department of Geosciences University of Trieste Trieste, Italy Carlo Laj Laboratoire des Sciences du Climat Unité mixte CEA-CNRS-UVSQ Gif-sur-Yvette, France G. S. Lakhina Indian Institute of Geomagnetism New Panvel (W), Navi Mumbai, India Gilles Lambaré CGG Massy, France Carène Larmat Geophysics Group, EES-17, MS D542 Los Alamos National Laboratory Los Alamos, NM, USA Stanislaw Lasocki Department of Seismology Institute of Geophysics Polish Academy of Sciences Warsaw, Poland Thorne Lay Earth and Planetary Sciences Department University of California Santa Cruz, CA, USA William H. K. Lee U.S. Geological Survey Menlo Park, CA, USA Juliette Legrand Department of Reference Systems and Planetology Royal Observatory of Belgium Brussels, Belgium

Bhaskar Kundu Department of Earth and Atmospheric Sciences NIT Rourkela Rourkela, India

Vincent Lesur Université de Paris, Institut de physique du globe de Paris, CNRS Paris, France

O. Kuras Geophysical Tomography Team British Geological Survey Keyworth, Nottingham, UK

Kuo-wan Lin U.S. Geological Survey Denver Federal Center Lakewood, CO, USA

CONTRIBUTORS

Enru Liu China University of Mining and Technology Xuzhou, China Mian Liu Department of Geological Sciences University of Missouri Columbia, MO, USA M. H. Loke Geotomo Software Sdn. Bhd. Penang, Gelugor, Malaysia William Lowrie Institute of Geophysics Swiss Federal Institute of Technology Zürich, Switzerland Susan Macmillan Geomagnetism Team British Geological Survey Edinburgh, UK Raul Madariaga Ecole Normale Supérieure, Université PSL Paris, France Peter Malin GFZ German Research Center for Geosciences Geomechanics and Scientific Drilling Potsdam, Germany Peter Eric Malin Earth and Ocean Sciences Duke University Durham, NC, USA Kumarendra Mallick National Geophysical Research Institute Hyderabad, India Prantik Mandal Earthquake Seismology CSIR-National Geophysical Research Institute Hyderabad, Telangana, India Mioara Mandea CNES – Centre National d’Etudes Spatiales Paris, France Anne Mangeney Seismological Laboratory, Institut de Physique du Globe Université de Paris, UMR CNRS/7154 Paris, France

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Ajay Manglik National Geophysical Research Institute (Council of Scientific and Industrial Research) Hyderabad, India Maria-Daphne Mangriotis School of Geosciences and Grant Institute of Earth Science University of Edinburgh Edinburgh, UK and Institute of GeoEnergy Engineering Heriot-Watt University Edinburgh, UK

Kristin D. Marano U.S. Geological Survey Denver Federal Center Lakewood, CO, USA

Jean-Claude Mareschal Centre GEOTOP-UQAM University of Québec Montréal, QC, Canada

Ludovic Margerin Institut de Recherche en Astrophysique et Planétologie Observatoire Midi-Pyrénées/C.N.R.S. Toulouse, France

Yoshihisa Maruyama Graduate School of Engineering Chiba University Inage-ku, Chiba, Japan

Ali Masoudi ORC University of Southampton Southampton, UK Robert McCaffrey Department of Geology Portland State University Portland, OR, USA

James S. McClain Department of Earth and Planetary Sciences University of California, Davis Davis, CA, USA

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CONTRIBUTORS

Suzanne McEnroe Norwegian Geological Survey Trondheim, Norway Art McGarr Earthquake Science Center U.S. Geological Survey Menlo Park, CA, USA Allen K. McNamara Department of Earth and Environmental Sciences Michigan State University East Lansing, MI, USA

Elvira Mulyukova Department of Earth and Planetary Sciences Yale University New Haven, CT, USA Roger M. W. Musson British Geological Survey Edinburgh, UK and School of Geosciences University of Edinburgh Edinburgh, UK

Rolf Meissner (Deceased)

Toshiyasu Nagao Institute of Oceanic Research and Development Tokai University Shizuoka, Japan

Stelios P. Mertikas Geodesy and Geomatics Engineering Laboratory, School of Mineral Resources Engineering Technical University of Crete Chania, Greece

Nandini Nagarajan National Geophysical Research Institute Hyderabad, India

Uwe Meyer Sub-Department Geophysical Reconnaissance – Resources and Near Surface Processes Federal Agency for Geosciences and Mineral Resources Hannover, Germany Andrew J. Michael USGS Moffett Field, CA, USA Dinesh Chandra Mishra (Deceased) Asanulla R. Mohamed National Centre for Earth Science Studies Trivandrum, India Jean-Paul Montagner Seismological Laboratory, Institut de Physique du Globe University Paris-Diderot, UMR CNRS/7154 Paris, France Paul Morgan Colorado Geological Survey Denver, CO, USA Wim A. Mulder Faculty of Civil Engineering and Geosciences, Department of Geoscience and Engineering Delft University of Technology Delft, The Netherlands and Shell Global Solutions International BV Amsterdam, The Netherlands

Shailesh Nayak National Institute of Advanced Studies (NIAS) Bengaluru, India J. W. Neuberg Institute of Geophysics and Tectonics, School of Earth and Environment University of Leeds Leeds, UK Guust Nolet IRD Geoazur and Université Côte d’Azur Sophia Antipolis, France Axel Nothnagel VLBI Research Group, Institute of Geodesy and Geoinformation University of Bonn Bonn, Germany Emile A. Okal Department of Earth and Planetary Sciences Northwestern University Evanston, IL, USA Beata Orlecka-Sikora Department of Seismology Institute of Geophysics Polish Academy of Sciences Warsaw, Poland Jephraim Oro GEM Foundation Pavia, Italy

CONTRIBUTORS

Gordon R. Osinski Department of Earth Sciences University of Western Ontario London, ON, Canada and Institute for Earth and Space Exploration University of Western Ontario London, ON, Canada Simanchal Padhy Seismological Observatory CSIR-National Geophysical Research Institute Hyderabad, India

Dibyashakti Panda Department of Earth and Atmospheric Sciences NIT Rourkela Rourkela, India Giuliano F. Panza Department of Geosciences University of Trieste Trieste, Italy and Earth System Physics Section/Sand Group The Abdus Salam International Centre for Theoretical Physics Trieste, Italy

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Ingo A. Pecher GNS Science Lower Hutt, New Zealand and Institute of Petroleum Engineering Heriot-Watt University Edinburgh, UK W. R. Peltier Department of Physics University of Toronto Toronto, ON, Canada Leif Persson Department of Underwater Research FOI Stockholm, Sweden William K. Peterson Laboratory for Atmospheric and Space Physics University of Colorado Boulder, CO, USA J. D. A. Piper Geomagnetism Laboratory, Department of Earth and Ocean Sciences University of Liverpool Liverpool, UK

Mark A. Parsons Rensselaer Polytechnic Institute Troy, NY, USA

Donald R. Prothero Department of Geology Occidental College Los Angeles, CA, USA

Vincenzo Pasquale Department of Earth, Environment and Life Sciences University of Genoa Genoa, Italy

Ivan Pšenčík Department of Geophysics, Faculty of Mathematics and Physics Charles University Praha, Czech Republic and Institute of Geophysics Academy of Sciences of Czech Republic Praha, Czech Republic

Greig A. Paterson Geomagnetism Lab, Department of Earth, Ocean and Ecological Sciences University of Liverpool Liverpool, UK

Prasanta K. Patro CSIR-National Geophysical Research Institute Hyderabad, India Nikolaos K. Pavlis Geodesy and Geophysics Basic and Applied Research National Geospatial-Intelligence Agency (NGA) Reston, VA, USA

Alfio Quarteroni MOX, Department of Mathematics Politecnico di Milano Milan, Italy and CMCS-MATHICSE, EPFL Lausanne, Switzerland Vincent Quitoriano U.S. Geological Survey, Denver Federal Center Lakewood, CO, USA

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CONTRIBUTORS

T. Radhakrishna National Centre for Earth Science Studies Trivandrum, India and GITAM University Nagadenehalli, Bengaluru, India R. Rajesh CSIR-NGRI Hyderabad, India V. Rama Murthy (Deceased) G. Ramillien GET, Observatoire Midi Pyrénées Toulouse, France Sunita Rani Department of Mathematics Guru Jambheshwar University of Science and Technology Hisar, India Hojjatollah Ranjbar Department of Mining Engineering Shahid Bahonar University of Kerman Kerman, Iran E. Pattabhi Rama Rao Earth System Science Ogranisation–Indian National Centre for Ocean Information Services (ESSO–INCOIS) Hyderabad, India Ying Rao College of Geophysics China University of Petroleum (Beijing) Beijing, China Y. J. Bhaskar Rao Geochronology Division CSIR-National Geophysical Research Institute Hyderabad, India James W. Rector III Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, CA, USA

David A. Rhoades GNS Science Lower Hutt, New Zealand Paul G. Richards Lamont-Doherty Earth Observatory Columbia University Palisades, NY, USA Paul H. Roberts Institute of Geophysics and Planetary Physics University of California Los Angeles, CA, USA Johan O. A. Robertsson Institute of Geophysics, Department of Earth Sciences ETH-Zürich Zürich, Switzerland Fabio Romanelli Department of Geosciences University of Trieste Trieste, Italy Barbara Romanowicz Berkeley Seismological Laboratory Berkeley, CA, USA and Collège de France Paris, France Philippe Roux Observatoire de Grenoble Université Joseph Fourier and CNRS Grenoble, France Kabir Roy Chowdhury Department of Earth Sciences Utrecht University Utrecht, The Netherlands D. F. Rucker hydroGEOPHYSICS, Inc. Tucson, AZ, USA Ladislaus Rybach Institute of Geophysics ETH Zurich Zurich, Switzerland

J. N. Reddy Department of Mechanical Engineering Texas A&M University College Station, TX, USA

Leonardo Sagnotti Istituto Nazionale di Geofisica e Vulcanologia Rome, Italy

Thomas Reinsch GFZ German Research Centre for Geosciences Potsdam, Germany

Kalachand Sain Wadia Institute of Himalayan Geology Dehradun, India

CONTRIBUTORS

Malcolm Sambridge Seismology and Mathematical Geophysics Research School of Earth Sciences, The Australian National University Canberra, ACT, Australia Francisco J. Sánchez-Sesma Instituto de Ingeniería Universidad Nacional Autónoma de México, Ciudad Universitaria México City, Mexico William A. Sandham Scotsig Glasgow, UK John H. Sass (Deceased) Nittala Satyavani Seismic Group National Geophysical Research Institute Hyderabad, India David P. Schaff Lamont-Doherty Earth Observatory Columbia University Palisades, NY, USA Magdalena Scheck-Wenderoth Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences Potsdam, Germany and RWTH Aachen University Aachen, Germany John F. Schneider GEM Foundation Pavia, Italy Harald Schuh Department 1 “Geodesy” at Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences Potsdam, Germany

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Johannes Schweitzer NORSAR Kjeller, Norway Anna Scotti Department of Mathematics Politecnico di Milano Milan, Italy Mahmoud Mohamed Selim Saleh Department of Nature and Applied Sciences Al-Aflaj Community College, AL-Kharj University Al-Aflaj, Riyadh, Saudi Arabia Mrinal K. Sen Department of Geological Sciences, UT Institute for Geophysics, Jackson School of Geosciences The University of Texas at Austin Austin, TX, USA A. M. Celâl Şengör Faculty of Mines, Department of Geology and Eurasia Institute of Earth Sciences Istanbul Technical University Istanbul, Turkey Nikolai M. Shapiro Institut de Physique du Globe de Paris Paris, France Krishna Kant Sharma Department of Applied Geology University of Madras Chennai, India John Shaw Geomagnetism Lab, Department of Earth, Ocean and Ecological Sciences University of Liverpool Liverpool, UK Randy Showstack American Geophysical Union Washington, DC, USA and Independent Journalist Washington, DC, USA

Gerard T. Schuster Division of Environmental and Earth Sciences King Abdullah University of Science and Technology Thule, Saudi Arabia

Lydie Sichoix Observatoire Géodésique de Tahiti University of French Polynesia Faaa-Tahiti, French Polynesia

Martha Schwartz Department of Earth Sciences University of Southern California Los Angeles, CA, USA

Michael G. Sideris Department of Geomatics Engineering University of Calgary Calgary, AB, Canada

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CONTRIBUTORS

Kumar Hemant Singh Department of Earth Sciences Indian Institute of Technology Bombay Mumbai, India

Frank D. Stacey Division of Exploration and Mining CSIRO Kenmore, Australia

Sarva Jit Singh Indian National Science Academy New Delhi, India

Vaughan Stagpoole GNS Science Lower Hutt, New Zealand

Satish C. Singh Laboratoire de Géoscience Marines Institut de Physique du Globe de Paris Paris, France Lee Slater Department of Earth and Environmental Sciences Rutgers University Newark Newark, NJ, USA Daniel Slosky U.S. Geological Survey Denver Federal Center Lakewood, CO, USA Alan G. Smith (Deceased) Anselm Smolka GEM Foundation Pavia, Italy Gennady Sobolev Institute of Physics of the Earth Russian Academy of Sciences Moscow, Russia Semere Solomon Det Norske Veritas DNV Research and Innovation Høvik, Norway Xiaodong Song Department of Geology University of Illinois at Urbana-Champaign Urbana, IL, USA

Seth Stein Department of Earth and Planetary Sciences Northwestern University Evanston, IL, USA France St-Laurent LaSalle, QC, Canada Paul L. Stoffa Department of Geological Sciences, UT Institute for Geophysics, Jackson School of Geosciences The University of Texas at Austin Austin, TX, USA Dmitry A. Storchak International Seismological Centre (ISC) Thatcham, UK Eléonore Stutzmann Seismological Laboratory, Institut de Physique du Globe Université de Paris, UMR CNRS/7154 Paris, France Peter Suhadolc Department of Geosciences University of Trieste Trieste, Italy Neil Suttie School of Archaeological and Forensic Sciences University of Bradford Bradford, UK Christopher J. Swain School of Earth and Planetary Sciences Curtin University Perth, WA, Australia

Ravi P. Srivastava Equinor ASA Bergen, Norway

Laszló Szarka Geodetic and Geophysical Research, Institute of the Hungarian Academy of Sciences Sopron, Hungary

Vinee Srivastava Department of Earth and Environmental Sciences IISER Bhopal Bhopal, India

Sampat Kumar Tandon Department of Earth and Environmental Sciences IISER Bhopal Bhopal, India

CONTRIBUTORS

Donald H. Tarling School of Earth, Ocean and Environmental Sciences University of Plymouth Plymouth, UK E. Thébault Laboratoire Magma et Volcans, Université Clermont Auvergne (UCA), UMR CNRS 6524, Campus des Cézeaux, 6, Avenue Blaise Pascal 63178 AUBIERE, FRANCE Robert Thériault Québec Ministry of Energy and Natural Resources Québec, QC, Canada Eric M. Thompson U.S. Geological Survey Denver Federal Center Lakewood, CO, USA Clifford H. Thurber Department of Geoscience University of Wisconsin-Madison Madison, WI, USA H. Thybo Eurasia Institute of Earth Sciences Istanbul Technical University Istanbul, Turkey and Centre for Earth Evolution and Dynamics (CEED) University of Oslo Oslo, Norway

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Srinivasa Kumar Tummala Earth System Science Ogranisation–Indian National Centre for Ocean Information Services (ESSO–INCOIS) Hyderabad, India and ICG/IOTWMS Secretariat UNESCO IOC Perth Regional Programme Office West Perth, WA, Australia James A. Tyburczy School of Earth and Space Exploration Arizona State University Tempe, AZ, USA Seiya Uyeda Japan Academy Tokyo, Japan Franco Vaccari Department of Geosciences University of Trieste Trieste, Italy Petr Vaníček Department of Geodesy and Geomatics Engineering University of New Brunswick Fredericton, NB, Canada Nimisha Vedanti CSIR-National Geophysical Research Institute Hyderabad, India

Virendra M. Tiwari Gravity and Magnetic Studies Group CSIR- National Geophysical Research Institute (CSIR) Hyderabad, India

Massimo Verdoya Department of Earth, Environment and Life Sciences University of Genoa Genoa, Italy

Harold Tobin Department of Earth and Space Sciences University of Washington Seattle, WA, USA

Lev P. Vinnik Institute of Physics of the Earth Moscow, Russia

Shinji Toda Disaster Prevention Research Institute Kyoto University Uji, Kyoto, Japan

Harald von der Osten-Woldenburg National Heritage Department Regional Government of Baden-Wuerttemberg Esslingen am Neckar, Germany

Tamás Tóth Geomega Limited Budapest, Hungary

John M. Wahr (Deceased)

B. T. Tsurutani Jet Propulsion Laboratory California Institute of Technology Pasadena, CA, USA

David J. Wald U.S. Geological Survey Denver Federal Center Lakewood, CO, USA

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CONTRIBUTORS

Yanghua Wang Resource Geophysics Academy, Department of Earth Science and Engineering Imperial College London London, UK

C. Bruce Worden U.S. Geological Survey Denver Federal Center Lakewood, CO, USA

Steven N. Ward Institute of Geophysics and Planetary Physics University of California at Santa Cruz Santa Cruz, CA, USA

Zhongliang Wu Institute of Earthquake Forecasting China Earthquake Administration Beijing, China

A. B. Watts Department of Earth Sciences University of Oxford Oxford, UK

Guangyin Xu Institute of Geophysics China Earthquake Administration Beijing, China

Florian Wellmann Computational Geoscience and Reservoir Engineering RWTH Aachen University Aachen, Germany Lianxing Wen Department of Geosciences State University of New York at Stony Brook Stony Brook, NY, USA Kathryn A. Whaler School of GeoSciences University of Edinburgh Edinburgh, UK Joanne Whittaker Institute for Marine and Antarctic Studies (IMAS) University of Tasmania Hobart, TAS, Australia Helmut Wiesemeyer Millimeter and Submillimeter Astronomy Max-Planck-Institute for Radio Astronomy Bonn, Germany P. B. Wilkinson Geophysical Tomography Team British Geological Survey Keyworth, Nottingham, UK Ray Wood CRP-OCS Ltd Haumoana, New Zealand

Fumio Yamazaki National Research Institute for Earth Science and Disaster Resilience Tsukuba, Ibaraki, Japan Takashi Yoshino Institute for Planetary Materials Okayama University Misasa, Japan Yanxiang Yu Institute of Geophysics China Earthquake Administration Beijing, China Xiaohui Yuan Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences Potsdam, Germany Colin A. Zelt Department of Earth Science Rice University Houston, TX, USA Mark A. Zumberge Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography University of California, San Diego La Jolla, CA, USA

Preface to the Second Edition

The oldest surviving encyclopedia is Naturalis Historia by Pliny the Elder, a Roman statesman in the first century AD. It has 37 chapters covering the history, geography, geology, and various other aspects of the world. In today’s digital world of easy access to information, encyclopedias continue to be a valued source of reliable content contributed by experts. Subject encyclopedias contain in-depth entries offering essential and up-to-date information on the respective field of study, reflecting the current state of knowledge in the discipline as well as current trends. Encyclopedias published in electronic format are much easier to use than printed versions, given their search functions. All information about the Earth’s interior comes from field observations and measurements made within the top few kilometers of the surface, from laboratory experiments and from the powers of human deduction, relying on complex numerical modeling. Solid Earth geophysics encompasses all these endeavors and aspires to define and quantify the internal structure and processes of the Earth in terms of physical principles, corresponding mathematical formulations, and computational procedures. The role of solid Earth geophysics has gained prominence with increasing recognition of the fact that understanding Earth processes is central to the continued well-being of humanity. Apart from the exploration of natural resources, geophysical investigations provide crucial insights regarding the mutual relationships between climate and tectonics and on the effects of global change in terms of a broad range of natural hazards. The field continues to grow rapidly, both in fundamental and applied aspects, parallel to advances in allied fields of science and technology. The second edition of the Encyclopedia of Solid Earth Geophysics (ESEG) serves as a comprehensive compendium of information on important topics in solid earth geophysics and provides systematic and up-to-date

coverage of central concepts and key topics of interest, while making no claims to represent every aspect of this multi-disciplinary and multi-faceted field. The earliest version of the Encyclopedia of Solid Earth Geophysics, edited by Prof. David E. James, was published by Van Nostrand Reinhold publishing company in 1989. The extraordinary growth and diversification of the field over the next 20 years called for a major revision. A completely new work, namely Encyclopedia of Solid Earth Geophysics, published by Springer in 2011, was edited by Harsh K. Gupta. It brought together over 200 entries covering established and new concepts in geophysics across subfields such as gravity, geodesy, geoelectricity, geomagnetism, seismology, seismics, deep earth interior and processes, plate tectonics, geothermics, and computational methods in a consistent format. Less than 10 years later, the need for an updated second edition was driven by the desire to expand the scope of the entries, which would include the new knowledge derived through novel observations and techniques. Consequently, this edition includes 256 entries written by 347 authors, and includes 44 new topics, such as Fiber Optics in Geophysics, Tracking Earth’s Water in Motion, A New Era of Earth Observatories and Others Based on Satellite Observations, Deep Continental Boreholes Like the KTB, Seismic Arrays in Boreholes, Environmental Seismology, High-Frequency Seismology, Floating Seismographs, and several other new topics that have established themselves in recent years. The entries, written by leading experts, are intended to provide a holistic treatment of solid Earth geophysics and guide researchers to more detailed sources of knowledge should they require them. A basic understanding of solid Earth geophysics is essential for professionals in many allied disciplines, such as civil engineering, environmental sciences, and the mining, exploration and software Industries, and for NGOs working on large-scale social agendas,

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PREFACE TO THE SECOND EDITION

etc. This encyclopedia offers readers an authoritative and up-to-date reference source of extraordinary scope, drawing its unique strength from the contributions prepared by experts from around the globe. 2021 is the diamond jubilee year of the creation of the National Geophysical Research Institute, Council of Scientific & Industrial Research (CSIR-NGRI), where I spent

a major part of my academic life. It is a pleasure to commemorate the diamond jubilee of CSIR-NGRI with the publication of the second edition of ESEG. Harsh K. Gupta

Preface to the First Edition

All information about the Earth’s interior comes from field observations and measurements made within the top few kilometers of the surface, from laboratory experiments and from the powers of human deduction, relying on complex numerical modeling. Solid Earth Geophysics encompasses all these endeavors and aspires to define and quantify the internal structure and processes of the Earth in terms of the principles of physics, corresponding mathematical formulations and computational procedures. The role of Solid Earth Geophysics has gained prominence with increasing recognition of the fact that knowledge and understanding of Earth processes are central to the continued well being of the global community. Apart from persistent search for natural resources, this research line is linked to basic investigations regarding the mutual relationships between climate and tectonics and on the effects of global change in terms of a wide spectrum of natural hazards. Consequently, the pursuit of this science has seen spectacular progress all over the world in recent decades, both in fundamental and applied aspects, necessarily aided by advancements in allied fields of science and technology. The Encyclopedia of Solid Earth Geophysics aims to serve as a comprehensive compendium of information on important topics of Solid Earth Geophysics and provide a systematic and up-to-date coverage of its important aspects including primary concepts as well as key topics of interest. It, however, does not claim to chronicle each and every niche area that in reality is a part of this multidisciplinary and multi-faceted science. Neither does it attempt to describe the basic physics of matter and energy systems, which comprise the underlying tenets of geophysical research. The first edition of this Encyclopedia, edited by Prof. David James, was published in 1989 by the Van Nostrand Reinhold publishing company. The

extraordinary growth and diversification of this science over the last twenty years called for a complete revision. This is realized by identifying the necessary topics and bringing together over 200 articles covering established and new concepts of Geophysics across the sub-disciplines such as Gravity, Geodesy, Geoelectricity, Geomagnetism, Seismology, Seismics, Deep Earth Interior and Processes, Plate Tectonics, Geothermics, Computational Methods, etc. in a consistent format. Exceptional Exploration Geophysics and Geotechnical Engineering topics are included for the sake of completeness. Topics pertaining to near Earth environs, other than the classical ‘Solid Earth’, are not within the scope of this volume as it is felt that the growth of knowledge in these fields justify a dedicated volume to cover them. Articles written by leading experts intend to provide a holistic treatment of Solid Earth Geophysics and guide researchers to more detailed sources of knowledge should they require them. As basic understanding and application of Solid Earth Geophysics is essential for professionals of many allied disciplines such as Civil Engineering; Environmental Sciences; Mining, Exploration and software industries; NGOs working on large scale social agenda; etc., it would be useful to them to have access to a ready and up-to-date source of knowledge on key topics of Solid Earth Geophysics. Hopefully, this Encyclopedia would prove to be an authoritative and current reference source with extraordinary width of scope, drawing its unique strength from the expert contributions of editors and authors across the globe. I am grateful to Anny Cazenave, Kusumita Arora, Bob Engdahl, Seiya Uyeda, Rainer Kind, Ajay Manglik, Kalachand Sain and Sukanta Roy, members of the Editorial Board for their constant advice and guidance in developing

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the framework of this Encyclopedia and help with the editorial work. I am equally grateful to all the authors who readily agreed to contribute and honoured the guidelines and time schedule. Petra van Steenbergen, Sylvia Blago, Simone Giesler and D. Nishantini from Springer were very co-operative. It has been a pleasure working with Springer.

Ms. M. Uma Anuradha provided extraordinary assistance in the preparation of this volume. My wife Manju and daughters Nidhi & Benu supported me through the entire project. Harsh K. Gupta Editor-in-Chief

Acknowledgments

It has been a great pleasure working on the second edition of the Encyclopedia of Solid Earth Geophysics. My thanks go to Springer, particularly Petra Van Steenbergen and Sylvia Blago, for requesting it less than 10 years after the release of the first edition in 2011. I am also grateful to Anny Cazenave, Kusumita Arora, Bob Engdahl, Seiya Uyeda, Rainer Kind, Ajay Manglik, Kalachand Sain, and Sukanta Roy, the members of the editorial board, for their constant advice and guidance in developing the framework for this encyclopedia and support with the editorial work. I am equally indebted to the 347 authors who readily agreed to contribute and adhered to the guidelines and schedule. The manuscripts went through several iterations during their development, and at each stage, they were carefully reviewed by prominent experts in their respective fields,

numbering over 200 from all over the world. Please accept my heartfelt thanks. Petra van Steenbergen, Sylvia Blago, and Johanna Klute from Springer were very co-operative throughout this project, while Santhiya Rajarathinam helped with printing the manuscripts. It has been a pleasure working with Springer on this endeavor. M. Uma Anuradha provided extraordinary assistance in the preparation of this volume. I gratefully acknowledge the support provided by the CSIR-National Geophysical Research Institute, Hyderabad, and the National Academy of Sciences, India. Last but certainly not least, my wife Manju and daughters Nidhi and Benu gave their unflagging encouragement throughout the entire project. Harsh K. Gupta

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Absolute Age Determinations: Radiometric Richard W. Carlson Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, DC, USA

Definition Radiometric dating uses the decay of naturally occurring radioactive elements to determine the absolute age of geologic events.

for the ore (Rutherford 1906). By 1913, radioactive dating of rocks had made important inroads into geologic investigations of Earth history (Holmes 1913). The discovery of isotopes, also in 1913, and the improvement of the instruments used to measure their abundance (mass spectrometers) over the next decades, allowed radioactive dating to be applied to an increasingly broad range of natural materials, and hence, geologic processes. This, in turn, facilitated the transition of isotopic measurements from physics to geology departments and established radiometric geochronology as a tool of widespread use in the geosciences in the years following World War II.

Radioactivity and the Systematics of Its Use as a Chronometer Introduction Time is an essential datum in geology. Some geological processes, particularly those responsible for generating natural hazards, occur over time intervals that can be measured with conventional clocks. Most geologic processes, however, occur on timescales beyond human experience – thousands, millions, and even billions of years (see ▶ “Continental Drift”, ▶ “Continental Rifts”, ▶ “Geodynamics”, ▶ “Lithosphere, Continental”, ▶ “Paleomagnetism, Magnetostratigraphy”, ▶ “Paleoseismology”, and ▶ “Plate Tectonics, Precambrian”). For these, the chronometer of choice is radiometric dating, where the decay of naturally occurring radioactive elements is translated into time (Dickin 2005; Faure and Mensing 2005). In its application to geochronology, the most important aspect of radioactive decay is that it occurs at a known, constant rate, independent of environmental factors such as changing temperature and pressure, at least within the ranges of these parameters found outside the interior of stars. Only 10 years after the discovery of radioactivity, the renowned physicist Ernest Rutherford measured the amount of helium in a uranium ore and derived an age of 500 million years

© Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

Radiometric dating is based on the principle of nuclear transmutation, common to alchemy. Some elements have isotopes whose nuclei contain an unstable number of neutrons and protons. The instability is most often remedied by the ejection of material from the nucleus. Alpha decay involves the ejection of two protons and two neutrons (a 4-helium (4He) nucleus or alpha particle). Beta decay occurs through loss of an electron from the nucleus, turning a neutron into a proton. Another form of radioactive decay involves the capture of an electron from those surrounding the nucleus, turning a proton into a neutron. All forms of radioactive decay also release energy, and are thus a heat source in Earth’s interior (see ▶ “Radiogenic Heat Production of Rocks” and ▶ “Energy Budget of the Earth”). The result of radioactive decay is that an isotope of one element is transformed into an isotope of another element. While the path from lead to gold does not occur naturally, if one waits long enough, radioactive decay will eventually transform all uranium and thorium into lead. Table 1 lists the major radioactive nuclides that have been used for dating various geologic events. Naturally occurring radioactive isotopes are produced through four mechanisms:

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Absolute Age Determinations: Radiometric

Absolute Age Determinations: Radiometric, Table 1 Radioactive elements commonly used for absolute age determinations Parent isotope 7-Berylium (7Be) 210-Lead (210Pb) 226-Radium (226Ra) 14-Carbon (14C) 231-Protactinium (231 pa) 234-Uranium (234U) 36-Chlorine (36Cl) 26-Aluminum (26Al) 230-Thorium (230Th) 60-Iron (60Fe) 10-Berylium (10Be) 53-Manganese (53Mn) 107-Paladium (107Pd) 182-Hafnium (182Hf) 129-Iodine (129I) 244-Plutonium (244Pu) 146-Samarium (146Sm) 235-Uranium (235U) 40-Potassium (40K)

Production mechanism Cosmogenic Uranium decay Uranium decay Cosmogenic, bomb Uranium decay Uranium decay Cosmogenic, bomb Cosmogenic, stellar Uranium decay Stellar Cosmogenic, stellar Cosmogenic, stellar Stellar Stellar Stellar, cosmogenic Stellar Stellar Stellar Stellar

238-Uranium (238U) 232-Thorium (232Th) 176-Lutetium (176Lu) 187-Rhenium (187Re) 87-Rubidium (87Rb) 147-Samarium (147Sm) 190-Platinum (190Pt)

Stellar Stellar Stellar Stellar Stellar Stellar Stellar

1. Stellar nucleosynthesis where the very high temperatures and pressures present in stellar interiors fuse nuclei, creating new elements (Truran and Heger 2005). 2. Some radioactive elements decay into other radioactive elements. For example, 238-uranium requires eight alpha decays and six beta decays before it reaches stable 206-lead. Along the way, the decay steps include some isotopes with long enough decay lives to be useful in geochronology. These include 234U, 230Th, 226Ra, 222Rn, and 210Pb. 3. High-energy cosmic ray particles collide with atoms in Earth’s atmosphere or surface rocks with enough energy to cause nuclear reactions. For example, 14-carbon is created when 14-nitrogen in the atmosphere captures a neutron released by a cosmic ray interaction with some other atom. Other cosmicray-produced nuclides include 10Be and 26Al that are made by spallation, which occurs when an energetic cosmic ray proton simply breaks off a fragment from the nucleus of an atom with which it collides. Radioactive isotopes produced in these reactions are known as cosmogenic isotopes. 4. Above-ground detonations of nuclear bombs introduced into the environment substantial quantities of a number of radioactive species that have been used to investigate a number of atmosphere – Earth’s surface exchange processes, and for tracing ocean water circulation.

Daughter isotope 7-Lithium (7Li) 210-Bismuth (210Bi) 222-Radon (222Rn), 4He 14-Nitrogen (14N) 227-Thorium (227Th), 4He 230-Thorium (230Th), 4He 36-Argon (36Ar) 26-Magnesium (26Mg) 226-Radium (226Ra), 4He 60-Nickel (60Ni) 10-Boron (10B) 53-Chromium (53Cr) 107-Silver (107Ag) 182-Tungsten (182W) 129-Xenon (129Xe) Various fission products 142-Neodymium (142Nd) 207-Lead (207Pb), 4He 40-Argon (40Ar) 40-calcium (40Ca) 206-Lead (206Pb), 4He 208-Lead (208Pb), 4He 176-Hafnium (176Hf) 187-Osmium (187Os) 87-Strontium (87Sr) 143-Neodymium (143Nd), 4He 186-Osmium (186Os)

Half-Life (Million years) 53 days 22.3 years 1,622 years 5,730 years 0.033 0.25 0.31 0.73 0.75 1.5 1.6 3.7 6.5 9 15.7 80 103 704 1,270 4,469 14,010 35,700 41,600 48,800 106,000 450,000

The probability that a radioactive isotope will decay over any time interval is described by the equation:
dN=dt ¼ lN

ð1Þ

where N is the number of atoms of the radioactive species (parent isotope), t is time, and l is the rate constant of the decay. Rate constants often are converted to “half-life”; the time needed for half of the radioactive species to decay. The half-life of a radioactive species is equal to ln(2)/l. Integrating Eq. (1) gives: N ¼ N0 e
lt

ð2Þ

where N0 is the number of atoms present when t ¼ 0. This equation can be used to determine ages if one has independent information on the initial abundance of the parent isotope. For example, in 14C dating, if one assumes that the ratio of 14C to stable 12C is constant in the atmosphere, then one need only measure the present day 14C/12C ratio in a material that obtained its carbon from the atmosphere and use Eq. (2) in order to determine the material’s age. The assumption of a constant atmospheric 14C/12C ratio is now known to be invalid because the production rate of 14C in the atmosphere depends

Absolute Age Determinations: Radiometric

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on issues as diverse as sunspot activity and the varying strength of Earth’s magnetic field (see ▶ “Paleomagnetic Field Intensity”). Various schemes have been used to correct 14C chronology for variations in the atmospheric 14C production rate including comparison with the carbon in tree rings dated simply by counting the annual growth rings (Friedrich et al. 1999) or, for longer time intervals, to growth rings in corals dated with 230Th and 234U (Fairbanks et al. 2005). Although the variations in 14C production rate complicate 14C dating, they potentially provide information on the variability of Earth’s magnetic field, or sunspot activity, and how these parameters may affect Earth’s climate (Bard and Frank 2006). Because the initial abundance of the parent isotope generally is not known, most radioactive dating schemes measure the increase in the abundance of the decay product (daughter). If no atoms are lost, the abundance of the daughter isotope (D) increases with time at the rate described by:


 D ¼ D0 þ N elt
1

ð4Þ

Because mass spectrometers can measure isotopic ratios of some elements to precisions of 0.0005% whereas elemental abundances can only be determined to about 0.1–1% precision, Eq. 4, using the 238U-206Pb decay scheme as an example, can be expressed as: 

206

Pb=204 Pb

 m

¼



206

Pb=204 Pb

 0

þ



238

U=204 Pb


m

elt
1



ð5Þ

Thus, in an ideal situation where, for example, a mineral forms that contains radioactive 40 K, but no 40Ar, the age of the mineral can be determined simply by measuring the abundance of 40K (N) and 40Ar (D) in the mineral after some decay interval has passed. In nature, perfect separation of parent from daughter isotope is rare, so if some atoms of the decay product are present initially (D0), then:

where “m” stands for the measured ratio and l is the decay constant of 238U. Pb-204 is a stable isotope of lead. If two or more samples with different U/Pb ratios are formed at the same time with the same initial Pb isotope composition ((206Pb/204Pb)0), after some time has passed, plotting the measured 206Pb/204Pb for each sample against its 238U/204Pb ratio will produce a line, called an isochron, whose slope and y-intercept correspond to (elt – 1), and the initial 206Pb/204Pb ratio, respectively (Fig. 1a). Thus, the slope of an isochron provides the time when the samples on the line had the same Pb isotope composition. The U–Pb system is unique in that it contains two isotopes of U that decay into two isotopes of Pb. Thus, an analogous Eq. (5) can be written for the decay of 235 U to 207Pb. More importantly, the two equations can be combined to give:

a

b


 D ¼ N0
N ¼ N elt
1

ð3Þ

1.1421

0.518

1.1420

Whole rock

Light

142

0.512

0.510

1.1419

1.1418

Plagioclase 1.1417

0.508

0.506 0.00

Binda Slope corresponds to: 146Sm/144Sm = 0.00728

Heavy

143

Nd/144Nd

0.514

Pyroxene

Nd/144Nd

0.516

Eucrite binda Age = 4544 ± 88 Ma Initial 143Nd/144Nd = 0.50677

0.05

0.10

0.15

0.20

0.25

0.30

0.35

147Sm/144Nd

Absolute Age Determinations: Radiometric, Fig. 1 Isochron diagrams for the long-lived 147Sm–143Nd and short-lived, now extinct, 146 Sm–144Nd systems. The data shown are for minerals separated from the Binda eucrite, a type of meteorite that is a volcanic rock erupted on some small planetesimal. The meteorite consists primarily of the minerals plagioclase and pyroxene, which have different ratios of Sm to Nd

1.1416 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 144Sm/144Nd

when they form, but the same initial Nd isotopic composition. As Sm decays, it changes the isotopic composition of Nd. Using an equation similar to 5, the slope on the 147Sm-143Nd diagram directly provides the age of the sample, but on the 146Sm-142Nd diagram the slope provides the initial abundance of 146Sm (actually the 146Sm/144Sm ratio as illustrated in an equation analogous to Eq. 8). Data from Boyet et al. (2010)

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Absolute Age Determinations: Radiometric

 

207

Pb=204 Pb

206 Pb=204 Pb

 m






207



Pb=204 Pb


206 Pb=204 Pb  
 235 U=238 U el 5t
1

¼

m

 0 0

ð6Þ

ð e l 8t
1Þ

where l5 and l8 are the decay constants for 235U and 238U, respectively. Because the 235U/238U ratio is nearly constant in nature at any given time, this equation allows an age to be determined from measurements of lead isotopic composition alone, without the need for concentration determinations of U or Pb. Such ages, called Pb–Pb ages, are obtained by plotting (207Pb/204Pb)m versus (206Pb/204Pb)m, which will produce a line for two or more samples whose slope is equal to the right hand side of Eq. (6). This equation can then be solved numerically to determine “t”, the time when the samples formed. A very active application of radiometric chronology uses the variety of short-lived (half-lives of 105–108 year) radionuclides that were present when the solar system formed, but have long since decayed away (Carlson and Boyet 2009). For these systems, the “t” in Eq. (2) cannot be referenced to today because e–l t would always be close to zero. Instead, the “t” for these extinct systems must be a time interval referenced to a time when the parent isotope still existed. In this case, using the decay of 26Al to 26Mg as an example, Eq. (5) can be rewritten as: 

26

Mg=24 Mg

 Dt

¼



   Mg=24 Mg þ 26 Al=27 Al 0   0 ð7Þ 24
lDt 27 Al= Mg e 26

m

Plotting the two measurable parameters of this equation (27Al/24Mg, 26Mg/24Mg) for two or more samples will produce a line whose slope gives not the age, but the (26Al/27Al) at the time when the samples formed (Fig. 1b). The time interval between the reference time and the time of sample formation is then:    i  h Dt ¼
ln 26 Al=27 Al = 26 Al=27 Al =l Dt

0

ð8Þ

To convert these relative ages into absolute ages requires cross-calibration with a chronometer that provides absolute ages. For example, a certain type of mineral inclusion within the Allende meteorite provided a Pb–Pb age of 4567.6  0.4 Ma and an Al-Mg isochron whose slope corresponds to 26Al/27Al ¼ 4.96 ( 0.25)  10
5 (Jacobsen et al. 2008). This provides a “pinning point” for the 26Al/27Al ratio present in the solar system at a known time. Thus, if analyses of some other sample produced an Al–Mg isochron whose slope corresponded to, for example, 26Al/27Al ¼ 2.5  10
5, this sample would be one half-life of 26Al, or 750,000 years,

younger than the other. Extinct radioactive dating systems thus provide not absolute, but relative, ages. If these relative ages are anchored to an absolute age, as done in the example above, then the extinct system can be used to indicate that the hypothetical sample mentioned above is 4566.8 million years old. A good example of the use of extinct radionuclides is the attempt to date the oldest volcanism in the solar system. Several types of meteorites are interpreted to be lavas produced on small planetesimals. One of these, the D’Orbigny meteorite, has provided Al–Mg, Mn–Cr, Hf–W, and Pb–Pb ages of 0.5 Ma, 4564.0  0.6 Ma, 4562.8  1.5 Ma, and 4564.42  0.12 Ma, respectively. These results offer the promise of temporal precisions of 0.002% for the earliest events in solar system history.

The Application of Radiometric Dating The relatively simple equations given in the previous section provide the basis by which measurements of radioactive decay products can be translated into absolute ages. Applying these chronometers to natural processes requires matching the distinct properties of the many radioactive systems listed in Table 1 to the problem being investigated. Systems with short half-lives can be used to determine precise ages of geologic events only a few to a few thousand years old. Many of these short-lived nuclides are created either in the atmosphere or upper meter of Earth’s surface, which makes them particularly amenable to dating a variety of near-surface processes. The concentrations of 10Be or 26Al in quartz in rocks or sediments can be used to date the time when the rock was exposed to cosmic rays, which can provide information on both uplift and erosion rates (von Blanckenburg 2005) (see ▶ “Paleoseismology”). The rates of water movement from surface through subsurface aquifers can be traced with a number of radiometric chronometers (Bethke and Johnson 2008). The presence of 10Be in the lavas erupted at convergent margins tracks the subduction of surficial sediments to the 100 km + depths of magma generation (Morris 1991) (see ▶ “Subduction Zones”). Carbon-14 is used to date a huge variety of events from the eruption of young volcanoes, movement along faults (see ▶ “Earthquakes and Crustal Deformation”, ▶ “Earthquake, Archaeoseismology” and ▶ “Paleoseismology”), the rates of ocean water circulation and ocean uptake of atmospheric CO2, and the death of a living organism with its applications in archeology and paleoecology (Broecker 2005). The longer-lived radionuclides can be applied to a wide variety of issues concerning the evolution of the solid earth, from dating the formation of crustal rocks (see also ▶ “Paleomagnetism, Magnetostratigraphy”), to determining the time of continental assembly and breakup (see ▶ “Continental Drift” and ▶ “Plates and Paleoreconstructions”), to defining the age of the Earth.

Absolute Age Determinations: Radiometric

The key point in the application and proper interpretation of radiometric dating is understanding exactly what event is being dated. What “starts” a radiometric clock is some event that changes the composition of a sample and fixes this new compositional state until the sample is dated in the laboratory. For example, 14C often is used to determine the age of young volcanic eruptions. This system works in this application not by dating the rock itself, but by dating charcoal formed when the lava flowed over, and burnt, a bush or tree. What the 14C age actually dates is the time when the plant stopped exchanging carbon with Earth’s atmosphere. In most cases, this happened because the plant was killed by the volcanic eruption, in which case, the 14C age accurately dates the eruption. However, if a 100-year-old lava flow were to flow over a 10,000- year-old bristlecone pine tree trunk lying long dead on the ground, the 14C date of the charcoal would be 10,000 years, not 100 years. The precision of a radiometric age depends on the amount radioactive decay has changed the isotopic composition of the parent or daughter element. This depends primarily on the time passed relative to the half-life of the system, and in systems where measurement of the daughter element is involved, on the degree to which parent and daughter element are fractionated from one another. Thus, one would use 14C with its 5,730 year half-life to date an organic remain from ancient Rome, but it would be useless to date a million year old limestone because the 14C “clock” stopped measuring time when the 14C decayed away after a few half-lives into the history of the limestone. Similarly, 147Sm, with a 106 billion year half-life, is used to date a variety of materials ranging in age from millions to billions of years old, but would be useless to date very young processes because only 7  10
15 g of 143Nd is produced every thousand years per ppm of Sm, and this amount simply is too small to measure accurately. Besides choosing the proper radiometric system for the application, understanding whether a radiometric age is providing an accurate age for the event of interest demands an understanding of the processes that set, and reset, a radiometric clock. For Eq. (5) to provide an accurate age demands that the samples used to construct the isochron: (1) all formed at exactly the same time, (2) all had exactly the same initial daughter isotope composition, (3) experienced no change in parent/daughter element ratio between formation of the sample and its measurement in the laboratory, and (4) nothing other than the decay of the parent modified the daughter isotopic composition. In nature, all of these requirements have the potential to fail, but, in some cases, these “failures” can still provide useful information. A good candidate to meet the requirements above is a volcanic rock. Under conditions ideal for an accurate age, a well-mixed magma will crystallize, over a short time interval after eruption, a variety of minerals with different parent/

5

daughter elemental ratios, but the same initial daughter isotopic composition that they inherit from the magma. Over time, if the minerals experience no chemical exchange with their surroundings, the radioactive decay products will build up allowing their measurement to provide the accurate age of eruption of this volcanic rock. The Apollo 15 basalt 15,386 provides evidence that when these criteria are met, ages from many radioactive systems can agree. For this sample, the age (in billions of years) obtained from different radiometric systems are: K-Ar 3.89  0.04, Rb-Sr 3.88  0.01, Sm-Nd 3.85  0.08, and the U–Pb age for zircon from a compositionally similar rock is 3.89. For volcanic rocks this old, the assumption that the minerals all formed at the same time, at least relative to the ancient age of the rock, likely will be accurate. When one moves to younger volcanic systems, and the use of chronometers with better temporal resolution, the issue of the duration of crystallization becomes important. Several cases have now been presented where different minerals in the same volcanic rock give ages that range from the age of eruption to significantly older. One example is the Coso Volcanic field in California where zircons in one of the rhyolites range in age from the same as a K-Ar date on sanidine from the rhyolite, interpreted as the eruption age, to as much as 200,000 years older (Miller and Wooden 2004). This result suggests that emplacement and crystallization of distinct magma batches in the crustal magma chamber that eventually fed this eruption is a prolonged process occurring over many thousands of years. Similar results have now been reported in many young volcanic rocks (Cooper and Reid 2008). Although the age range of the minerals mentioned above is interpreted as reflecting a prolonged magmatic event, another way that an old crystal can get into a young magma is if a crystal from an older rock underlying the volcano is picked up by the magma during its ascent. Finding such “xenocrystic” material, particularly within more viscous, lower melting temperature, silica-rich magmas, is so common that it can be used to detect and even determine the age of older crust buried beneath a volcano, in some cases enabling the identification of older basement where none was previously known to exist (Smyth et al. 2007). This brings up what is possibly the most insidious cause of an inaccurate radiometric age: the generally poorly-known temperature history of the sample. Because the rate of chemical diffusion within solids increases with temperature, elevated temperatures facilitate chemical changes. Heating a mineral can cause gases, such as He and Ar that are present in a crystal lattice as a result of their production through radioactive decay, to diffuse out of the crystal and escape along grain boundaries. At a temperature of 1,000 °C, diffusion will cause a Sr atom in a mineral to move about 2 mm in a million years. If a small grain of a high Rb/Sr ratio mineral, such as mica, is next to a grain of a low Rb/Sr ratio, and high

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Sr content, mineral like plagioclase, the ingrowth of radiogenic 87Sr in the mica will be diluted by diffusional exchange with the unradiogenic Sr in the plagioclase. Diffusion thus makes all radioactive chronometers “thermochronometers” that date the time when the material of interest reached a low-enough temperature to stop diffusional resetting of the radiogenic ingrowth. Because rates of diffusion are a function both of the element that is diffusing and the structure of the crystal through which it is diffusing, different radiometric systems, and different minerals, can have different sensitivities to temperature excursions. This gives rise to the concept of “closure” or “blocking” temperature, above which the diffusion rate is too high to allow for the buildup of the radioactive decay product. In other words, the radiometric clock does not start until the mineral falls below its closure temperature. In the Coso example above, the K–Ar age obtained for the sanidine is interpreted as the eruption age because even if this mineral formed well prior to eruption, the 200–400 °C closure temperature of the K–Ar system in sanidine (Fig. 2) is so low that the radiogenic 40Ar being produced in the sanidine was diffusing out as fast as it was being produced because of magma temperatures in the range of ~750 °C. In contrast, the closure temperature of the U–Pb system in zircon is higher than the magma temperature, so as soon as the zircon first crystallized, its U–Pb clock started to record time. Temperature-induced diffusional resetting of radiometric systems may make interpretations of ages a bit more difficult, but it creates a new field – thermochronometry – that constructs time-temperature histories for various geologic processes (Reiners and Ehlers 2005). Though many radiometric systems can be used for thermochronometry, the oldest radiometric dating technique, the decay of U and Th to He, has been reenergized (Farley 2002) because the low closure temperature of the U,Th – He system can be used to address geological processes such as the rate of mountain uplift or surface erosion by telling the time when a rock arrived close enough to Earth’s surface to cool below the closure temperature of the radiometric clock (Reiners and Brandon 2006). While the appreciation and use of thermochronometry has been the primary growth field in radiometric geochronology in the last decade, other efforts are directed at finding ways to avoid reset ages. The traditional way to do this is to move to sample sizes much larger than diffusion lengths. This is the basis for “whole rock isochrons” where kilogram rock samples are homogenized and measured as an individual point on an isochron. Although this approach has produced some accurate ages, it suffers two main weaknesses. First, deciding whether all the rocks included on an isochron are the same age is not always easy, particularly for ancient, highly deformed, gneisses where this approach often is applied. Second, in order to obtain a sufficient range in parent/daughter ratio to precisely determine an isochron, compositionally distinct rocks are usually

Absolute Age Determinations: Radiometric Closure temperatures of various chronometers (U.Th)/He apatite Fission track apatite (U.Th)/He zircon (U.Th)/He titanite Ar-Ar potassium-feldspar Fission track zircon Ar-Ar biotite Fission track titanite Rb-Sr biotite Ar-Ar muscovite Rb-Sr muscovite U-Pb apatite U-Pb rutile Ar-Ar hornblende U-Pb titanite Sm-Nd garnet Th-Pb monazite U-Pb zircon 100

200

300 400 500 600 Temperature (°C)

700

800

Absolute Age Determinations: Radiometric, Fig. 2 Closure temperatures for various radiometric systems in different minerals. The fission track technique counts the density of damage tracks made in crystals due to the fission of 238U. Coupled with the measurement of U concentration in the crystal, the density of fission tracks can be converted into a radiometric age. Figure modified from (Pollard, 2002 – http://pangea.stanford.edu/~dpollard/NSF/main.html) which was adapted from a figure produced by P. Fitzgerald, S. Baldwin, G. Gehrels, P. Reiners, and M. Ducea

included on the isochron. Judging from modern igneous rocks, large compositional distinctions often are associated with variable initial isotope composition, which would violate the requirement that all the samples used to construct an isochron have the same initial isotope composition. Where whole rock isochrons can be useful is in detecting when much older components are involved in the genesis of younger rocks. A good example of this are the Sm-Nd data for lunar basalts where mineral isochrons of individual samples provide eruption ages generally less than 4 billion years, but the whole rock data scatter about a line corresponding to an age near 4.45 Ga (Carlson and Boyet 2009). The precision of this older age is debatable, as is its interpretation, but it could imply that the younger basalts are made by melting source materials in the lunar interior that formed from some global differentiation event on the Moon that occurred some time near 4.45 Ga. Memory of this event was erased from the minerals in the

Absolute Age Determinations: Radiometric

basalts, that know only of their crystallization from the magma, but the magmas themselves may “remember” the events that led to the formation of their source materials in the lunar interior. The other approach to avoiding thermally reset ages is to use minerals whose crystal structure allows only very slow diffusion. One curious example is diamond. Diamond cannot be dated directly because it contains insufficient amounts of the radioactive elements, but it often contains as inclusions minerals that can be dated. Even though these minerals were formed, and have been stored, at temperatures above 1,000 °C and depths greater than 150 km in the mantle (see ▶ “Lithosphere, Continental: Thermal Structure”), the encapsulation by the diamond does not allow them to communicate through diffusion with their surroundings. As a result, dating of inclusions in diamond show that diamond formation in the mantle has occurred over a good part of Earth history, with the oldest diamonds being approximately 3.5 Ga (Pearson and Shirey 1999) (see ▶ “Lithosphere, Continental”). By far, the most commonly used and most versatile way to avoid reset ages is with the application of the U–Pb system in zircon (Hanchar and Hoskins 2003). The strengths of U–Pb zircon dating include: (a) very high U–Pb closure temperature, (b) resistance to chemical modification through alteration and metamorphism, (c) high U concentrations, but initially vanishingly small Pb contents, (d) the U–Pb system includes two independent decay schemes that allow tests to determine whether or not the age has been disturbed (Wetherill 1956), (e) among the long-lived radioisotopes, the relatively short half-life of 235U allows highly precise ages to be obtained for ancient rocks, (f) both the trace element concentrations and Hf isotope composition in zircon provide information on the nature of the source rock from which the zircon was derived, and (g) U–Pb ages can be obtained in samples as small as fragments of individual grains or even spots, tens of microns in diameter, ablated with either lasers or ion beams. Perhaps the only weakness of zircon is that it is not found in all rock types, occurring primarily in felsic igneous rocks. These characteristics explain the many varied applications of zircon U–Pb dating that include the discovery of the oldest dated material on Earth, 4.36 Ga zircons from quartzites in western Australia (Wilde et al. 2001), to high-precision calibration of the geologic timescale (Bowring and Schmitz 2003), to the previously described use of zircons to determine the duration of crystallization in the magma chambers of young volcanic rocks.

Summary and Conclusions Continually improving analytical capabilities coupled with expanding appreciation of the physical controls on what sets, and resets, radiometric clocks are allowing everincreasing expansion of this now century-old approach. The

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variety of radioactive species now in use provide the ability to determine absolute ages for geologic processes that occur over timescales that range from days to the age of the Earth and solar system.

Cross-References ▶ Continental Drift ▶ Continental Rifts ▶ Earthquake, Archaeoseismology ▶ Energy Budget of the Earth ▶ Geodynamics ▶ Lithosphere, Continental ▶ Lithosphere, Continental: Thermal Structure ▶ Paleomagnetic Field Intensity ▶ Paleomagnetism, Magnetostratigraphy ▶ Paleoseismology ▶ Plate Tectonics, Precambrian ▶ Radiogenic Heat Production of Rocks ▶ Subduction Zones

Bibliography Bard E, Frank M (2006) Climate change or solar variability: what’s new under the sun? Earth Planet Sci Lett 248:1–14 Bethke CM, Johnson TM (2008) Groundwater age and groundwater age dating. Annu Rev Earth Planet Sci 36:121–152 Bowring SA, Schmitz MD (2003) High-precision U-Pb zircon geochronology and the statigraphic record. Rev Mineral Geochem 53:305–326 Boyet M, Carlson RW, Horan M (2010) Old Sm-Nd ages for cumulate eucrites and redetermination of the solar system initial 146Sm/144Nd ratio. Earth Planet Sci Lett 291:172–181 Broecker WS (2005) Radiocarbon. In: Keeling RK (ed) Treatise on geochemistry. Elsevier, Amsterdam, pp 1–18 Carlson RW, Boyet M (2009) Short-lived radionuclides as monitors of early crust-mantle differentiation on the terrestrial planets. Earth Planet Sci Lett 279:147–156 Cooper KM, Reid MR (2008) Uranium-series crystal ages. Rev Mineral Geochem 69:479–544 Dickin AP (2005) Radiogenic isotope geology, 2nd edn. Cambridge University Press, Cambridge Fairbanks RG, Mortlock RA, Chiu T-Z, Cao L, Kaplan A, Guilderson TP, Fairbanks TW, Bloom AL, Grootes PM, Nadeau M-J (2005) Radiocarbon calibration curve spanning 0–50, 000 years BP based on paired 230Th/234U/238U and 14C dates on pristine corals. Quat Sci Rev 24:1781–1796 Farley KA (2002) (U-Th)/He dating: techniques, calibrations, and applications. In: Porcelli D, Ballentine CJ, Wieler R (eds) Reviews in mineralogy and geochemistry: noble gases in geochemistry and cosmochemistry. Mineralogical Society of America, Washington, DC, pp 819–844 Faure G, Mensing TM (2005) Isotopes: principles and applications, 3rd edn. Wiley, Hoboken Friedrich M, Kromer B, Spurk M, Hofmann J, Kaiser KF (1999) Paleoenvironment and radiocarbon calibration as derived from late glacial/ early Holocene tree-ring chronologies. Quat Int 61:27–39 Hanchar JM, Hoskins PWO (2003) Zircon. Rev Mineral Geochem 53:500

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8 Holmes A (1913) The age of the Earth. Harper and Brothers, London Jacobsen B, Yin Q, Moynier F, Amelin Y, Krot AN, Nagashima K, Hutcheon ID, Palme H (2008) 26Al-26Mg and 207Pb-206Pb systematics of Allende CAIs: canonical solar initial 26Al/27Al reinstated. Earth Planet Sci Lett 272:353–364 Miller JS, Wooden JL (2004) Residence, resorption and recycling of zircons in Devils Kitchen rhyolite, Coso Volcanic field, California. J Petrol 45:2155–2170 Morris JD (1991) Applications of cosmogenic 10Be to problems in the Earth sciences. Annu Rev Earth Planet Sci 19:313–350 Pearson DG, Shirey SB (1999) Isotopic dating of diamonds. In: Lambert DD, Ruiz J (eds) Application of radiogenic isotopes to ore deposit research and exploration. Boulder, Society of Economic Geologists, pp 143–172 Reiners PW, Brandon MT (2006) Using thermochronology to understand orogenic erosion. Annu Rev Earth Planet Sci 34:419–466 Reiners PW, Ehlers TA (2005) Low-temperature thermochronology: techniques, interpretation, and applications. Rev Mineral Geochem 58:622 Rutherford E (1906) Radioactive transformations. Scribner’s, New York Smyth HR, Hamilton PJ, Hall R, Kinny PD (2007) The deep crust beneath island arcs: inherited zircons reveal a Gondwana continental fragment beneath East Java, Indonesia. Earth Planet Sci Lett 258:269–282 Truran JWJ, Heger A (2005) Origin of the elements. In: Davis AM (ed) Treatise on geochemistry. Elsevier, Amsterdam, pp 1–16 von Blanckenburg F (2005) The control mechanisms of erosion and weathering at basin scale from cosmogenic nuclides in river sediments. Earth Planet Sci Lett 237:462–479 Wetherill GW (1956) Discordant U-Pb ages. 1. Trans Am Geophys Union 37:320–326 Wilde SA, Valley JW, Peck WH, Graham CM (2001) Evidence from detrital zircons for the existence of continental crust and oceans on Earth 4.4 Gyr ago. Nature 409:175–178

Anthropogenic Seismicity Related to Exploitation of Georesources

description of a damaging seismic event, called a rockburst, refers to the lead mine in Derbyshire in England in 1738. With rising demand for energy and minerals, anthropogenic seismicity has been appearing in association with diverse technological processes, and also in areas previously known as aseismic. Seismic events can be induced by underground and open-pit mining, both conventional and unconventional hydrocarbon exploitation, and geothermal energy production, as well as many other technological processes that perturb the boundary conditions in the affected rock mass (Fig. 1). Although most of the induced seismic events are weak, hazards posed by anthropogenic seismicity can be significant. The severity of anthropogenic seismicity depends on the inducing technological activity. Figure 2 compares the largest seismic events induced by different georesources exploitation activities, respectively (source: The Human-Induced Earthquake Database (HiQuake), www.inducedearthquakes.org). The circled areas represent mutual proportions in the seismic energy of these events. All events presented here had significant damaging impacts.

Mechanisms of Interaction Between Technological Activity and the Rock Mass A simplified mechanism of earthquake generation can be quantified by the Coulomb failure criterion (Fig. 3). Critical conditions leading to the seismic slip on a fault are met when the shear stress in the direction of slip, t, equalizes the frictional strength of the fault, m(sn
p) + c, where sn is the compressive

Anthropogenic Seismicity Related to Exploitation of Georesources Stanislaw Lasocki and Beata Orlecka-Sikora Department of Seismology, Institute of Geophysics Polish Academy of Sciences, Warsaw, Poland

Definition Anthropogenic seismicity: Earthquakes induced or triggered by technological, usually georesources exploitation-related, activity.

Overview Exploitation of georesources alters the stress and strength conditions in the engaged rock mass. When such exploitation takes place in brittle rocks and yields stresses exceeding the rock strength at any point, the rock can fracture, violently emitting seismic waves. This anthropogenic seismicity was first observed as related to underground mining. The first

Anthropogenic Seismicity Related to Exploitation of Georesources, Fig. 1 Georesources exploitation activities that induce seismicity

Anthropogenic Seismicity Related to Exploitation of Georesources

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Groundwater extraction M=5.1 Lorca, Spain 11/05/2011 Hydrofracturing M=5.2 China 16/12/2018 Geothermal energy M=5.5 Pohang South Korea 15/11/2017 Wastewater disposal M=5.8 Oklahoma, USA 3/09/2016 Underground mining M=6.1 Kuzbas, Russia 18/06/2013 Water reservoir impoundment M=7.1 Montana, USA 17/08/1959 Conventional hydrocarbon exploitation M=7.5 Neftogorsk, Russia 27/05/1995 Cases in debate: M=7.9 Zipingpu (Wenchuan), China 12/05/2008

Anthropogenic Seismicity Related to Exploitation of Georesources, Fig. 2 The strongest seismic events induced by georesources exploitation. The circled areas represent mutual

proportions in the seismic energy of these events. (Source: www. inducedearthquakes.org)

Anthropogenic Seismicity Related to Exploitation of Georesources, Fig. 3 Scheme of conditions leading to seismic slip on a fault

stress normal to the fault plane, p is the pore pressure, c is the cohesion , and m is the coefficient of friction. The term (sn
p) is called the effective normal stress. The Coulomb criterion indicates that a fault can be brought to failure by reducing the normal stress, increasing the pore pressure , increasing the shear stress, decreasing frictional strength of the fault, or combinations of these factors. Different georesources exploitation activities interact in different ways with a rock mass, inducing seismicity. The schematic mechanisms of these interactions are shown in Fig. 4 (McGarr et al. 2002). The mass transfer leading to empty spaces formation results in the possible simultaneous reduction of the normal stress and increase of shear stress during mining. The

volume contraction caused by the reservoir depletion in conventional extraction of hydrocarbons decreases the pore pressure. However, for certain orientations of nearby faults the contraction can also lead to a decrease of normal stress and an increase of shear stress on the fault, as is demonstrated in the lower part of Fig. 4. The pore-pressure increase is a primary seismicityinducing factor for all technology using pressurized fluid injection, including secondary hydrocarbon recovery from conventional deposits, hydraulic fracturing for unconventional hydrocarbon production, underground disposal of liquid or gaseous wastes including carbon capture and storage (CCS) projects, enhanced geothermal systems (EGS), and underground storage of fluids and gases.

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Anthropogenic Seismicity Related to Exploitation of Georesources

Anthropogenic Seismicity Related to Exploitation of Georesources, Fig. 4 Schematic mechanisms of inducing seismicity by georesources exploitation. (After McGarr et al. 2002)

Induced, Triggered, or Natural Anthropogenic seismicity, for which technological activity provides most of the stress/rock strength perturbations needed for its generation, is called induced seismicity . For seismicity known as triggered, anthropogenic stress/rock strength disturbances meet preexisting, preloaded faults and push these faults to failures. In such cases human activity accounts for only a small fraction of the energy associated with the earthquakes, whereas tectonic loading has the primary role. The stress perturbations introduced by georesources exploitation are rarely capable of producing major earthquakes. Most of the severely damaging or devastating seismic events linked to technological activities are triggered. Because tectonic loading of preexisting faults increases with time, the fault pushed to rupture by a technological activity would have ruptured over time solely from increased tectonic loading, which would have occurred when the increasing shear stress on the fault plane had equalized the frictional stress. Moreover, if earthquake triggering is the result of an increase of pore pressure that weakens the fault, the triggered earthquake is weaker than the earthquake which would have occurred without the triggering technological activity. However, this realization is by no means comforting for people affected by the triggered earthquake. When an earthquake has occurred within the range of technological operations related to the exploitation of georesources, the essential question is posed: did the technological activity

triggered this earthquake, or was it a natural event caused solely by tectonic forcing? An accurate answer to this question would determine the extent of responsibility of the company whose activities may have led to the earthquake. However, because the reaction of a rock mass to georesources exploitation is complex and diverse, in spite of numerous attempts (e.g., Davis and Frohlich 1993; Dahm et al. 2013; Zhang et al. 2016) there are no general recommendations that, when followed, would result in unequivocally answering this question. Moreover, even for individual earthquake cases studied in detail, debates on whether these are natural or human triggered may continue for years. Such are the sequence of earthquakes in Gazli, Uzbekistan (1976: M7, M7; 1978: M5.7; 1984: M7) possibly linked to conventional gas production; the case of the Coalinga, USA ML6.5 earthquake in 1983, which might have been connected with oil production; the M5.1 Lorca, Spain earthquake in 2011, suspected to have been triggered by long periods of water pumping; and many others. Presently, it is often only possible to state whether human influence in the occurrence of a specific earthquake can or cannot be ruled out. This question was answered in the case of the sequence of devastating earthquakes that occurred in the Emilia-Romagna region, Italy, in May 2012. The sequence, with two main shocks of magnitudes approaching 6.0, killed 27 people, left more than 20,000 people homeless, and damaged or destroyed hundreds of structures of historical significance. The total loss has been estimated as some 5.4 billion euros. The general region as the Apennine foreland is seismically active, and this region has also had a long

Anthropogenic Seismicity Related to Exploitation of Georesources

history of georesources exploitation. In proximity to the earthquake sequence location, there were three active licenses for hydrocarbon production, one active gas storage concession, and one geothermal license. In the aftermath of the devastating earthquakes of May 2012, the Department of Civil Protection of Italy appointed an International Commission, ICHESE, to evaluate the possibility of an impact of these technological activities on the occurrence of the earthquakes (Styles et al. 2014). Analysis of the operational data from the hydrocarbon production license, the operation closest to the location of the earthquake sequence , showed that all parameters fluctuated, that is, monthly production of oil, monthly production of gas, average monthly reinjected volume of water, and average injection wellhead pressure. In particular, a rapid simultaneous change in trend from decreasing to increasing of monthly extracted and injected volumes of fluid and of the wellhead pressures occurred 1 year before the seismic crisis, in April– May 2011 (Fig. 5). This increasing trend lasted until the crisis in May 2012. Statistical tests indicated that the seismicity rate became greater in the period after April–May 2011, when all production parameters were increasing, than in the period before April–May 2011, when those parameters were decreasing. This result pointed to a link between technological operations in the area and the seismicity before the seismic crisis. At the same time the time–space clustering analysis indicated a strong connection between the seismic events that preceded the earthquake of May 20 and the subsequent major shock sequence. Furthermore, it was ascertained that the directions of the P and T axes within the group of events that occurred in the periods of increasing trends of operational parameters differed from the directions of these axes within the group of events that occurred in the periods of decreasing trends. The statistical analyses pointed also to the correlation between the P and T axes directions of events from the ‘increasing trends’ Anthropogenic Seismicity Related to Exploitation of Georesources, Fig. 5 Fluctuations of average monthly reinjected volume of water and average injection wellhead pressure in the hydrocarbon production license, the closest to the location of the Emilia-Romagna earthquake sequence of May 2012. In April– May 2011 (marked by the vertical bar), the trend of these parameters changed from decreasing to increasing. This change yielded an increase of seismicity rate. (After Styles et al. 2014)

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group with these axes directions of major shocks from May– June 2012. These observations indicated that the seismic activity within the time–space cluster of events which concluded with the May 20th main event was correlated with an increase of extraction and injection activity at the studied hydrocarbon production license. The ICHESE commission also carried out geomechanical modeling of potential triggers for earthquake initiation. It has been recognized that the stress changes generated by the activities of reservoir depletion were not sufficient in themselves to initiate movement on the fault resulting in the earthquake of May 2012. Similar results from modeling were obtained later by various groups investigating the Emilia–Romagna seismic crisis. These results might be seen as arguing in favor of a tectonic origin of the whole earthquake sequence. However, the significant statistical correlations between the changes in seismic activity characteristics before the May 20, 2012 event and the increase in production parameters since April–May 2011 could not be ignored. Bearing in mind limitations of modeling, and possible discrepancies between the actual behavior of the Earth and the modeled behavior, the ICHESE commission has concluded that the combined anthropogenic actions of extraction and injection of fluids in a tectonically active region, already close to criticality, may have contributed to producing the conditions required to produce a significant earthquake (Styles et al. 2014). This example illustrates the possible difficulties in discrimination between natural and human-triggered seismicity.

Specific Features Anthropogenic seismicity is by definition related to georesource exploitation. These relationships, the dependence

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of seismic processes on technological processes, mean that some vital properties of anthropogenic seismicity differ from the properties of natural seismicity . Locations of anthropogenic seismic sources are in some ways linked to locations of the inducing factors. Therefore, these sources are mostly, although not exclusively, shallower than the sources of tectonic earthquakes. This shallowness results in strong shaking in epicentral areas caused even by events that are relatively weaker, and in fast decay of amplitudes of shaking with increasing epicentral distance. Mining seismicity forms well-defined time–space clusters that are correlated in time and space with the time-varying location of mining work. Peak concentrations of seismicity locate at close distances ahead of mining work at and above unmined areas. Seismicity that is located at greater distances occurs mostly above the goaf and is usually linked to tectonic or mining discontinuities or weak zones. However, the detailed distribution of seismic events is mine specific. Fluid-injectionand fluid-extraction- induced seismicity are primarily related to the distribution of stress field changes in the subsurface surrounding the open hole of a well, the changes being caused by a combined effect of changes in pore pressure and changes in elasticity of the bedrock. Nevertheless, the distance at which earthquakes are triggered by fluid injection and the decay of seismic activity depend on the type of rock (Goebel and Brodsky 2018). In sedimentary rocks the pore pressure is transmitted to the surroundings more readily than in crystalline rocks. Consequently, induced seismic events extend further in sedimentary than in crystalline rocks. Moreover, in sedimentary rocks the seismic activity decays steadily with the distance from the injection point, whereas in crystalline rock the activity rapidly disappears beyond a certain distance. Resulting from the complicated seismogenesis of anthropogenic seismicity, the source mechanisms of induced earthquakes, contrary to those of tectonic earthquakes, often have prevailing nonshearing components (e.g., Vavryčuk 2011). These components are on the one hand indicators of local modifications of the stress field induced by exploitation, but they depend also on local tectonics and the location of a source with respect to the geometry of technological operations. In mines the pure shearing mechanism is associated with either the slip across a preexisting fault or the slip associated with fracturing of intact rock. Events with such focal mechanisms result from stress changes induced by mining, but they are usually remote from excavations, and their focal mechanisms are controlled mostly by ambient stresses (McGarr 2005). The mechanisms composed of shear and implosive components result from strong interaction between the seismic events and the adjacent mine stope and involve combinations of fault slip and co-seismic stope closure. The mechanisms composed of deviatoric and explosive components are associated with time-dependent development of a fracture zone in the immediate environment of mining excavations (McGarr 2005). The most hazardous seismic phenomena in underground mines are rockbursts and tunnel collapses. Mining collapses are

Anthropogenic Seismicity Related to Exploitation of Georesources

very shallow, located on excavating levels not deeper than about 3 km. Being this shallow, they generate significant surface vibrations and rapid ground deformation. Typically, their mechanisms are mostly isotropic with a high (more than 50%) implosive element. It has been shown by Rudziński et al. (2016) that a collapse source can be modeled with a simple source model called tabular cavity collapse using diagonal moment tensor density with the nonzero elements m11 ¼ m22 ¼
1 and m33 ¼
3 (Fig. 6). Massive fluid injections and extractions cause spatiotemporal variations of crustal stresses on the reservoir scale, resulting also in diverse source mechanisms of injection-induced seismicity (Segall and Fitzgerald 1998). Tensile components of source mechanisms observed in injection-induced seismicity result from high pore pressure. Within hydrocarbon reservoirs, the least horizontal stress decreases with declining reservoir pressure caused by extraction, leading to a relative horizontal tension within the reservoir. In extensional environments, dilatant fracturing and normal faulting are always promoted near the edges of the reservoir or in regions of a high pore-pressure gradient. In regional compressional environments, production may trigger reverse faulting above and below the reservoir. In geothermal reservoirs, thermoelastic effects are also present and they even may dominate the poroelastic stresses , especially near injection wells and steam-producing fractures (Segall and Fitzgerald 1998). Inducing technological operations continue only in a finite (from the human point of view) time period, and the operational parameters change with time. These time constraints cause time limitations and changes in time of the induced seismic processes. The time–space location of anthropogenic seismicity is transient. In contrast with tectonic earthquakes, by detailed insight the anthropogenic seismic processes can rarely be considered as stationary. Moreover, interactions among induced seismic events caused by co-seismic stress changes have been also identified in mines and in a geothermal reservoir (Orlecka-Sikora et al. 2012; Catalli et al. 2013). These stress changes influence the space–time patterns of seismicity. The simplest, although quite efficient, approach to infer the time changes of seismic activity assumes that the event occurrences follow a nonhomogeneous Poisson process with a slowly changing event rate so that interval stationarity is an acceptable approximation. The event rate is then modeled by the Poisson distribution in which the probability of n events occurring in the time period [t
Δ, t) is Pr½N ¼ n ¼ ½lðtÞDn
lðtÞD , where Δ is sufficiently small to allow for assumn! e ing the event rate, l(t), to be constant in this interval (Lasocki 2017 and references therein). There are also attempts to include operational parameters of the inducing technology into models of induced events occurrence (e.g., Shapiro et al. 2010; Mena et al. 2013; Garcia-Aristizabal 2018). However, such models are so far very case-specific.

Anthropogenic Seismicity Related to Exploitation of Georesources

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Anthropogenic Seismicity Related to Exploitation of Georesources, Fig. 6 Left: observed seismograms recorded on broadband stations after a strong mining collapse that occurred in Rudna Mine, Poland, in March 2013 (gray line) and synthetic (black line) traces generated for a tabular cavity collapse. Right: sketch of the contraction of a tabular cavity collapse and its source on the Hudson source-type plot. (By courtesy of Lukasz Rudziński)

With a less granular view of the problem, when seismicity is induced by a number of activities distributed in time and space (e.g., seismicity in a broader area with many operating injection wells), the impact of time changes of these individual factors may average itself, and the seismicity can present itself as stationary in longer time periods. However, even in such cases the anthropogenic event occurrence processes often deviate significantly from Poissonianity. For a Poisson process the interevent time (the time laps between two consecutive events) distribution is exponential, but observed interevent time distributions conform to Weibull or to even more complex distribution forms (e.g., SHEER Project, D4.2 2017). The magnitude distribution models most commonly used in tectonic seismology stem from a statistical relationship known as the Gutenberg–Richter relationship . This relationship links linearly the common logarithm of the number of events observed in consecutive, same size magnitude bins, n, to magnitudevalues ofthe bin centers, mm,log n¼ a
b· log mm or the common logarithm of the observed number, N, of events whose magnitudes are greater than or equal to any m to m, log N ¼ a
b · log m. The relationship is obeyed above a certain completeness magnitude, mc, which is the lowest magnitude of events, which are statistically all recorded

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in given monitoring conditions. The Gutenberg–Richter relation leads to exponential models of magnitude distribution, either unlimited or with upper limits. For the same reason as does the event rate, the distribution of magnitudes of anthropogenic seismic events also changes in time. Moreover, the variety of combinations of technological and geological factors inducing anthropogenic earthquakes may result in significant deviations of the observed magnitude distribution from the models derived from the Gutenberg–Richter relation. Such deviations have been statistically ascertained (e.g., Lasocki and Orlecka-Sikora 2008; Urban et al. 2016; SHEER Project D4.2 2017). The multimodality of magnitude distribution was indicated in some cases, which suggested the observed magnitudes to be mixtures of outcomes from different fracturing processes. Usually the deviations of observed magnitude distributions from the Gutenberg–Richter relation-based models are not dramatic and do not preclude the use of these models as handy approximations. Then, the changes in time of the slope parameter, b, must be taken into account. However, these deviations have significant consequences for seismic hazard assessment (Lasocki and Orlecka-Sikora 2008; Lasocki 2017 and references therein).

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Collaborative Undertakings for Research into Anthropogenic Seismicity Research into anthropogenic seismicity and related hazards requires an insight into relationships between the cause—the inducing technological activity, and the effect—the seismicity. Because of the complexity of induced seismic processes and the diversity and time-variability of the technological conditions responsible for generating the seismic events, these relationships are complex and difficult to quantify and generalize. Much progress in the understanding of these processes requires both deep integration of research groups active in the field and an access to high-quality data sets comprehensively describing both the inducing technological activity and the seismicity. In response to the increasing need for integration among scientists and engineers interested in induced seismicity, in 2007 the Triggered and Induced Seismicity Working Group (TAIS WG) was established in the International Association of Seismology and Physics of the Earth’s Interior (IASPEI). The goal of TAIS WG has been to summarize the present state of knowledge on induced and triggered seismic processes and to discuss future trends in the field. The discussions conducted within the TAIS WG inspired the creation of open-access services for studying anthropogenic hazards associated with exploitation of georesources. This goal has been realized by international teams in the form of the Thematic Core Service Anthropogenic Hazards (TCS AH) in the framework of the European Plate Observing System Program (EPOS-ERIC) (https://tcs.ah-epos.eu, Lasocki and Orlecka-Sikora 2016). The TCS AH consortium runs an e-research infrastructure, an IS-EPOS platform connected with the international data nodes that provides access to large sets of relevant data and allows free experimentation in a virtual laboratory. As well as its value for scientific work, the free access for all world researchers to this international facility enables proliferating discoveries and societal applications. The TCS AH consortium promotes interdisciplinary collaborations among all stakeholders: the scientific community, industrial partners, and society. The roadmap for TCS AH includes integrating worldwide research to develop capacities for addressing anthropogenic hazards challenges; further integrating and developing research infrastructures; increasing public understanding of anthropogenic hazards; driving innovation by transferring ready-to-use solutions to the industry; and transferring the expert knowledge and experience from industry to science.

Summary In response to rising demands for minerals, exploitation of georesources is being carried out in more and more difficult stress conditions. As a result, the appearance of anthropogenic seismicity in the affected rock mass has been, and will be, increasing.

Anthropogenic Seismicity Related to Exploitation of Georesources

Most induced seismic events are weak; however, the hazard posed by anthropogenic seismicity can be significant. In particular, when perturbations in the rock mass caused by exploitation of georesources meet tectonically preloaded faults, the triggered seismic events can cause material loss, injuries, and even fatalities. Exploitation of georesources often takes place in densely populated areas. Therefore, the accompanying seismicity is of socioeconomic significance. Although the hazard posed by anthropogenic seismicity is frequently overrated, public perception of these hazards and risks cannot be neglected. Vital georesources exploitation activities can lose public confidence unless the accompanying seismic risks are accurately assessed and properly presented to the public. Anthropogenic seismicity results from and is in the first place related to the evoking georesource exploitation processes. In particular, because the exploitation activities change in time, anthropogenic seismicity processes and the related hazards are intrinsically time dependent. The link of anthropogenic seismicity to the respective georesource exploitation activities suggests that potentially it is possible to maintain the anthropogenic seismic risk at an acceptable level by modifying operational practices. However, research into these cause–effect relationships requires multidisciplinary data collection with mandatory inclusion of relevant technological data. Further advances in the recognition of anthropogenic seismicity problems can be achieved only through problemoriented international integration of the research groups active in the field and deep science–industry synergy.

Bibliography Catalli F, Meier M-A, Wiemer S (2013) The role of Coulomb stress changes for injection-induced seismicity: the Basel enhanced geothermal system. Geophys Res Lett 40:72–77. https://doi.org/10. 1029/2012GL054147 Dahm T, Becker D, Bischoff M, Cesca S, Dost B, Fritschen R, Hainzl S, Klose CD, Kühn D, Lasocki S, Meier T, Ohrnberger M, Rivalta E, Wegler U, Husen S (2013) Recommendation for the discrimination of human-related and natural seismicity. J Seismol 17:197–202 Davis SD, Frohlich C (1993) Did (or will) fluid injection cause earthquakes? Criteria for a rational assessment. Seismol Res Lett 64:207–224 Garcia-Aristizabal A (2018) Modelling fluid-induced seismicity rates associated with fluid injections: examples related to fracture stimulations in geothermal areas. Geophys J Int 186:793–807 Goebel THW, Brodsky EE (2018) The spatial footprint of injection wells in a global compilation of induced earthquake. Science 361(6405):899–904 Lasocki S (2017) Probabilistic assessment of mining-induced timedependent seismic hazards, Chapter 11.3. In: Feng X-T (ed) Rockburst mechanisms, monitoring, warning, and mitigation. Butterworth-Heinemann (Elsevier), Oxford, UK, pp 366–380 Lasocki S, Orlecka-Sikora B (2008) Seismic hazard assessment under complex source size distribution of mining-induced seismicity. Tectonophysics 456:28–37 Lasocki S., Orlecka-Sikora B. (2016) Integrated approach to geophysical hazards induced by exploration and exploitation of georesources – to

Archaeomagnetism facilitate the way of attaining excellence. The EPOS Newsletter issue 01, July 2016, Article 03. https://www.epos-ip.org/news-press/eposip-newsletter, http://www.rich2010.eu/rich-success-stories McGarr A (2005) Observations concerning diverse mechanisms for mining-induced earthquakes. In: Potvin Y, Hudyma M (eds) Rockbursts and seismicity in mines: controlling seismic risk. Australian Centre for Geomechanics, Nedlands, pp 107–111 McGarr A, Simpson D, Seeber L (2002) Case histories of induced and triggered seismicity. In: Lee WHK, Kanamori H, Jennings PC, Kisslinger C (eds) International handbook of earthquake and engineering seismology, part A. Academic Press, London, pp 647–661 Mena B, Wiemer S, Bachmann C (2013) Building robust models to forecast the induced seismicity related to geothermal reservoir enhancement. Bull Seismol Soc Am 103:383–393 Orlecka-Sikora B, Lasocki S, Lizurek G, Rudziński Ł (2012) Response of seismic activity in mines to the stress changes due to mining induced strong seismic events. Int J Rock Mech Min Sci 53:151–158 Rudziński Ł, Cesca S, Lizurek G (2016) Complex rupture process of the March 19, 2013, Rudna mine (Poland) induced seismic event and collapse in the light of local and regional moment tensor inversion. Seismol Res Lett 87:274–284 Segall P, Fitzgerald SD (1998) A note on induced stress changes in hydrocarbon and geothermal reservoirs. Tectonophysics 289:117–128 Shapiro SA, Dinske C, Langenbruch C, Wenzel F (2010) Seismogenic index and magnitude probability of earthquakes induced during reservoir fluid stimulations. Leading Edge 29:304–309 SHEER Project, D4.2 (2017) Report on modelling the probability distributions used in seismic hazard analysis. www.sheerproject.eu/ images/deliverables/SHEER-Deliverable-4.2.pdf Styles P, Gasparini P, Huenges E, Scandone P, Lasocki S, Terlizzese F, International Commission on Hydrocarbon Exploration and Seismicity in the Emilia Region (ICHESE) (2014) Report on the hydrocarbon exploration and seismicity in Emilia Region. http://unmig.sviluppoe conomico.gov.it/unmig/agenda/dettaglionotizia.asp?id¼175 Urban P, Lasocki S, Blascheck P, do Nascimento AF, Van Giang N, Kwiatek G (2016) Violations of Gutenberg–Richter relation in anthropogenic seismicity. Pure Appl Geophys 173:1517–1537 Vavryčuk V (2011) Tensile earthquakes: theory, modeling, and inversion. J Geophys Res Solid Earth 116:B12320 Zhang H, Eaton DW, Li G, Liu Y, Harrington RM (2016) Discriminating induced seismicity from natural earthquakes using moment tensors and source spectra. J Geophys Res Solid Earth 121:972–993

Archaeomagnetism Donald H. Tarling School of Earth, Ocean and Environmental Sciences, University of Plymouth, Plymouth, UK

Synonyms Magnetic dating, archaeological; Remanence dating

Definition Archaeomagnetism is the study of all remanent magnetization associated with materials found in, or associated with, an

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archaeological context. In practice, it is the application of paleomagnetic techniques to archaeological materials, predominantly in terms of their uses in dating or “sourcing” such materials. This generally means that magnetic surveys are not usually considered “archaeomagnetic,” other than when used to provide information on the actual directions and intensities of the remanent magnetizations of the materials causing the anomalies. As most applications of archaeomagnetic studies relate to magnetic dating, these aspects are described first.

Basic Features Most materials within archaeomagnetic contexts contain iron oxide minerals that are capable of carrying a magnetic remanence acquired at some past time. More rarely, iron sulphides and hydroxides can carry such remanences. Even more rarely, a remanence can be carried by pure iron, nickel and cobalt and their alloys, and in very specific conditions, iron sulphates. The commonness of magnetite (Fe3O4) and hematite (Fe2O3) makes these almost always the dominant minerals involved. Such ferromagnetic (sensu lato) minerals can acquire a thermal remanence in the Earth’s magnetic field (or of nearby magnetic objects) while cooling from higher than ambient temperatures, chemical remanences as a result of crystalline or compositional changes. Some archaeological materials can comprise iron objects, but these are generally poor recorders of early remanences, being generally magnetically “soft.” All archaeological materials can become comminuted and eventually deposited as sediments (v.i. Archaeological sediments) in which the individual magnetic grains can be magnetically aligned during their deposition in aqueous or aeolian conditions, resulting in a detrital magnetization (see ▶ “Paleomagnetism, Principles” for details). In an archaeological environment, it is unlikely that only one form of remanence is present. It can also be presumed that some of the originally acquired remanence will have decayed with time, and other time-dependent magnetizations, viscous magnetizations, will have been acquired as the materials lay within a geomagnetic field that gradually changes in both direction and intensity. In order to determine the direction and intensity of the geomagnetic field for some specific time in the past (such as when last heated or deposited), it is necessary to remove such viscous magnetization. Where such viscous magnetizations are not associated with chemical changes, they can be removed by partial demagnetization in alternating magnetic fields or heating in zero-magnetic field in nonmagnetic furnaces. Alternating field demagnetization randomizes the low coercivity components of remanence, that is, the viscous remanences, leaving the high coercivity components unchanged. Partial thermal demagnetization reduces the relaxation time of all contained minerals, that is, the time

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taken for the direction of individual grain magnetization to relax into the ambient field direction. As the ambient field is zero, the grains with low relaxation times (those that carry the viscous magnetizations) thereby become randomized, leaving the high relaxation grains still in original directions. (For further detail, see ▶ “Paleomagnetism, Principles”.) Such partial demagnetization procedures are necessary for magnetic dating using directions or intensity. Statistical analyses, initially Principle Component Analyses, are then made to determine the number, direction, intensity, and precision of the magnetic vectors present in any single sample (see ▶ “Paleomagnetism, Measurement Techniques and Instrumentation” for details). The isolated individual sample vectors defined within some 5° that are considered to have been acquired originally are then combined, usually using Fisherian Statistics and giving equal weight to each sample vector, to provide a mean site direction. (Where individual oriented hand samples have been subsampled, these subsamples are combined to obtain the sample values that then yield the site mean value.) Such treatments normally require different instrumentation for dating based on directions than those for dating based on intensity and are described separately. The theoretical bases are the same.

Archaeological Site Sampling Directional Analyses Generally between 7 and 20 individually oriented samples are taken. This enables orientation errors to be averaged, as well as measurement errors. Ideally such samples are distributed evenly throughout the structure being sampled, although concentration is made on materials that are most likely to yield meaningful results and where the physical stability (since acquiring remanence) is better established. Sample orientation in situ is preferably undertaken nonmagnetically (sun compass, gyro-theodolite, or by sightings) as the objects themselves commonly distort magnetic compass readings. It is also desirable that the nature and conditions at the site, with the in situ samples, is recorded so that, when completing the analyses, possible causes from anomalous determinations can be better evaluated (see ▶ “Paleomagnetism, Measurement Techniques and Instrumentation”). Paleo-intensity Analyses Samples for such studies do not need to be oriented while in situ nor need the samples be still in their original positions in which their remanence was acquired. However, such orientations can be useful is assisting assessments of samples that show more than one remanent magnetic vector. Generally, some three to five samples are selected for study from any one archaeological site (see ▶ “Paleomagnetic Field Intensity”).

Archaeomagnetism

Magnetic Dating Based on the Direction of Remanence This dating method depends on the longtime spatial variations (>~10 years) of the geomagnetic field (see ▶ “Geomagnetic Field, Secular Variation”). Analyses of the present-day geomagnetic variations suggest that, at a medium latitude location, the observed geomagnetic field in a circular area of some 550 km radius (~1 Mkm) can be modeled as that of a geocentric inclined dipole field within an error of ~0.5° (solid angle) in direction. At greater distances, the error increases to >1° (solid angle) rendering the model increasingly unreliable for comparing the directions of remanence at more distant sites. Such modeling enables “Master Curves” (Paleo-Secular Variation Curves) of directional change to be constructed using previously studied sites for specific regions, usually for a specific country or groups of small countries. In the UK, most English and Welsh observations were corrected to a central location, Meriden, while French observations were generally corrected to Paris. Master Curves are now more commonly calculated within a far more sophisticated statistical modeling, notably hierarchical Bayesian modeling (Lanos et al. 2005). In such analyses, the Gaussian errors for the date assignments as well as for directional properties are incorporated in the construction of the Master Curves. Such Bayesian statistics similarly incorporate harmonic analyses. Spherical Cap analyses have the objective to determine the geomagnetic field variations attributable to fields at the Earth’s core (Pavón-Carrasco et al. 2008). The latter method is therefore more appropriate for geomagnetic field modeling (see ▶ “Geomagnetic Field, Theory”). Relative Dating Nearby sites (within some 50–80 km) that acquired their remanence at the same time will exhibit identical (within 1°) directions of remanence. Consequently, nearby sites with differing directions of remanence can be considered to have acquired their remanence at different times. Conversely, similar directions can indicate similar ages for their remanence acquisition. However, identical geomagnetic directions can occur in the same location at different times. When such repeat directions occur at time intervals of a few 100 years, the archaeological context is often adequate to distinguish between alternative age assignments. Chronological Dating This dating method is dependent on the secular variation of the geomagnetic field being known for the region being investigated (see ▶ “Geomagnetic Field, Secular Variation”). As the current geomagnetic theories for the origin of the geomagnetic field are still evolving (see ▶ “Geomagnetic Field, Theory”), this usually requires previously well-dated

Archaeomagnetism

(by non-archaeomagnetic means) site values to be available. In newly studied areas, the method is clearly inhibited by the lack of such data, but as more and more observations are obtained, the past directions of the geomagnetic field become increasingly well defined. Archaeomagnetic dating is thus an unusual scientific dating technique as its precision increases as more observations accumulate. Magnetic Polarity and Polarity Excursion Dating This dating application is based on unusual aspects of the long-term geomagnetic behavior. On scales of a few thousand years, the geomagnetic field can undertake major excursions away from its normal direction, geomagnetic events, or excursions. On timescales of a few 100,000 years, the Earth’s magnetic field changes polarity – the North magnetic pole becoming the South magnetic pole, and vice versa for the South magnetic pole. Consequently, some geomagnetic excursions occur on archaeological timescales, and the younger polarity changes occur on archaeological-anthropological timescales (see ▶ “Geomagnetic Excursions” and ▶ “Geomagnetic Field, Polarity Reversals” for more detailed considerations).

Magnetic Dating Based on the Intensity of Remanence Relative dating can be based on the determination of paleointensities of the ancient field in a similar way as for direction (qv). Most current studies are directed toward defining the changes in geomagnetic intensity during archaeological time although sufficient quantities of high quality paleo-intensity determinations are now becoming available for specific regions, enabling chronological dating to be undertaken. As for directional studies, when past records of the Earth’s magnetic field intensity are available, a Master Curve can be constructed for a similar circular area of ~ 1 Mkm2, radius of ~550 km, as for directional studies.

Errors General Errors These apply to both directional and intensity dating techniques. 1. It is usually assumed that any effects of weathering and other environmental changes are minimal and that the magnetic grains carrying the remanence of interest are essentially unaffected. However, some weathering products, such as goethite and hematite, can carry high-stability remanence. It is desirable that, as far as practicable, unweathered materials are collected.

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2. During partial demagnetization, it is assumed that no chemical or structural alterations occur to the existing magnetic minerals and that no new magnetic minerals are formed. Such changes are far more likely during thermal demagnetization than by alternating magnetic field techniques and are usually monitored by repeated measurements of the initial susceptibility. This is, however, a coarse measure and magnetic domain changes (see ▶ “Magnetic Domains”) which can occur with no clear change in susceptibility. 3. Both directions and intensities can be affected by magnetic anisotropy (shape and crystalline) that may be associated with the sample shape, the shape of contained minerals and the orientations of their crystallographic axes (see ▶ “Magnetic Anisotropy”). 4. Lightning strikes at times subsequent to the original acquisition of remanence, and have electromagnetic effects that can drastically alter the magnetic remanence. Although mostly associated with electrical charges that usually travel through surface waters, these effects can penetrate. Such effects can usually be recognized by their high intensities and tendency to be in random directions. 5. The local geomagnetic field direction and intensity may well be distorted if there were nearby magnetized objects, natural or manufactured, at the time of remanence acquisition. 6. The behavior of the geomagnetic field is still poorly defined and, particularly at high magnetic latitudes, can show major diurnal variations. It also appears that secular variations are not necessarily smooth, but may occur in “jerks” (see ▶ “Geomagnetic Field, Secular Variation”) although these appear to be mostly within the current experimental error of the technique. 7. The geomagnetic field over time can return to previous direction and intensity values. When such “crossover” points occur in the Master Curves at sufficiently different times, then the archaeological age constraints on the site may enable the most likely chronological age to be applied. However, most extant Master Curves are likely to still be distorted by some studies for which the archaeological or chronological age has subsequently been modified. However, such “anomalous” sites becoming increasingly recognized as databases increase. Errors Specific to Directional Dating In addition to the general errors, directional studies are sensitive to the oriented samples having remained in situ since they acquired their remanences. In a solid structure, such loose materials can be readily identified, but individual parts of some structures can have undergone motions relative to the rest of the structure, for example, kiln-wall “fall-out” or “fallin.” On hillside, it is also possible that the whole structure may have undergone motion away from where the remanence was

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acquired. Thus, a knowledge of the archaeological controls on possible motions of the site is essential. Errors Specific to Intensity Dating In addition to the general errors, paleo-intensity studies are sensitive to the “cooling rate” of the acquired magnetic properties. The naturally acquired remanence will commonly have been acquired over times that can be of the order of several days, while the laboratory studies are necessarily on far shorter times scales. The laboratory redox conditions during cooling are normally very different. Overall Error Assessment While there are many potential sources of error in archaeomagnetic dating, in most archaeological sites where the conditions are clear and the materials largely pristine, the errors arising appear to be only rarely of sufficient magnitude to warrant rejection.

Specialized Applications Murals and Plaster Remarkably, it has been shown that pigments can carry a remanence relating to the time when they were applied. Where such pigments have been applied to walls, their remanence has been shown to relate to the time that the pigments were applied, particularly in tempera paintings – either originally or during subsequent retouching. It seems most likely that this remanence is associated with hematite and was acquired “detritally,” that is, by rotation of the grains while still fluidized (Lanza et al. 2009). Similar observations have also been shown for plaster work (Soler-Arechalde et al. 2006). Object Reconstruction Specific artifacts, such as pottery, when fired, obtain a uniform magnetization in the ambient geomagnetic field as they cool within the furnace. If broken, individual shards retain this original remanence and can be reassembled to assist in defining the original shape of the pot (Burnham and Tarling 1975).

Archaeological Sediments As sediments are deposited within an archaeological site, any contained already magnetized grains are influenced by the ambient geomagnetic field. As these alignment forces are weak (compared with the effects of gravity and flow motions) only a percentage of the magnetic grains are aligned and such alignments can be gravitational flattening during the deposition process. However, the more magnetically stable grains (usually with single-domain dimensions of ~1 mm) can rotate into full alignment while they are still fluidized immediately after deposition. However, such alignments can be modified by meniscus

Archaeomagnetism

forces as the water table subsequently rises or falls. Sedimentary minerals also undergo a series of chemical reactions as they adjust to their new environment. Consequently, it is often difficult to establish the true direction of the geomagnetic field at the time of deposition with sufficient certainty to assign ages based on the magnetic determinations. However, there are archaeological sites, such as basal sediments, deposited immediately after a ditch had been cut, that appear to be recording the geomagnetic field direction at that time.

Magnetic “Sourcing” Magnetic sourcing attributes an object to a particular location on the basis of its magnetic properties. For example, an obsidian arrowhead can sometimes be sourced to the obsidian outcrop from which it was originally fashioned. The physical basis for this is that the compositions and grains sizes of the magnetic minerals within a rock can be specific to a particular outcrop. Each outcrop also tends to have a slightly different cooling history and became magnetized by a specific geomagnetic field strength. Several magnetic properties can be rapidly and cheaply measured, such as its initial (low field) susceptibility, its saturation remanence, and its intensity of natural remanence, are sufficient to identify a specific outcrop. Generally, the range of magnetic properties in most sedimentary rocks is too great to allow such assignments, but obsidians, rapidly cooled volcanic glasses, commonly have a range of magnetic properties that are either unique to that outcrop or to only two or three outcrops.

Summary The study of the magnetic properties of archaeological materials, archaeomagnetism, is particularly significant in geophysics as it enables secular variations in the direction and strength of the geomagnetic field over timesscale that are far longer than those for direct measurement (~400 years). In comparable magnetic studies of rocks, paleomagnetism, these secular variations are commonly averaged out during the processes by which rocks acquire their remanent magnetization. Conversely, archaeomagnetic records provide an additional dating tool for archaeologists. As a comparative dating tool, archaeomagnetism enables relative dating between nearby samples (100–200 km) within ~ 10–20 years. As a dating tool, it is mainly dependent on the validity of the ages used to compile the regional Master Curve. It is thus unique, as a scientific dating method, in that as more data are acquired, age “outliers” can be identified and their age reassessed. Consequently, the Master Curves are continually improving and, ultimately, could have the same precision as for relative dating. There are also interesting applications in using such observations in conservation studies and establishing past environmental conditions.

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

Cross-References ▶ Geomagnetic Excursions ▶ Geomagnetic Field, Polarity Reversals ▶ Geomagnetic Field, Secular Variation ▶ Geomagnetic Field, Theory ▶ Magnetic Anisotropy ▶ Magnetic Domains ▶ Paleomagnetic Field Intensity ▶ Paleomagnetism, Measurement Techniques and Instrumentation ▶ Paleomagnetism, Principles

Bibliography Batt CM, Zanarini I (eds) (2008) Archaeomagnetic applications for the rescue of cultural heritage (AARCH). Phys Chem Earth 33(6–7):403–608 Burnham RJP, Tarling DH (1975) Magnetisation of shards as an assistance to the reconstruction of pottery vessels. Stud Conserv 20:152–158 Lanos P, Le Goff M, Kovacheva M, Schnepp E (2005) Hierarchical modeling of archaeomagnetic data and curve estimation by moving average technique. Geophys J Int 160:440–476 Lanza R, Zanella E, Sandino S (2009) Magnetic remanence of haematitebearing minerals. Geophys Res Lett 36:L24302 Pavón-Carrasco FJ, Osete ML, Torta JM, Gaya-Piqué LR, Lanos P (2008) Initial SCHA.DI.00 regional archaeomagnetic model for Europe for the last 2000 years. Phys Chem Earth 33(6–7):597–608 Soler-Arechalde AM, Sánchez F, Rodriguez M, Caballero- Miranda A, Goguitchaichvil A, Urrutia-Fufugauchi J, Manzanilla L, Tarling DH (2006) Archaeomagnetic investigation of oriented pre-Columbian lime-plasters at Teotihuacan, Mesoamerica. Earth Planets Space 58:1433–1439

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India Harsh K. Gupta National Geophysical Research Institute, Council of Scientific and Industrial Research, Hyderabad, India

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whether the M 7 Gazli earthquakes of May 1976 and 19 March 1984 were induced due to the production of large quantities of gas at the Gazli oil field in Uzbekistan. There is an argument that the Sichuan, China, M 7.9 earthquake of 12 May 2008, that claimed over 80,000 human lives, was triggered due to filling of the nearby Zipingpu reservoir. It has been also proposed that flooding of a river near San Andreas Fault in California caused at least two M~6 earthquakes. A good account of triggered/induced seismicity can be found in a review by (McGarr et al. 2002). Hudyma and Potvin (2005) and Drzewiecki and Piernikarczyk (2017) have extensively dealt with mining-induced seismicity. Gupta (2002) has reviewed artificial water reservoir-triggered seismicity. Here we present cases of artificial water reservoir-triggered earthquakes that occurred all over the world, with a special emphasis on earthquakes in the Koyna region near the west coast of India. This is a classical case where triggered earthquakes have been occurring since the impoundment of reservoir in 1962 till now in 2019.

Triggered vis-a`-vis Induced Earthquakes For a long time, the adjectives “induced” and “triggered” were used interchangeably whenever one talked of artificially simulated earthquakes. McGarr and Simpson (1997) have addressed this question and suggested that it would be important to draw a distinction between the two. They proposed that the adjective “triggered seismicity” should be used only when a small fraction of stress change or energy associated with earthquakes is accounted for by the causative activity. The term “induced seismicity” should be used where the causative activity is responsible for a substantial part of the stress change. In the case of triggered seismicity, tectonic loading plays an important role. The stress-level changes associated with filling of some of the deepest artificial water reservoirs are only of the order of 1 MPa or so, whereas the stress drop associated with the earthquakes is much larger. Therefore, all cases of earthquakes occurring subsequent to filling of the artificial water reservoirs fall in the category of “triggered earthquakes,” and hence it is appropriate to call it “reservoir-triggered seismicity” (RTS).

Artificial Water Reservoir-Triggered Earthquakes Definition and Introduction Under certain suitable geological conditions, anthropogenic activity can trigger or induce earthquakes. The triggered/induced earthquakes are known to have occurred due to gold and coal mining, petroleum production, filling of artificial water reservoirs, highpressure liquid injection into ground, geothermal energy, and natural gas production. The largest scientifically accepted triggered earthquake of M 6.3 occurred on 10 December 1967 in the vicinity of Koyna Dam near the west coast of India. It is debated

The generation of hydroelectric power, flood control, and irrigation has led to the creation of huge artificial water reservoirs globally. For the first time, triggering of earthquakes was pointed out by Carder (1945) at Lake Mead in the USA. Figure 1 depicts Lake Mead water levels and local seismicity for the period 1936 through 1944. The rise in water level and corresponding bursts of seismic activity are numbered. The correspondence is indeed remarkable. Damaging triggered earthquakes exceeding M 6 occurred at Hsingfengkiang, China (1962); Kariba, Zambia-Zimbabwe border (1963);

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Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Fig. 1 Lake Mead water levels and the local seismicity. For 1936 and 1937, only the felt shocks are plotted. The

Kremasta, Greece (1966); and Koyna, India (1967). Koyna earthquake of M 6.3 that occurred on 10 December 1967 is so far the largest scientifically accepted triggered earthquake. It claimed over 200 human lives, injured about 1500, and rendered thousands homeless. The occurrence and potential of triggered earthquakes has caused major modification of civil works and engineering projects. Anticipating a large triggered earthquake, the Hsingfengkiang Dam was strengthened twice before the occurrence of M 6.1 earthquake on 20 March 1962 (Shen et al. 1974). The disposal of waste fluid through injection into the ground at the Rocky Mountain Arsenal had to be discontinued due to triggered earthquakes (Evans 1966). The possibility of high-magnitude-triggered seismicity was responsible for terminating the Auburn Dam project in California (Allen 1978). There is a general reluctance on the part of Engineering Community, globally, to accept the significance or even existence of the phenomenon of triggered seismicity (Allen 1982; Simpson and Leith 1988). What Allen (1982) said some four decades ago, “From a purely economic point of view, not to speak of public safety, the problem of reservoir induced earthquakes deserves far more attention than it currently is receiving in most parts of the world,” is still true.

Reservoir-Triggered Seismicity (RTS): Global Distribution Globally there are hundreds of sites where RTS is reported to have occurred. Gupta (2011), on the basis of magnitude of the largest triggered earthquake, grouped these sites in the following categories (Table 1): 1. Sites where largest earthquake exceeded M 6.0 (4 sites) 2. Sites where the largest earthquake M was 5.0 to 5.9 (10 sites) 3. Sites where the largest earthquake M was 4.0 to 4.9 (29 sites)

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rises in water levels and the corresponding bursts of seismic activity are numbered. General trend of tremor-frequency variation is shown by dotted lines (After Carder 1945)

Over the years, RTS has been reported from many more sites. Among these, a mention must be made of Polyphyto Dam, Greece, where the largest triggered earthquake of M 6.5 occurred on 13 May 1995 (Pavlou 2019), and Tranh River No.2, Vietnam, where the largest triggered earthquake was of M 4.6 on 22 October 2012 (Trieu et al. 2014).

Important Factors for RTS Several studies examined the correspondence among possible correlates like rate of loading, highest water level reached and the duration of retention of high water levels, and the occurrence of RTS. The most important correlate is the depth of water column in the reservoir (Baecher and Keeney 1982). Figure 2 demonstrates that when the water column depth exceeds 150 m, about one in five reservoirs has experienced RTS. A review of recent global examples gives support to this observation. It must also be noted that a reservoir with a water volume exceeding 1 km3 and/or water depth exceeding 100 m is called a large reservoir. Globally there are more than one thousand such reservoirs, and only a small percentage has evidenced RTS.

Common Characteristics of RTS Sequences By the early 1970s, over a dozen cases of RTS were known. In a couple of detailed studies, Gupta et al. (1972a, b) discovered several common characteristics of RTS sequences, which discriminated them from the normal earthquake sequences occurring in close by regions but not associated with reservoirs. These characteristics are: 1. In the earthquake frequency-magnitude relation (log N ¼ A – b M, where N is the number of earthquakes with magnitude  M and A and b are constants), the foreshock

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

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Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Table 1 Reported cases of reservoir-triggered seismicity (RTS) where M  4 earthquake occurred (Gupta 2011) Name of the Height of Reservoir volume Year of dam/reservoir Country dam (m) (106 m3) impounding Sites where earthquakes having magnitude 6.0 were triggered Hsingfengkiang China 105 13,896 1959 (PRC) Kariba, Zambia Zimbabwe 128 175,000 1958 Koyna India 103 2,780 1962 Kremasta Greece 160 4,750 1965 Sites where earthquakes having magnitude between 5.0 and 5.9 were triggered Aswan Egypt 111 1,64,000 1964 Benmore New 110 2,040 1964 Zealand Charvak Uzbekistan 148 2,000 1971 Eucumbene Australia 116 4,761 1957 Geheyan China 151 3,400 1993 Hoover USA 221 36,703 1935 Marathon Greece 67 41 1929 Oroville USA 236 4400 1967 Srinagarind Thailand 140 11,750 1977 Warna India 80 1,260 1985 Sites where earthquakes having magnitude between 4.0 and 4.9 were triggered Akosombo Main Ghana 134 148,000 1964 Bajina Basta Yugoslavia 90 340 1966 Bhatsa India 88 947 1981 Bratsk Russia 100 169 Camarillas Spain 49 37 1960 Canelles Spain 150 678 1960 CapivariBrazil 58 180 1970 Cachoeira Clark hill USA 60 3517 1952 Dahua China 74.5 420 1982 (PRC) Danjiangkou China 97 16,000 1967 (PRC) Foziling China 74 470 1954 (PRC) Grandwal France 88 292 1959 Hoa Binh Vietnam 125 1988 Kastraki Greece 96 1000 1968 Kerr USA 60 1505 1958 Komani Albania 130 1600 1985 Kurobe Japan 186 149 1960 Lake Baikal Russia Lake Pukaki New 106 9000 1976 Zealand Manicouagan 3 Canada 108 10,423 1975 Marimbondo Brazil 94 6150 1975 Monteynard France 155 275 1962 Nurek Tajikistan 317 1000 1972 P. Colombia/V. Brazil 40/56 1500/2300 1973–1974 Grande Piastra Italy 93 13 1965 Pieve di Cadore Italy 116 69 1949 Shenwo China 50 540 1972 (PRC)

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2, 12 26

1973

4.7

16

1973

4.5

16

1963 1989 1969 1971 1986 1961 1978

V 4.9 4.6 4.9 4.2 4.9 4–4.8a 4.6

1, 2, 4, 5 22c 2 1, 2, 9 27 2, 13 23 21

1975 1975 1963 1972 1974

4.1 IV 4.9 4.6 4.2

2 18 1, 2, 4, 5 1, 2, 14 19

1966 1950 1974

4.4 V 4.8

2, 4, 5 2, 15 16 (continued)

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22

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Table 1 (continued) Name of the dam/reservoir Vouglans Karun-III

Country France Iran

Height of dam (m) 130 185

Reservoir volume (106 m3) 605 2970

Year of impounding 1968 2005

Year of the largest earthquake 1971 2005

Magnitude/ intensity 4.4 4.3

References 2, 4, 5 29

References: 1 ¼ (Gupta and Rastogi 1976); 2 ¼ (Packer et al. 1979); 3 ¼ (Shen et al. 1974); 4 ¼ (Rothe 1970, 1973); 5 ¼ (Bozovic 1974); 6 ¼ (Gough and Gough 1970b); 7 ¼ (Toppozada 1982); 8 ¼ (Adams 1974); 9 ¼ (Simpson 1976); 10 ¼ (Carder 1945); 11 ¼ (Bufe et al. 1976); 12 ¼ (Talwani 1976); 13 ¼ (Hagiwara and Ohtake 1972); 14 ¼ (Soboleva and Mamadaliev 1976); 15 ¼ (Caloi 1970); 16 ¼ (Oike and Ishikawa 1983); 17 ¼ (Berrocal (personal communication), 1990); 18 ¼ (Veloso et al. 1987); 19 ¼ (Berrocal et al. 1984); 20 ¼ (Rastogi et al. 1986); 21 ¼ (Reyners 1988); 22a ¼ (Chen et al. 1996), 22b ¼ (Pavlenov and Sherman 1996), 22c ¼ (Tung 1996), 23 ¼ (Djadkov 1997), 24 ¼ (Rastogi et al. 1997); 25 ¼ (Plotnikova et al. 1992); 26 ¼ (Guang 1995); 27 ¼ (Muco 1991); 28 ¼ (Chung and Liu 1992); 29 ¼ (Kangi and Heidari 2008)

OBSERVED FREQUENCY OF INDUCED SEISMICITY IN PERCENT

60

50

40

30 5 out of 19 (26%)

20 5 out of 29 (17%)

10 5 out of 78 (6%)

0

0

30

60

90

120

150

250

RESERVOIR WATER DEPTH IN METERS

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Fig. 2 Height of water column is the most important correlate (After Stuart-Alexander and Mark 1976)

and aftershock b values of the RTS sequences are higher than the b values for natural earthquake sequences in the regions concerned and the regional b values. 2. In addition to high b values, the magnitude ratio of the largest aftershock to the mainshock is also high. 3. Aftershock activity decays slowly compared to normal earthquake sequences. 4. The foreshock-aftershock sequence pattern belongs to Type II of Mogi’s model (Mogi 1963), whereas the natural earthquake sequence pattern belongs to Type I of Mogi’s model (Fig. 3 I).

The above mentioned observations are governed by the mechanical properties of the media, and their deviation from the normal implies changes in these properties consequent to impoundment of the artificial water reservoir. The sketch shown in Fig. 3 illustrates the changes. “A” in this figure is a homogenous media rock volume. When the stress exceeds the strength

of the rock, there would be a major event releasing most of the strain, followed by peripheral adjustment aftershocks. In such an earthquake sequence, there would not be any foreshocks before the major event. The aftershock activity would be over in a short time, the ratio of the largest aftershock to the main event would be low, and the b value would also be low. This is typically the situation with the earthquake sequences in stable continental regions not associated with the reservoir loading. Due to filling of the water reservoir, the heterogeneity of the media increases (“B” in Fig. 3 II), and the rock volume gets fragmented. This results in the release of accumulated stresses by smaller rock volumes. In such a situation, the events would start occurring, as and when the strength of an individual rock volume is exceeded. The main event would correspond to the largest rock volume, and there would be events before it and after it, changing the pattern from Type I of Mogi’s model to Type II. The ratio of the magnitude of the largest aftershock to the mainshock would also be high, and the b value of the foreshock sequence as well as the aftershock sequence would be high. This is what is observed with RTS sequences. These common characteristics were identified in the 1970s, based on around two dozen cases of RTS. Over the years, these are found to hold good for all cases of RTS investigated subsequently.

Mechanism of Triggered Earthquakes In the following we give a gist of major milestones in comprehending the phenomenon of triggered earthquakes. The foundation of understanding the phenomenon of triggered earthquakes was laid by the study of the waste fluid injection-induced earthquakes in the vicinity of the Rocky Mountain Arsenal well near Denver, Colorado, USA, in the early 1960s (Evans 1966). There are three main effects of reservoir loading relevant to triggering of earthquakes as pointed out by Gough and Gough (1970b) and several others: 1. The elastic stress increases following filling of the reservoir. 2. The increase in pore fluid pressure in saturated rocks, basically due to decrease in pore volume due to compaction, in response to increase in elastic stress. 3. Pore pressure changes related to fluid migration.

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India Artificial Water ReservoirTriggered Earthquakes, with Special Emphasis on Koyna, India, Fig. 3 (I) Depicts (Mogi 1963) classification of the earthquake sequences into three broad categories. (II) Cartoon for (A) homogenous rock mass and (B) fragmented rock mass. For details see the text

I

n

Main Shock

23

Structure of Material

Distribution of External Stress

Type 1

Homogeneous

n

Uniform

Main Shock

Type 2 Heterogeneous in Some Degree

Not Uniform

Extremely Heterogeneous

Very Concentrated

n Type 3

II

A

Gough and Gough (1970a, b) provided the first definitive quantitative analysis of the role of the load of the water reservoir in triggering earthquakes at Lake Kariba. The role of reservoir load in triggering earthquakes was also considered by Bell and Nur (1978) and Roeloffs (1988). They pointed out that reservoir-load-induced stresses at seismogenic depths are very small and can only perturb the ambient stress field. Gupta et al. (1972a, b) identified the rate of increase of reservoir water levels, maximum water levels reached, and the duration of retention of high water levels as factors affecting the frequency and magnitude of triggered earthquakes. The influence of pore fluid pressure in inducing earthquakes in very simple 1D reservoir models was presented by (Snow 1972). More sophisticated models were dealt by Withers and Nyland (1976), Bell and Nur (1978), and Roeloffs (1988) based on Biot’s (1941) consolidation theory, which later on is generalized by Rice and Cleary (1976) by recognizing that the fluids too may be compressible. Very interesting results related to modeling of pore

B

pressure diffusion have been reported for Acu Reservoir in Brazil (Do Nascimento et al. 2004). Pore pressure diffusion plays a very important role in the triggering of the earthquakes. However, there are a very few direct measurements of diffusivity, and it is mostly inferred from the temporal migration of seismicity. Talwani et al. (1999) have reported in situ measurements of hydrological diffusivity and Skempton’s coefficient at the Bad Creek Reservoir in South Carolina, USA. At an observation well located 250 m away from the reservoir, a change in the water level of the well had a direct correspondence with the changes in the water level of the reservoir, initially with a delay of 98 h, which later stabilized at 72 h. This led to a frequency-independent estimate of diffusivity of ~ 0.076 m2 s
1 and Skempton’s coefficient of 0.66 for an undrained response of the reservoir. Later, Talwani et al. (2007) analyzed more than 90 case histories of triggered seismicity and found diffusivity to vary between 0.1 and 10 m2 s
1. This range of diffusivity values corresponds to a

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24

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

range of intrinsic permeability between 5  10
16 and 5  10
14 m2. Talwani et al. (2007) referred to this range of the permeability of fractures as seismogenic permeability. Seismogenic permeability is an intrinsic property of fractures where pore pressure diffusion is associated with seismicity. The in situ measurements of physical properties and examination of physical mechanism controlling triggered seismicity at Monticello Reservoir, South Carolina, USA, provided the much-needed field verification of the theoretical and model developments of the concept of triggered seismicity (Zoback and Hickman 1982). The effect of changes in lake levels and other derived parameters on triggered earthquakes have been dealt by Simpson and Negmatullaev (1981) for the Nurek Dam and by Gupta (1983) for the Koyna Dam. The part played by pore pressure diffusion in triggering earthquakes has been dealt by (Talwani and Acree 1984/ 1985). The effect of inhomogeneities in rock properties on triggering earthquakes was addressed by (Simpson and Narasimhan 1990). Most of the theoretical models discussed the effect of pore fluid pressure in isotropic rocks. Chen and Nur (1992) point out that deviatory effect of pore fluid pressure in anisotropic rocks has wide applications in comprehending triggered seismicity, earthquake precursors, and aftershocks. This approach needs to be applied to a few cases of RTS. Other interesting studies include the notion of fault stability as a measure of the interplay of frictional stresses mobilized and the resolved shear stresses acting on a fault (Chander and Kalpna 1997). Kalpna and Chander (2000) have developed an algorithm for the simulation of stresses and pore pressure for more realistic laterally finite 3D models of reservoirs.

Koyna, India Koyna reservoir located close to the west coast of India continues to be the most significant site of triggered earthquakes. Earthquakes began to occur soon after the impoundment of the Shivaji Sagar Lake created by the Koyna Dam in 1962. So far, globally the largest triggered earthquake of M 6.3 on 10 December 1967, 22 earthquakes of M  5, about 200 earthquakes of M  4, and several thousand smaller earthquakes have occurred in the region. Talking to the residents in the region in the 1960s revealed that they had not experienced any earthquake in the region in their living memory. There is a seismic station operating at Pune, about 120 km away from Koyna. A scrutiny of the seismograms revealed no earthquakes that could be assigned to the Koyna region. Another water reservoir, Warna, was impounded in 1985. This reservoir is located 35 km SSE of Koyna (Fig. 4). After the impoundment of Warna reservoir, the triggered activity got enhanced. For convenience, we shall call the Koyna and the Warna reservoir

region as the Koyna region. Major bursts of seismic activity associated with the Koyna reservoir occurred in 1967, 1993, 1980, 1993–1994, 2005, and 2009.

How Long Triggered Earthquakes Will Continue at Koyna? Most RTS sites are located in stable continental regions (SCRs), where strain is of the order of 10
9 to 10
10 as compared to 10
5 at plate boundary and 10
7 to 10
9 for intraplate regions (Gupta and Johnston 1998). RTS typically gets initiated soon after the filling of the reservoir and may continue for a year to several years, with the largest event occurring within a few years of impoundment. The Hsingfengkiang Reservoir in China was impounded in 1959, and the largest RTS event of M 6.1 occurred on 19 March 1961. The frequency of RTS events dropped from thousands every month in the 1960s to a few every month by 1978 (Ishikawa and Oike 1982; Gupta 1992). Lake Kariba, located at Zambia-Zimbabwe border, was filled in 1958. The frequency of earthquakes increased several folds with the largest RTS event of M 6.2 occurring on 13 April 1963 (Gough and Gough 1970a, b). By 1974, RTS events dropped to less than 10% of 1963 level. At Kremasta, Greece, impoundment took place in 1965, and there was a very rapid filling of the reservoir in January 1966, and the largest RTS event of M 6.2 occurred on 5 February 1966 (Galanopolous 1967; Stein et al. 1982). The RTS at Kremasta was short lived. However, at Koyna RTS was initiated in 1962 and has continued till now with the latest M 4 earthquake occurring on 20 June 2019. Gupta et al. (2002) have examined in detail the question as to how long the triggered earthquakes would continue at Koyna. The maximum credible earthquake for the Indian shield region has been estimated to be M 6.8. It is hypothesized that the region between Koyna and Warna was stressed close to critical before the impoundment of the Koyna reservoir and was capable of generating an M 6.8 earthquake. As demonstrated through the study of b values in earthquake magnitude-frequency relation, foreshockaftershock patterns, the ratio of the magnitude of the largest aftershock to the mainshock, and the decay of aftershock activity in earthquake sequences at Koyna, the heterogeneity of the media has increased. None of the M~5 earthquake has occurred at the same location. M~5 earthquakes occur in Koyna region when the previous water maximum in the reservoir has been exceeded. Long time ago, Kaiser (1953) had reported that acoustic emission, under monotonically increasing stress, shows an appreciable increase after the applied stress exceeds the previously applied stress maxima. This approach has been successfully used by Yoshikawa and Mogi (1981) to estimate crustal stresses from cored samples. For the Nurek Dam also, it was reported that major triggered

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

25

A

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Fig. 4 Updated from Gupta et al. (2017). Koyna-Warna region near west coast of India. Location of the Koyna main earthquake of 10 December 1967; earthquakes of M ~ 5 during August 1967 through June 2017; smaller earthquakes from August 2005 through June 2017; surface and borehole seismic

stations; green curve indicates the WGE (Western Ghat Escarpment). (I) Koyna location in India; (II) Distribution of M  3.7earthquakes for 1967–2015 (USGS) in the vicinity of Koyna and an outer circle of 100 km radius indicating that there is almost no seismic activity outside the Koyna region

events occurred when the water level in the reservoir reached the previous maximum and exceeded it (Simpson and Negmatullaev 1981). Gupta et al. (2002) had concluded that as of 2002, about one-half of an M 6.8 earthquake energy has been released in the Koyna region, and the activity should continue for another three to four decades. Due to the increase in heterogeneity, no fault segment, long enough to generate a M > 6 earthquake, is left intact, so an earthquake like the 10 December 1967 may not occur. The occurrence of M~5 earthquakes would be governed by the fact whether the previous water maximum has been exceeded, and other factors such as the rate of loading, the highest water levels reached, and the duration of the retention of high water levels.

Short-Term Earthquake Forecast at Koyna A case was made that short-term earthquake forecast may be feasible at Koyna (Gupta 2001). This was based on the fact that the shallow (depth  10 km) seismic activity in the vicinity of the Koyna is confined to an area of 20  30 km2, and there is no other seismic source within 50 km radius. Every year, following the rainy season, the reservoirs get filled, and there is an enhancement of seismic activity. From among earthquake precursors, foreshocks are important and have a potential in forecasting earthquakes (Dodge et al. 1996; Helmestetter et al. 2003). Ohnaka (1992) has noted that immediate foreshock activity is a part of the nucleation process leading to mainshock

26

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Table 2 Events preceding M  4.0 earthquakes in Koyna region Main earthquake 30 August 2005 M 4.8 13 November 2005 M 4.0 26 December 2005 M 4.2 17 April 2006 M 4.7

Duration of nucleation period (hours) 110

No. of events before the main earthquake M M M Largest 1.0–1.9 2.0–2.9 3.0–3.9 earthquake 13 8 0 2.6

50 h prior to the beginning of nucleation period 2

160

18

8

1

3.0

3

150

22

8

0

2.9

6

380

40

10

0

2.7

2

dynamic rupture. Gupta et al. (2007) have reported successful earthquake forecast in Koyna based on identification of nucleation in real time. The seismic activity in the Koyna region is monitored by a closely spaced network of seven modern seismic stations. Table 2 from Gupta et al. (2007) is the learning phase where four earthquakes of M 4.0 to M 4.8 were found to be preceded by well-defined nucleation where several events occurred in a tight cluster of less than 10 km radius (Fig. 5a–d). It was realized that if the formation of nucleation could be identified in real time before the occurrence of the mainshock, it may be possible to make a short-time forecast. During the middle of May 2006, an interesting situation was developed. The events in the Koyna region for the period 11–16 May 2006 are depicted in Fig. 6a (Gupta et al. 2007). By the afternoon of 16 May 2006, some 50 events of M  1.0, the largest being M 2.7, had occurred in the preceding 107 h in a small area, the focal depth being between 2 and 8 km. It was inferred that the region is going through a nucleation phase. Based on the experience of previous nucleation, the following forecast was made at 19:05 IST and communicated to the Editor of Current Science, Secretary of the Ministry of Earth Sciences of the Government of India, and the President of the Geological Society of India: “On the basis of the data available from seven seismic stations operating in the Koyna region, we have identified a nucleation which started on 12 May 2006. This may lead to the occurrence of an M~4 earthquake in the next 15 days. This shallow earthquake (focal depth less than 8 km) will occur within a radius of 10 km centered at 17.1°N, 73.8°E. On the basis of our previous experience of studying nucleation-preceding earthquakes in the Koyna region, we expect this earthquake to occur over the next 15 days time (till 31 May 2006), with a 50% probability.” An earthquake of M 4.2 occurred on 21 May. Table 3 gives the comparison of forecasted parameters and that of 21 May 2006 earthquake. Since 2006, eight similar short-term forecasts have been made in the Koyna region, and all have come true.

Near-Source Studies at Koyna The correspondence between the water level variations in Koyna and Warna reservoirs and observed RTS is well established. However, in the absence of the knowledge of physical properties of rocks and fluids in the fault zone, the triggering mechanism is not comprehended. This necessitates near-source studies of earthquakes in the Koyna region. The fact that the RTS is confined to a small area of 30  20 km, most of the earthquakes occur within a depth of 2–9 km; there is no other source of earthquakes within 100 km of Koyna Dam, and the region is easily accessible for all kind of field observations, and temperatures are estimated to be ~ 130 °C at a depth of 6 km, making Koyna an ideal site for setting up of a deep borehole laboratory. The project of setting up of a borehole laboratory for near-field study of earthquakes was discussed in an International Continental Drilling Program (ICDP) sponsored discussion meeting during 21–26 March 2011, where global experts participated and supported the proposed program and suggested certain field work and observations to be carried out before putting the borehole (Gupta and Nayak 2011). In the following 3 years, the recommended investigations were carried out. These included (1) drilling of nine boreholes into the basalt column and penetrating the granitic basement to a depth of 300–500 m, (2) airborne gravity gradient and magnetic surveys covering the entire RTS area, (3) magnetotelluric surveys to map the subsurface and estimate the thickness of basalt cover, (4) LiDAR surveys for getting a bare-earth model and getting a high-resolution topography, (5) instrumenting of six boreholes with three component seismometers for better estimate of the focal parameters of the earthquakes, and (6) heat flow measurements to estimate temperatures at depths of 6 km. The work carried out during 2011–2014 was reported in the 2nd ICDP workshop held during 16–18 May 2014 at Koyna and Karad (Gupta et al. 2014). The results were also presented elsewhere (Gupta et al.

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

27

Time (hrs)

a

–50 0

0

50

100

A 17.4

Depth (km)

2 4 6

17.2 Mag4.8

8 10

17.0

b

–50 0

0

50

100

150

73.8

73.6

73.8

73.6

73.8

73.6

73.8

17.4

2

Depth (km)

73.6

4 Mag4.0

6

17.2

8 10

c

17.0

–50 0

0

50

100

150 17.4

Depth (km)

2 4 6

17.2

Mag4.2

8 10

Depth (km)

d

–50 0

17.0

0

50

100

150

200

250

300

350

400 17.4

2 4

Mag4.7 17.2

6 8 10

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Fig. 5 Seismic activity for the period 50 h before the start of nucleation till the occurrence of M  4 earthquakes in Koyna region during August 2005–April 2006. Foreshocks clustering before the mainshocks in a region of 10 km radius (right). Temporal distribution of events within the circle with the depth. ○, O, ,

17.0

and

are events in M 1.0–1.9, 2.0–2.9, and 3.0–3.9 ranges and main

earthquake, respectively (left). The onset of identified nucleation period is given “0” time mark. Nucleation period lasted for 110, 160, 150, and 380 h for (a) 30 August, (b) 13 November, and (c) 26 December 2005 and (d) 17 April 2006 earthquakes, respectively (Table 2)

28

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Fig. 6 Symbols as in Fig. 5. (a) Events that occurred during 107 h (11–16 May) preceding the identification of nucleation. (c) Temporal-depth plot of the events within the 10-km-

2016; Gupta 2017, 2018). Many of the 2nd ICDP workshop participants had participated in the 1st ICDP workshop as well. Satisfied with the progress made, a go ahead was given to put a 3-km-deep Pilot Borehole.

Pilot Borehole Location Five possible locations were examined for suitability of placing the Pilot Borehole (Gupta et al. 2017). Finally, based on

radius circle. Nucleation was inferred to have started 107 h before, where 0 h is put on the time axis. (b) Seismic activity from 13:35 UTC on 16 May till the occurrence of the M 4.2 earthquake on 21 May 2006, on the right side of (c) Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India, Table 3 Forecast of 21 May 2006 M 4.2 earthquake in Koyna on 16 May 2006 Epicenter Magnitude Time Focal depth

Forecast parameters Within 10 km radius of 17.1° N, 73.8° E ~4 Within 15 days of 16 May 2006, i.e., until 31 May 2006 Less than 8 km

Occurrence 17.171° N, 73.777° E 4.2 20:29:01 UTC On 21 May 2006 4.7 km

Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

vicinity of site D (Fig. 7) to the Donachiwada fault, which hosted the 10 December 1967 M 6.3 earthquakes and many other events of M > 4, logistic convenience, and the trend of earthquakes for the period 2005–2015, it was chosen to place the Pilot Borehole. Drilling of the Pilot Borehole started on 20 December 2016, and the borehole was completed on 11 June 2017. Granitic basement was reached at a depth of 1247 m. No sediments were found below the basalt. Zones of huge fluid losses were encountered at several depths. An online gas analysis facility has been set up. Measurements of in situ stresses were made at depths of 1600 m and deeper. It is found that 10
16 to 10
14 permeability of the granites would not produce significant pore fluid pressure changes to trigger earthquakes. However, core samples of granites at 1522 m depth provided evidence of fractured zones that would contribute to water channelization and triggering of earthquakes (Hazarika et al. 2017). Measurement of elastic properties of granitoids from Koyna region showed low and variable rock strength compared with granitic rocks of aseismic areas (Goswami et al. 2017). The 3rd ICDP postoperation workshop was held during 14 to 16 October 2017 to

Artificial Water ReservoirTriggered Earthquakes, with Special Emphasis on Koyna, India, Fig. 7 (a) Earthquakes of M  2.0 in the vicinity of block D during August 2005–December 2015. A 2-km swath is indicated by dotted lines. The surface expression of the Donachiwada fault is indicated by dashed lines. (b) Depth section of hypocenters for the period 2005–2015 for the 2-km-wide swath. (c) Earthquakes in the 2-km swath for the period 2016–2017

29

review the work carried out to develop plans for the Main Borehole of ~ 6 km depth, its instrumentation, and future course of action (Podugu et al. 2018). In a recent study, detailed geophysical logging revealed seven anomalous zones below 2100 m depth (Goswami et al. 2019), characterized by shear wave velocity anomaly of up to 25%. All the data acquired are being analyzed for finalizing the specifications of the proposed ~ 6 km deep Main Borehole.

Summary In this article we provide a thumbnail review of studies of global occurrence of artificial water reservoir-triggered earthquakes, with a special emphasis on Koyna, India. Considering the small changes in the stress regime caused by filling of the deepest reservoirs compared to the stress drop of the associated earthquakes, it is appropriate to replace “reservoirinduced seismicity” by “reservoir-triggered seismicity.” Koyna continues to be the most significant sight of RTS globally. The latest M 4 earthquake occurred on 20 June 2019.

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Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India

At Koyna, earthquakes of M 4–5 are often preceded by well-defined clusters of foreshocks of M  3, referred as nucleation that is found to last typically 100–400 h. Identification of nucleation in real time has led to successful shortterm forecasts of M ~ 4 earthquakes. The study of RTS provides exceptionally good opportunity to comprehend physics of earthquakes, finding safer sites of creating artificial water reservoirs and bringing us closer to accurate short-term earthquake forecasts. Setting up of a deep borehole laboratory for near-field study of earthquakes is underway at Koyna. The first phase of setting up of a 3-km-deep Pilot Borehole has been successfully completed. One of the major discoveries has been that it is not the fluid flow through permeability, but through deep fractures discovered at depths of 3 km in the seismic zone that trigger earthquakes.

Cross-References ▶ Earthquake Precursors and Prediction ▶ Earthquake Prediction, M8 Algorithm ▶ Earthquake, Aftershocks ▶ Earthquake, Foreshocks ▶ Earthquakes, Energy ▶ Seismicity, Intraplate ▶ Seismological Networks

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Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India Gupta HK (2002) A review of recent studies of triggered earthquakes by artificial water reservoirs with special emphasis on earthquakes in Koyna, India. Earth Sci Rev 58:279–310 Gupta HK, Shashidhar D, Periera M, Mandal P, Rao NP, Kousalya M, Satyanarayana HVS, Dimri VP (2007) Earthquake forecast appears feasible at Koyna, India. Curr Sci 93:843–848 Gupta HK (2011) Artificial water reservoir triggered earthquakes. Encyclopedia of solid earth geophysics. Springer+ Business media 1:15–24 Gupta HK, Nayak S (2011) Deep scientific drilling to study reservoir triggered earthquakes in Koyna, western India. Sci Drill 12:53–54 Gupta HK, Rao NP, Roy S, Arora K, Tiwari VM, Patro BPK, Satyanarayana HVS, Shashidhar D, Mallika K, Akkiraju VV, Goswami D, Vyas D, Ravi G, Srinivas KNSSS, Srihari M, Mishra S, Dubey CP, Raju CV, Borah U, Reddy KC, Babu N, Rohilla S, Dhar U, Sen M, Rao YJB (2014) Investigations related to scientific deep drilling to study reservoir-triggered earthquakes at Koyna, India. Int J Earth Sci 104:1511–1522 Gupta HK, Arora K, Rao NP, Roy S, Tiwari VM, Patro PK, Satyanarayana HVS, Shashidhar D, Mahato CR, Srinivas KNSSS, Srihari M, Satyavani N, Srinu Y, Gopinadh D, Raza H, Jana M, Akkiraju VV, Goswami D, Vyas D, Dubey CP, Raju DCV, Borah U, Raju K, Reddy KC, Babu N, Bansal BK, Nayak S (2016) Investigations of continued reservoir triggered seismicity at Koyna, India. J Geol Soc London, Special Publications 445:151–188 Gupta HK, Shashidhar D, Mahato CR, Satyanarayana HVS, Mallika K, Rao NP, Maity BS, Navitha K (2017) Location of the pilot borehole for investigations of reservoir triggered seismicity at Koyna, India. Gondwana Res 42:133–139 Gupta HK (2017) Koyna, India, an ideal site for near field earthquake observations. Geol Soc India 90(6):645–652 Gupta HK (2018) Review: reservoir triggered seismicity (RTS) at Koyna, India over the past 50 years. Bull Seismol Soc Am 108(5B):2907–2918 Hagiwara T, Ohtake M (1972) Seismic activity associated with the failing of the reservoir behind Kurobe Dam, Japan, 1963–1970. Tectonophysics 15:241–254 Hazarika P, Yadav A, Roy S (2017) Influence of permeability in modeling of reservoir triggered seismicity in Koyna region, western India. J Geol Soc India 90:728–732 Helmestetter A, Sornette D, Grasso JR (2003) Mainshocks are aftershocks of conditional foreshocks: how do foreshock statistical properties emerge from aftershock laws? J Geophys Res 108B:2046 Hudyma MR, Potvin Y (2005) Mining-induced seismicity in underground mechanised hard rock mines-results of a world–wide survey. In: Seminar on advanced geomechanics in mines, 3rd August 2005. Australian Centre for Geomechanics, Perth, p 47 Ishikawa M, Oike K (1982) On reservoir-induced earthquakes in China, Zisin 35(2):171–181 Kaiser J (1953) Erkenntnisse and Folgerungen aus der Messung von Gerauschen bei Zugbeanspruchung von metallischen Werkstoffen. Archiv fur das Eisenhuttenwesen 24:43–45. (in German) Kalpna, Chander R (2000) Green’s function based stress diffusion solutions in the porous elastic half space for time varying finite reservoir loads. Phys Earth Planet Inter 120:93–101 Kangi A, Heidari N (2008) Reservoir-induced seismicity in Karun III dam (Southwestern Iran). J Seismol 12:519–527 McGarr A, Simpson D (1997) Keynote lecture: a broad look at induced and triggered seismicity “Rockbursts and seismicity in mines.”. In: Gibowicz SJ, Lasocki S (eds) Proceedings of 4th international symposium on rockbursts and seismicity in mines Poland, 11–14 Aug 1997. A. A. Balkema, Rotterdam, pp 385–396 McGarr A, Simpson D, Seeber L (2002) Case histories of induced and triggered seismicity. International handbook of earthquake and engineering seismology. International Geophysics 81A:647–661 Mogi K (1963) Some discussions on aftershocks, foreshocks and earthquake swarms – the fracture of a semi-infinite body caused by an

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Talwani P (1976) Earthquakes associated with Clark Hill reservoir, South Carolina – a case of induced seismicity paper presented at the 1st international symposium on induced seismicity. Eng Geol 10:239–253 Talwani P, Acree S (1984/1985) Pore pressure diffusion and the mechanism of reservoir-induced seismicity. PAGEOPH 122:947–965 Talwani P, Cobb JS, Schaeffer MF (1999) In situ measurements of hydraulic properties of a shear zone in northwestern South Carolina. J Geophys Res 104(B7):14993–15003 Talwani P, Chen L, Gahalaut K (2007) Seismogenic permeability, ks. J Geophys Res 112:B07309. https://doi.org/10.1029/2006JB004665, 1–18 Toppozada TR (1982) UNDP/Tokten report on Aswan earthquakes Trieu CD, Trong CD, Dung LV, Tuan TA, Van DQ, Long HV (2014) Triggered earthquake study in Tranh River no.2 (Vietnam) hydropower reservoir. J Geol Soc India 84:319–325

Tung NT (1996) The induced seismicity at Hoa Binh Reservoir region, Abstract Volume. IASPEI Reg. Assembly in Asia, Tangshan, Aug 1–3, 1996 Veloso JAV, Assumpcao M, Concalves ES, Reis JC, Duarte VM, da Motta CBG (1987) Registro de sismicidade induzida em reservatorios da CEMIG e FURNAS. An 50 Congr Bras Geol Eng 1:135–146 Withers RJ, Nyland E (1976) Theory for the rapid solution of ground subsidence near reservoirs on layered and porous media. Eng Geol 10:169–185 Yoshikawa S, Mogi K (1981) A new method for estimating the crustal stress from cored rock samples; laboratory study in the case of uniaxial compression. Tectonophysics 74:323–339 Zoback MD, Hickman S (1982) Physical mechanisms controlling induced seismicity at Monticello reservoir, South Carolina. J Geophys Res 87:6959–6974

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Biogeophysics Lee Slater1 and Estella Atekwana2 1 Department of Earth and Environmental Sciences, Rutgers University Newark, Newark, NJ, USA 2 College of Earth, Ocean, and Environment, University of Delaware, Newark, DE, USA

These geophysical signatures may arise from (1) microbial cells and extracellular structures themselves, (2) growth of microorganisms (production of biomass) and biofilm formation, (3) generation of metabolic by-products and the interactions of these metabolites with the host material, and (4) microbially mediated processes. The geophysical signatures arising from each of these four source mechanisms are described below.

Definition

Geophysical Detection of Cells and Biofilms

Biogeophysics

Subdiscipline of exploration geophysics focusing on the geophysical signatures resulting from microbial communities and their interactions with geologic media.

Introduction Geophysical imaging techniques have the potential to measure not just the subsurface physical and chemical properties, as geophysics is conventionally used, but also microbes, microbial processes, and microbe-mineral interactions. “Biogeophysics” is defined here as a rapidly evolving discipline of exploration geophysics concerned with the geophysical signatures of microbial communities and their interactions with geologic media that combines the fields of microbiology, biogeoscience, and geophysics (Atekwana and Slater 2009) (Fig. 1). Within this context, biogeophysics examines the links between dynamic subsurface microbial processes, microbial-induced alterations to geologic materials, and geophysical signatures. We note that the term biogeophysics is also used in other disciplines (a) to describe research into the origins of life and (b) to describe fluxes of energy, water, and momentum between earth surface and the atmosphere. Our use of the term describes how it has been adopted by the exploration geophysics community in recognition of the geophysical signatures of microbial activity. © Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

Microbial cells and biofilms exhibit distinct electrical properties and certain (magnetotactic) bacteria also display unique magnetic characteristics. The membrane potential (the voltage drop across the membrane due to the negatively charged molecules inside cells) of live cells results in an accumulation of mobile electric charge carriers at membrane surfaces. When live cells are placed in time-oscillating electric fields, these charges move on the surface of the membrane, giving rise to high polarizations for cellular suspensions that are readily measured with dielectric spectroscopy. As the mobility of these surface charges is relatively small, this effect is manifest at low frequencies, such that the relative dielectric permittivity of live cell suspensions can be as high as 106 (Prodan et al. 2004; Stoy et al. 1982). The outer and inner cell radii, diffusion constants, and membrane potential all have a very distinct effect on broadband dielectric spectroscopy data (Prodan et al. 2008). Polarization enhancement is also observed when microbial cells are present in high concentrations in porous media (Ntarlagiannis et al. 2005). This polarization signature of porous media has been modeled in terms of the growth curve for microbial populations (Revil et al. 2012; Mellage et al. 2018). The mechanism may be enhanced when cells are more preferentially adsorbed onto mineral surfaces (Abdel Aal et al. 2009). The magnetic properties of soils and rock are altered by a diverse group of prokaryotes that exert a significant control over magnetite formation in two ways that differ mechanistically: biologically induced mineralization (BIM) and

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Biogeophysics

acoustic properties of porous media, with biofilm growth in soil columns resulting in spatial complexity of acoustic amplitudes (Davis et al. 2009). Such variations likely arise from a nonuniform distribution of microbial activity or heterogeneity in the biomass distribution and biofilm morphology (e.g., variations in biofilm thickness, roughness, hydration, etc.).

Geophysical Detection of Metabolic By-Products and Mineral Weathering

Biogeophysics, Fig. 1 Primary disciplinary themes encompassing biogeophysics

biologically controlled mineralization (BCM) (Jimenez-Lopez et al. 2010). Biologically induced magnetite formation is indistinguishable from magnetite formed by inorganic processes; however, biologically controlled magnetite formation is distinct and produced by magnetotactic bacteria (Jimenez-Lopez et al. 2010). These magnetotactic bacteria biomineralize intracellular, membrane-bound, single-magnetic-domain crystals of the magnetic minerals magnetite and/or greigite called magnetosomes (Bazylinski and Frankel 2004). Magnetotactic bacteria are typically found in a wide variety of aquatic environments (e.g., fresh water lakes) with the highest numbers occurring mostly at oxic-anoxic interfaces. The presence of magnetotactic bacteria in sediments is often used as a proxy for paleoenvironmental/climatic conditions, and the presence of magnetosomes in sediments and other secondary magnetic minerals produced by microbial activity likely impacts the magnetic properties of the subsurface recorded with magnetic geophysical techniques as has been demonstrated at hydrocarbon contaminated sites (Rijal et al. 2010). Biofilms (an attached state of cell growth, whereby cells are closely packed and firmly attached to each other) further alter the geophysical properties of soils. Low frequency electrical measurements respond to self-limiting microbial growth/cell attachment and biofilm formation in porous media, as well as death and lyses of cells (Davis et al. 2006). Electrical polarization may result from the electrical properties of the biofilm itself or from the modification of grain surfaces due to cell attachment. Biofilms also alter the

Microbial metabolism enhances the weathering of minerals through the attachment, growth, and colonization of mineral surfaces by microorganisms (Bennett et al. 1996). Metabolic by-products, including biogenic gases (e.g., CO2, H2S, CH4, etc.), organic acids, and biosurfactants all affect electrical properties of porous media. Microbial production of organic acids and biosurfactants adds ions to solution, increasing electrolyte concentration of pore fluids (Cassidy et al. 2001). Organic acids enhance mobility of sparingly soluble metals and also increase the number of reaction sites, thus accelerating mineral weathering (Hiebert and Bennett 1992). Enhanced mineral dissolution catalyzed by increased organic acid concentration can lead to physical changes in grain surface morphology, surface area, surface roughness, and the generation of secondary porosity and increased permeability (McMahon et al. 1992). These changes in porosity affect acoustic wave propagation by altering grain contact coupling and changes in surface area/roughness drive changes in electrical polarization. Biogenic gases reduce bulk electrical conductivity and enhance attenuation of seismic signal amplitudes. The effects of metabolic by-products on geophysical properties have been recorded in hydrocarbon-contaminated environments (Abdel Aal et al. 2004; Werkema et al. 2003). Such sites are natural bioreactors where excess organic substrates stimulate microbial activity. Enhanced mineral weathering in hydrocarbon-contaminated aquifers (Hiebert and Bennett 1992; McMahon et al. 1995) increases pore fluid conductivity and thereby bulk conductivity sensed with a range of electromagnetic methods (Abdel Aal et al. 2004; Atekwana et al. 2004; Sauck et al. 1998). Elevated bulk conductivity is therefore found where intrinsic bioremediation and enhanced mineral weathering are occurring. These zones of highest bulk electrical conductivity may coincide with the highest percentages of oil-degrading microbial populations, with spatial heterogeneity in the microbial community structure and shifts in the microbial community concomitant with vertical changes in bulk electrical conductivity (Allen et al. 2007). Biogenic methane production by archaea in anaerobic soils can result in extensive free phase gas production that reduces dielectric permittivity and electrical conductivity (Comas and Slater 2007; Parsekian et al. 2010).

Biogeophysics

Geophysical Detection of Microbially Mediated Redox Processes Microbial metabolic activity is a critical driver of redox chemistry because microbes derive energy from oxidationreduction reactions (the transfer of electrons from one reactant to the other). Terminal electron acceptors (TEAs) govern nutrient utilization by microbes during the breakdown of organic carbon (Cozzarelli et al. 1999) Microbial respiration consequently results in reduced conditions and strong redox gradients in the Earth typically develop in the presence of heterotrophic bacteria. Significant changes in Eh and pH result in new mineral stability fields in which some minerals become unstable and are dissolved and mobilized, whereas others may precipitate from solution. As TEAs are consumed, changes in pore fluid chemistry may drive changes in pore fluid conductivity, thereby affecting bulk electrical conductivity. Strong electrical potentials associated with current sources in the Earth are correlated with redox gradients recorded at sites where microbial degradation of hydrocarbons is occurring (Minsley et al. 2007; Naudet et al. 2003). Such large potentials require geobatteries traditionally invoked to explain very large (~1.0 V) potential gradients due to internal current sources recorded over electronically conductive ore bodies straddling the large redox gradient provided by the water table (Sato and Mooney 1960). Biogeobatteries may occur in conjunction with a strong redox gradient between highly reducing conditions below the water table within a contaminant plume and an oxidized zone above the water table if microbial activity can generate the required electron bridge (Revil et al. 2010; Heenan et al. 2017). Possible mechanisms facilitating electron migration include iron oxides, clays, and conductive biological materials (Revil et al. 2010). Metal-reducing organisms, such as Shewanella and Geobacter, produce electrically conductive appendages called bacterial nanowires that may facilitate electron transfer to solid phase electron acceptors (Reguera et al. 2005). However, the ability of biofilms to facilitate electron transport over the scale of the groundwater interface is unknown although new evidence suggests that such electron transfer can take place at mm scales (Nielsen et al. 2010). Nonetheless, more studies are needed to confirm this finding.

Geophysical Detection of Microbe-Mineral Transformations Biogeochemical processes result in mineral precipitation that alters the physicochemical properties of the grain surfaces. When microbial-induced precipitation is extensive,

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the volumetric physical properties of porous media are modified. In anaerobic environments, iron-reducing bacteria and sulfate-reducing bacteria can use Fe(III) and sulfate, respectively, as terminal electron acceptors. Ferrous iron produced by iron-reducing bacteria promotes the precipitation of electrically conductive secondary minerals such as siderite (FeCO3), magnetite (Fe3O4), and goethite (FeOOH) (Fredrickson et al. 1998). Magnetic susceptibility is a simple and powerful geophysical parameter for monitoring iron mineral transformations coupled to microbial induced oxidation of hydrocarbons (Atekwana et al. 2014; Lund et al. 2017) Sulfide produced during microbial sulfate reduction can react with iron (II) produced by iron-reducing bacteria to precipitate iron sulfide minerals. Strong seismic and electrical signatures are generated as a result of microbe-induced ZnS and FeS precipitation (Williams et al. 2005). Decreases in seismic wave amplitude result from the development of differential elastic moduli associated with accumulation of metal sulfide-encrusted microbes within pores (Williams et al. 2005). Electrical signals result from the formation, movement, and dissolution of electronically conductive biominerals that profoundly enhance the polarization of a porous medium. These electrical signals may be diagnostic of both the concentration and the distribution of the biominerals throughout a porous medium (Slater et al. 2007). Geophysical signals also result when microbial processes involve the precipitation of semimetallic or nonmetallic minerals, e.g., metabolically induced calcite precipitation by bacterial hydrolysis of urea (Ferris et al. 1995). Relative to metallic minerals, smaller electrical signals result from changes in pore volume/pore tortuosity and/or surface area/surface roughness driven by precipitation of nonmetallic minerals (Wu et al. 2010). Calcite precipitation induced by bacteria has been shown to form cements in porous media and affect subsurface fluid flow (Ferris et al. 1995). Such cements profoundly change the elastic properties of soils and rocks, particularly when ureolysis is stimulated to form calcite cement that acts to stiffen the soil matrix (DeJong et al. 2006). Shear waves are well suited to monitoring changes in the particle soil matrix due to precipitation as shear velocity (Vs) is largely unaffected by pore fluid composition and directly dependent on void ratio, coordination number (average number of surrounding particles a given particle is in contact with), and confining stress (DeJong et al. 2010). Large changes in shear wave velocity accompany stiffening result from initial binding of microbes to the soil matrix, suggesting that relatively small volumes of precipitates can generate large geophysical signals. Electrical signals can also be used to monitor soil strengthening by microbial precipitation of calcite (Saneiyan et al. 2019).

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Summary Twenty years ago, it seemed inconceivable to suggest that microbial processes could potentially impact geophysical signatures. However, today it is clear that geophysical signatures result from a range of microbial processes covering the scale of an individual cell to the scale of contaminant plumes in the Earth. A pressing question in biogeophysics remains how to better quantify the geophysical signatures of microbial processes through the development of appropriate modeling frameworks. Success in this venture will require new multidisciplinary research between geophysicists, biogeochemists, and microbiologists.

Cross-References ▶ Electrical Properties of Rocks ▶ Electrical Resistivity Surveys and Data Interpretation ▶ Geomagnetic Field, Measurement Techniques ▶ Geomagnetic Field, Theory ▶ Geophysical Well Logging ▶ Magnetic Anomalies: Interpretation ▶ Magnetic Methods, Principles ▶ Propagation of Elastic Waves: Fundamentals ▶ Remanent Magnetism ▶ Seismic Properties of Rocks ▶ Seismic Viscoelastic Attenuation

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Biogeophysics Bazylinski DA, Frankel RB (2004) Magnetosome formation in prokaryotes. Nat Rev Microbiol 2(3):217–230 Bennett PC, Hiebert FK, Choi WJ (1996) Microbial colonization and weathering of silicates in a petroleum-contaminated groundwater. Chem Geol 132(1–4):45–53 Cassidy DP, Werkema DD, Sauck WA, Atekwana E, Rossbach S, Duris J (2001) The effects of LNAPL biodegradation products on electrical conductivity measurements. J Environ Eng Geophys 6(1):47–52 Comas X, Slater L (2007) Evolution of biogenic gases in peat blocks inferred from noninvasive dielectric permittivity measurements. Water Resour Res 43(5):W05424. https://doi.org/10.1029/2006w r005562 Cozzarelli IM, Herman JS, Baedecker MJ, Fischer JM (1999) Geochemical heterogeneity of a gasoline-contaminated aquifer. J Contam Hydrol 40(3):261–284 Davis CA, Atekwana E, Slater LD, Rossbach S, Mormile MR (2006) Microbial growth and biofilm formation in geologic media is detected with complex conductivity measurements. Geophys Res Lett 33(18):L18403. https://doi.org/10.1029/2006gl027312 Davis CA, Pyrak-Nolte LJ, Atekwana EA, Werkema DD, Haugen ME (2009) Microbial-induced heterogeneity in the acoustic properties of porous media. Geophys Res Lett 36:L21405. https://doi.org/10. 1029/2009gl039569 DeJong, JT, Mortensen BM, Martinez BC, Nelson DC (2010) Biomediated soil improvement. Ecol Eng 36(2):197–210 DeJong JT, Fritzges MB, Nusslein K (2006) Microbially induced cementation to control sand response to undrained shear. J Geotech Geoenviron 132(11):1381–1392. https://doi.org/10.1061/(asce)10 90-0241(2006)132:11(1381) Ferris FG, Fratton CM, Gertis JP, Schultzelam S, Lollar BS (1995) Microbial precipitation of a strontium calcite phase at a groundwater discharge zone near rock-creek, British-Columbia, Canada. Geomicrobiol J 13(1):57–67 Fredrickson JK, Zachara JM, Kennedy DW, Dong HL, Onstott TC, Hinman NW, Li SM (1998) Biogenic iron mineralization accompanying the dissimilatory reduction of hydrous ferric oxide by a groundwater bacterium. Geochim Cosmochim Acta 62(19–20):3239–3257 Heenan JW, Ntarlagiannis D, Slater LD, Beaver CL, Rossbach S, Revil A, Atekwana EA, Bekins B (2017) Field-scale observations of a transient geobattery resulting from natural attenuation of a crude oil spill. J Geophys Res Biogeosci 122:918–929. https://doi.org/10. 1002/2016JG003596 Hiebert FK, Bennett PC (1992) Microbial control of silicate weathering in organic-rich ground water. Science 258(5080):278–281 Jimenez-Lopez C, Romanek CS, Bazylinski DA (2010) Magnetite as a prokaryotic biomarker: a review. J Geophys Res 115:G00G03. https://doi.org/10.1029/2009JG001152 Lund AL, Slater LD, Atekwana EA, Ntarlagiannis D, Cozzarelli I, Bekins BA (2017) Evidence of coupled carbon and iron cycling at a hydrocarbon-contaminated site from time lapse magnetic susceptibility. Environ Sci Technol 51(19):11244–11249 McMahon PB, Chapelle FH, Falls WF, Bradley PM (1992) Role of microbial processes in linking sandstone diagenesis with organicrich clays. J Sediment Petrol 62(1):1–10 McMahon PB, Vroblesky DA, Bradley PM, Chapelle FH, Gullett CD (1995) Evidence for enhanced mineral dissolution in organic acidrich shallow ground-water. Ground Water 33(2):207–216 Mellage A, Smeaton CM, Furman A, Atekwana EA, Rezanezhad F, Van Cappellen P (2018) Linking spectral induced polarization (SIP) and subsurface microbial processes: Results from sand column incubation experiments. Environ Sci Technol 52(4):2081–2090 Minsley B, Sogade J, Morgan FD (2007) Three-dimensional selfpotential inversion for subsurface DNAPL contaminant detection at

Body Waves the Savannah river site, South Carolina. Water Resour Res 43: W04429. https://doi.org/10.1029/2005WR003996. Naudet V, Revil A, Bottero J-Y, Bégassat P (2003) Relationship between self-potential (SP) signals and redox conditions in contaminated groundwater. Geophys Res Lett 30(21):2091 Nielsen LP, Risgaard-Petersen N, Fossing H, Christensen PB, Sayama M (2010) Electric currents couple spatially separated biogeochemical processes in marine sediment. Nature 463(7284): 1071–1074 Ntarlagiannis D, Williams KH, Slater L, Hubbard S (2005) The low frequency electrical response to microbially induced sulfide precipitation. J Geophys Res 110:G02009 Parsekian AD, Slater L, Comas X, Glaser PH (2010) Variations in freephase gases in peat landforms determined by ground-penetrating radar. J Geophys Res 115:G02002. https://doi.org/10.1029/ 2009JG001086. Prodan C, Mayo F, Claycomb JR, Miller JHJ (2004) Low-frequency, low-field dielectric spectroscopy of living cell suspensions. J Appl Phys 95(7):3754–3756 Prodan E, Prodan C, Miller JH (2008) The dielectric response of spherical live cells in suspension: an analytic solution. Biophys J 95(9):4174–4182. https://doi.org/10.1529/biophysj.108. 137042 Reguera G, McCarthy KD, Mehta T, Nicoll JS, Tuominen MT, Lovley D r (2005) Extracellular electron transfer via microbial nanowires. Nature 435:1098–1101 Revil A, Mendonça CA, Atekwana EA, Kulessa B, Hubbard SS, Bohlen KJ (2010) Understanding biogeobatteries: Where geophysics meets microbiology. J Geophys Res 115. https://doi.org/10.1029/ 2009jg001065 Revil A, Atekwana E, Zhang C, Jardani A, Smith S (2012) A new model for the spectral induced polarization signature of bacterial growth in porous media. Water Resour Res 48:W09545. https://doi.org/10. 1029/2012WR011965 Rijal ML, Appel E, Petrovský E, Blaha U (2010) Change of magnetic properties due to fluctuations of hydrocarbon contaminated groundwater in unconsolidated sediments. Environ Pollut 158(5): 1756–1762 Saneiyan S, Ntarlagiannis D, Ohan J, Lee J, Colwell F, Burns S (2019) Induced polarization as a monitoring tool for in-situ microbial induced carbonate precipitation (MICP) processes. Ecol Eng 127:36–47 Sato M, Mooney HM (1960) The electrochemical mechanism of sulfide self-potentials. Geophysics 25(1):226–249 Sauck WA, Atekwana E, Nash MS (1998) High conductivities associated with an LNAPL plume imaged by integrated geophysical techniques. J Environ Eng Geophys 2(3):203–212 Slater L, Ntarlagiannis D, Personna YR, Hubbard S (2007) Pore-scale spectral induced polarization signatures associated with FeS biomineral transformations. Geophys Res Lett 34(21):L21404. https://doi. org/10.1029/2007gl031840 Stoy RD, Foster KR, Schwan HP (1982) Dielectric properties of mammalian tissues from 0.1 to 100 MHz: a summary of recent data. Phys Med Biol 27:501–513 Werkema DD Jr, Atekwana EA, Endres AL, Sauck WA, Cassidy DP (2003) Investigating the geoelectrical response of hydrocarbon contamination undergoing biodegradation. Geophys Res Lett 30(12):1647–1651 Williams KH, Ntarlagiannis D, Slater LD, Dohnalkova A, Hubbard SS, Banfield JF (2005) Geophysical Imaging of Stimulated Microbial Biomineralization. Environ Sci Technol 39(19): 7592–7600 Wu Y, Hubbard S, Williams KH, Ajo-Franklin J (2010) On the complex conductivity signatures of calcite precipitation. J Geophys Res. https://doi.org/10.1029/2009JG001129

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Body Waves Mahmoud Mohamed Selim Saleh Department of Nature and Applied Sciences, Al-Aflaj Community College, AL-Kharj University, Al-Aflaj, Riyadh, Saudi Arabia

Synonyms Energy waves generated by an earthquake or an artificial explosion

Definition Waves consist of a disturbance in materials (media) that carries energy and propagates. However, the material that the wave propagates in generally does not move with the wave. The movement of the material is generally confined to small motions, called particle motion, of the material as the wave passes. After the wave has passed, the material usually looks just like it did before the wave, and, is in the same location as before the wave. When stress builds up in the Earth to such a level that rapid slip occurs on a fracture (i.e., an earthquake takes place) or when an explosion or mechanical device is used to initiate a seismic disturbance artificially, a complex field of seismic waves is generated. This wave field propagates much like the waves that travel away from the point at which a stone is thrown into a pond. Waves that travel through the interior of the Earth are called body waves. They follow ray paths bent by the varying density and modulus (stiffness) of the Earth’s interior. The density and modulus, in turn, vary according to temperature, composition, and phase. These waves are usually observed at higher frequency.

Types of Body Waves There are two main kinds of body waves: P-waves and S-waves, so-named because they are the primary and secondary waves detected by a seismograph (Fig. 1). The P-wave is always the first wave to arrive at a seismometer, closely followed by its reflection from the surface and S-waves arrive next. P-waves, or compressional waves, are longitudinal waves (wave motion in the direction the wave is traveling). S-waves are transverse waves or shear waves, involving a back-and-forth shearing motion at right angles to the direction the wave is traveling (Fig. 2). Body waves are reflected and transmitted at interfaces where the seismic velocity and/or density changes, obeying Snell’s Law. At such an interface,

B

38

Body Waves

Body Waves, Fig. 1 P- and S-wave recorded in a threecomponent seismograph.

0.2

Horizontal (East/West)

0

Velocity (mm/s)

−0.2 S-wave

Horizontal (North/South)

0.2 0 −0.2 P-wave

Vertical

0.2 0 10 s

−0.2

Body Waves, Fig. 2 P- and S-waves motion. The arrow shows the direction that the wave is moving

P-wave Compressions Undisturbed medium

Dilatations S-wave

Double ampllitude Wavelength

Body Waves

39

or discontinuity, some of the energy of an incident body wave is reflected as a P-wave, some as an S-wave, some is transmitted as a P-wave and some as an S-wave. The notation for the various seismic ray paths within the Earth are as follows: When an earthquake occurs, seismic waves are emitted from the focus (hypocenter); there are several paths that it can take through the Earth before emerging again at the surface. These paths (refer to Fig. 3) are symbolized by the letters: p ¼ P-wave arrival from a path that traveled upward from the focus (hypocenter); pP ¼ P-wave arrival from a path that traveled upward from the focus, reflected off the surface of the Earth, then arrived back at the surface; P ¼ P-wave arrival from a path that traveled downward from the focus (hypocenter); PP ¼ P-wave reflected off the surface once; PPP ¼ P-wave reflected off the surface twice; c ¼ a reflection off the outside of the outer surface of the outer core – note that this is the principle cause of multiple arrivals of P- and S-waves right at the epicenter; K ¼ a travel path (refraction) through the outer core; KK ¼ one reflection on the inside outer surface of the outer core; KKK ¼ two reflections off the inside outer surface of the outer core; i ¼ a reflection off the outside of the outer surface of the inner core; I ¼ a travel path (refraction) through the inner core. These letters can be used to indicate the path of a seismic wave through the Earth (Fig. 3). For example PKiKP indicates that the wave traveled downward from the focus, refracted through the outer core, reflected off the surface of the inner core, traveled through the outer core, and then traveled through the mantle to arrive at the surface. SKiKS is the same path, but an S-wave. Because liquids have no resistance to shear and cannot sustain a shear wave, S-waves cannot travel through liquid material. The Earth’s outer core is

PKP

PcP

Focus PPP

pP PKKP

P

Wave Propagation

B

Seismic body waves propagate through the Earth’s interior. Because of the elastic properties of the Earth’s materials (rocks) and the presence of the Earth’s surface, four main types of seismic waves propagate within the Earth. Compressional (P) and Shear (S) waves propagate through the Earth’s interior. Waves on a Seismogram If we look at a seismogram, we expect to see the first wave to arrive to be a P-wave (the fastest), then the S-wave, and finally, the Love and Rayleigh (the slowest) waves. As you might expect, the difference in wave speed has a profound influence on the nature of seismograms. Since the travel time of a wave is equal to the distance the wave has traveled divided by the average speed the wave moved during the transit, we expect that the fastest waves arrive at a seismometer first. The fact that the waves travel at speeds that depend on the material properties (elastic moduli and density) allows us to use seismic-wave observations to investigate the interior structure of the planet. We can look at the travel times, or the travel times and the amplitudes of waves to infer the existence of features within the planet, and this is an active area of seismological research. To understand how we “see” into Earth using vibrations, we must study how waves interact with the rocks that make up Earth. Several types of interaction between waves and the subsurface geology (i.e., the rocks) are commonly observable on seismograms. We will examine the two simplest types of interaction, refraction and reflection. Refraction As a wave travels through Earth, the path it takes depends on the velocity. Perhaps you recall from high school a principle called Snell’s law, which is the mathematical expression that allows us to determine the path a wave takes as it is transmitted from one rock layer into another. Snell’s law describes the relationship between the angles and the velocities of the waves. Snell’s law equates the ratio of material velocities V1 and V2 to the ratio of the sines of incident (θ1) and refracted (θ2) angles, as shown in the following equation:

PKIKP PKiKP

believed to be liquid because S-waves disappear at the mantle–core boundary, while P-waves do not.

PP

Body Waves, Fig. 3 P- and S-wave paths in the Earth’s interior

sin y1 sin y2 ¼ V L1 V L2

ð1Þ

where V L1 is the longitudinal wave velocity in material 1 and V L2 is the longitudinal wave velocity in material 2.
Note  that in Fig. 4, there is a reflected longitudinal wave V L10 shown. This wave is reflected at the same angle as the incident wave because the two waves are traveling in the same

40

Body Waves VL1′

VL1 θ1 θ1

θ2

Rock Type 1

VL2


 Body Waves, Fig. 4 Reflected V L10 and refracted ðV L2 Þ longitudinal waves

material, and hence have the same velocities. This reflected wave is unimportant in our explanation of Snell’s law, but it should be remembered that some of the wave energy is reflected at the interface. When a longitudinal wave moves from a slower to a faster material, there is an incident angle that makes the angle of refraction for the wave 90°. This is known as the first critical angle. The change in direction depends on the ratio of the wave velocities of the two different rocks. When waves reach a boundary between different rock types, part of the energy is transmitted across the boundary. The actual interaction between a seismic wave and a contrast in rock properties is more complicated because an incident P-wave generates transmitted and reflected P- and S-waves and also an incident S-wave generates transmitted and reflected P- and S-waves, so five waves are involved. The transmitted wave travels in a different direction, which depends on the ratio of velocities of the two rock types (Figs. 5 and 6). Part of the energy is also reflected backwards into the region with rock type 1, but we have not shown that on these figures. Refraction has an important effect on waves that travel through Earth. In general, the seismic velocity in Earth increases with depth (there are some important exceptions to this trend) and refraction of waves causes the path followed by body waves to curve upward (Fig. 7). Reflection The second wave interaction with variations in rock type is reflection. I am sure you are familiar with reflected sound waves: we call them echoes. And your reflection in a mirror or pool of water is composed of reflected light waves. In seismology, reflections are used to prospect for petroleum and investigate Earth’s internal structure. In some instances, reflections from the boundary between the mantle and crust may induce strong shaking that causes damage about 100 km from an earthquake (we call that boundary the “Moho” in honor of Mohorovicic, the scientist who discovered it). A seismic reflection occurs when a wave impinges on a change in rock type (which usually is accompanied by a change in seismic-wave speed). A part of the energy carried

Rock Type 2

Body Waves, Fig. 5 Velocity in rock type 2 is greater than velocity in rock type 1

Rock Type 1

Rock Type 2

Body Waves, Fig. 6 Velocity in rock type 2 is less than velocity in rock type 1 Seismometer

Earthquake Constant-Velocity Sphere

Seismometer

Earthquake Earth

Body Waves, Fig. 7 The overall increase in seismic-wave speed with depth in the Earth produces an upward curvature to rays that pass through the mantle. A notable exception is caused by the decrease in velocity from the mantle to the core. This speed decrease bends waves backwards and creates a “P-wave shadow zone” between about 100° and 140° distance (1° ¼ 111.19 km)

by the incident wave is transmitted through the material (that is the refracted wave described above), and a part is reflected back into the medium that contained the incident wave (Fig. 8). The amplitude of the reflection depends strongly on the angle that the incident wave makes with the boundary and the contrast in material properties across the boundary. For some angles, all the energy can be returned into the medium containing the incident wave. An incident P-wave generates transmitted and reflected P- and S-waves and so five waves are involved. Likewise, when an S-wave interacts with a boundary in rock properties, it too generates reflected and refracted P- and S-waves. Because major boundaries between different rock types within the Earth are normally

Body Waves

41

ont efr r av cula W endi Perp e angl

Reflected wave

Rock Type 1

t3

Ray

path

t1 t0

Rock Type 2

B

t2

Compressional (P) motion

Source

Body Waves, Fig. 8 When a wave encounters a change in material properties (seismic velocities and or density), its energy is split into reflected and refracted waves

approximately parallel to the Earth’s surface, S-wave particle motion is commonly in the SV (perpendicular to the ray path and vertical) and SH (perpendicular to the ray path and horizontal) directions. P-waves travel faster than S-waves, so there will be separate wave front representations for the P- and S-waves (Fig. 9). If the physical properties of the material through which the waves are propagating are constant, the wave fronts will be circular (or spherical in three dimensions). If the physical properties vary in the model, the wave fronts will be of more complex shapes. In the transverse or shear wave, the particles oscillate at a right angle or transverse to the direction of propagation. Shear waves require an acoustically solid material for effective propagation, and therefore, are not effectively propagated in materials such as liquids or gases. Shear waves are relatively weak when compared to longitudinal waves. In fact, shear waves are usually generated in materials using some of the energy from longitudinal waves.

Velocity Among the properties of waves propagating in isotropic solid materials are wavelength, frequency, and velocity. The wavelength (l) is directly proportional to the velocity of the wave (C) and inversely proportional to the frequency (f) of the wave. The phase velocity of a seismic wave can be written as: C ¼ o=k,

ð2Þ

where o ¼ 2πf is the angular frequency and k ¼ l/2π is wave number. Velocity of P- and S-waves Seismic waves travel fast, of the order of kilometers per second (km/s). The precise speed that a seismic wave travels depends on several factors, the most important of which is the composition of the rock. We are fortunate that

Shear (S) motion

Body Waves, Fig. 9 Wave fronts and ray paths in seismic-wave propagation

the speed depends on the rock type because it allows us to use observations recorded on seismograms to infer the composition or a range of compositions of the planet. But the process is not always simple, because sometimes different rock types have the same seismic-wave velocity, and other factors also affect the speed, particularly temperature and pressure. Temperature tends to lower the speed of seismic waves and pressure tends to increase the speed. Pressure increases with depth in Earth because the weight of the rocks above increases with increasing depth. Usually, the effect of pressure is larger and in regions of uniform composition, the velocity generally increases with depth, despite the fact that the increase of temperature with depth works to lower the wave velocity. When the different seismic-wave types are described below, I will quote ranges of speed to indicate the range of values we observe in common terrestrial rocks. But you should keep in mind that the specific speed throughout Earth will depend on composition, temperature, and pressure. P-waves

P-waves are the first waves to arrive on a complete record of ground shaking because they travel the fastest (their name derives from this fact –P is an abbreviation for primary, first wave to arrive). They typically travel at speeds between ~1 and ~ 14 km/s. The slower values correspond to a P-wave traveling in water, the higher number represents the P-wave speed near the base of Earth’s mantle. The velocity of a wave depends on the elastic properties and density of a material. If we let k represent the bulk modulus of a material, m the shear modulus, and r the density, then the P-wave velocity, which we represent by CP, is defined by:

42

Body Waves

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi l þ 2m CP ¼ r

Attenuation of frequencies

ð3Þ

A modulus is a measure of how easy or difficult it is to deform a material. For example, the bulk modulus is a measure of how a material changes volume when pressure is applied and is a characteristic of a material. For example, foam rubber has a lower bulk modulus than steel. P-waves are sound waves, it is just that in seismology we are interested in frequencies that are lower than humans’ range of hearing (the speed of sound in air is about 0.3 km/s). The vibration caused by P-waves is a volume change, alternating from compression to expansion in the direction that the wave is traveling. P-waves travel through all types of media – solid, liquid, or gas. S-waves

Secondary, or S-waves, travel slower than P-waves and are also called “shear” waves because they do not change the volume of the material through which they propagate – they shear it. S-waves are transverse waves because they vibrate the ground in a direction “transverse,” or perpendicular, to the direction that the wave is traveling. The S-wave speed, call it CS, depends on the shear modulus and the density rffiffiffi m CS ¼ r

ð4Þ

Even though they are slower than P-waves, the S-waves move quickly. Typical S-wave propagation speeds are of the order of 1–8 km/s. The lower value corresponds to the wave speed in loose, unconsolidated sediment, the higher value is near the base of Earth’s mantle. An important distinguishing characteristic of an S-wave is its inability to propagate through a fluid or a gas because fluids and gases cannot transmit shear stress and S-waves are waves that shear the material. The velocity of seismic waves depends on the elastic properties and density of a material in which the wave is traveling. P-waves are sound waves, it is just that in seismology we are interested in frequencies that are lower than humans’ range of hearing (the speed of sound in air is about 0.3 km/s). The vibration caused by P-waves is a volume change, alternating from compression to expansion in the direction that the wave is traveling. In general, earthquakes generate larger shear waves than compressional waves and much of the damage close to an earthquake is the result of strong shaking caused by shear waves.

Amplitude

1 0.8 0.6 0.4 0.2 0 0

50

100 Frequency

150

200

Body Waves, Fig. 10 Amplitude spectrum in red is for signal after attenuation

will be visible as cracks, fault offsets, and displacements of the ground after the disturbance has passed. A source of energy creates the initial disturbance and the resulting waves propagate (travel) outward from the disturbance. Because there is finite energy in a confined or short-duration disturbance, the waves generated by such a source will spread out during propagation and become smaller (attenuate) with distance away from the source or with time after the initial source, and thus, will eventually die out (Fig. 10). Knowledge of attenuation can be very useful in seismic data processing, as its removal increases resolution. Attenuation often serves as a measurement tool that leads to the formation of theories to explain physical or chemical phenomenon that decreases the body wave’s intensity. The amplitude change of a decaying plane wave can be expressed as: A ¼ A0 eaz

ð5Þ

In this expression, A0 is the unattenuated amplitude of the propagating wave at some location. The amplitude A is the reduced amplitude after the wave has traveled a distance z from the source. The quantity α is the attenuation coefficient of the wave traveling in the z-direction. The more common unit of the attenuation value is Decibels. Attenuation is generally proportional to the square of the frequency. Quoted values of attenuation are often given for a single frequency, or an attenuation value averaged over many frequencies may be given. The quoted values of attenuation only give a rough indication of the attenuation and should not be automatically trusted. Generally, a reliable value of attenuation can only be obtained by determining the attenuation experimentally for the particular material being used.

Attenuation Summary: Results of Regional and Global Studies Near the source of a strong disturbance, such as a large explosion or earthquake, the wave-generated deformation can be large enough to cause permanent deformation, which

The study of seismic body waves permits us to derive important knowledge about the internal constitution of the Earth.

Body Waves

The problems related to propagation of elastic seismic body waves has been discussed in a number of books and in numerous papers (e.g., Bromwich 1898; Sezawa 1927; Haskell 1953; Ewing et al. 1957; Brekhovskikh 1960; Bullen 1963; Bath 1968; Achenbach 1975 and others). Attenuations of seismic waves have been investigated by many authors (e.g., McDonald et al. 1958; Knopoff and Macdonald 1958; Kolsky 1963; White 1965; Kuster and Toksoz 1974; Shoenberger and Levin 1974; Kennett 1983; Hong and Kennett 2003 and others). The asymptotic theory of bodywave propagation in anisotropic media is also well developed (e.g., Crampin 1984; Thomsen 1986; Tsvankin 1997; Cerveny 2001; Chapman 2004). The effect of initial stresses present in the medium is not considered in the above studies. Recently, Dey and Dutta (1998) have studied the problem of propagation and attenuation of seismic waves, taking the effect of initial stresses present in the medium into account. Biot incremental deformation theory has been used (Biot 1965) in their study. Selim and Ahmed (2006) have studied the velocities of propagation, damping, and attenuations of seismic body waves in compressible and dissipative medium under the effect of both initial and couple stresses. Body-Wave Modeling Seismological methods for determining Earth structure are often classified as being either active or passive in nature. Passive methods involve waiting for an earthquake to occur and provide the seismic source for recording. In contrast, controlled-source seismology refers to active methods where the experimenter provides the source by an explosion or a mechanical device such as a hammer, a weight that is dropped, or a vibrator. Active methods can, in turn, be divided into two basic classifications. The first is seismic reflection profiling, which is a clearly defined approach in which the goal is to produce an image of the subsurface in which structures can be seen directly, in the way that an X-ray image reveals features inside an object. Other active seismic methods infer seismic velocities and the presence of discontinuities in velocity and structure (such as faults) using a variety of approaches that analyze the arrival times and sometimes the shape of seismic waves traveling along different paths through the Earth. As a consequence of advances in seismic instrumentation and national programs increasing the number of instruments available, there have been many recent developments in these techniques that do not directly image the Earth. A convenient aspect of the theoretical basis for virtually all active-source techniques is that they are largely independent of scale. Thus, detailed studies to address environmental problems and regional studies to determine deep Earth structure employ the same basic techniques of analysis (Hancock and Skinner 2000). Recently, several efforts have been made to complete our knowledge of the structure of the

43

Earth’s upper mantle by detailed observation of seismic body waves, especially by their amplitude–distance curves.

Cross-References ▶ Deep Seismic Reflection and Refraction Profiling ▶ Earth’s Structure, Global ▶ Earthquake Source Theory ▶ Earthquakes and Crustal Deformation ▶ Earthquakes, Energy ▶ Propagation of Elastic Waves: Fundamentals ▶ Seismic Phase Nomenclature: The IASPEI Standard ▶ Seismic Ray Theory ▶ Seismic Signals in Well Water Observations ▶ Seismic Tomography ▶ Seismic Viscoelastic Attenuation ▶ Seismic Wave Propagation in Real Media: Numerical Modeling Approaches ▶ Seismic Waves, Scattering ▶ Seismic, Velocity, and Density Relationships ▶ Seismic, Waveform Modeling and Tomography ▶ Seismology, Global Earthquake Model ▶ Traveltime Tomography Using Controlled-Source Seismic Data

Bibliography Achenbach JD (1975) Wave propagation in elastic solids. NorthHolland, Amsterdam, p 425 Bath MA (1968) Mathematical aspects of seismology. Elsevier, Amsterdam, p 414 Biot MA (1965) Mechanics of incremental deformation. Wiley, New York, p 181 Brekhovskikh LM (1960) Waves in layered media. Academic, New York Bromwich TJLA (1898) On the influence of gravity on the elastic waves and in particular on the vibration of an elastic globe. Proc Lond Math Soc 30:98–120 Bullen KE (1963) An introduction to the theory of seismology. Cambridge University Press, Cambridge Cerveny V (2001) Seismic ray theory. Cambridge University Press, Cambridge Chapman C (2004) Fundamentals of seismic wave propagation. Cambridge University Press, Cambridge Crampin S (1984) An introduction to wave propagation in anisotropic media. Geophys J R Astron Soc 76:17–28 Dey S, Dutta D (1998) Propagation and attenuation of seismic body waves in initially stressed dissipative medium. Acta Geophysica Polonica XLVI(3):351–365 Ewing WM, Jardetzky WS, Press F (1957) Elastic waves in layered media. McGraw-Hill, New York, p 380 Hancock P, Skinner BJ (2000) Controlled-source seismology. In: The Oxford companion to the Earth. Encyclopedia.com Haskell NA (1953) The dispersion of surface waves in multilayered media. Bull Seismol Soc Am 43:17–34 Hong TK, Kennett BLN (2003) Scattering attenuation of 2D elastic waves: theory and numerical modeling using a wavelet-based method. Bull Seismol Soc Am 93(2):922–938

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44 Kennett BLN (1983) Seismic wave propagation in stratified media. Cambridge University Press, Cambridge, p 342 Knopoff L, Macdonald JF (1958) Attenuation of small amplitude stress waves in solid. Rev Mod Phys 30:1178 Kolsky H (1963) Stress waves in solid. Dover, New York Kuster GT, Toksoz MN (1974) Velocity and attenuation of seismic waves in two-phase media, part I: theoretical formulations. Geophysics 39(5):587–618 McDonald FJ, Angona FA, Mills RL, Sengbush RL, Van Nostrand RG, White JE (1958) Attenuation of shear and compressional waves in Pierre shale. Geophysics 23(3):421–439 Selim MM, Ahmed MK (2006) Propagation and attenuation of seismic body waves in dissipative medium under initial and couple stresses. Appl Math Comput 182:1064–1074 Sezawa K (1927) Dispersion of elastic waves propagated on the surface of stratified bodies and on curves surfaces. Bull Earthquake Res Inst (Tokyo) 3:1–18 Shoenberger M, Levin FK (1974) Apparent attenuation due to intrabed multiples. Geophysics 39(3):278–291 Thomsen L (1986) Weak elastic anisotropy. Geophysics 51:1954–1966 Tsvankin I (1997) Anisotropic parameters and p-wave velocity for orthorhombic media. Geophysics 62:1292–1309 White JE (1965) Seismic waves, radiation, transformation and attenuation. McGraw-Hill, New York, p 302

Borehole Seismic Networks and Arrays Marco Bohnhoff and Peter Malin GFZ German Research Center for Geosciences, Geomechanics and Scientific Drilling, Potsdam, Germany

Synonyms Borehole strings and chains; Buried grids; Downhole networks and arrays; Vertical arrays; VSP

Definition In the broadest sense, borehole networks and arrays are assemblies of geophysical instruments placed in the subsurface via boreholes. This article focuses on their seismological applications. While common usage often blurs the distinction between a network and an array, this article explicitly distinguishes between their operational theories and engineering practices, and in the case histories and examples provided. To strictly qualify as an array, the assembly of instruments must include matching or similar sensors located at different depths in one well or lateral separated in different wells. The combined natural-frequency responses and spacing of the sensors must also include a range over which they can be used to improve signal detection and analysis via time series processing. Outside of this range a borehole assembly acts as a network of independent detectors.

Borehole Seismic Networks and Arrays

Introduction The most common borehole geophysical instruments measure elastic and permanent displacements, electromagnetic fields, and fluid pressures. These sensors are used to variously record underground accelerations, velocities, tilts, tensor and volumetric strains, electrical conductivities, and porepressures. While accelerometers, tiltmeters, and stainmeters are common borehole sensors, most borehole installations are currently done with seismometers. The primary driver behind this application is the significant improvement in seismic signal quality with installation depth (e.g., Prevedel et al. 2015). Boreholes not only increase the distance of seismometers from near-surface environmental and cultural noise, they are also placed below the signaltrapping-and-attenuating properties of near-surface materials. As a result, borehole seismometers can dramatically improve the detection and location of natural and induced microearthquakes and enhance the quality of controlled-source data (e.g., Kwiatek et al. 2019). It has allowed the observation of phenomena such as fault zone guided waves (Ellsworth and Malin 2011). Recently, it has revealed the existence of resonances in fluid-filled fractures (Sicking and Malin 2019). This entry describes some of the early developments in borehole seismology, its current instrumentation and installation methods, its noise-suppression and signal-detection advantages, and the differences between networks and arrays (e.g., Bohnhoff et al. 2018). It also presents representative examples and case histories that illustrate the key aspects of this method. The article concludes with a brief discussion of the new borehole fiber-optic sensor-cables method for borehole seismology. Such cables are showing promise in the measurement of seismic waves, rock strains, and subsurface temperatures and pore pressures (e.g., Lellouch et al. 2019). As their installation and analysis methods progress, it is likely that these types of sensors will become standard choices for borehole networks and arrays.

Early Developments The general use of boreholes for making subsurface seismic measurements appears to have begun nearly 100 years ago. McCollum and LaRue (1931) report that “during the past year or more, use has been made of existent wildcat wells for the purpose of making a seismic exploration [of salt dome structures].” By the 1940s, use of a single borehole seismograph to profile the flanks of a salt dome was an established practice (Gardner 1949; Musgrave et al. 1960). These developments parallel the first test of the borehole electric log by Conrad and Marcel Schlumberger in 1927 (Johnson 1962). The diversity of what became known as well logging tools grew rapidly, as did their use in make

Borehole Seismic Networks and Arrays

point measurements of near-well rock properties and structures. The development of borehole seismic networks and arrays can be viewed as efforts to penetrate further into the rock mass for the same purposes. From this effort came the finding that placing seismometers even a few tens of meters underground significantly increased the quality of seismic data. These include increases in signal-to-noise ratios, recorded signal frequency bandwidths, and the detection and resolution of remote lateral velocity variations and structures.

Current Instrumentation and Installation Examples of the type of seismic instruments used to achieve these advantages are shown in Fig. 1. In the case of traditional three-component, mass and spring sensors, these range from greater than ~10–12 Hz omni directional seismometers in fixed orthogonal mounts, to lower frequency ~1–10 Hz, gimble mounted, vertical and horizontal components. In both cases, the installed orientation of the components needs to be determined, either by special downhole tools or by calibrations shots or distant earthquakes. For a fixed depth installation, the latter two methods are most cost-effective. In addition to the borehole sensors and their communication cables, a critical component of their deployment is the method used to couple them to the surrounding rock. This can be by cementing of the entire well, including instruments and cables. The disadvantage is that nothing is retrievable, and the well is permanently blocked. With this type of installation, the use of mechanically robust and electrically isolated sensors is essential: no active components requiring electrical power or corrodible materials, and triple barriers from sensorcompromising borehole fluids and gases. For some applications an adequate degree of coupling can also be achieved by lowering the sensors on a heavy but flexible backbone – e.g., oil production tube – the spring action of which, when standing on the borehole bottom, pushes the sensors against the wall. This system is retrievable and works well for signals with accelerations significantly less than 1 g. Its disadvantage is the need for heavy surface equipment to handle the weight of the backbone plus sensors and cable. A more robust coupling alternative to a flexible backbone is the use of collapsible bladders or the equivalent of a firehose, the filing of which pushes the sensor against the borehole wall. Neither of these methods are suitable for moving the array along the well for profiling purposes. For relocatable networks or arrays, passive coupling can be accomplished with bow-spring or magnetic clamps, the continuously coupled sensors pulled into the borehole by a heavy weight or mechanical tractor. The most sophisticated moveable couplers include electrically or hydraulically driven arms, the controlled forces of which guarantee the best connection to the borehole wall.

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Noise-Suppression and Signal-Detection Advantages Whether in the form of a borehole network or array, placing seismic receivers underground significantly improves the quality of seismic signals and the amount of information they contain about seismic events, velocities, and geological structures (Figs. 2 and 3). Perhaps the most significant improvements are the increases in the signal-to-noise ratio and bandwidth of upward propagating seismic waves. As shown in Fig. 2a, several factors signal-to-noise improvement can be achieved by installing sensors just a few tens of meters of downhole. These improvements are most dramatic for signal frequencies above a few Hz, as these are most affected by attenuation and scattering near the earth’s surface. Since these effects apply equally to downward propagating noises generated at the surface, their damping-out with depth adds to the improvement as well. The combined effects on signal-to-noise and bandwidth can be seen in Fig. 2 by comparing the frequency-dependent improvements at depth with surface values: while it is not common to record 100 Hz seismic signals at ground levels, they are usually well recorded at a depth of 250–500 m. These signal improvements have significant effects on the detection and location of small seismic events – both natural and fabricated. As shown in Fig. 2b, by lowering the sensors of an established seismic monitoring network tens to hundreds of meters below their current positions, the network’s event-detection threshold can be lower by one to three magnitude units. Further improvements relate to more accurate determination of lateral velocity changes, observation of nearvertical structures, and the location of other subsurface sources – e.g., well drilling bits (Fig. 3).

Networks Versus Arrays Data from multiple borehole sensors can be processed in two modes: as travel-time/ray path signals or as fully sampled waveforms. This distinction depends on the separation of the sensors, laterally or vertically, and the apparent wavelength of the signals to be processed. A simple rule that separates these two domains is given by Rayleigh’s criterion – the rule of thumb being that sensors separated by less than a quarter of the apparent wavelength observed can be used in practical signal enhancement procedures. Otherwise the data must be handled as independent arrivals. In the ray domain, borehole seismometer data have been used in two fashions: back-projecting rays to their sources based on P-wave polarization and S-P times and Kirchhoff migration studies of event amplitudes. In the waveform domain, signal-to-noise improvement have been accomplished

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Borehole Seismic Networks and Arrays, Fig. 1 (a) An example of analog borehole seismographs for deployment in a hybrid, four-level vertical borehole array and seven-site lateral network configuration. These sensor arrays of 1, 2, and 15 Hz seismographs were deployed at 75 m spacing in each of the seven matching, 300 m deep, GONAF stations surrounding the eastern Marmara Sea (Prevedel et al. 2015; Bohnhoff et al. 2017.) While the seven stations are too far apart to be used as a lateral array for signal improvement methods such as beam forming, the arrays in each of the seven can be used to separate up-coming signals from down-going surface noise. These deployments can also be used to determine near-surface site effects as they impact earthquake locations (Raub et al. 2016). (b) An example of a multilevel Vertical Seismic Profiling array, with clamping arms extended. The

Borehole Seismic Networks and Arrays

availability of electrical power for the clamps allows for the inclusion of digitizing and telemetry electronics in each of the 15 Hz-seismometerbased sounds. The spacing between levels can be adjusted from just a few meters to tens by changing the interconnects between levels. (c) An example of a marine hydrophone array – known as a streamer – rolled up on a reel. When deployed as a vertical array, the end of the streamer is anchored on the ocean floor and held vertical with a surface or nearsurface buoy. (d) An example of a Fiber-Optic Distributed Acoustic Sensing (DAS) system. ((a) Modified from Prevedel et al. 2015; (b) courtesy of Section 4.2 ‘Geomechanics and Scientific Drilling’ at the GFZ German Geosciences Research Center, Potsdam, Germany; (c) https://commons.wikimedia.org/wiki/File:Streamer-detail_hg.jpg; (d) courtesy of J.-A. Chavarria, Horizon 2020)

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Borehole Seismic Networks and Arrays, Fig. 2 (a) A typical range of signal-to-noise and bandwidth improvement for different frequency bands as a function of seismometer depth. This diagram, obtained from a variety of geological sites, shows the limited improvements resulting for burial of low frequency signals versus the significant one for high frequency ones. It also shows the increase in bandwidth versus depth. The width of the frequency band improvements is intended to cover the effects of different rock types and surface weathering and fracturing.

(b) Improvement in network detection and location capacity as a function of placing sensors below existing stations. The sequence of curved shapes is intended to show how elimination of near surface attenuation and noise by sensor burial lowers the network’s threshold. In the example indicated on the right side of the diagram, a surface-based network that detects and locates ~M3 events at a given distance d can do the same for events one to two magnitude units lower by placing the networks sensors downhole modified from Bohnhoff et al. 2018

Borehole Seismic Networks and Arrays, Fig. 3 (Left) A schematic comparison between a surface array or network recording a subsurface seismic event (red dot) (upper left) and a borehole array or network (lower left). Blue triangles are sensors. The main advantages of the borehole installation are avoiding upward propagation through the attenuation and scattering of the near surface layers (e.g., Blakeslee and Malin 1990), attenuation and trapping of surface noise by these same layers, the appropriate geometry to record signals from vertical and subvertical structures,

and sensitivity to lateral velocity gradients. (Right) Improvements in event location at the SAFOD drill site: (1) are earthquake locations derived from (near) surface sensors (Zoback et al. 2011; E. Shalev, Personal Communication, 2003), (2) shows the refinement by the addition of the 32-level Pilot Hole vertical array (PHA, blue triangles) (Thurber et al. 2003), and then (3) by the subsequent installation of the Main Hole seismometer (MHS, large blue triangle). (Background color shows electrical resistivity structure modified from Unsworth et al. (1997).)

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Prime examples of borehole seismic networks made up of single-seismometer stations are government run local, national, and global monitoring systems. These include the high-resolution networks at Parkfield in California (Parkfield HRSN 2014) and Taipei in Taiwan (Huang et al. 2010). The most extensive national network is the constantly growing Japanese Hi-Net, with many hundreds of stations placed more than 100 m underground (Takahashi 1982; Hamada et al. 1985; Obara 2003). Global borehole network stations have served both research and international security needs. The borehole-based Seismic Research Observatory made

significant contributions to the study of the Earth’s deep structure (https://www.fdsn.org/networks/detail/SR/). Many of the stations of the Global Seismic Network and Comprehensive Test Ban Treaty nuclear verification network are borehole-based (e.g., Bent 2013; https://www.iris.edu/hq/ programs/gsn; Richards 2016; https://www.ctbto.org/). In the ray domain, borehole seismometer strings like the one in the San Andreas Fault Observatory at Depth (SAFOD; Zoback et al. 2011) have been used in two forms: (1) back-projecting rays to their earthquake sources based on P-wave polarization and S-P times (Oye et al. 2004) and (2) Kirchhoff Migration studies of secondary arrivals originating from subvertical faults and other structures (Fig. 4a, b; Chavarria et al. 2003). The ability of deep seismometer installations to accurately locate earthquake hypocenters is illustrated by the target event for the San Andreas Fault Observatory at Depth project (Fig. 3, right; Zoback et al. 2011; Thurber et al. 2004; Zhang et al. 2009; Unsworth et al. 1997). Initially, a large network of several dozen surface stations was used to establish a preliminary target event location using travel time tomography (Thurber et al. 2003). Shortly thereafter, arrival times from both earthquakes and explosions recorded on a 32-level string of borehole seismometers installed in the 2 km deep SAFOD Pilot Hole were added to the inversion (Thurber et al. 2004). The result was that the location of the target event moved some 700–800 m, mostly upwards, and less so to the

Borehole Seismic Networks and Arrays, Fig. 4 (a) The vertical component seismograms from the 32-level SAFOD 2 km deep Pilot Hole seismometer string. The signals are upcoming waves from an earthquake located on the San Andreas Fault. (b) A data set of both earthquakes and explosions were Kirchhoff migrated by Chavarria et al. (2003) to find potential reflection structures along the path of the SAFOD Main Hole. (c) Interferometric processing of SAFOD Main Hole Seismic

While Drilling data recorded on the Pilot Hole string to yield a clear reflection from fault structures within the San Andreas Fault Zone (Vasconcelos et al. 2008). The signals indicated by the red arrows are from structures inside the SAFZ system. As part of their analysis, Vasconcelos et al. (2008) also used Pilot Hole data to complete a wave field migrated image of the SAFZ structure. ((a and b) Modified from Chavarria et al. 2003; (c) modified from Vasconcelos et al. 2008)

with various signal stacking and correlation procedures. It is important to note that in both domains it is essential that the orientation of the three component sensors be accurately known. Even when processed as a network of sensors along a borehole versus as a signal-enhancement array, vertically separated seismometers have advantages such as identifying down-going surface noise from upcoming earthquake signals. For impulsive signals, this can be done by noting arrival orders. For long noise trains, it can be done by noting the lack of noise reaching the deepest sensors.

Illustrative Examples

Borehole Seismic Networks and Arrays

southwest (Zoback et al. 2011; E. Shalev, Personal Communication, 2003). Afterwards, drilling of the SAFOD Main Hole (MH) was stopped short of this target to allow further location improvement by placing a seismometer at its bottom. The results were a further 200 m move mostly to the southwest. An unexpected advance in fault zone velocity structure imaging at Parkfield came from borehole seismometer placed within tens of meters of the SAF (Ellsworth and Malin 2011; Malin: Fault Zone Guided Waves, this volume). These revealed head waves refracted along the faster side of the SAFZ (Ben-Zion and Malin 1991). In this same data set, there were clear, normally dispersed, fault zone guided waves (Fg). These fit Love-type trapped waves (FL) propagating in a several km deep, ~100–120 m thick, low velocity channel (Li and Malin 2008). This picture was further refined by data from the SAFOD MH sensor, which contained Rayleigh-type trapped waves (FR) and a new, leaky mode, fault guided wave FF. The latter phase fits partially trapped PSV waves travel in a several km deep, ~30 m wide fault core of very low P velocity. A unique example of a local borehole-based network is the Geophysical Observatory on the North Anatolian Fault (GONAF) near Istanbul, Turkey (Bohnhoff et al. 2017). This seismic observatory is a hybrid network of borehole array stations, each station consisting of a four-level vertical array of seismometers spaced ~75 m apart, going up from ~300 m (Fig. 1a). The different levels has been used to follow up-coming and surface-reflected microearthquake signals for site effect studies (Raub et al. 2016) and to discriminate between down going surface noise and actual microearthquake signals (Malin et al. 2018). The former study uncovered a reversal in seismic velocities that grossly affected the motion of upcoming signals. In the latter, the noise suppression was used to lower the magnitude threshold by ~2–3 units, allowing detection of many microearthquakes preceding a 4.2 mainshock. Some of the networks mentioned above can also be operated as arrays in a narrow range of signal frequencies, as in the case of surface wave analysis (Luo et al. 2008). However, lateral arrays and multilevel borehole sensor strings have been developed with the specific purpose of signal enhancement in mind (Fig. 1b). Early work in this type of Vertical Seismic Profiling in boreholes was begun in the 1950s by Y. I. Galperin (Galperin 1961; Galperin et al. 1986) and extensively expanded upon in the following decades (Balch and Lee 1984; Balch et al. 1986). The array aspects of lateral borehole networks have been extensively developed by the nuclear verification community (Richards 2016), primarily in the area of beam forming. These methods are now being adapted by energy industry seismic contractors (McClellan et al. 2018). Full waveform signal-enhancement processing has also been applied to Seismic-While-Drilling data that was

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collected with the Pilot Hole string during the drilling of the SAFOD MH. Using vibroseis-type deconvolution and seismic reflection signal processing methods, Coates et al. (2006) were able to obtain an image showing steeply dipping Tertiary Great Valley strata immediately south of the SAFOD site – an unexpected juxtaposition based on surface and area borehole geology. Because of their subvertical attitude and signal loss to near-surface attenuation and noise, this structure would have been difficult to image with standard surface reflection profiling. Further signal processing of the Seismic While Drilling data was done by Vasconcelos et al. (2008) (Fig. 4c). They used an interferometric approach to signal stacking to obtain an image of the internal structure within the fault zone. In addition to downhole seismometers and accelerometers, multisensor hydrophone (streamer) arrays have been adapted for vertical deployments in boreholes on land and in the open ocean (Fig. 1c; Marzetta et al. 1988; Toksöz et al. 1992; Ikelle and Amundsen 2018). On land, early tests revealed that incoherent tube waves in long boreholes could completely cover secondary arrivals, initially limiting their usefulness. This problem was by and large solved by various signal processing and mechanical tube-wave-damping methods. Moreover, such arrays allowed detection of new seismic source processes related to migrating fluids and gases due to wellbore leakage (Bohnhoff and Zoback 2010; Bohnhoff et al. 2010; Martinez-Garzon et al. 2013). In the ocean, while near-bottom sensors are significantly quieter than near-surface ones, ocean currents can cause vertical arrays to tilt in various directions over minutes, hours, and days, making signal processing a challenge. On land, the elimination of near-surface attenuation and suppression of surface noise has also allowed observation of what appear to be Krauklis wave resonances in subsurface fractures (Sicking and Vermilye 2019; Liang et al. 2017; Tary et al. 2014a, b). Figure 5a shows a spectrogram obtained by the stacking of frequency-time-amplitude data from a 12-level 2.5 km deep borehole seismometer string at a geothermal development site in Finland. All these previous studies collected point-observation data using conventional borehole seismographic instruments: passive spring-and-mass and hydrophone sensors, force feedback, and similar nodal devises. Now on the near-horizon for broad application are borehole fiber-optic Distributed Acoustic Sensing (DAS) systems (Fig. 1d). A high-level account of this new technology was recently published by Lellouch et al. (2019). DAS systems have now effectively achieved continuous sampling of broadband seismic wavefields. Lellouch et al. (2019) have compared borehole DAS observations of earthquake and explosion data from multichannel VSP instruments at the SAFOD site. Their analysis included travel time picking, slant stack beam forming, and interferometry. Due to its redundancy, it was

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Borehole Seismic Networks and Arrays, Fig. 5 (a) Spectrogram of resonances from major subsurface fractures recorded by a 12-level borehole seismometer string during hydraulic stimulation of a geothermal well in Finland. The string was placed between 1970 and 2420 m below ground and ~3600 m above the stimulation zone. While the resonance could be seen on individual-level spectrograms, the current plot is the stack of spectrograms from 11 levels. Based on the work of Tary et al. (2014a, b) and Liang et al. (2017), and correlations with stimulation pumping and induced earthquakes, Sicking and Malin (2019) have interpreted these signals as coming from interfering

Krauklis wave on major subsurface fractures (Figure courtesy of P. E. Malin 2019). (b) Broadband Fiber-Optic DAS recordings of M1.33 and M2.4 earthquakes on the San Andreas Fault at Parkfield. These data were collected with a DAS cable along the first 800 m of the SAFOD Main Hole. The data have been bandpass filtered between 0 and 120 Hz. Compared to the point observation data in Fig. 4a, b, these data meet the 1/4 wavelength criterion for high-frequency signal enhancement. The result is a nearly continuous wavefield image that provides a much richer look at the velocity structure surrounding the drill site and SAF. (Figure from Lellouch et al. 2019)

possible to detect both S-waves and clear P-S reflections in the DAS data (Fig. 5b). In addition to the technical description of DAS operation, their paper includes important velocity and structural results from SAFOD.

intrusive remote exploration of the crust and refinement of seismic hazards and risks. With further technological and methodological improvements, downhole seismic networks and arrays may become the norm of high-quality seismic observations. This evolution is currently taking place with the advent of Fiber-Optic DAS systems. Combined with lowered costs in drilling and the needs for a more fundamental understanding of the earth and its resources and hazards, borehole seismic networks and arrays are likely to be the next step in the history of seismology.

Conclusions Progressing from a single spring-mass seismometer in shallow water well to continuous Fiber-Optical DAS cables over the past 100 years, borehole seismic networks and arrays have opened up a previously unobtainable view of the earth’s interior. It facilitates the detection and monitoring of deep and remote, and otherwise inaccessible, seismic events and tectonic processes. This includes the signals of very small earthquakes, very large explosions, and, now, evidently, resonances in fractures stimulated by hydraulic pumping. Used to image geological resources, tectonic structures and potentially hazardous faults, borehole seismology allows for less

Cross-References ▶ Body Waves ▶ Continental Crustal Structure ▶ Earthquake, Focal Mechanism ▶ Earthquake, Magnitude

Borehole Seismic Networks and Arrays

▶ Earthquakes and Crustal Deformation ▶ Earthquakes, Strong-Ground Motion ▶ Energy Partitioning of Seismic Waves ▶ Fault Zone Guided Waves ▶ Propagation of Elastic Waves: Fundamentals ▶ Seismic Diffraction ▶ Seismic Imaging, Overview ▶ Seismic Instrumentation ▶ Seismic Phase Nomenclature: The IASPEI Standard ▶ Seismic Signals in Well Water Observations ▶ Seismic Wave Propagation in Real Media: Numerical Modeling Approaches ▶ Seismic Waves, Scattering ▶ Seismic, Reflectivity Method ▶ Seismogram Interpretation ▶ Shear-Wave Splitting: New Geophysics and Earthquake Stress-Forecasting ▶ Single and Multichannel Seismics ▶ Slow Earthquake ▶ Surface Waves

Bibliography Balch AH, Lee MW (1984) Vertical seismic profiling: technique, applications, and case histories. International Human Resources Development Corporation/Reidel, Boston/Dordrecht/Lancaster Balch AH, Lee MW, Paillet FL (1986) Vertical seismic profiling – technique, application, and case histories. J Acoust Soc Am 80(3). https://doi.org/10.1121/1.393897 Bent A (2013) Global seismograph network (GSN). In: Bobrowsky PT (ed) Encyclopedia of natural hazards. Encyclopedia of earth sciences series. Springer, Dordrecht Ben-Zion Y, Malin PE (1991) San Andreas Fault zone head waves near Parkfield, California. Science 251:1592–1594 Blakeslee SN, Malin PE (1990) A comparison of earthquake coda waves at surface versus subsurface seismometers. J Geophys Res Solid Earth 95:309–326. https://doi.org/10.1029/JB095iB01p00309 Bohnhoff M, Zoback MD (2010) Oscillation of fluid-filled cracks triggered by degassing of CO2 due to leakage along wellbores. J Geophys Res 115:B11305. https://doi.org/10.1029/2010JB000848 Bohnhoff M, Zoback MD, Chiaramonte L, Gerst JL, Gupta N (2010) Seismic detection of CO2 leakage along monitoring wellbores. Int J Greenhouse Gas Control 4(4):687–697. https://doi.org/10.1016/j. ijggc.2010.01.009 Bohnhoff M, Dresen G, Ceken U, Kadirioglu FT, Karal RF, Kilic T, Nurlu M, Yanik K, Acarel D, Bulut F, Ito H, Johnson W, Malin PE, Mencin D (2017) GONAF – a borehole geophysical observatory around the north Anatolian fault in the eastern Sea of Marmara. Sci Drill 22:19–28 Bohnhoff M, Malin P, ter Heege J, Deflandre J-P, Sicking C (2018) Suggested best practice for seismic monitoring and characterization of non-conventional reservoirs. First Break 36:59–64 Chavarria JA, Malin PE, Shalev E, Catchings RD (2003) A look inside the San Andreas Fault at Parkfield through vertical seismic profiling. Science 302:1746–1748 Coates R, Haldorsen JBU, Miller D, Malin P, Shalev E, Taylor ST, Stolte C, Verliac M (2006) Oilfield technologies for earthquake science. Oilfield Rev 18(2):24–33

51 Eisner L, Hulsey BJ, Duncan P, Jurick D, Werner H, Keller W (2010) Comparison of surface and borehole locations of induced seismicity. Geophys Prospect 58(5):809–820. https://doi.org/10.1111/j.13652478.2010.00867.x Ellsworth WL, Malin PE (2011) Deep rock damage in the San Andreas Fault revealed by P- and S-type fault-zone-guided waves. Geol Soc London Spec Publ 359:39–53. https://doi.org/10.1144/SP359.3 Galperin YI (1961) Three component downhole seismic acquisition. Moscow (in Russian) Galperin EI, Nersesov IL, Galperina RM (1986) Seismological observations in boreholes. In: Borehole seismology and the study of the seismic regime of large industrial centres. Seismology and exploration geophysics, vol 2. Springer, Dordrecht Gardner LW (1949) Seismograph determination of a salt dome boundary using a well detector deep on the dome flank. Geophysics 14:29–38 Hamada K, Ohtake M, Okada Y, Matsumura S, Sato H (1985) A highquality digital network for microearthquake and ground tilt observations in the Kanto-Tokai area, Japan. Earthquake Pred Res 3:447–469 Haring MO, Schanz U, Ladner F, Dyer BC (2008) Characterization of the Basel 1 enhanced geothermal system. Geothermics 37:469–495 High-Resolution Seismic Network (HRSN) (2014) High resolution seismic network. UC Berkeley Seismological Laboratory, Berkley. https://doi.org/10.7932/HRSN Huang W-G, Huang B-S, Wang I-H, Chen K-C, Wen K-L, Tsao S, Hsieh Y-C, Chen C-H (2010) Seismic observations in the Taipei Metropolitan Area using the downhole network. Terr Atmos Ocean Sci 21(3). https://doi.org/10.3319/TAO.2009.12.11.03(TH) Ikelle LT, Amundsen L (2018) Introduction to petroleum seismology. Society of Exploration Geophysicists, Tulsa. https://doi.org/10.1190/ 1.9781560801702.ch5 Johnson HM (1962) A history of well logging. Geophysics 27(4):427–541. ISSN: 0016-8033; 1942-2156. https://doi.org/10. 1190/1.1439054 Kwiatek G, Saarno T, Ader T, Bluemle F, Bohnhoff M, Chendorain M, Dresen G, Heikkinen P, Kukkonen I, Leary P, Leonhardt M, Malin P, Martínez-Garzón P, Passmore K, Passmore P, Valenzuela S, Wollin C (2019) Controlling fluid-induced seismicity during a 6.1-km-deep geothermal stimulation in Finland. Sci Adv 5(5):eaav7224. https:// doi.org/10.1126/sciadv.aav7224 Lellouch A, Yuan S, Spica Z, Biondi B, Ellsworth WL (2019) Seismic velocity estimation using passive downhole distributed acoustic sensing records – examples from the San Andreas Fault Observatory at Depth. J Geophys Res Solid Earth. https://doi.org/10.1029/ 2019JB017533 Li Y-G, Malin PE (2008) San Andreas Fault damage at SAFOD viewed with fault-guided waves. Geophys Res Lett 35:L08304. https://doi. org/10.1029/2007GL032924 Liang C, O’Reilly O, Dunham EM, Moos D (2017) Hydraulic fracture diagnostics from Krauklis-wave resonance and tube-wave reflections. Geophysics 82(3):D171–D186 Luo Y, Xia J, Miller RD et al (2008) Rayleigh-wave dispersive energy imaging by high resolution linear radon transform. Pure Appl Geophys 165(5):903–922 Malin PE, Bohnhoff M, Blumle F, Dresen G, Martinez-Garzon P, Nurlu M, Ceken U, Kadirioglu FT, Kartal RF, Kilic T, Yanik K (2018) Microearthquakes preceding a M4.2 Earthquake Offshore Istanbul. Sci Rep 8:Article number: 16176. https://doi.org/10.1038/ s41598-018-34563-9 Martinez-Garzon P, Bohnhoff M, Zambrano-Narvaez G, Chalaturnyk R (2013) Microseismic monitoring of CO2 injection at the Penn West Pembina Cardium EOR Project, Canada. Sensors 13:11522–11538. https://doi.org/10.3390/s130911522 Marzetta T, Orton M, Krampe A, Johnston L, Wuenschel P (1988) A hydrophone vertical seismic profiling experiment. Geophysics 53(11):1437–1444. https://doi.org/10.1190/1.1442423

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52 McClellan JH, Eisner L, Liu E, Iqbal N, Al-Shuhail AA, Kaka SI (2018) Array processing in microseismic monitoring detection, enhancement, and localization of induced seismicity. IEEE Signal Process Mag. https://doi.org/10.1109/MSP.2017.2776798 McCollum B, LaRue WW (1931) Utilization of existing wells in seismograph work. Bull Am Assoc Pet Geol 15:1409–1417 Musgrave AW, Woolley WC, Gray H (1960) Outlining of salt masses by refraction methods. Geophysics 25:141–167 Obara K (2003) Hi-Net: high sensitivity seismograph network, Japan. In: Methods and applications of signal processing in seismic network operations. Lecture notes in earth sciences, vol 98. Springer, Berlin Oye V, Chavarria J-A, Malin P (2004) Determining SAFOD area microearthquake locations solely with the Pilot Hole seismic array data. Geophys Res Lett 31:L12S10. https://doi.org/10.1029/ 2003GL019403 Prevedel B, Bulut F, Bohnhoff M, Raub C, Kartal R, Fatih Alver F, Malin PE (2015) Downhole geophysical observatories: best installation practices and a case history from Turkey. Int J Earth Sci 104:1537. https://doi.org/10.1007/s00531-015-1147-5 Raub C, Bohnhoff M, Petrovic B, Parolai S, Malin PE, Yanik K, Kartal RF, Kiliç T (2016) Seismic wave propagation in shallow layers at the GONAF-Tuzla site, Istanbul, Turkey. Bull Seismol Soc Am 106:912–927. https://doi.org/10.1785/0120150216 Richards PG (2016) The history and outlook for seismic monitoring of nuclear explosions in the context of the Comprehensive NuclearTest-Ban Treaty. Nonprolif Rev. https://doi.org/10.1080/10736700. 2016.1272207 Sicking CJ, Malin PE (2019) Fracture seismic: mapping subsurface connectivity. Geosciences 2019(9):508 Sicking CJ, Vermilye J (2019) Resonance frequencies in passive recordings map fracture systems: Eagle Ford and New Albany Shale examples. Extended abstract, unconventional resources technology conference (URTeC). https://doi.org/10.15530/urtec-347 Takahashi H (1982) The deep borehole observatories and their contribution for revealing the characteristics of microearthquake activity in

Borehole Seismic Networks and Arrays the Kanto district. Rep Natl Res Cent Disaster Prev 28:1–104. (in Japanese with English abstract) Tary JB, Van der Baan M, Eaton DW (2014a) Interpretation of resonance frequencies recorded during hydraulic fracturing treatments. J Geophys Res Solid Earth 119(2):1295–1315 Tary JB, Van der Baan M, Sutherland B, Eaton DW (2014b) Characteristics of fluid induced resonances observed during microseismic monitoring. J Geophys Res 119:8207–8222 Thurber C, Roecker S, Roberts K, Gold M, Powell L, Rittger K (2003) Earthquake locations and three-dimensional fault zone structure along the creeping section of the San Andreas Fault near Parkfield, CA: preparing for SAFOD. Geophys Res Lett 30(3):1112. https://doi. org/10.1029/2002GL016004 Thurber C, Roecker S, Zhang H, Baher S, Ellsworth W (2004) Fine-scale structure of the San Andreas Fault zone and location of the SAFOD target earthquakes. Geophys Res Lett 31(12):L12S02. https://doi. org/10.1029/2003GL019398 Toksöz MN, Cheng CH, Cicerone RD (1992) Fracture detection and characterization from hydrophone vertical seismic profiling data. Int Geophys Ser 51:389–414. https://doi.org/10.1016/S0074-6142(08) 62831-4 Unsworth MJ, Malin PE, Egbert GD, Booker JR (1997) Internal structure of the San Andreas Fault at Parkfield, California. Geology 25:359–362 Vasconcelos I, Sneider R, Sava P, Taylor T, Malin P, Chavarria A (2008) Drill bit noise illuminates the San Andreas Fault. EOS Trans Am Geophys Union 89(38):349–360 Zhang H, Thurber C, Bedrosian P (2009) Joint inversion for Vp, Vs, and Vp/Vs at SAFOD, Parkfield, California. Geochem Geophys Geosyst 10:Q11002. https://doi.org/10.1029/2009GC002709 Zoback M, Hickman S, Ellsworth W, the SAFOD Science Team (2011) Scientific drilling into the San Andreas Fault zone – an overview of SAFOD’s first five years. Sci Drill 11:14–28. https:// doi.org/10.2204/iodp.sd.11.02.2011

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Characteristic Earthquakes and Seismic Gaps David D. Jackson and Yan Y. Kagan Department of Earth and Space Sciences, University of California Los Angeles, Los Angeles, CA, USA

Definition Fault slip

Rupture Elastic rebound Segment

Characteristic earthquake Recurrence interval

Seismic cycle

Relative motion, either steady or sudden as in earthquakes, between rock units on either side of a fault. Sudden fault slip in an earthquake. Sudden release, as by an earthquake, of slowly accumulated strain energy. A section of a fault or plate interface bounded by features thought to serve as strong barriers to earthquake rupture. Features postulated to form such barriers include changes in fault orientation or in rock type across parts of the fault, and intersections with other faults. An earthquake rupturing an entire fault segment. Alternately, one of a sequence of earthquakes rupturing the same area of fault. The time between characteristic earthquakes on a given segment or fault area. Quasiperiodic Occurring at approximately equal recurrence intervals. A sequence of events on a segment starting with a large earthquake, followed by aftershocks, then by steady stress accumulation, and culminating with another large earthquake. The term “cycle” is sometimes but not always meant to imply quasiperiodic recurrence.

© Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

Seismic gap

A segment for which the time since the previous characteristic earthquake approaches or exceeds the average recurrence interval.

Introduction The seismic gap hypothesis holds that most long-term geologic slip on faults or plate boundaries is accomplished by characteristic earthquakes on segments. Such quakes are presumed to reduce the stress substantially, necessitating a substantial recurrence time for elastic stress to recover before the next characteristic earthquake. The dates and rupture extent of past earthquakes may be determined by modern seismic networks; by historic reports of faulting, damage, or strong shaking; or by paleoseismic investigation of trenches across faults. The average recurrence time may be determined either from a sequence of dates of past characteristic earthquakes, or by the time required for steady slip on a fault to accumulate the slip experienced in a characteristic earthquake (Working Group on California Earthquake Probabilities 1988, 1990).

History Well before plate tectonics became accepted in the 1960s, Gilbert (1884) argued that large earthquakes should be separated by substantial time intervals. Reid (1910, 1911) proposed that faults slowly accumulate energy later released suddenly by earthquakes (“elastic rebound”), and that the time to a future earthquake could be estimated by surveying the strain released by a previous one. Fedotov (1965) noticed that most of the Kamchataka–Kurile trench in the northwest Pacific had been ruptured by earthquakes in the early twentieth century, with the exception of one zone southeast of Hokkaido. He concluded that a future earthquake was likely in that area, a forecast realized in 1973 when the Nemuro-oki earthquake struck with magnitude about 7.5 (Kasahara 1981, 182).

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Plate tectonics now provides a compelling, steady source of the strain energy driving elastic rebound, and that fact led many to conclude that occurrence of large earthquakes must be separated by enough time to allow recharge. Scholz (1990, 260) remarked that “A tenet of plate tectonics is that the rates of plate motion must be steady over geologic time periods and must be continuous along plate boundaries. If it is assumed further that a significant portion of this motion must be released seismically, then it follows that segments of plate boundaries that have not ruptured for the longest time are those most likely to rupture in the near future. These places are called seismic gaps.” The seismic gap hypothesis was celebrated when Nishenko 1989 published a list of 13 earthquakes that fit, at least approximately, the descriptions of previously expected earthquakes. The seismic gap model has been applied to long-term forecasting of earthquakes in many regions (Sykes 1971; Kelleher 1972; Kelleher et al. 1973; McCann et al. 1979; Working Group on California Earthquake Probabilities 1988; Nishenko 1991). The model and its definition have evolved along with the quality and quantity of data that go into it. Fedotov (1965) defined the Nemuro-oki gap as the last remaining unbroken segment. McCann et al. (1979) used elapsed time and a color code to label segments around the Pacific Rim: more than 100 years had elapsed in red gaps, between 30 and 100 years in orange gaps, and less than 30 years in apparently safe green zones. In the most comprehensive forecast ever using the seismic gap model, Nishenko (1991) actually made testable probabilistic forecasts for about 125 plate boundary segments around the Pacific Rim. In that study he estimated the mean recurrence time and the elapsed time for each segment, and assumed that their ratio obeys a log-normal probability density function. With that information, he calculated the probability that each zone would be ruptured by a characteristic earthquake whose magnitude he also listed for each zone within 5-, 10-, and 30-year periods. A very similar model was used by the Working Group on California Earthquake Probabilities (1988) in its official earthquake probability estimates.

Assumptions All the applications mentioned above share several important assumptions. First, their authors assume that faults and plate boundaries are segmented and that rupture does not cross segment boundaries. Second, they assume that each characteristic earthquake ruptures to both ends of its segment, reducing the stress to a uniform base level and beginning the process of stress recharge. Third, they assume that the time to the next characteristic earthquake depends almost entirely on the time of the previous one: not on other

Characteristic Earthquakes and Seismic Gaps

earthquakes, nonelastic stress redistribution, or other causes. To make useful forecasts, scientists must obviously be able to distinguish characteristic earthquakes from all others in order to know the elapsed time since the last one.

Small Characteristic Earthquakes Recent studies in California (e.g., Nadeau and McEvilly 1999), Japan (e.g., Igarashi et al. 2003; Okada et al. 2003), and elsewhere have identified sequences of small earthquakes fitting the alternative definition of “characteristic earthquake” above. In each sequence, the events are approximately the same size, and rupture approximately the same fault area. They recur at nearly equal time intervals or in some cases variable intervals consistent with variations in their size or in fault slip rate. In most cases, the slipped areas appear to be surrounded by areas where displacement occurs by steady slip rather than earthquakes. Because of that special circumstance, these small repeating earthquakes are not relevant to the discussion of seismic gaps.

Modified Seismic Gap Hypothesis As time, earthquake experience, and theoretical sophistication have accumulated, earth scientists have modified the seismic gap theory to rely less on the assumptions above. The Working Group on California Earthquake Probabilities (1990) allowed adjustments to account for stresses from earthquakes not on the relevant segment. The 1992 Landers, California earthquake (magnitude about 7) presented a particularly important observation. In that event, rupture jumped segment boundaries and even faults, making use of up to five faults mapped as separate before 1992. In a seismic hazard model produced by the Southern California Earthquake Center (Working Group on California Earthquake Probabilities 1995), the seismic gap model was modified to allow rupture to jump segment boundaries with a modest probability. Later uses of the seismic gap model in California for official hazard estimates employ increasingly complex assumptions, especially about conditions under which rupture is likely to involve more than one segment (Working Group on California Earthquake Probability 2002, 2008). The more complex versions of the model raise interesting questions. What truly constitutes a segment? How effective are weak barriers in stopping rupture, and what controls their strength? Are the boundaries fixed in space, or can they move as stress conditions change? When rupture crosses a boundary, does it consistently continue to the next? If not, does it reset the stress and the clock on the partially ruptured segment? Do the elapsed times on adjacent segments control the

Characteristic Earthquakes and Seismic Gaps

probability that rupture will jump the barrier between them? If so, which segment is most important? Modelers must answer these questions, implicitly or explicitly, to forecast using the modified gap models. So far, there is no clear consensus on the answers.

Challenges to the Seismic Gap Model Despite some reported successes, the seismic gap hypothesis has often been questioned. Critics point to the difficulty of verifying the rather strong assumptions behind the hypothesis, and to its limited success in forecasting earthquakes. The basic assumption that faults and plate boundaries are segmented has provoked significant debate. Even the few apparent successes (e.g., Loma Prieta, CA, 1989; Parkfield, CA 2004; Chile, 2010), are equivocal at best. The rupture of the Loma Prieta earthquake was about the length of the nearest segment mapped by the Working Group on California Earthquake Probabilities (1988), but it shifted south and spilled over the southern segment boundary. Moreover, the event occurred near, but not actually on the San Andreas Fault for which the segment was defined. Before 2004, the Parkfield segment was defined in several different ways, so its location at Parkfield does not confirm the segmentation hypothesis (Jackson and Kagan 2006). The 2010 Chile earthquake went well beyond the segment boundaries specified by Nishenko (1991). In addition, several events have clearly violated preassigned boundaries. The 1992 Landers quake was mentioned above, and the great Sumatra tsunami earthquake of 2004 breached several boundaries along its 1300-km rupture zone (Nalbant et al. 2005). The assumption that earthquakes rupture to both ends of segment boundaries also lacks verification. A basic problem is that the locations of segment boundaries are usually estimated inaccurately from the extent of past earthquake ruptures. For earthquakes identified by paleoseismic investigations, rupture can generally be pinpointed at only a few widely spaced locations. For historical earthquakes, rupture extent is typically estimated with great uncertainty from the extent of damage or reported shaking. Even for modern instrumentally recorded earthquakes, the extent of the aftershock zone or fault scarp may not accurately represent the rupture at depth where the elastic rebound occurs. In many cases, the extent of rupture for older earthquakes is assumed to be similar to that of the most recent event, a clear case of circular reasoning. The connection between plate tectonics and earthquake recurrence referred to in the words of Scholz above depends on the assumption that characteristic earthquakes release most of the slowly accumulated fault slip. However, that assumption fails in many examples. For instance, the cumulative slip of the

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Parkfield, CA earthquakes since 1857, often regarded as an archetypical characteristic sequence, accounts for only a small fraction of expected slip at the long-term geological rate (Jackson and Kagan 2006). In such cases, the alternative definition of characteristic earthquakes listed above, and the times of past events, may provide valuable information on the causes of some earthquakes but the direct link to plate tectonics is lost. Proponents of the seismic gap theory cite many examples in which identified gaps have been filled by earthquakes. The positive examples are appealing but insufficient for two reasons. First, the definitions of gaps and forecasted earthquakes were quite general, making the target easier to hit at random. Second, they included only successes; a fair evaluation needs to consider failures as well. Kagan and Jackson (1991, 1995), and Rong et al. (2003) applied statistical tests to the seismic gap theory as articulated by McCann et al. (1979) and Nishenko (1991). Earthquakes were actually more frequent in McCann’s green zones than in the red ones, opposite to what the gap theory assumes. The 1991 gap model implied far more earthquakes, in different places, than actually occurred. Kagan and Jackson (1995) and Rong et al. (2003) also tested a simple alternative to the gap model, assuming that earthquakes occur randomly in time and near past earthquakes. The alternative model fits the total number and the locations of future earthquakes much better than the 1991 gap model. These statistical tests necessarily applied to the earlier versions of the gap hypothesis, in which segments were assumed to be independent of one another. Since then, more complex versions of the model have been applied. Most of these applications involve one or a few purported gaps with estimated recurrence times of decades or centuries. Published models generally do not provide probabilities for multisegment ruptures, and they cannot be effectively evaluated until several seismic cycles have elapsed. To be rigorously testable, such a model must forecast a few tens of well-defined earthquakes. Unfortunately, no systematic, well-specified version of the more complex seismic gap model has been applied broadly enough to be tested.

Conclusions The intuitively appealing seismic gap model encompasses the virtually unassailable principle that earthquakes release strain energy accumulated over a long time. Although many large events have occurred in previously identified gaps, the same is true of locations outside them. Simple versions of the gap theory, in which characteristic earthquakes occur on independent segments, are no longer tenable. Modified versions of the gap theory have not yet been formulated in a rigorously testable way.

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Cross-References ▶ Earthquake Precursors and Prediction ▶ Earthquake Rupture: The Inverse Problem ▶ Earthquakes and Crustal Deformation ▶ Earthquakes, Energy ▶ Earthquakes, Location Techniques ▶ GPS, Tectonic Geodesy ▶ Great Earthquakes ▶ Paleoseismology ▶ Seismic Hazard ▶ Seismicity, Subduction Zone ▶ Seismology, Global Earthquake Model ▶ Statistical Seismology

Bibliography Fedotov SA (1965) Regularities of the distribution of strong earthquakes in Kamchatka, the Kurile Islands and northeastern Japan. Tr Inst Fiz Zemli Akad Nauk SSSR 36(203):66–93 (in Russian) Gilbert GK (1884) A theory of the earthquakes of the Great Basin, with a practical application. Am J Sci 27(157):49–54. Ser. 3 Igarashi T, Matsuzawa T, Hasegawa A (2003) Repeating earthquakes and interplate aseismic slip in the northeastern Japan subduction zone. J Geophys Res 108(B5):2249. https://doi.org/10.1029/ 2002JB001920 Jackson DD, Kagan YY (2006) The 2004 parkfield earthquake, the 1985 prediction, and characteristic earthquakes: lessons for the future. Bull Seismol Soc Am 96:S397–S409 Kagan YY, Jackson DD (1991) Seismic gap hypothesis: ten years after. J Geophys Res 96(21):21419–21431 Kagan YY, Jackson DD (1995) New seismic gap hypothesis: five years after. J Geophys Res 100(B3):3943–3959 Kasahara K (1981) Earthquake mechanics. Cambridge University Press, Cambridge Kelleher JA (1972) Rupture zones of large South American earthquakes and some predictions. J Geophys Res 77:2087–2103 Kelleher JA, Sykes LR, Oliver J (1973) Possible criteria for predicting earthquake locations and their applications to major plate boundaries of the Pacific and Caribbean. J Geophys Res 78:2547–2585 McCann WR, Nishenko SP, Sykes LR, Krause J (1979) Seismic gaps and plate tectonics: seismic potential for major boundaries. Pure Appl Geophys 117:1082–1147 Nadeau RM, McEvilly TV (1999) Fault slip rates at depth from recurrence intervals of repeating microearthquakes. Science 285:718–721 Nalbant S, Steacy S, Sieh K, Natawidjaja D, McCloskey J (2005) Seismology: earthquake risk on the Sunda trench. Nature 435(7043):756–757 Nishenko SP (1989) Earthquakes: hazards and predictions. In: James DE (ed) The encyclopedia of solid earth geophysics. Van Nostrand Reinhold, New York, pp 260–268 Nishenko SP (1991) Circum-Pacific seismic potential – 1989–1999. Pure Appl Geophys 135:169–259 Okada T, Matsuzawa T, Hasegawa A (2003) Comparison of source areas of M4.8
0.1 repeating earthquakes off Kamaishi, NE Japan: are asperities persistent features? Earth Planet Sci Lett 213:361–374 Reid HF (1910) The California earthquake of April 18, 1906. The mechanics of the earthquake, vol 2. Carnegie Institution of Washington, Washington, DC Reid HF (1911) The elastic-rebound theory of earthquakes. Univ Calif Dep Geol Bull 6:413–444

Continental Crustal Structure Rong Y-F, Jackson DD, Kagan YY (2003) Seismic gaps and earthquakes. J Geophys Res 108(B10):2471, ESE-6, 1–14 Scholz C (1990) The mechanics of earthquakes and faulting. Cambridge University Press, Cambridge Sykes LR (1971) Aftershock zones of great earthquakes, seismicity gaps, and earthquake prediction for Alaska and the Aleutians. J Geophys Res 76:8021–8041 Working Group on California Earthquake Probabilities (1988) Probabilities of large earthquakes occurring in California, on the San Andreas Fault. U.S. Geological Survey Open-File Report, 88–398, 62 ~ pp Working Group on California Earthquake Probabilities (1990) Probabilities of large earthquakes in the San Francisco Bay Region, California, USGS Circular 1053 Working Group on California Earthquake Probabilities (1995) Seismic hazards in southern California: probable earthquakes, 1994–2024. Bull Seismol Soc Am 85:379–439 Working Group on California Earthquake Probabilities (2002) Earthquake probabilities in the San Francisco Bay region: 2002 to 2031, USGS Circular 1189 Working Group on California Earthquake Probabilities (2008) The Uniform California Earthquake Rupture Forecast, Version 2 (UCERF 2): U.S. Geological Survey Open-File Report 2007–1437 and California Geological Survey Special Report 203. http://pubs.usgs.gov/of/ 2007/1437/

Continental Crustal Structure Rolf Meissner (Deceased) and Hartmut Kern Kiel, Germany

Definition Seismic reflection and refraction profiling supplying velocitydepth functions and relations to petrology is the key to probe the continental crust.

Introduction In the last 20 years, much progress has been made in the development of geophysical techniques that increased our knowledge about the variability of crustal structure substantially (see ▶ “Deep Seismic Reflection and Refraction Profiling”). A big misunderstanding, still often presented today, is the use of the phrase “the” crust. This is a marked oversimplification. Continental crust has been found to be extremely different (Christensen and Mooney 1995; Mooney 2015; Rudnick and Gao 2014; Hacker et al. 2015). There are at least three main types to be distinguished (see ▶ “Lithosphere, Continental”). 1. Thick, old cratons, shields, and platforms 2. Orogens, often related to continental subduction 3. Extensional areas, including shelves and rifts

Continental Crustal Structure

Thick Precambrian cratons, disrupted by early plate tectonics, are presently found in all continents; they contribute more than 70% to the continental lithosphere. They are characterized by a thick crust (45–50 km) with high velocities (>7.2 km/s) in the lower part. The two other crustal types were generated by modern plate tectonics: the orogens by convergence and plate collisions, most of them showing deep crustal roots, and the extensional areas by divergence and wrench tectonics, often accompanied by thermal and magmatic processes. They generally show a shallow crustal thickness without a high-velocity lower crust. Special forms are rifts, basins, and igneous provinces. Type sections of the three main crustal structures are presented in Fig. 1. In general, the continental crust has lower seismic velocities and consists of lighter material than the underlying mantle; VP is smaller than 7.8 km/s, VS smaller than 4.3 km/s, and r below 3.1 g/cm3. Crustal thickness is between 20 km (shelves) and more than 70 km (some orogens) (Mooney et al. 2005). Mohorovičić was the first to determine the base of the crust, today called the “Moho” for short. It is observed worldwide and is clearly recognized by seismic methods. It is worthy to note that the position of the seismic Moho is not always identical with the petrological Moho (defined as the boundary between non-peridotitic crustal rocks and olivine-dominated rocks). In case that mafic rocks are transformed into eclogites at the base of orogenically thickened crust, P-wave velocities of eclogites largely overlap with those of peridotites so that seismic field studies may not detect eclogites as crustal rocks (e.g., Mengel and Kern 1992; O’Reilly and Griffin 2013). An intracrustal boundary – the “Conrad” – seems to separate upper sialic from more mafic lower crust in some areas. Transport of crustal material into the mantle and delamination of the lithosphere and the lowermost part of a thickened crust may give rise to a reduction of crustal thicknesses.

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Methods Reliable studies of the continental crust were started in the last 40 years by seismic methods, applying first refraction studies with man-made explosions and lines of simple geophones. Wide-angle reflection seismics followed, and steep-angle reflection seismics became more and more important (see ▶ “Deep Seismic Reflection and Refraction Profiling”). In the last 30 years, digital technology, computer programs for ray tracing and tomography, and large, national, and international research programs with an increased number of recording devices and energy sources were initiated. Large, national reflection programs, often with a substantial support by the industry or large funding agencies, were developed in nearly all industrialized countries, spearheaded by COCORP in the USA. Both methods – refraction-wide-angle and steep-angle reflection – complement each other, the former supplying velocity-depth functions and relations to petrology and the latter providing reflectivity images of the subsurface that reveal past tectonic processes. Both methods form the backbone of crustal interpretation. They confirmed identical Mohos, either by a jump of velocities or by termination of (crustal) reflectivity because the mantle is less reflective (Meissner and Brown 1991). Vibroseis and dynamite shooting are used for reflection programs, the latter one generally providing a higher energy output, which is sometimes used for piggy-back wide-angle work. In the last 20 years, a relatively cheap seismological method, the “receiver function method” (see ▶ “Seismic, Receiver Function Technique”), supported interpretation of seismic data, making use of global seismicity and a wave change at seismic boundaries. This “receiver function method” is based on seismic phase conversions of teleseismic waves at the Moho and at boundaries in the crust and mantle. Non-seismic geophysical techniques are complementary to seismic methods: gravity, for instance, for investigating isostatic problems or selecting between competing seismic interpretations or – together with magnetics – looking for inhomogeneities or controlling structure. Electromagnetic and magnetotelluric methods help to find and to interpret conductivity anomalies. Measurements of heat flow help to observe heat sources and decipher the thermal or magmatic history of a specific area. Depending on the heat flow, temperature-depth functions and viscosities that control rheology can be estimated. The brittle upper crust extends to a depth of 12–20 km and is a zone where generally earthquakes occur. It is separated from the more ductile, aseismic part of the middle and lower crust (some exceptions). Various experimental studies of creep, stress, and strength support seismic interpretation.

Cratons Continental Crustal Structure, Fig. 1 Type sections of the three main crustal structures (a–c) and ranges of P-wave velocities in the upper, middle, and lower crust. Numbers are velocities in km/s

Cratons are relatively stable Precambrian continental units with a rather deep lithosphere (Fig. 1a). They are called

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shields when a sediment cover is lacking and platforms when covered by sediments. Today, cratons are found in all continents, possibly created from a few early supercontinents that were later disrupted by plate tectonics. No consensus exists about the number of cratons. They comprise up to ten in Africa and even more in North America. The latter were finally combined into the big “North American Craton” forming the whole interior of North America (without its margins in the west and east). Even “stable” cratons are influenced by Proterozoic and Phanerozoic processes like rifts or plumes. In North America there is a 2500-km-long Proterozoic midcontinental rift with a thick cover of sediments and volcanic intrusions and a reduced crustal thickness. There are also Paleozoic rifts in the southern part of the East European Craton, and there is an unusual 60-km-thick crust in the cold Baltic Shield, well within the eclogite stability field in the lowermost part. However, high velocities (> 7.8 km/s) corresponding to eclogitic rocks are not observed in the lowermost crust of the Baltic Shield (Korja et al. 2009). The crystalline crust of cratons is similar worldwide. Cratons are characterized by low heat flow values. Their thickness is in the range 35–55 km with a “classical” three-velocity structure (Pawlenkowa 1996), as shown by “deep seismic sounding” (wide-angle plus refraction studies). In general, cratons have an upper crust with VP around 6 km/s, a middle crust around 6.5 km/s, and a roughly 10-kmthick lower crust with velocities around 7.0–7.2 km/s, sometimes 6.7–7.5 km/s (Fig. 1a). A typical example is the EUROBRIDGE profile (Guterch et al. 1997) crossing the crust of the East European platform (Fig. 2). Reflection seismic experiments generally show no special enhancement of reflectivity in the (mafic) lower crust, as shown by the large transects of LITHOPROBE in Canada,

Continental Crustal Structure

by the COCORP data in the USA, or by the BABEL surveys in the northern Baltic Sea (some exceptions).

Orogens Collisions of plates or terranes create orogens by compressing material upward (mountain ranges) and downward (roots), and dynamic processes initiate oceanic or continental subduction. While the elevation of an orogen apparently depends on the time and speed of the colliding plates (e.g., Himalaya, Tibet), processes of crust/mantel interaction are quite complex. Magmatic processes are found behind subducting plates in many orogens (Alps, Sierra Nevada, Andes). In continental collision zones, rock types of the lower crust resemble those of the neighboring plates, guided into the orogens. There are high-velocity (cratonic) crustal rocks with a mafic lower crust in the Uralides (from the East European platform) and sialic, extensional crusts with a reflecting lower crust in the Alps (mostly from the Variscides). Continental subduction often follows oceanic subduction. Remnants of these processes may be reflected by the crustal “roots” (see ▶ “Seismic, Reflectivity Method”). Figure 3 shows simplified cross sections through the Alps and Pyrenees. Orogens are more or less isostatically compensated, and its Bouguer anomaly is strongly negative (Ebbing 2004).

Extensional Areas Extensional processes cover many dimensions: from rifts, basins, (the basin and range of North America), and continental shelves up to large Phanerozoic areas like

Continental Crustal Structure, Fig. 2 Typical cratonic crust from the East European platform (after Guterch et al. 1997)

Continental Crustal Structure

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Continental Crustal Structure, Fig. 3 Typical sections of orogenic crust (Alps and Pyrenees)

Europe and west of the Teisseyre-Tornquist line (see ▶ “TransEuropean Suture Zone”). This large extensional area was created in post-Avalonian and post-Variscan time at the end of the Paleozoic. A surprisingly uniform crust has been formed, showing three prominent differences compared to the cratons: their thickness is only 30 km (Fig. 1c), their velocities is lower than 6.8 km/s, and in many places, they show prominent reflectivity (lamellae) in the lower crust. Apparently, all three characteristic crustal patterns are genetically interrelated. While a former lower crust (in the basin and range province and in the Variscides) got lost in the previous orogeny (by delamination or subduction),

subsequent moderate, but widespread, extension dominated. Mafic-ultramafic dykes intruded into the lower crust and formed sill-like intrusions, giving rise to high impedance contrasts after cooling. Importantly, sills are mostly rather thin and do not generally increase the lower-crustal overall (sialic) seismic velocities. Figure 4 gives an example of strong lower crust reflectivity and low average velocities. In contrast, the formation of rifts and basins is a result of local (extensional) stress and/or strong local heat anomalies. As a consequence, large volumes of mantle-derived basaltic magmas intrude into the crust and after cooling the mafic rocks (gabbros or metamorphic equivalents) cause the

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Continental Crustal Structure

Continental Crustal Structure, Fig. 4 Multiple reflections (laminations) in the Moldanubian lower crust along the DEKORP II line, along with ranges of Vp velocities at different depths

observed high velocities at the base of most rifts and basins. Examples are the Oslo Graben, the Danish Basin and other adjacent rifts and basins north of the Variscan front, the great East African Rift, the Donetz Basin, and the midcontinental North American rift. In general, the massive intrusions created a transparent lower crust, but some basins with high lower crust velocities (above 7.2 km/s) show some thin lamellae (apparently ultramafic) on top of the Moho. Also in some areas of continental/ocean transition, high-velocity lamellae are observed in the lower crust (Roberts et al. 2009). In general, lower crustal reflectivity seems to depend on the volume and thickness of the mafic intrusions compared to the host rock. Whereas sill-like intrusions are suggested to generate the observed lower crustal reflectivity, massive intrusions in the lower crust reduce reflectivity and enhance the average velocities to values of 6.8–7.2 km/s. The volcanic Messum ring complex in Namibia shows no lamellae in the high-velocity interior but a good reflectivity in the surrounding (Bauer et al. 2003). In contrast, the Viking Graben in the North Sea shows lower crust reflectivity and no reflection around it (McBride et al. 2004). Most continental shelves show extension and a stretched continental crust with a thickness of only 20–30 km, often containing aborted rifts (Korja et al. 2009). Reflectivity in the lower crust is found in the West Atlantic margin (Faroe or Hatton) and along the Greenland coast. Often, seaward dipping reflections, caused by intrusive and extrusive magmatism, mark the onset of oceanic crust with high velocities. BIRPS WAM – profile from the British Isles toward the

Atlantic Ocean – shows a preferred lower crust reflectivity below the (continental) Mesozoic basins, while the oceanic crust is free from reflections.

Constraints on Crustal Petrology The measured seismic velocity structures are very helpful in providing a rough characterization of the Earth’s crust in different tectonic environments, but they are nonunique so that inferences about composition cannot be drawn from wave velocities (at least P-wave velocities). The seismic properties at depth are determined by a number of lithologic and physical factors that control the in situ rock properties in a very complex manner. Velocities are controlled by the intrinsic properties of the rocks (mineralogical composition, chemical composition, metamorphic grade, crystallographic preferred orientation of constituent minerals, etc.) and by the physical environment of the crust (temperature, pressure, porosity, fluid content, etc.) (see ▶ “Seismic Properties of Rocks”). Direct information about the composition and structure of the deep crust can be obtained either from crustal terrains exposed at the surface (e.g., Fountain and Salisbury 1981; Kern and Schenk 1988) or from xenoliths brought to the surface by magmas (Downes et al. 1990). Deeply eroded Precambrian terrains (e.g., Sri Lanka) and upthrust tectonic slices in orogenic belts (e.g., Ivrea Zone, N. Italy; Serre Mountains, S. Calabria; Kapuskasing Zone, Canada)

Continental Crustal Structure

provide perhaps the best geologic guides to structural style and composition at depth. Such rocks are important in providing direct data although they contain a mixed message as they are no longer the in situ deep crust. A unique ability to correlate the seismic data with the structure and composition of the in situ deep crust can be provided by coupling experimentally determined or calculated P- and S-wave velocities for relevant crustal rocks collected from surface outcrops or from xenoliths, simulating in situ conditions: (1) by laboratory seismic measurements at actual PT conditions (e.g., Christensen and Wepfer 1989; Kern et al. 1999; Ji et al. 2015) and (2) by calculations from modal analyses and elastic properties of the rock-forming minerals and their pressure and temperature derivatives (e.g., Jackson et al. 1990; Barruol and Kern 1996; Almquist and Mainprice 2017). High P-wave velocities (>6.9 km/s) generally defining the lowermost crust (below about 20–25 km depth) are typical for metamorphic rocks in the granulite facies. These are mafic granulites, mafic amphibolite facies rocks, anorthosites, and high-grade metapelites. Intermediate P-wave velocities (6.5–6.9 km/s) represent the upper-lower to mid-crust and are characteristic for intermediate granulites and metapelites. Low P-wave velocities (6.2–6.5 km/s) occurring at midcrustal levels in Paleozoic and more recent fold belts and in the uppermost crust in shields and platforms are likely to be composed of felsic rocks in the amphibolite and granulite facies. It should be noted, however, that not all felsic (granitic) upper crustal rocks are intermediate- to high-metamorphic grade. The Superior Province (Canada) is an example of low metamorphic shield rocks. Mineralogically, feldspar (K-feldspar, plagioclase) is the most abundant mineral, followed by quartz and hydrous minerals (such as mica and amphiboles). The bulk chemical composition of the upper crust is felsic, and the middle crust is intermediate in composition. The average composition of the lower crust is in general mafic in cratons, but it tends to be intermediate or even felsic in some regions. The continental crust as the whole (average of all crustal types) is suggested to be intermediate in average bulk composition (Rudnick and Fountain 1995; Mooney 2015). Importantly, most of the rocks constituting the Earth’s crust exhibit marked anisotropy of elastic properties (e.g., Kern et al. 2001; Ji et al. 2002; Almquist and Mainprice 2017). Typical values of intrinsic P-wave anisotropy (quartzmica-schists, felsic gneisses, granulite-facies metapelites, amphibolites) range from 5.4% to 10.7%. Anisotropy may be caused by crystallographic (CPO) and morphological (or shape) preferred orientation (SPO), by oriented microcracks or by thin layers of otherwise isotropic materials with different properties (see ▶ “Magnetic Anisotropy”).

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Whereas oriented cracks may contribute a large fraction of observed seismic anisotropy in the uppermost crust (Crampin 1987; Crampin and Gao 2018), CPO- and SPO-related seismic anisotropy is a major contribution in ductily deformed and foliated crustal structures (shear zones, gneiss sections) of the deeper crust. It is mainly due to the alignment of mica and hornblende minerals and their strong single crystal anisotropies (Barruol and Kern 1996; Meissner and Kern 2008). In addition to mineral and chemical composition and rock fabric, pore fluids may have an important bearing on in situ seismic properties. Saturation of pore space (Ppore ≈ 0) increases Vp, whereas Vs remains unaffected (Nur and Simmons 1969). In case of pore pressure approaching lithostatic pressure (Ppore ≈ Plith), microcracks and grain boundaries are more or less kept open. As a consequence, both P- and S-wave velocities are smaller than in dry rocks, due to a reduction of the effective pressure (Peff. ¼ Plith. – n  Ppore; n ≈ 1). Importantly, Vp anisotropy is significantly higher in dry rocks than in water-saturated rocks at atmospheric pressure conditions (Popp and Kern 1994). As pressure is increased, differences progressively decrease. The corresponding S-wave velocities and S-wave splitting data are only weakly affected by intergranular fluids. Seismic anisotropy is becoming increasingly important also in understanding the evolution of the Earth’s crust (and upper mantle). Since the measured seismic anisotropy has a structural (macro- and/or micro-) origin, seismic azimuthal anisotropy manifested by shear wave splitting, in particular, can provide important geophysical evidence of deformation because the orientation and magnitude of anisotropy are, in general, strongly related to the internal strain-induced rock fabric. Although variations of lithology are suggested to be most important for the generation of seismic reflections in the lower crust (laminated lower crust) (e.g., Meissner et al. 2006), seismic reflectivity (Warner 1960) may be enhanced by the fabric-related anisotropy of ductily deformed structures (Kern and Wenk 1990; Vasin et al. 2017). It should be noted, however, that in crustal rocks, the variability due to variation in composition is generally higher than the variation caused by velocity anisotropy (except for schists). The reverse is true in the olivine-rich upper mantle rocks. This probably implies that on a regional scale, fabric-related anisotropy is more important in the oceanic and continental upper mantle than in the continental crust.

Summary Seismic refraction and reflection surveys have revealed three main types of continental crust: (1) the thick old cratons, constituting large parts of all continents; (2) the orogens, generated by plate collisions; and (3) the extensional areas with a relatively thin crust, exhibiting ample reflectivity

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(lamellae) in the lower part. Crustal thickness is between 20 km (shelves) and more than 70 km (some orogens). The Mohorovičić discontinuity (Moho), defined by a jump in velocity and density, separates the lowermost crust from the upper mantle. High P-wave velocities (>6.9 km/s) are typical for the lowermost crust of platforms and orogens; they stand for high-grade mafic rocks in the granulite facies. In the thin lower crust of extensional areas, high-velocity mafic rocks are almost lacking. The upper-lower to mid-crust is characterized by intermediate P-wave velocities (6.5–6.9 km/s) and is likely to be composed of intermediate granulites and metapelites. Low P-wave velocities (6.2–6.5 km/s) of mid-crustal levels in Paleozoic and more recent fold belts as well as in the uppermost crust in shields and platform point to felsic rocks in the amphibolite and granulite facies. Seismic anisotropy is an important property of most rocks constituting the Earth’s crust. It is basically caused by crystallographic and morphological (or shape) preferred orientation, by oriented microcracks, or by thin layers of otherwise isotropic material with different properties.

Cross-References ▶ Continental Drift ▶ Geodynamics ▶ Lithosphere, Continental ▶ Propagation of Elastic Waves: Fundamentals ▶ Seismic Imaging, Overview ▶ Shear-Wave Splitting: New Geophysics and Earthquake Stress-Forecasting ▶ Seismic Tomography ▶ Seismicity, Subduction Zone ▶ Subduction Zones Acknowledgments We greatly appreciate the assistance of Gudrun Reim (Kiel) in preparing the figures, and we thank Walter Mooney (Menlo Park) for his review, leading to many improvements of the manuscript.

Bibliography Almquist BSG, Mainprice D (2017) Seismic properties and anisotropy of the continental crust: predictions based on mineral texture and rock microstructure. Rev Geophys 55. https://doi.org/10.1002/ 2016RG000552. 524 Barruol G, Kern H (1996) Seismic anisotropy and shear-wave splitting in lower-crustal and upper-mantle rocks from the Ivrea Zone – experimental and calculated data. Phys Earth Planet Inter 95:175–194 Bauer K, Trumbull RB, Victor T (2003) Geophysical images and a crustal model of the intrusive structures beneath the Messum ring complex, Namibia. Earth Planet Sci Lett 216:65–80

Continental Crustal Structure Christensen NI, Mooney WD (1995) Seismic velocity structure and composition of the continental crust: a global view. J Geophys Res 100:9761–9788 Christensen NI, Wepfer WW (1989) Laboratory techniques for determining seismic velocities and attenuations, with applications to the continental lithosphere. In: Pakiser LC, Mooney WD (eds) Geophysical framework of the continental United States. Geophysical Society of America Memoir, 172. Geological Society of America, Boulder Crampin S (1987) The geological and industrial implications of extensive-dilatancy anisotropy. Nature 328:491–496 Crampin S, Gao Y (2018) Evidence supporting new geophysics. Earth Planet Phys 2:173–188. https://doi.org/10.26464/epp2018018 Downes H, Dupuy C, Leyreloup A (1990) Crustal evolution of the Herzynian belt of Western Europe: evidence from lower crustal granulitic xenoliths. Chem Geol 68:291–303 Ebbing J (2004) The crustal structure of the Eastern Alps from a combination of 3D gravity modeling and isostatic investigations. Tectonophysics 350:89–104 Fountain D, Salisbury MH (1981) Exposed cross sections through the continental crust; Implications for the crustal structure, petrology and evolution. Earth Planet Sci Lett 56:263–277 Guterch A, Grad M, Thybo H, Keller GR, POLONAISE Working Group (1997) POLONAISE ’97 – an international seismic experiment between Precambrian and Variscan Europe in Poland. Tectonophysics 314:101–121 Hacker BR, Keleman PB, Behn MD (2015) Continental lower crust. Annual Review of Earth and Planetary Sciences 43:167–205 Jackson I, Rudnick RL, O’Reilly SY, Bezant C (1990) Measured and calculated elastic wave velocities for xenoliths from the lower crust and upper mantle. Tectonophysics 173:207–210 Ji SC, Wang Q, Xia B (2002) Handbook of seismic properties of minerals, rocks, and ores. Polytechnic International Press, Montreal, 630 pp Ji S, Shao T, Michibayashi K, Oya S, Satsukawa T, Wang Q, Zhao W, Salisbury M (2015) Magnitude and symmetry of seismic anisotropy in mica- and amphibole- bearing metamorphic rocks and implications for tectonic interpretation of seismic data from southeast Tibetan Plateau. J Geophys Res 120. https://doi.org/10.1002/ 2015JB012209 Kern H, Schenk V (1988) A model of velocity structure beneath Calabria, South Italy, on laboratory data. Earth Planet Sci Lett 87:325–337 Kern H, Wenk H-R (1990) Fabric related velocity anisotropy and shear wave splitting in rocks from the Santa Rosa mylonite Zone, California. J Geophys Res 95:11213–11223 Kern H, Gao S, Jin Z, Popp T, Jin S (1999) Petrophysical studies on rocks from the Dabie ultrahigh-pressure (UHP) metamorphic belt, Central China: implications for the composition and delamination of the lower crust. Tectonophysics 301:191–215 Kern H, Popp T, Gorbatsevich F, Zharikov A, Lobanov KV, Smirnov YP (2001) Pressure and temperature dependence of Vp and Vs in rocks from the superdeep well and from surface analogues at Kola and the nature of velocity anisotropy. Tectonophysics 338:113–134 Korja A, Hyönen T, Tira T, Heikkinen P (2009) Examining threedimensional crustal heterogeneity in Finland. Eos Trans AGU 90(15):129–130 McBride JH, White RS, Smallwood JR, England RW (2004) Must magmatic intrusion in the lower crust produce reflectivity? Tectonophysics 388:271–297 Meissner R, Brown L (1991) Seismic reflections from the Earth’s crust: comparative studies of tectonic patterns. Geophys J Int 105:1–2 Meissner R, Kern H (2008) Earthquakes and strength in the laminated lower crust – can they be explained by the corset model? Tectonophysics 448:49–59

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Meissner R, Rabbel W, Kern H (2006) Seismic lamination and anisotropy of the lower continental crust. Tectonophysics 416:81–99 Mengel K, Kern H (1992) Evolution of the petrological and seismic Moho – implications for the continental crust-mantle boundary. Terra Nova 4:109–116 Mooney W (2015) Crust and lithospheric structure – global crustal structure. In: Treatise of geophysics, vol I, 2nd edn, pp 339–390 Mooney WD, Rao VV, Chulik GS, Detweiler ST (2005) Comparison of the deep crustal structure and seismicity of North America with the Indian Subcontinent. Curr Sci 88:1639–1651 Nur A, Simmons C (1969) The effect of saturation on velocity in lowporosity rocks. Earth Planet Sci Lett 7:183 O’Reilly S, Griffin W (2013) Moho vs crust-mantle boundary: evolution of an idea. Tectonophysics 609:535–546 Pawlenkowa N (1996) Crust and mantle structure in Northern Eurasia from seismic data. In: Dmowska R, Saltzmann B (eds) Advances in geophysics, vol 37. Academic, San Diego Popp T, Kern H (1994) The influence of dry and water-saturated cracks on seismic velocities of crustal rocks – a comparison of experimental data with theoretical model. Surv Geophys 15:443–465 Roberts AW, White RS, Christie PAF (2009) Imaging igneous rocks on the North Atlantic rifted continental margin. Geophys J Int 179:1029–1038 Rudnick RL, Fountain DM (1995) Nature and composition of the continental crust: a lower crustal perspective. Rev Geophys 33:267–309 Rudnick R, Gao S (2014) Composition of the continental crust. In: Treatise on geochemistry, vol 4, 2nd edn. Elsevier, Amsterdam, pp 1–51 Vasin RN, Kern H, Lokajicek T, Svitek T, Lehmann E, Mannes DC, Chaousche M, Wenk H-R (2017) Elastic anisotropy of Tambo gneiss from Promontogno, Switzerland: a comparison of crystal orientation and microstructure-based modelling and experimental measurements. Geophys J Int 209:1–20 Warner M (1960) Absolute reflection coefficients from deep seismic reflections. Tectonophysics 173:15–23

Continental Drift Alan G. Smith (Deceased)

Definition Continental drift

The name given to the relative movement between continents.

Introduction Apparently, continental drift was first postulated by Abraham Ortelius, who made one of the first atlases, in the third edition of his Thesaurus Geographicus (Ortelius 1596) to account for the similarity between the coastlines of western Africa, western Europe, and the eastern Americas. During the next three

centuries or so, several other thinkers used this morphological similarity to come to the same conclusion.

Alfred Wegener Although Ortelius speculated that earthquakes and floods had torn the continents apart, Alfred Wegener, an Austrian meteorologist, was the first to systematically gather the geological evidence for continental drift in his synthesis (Wegener 1929a), also translated into English (Wegener 1929b), parts of which had been published as early as 1912 (Wegener 1912). Wegener was puzzled in particular by the present-day distribution of former ice-age deposits, or tillites, of PermoCarboniferous age, now known to be about 300 million years old that are found today in South America, Africa, Antarctica, India, and Australia. Wegener’s meteorological background led him to assume that the present-day climatic zones, with cold polar regions and a hot tropical belt, was a fundamental property of the Earth’s atmosphere that had been established before these glacial deposits had formed and had persisted to the present-day. He realized that if all the southern continents had been joined together to form a supercontinent lying near the south geographic pole, then the present-day distribution of all the Permo-Carboniferous tillites would have a logical explanation. This supercontinent is known as Gondwana. But Wegener went further and postulated that the northern continents had also been joined to form a northern supercontinent known as Laurasia, which, with Gondwana, formed a huge continental area incorporating all the major continents, known as Pangea. Wegener’s solution to the tillite distribution had a compelling elegance about it, partly because it also placed the Carboniferous forests of Europe and North America, (whose compressed remains gave the name to the Carboniferous period) in what would have been the tropical region of that time. However, elegance is not a scientific proof, and Wegener’s ideas were rejected by most geophysicists and many geologists, principally because there were no physical measurements that demonstrated drift and there was also no known physical mechanism that could move continents independently of one another (e.g., Jeffreys 1929). Wegener (1929a) made the reassembly by positioning the continents at where he presumed the edges of the continents lay, that is, somewhat beyond the present-day coastlines, but his maps were rather schematic. A more convincing Gondwana map, supported by much geological evidence, was made by Du Toit (1937), but the best precomputer map showing the excellence of the fit between the continental edges of South America and Africa was made by Carey (1958). Jeffreys (1964) denied that there was a fit, and it was his denial that in part led Bullard and others to examine how well the continental edges fitted

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together (Bullard et al. 1965; Everett and Smith 2008). These ideas were supported by the changes in the Earth’s magnetic field preserved in rocks (see ▶ “Paleomagnetism, Principles”) determined by workers such as Irving (1964), but the results were not fully accepted by the geoscience community until after the plate tectonic revolution.

Continental Rifts Wegener AL (1912) Die Entestehung der kontinente (the origin of continents). Geol Rundsch 3:276–292 Wegener AL (1929a) Die Entestehung der kontinente und Ozeane. (The origin of continents and oceans). Brunswick, Vieeweg Wegener AL (1929b) The origin of continents and oceans (English translation). Methuen, London

Continental Rifts Conclusions It is now established that the Earth’s lithosphere is divided into a number of rigid Plate Tectonics, Precambrian in relative motion. Tracing these motions back in time shows that Gondwana, Laurasia, and Pangea did exist in the past, vindicating Wegener’s idea completely.

A. M. Celâl Şengör Faculty of Mines, Department of Geology and Eurasia Institute of Earth Sciences, Istanbul Technical University, Istanbul, Turkey

Synonyms Cross-References ▶ Lithosphere, Continental ▶ Lithosphere, Continental: Thermal Structure ▶ Lithosphere, Mechanical Properties ▶ Lithosphere, Oceanic ▶ Paleomagnetism, Principles ▶ Plate-Driving Forces ▶ Plate Motions in Time: Inferences on Driving and Resisting Forces ▶ Plate Tectonics, Precambrian

Bibliography Bullard E, Everett JE, Smith AG (1965) The fit of the continents around the Atlantic. Philos Trans R Soc Lond A258:41–51 Carey SW (1955) The orocline concept in geotectonics – part I, Publication no. 28. The papers and proceedings of the Royal Society of Tasmania, vol 89, pp 255–288 Carey SW (1958) A tectonic approach to continental drift. In: Carey SW (ed) Continental drift: a symposium. University of Tasmania, Hobart, pp 177–355 Du Toit AL (1937) Our wandering continents. Oliver & Boyd, Edinburgh Everett JE (1965) The fit of the continents around the Atlantic. PhD, thesis, Cambridge Everett JE, Smith AG (2008) Genesis of a geophysical icon: the Bullard, Everett and Smith reconstruction of the circum-Atlantic continents. Earth Sci Hist 27(1):1–11. http://pubs.usgs.gov/gip/dynamic/histori cal.html Irving E (1964) Palaeomagnetism and its application to geological and geophysical problems. Wiley, New York, p 399 Jeffreys H (1929) The Earth. Cambridge University Press, Cambridge Jeffreys H (1964) How soft is the Earth? Q J R Astron Soc 5:10–22 Ortelius A (1596) Thesaurus geographicus, recognitus et auctus. Ex Officina Plantiniana, Antwerp Ortelius A (1603) Theatrum orbis terrarum. Abridged edition printed by James Shawe, London Smith AG, Hallam A (1970) The fit of the southern continents. Nature 225:139–144

Graben; Taphrogen

Definition A continental rift (Gregory 1894; Quennell 1982, 1985) is a fault-bounded elongate trough under or near which the entire thickness of the lithosphere (▶ “Lithosphere, Continental” and ▶ “Lithosphere, Mechanical Properties”) has been reduced in extension during its formation. Just as old mountain ranges may no longer have any topographic expression because of tectonic and/or erosional events, some, especially old, rifts may no longer appear as troughs for the same reasons, but their original trough shape is recognized by their stratigraphically younger fills, or metamorphically lower grade of their downdropped central blocks (▶ “Sedimentary Basins”).

Introduction Rifts form one of the three main categories of lithosphericscale structures resulting from differential motion of parts of the lithosphere. Lithospheric shortening creates orogens, simple shear creates keirogens, and stretching creates taphrogens that are collections of rifts (Fig. 1). However, at present only 20.5% of the active plate boundaries show normal convergence and 21% normal divergence. Some 8% show pure dextral strike-slip and some 6% pure sinistral strike-slip. The remaining 58.5% display some deviation from the three end-members, with relative motion vector angles to boundary strikes varying between 0° and 67° (Woodcock 1986). Plate boundaries must have shown a similar behavior in the past, so only about half of all the rifts a geologist may encounter is likely to show normal extension (▶ “Plate Motions in Time: Inferences on Driving and Resisting Forces” and ▶ “Plates and Paleoreconstructions”).

Continental Rifts

65

Continental Rifts, Fig. 1 A ternary diagram of strain showing the three lithospheric megastructure categories resulting from various strains

Shortening orogens

Stretching orogens

Transpressional orogens

Locus of pure vertical stretching

Shortening taphrogens (not observed on earth, but may exist on Venus as coronae)

xis

Shortening keirogens

ga

hin

tc tre

S

Stretching taphrogens

Simple shear keirogens Transtensional taphrogens

Rifts are important structures from the viewpoints of our understanding of the behavior of our planet and our exploitation of its resources. They record evidence of continental fragmentation in diverse tectonic settings including all three types of plate boundaries and in plate interiors (▶ “Plate Motions in Time: Inferences on Driving and Resisting Forces” and ▶ “Plates and Paleoreconstructions”). Also, at different stages of their evolution they present opportunities of studying the continental crust (and in extreme cases even the upper mantle) from its surficial sedimentary rocks down to the crust-mantle interface (▶ “Lithosphere, Continental”). Especially the lacustrine sedimentary sections of rifts are useful for studies on the past climates (e.g., Olsen and Kent 1999; Kravchinsky et al. 2003; Felton et al. 2007), and they have enabled geologists to refine stratigraphic correlations down to thousands of years as far back as in the early Mesozoic (e.g., Olsen and Kent 1999). Rifts house important economic reserves such as hydrocarbons (the Sirt rift in Libya being the most productive with about 45 billion barrels of oil and 33 trillion cubic feet of gas; for the hydrocarbon potential of rifts, see: Harding 1983; MacKenzie and McKenzie 1983: a classic on hydrocarbon genesis in rifts; Ziegler 1994a; Lambiase 1995), hydrothermal and stratiform copper, copper-nickel, molybdenum, giant lead-zinc, and even uranium deposits (for metallogenesis in rifts, see Sawkins 1990). Most of the geothermal areas in the world are located in active rifts (e.g., Jaffé 1971, Fig. 1) (▶ “Geothermal Heat Pumps”). At present 1/5 of the entire fresh-water reserve of the Earth resides in a rift basin, namely, Lake Baykal. Rifts are thus significant not only for increasing our geological knowledge, but also for contributing to our well-being. It is perhaps of some interest in this context to remember that our genus and species were born in a rift, namely, in the East African. For good overviews of rifts and rifting, see Burke and Dewey (1973), Burke and Wilson (1976), Coward et al.

Transtensional keirogens

(1987), McKenzie (1978 a “must” reading for understanding the evolution of rifts), Manspeizer (1988), Ziegler (1992, 1994b), Landon (1994), Şengör (1995), Şengör and Natal’in (2001: a detailed catalogue of the world’s rifts with tabulated summaries of origin and history and references), and Ring and Wernicke (2009). For the geology of some of the classic rift regions of the world, with increasing age, see: Africa as a whole: Kampunzu and Lubala (1991), Burke (1996), and Burke and Gunnell (2008). East Africa: Frostick et al. (1986), Frostick (1997: a good introduction for beginner), Schlüter (1997; this book has a bibliography of some 1300 items), Morley (1999; has an excellent collection of seismic sections). Basin-and-Range: Wernicke (1990), Beratan (1996), Faulds and Stewart (1998), Snow and Wernicke (2000), Dickinson (2002: best introduction for an outsider), and McQuarrie and Wernicke (2005; best kinematic reconstruction of any rift I know). Rio Grande rift, USA: Ingersoll (2001). The Upper Rhine rift: see the special issue on it in the International Journal of Earth Sciences (Geologische Rundschau), v. 94 (2005); also: Hüttner (1991), Schumacher (2002), Rotstein et al. (2006). The Lake Baykal rift: Crane et al. (1991), Kuz’min et al. (2001), ten Brink and Taylor (2002), Kravchinsky et al. (2003). The West Siberian Basin: Surkov et al. (1994). The North Sea Basin with the North German (Lower Saxony) Basin: Blundell and Gibbs (1990), Evans et al. (2003; this lavishly illustrated and richly documented mammoth book makes all earlier publications on the subject either obsolete or redundant; also available as a two CD-ROM set), Wilson et al. (2004), Littke et al. (2008). The Mediterranean region: Durand et al. (1999), Corti et al. (2006; an excellent paper about a little-known rift cluster). Willis (1936, 1–30) presents a good historical review of ideas pertaining to rifting. For some of the important early ideas on rifting, see Şengör (2018). For an excellent recent review of ideas and studies on the East African rifts, see Mohr (2009). For a history of uplift-related rift ideas, see Şengör

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(2003). For a historical review of recent developments in our understanding of extensional tectonics, see Wernicke (2009). Quennell (1982, 1985) are anthologies of some historical papers on rift studies. They are not satisfactory, but are the only ones of their kind.

Rift, Graben, and Taphrogen “Rift” and “graben” are two words used interchangeably in the international literature designating elongate troughs bounded by normal faults. However, not all such troughs are parts of structures that rupture, or even thin by rock creep, the entire lithosphere. Some normal-fault-bounded troughs result from sliding and stretching of surficial rocks over lubricating sedimentary horizons such as evaporites or shales. The normal-fault bounded basinets that formed during a landslide associated with the 27 March 1964 Prince William Sound earthquake in Alaska or the similar troughs in the Canyonlands National Park in Utah formed by sliding and stretching the sedimentary section above the Upper Carboniferous evaporites are examples. Such structures have nothing to do with lithospheric stretching and thus they do not fit the definition of rift given above. Şengör (1995) therefore suggested that the word rift be confined to those structures that actually disrupt or otherwise (by creep) thin the lithosphere and graben be reserved for those that do not. His usage is what is recommended and followed in this entry. In rifts when the extension factor ß, the ratio of extended width to unextended width (McKenzie 1978), goes beyond 3 the continental lithosphere generally ruptures completely and ocean generation begins by sea-floor spreading (Le Pichon and Continental Rifts, Fig. 2 A highly schematic block diagram of an “ideal” rift drawn to emphasize the main architectural elements of rifts. The straight marginal faults, for example, do not mean that all rifts must have straight faults or that they are single faults instead of fault families

Sibuet 1981) (▶ “Seafloor Spreading”) either by dyke injection or by cold extrusion of subcontinental mantle in the form of ultramafic “core complexes” as seen in such places as along the boundary between the Upper Penninic and the Lower Austroalpine units of the Alps (Bernoulli et al. 2003 and the references therein) or along the South Galicia Bank (Boillot et al. 1987). For obvious reasons, this cannot happen to grabens; however, much they may be extended. Large regions of localized extension exist on our planet where many rifts and grabens occur together. It is inappropriate to call such regions simply rifts. It would be like calling whole orogens simply “thrusts.” These large regions of extension are the extensional counterparts of orogens (Figs. 1 and 7) and therefore it is appropriate to use for them the term “taphrogen” derived from Krenkel’s term “taphrogeny.”

General Properties of Rifts Structure of Rifts Figure 2 is an idealized and highly simplified block diagram of a rift. Rifts have two fundamental structural entities: a main hanging wall block and one or two footwall blocks depending on whether the rift is symmetric (as in Fig. 2) or asymmetric (see Fig. 3). In symmetric rifts, the extension represented by the rift is almost entirely localized between two master bounding normal (or oblique extensional) fault systems and it is these two systems that take up much of the extension, although the hanging wall block delimited by them is also intensely broken up by smaller normal faults with strikes largely parallel with those of the major bounding fault families (Fig. 4). The hanging wall is commonly dropped down Sedimentary infill

Branch rift Master fault Intra-rift horst Shoulder uplift

Lower crust 30 henosphere A st

Deepest earthquakes

er ph th

os

(brittle-ductile transition zone)

Li

Crust

Middle crust

e

Upper, brittle crust

an tle

Zore of seismicity

M

km 0

10

10

0 km 10 50–150

Continental Rifts

67 Bounding fault systems

Footwall block

Footwall block

Hanging wall block

Symmetric rift Bounding fault system Facing direction of rift Hanging wall block

Footwall block

Inceptive asymmetric rift Hanging wall block horse Increasing metamorp

hic

ad gr

Footwall block

Hanging wall block

e

Advanced asymmetric rift Sedimentary cover

Footwall bloc k

Hanging

ck wall blo

Dead asymmetric rift

Continental Rifts, Fig. 3 Some properties of symmetric and asymmetric rifts

Continental Rifts, Fig. 4 A part of the eastern (“Gregory”) rift in Kenya showing the intense faulting of the down-dropped hanging wall block between the master fault families. Google Earth image

because of crustal thinning affecting it and in the basin thus formed sediments accumulate whose thicknesses can be in excess of 15 km in large rifts such as the Dneper-Donetz rift in the Ukraine (Ulmishek et al. 1994), which is very close to the maximum possible thickness (17 km) of sediment accumulation on continental crust (Dewey 1982). Most rifts of average size, however, have sediment thicknesses ranging from 2 to 5 km. Individual rift lengths vary from about a 1000 km to 100 km and widths from about 150 to 10 km. Rift size is a function not only of the amount of extension, but also the thickness of the lithosphere it disrupts. The thicker the lithosphere, the wider the rift. In Fig. 5, the distribution of continental rifts on Earth is shown. In areas of thick lithosphere (e.g., in the North American craton, eastern Europe and Siberian craton), the rifts are very wide, whereas in areas of thin lithosphere such as the western USA or in the Aegean they are very narrow, in places less than 10 km wide. The bounding faults of the rifts have long been thought to be listric (shovel-shaped), that is, flattening with depth, although studies on earthquakes have not supported this. Wherever precise hypocenter location is possible and where the earthquake has caused a surface rupture (▶ “Earthquake, Focal Mechanism”, ▶ “Earthquakes and Crustal Deformation”, and ▶ “Earthquakes, Location Techniques”), it is seen that the associated fault is planar down to depths of 8 to 10 km. Eyidogan and Jackson (1985) documented that the 28 March 1969 Demirci earthquake in western Turkey consists of two discrete subevents: one along a northerly dipping planar normal fault and another one on a horizontal surface north of, but at the same depth as, the hypocenter of the first one. The horizontal rupture is believed to be located in a

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Lake Baykal WST D Ae TRC Basin-and-Range taphrogen Hormuz taphrogen East African rift valleys 0

2,000 km

Cainozoic rifts Mesozoic rifts

0

1,000 km

Palaeozoic rifts Precambrian rifts

Continental Rifts, Fig. 5 Distribution of continental rifts in the world. (Updated from Şengör and Natal’in 2001)

region that deforms ductiley under long-term strain rates, but ruptures when that rate is suddenly increased as in an earthquake (Fig. 2). This suggests a kinked cross-section for the master fault. Such “kinked” faulting events may be the explanation of why hanging wall blocks in some rifts are so much more disrupted than the rift shoulders. In areas of thicker lithosphere, however, hypocenters along the master faults of rifts may be as deep as 30 km as exemplified by the 10 March 1989 Malawi earthquake in east Africa (Fig. 2; Jackson and Belkinsop 1993) (▶ “Seismicity, Intraplate”). The straight fault hypothesis is difficult to generalize especially where the master fault families bounding rifts are strongly curved in map view such as those that created the Lake Tanganyika rift system (Burgess et al. 1988). It is possible that individual segments of large faults rupture as straight fractures, but eventually the entire fault acquires a listric geometry. This seems supported by the fact that in Lake Tanganyika, where the rift floor has deeply subsided, faulting within the hanging wall block is less intense (Burgess et al. 1988). This is incompatible with a major kink of the master fault at depth. Wernicke and Burchfiel (1982) pointed out that where numerous, parallel-striking planar normal faults have rotated in unison around horizontal axes, one listric fault at the head of the series is a kinematic necessity to allow the others to rotate.

Asymmetric rifts are said to “face” in the direction of the dip of their master faults (Fig. 3). Many major rifts are made up of half rifts that change their facing along the strike. In such cases, “transfer” or “accommodation zones” form along their boundaries that take the form of areas of complex oblique faulting and crustal block rotations (see especially the excellent discussions on transfer and accommodation zones in Faulds and Stewart 1998; also Derer et al. 2005). Smaller and shallower transfer faults, essentially tear faults disrupting the hanging wall block with complicated displacement histories, form where the hanging wall is differentially stretched along the strike of the rift (cf. Şengör 1987). Rift shoulders are commonly uplifted. In symmetric rifts, both shoulders go up because of isostatic adjustment of the crustal wedges that are formed by the master faults dipping towards the rift (▶ “Isostasy”). Such uplift not only rotates the land surface on the shoulders away from the rift thus guiding much of the regional drainage in the same direction, but it rotates also the bounding normal faults (Buck 1988). Each increment of faulting further rotates the shoulder and further rotates the previously formed fault segment. In Fig. 3, in the case of a young asymmetric rift the fault is shown to be straight. As the rift ages and extension advances, the fault surface curves away from the rift as a result of incremental rotations and finally its oldest parts acquire a flat position,

Continental Rifts

possibly with “horses” of the hanging wall block stuck on it as “extensional allochthons” (Fig. 3). This seems to be the origin of many near-horizontal normal faults in areas of thin lithosphere and large amounts of extension such as the extensional metamorphic core complexes of the North American Cordillera (Armstrong 1982), or those of the Tethysides (Verdel et al. 2007), although the existence of “primary” low-angle normal faults is also known both from detailed field studies (e.g., Miller and John 1988) and from the seismicity (Eyidogan and Jackson 1985; Abers et al. 1997). The amount of extension in many rifts, especially those that are symmetric, does not go that far, however, and their bounding faults rarely dip less that 30°. In places where intracontinental rifting along a narrow rift valley has advanced far, as, for example, along the Wonji Fault Belt in Ethiopia, it has been noted that faulting is driven by dyke injection below the brittle upper carapace of about 10 km thickness and that segments with active deformation at the surface correlate with those of active magmatism (Kendall et al. 2005; Casey et al. 2006). Cloos (1939) believed that the shoulder uplift in symmetric rifts resulted from the original doming that had led to keystone drop of the hanging wall block, but it was quickly realized that the observed angles of slope on crustal domes rarely exceed 1°, whereas to give rise to substantial rifts one needs about 10° dome slopes, which are excessive. Table 1 lists values of extension and shoulder uplift in some selected rifts. In none, except possibly in the Española Basin, is the shoulder uplift sufficient to cause the estimated amount of stretching. Sedimentation in Rifts Because rifts form elongate basins, they are ideal sediment receptacles. The type of sediment to be found in a rift varies according to the size and geometry of the rift, the climatic zone(s) in which it is located and its evolutionary history. Although most rifts are filled slowly by sedimentation in the sea, lakes, rivers and, in a subordinate degree, subaerially by wind, some are catastrophically invaded by the sea by breaching the barriers of sub-sea-level rift complexes (e.g., the Permian North Sea or the medial Cretaceous South Atlantic). In such cases, either vast evaporite deposits (as in the Aptian of the South Atlantic: Burke and Şengör 1988) or thick shales (as in the late Permian of the North sea: Evans et al. 2003) are laid down in rift troughs. Subaerial rifts are normally fed by rivers that locally create lakes within them because of endorheic conditions. Near the fault-bounded shoulders, stream valleys are commonly short (as the crest of the shoulder often is also the water-divide) but the streams in them are energetic because of the steep slopes. They thus carry clastic sediments of a variety of sizes that are horizontally sorted in large alluvial fans at the feet of the faultescarpments. Along these escarpments, many alluvial fans

69 Continental Rifts, Table 1 Amounts of extension and shoulder uplift at selected continental rifts Name of rift N. Kenya Gregory Malawi

Amount extended (km) 35–40 10 7.5 (from transfer fault offset)

Suez N Central S

16.3 17.7–32.6 29.3 (all from fault geometry) 17 (from crustal configuration), 5–7 (from fault geometry) 28–36 (from crustal configuration), 11–13 (from fault geometry) 100–130 (from crustal configuration), 30 (from fault geometry) 100–105 (from crustal configuration), 15 from fault geometry) 100a 15–20 (from crustal configuration), 10 (from fault geometry) 5.5 10 (from fault geometry) 16 (from fault geometry)

Upper Rhine

Oslo

Viking

Central (North Sea) Benue Baykal

Española basin Albuquerque Basin N Albuquerque Basin S

Uplift of shoulder area (km) >1.4–1.7 2 2.8 (strike-slip affected?) 1.5 1 ~500 m 2.2

1

3

1.5

? 2–3

1.5 ~1 0.9

a

Benkhelil et al. (1988) established at least 10 km of shortening during the following compression

form and coalesce and ongoing subsidence leads to the accumulation of large thicknesses of fanglomerates, sandstones, and shales near the faulted rift margin(s). Because of rotation towards the fault plane of the basin floors, the sedimentary thicknesses systematically decrease away from rift shoulders towards rift median lines. Steep slopes and catastrophic rain in both tropical and desert climates frequently lead to slope failures resulting in landslides complicating the sedimentary record. Permanent or ephemeral lakes may occupy the middle parts of rift floors. Ephemeral lakes commonly occur in dry climates in the form of playas and deposit evaporites and clays that may interfinger with marginal alluvial fan deposits. Permanent and deep lakes are important sediment repositories in rifts. They may accumulate very thick sedimentary sections in them exhibiting complex facies dictated by the evolution of the geomorphology of the rift and its surroundings. Such lakes can be very deep (Lake Baykal in Sibearia: with 1637 m the world’s deepest lake in addition to an 8 km rift sedimentary fill; Lake Tanganyika in East Africa: 1457 m, but with half the sediment thickness of Lake Baykal). They are

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well-aerated in cold climates because the cold upper layers of the lake waters make convection and overturning possible (as in Lake Baykal), but they are stratified with anoxic deeps in hot regions because hot surface waters cannot sink and thus inhibit overturning (as in Tanganyika). As the anoxic conditions in Lake Tanganyika exemplify, deep rift lakes in hot climates may generate good hydrocarbon source rocks in reducing environments. This is corroborated by ancient rift fills that formed under similar conditions such as the Permian oil-bearing sequences in the Junggar and Turfan rift basins in Central Asia (e.g., Graham et al. 1990; Miao et al. 2006) or the organic-rich (up to 8% organic C) Triassic-Jurassic lacustrine sediments of the Newark-type rifts of the eastern United States (Manspeizer and Olsen 1981). Rift valley floors are seldom level along the strike of the master bounding faults and this commonly induces longitudinal drainage in rifts. In fact most of world’s major rivers flow in two kinds of valleys: those that are located in rift troughs or in depressions inherited from them and those that are placed in foreland basin troughs (cf. Audley-Charles et al. 1977). The Rhine in Europe, the Irrawady in Asia, flow along their almost entire lengths in active rift troughs. Along the western arm of the East African rift valleys, Lake Tanganyika receives (surface elevation at 773 m a.s.l.) the excess waters of Lake Kivu (at 1460 m) by means of the south flowing river Ruzizi that came into being only 10,600 years ago and may have stopped flowing between 8 and 6000 years ago (Felton et al. 2007) showing the sensitive dependence of rift sedimentation on the global climate. Farther north, just beyond Kivu, Lake Edward (at 912 m) is connected to Lake Albert (at 619 m) through the northerly flowing river Semliki. Lake Albert in turns feeds the Nile northwards via the river Albert Nile (White Nile or “Bahr al Jebel,” that is, “the river of the mountain”). In the sedimentary record, the rift facies is commonly characterized by thick (2–15 km) clastic sequences with rapid thickness and facies changes both laterally and vertically. If the rifts are underwater and far from clastic sources, they may house carbonate and even siliceous sedimentary sections as illustrated by the Alpine Triassic-early Jurassic facies. There, a continuous passage from late Triassic shallow water carbonates via a horizon of extensional-faulting-related neptunian dykes filled with progressively deepening facies and finally covered entirely by radiolarian cherts indicates a stretching basin floor as it deepened. That no shoulder uplift is indicated by any unconformities suggests that the Alpine rift was asymmetric with a major south-dipping detachment and that the rocks on the hanging wall block never experienced normal-fault-related upheaval. If catastrophic marine invasion occurs, then sediments at once blanket the preexisting topography. For good reviews of sedimentation in rifts, see Frostick et al. (1986), Lorenz (1988), Olsen and Schlische (1990), Leeder (1995), Şengör (1995), Beratan (1996), also the section entitled “Rift Basins” in Einsele (2000, 543–560).

Continental Rifts

Magmatism in Rifts Rift magmatism is extremely variable, because the tectonic environments in which rifts form are so diverse (see below). Rifts that form above mantle plumes (▶ “Mantle Plumes”) in plate interiors away from plate boundaries may begin their activity with restricted volumes of alkalic basalts indicating limited partial melting of mantle material at depth and may evolve towards abundant tholeiitic effusions as the temperature at depth increases and more and more mantle rock is melted, as, for example, illustrated, by the evolution of the Afar plume in east Africa. Most geologists take alkalic to peralkalic magmatism as indication of extensional tectonics. The frequent association of nepheline syenites, kindred alkaline igneous rocks, and carbonatites (“alkalic rocks and carbonatites”: ARCs) with intracontinental rifts has been taken by Burke et al. (2003, 2008) to argue that deformed rocks of the same compositions (“deformed alkalic rocks and carbonatites”: DARCs) that have been shown to characterize suture zones in regions in Asia, Europe, Africa and North America are nothing but ARC rocks that erupted into intracontinental rifts that became involved in a Wilson cycle of ocean opening and closing. This has been shown to form a powerful tool to map suture zones where other suture indicators such as ophiolites or subduction-related rocks may be missing. However, to think that rifts generate only alkalic magmatic rocks is clearly too simplistic. Those rifts that form along magmatic arc systems, for instance, inevitably have calc-alkalic magmas and those that form above recently thickened continental crust, may generate vast quantities of calc-alkalic andesitic/rhyolitic melts. Rifts forming in backarc regions, such as those in the Aegean, contain volcanic rocks showing evidence for a metasomatized mantle beneath them (e.g., the so-called Kulaites, that is, hornblende basalts, in western Turkey: Richardson-Bunbury 1996). Large amounts of silica-rich magmatic rocks may be generated if mantle plumes manage to melt large quantities of continental crust while ascending or if fractional crystallization is allowed to proceed undisturbed (e.g., the Cretaceous tin granites in Nigeria or the Cainozoic granites in Scotland; see Wiart and Oppenheimer 2004 for an active example in Afar). Some rifts such as Tanganyika or Baykal are almost entirely devoid of magmatism, while others, such as the late Palaeozoic Oslo Rift, have floors made up entirely of magmatic rocks. Some Atlantic-type continental margins, which grew out of disrupted rifts, are called “volcanic,” whereas others “nonvolcanic,” because they descended from rifts with similar magmatic characteristics. It is naive to think that lithospheric extension generates only certain kinds of magmatic rocks without considering the tectonic environments that gave rise to the extension and in which the extension proceeds (▶ “Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity”). It is therefore imperative to look at the diversity of the environments which generate rifts.

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71

For good reviews of magmatism in rifts, see Kampunzu and Lubala (1991), Kinnaird (1998), Ulrych et al. (1999), Burke et al. (2003, 2008), Wilson et al. (2004), Yirgou et al. (2006); also the special section on magmatism and extension in Journal of Geophysical Research 100 (1995). Metamorphism in Rifts Continental rifts are not commonly thought of as prime loci of metamorphic events, but five different kinds of metamorphic rocks do occur in them (▶ “Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity”). In increasing importance: (1) contact metamorphic rocks around rift-related magmatic rocks, (2) hydrothermal metamorphism around vents, (3) Buchan-type metamorphism due to rise of the asthenosphere below highly stretched rift floors and associated abundant intrusions, (4) burial metamorphism due to large sediment thicknesses, and (5) metamorphic rocks that form within the continental crust and are brought to surface by rifting. There is little to say about the first kind of metamorphic rocks in rifts as they are no different from any other contact metamorphic rock. Hydrothermal metamorphism is important in many rifts, because of the high heat flow and the continuous opening of extensional fractures allowing hot waters to circulate (e.g., Crane et al. 1991). Buchan-type metamorphism in rifts recently attracted much attention as a result of the early metamorphic history of the Pyrenees (Wickham and Oxburgh 1985), Mouthumet and the Montaigne Noir massifs in southern France during the late Palaeozoic (Şengör 2013). Rift-related metamorphism creating high-grade gneisses has been reported on the basis of condensed metamorphic zones and absence of associated plutonism also elsewhere (e.g., St-Onge and King 1987). Burial metamorphism is not common in rifts unless they accumulate very large thicknesses of sediment quickly, sufficient to raise the temperature and pressure conditions to initiate metamorphism. If we assume a normal geothermal gradient (~3 °C/100 m; see ▶ “Heat Flow, Continental”), an accumulation of about 4 km of sediment would be sufficient to initiate metamorphism by raising the temperature beyond 120 °C. Jackson (1987) argued, however, that because earthquake hypocenters in rifts are seldom deeper than 10 km, at least beyond that depth rocks must be metamorphic (the 600 °C is a thermal limit above which earthquakes do not occur). In a few places, there are earthquakes down to 30 km, as in the east African Rift System. It is true however, that rift sediments are seldom metamorphosed beyond greenschist grade, because they are seldom stretched enough and seldom accumulate sufficient thicknesses of sediment to heat the rocks beyond 320 °C. Smith (1976) pointed out, for example, that at the bottom of the about 16 km-thick Gulf Coast sediment pile, the temperature may be just about 320 °C under 4 kb pressure. But this ignores both the radioactive heating of the pile (▶ “Radiogenic Heat Production of Rocks”) and raising the geotherm by stretching. However,

Continental crust Feldspar-dominated

Geotherm after stretching

Mantle lithosphere Olivine-dominated Base of thermal lithosphere

a

C Original regional geotherm

Isotherm x

b Bovine headbasin

Regional geotherm “flatter” than before rifting

c Continental Rifts, Fig. 6 Thermal regime of rifts with a simple geotherm. (a) Immediately after the stretching both the crust and the lithospheric mantle thin and isotherms become crowded under the rift increasing the heat flow in the rift. Although no particle in the rift basement will become any hotter. (b) After stretching ceases, isotherms begin to subside. (c) When the lithosphere “recovers” the rift will end up with more lithospheric mantle beneath it and will have a heavier substratum that will subside and pull its surroundings with it creating an “intracratonic basin” overlying the rift. This is called a “bovine head” cross-section because of its resemblance to a long horn ox. While the basin subsides, its margins will be elastically uplifted (solid line above c), but may relax visco-elastically in time (dotted line above c). Here the margins are shown only on one side for graphic convenience, but naturally the events will take place all around the basin

even when the crust is stretched, every particle in the stretched crust must cool, unless the rift trough is rapidly sedimented (Fig. 6; McKenzie 1978; Jackson 1987). I am unaware of any well-documented metamorphic terrane that formed as a consequence solely of rift burial. By contrast, large normal faults of the kind shown for the advanced asymmetric rifts in Fig. 3 may pull up rocks from the lower continental crust that may have been metamorphosed even up to granulite facies as shown by many extensional metamorphic core complexes. But in this case the metamorphism is not caused by rifting, but it is exposed by rifting. In fact, as the rocks ascend in the footwall, they get retrograded, which is directly caused by rifting.

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For metamorphism in rifts, see: Armstrong (1982), Wickham and Oxburgh (1985), Jackson (1987), Lister and Davis (1989), Block and Royden (1990), Verdel et al. (2007), and Şengör (2013).

Kinds of Rifts A classification of rifts must serve to answer the questions: how many different sorts of environments lead to lithospheric extension and what kinds of rifts form in these environments? In essence one must specify where and how taphrogens commonly form. If taphrogeny stops before producing ocean, it causes subsidence and creates large basins (called “koilogens” by Spizaharsky and Borovikov 1966, pp. 113ff.) overlying taphrogens (Fig. 6; cf. McKenzie 1978; Şengör 1995). A comprehensive synthesis of various taphrogen types does not yet exist. Below is a restatement of Şengör’s (Şengör 1995; Şengör and Natal’in 2001) classification of types of rifts that make up the main elements of taphrogens. That classification is hierarchical and goes from pure geometry to dynamics (Fig. 8). Its hierarchical nature allows the environment and the path of formation of a given rift to be determined. For numerous examples of each of Şengör’s categories, see Şengör and Natal’in (2001). In this section, references already given in Şengör (1995) and in Şengör and Natal’in (2001) are not repeated. Şengör’s classification has three different categories that do not completely overlap, namely geometric, kinematic, and dynamic. In the following, the three different categories are identified with their initials, that is, g, k, and d, respectively.

Geometric Classification of Rifts (See Figs. 7 and 8) Rifts or groups of rifts forming individual taphrogenic units display five kinds of fundamental map patterns. From simplest to more complex, these are: g1) Solitary rifts: Solitary rifts form small, fairly insignificant, and very rare taphrogens and are extremely difficult to ascertain in the geological record, because it is commonly hard to tell whether a given rift fragment is isolated or part of a larger taphrogen (for a comparable difficulty in shortening structures, imagine a big solitary fold or a large thrust fault!). g2) Rift stars: Rift stars form when more than two rifts radiate away from a common center, building a roundish taphrogen. Rift stars are very common features of the structural repertoire of our planet today. g3) Rift chains: When several rifts are aligned end-to-end along linear/arcuate belts of rifting, they form rift chains. The East African Rift System constitutes the best known active rift chain in the world. Solitary rifts or rift stars or combinations of these may be connected to form different kinds of rift chains (e.g., Fig. 5 rift chains a or b). g4) Rift clusters: When several subparallel rifts occur in roughly equant areas, they are said to form a rift cluster. The two best-known active rift clusters in the world are the Basin-and-Range extensional area in North America and the Aegean Sea and the surrounding regions (see Fig. 5). g5) Rift nets: Rift nets constitute a rare pattern, which comes about when rifts form a roughly checkered pattern as in the Proterozoic basement of the East European platform or in the late Mesozoic in central North Africa (cf. Şengör and

Continental Rifts, Fig. 7 The geometric classification of taphrogens. (After Şengör 1995)

Solitary rift (g1) Rift star (g2)

Rift chain (a) (g3)

Rift chain (b) (g3)

Rift cluster (g4) Rift net (g5)

Continental Rifts

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Rifts Passive (d2)

Active (d1)

Intraplate rifts (k1) Solitary rifts (g1) Rift stars (g2) Rift chains (g3)

Divergent plate boundary rifts (k2)

Rifts with previous doming (k21)

Conservative plate boundary rifts (k3)

Convergent plate boundary rifts (k4)

Rift stars (g2) Rift chains (g3)

Transtensional rifts (k31) Rift chains (g3) Rift clusters (g4)

Pull-apart rifts (k32) Solitary rifts (g1) Rift clusters (g4) Rift chains? (g3)

Sphenochasms (k33) Solitary rifts (g1)

Subduction-related rifts (k41)

Solitary rifts (g1) Rift chains (g3)

Solitary rifts (g1) Rift clusters (g4)

Rifts without previous doming (k22)

Rift stars (g2) Rift chains (g3) Rift nets? (g5)

Exstensional arc rifts (k411)

Triple junction rifts (k5)

Neutral arc rifts (k412)

Compressional arc rifts (k413)

Solitary rifts (g1) Rift clusters (g4) (Rift chains?) (g3)

Solitary rifts (g1) Rift clusters (g4) (Rift chains?) (g3)

Continental collisionrelated rifts (k42)

Impactogens (k421) Solitary rifts (g1) (Rift chains?) (g3)

Intracontinental convergence related rifts (k22) Rift clusters (g4) (Solitary rifts?) (g1)

Pack-ice type rifts (k423) Solitary rifts (g1) Rift clusters (g4) Rift chains (rare) (g3)

Continental Rifts, Fig. 8 The classification of rifts and taphrogens. (After Şengör 1995)

Natal’in 2001). They resemble chocolate-bar boudinage, as seen in the Proterozoic basement of Eastern Europe (Fig. 5), and may have a similar origin, but more commonly rift nets form in complex and rapidly shifting stress environments in which dominant extension directions change fast. Many rift nets in fact may represent two superimposed rift clusters. Kinematic Classification of Rifts (See Fig. 8) Because rifts are ubiquitous in all stages of the Wilson Cycle of ocean opening and closing, the kinematic characteristics of the plate boundaries have been taken as a basis for classifying them according to the environment of the overall displacement and strain in which they form. There are three types of plate boundaries plus the plate interiors, to which four types of rift families correspond. In addition, incompatibilities arise around some unstable intracontinental triple junctions leading to complex rifting that should be treated separately from the other four classes, thus creating a fifth kinematic class, called “triple-junction rifts.”

k1) Intraplate rifts: Rifts surrounded entirely by underformed lithosphere occupy this category. Such rifts are usually solitary, small, and rare (the difficulty in forming them is analogous to that forming a solitary fold or a nappe surrounded by entirely underformed terrain) and are not easy to identify in the geological history. The Lake George and Lake Champlain rifts in the northeastern United States are active examples (▶ “Seismicity, Intraplate”). k2) Rifts associated with divergent plate boundaries: These rifts result from plate separation along nascent extensional boundaries. All the Cainozoic rifts in east Africa belong here. This category of rifts may be further subdivided into two classes as follows: k21) Rifts that form following an episode of doming: The divergent boundary along which rifts form is in this case preceded by an episode of lithospheric doming. The East African Rift Valleys are the best-known examples of such rifting (▶ “Isostasy, Thermal”). k22) Rifts that form with no pre-rift doming: In this case rifts form without a prelude of uplift, as is the case in the

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Salton Trough and the Gulf of California. A good fossil example is the rifting of the Alpine Neo-Tethys in the earlier Mesozoic. k3) Rifts that form in association with conservative plate boundaries: Conservative, that is, transform fault, boundaries are those along which neither extension nor shortening is required by the regional slip vectors. However, various reasons conspire to induce both extension and shortening to occur along considerable stretches of these boundaries. Rifts along conservative plate boundaries form in three different settings: k31) Transtensional conservative boundaries: If a conservative boundary is opening up all along its length because of a component of extension, it is called transtensional. Many active rifts have a transtensional component and fossil examples of such rifts may be recognized largely through the structures they contain as shown by Olsen and Schlische (1990). Dewey (2002) gave an exhaustive analysis of the strain that develops in such rifts. k32) Pull-apart basins along conservative boundaries: Major strike-slip faults commonly have bends along them that either facilitate (“releasing bends”) or obstruct (“restraining bends”) slip along them. Extensional basins form along the releasing bends, in which the magnitude of extension equals the magnitude of cumulative strike-slip offset along the strike-slip fault since the formation of the releasing bend. Such basins are called “pull-apart basins.” Crowell’s (1974) faultwedge basins are nothing more than special cases of pull-apart basins. k33) Sphenochasms: Not all basins created by secondary extension associated with strike-slip faults are pullapart basins. Some represent tears caused by either an asperity or differential drag along the strike-slip fault in one of the fault walls, in which the amount of extension changes from a maximum along the fault to zero at the pole of opening of the tear-basin. S. W. Carey called such wedge-shaped rifts that open towards a major strike-slip fault sphenochasms. k4) Rifts that form in association with convergent plate boundaries. A large family of rifts forms in association with convergent plate boundaries. In this group, a firstorder subdivision is between rifts associated with subduction zones and rifts associated with continental collision, although this may artificially split some genetic groups, such as those rifts that presumably form because of the tension generated by excessive crustal thickening. The usefulness of the present grouping is that it enables a rapid overview of the presently active rift environments and comparison with ancient ones. k41) Rifts associated with subduction zones: Environments of rifting associated with subduction zones

Continental Rifts

correspond to three different types of arc behavior, namely, extensional, neutral, and compressional arcs. k411) Rifts associated with extensional arcs: An extensional arc generally splits along the magmatic axis (if such an axis is already in existence) forming a small rift chain. Such a situation is today known from both the Okinawa rift and the Izu-Bonin arc system. Such rifts commonly do not get preserved intact, both because of the complications of the tectonic evolution of arcs involving common changes of behavior and because of later collisions with other arcs or continents. In extensional arcs, rifts also develop orthogonal to arc trend owing to the extension of the arc as it bows out in front of an expanding marginal basin (as, for instance, in Crete). k412) Rifts associated with neutral arcs: Neutral arcs have neither shortening nor extension across them. Therefore, the only rifts that may form in neutral arcs are those associated with arc-parallel strike-slip faults, which may be classified in the same way as the rifts that form along conservative plate boundaries. More complex rift basins may originate along such arc-parallel strike-slip faults, if the sliver plate in the forearc area and its various pieces rotate about vertical axes. Pull-apart basins in arcs are difficult to recognize, but the Sumatra Fault has several welldeveloped examples along it. Sphenochasms along strike-slip faults in arcs are rarer still. Davis et al. (1978) have discussed two possible examples, the more recent of which may have created the “Columbia Embayment” by motion along the Straight Creek fault in the latest Cretaceous and the earliest Cainozoic. k413) Rifts associated with compressional arcs: In compressional arcs crust commonly thickens and lithosphere thins, both by heating and by eventual delamination. The arc becomes shortened across, and elongated along, its trend. This elongation commonly generates rifts at high angles to the trend of the arc. Rifts on the Altiplano in the Andes are examples of such high-angle rifts. k42) Rifts associated with zones of continental collision: Three different environments of rifting form associated with the collision of continents: (1) Lines of extension that radiate from points at which collision commences, (2) Regions of extension abutting against sutures, and (3) Nodes of extension in areas of complex deformation in fore- and hinterlands shattered by collisions. Impactogens (k421), rifts forming in intracontinental convergence belts (k422), and pack-ice-type rifts (k423) correspond with these three environments, respectively.

Continental Rifts

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k5) Triple Junction Rifts: Triple junction rifts form at or near unstable intracontinental triple junctions, at which plate evolution dictates the generation of “holes” because continental lithosphere cannot be continuously subducted. Dynamic (Genetic) Classification of Rifts (See Fig. 6) Rifts are also classified according to the origin of forces (▶ “Geodynamics”) that lead to rifting. Şengör and Burke (1978) proposed that stresses which cause rifting may be imposed on the lithosphere directly by the mantle beneath it (cf. Şengör 2001) or they may result from two-dimensional plate evolution. Accordingly, they termed these two modes of rifting “active” and “passive.” Gans (1987) tried to replace these terms with “open-system” and “closed-system” rifting, respectively, which, however, did not find general acceptance.

There is only one kind of rift this classification does not consider: rifts that form by propagating from an already existing rift. Since propagation may take many forms, it might be sufficient to indicate such rifts with the notation d2 to indicate their passive mode of opening. The regional geologist may use Fig. 8 as a “flow chart” to follow the evolutionary histories of the various kinds of rift basins he encounters.

Mantle plume head

b

Mantle lith. to be detached

c

ing asthenosp nd he ce s A

re

d1) Active rifting: “Active rifting” is rifting caused by mantle upwelling (▶ “Earth, Density Distribution” and ▶ “Mantle Convection”) associated with plumes in the mantle (▶ “Mantle Plumes”). Two views have been advanced to explain the origin of the extension in domes rising above hot-spot jets: one ascribes the rifting to basal shear stresses induced by a spreading plume head beneath a dome. The other holds the potential energy of the rising dome responsible for driving the rifting. All of these factors probably do contribute to maintaining the active rifting process at its habitually slow pace of considerably less than 1 cm/a (▶ “Geoid”). Plume-caused taprogens may be also termed deeplyrooted rifts and would be the extensional counterparts of subduction-related orogens (Fig. 9a, b). These can be contrasted to collisional orogens that have lost their subductive “anchors” and the passive rifts with no deep mantle connexions (Fig. 9c, d). d2) Passive rifting: In the passive rifting mode, extension is caused by the two dimensional motions of the lithospheric plates and not by an autonomous interference from the mantle (Fig. 11d). In this mode of rifting, there is no prerifting doming associated with a hot-spot (Şengör and Burke 1978). Kinematic mechanisms reviewed above under the headings k22, k31, k32, k33, k411, k412, k413, k421, k422, k423, and k5 all may form rifts in a “passive rifting mode.”

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a

d Continental crust

Oceanic crust

Mantle lithosphere

Continental Rifts, Fig. 9 Deeply rooted mega-structures ((a) subduction-controlled orogens, (b) mantle-plume controlled, i.e., active rifts) and rootless mega-structures ((c) collisional orogens, (d) plate motion-controlled, i.e., passive rifts)

Concluding Note Although they constitute one of the three great continental structure types and have serious relevance to human wellbeing, rifts remain poorly known. This is largely because there are still important technological difficulties in studying them directly because of the large sediment thicknesses they contain and because their diversity is little appreciated. They are so ubiquitous in all tectonic environments that they tend to be overlooked in favor of other, rarer, structures that may appear more prominent. It is commonly not recognized that all kinds of structures such as magmatic arcs, collisional orogens, keirogens, intracratonic basins all have associated rifts and, consequently, people wishing to know about rifts tend not to concern themselves with them. A stereotypic rift that commonly bedevils textbooks is one that disrupts a craton

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and has alkalic or peralkalic vulcanism with a thick, mainly nonmarine sedimentary fill, although there are many more rifts in the world than those with such characteristics. What is now needed in rift studies is a comprehensive synthesis that takes into account this diversity.

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77 McKenzie D (1978) Some remarks on the development of sedimentary basins. Earth Planet Sci Lett 40:25–32 McQuarrie N, Wernicke BP (2005) An animated tectonic reconstruction of southwestern North America since 36 MA. Geosphere 1:147–172. https://doi.org/10.1130/GES00016.1 Miao JY, Kou HS, Zhou LF, Han ZY (2006) Sedimentary environments of organic matter from Middle Permian source rocks in northern Xinjiang, China. Chin J Geochem 25:258–265 Miller JMG, John BE (1988) Detached strata in a tertiary low-angle normal fault terrane, southeastern California: a sedimentary record of unroofing, breaching, and continued slip. Geology 16:645–648 Mohr P (2009) Africa beckoning. Millbrook Nova Press, Galway Morley CK (1999) Geoscience of rift systems – evolution of East Africa. AAPG studies in geology 44. The American Association of Petroleum Geologists, Tulsa, 242 pp. + 5 appendices+ 16 pp. Index Olsen PE, Kent DV (1999) Long-period Milankovich cycles from late Triassic and early Jurassic of eastern North America and their implicatiodns for the calibration of the early Mesozoic time-scale and the long-term behaviour of the planets. Philos Trans R Soc Lond A357:1761–1786 Olsen PE, Schlische RW (1990) Transtensional arm of the early Mesozoic Fundy rift basin: penecontemporaneous faulting and sedimentation. Geology 18:695–698 Quennell AM (1982) Rift valleys Afro-Arabian. Benchmark papers in geology, vol 60. Hutchison Ross, Stroudsburg Quennell AM (ed) (1985) Continental rifts. Benchmark papers in geology series. Van Nostrand Reinhold, New York Richardson-Bunbury JM (1996) The Kula volcanic field, western Turkey: the development of a Holocene alkali basalt province and the adjacent normal-faulting graben. Geol Mag 133:275–283 Ring U, Wernicke B (eds) (2009) Extending a continent: architecture, rheology and heat budget. Geological Society Special Publication 321, 272 pp Rotstein Y, Edel J-B, Gabriel G, Boulanger D, Schaming M, Munschy M (2006) Insight into the structure of the Upper Rhine Graben and its basement from a new compilation of Bouguer Gravity. Tectonophysics 425:55–70 Sawkins FJ (1990) Metal deposits in relation to plate tectonics. Springer, Berlin Schlüter T (1997) Geology of East Africa with contributions by Craig Hampton: Beiträge zur Regionalen Geologe der Erde, 27. Gebrüder Borntraeger, Berlin Schumacher ME (2002) Upper Rhine Graben: role of preexisting structures during rift evolution. Tectonics 21:1006–1022. https://doi.org/ 10.1029/2001TC900022 Şengör AMC (1987) Cross-faults and differential stretching of hangingwalls in regions of low-angle normal faulting: examples form western Turkey. In Coward MP, Dewey JF, Hancock PL (eds) Continental extensional tectonics. Geological Society Special Publication 321 (Albert M. Quennell volume), pp 575–589 Şengör AMC (1995) Sedimentation and tectonics of fossil rifts. In: Busby CJ, Ingersoll RV (eds) Tectonics of sedimentary basins. Blackwell, Oxford, pp 53–117 Şengör AMC (2001) Elevation as indicator of mantle plume activity. In: Ernst R, Buchan K (eds). Geological Society of America Special Paper 352, pp 183–225 Şengör AMC (2003) The large wavelength deformations of the lithosphere: materials for a history of the evolution of thought from the earliest times to plate tectonics. Geological Society of America Memoir 196, xvii+347 pp. + 3 folded plates in pocket Şengör AMC (2013) The Pyrenean Hercynian Keirogen and the Cantabrian Orocline as genetically coupled structures. J Geodyn 65:3–21. (Ali Koçyiğit Festschrift) Şengör AMC (2018) Some notes towards a history of ideas on rifts and extensional tectonics in general. Earth Sci Hist 37:144–156 Şengör AMC, Burke K (1978) Relative timing of rifting and volcanism on Earth and its tectonic implications. Geophys Res Lett 5:419–421

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78 Şengör AMC, Natal’in BA (2001) Mantle Plumes: Their Identficaniton Through Time. Rifts of the world. In: Ernst R, Buchan K (eds). Geological Society of America, Boulder, Colorado 352, pp 389–482 Smith AG (1976) Orogeny: a review. Tectonophysics 33:215–285 Snow JK, Wernicke B (2000) Cenozoic tectonism in the Central Basin and range: magnitude, rate and distribution of upper crustal strain. Am J Sci 300:659–719 Spizaharsky TN, Borovikov LI (1966) Tectonic map of the Soviet Union on a scale of 1: 2 500 00. In: Scientific communications read to the Commission for the Geological Map of the World, Delhi, 22nd International Geological Congress, pp 111–120 St-Onge MR, King JE (1987) Evolution of regional metamorphism during back-arc stretching and subsequent crustal shortening in the 1.9. Ga Wopmay Orogen, Canada. Philos Trans R Soc Lond A321:199–218 Surkov VS, Smirnov LV, Zhero OG (1994) Early Mesozoic rifting and evolution of the West Siberian Basin. In: Roure F, Elluz N, Shein VS, Skvortsov I (eds) Geodynamic evolution of sedimentary basins, international smposium, Moscow. Éditions Technip, Paris, pp 135–143 ten Brink US, Taylor MH (2002) Crustal structure of central Lake Baikal: insights into intracontinental rifting. J Geophys Res 107:ETG-21–ETG2-15. https://doi.org/10.1029/2001JB000300 Ulmishek GF, Bogino VA, Keller MB, Poznyakevich ZL (1994) Structure, stratigraphy, and petroleum geology of the Pripyat and DnieperDonets Basins, Byelarus and Ukraine. In: Landon SM (ed) Interior rift basins. American Association of Petroleum Geologists Memoir 59, pp 125–156 Ulrych J, Cajz V, Adamovič J (eds) (1999) Magmatism and rift basin evolution. GeoLines 9:1–135 Verdel C, Wernicke BP, Ramezani J, Hassanzadeh J, Renne PR, Spell TL (2007) Geology and thermochronology of tertiary Cordilleran-style metamorphic core complexes in the Saghand region of central Iran. Geol Soc Am Bull 119:961–977 Wernicke B (ed) (1990) Basin and range extensional tectonics near the latitude of Las Vegas, Nevada. Geological Society of America Memoir 176, xii+511 pp. + numerous separate plates Wernicke B (2009) The detachment era (1977–1982) and its role in revolutionizing continental tectonics. In: Ring U, Wernicke B (eds). Geological Society Special Publication 321, pp 1–8 Wernicke B, Burchfiel BC (1982) Modes of extensional tectonics. J Struct Geol 4:105–111 Wiart P, Oppenheimer C (2004) Large magnitude silicic volcanism in north Afar: the Nabro Volcanic Range and Ma’alalta volcano. Bull Volcanol 67:99–115 Wickham S, Oxburgh ER (1985) Continental rifts as a setting for regional metamorphism. Nature 318:330–333 Willis B (1936) East African plateaus and rift valleys. Studies in comparative seismology. Carnegie Institution of Washington, Washington, DC Wilson M, Neumann E-R, Davies GR, Timmermann MJ, Heremans M, Larsen BT (eds) (2004) Permo-carboniferous magmatism and rifting in Europe. Geological Society Special Publication 223, 498 pp Woodcock NH (1986) The role of strike-slip fault systems at plate boundaries. Philos Trans R Soc Lond A317:13–29 Yirgou G, Ebinger CJ, Maguire PKH (eds) (2006) The Afar volcanic province within the East African rift system. Geological Society Special Publication 259, 331 pp Ziegler PA (ed) (1992) Geodynamics of rifting. Elsevier, Amsterdam Ziegler PA (1994a) Hydrocarbon habitat in rift basins. In: Roure F, Elluz N, Shein VS, Skvortsov I (eds) Geodynamic evolution of sedimentary basins, international smposium, Moscow. Éditions Technip, Paris, pp 85–94 Ziegler PA (1994b) Geodynamic processes governing development of rift basins. In: Roure F, Elluz N, Shein VS, Skvortsov I (eds) Geodynamic evolution of sedimentary basins, international smposium, Moscow. Éditions Technip, Paris, pp 19–67

Core Dynamo

Core Dynamo Ulrich R. Christensen Max-Planck-Institut für Sonnensystemforschung, Göttingen, Germany

Synonyms Geodynamo

Definition Dynamo: Process for generating electrical current and magnetic field by electromagnetic induction in a moving conducting medium Geodynamo: Dynamo process in the fluid outer core that generates Earth’s main magnetic field

Introduction It has been firmly established that the geomagnetic field must be generated by a magnetohydrodynamic dynamo operating in the Earth’s outer core. Thermal and compositional buoyancy forces drive a convective circulation of the liquid iron alloy. Assuming that a magnetic field is present, electrical currents are induced in the moving fluid. When the magnetic field associated with these currents has the strength and geometry that is suitable for the induction process and no external source for the field is required, this is called a selfsustained dynamo. To first approximation the Earth’s core is a sphere with uniform electrical conductivity, in contrast to technical dynamos, where the currents are guided by a highly inhomogeneous distribution of electrical conductors. In the former case we speak of a homogeneous dynamo. From a theoretical point of view, homogeneous dynamos are more difficult to understand. Simple motions, such as differential rotation (as in an electrical generator), are unable to drive a homogeneous dynamo and flow patterns of a certain complexity are required. Until late into the twentieth century dynamo theory has been concerned mainly with the conceptual understanding of how, in principle, a magnetic field can be generated in such an environment. Starting with models by Glatzmaier and Roberts (1995), Kageyama and Sato (1995), and Kuang and Bloxham (1997), realistic self-consistent numerical simulations of the geodynamo became available and have been successful in reproducing many of the observed properties of the geomagnetic field. The more fundamental aspects of the geodynamo are discussed elsewhere (▶ “Geomagnetic Field, Theory”). Here, the progress in

Core Dynamo

understanding the geodynamo based on numerical modeling and comparing its results with specific properties of the geomagnetic field is addressed.

Dynamo Model Concept and Equations Model Setup In contrast to earlier kinematic dynamo models, where the flow is prescribed, modern geodynamo models aim at a fully self-consistent treatment of fluid flow and magnetic field generation in the Earth’s outer core. There are some basic requirements for a realistic model: (i) The mechanism for driving flow by thermal or compositional buoyancy must be part of the model. (ii) Because an axisymmetric field cannot be generated by a dynamo (Cowling’s theorem), the model must be fully three-dimensional. (iii) The model must be rotating, because Coriolis forces are important to generate a flow pattern that is conducive for the dynamo process. The basic setup of geodynamo models is that of a rotating spherical shell of outer radius ro and inner radius ri, filled with a fluid of uniform conductivity, which represents Earth’s outer core (Fig. 1). For this system the coupled equations for convection-driven flow and electromagnetic induction are solved. A detailed account on fundamental aspects of convection in rotating spheres is given in Jones (2015) and modeling aspects and the commonly employed numerical schemes are in discussed in Christensen and Wicht (2015). Dynamo Equations The relevant magnetohydrodynamic equations are usually written in nondimensional form. A possible scheme for

79

scaling the equations is to use the shell thickness D ¼ ro  ri as length scale, the viscous diffusion time D2/v as time scale (v is kinematic viscosity), (rΩ/s)1/2 for the scale of the magnetic field B, and the imposed temperature contrast ΔT between inner and outer boundary for temperature T (r is density, s electrical conductivity, and Ω rotation rate). The Navier-Stokes equation for the velocity u, augmented by rotational and electromagnetic forces, is E


 @u þ u  ∇u þ 2bz  u þ ∇P @t ¼ E∇2 u þ

Ra E r 1 Tþ ð∇  BÞ  B: Pr r o Pm

ð1Þ

The terms on the left hand side describe in order the inertial force, the Coriolis force (with ẑ the unit vector parallel to the rotation axis), and the gradient of the non-hydrostatic pressure P. The terms on the right hand side stand for viscous friction, thermal buoyancy forces, and the Lorentz force. The magnetic induction equation, obtained from Maxwell’s equations and Ohm’s law for a moving incompressible conductor, is @B 1 2 þ ðu  ∇ÞB ¼ ðB  ∇Þu þ ∇ B, @t Pm

ð2Þ

where the second term on the LHS describes magnetic field advection and the terms on the RHS magnetic field generation and diffusion, respectively. Magnetic diffusion is a consequence of the ohmic resistance that damps the electrical currents associated with the magnetic field. The advection-diffusion equation for temperature is @T 1 þ u  ∇T ¼ ∇2 T þ ϵ, @t Pr

ð3Þ

with a heat source term ϵ on the RHS. For compositional convection an equivalent equation holds where the concentration of light components replaces temperature. The set of equations is completed by the condition of incompressibility, which seems to be justified for Earth’s core where density differences are moderate, and the condition that B is solenoidal: ∇  u ¼ 0,

Core Dynamo, Fig. 1 Columnar convection in a rotating spherical shell. The inner core tangent cylinder is shown by broken lines. Under Earth’s core conditions the columns would be thinner and more numerous. (From Christensen et al. (2010). Copyright: Cambridge University Press)

∇  B ¼ 0:

ð4Þ

The equations are complemented by boundary conditions for the velocity, usually u ¼ 0, fixed temperature or fixed heat flux at ro and ri, and a continuity condition for the magnetic field at ro that links the field in the dynamo with an external potential field that decays with radius. Simple models assume the inner core to be insulating, but in many cases it is taken as an electrical conductor with the same conductivity as in the

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Core Dynamo

outer core. It has been suggested that the finite conductivity of the inner core plays an important role for preventing frequent dipole reversals of the geodynamo, because a dipole reversal can only be completed successfully if the new polarity persists for the time scale of diffusive dipole decay in the inner core, r 2i =ðp2 lÞ , which is several thousand years. However, this is contested by dynamo models that show little differences between simulations with a conducting or an insulating inner core.

is approximately frozen into the moving fluid (Alfvén’s theorem), which is the case when the diffusion term in Eq. 2 can be neglected. In this case the variation of the radial magnetic field component Br at the core-mantle boundary can be written @ Br =@t þ ∇h  ðuBr Þ ¼ 0:

ð5Þ

Parameters The four nondimensional control parameters in these equations are defined in Table 1. In dynamo models the values of most of these parameters fall short of their extreme values in the Earth’s core. For practical reasons it is not possible to resolve the very small spatial structures that occur at realistic parameter values. The Rayleigh number Ra describes the ratio between buoyancy driving convection and retarding effects. In the core it is much larger than the critical value for the onset of convection, Rac, while models are more moderately supercritical (Table 1). The discrepancy is by at least eight orders of magnitude for the Ekman number E, the ratio of viscous forces to Coriolis forces. The very small core value of E indicates that viscous forces are negligible, except in very thin Ekman layers at the boundaries to the solid mantle and inner core. The magnetic Prandtl number Pm, the ratio of viscosity to magnetic diffusivity, is very small for liquid metals, but must be set to a value of order one to obtain a self-sustained dynamo in present models. Only the (hydrodynamic) Prandtl number Pr, the ratio of viscosity to thermal diffusivity, is of order one in the core and in dynamo models. In terms of physical parameters, the viscosity and the thermal diffusivity are far too large in the models, and in most cases, the rotation rate is too small. Several dimensionless diagnostic numbers can be formed with the characteristic flow velocity U and magnetic field strength B in the dynamo (Table 1). The velocity in Earth’s core can be estimated from the geomagnetic secular variation under the assumption that the magnetic flux at the core surface

where the suffix h indicates the horizontal part of the divergence operator. Additional assumptions are needed to infer the velocity at the top of the core (below the Ekman layer) from the observed Br and its time derivative, but the flow pattern obtained from these inversions are broadly similar and the characteristic velocity is approximately 0.5 mm/s (Holme 2015). The characteristic field strength inside Earth’s core is probably in the range of a few mT (milliTesla), as will be discussed further below. The most important diagnostic number is the magnetic Reynolds number Rm, which describes the ratio of magnetic advection and induction to magnetic diffusion. In order for induction effects in a dynamo to overcome the diffusive dissipation of magnetic field, it is expected that Rm must be larger than one. Practically, it turns out that self-sustained magnetic field generation occurs in geodynamo models when Rm exceeds approximately 40. Using U ≈ 0.5 mm/s the magnetic Reynolds number is of order 1000, safely above the critical limit. Also, Rm  1 is a condition for Eq. 5 to apply. Still, the value of Rm in the Earth’s core is moderate and can be handled in direct numerical simulations. This contrasts with the much larger values of Rm in other cosmic dynamos, which requires that magnetic induction and diffusion effects at small unresolved scales are parameterized by applying concepts of the mean-field dynamo theory. The ability to solve for magnetic induction and diffusion in Earth’s core directly without parameterizations is probably the main reason for the success of geodynamo models. The (hydrodynamic) Reynolds number Re is much smaller in the models than it is in the core where it is of the order 109, indicating a highly turbulent regime. Consequently, dynamo

Core Dynamo, Table 1 Dynamo parameters. α thermal expansivity, go gravity at core surface, ΔT superadiabatic temperature contrast across core, k thermal diffusivity, n kinematic viscosity, Ω rotation rate, D outer

core thickness, l ¼ 1/(mos) magnetic diffusivity, U characteristic flow velocity, B characteristic magnetic field strength, mo magnetic permeability, r density

Definition Core Models

Rayleigh no. Ra ¼ αgoΔTD3/(kn) 104 Rac (1–100) Rac

Definition Core Models

Magn. Reynolds no. Rm ¼ UD/l 103 40–2000

Control parameters Ekman no. E ¼ n/(ΩD2) 1015–1014 107–102 Diagnostic numbers Reynolds no. Re ¼ UD/n 109 0 Þ

ð6Þ

Wahr and de Vries (1989), Forte et al. (1994). The value of the constant A0 is about 1,300 m2 s2, according to Defraigne et al. (1996) and about 2,300 m2 s2 according to Forte et al. (1994); our compromise will be A0 ¼ 1,800 m2 s2. By Eq. 5i, p0a  rsa F0a : Therefore ea ¼ p0a =psa is 104 on the CMB equator and 105 at the ICB equator. (b) Deviations created by convection. We use the notation r ¼ r a þ rc ,

ð7aÞ

and similarly for other variables. The Earth radiates energy into space at a rate estimated to exceed 43 TW (e.g., §4.1.5 of Schubert et al. 2001). We take an extreme position by supposing the entire 43 TW emerges from the core. The outward heat flux in the FOC is the sum of the convective heat flux qc and the adiabatic heat flux qa ¼ – KdT s/dr, where K is thermal conductivity. Though the latter may be as much as 5 TW, we ignore it. Then qc ¼ 0.28 W m2 at the CMB. We take qc ¼ rcp T c V r , 1

ð7bÞ

1

where cp (≈800 J kg K ) is the specific heat at constant pressure, p. The overline denotes a horizontal average over the flow, which is undoubtedly highly turbulent. For Vr ¼ 104 m s1, it follows that T c ¼ 3:5  b a  4, 000 K (e.g., Kawai and Tsuchiya 104 K:As T 2009), εc ¼ Tc/Ta is less than 107, which is 3–4 orders of magnitude smaller than εa. Even this is an overestimate; it has never been suggested that Qc, is larger than 15 TW. The smallness of εΩ, εa, and εc means that all these effects can be treated as perturbations of the reference state. As p0a ðr, y, fÞ  pc ðr, y, fÞ, it is sensible to consider first the effect of p0a by introducing the “adiabatic topographic torques” on the mantle, ICB, and FOC: bT ¼ G a

þ

pa r  n dA, bS þ e T ¼  p r  n dA, G a e a ðS GTa ¼  r  ∇pa dV:

ð8a, b, cÞ

V

If the CMB  (ICB)  were an equipotential or if it were T T b G e spherical, G would vanish according to Eq. 8a, b, a a but generally these torques are nonzero. They can be evaluated only if pb ðpeÞ is known on the CMB (ICB). It will be

T

Ga are intimately shown in the next section that b Ga and e related to what we shall call “adiabatic gravitational torques.” We therefore postpone further discussion and estimation of adiabatic topographic torques. T Consider the torque b G associated with the convective T

c

motions in the FOC: T b Gc ¼

þ

T b Gc,z ¼

bS þ• bS •

b h r  ∇pc dA •, b h @ f pc dA •:

ð9a, bÞ

Reasons will be given in the section on “The Magnetic Torque” why core flow may be well described by the magnetostrophic approximation Eq. 20a, and why, deep in the core, Coriolis and Lorentz forces are comparable, implying a magnetic field strength ℬ there of about 2 mT, or about four times greater than the typical field strength ℬo on the CMB. The Lorentz force is therefore 16 times less on the CMB than in the bulk of the core. Also, as gf, is small, @ fpc ≈  2Ωr0 ro Vθ cos θ sin θ should be a good approximation to the f-component of Eq. 20a on the CMB. Therefore (Hide et al. 1993) T b Gc,z ¼ 2Or0 r o

þ bS •

b h ðy, fÞ V y ðr o , y, fÞ cos y sin y dA • : ð9cÞ

T In principle, b Gc,z can be estimated by extracting bh from seismological analysis, and by using the Vθ (ro, θ, f) inferred from the core surface motion. In practice, this is difficult and has generated controversy. Eq. 9c suggests that

  T b Gc,z ¼ O 2OrV H r 3o ,

ð9dÞ

which is 1018 Nm for a bump height o f H ¼ 100 m. Such a bump height is well within the bounds set by recent seismic investigations (e.g., Tanaka 2010). Equation 9b indicates, however, that Eq. 9d may be a serious overestimate because pc is a single-valued function and @ f pc is as often positive as negative in the integrand of Eq. 9b. Though
2ΩrV ro is a reasonable estimate of @ f pc at most points on the CMB, considerable cancellation is likely when evaluating the integral in Eq. 9b. There is even an outside chance that the cancellation might be complete; see Anufriev and Braginsky (1978b). Reliable estimation of the convective topographic torque must probably await careful experiments and allied theory. No argument has so far convincingly demonstrated that topography can create torques of the target magnitude of 1018 Nm but, equally, none have shown that it cannot.

Core-Mantle Coupling

91

The Gravitational Torque The gravitational torque on a body V of density r(x) in a gravitational field g(x) is ð GG ¼

r r  g dV:

ð10aÞ

By Eqs. 8a, b and 10d and the continuity of g and p, we also have GþT b Ga ¼

V

This volume integral can be usefully transformed into a surface integral by drawing on the analogy between the theories governing Newtonian gravitation and electrostatics, the only difference between these theories being one of sign: like charges repel but all bodies attract gravitationally. It can be shown that rgi ¼ ∇ j SG ij , ¼

where

SG ij

  1 1 gi g j  g2 dij 4pG 2

þ r bS n h io 1  pa n þ ð4pGÞ1 ðn  ga Þga  g2a n dA, 2 ð12bÞ

þ GþT e Ga ¼  r bS n h io 1  pa n þ ð4pGÞ1 ðn  ga Þga  g2a n dA, 2 ð12cÞ

ð10b, cÞ

from which, in agreement with Eq. 12a,

is the gravitational stress tensor, the gravitational pressure – g2/8πG being also the gravitational energy density. Equations 10b, c enable 10a to be written as a surface integral:

GþT b e GþT ¼ 0: Ga þ G a

1 4pG

GG ¼ 

þ

h i 1 r  ðn  gÞ g  g2 n dA, 2 S

ð10dÞ

where n points out of V. See RA12 and Appendix B of Braginsky and Roberts (1995) for derivations of these results. When they are applied below, n on the CMB and SIC will, as previously defined, be oriented approximately parallel to r. By Eq. 10a, the gravitational torque on the FOC due to the mantle and SIC is ð GG ¼ V

G ðra þ rc Þ r  ðga þ gc Þ dV ¼ GG a þ Gc ,

ð11aÞ

GG c

¼

GþT b Ga ¼ 0,

GþT e Ga ¼ 0:

ð13a, bÞ

If this minimum energy, torque-free state is perturbed, the restoring GT-torques (as we shall call them) set up a “gravitational oscillation” of the SIC relative to the mantle. To give a simple example, suppose that the gravitational anomaly defined by Eq. 6 is created by sources entirely within b GþT ¼ G bG . the mantle, the CMB having no bumps, so that G

ðV

ra r  ga dV, ð11b, cÞ r  ðrc ga þ ra gc þ rc gc Þ dV:

a,z

Similarly, suppose the SIC is spherical but has internal sources that above the ICB produce the gravitational anomaly F00a ¼ A00 ðr i =r Þ3 sin 2 y cos 2ðf  ’Þ,

ð ¼

GþT GþT b and f If the torques b Ga and e Ga are nonzero, V V evolve. The consequent change in the relative orientation of the mantle and SIC modifies ga and pa in Eq. 12b, c so that the system evolves toward a configuration of minimum energy E. In this configuration

a,z

G where GG a and Gc are the adiabatic and convective parts of G Γ :

GG a

ð12dÞ

ðA00 > 0Þ,

ð14aÞ

where ’ is the angular displacement of the system from the stable state, ’ ¼ 0, in which Eq. 13a, b hold. Eq. 12b, c show that

V

  It was pointed out earlier that rc ¼ O 104 r0a , gc ¼  4 0    4 G O 10 ga , etc. Apparently therefore GG c ¼ O 10 Ga , making it sensible to focus first on the adiabatic torque. Consistent with hydrostatic balance in the FOC (see Eq. 5f), Eqs. 8c and 11b, give ð T GGþT ¼ GG a a þ Ga ¼

r  ðra ga  ∇pa Þ dV ¼ 0: ð12aÞ V

b G ¼ G e G ¼ GG sin 2’, G a,z a,z 0 GG 0

¼

8A0 A00 r 3i =3Gr2o ð>

0Þ:

where

ð14b, cÞ

These torques vanish for the stable minimum energy states ’ ¼ 0, π (and also for the unstable ’ ¼
12 p). Small departures from a stable state satisfy G b 2’ e’ b Þ, Cd t b ¼ 2G0 ð’

G e 2’ e’ b Þ, ð14d, eÞ Cd t e ¼ 2G0 ð’

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Core-Mantle Coupling

    where Cb ¼ 7:12  1037 kg m2 and Cb ¼ 5:86  1034 kg m2 are the polar moments of inertia of mantle and SIC, respectively. The frequency, oG, of the oscillation is

h   i1=2 h i1=2 G e bCe = C  2G = C : oG ¼ 2 Cb þ Ce GG 0 0

ð14f Þ

Although an anomaly of the form of Eq. 14a could be produced by density variations within the SIC, it is more plausibly created by SIC topography for the following reasons. It is generally believed that the SIC is the result of freezing of the FOC, an ongoing process even today (Jacobs 1953). An alloy generally changes its composition when it changes phase. The rather large density jump at the ICB, e¼e D rðr i Þ  rðr i Þ  600 kg m3 , is hard to attribute to contraction on freezing but can be readily explained as a small reduction in X on freezing. Phase equilibrium at the ICB implies T sa ðFa , Sa , Xa Þ ¼ T m ðpa , Xa Þ, where Tm is the melting temperature. This implies that the ICB is an equipotential surface. Since F ¼ Fsa ðr Þ þ F0a ðr, y, fÞ where j F0a =Fsa j 1, Taylor expansion, using Eq. 4b, shows that Fsa ðr i Þ þ geheðy, fÞ þ F0a ðr i , y, fÞ is approximately constant, where ge ¼ gsa,r ðr i Þ ¼ @ r Fsa ðr i Þð> 0Þ is gravity at the ICB. It follows that he ¼ F0a ðr i , y, fÞ=e g ¼ ee sin 2 y cos 2f, ee ¼ A0 ðr i =r 0 Þ2 =e g:

ð14g, hÞ

This shows how the gravitational anomaly in the mantle imposes its n ¼ m ¼ 2 preference on the SIC. It makes the otherwise ad hoc assumption of Eq. 14a seem perfectly reasonable. The condition that he creates Eq. 14a, for ’ ¼ 0 and r  r i  he is 00

0

A =A ¼

e 2 ge  4pGr3i D=5r o

0:0034:

ð14iÞ

The maximum bump height on the ICB is j ee j 50 m . Nonzero ’ corresponds to a rotated SIC. Such a rotation is to be expected; core turbulence continually subjects the SIC to (topographic) torques that continuously change its orientation. Two relaxation processes act to restore the ICB to its equipotential: (i) flow within the SIC, (ii) new freezing/melting on the ICB. If either were instantaneous, there would be no torque between the mantle and SIC, but both appear to act on much longer time scales than core turbulence, so that SIC topography is almost “frozen” to the SIC as it turns. Concerning (i), the viscosity of the SIC is plausibly much less than the viscosity of the mantle. According to Schubert et al. (2001) vb  5  1019 m2 s1 in the deep mantle but, according to Mound and Buffett (2006), ve≳1013 m2 s1 : Whereas mantle anomalies are essentially “frozen in,” slow motions within the SIC created by stresses exerted by the

FOC on the ICB can gradually restore equilibrium (Yoshida et al. 1996; Buffett 1997). Concerning (ii), the thermodynamic disequilibrium created by the misalignment of the ICB from its equipotential surface is slowly removed by new freezing of the FOC or new melting of the SIC. This processes has not been fully explored (but see Fearn et al. 1981; Morse 1986). The possible significance of melting/ freezing processes on SIC structure has been recently investigated by Alboussière et al. (2010) and Monnereau et al. (2010). Interest in gravitational torques and oscillations was sparked by Buffett (1996). We follow him but by a different method, making use of Eq. 14c, i: 6e 0 4 e  6:7  1019 Nm GG 0 ¼ 32pr i DA =15r o g 2

ð15aÞ

By Eq. 14f, the frequency of the oscillation is oG ≈ 4.8  108 s1, with a period of tG ¼ 2p=oG  4:1 years:

ð15bÞ

According to Buffett et al. (2009), gravitational oscillations are mainly responsible for the LOD variations shown in Fig. 1. Mound and Buffett (2006) obtain GG 0  1:5  1020 Nm. Equations 14d, e imply, for some t0, b b ¼O b þ dt f O    0 ef b b 0 þ 2GG =Co b G f ¼O 0

max

sin oG ðt  t0 Þ:

ð15cÞ

The amplitude of the gravitational oscillation is therefore related to that of the variation, ΔP, in LOD by   ef b f

max

b G DP=GG P2  1:2¯, ¼ pCo 0 0

ð15dÞ

for ΔP ¼ 1 ms. This gives  a maximum angular velocity ef b difference of oG f  2¯ year1 . Furthermore the max

peak-to-peak variation in the radial gravitational acceleration at the Earth’s surface, r ¼ rE, is Dg00r ðr E Þ ¼


4  2 12A00 r i bf e f ’ 4 nGal: ri rE max

ð15eÞ

This value is too small by roughly a factor of 5 to be detectable by the GRACE satellite system (e.g., Wahr et al. 2006). One limitation of this analysis is the neglect of electromagnetic stresses at the boundaries when they are in motion relative to the core fluid. It has been implicitly assumed that the SIC is completely decoupled from the fluid within the tangent cylinder (TC), the imaginary cylinder that touches the

Core-Mantle Coupling

93

ICB at its equator. This is particularly significant because the fluid dynamics inside and outside the TC are quite dissimilar. See Hide and James (1983), Heimpel and Aurnou (2007), and the next section. Because the SIC is as good an electrical conductor as the FOC (or better), it may be tightly coupled magnetically to C N and CS, as suggested by Braginsky (1970); see the next section. To examine the effect of this coupling, we make the extreme assumption that all the fluid in the TC is completely locked to the SIC. Because the mantle is a poor electrical conductor, this fluid is not well coupled to the mantle, so that the entire fluid column within the TC can co-rotate about Oz almost freely with the SIC. This suggests that, instead of Eq. 1g, h, a more useful division of the total angular momentum of the Earth might be based on b þ GTC þ GXTC ¼ 0, G b þ MTC þ MXTC ¼ constant, M

ð16a, bÞ

where TC refers to the TC and SIC locked together, and XTC refers to the part of the FOC exterior to the TC. The moment of inertia of the fluid within the TC is 2.12  1035 kg m2 e gives CTC ¼ 2.71  1035 kg m2. which, when added to C, e Using this instead of C in Eq. 14f, tG is lengthened from 4.1 years to ¼  8:9 years, h   i1=2 b TC ¼ 2 Cb þ CTC GG = CC : 0 tG TC

where oG TC

2p=oG TC

the frozen flux theorem and Alfvén waves. Davidson (2001) contains the necessary background. It may be useful to remind readers that the magnetic torque about O on a body V carrying a current of density J is the integrated moment of the Lorentz force, J  B: ð

ð

GM ¼

r  ðJ  BÞ dV ¼ V

The Magnetic Torque This section assumes that readers are familiar with preMaxwell EM theory and the fundamentals of MHD, including

ð17aÞ

The Lorentz force can be expressed as a divergence: ð J  B Þ i ¼ ∇ j SM ij ,   1 2 1 where SM B ¼ m B B  d i j ij ij 0 2

ð17b, cÞ

is the magnetic stress tensor. An alternative form of Eq. 17a is therefore GM ¼ m1 0

þ

h i 1 r  ðn  BÞB  B2 n dA, 2 S

ð17dÞ

where the unit normal n to S points out of V. Therefore þ M 1 b Gz ¼ m0 sBbr Bbf dA, bS þ e M ¼ m1 sBer Bef dA: G z 0 bS

ð16c, dÞ

See also Mound and Buffett (2006). Even though rc/ra, pc/pa, etc., are of order 104, this does not mean that GGþT =GGþT is as small as that. In fact, Eq. 13a c a GþT shows that Ga ¼ 0 in the minimum energy state. The adiabatic GT-torques dominate the convective torques only if ’ is sufficiently large. Stated another way, a convective torque can be nullified by a small departure from the minimum energy state. Earlier, the torque on the SIC created by core turbulence was held responsible for causing ’ to deviate from zero. This torque is essentially (magneto-)convective, and is nullified by the GT-torque for a tiny change in ’. Another way of estimating how tiny this ’ is equates the magnitudes of the GT-torque, taken as 1.3  1020 ’ Nm (see Eqs. 14b, c and 15d), and the convective torque, taken to have its target value of 1018 Nm. This gives ’ ≈ 0.5°. Within this angle, the mantle and SIC are gravitationally locked together, over short time scales compared with those of the relaxation processes in the SIC described above. See Buffett and Glatzmaier (2000).

r ½Br J  J r B dV: V

ð17e, f Þ

These results can be used as they stand to assess the magnetic coupling between the inferred core surface flow and the mantle. See Stix and Roberts (1984), Love and Bloxham (1994), Holme (1998). Here, however, we are more interested in forging a link between the observed changes in LOD and torsional waves. To explain what the latter are, it is necessary to consider some dynamical issues. Most studies of core MHD are based on the Boussinesq approximation; see, e.g., Braginsky and Roberts (2007). This assumes constant density, r0(≈ 104kg m3), and expresses conservation of mass and momentum as ∇  V ¼ 0,

@ t V þ V  ∇V þ 2O  V

¼ ∇ ðpc =r0 Þ þ C ge þ J  B=r0 þ vT ∇ 2 V:

ð18a, bÞ

The accelerations in Eq. 18b are from inertia (@ tV and V  ∇V), rotation (2 Ω  V), pressure (pc), buoyancy (Cge), magnetic field (J  B/r0), and viscosity (vT∇2V). Thermal and compositional buoyancy, combined in the codensity C (Braginsky and Roberts 1995), maintains the flow and the magnetic field; see Core Dynamo. The Coriolis force is generally more significant than the inertial and viscous forces. This is indicated by the smallness of the Ekman and Rossby numbers:

C

94

Core-Mantle Coupling

Ro ¼ V =OL:

See Eq. 2c for the definition of E. From V ¼ 104 m s1, L ¼ r o and vT ¼ 102 m2 s1 follow Ro ≈ 106 and E ≈ 1011. This suggests that the inertial and viscous terms can be safely omitted from Eq. 18b, except on small length scales. If the inertial and viscous forces are ejected from Eq. 18b, it becomes 2r0 V  VN ¼ ∇pc þ r0 C ge þ J  B,

on the CMB and ICB:

C ðsÞ

ð20bÞ

The full boundary conditions of continuous B and n 3 E still apply. Because @ tV has been ejected from Eq. 20a, there are no Alfvén waves. Instead, the system evolves on the much longer ageostrophic time scale,

ðJ  BÞf dA ¼ 0

ð21bÞ

If s < ri, there are two cylinders, C N ðsÞ and C S ðsÞ, of radius s to the north and south of the SIC for which ð

ð

C N ðsÞ

ðJ  BÞf dA ¼ 0,

C S ðsÞ

ðJ  BÞf dA

¼ 0:

ð20aÞ

where N stands for “non-geostrophic,” and “geostrophic” is defined below. Equations 18a and 20a define the “magnetostrophic approximation,” often used to describe the quasi-steady macroscales of core MHD. As the viscous term has been ejected, the only boundary condition that VN must obey is n  VN ¼ 0,

ð

ð19Þ

ð21c, dÞ Equations 20b–d are examples of “Taylor’s constraint” (Taylor 1963). The cylinders C ðsÞ are termed “Taylor cylinders.” Of these, C ðr i Þ is the tangent cylinder (TC). It is obviously possible to assign a J which creates a B that contradicts Eqs. 21b–d, at least initially. This shows that Eq. 20a is an oversimplification. That approximation rested on discarding the inertial force in comparison with the Coriolis force. Consider however the class of “geostrophic flows”: v ¼ uðs, tÞ 1f :

ð22aÞ

The corresponding Coriolis acceleration is t ¼ 2OL N

2

=V 2A

¼ t =L,

ð20c, dÞ

where t ¼ L 2 =  105 years

2r0 V  v ¼ ∇X , where

is the free decay time for magnetic fields of scale L, and Λ is the “Elsasser number”: L ¼ V 2A =2O,

where

VA

¼ ℬ=√ðm0 r0 Þ

ð20e, f Þ

is the Alfvén velocity. Elsasser (1946) suggested that ℬ is determined by a balance of Lorentz and Coriolis forces. Taking J  sV ℬ, this implies Λ ¼ 1, ℬ ≈ 2 mT, and VA ≈ 2 cm s1. It also gives tN ≈ t ≈ 2  105 years for L ¼ r o. In cylindrical coordinates (s, f, z), the f-component of Eq. 20a is 2Or0 V N s

¼ @ f pc þ ðJ  BÞf :

ð21aÞ

Integrate this over the surface, C ðsÞ, of the circular cylinder of radius s (>ri) coaxial with Oz. The left-hand-side vanishes by mass conservation, as can be verified by integrating Eq. 18a over the interior, L (s), of C ðsÞ and applying Eq. 20b to Nb ðsÞ and b S ðsÞ , the spherical caps of C ðsÞ on the CMB that complete the boundary of I (s). It follows that

ð X ¼ 2Or0 uðs, tÞ ds

ð22b, cÞ

can be absorbed into pc. Coriolis forces are therefore totally ineffective when they act on geostrophic flows. Other forces previously abandoned in comparison with Coriolis forces become influential, especially the inertial forces, which must be restored when analyzing the geostrophic part of core flow. This recovers the Alfvén wave, or something very like it, called the “torsional wave.” They share a common time scale: tA ¼ r o =V A  6 years:

ð22dÞ

That this is also the time scale tLOD of the semi-decadal variations of in Fig. 1a, b may not be a coincidence, as argued by Gillet et al. (2010). The geostrophic part v of V can be extracted from V by taking the “geostrophic average,” hVfi, of Vf: for s > ri, this average is defined by 1 u ¼ V f ðsÞ A^ V ¼ V  u1f , N

ð CðsÞ

V f dA,

so that (22e)

Core-Mantle Coupling Core-Mantle Coupling, Fig. 2 Schematics showing (a) geostrophic flows in the core, Vg, and (b) plan view of an initially cylindrical magnetic field (dashed line) distorted by v. The restoring Lorentz torques on the distorted magnetic field, Bs (solid line), lead to the cylindrical propagation of torsional waves. (Adapted from Dumberry 2007)

95

a

B

CM

ICB

BS

u u

where AbðsÞ ¼ 4psz1 is the area of C ðsÞ , and z1 ðsÞ ¼   √ r 2o  s2 is the semi-length of its sides. The axial angular momentum of the FOC is carried by u. Therefore, insofar as the rotation of the SIC is locked to that of the fluid in the TC, e z , of the entire core can be the angular momentum, Mz þ M derived from the zonal part of the inferred core surface flow. e z. Therefore Eq. 1h can be tested: The LOD M  record provides  b z ¼  Mz þ M ez , Results have been gratifying; see Jault M et al. (1988), Jackson (1997) and Fig. 1c above. The previous section indicates however that the mantle and SIC are not locked together but take part in a gravitational oscillation having a period (4–9 years) similar to the torsional wave period tA ≈ 6 years. The implied convolvement of gravitational and magnetic coupling complicates the task of extracting information about either; see Buffett et al. (2009). In a torsional wave, the geostrophic cylinders are in relative angular motion about their common (polar) axis; see Fig. 2a. The response b of B to the motion v can, as for an Alfvén wave, be visualized using the frozen flux theorem, the field lines behaving like elastic strings threading the cylinders together and opposing their relative motion; see Fig. 2b. The resulting torque on a cylinder supplies the restoring force for a wave, the mass of the cylinders providing the inertia. Whenever J and B contradict Eqs. 21b–d, a torsional wave is launched that travels in the
s-indirections. The canonical torsional wave equation is  @2z 1 @ 2b @z 2 s AðsÞV A ðsÞ , ¼ @s @t2 s2 AbðsÞ @s

b

B

CM

ð23Þ

where ζ(s, t) ¼ u/s is the angular velocity of C ðsÞ and VA(s) ¼ ℬs(s)/(m0r0)1/2 is the Alfvén based on the mean D velocity  N 2  E 2 N 2 Bs over C ðsÞ :ℬs ðsÞ ¼ Bs . Equation 23 is called canonical because it displays the essential nature of torsional waves clearly. It is however not only incomplete but also ignores magnetic coupling to the mantle and SIC. Equation 23 presupposes that the field, BN, on which the waves ride is axisymmetric. In this case, Eq. 23 has a severe

singularity at s ¼ 0 which excludes physically acceptable solutions. This difficulty can be evaded by supposing that the TC rotates as a solid body, as suggested by Braginsky (1970), and by applying Eq. 23 only in XTC. For general, non-axisymmetric BN, the regular singularity at s ¼ 0 is harmless, but unfortunately Eq. 23 is then incomplete. The terms missing from Eq. 23 represent the transmission of torque from one Taylor cylinder to another by the potential field outside the core. As the terms are troublesome if retained, they are usually abandoned, with an unsubstantiated claim that they are too small to be worth keeping. Whether they are retained or abandoned, the wave equation admits normal mode solutions, i.e., solutions in which ζ is proportional to exp.(ιot), where every o is real, as shown in RA12. When magnetic coupling to the mantle is included, o acquires a negative imaginary part, representing the ohmic losses in the mantle. The inclusion of magnetic coupling is highly relevant to our focus here, but it clearly adds another layer of complexity. In view of these theoretical obstacles, the reader may wonder whether computer simulation would not provide a simpler approach. It is however difficult to extract evidence of torsional waves from geodynamo simulations. This is because it is not yet possible to attain geophysically realistic magnetic Prandtl numbers, Pm, in numerical integrations. The importance of viscous damping of torsional waves can be assessed by the ratio of the torsional wave time scale, tA ¼ ro/VA, to the  spin-up time scale, tSU ¼ √ r 2o =2Ov , which is their viscous decay time; see Roberts and Soward (1972). This ratio, tA =tSU ¼ 2ðOvÞ1=2 =V A ¼ √ðPm=LÞ,

ð24Þ

is small for the Earth (~103) but inconveniently large in simulations (≳0.1). See Wicht and Christensen (2010) for recent simulations of torsional waves. Space limitations do not permit the mathematical theory of torsional waves to be presented here; see Braginsky (1970), Roberts and Soward (1972), Jault (2003), Dumberry (2007) and RA12. The principal aim in what follows is to outline the

C

96

underlying physics, that has been employed in theoretical studies, and that will have to be incorporated in numerical models in the future, when computer technology has advanced far enough to permit core MHD to be modeled more faithfully. Our discussion here sidesteps interesting and as yet incompletely answered questions, the first of which is whether torsional waves are involved in any essential way with the semi-decadal LOD periodicity; might not the gravitational oscillation described in the last section be mainly responsible? The answer is unclear but core field strengths of 2 mT suggest that torsional waves very effectively link together geostrophic motions throughout the core, and that torsional waves therefore necessarily accompany gravitational oscillations. This obviously does not imply that torsional waves couple well to the mantle; if the mantle were an electrical insulator, there would be no magnetic coupling of core and mantle at all. The waves would then be detectable only through the inferred core surface motion. Strong magnetic coupling of the waves to the mantle raises difficult questions about both the excitation and dissipation of the waves. As shown below, the waves are mainly and heavily damped by ohmic dissipation in the mantle. What maintains the waves against this energy loss? No definitive answer has been given to this very significant question. Core turbulence may be directly responsible, or indirectly through the Reynolds magneto-stresses it exerts on the SIC or on the TC. Buffett et al. (2009) find evidence for wave generation at the TC. Core turbulence would presumably excite many damped normal modes. This might help to explain some of the recently discovered short time scales of core surface motions (Olsen and Mandea 2008; Finlay et al. 2010). It has not been established unequivocally that the 6 year period is the fundamental mode. Though often questioned, evidence of a 60 periodicity exists, This longer period signal might be representative of the fundamental mode but would require current estimates of ℬs(s) to be revised downward. These matters are beyond the scope of this review. We shall merely sketch, mostly in a qualitative way, the approximate theory that is currently the most complete. We also aim to expose its strengths and weaknesses. Underlying the entire theory is the idea that core flow can be neatly separated into geostrophic motions of short time scale, tA, and non-geostrophic motions of long time scale tN obeying Eq. 20a. In other words, the theory focusses on the large length scales of core flow, in the belief that this includes the fundamental torsional wave mode of greatest interest. The torsional wave therefore rides on a flow satisfying Taylor’s constraints Eqs. 21a–c. Because the Lehnert number, l ¼ tA/ tN ¼ VA/2Ωro ¼ oA/2Ω ≈ 3  105, is small, the time dependence of BN can be ignored in torsional wave theory. It was pointed out earlier that torsional waves are geostrophic motions in which the inertial force is crucial. The first

Core-Mantle Coupling

step in deriving the torsional wave equation is therefore to restore the time-dependent inertial force to Eq. 21a, obtaining r0 s@ t V f þ 2Or0 sV s ¼ @ f pc þ sðJ  BÞf ,

ð25aÞ

from which @ tζ is extracted, by taking the geostrophic average; see Eq. 22e. The evaluation of hJ  Bif is simplified because E is small and the Lundquist number, Lu ¼ t/tA ¼ VAro/ ≈ 3  104, of the waves is large. In a first approximation, E ¼ Lu1 ¼ 0. Then viscous and ohmic diffusion and the associated boundary conditions are discarded. In particular, the electric field created in the FOC by the waves simplifies to e ¼ v  BN  VN  b  v  b,

ð25bÞ

where b is the magnetic field of the waves. The term –VN  BN does not appear because it already acts on the Taylor state. The last term in Eq. 25b is also ignored because the wave amplitude is assumed to be small. The ratio of –VN  b to – v  BN is of order AN[b/u √ (m0r0)]1, where A ¼ V/VA is the Alfvén number, which for the non-geostrophic flow is about 0.1 (see above). As in an Alfvén wave, u ¼ O[b/ √ (m0r0)], so that jVN  b j ≈ 0.1 j v  BNj. This, combined with the fact that the inclusion of –VN  b in e adds severe complications, encourages the neglect of –VN  b in Eq. 25b, leaving e ¼ v  BN :

ð25cÞ

The mantle and core are linked across a boundary layer of Ekman-Hartmann type; e.g., see Dormy et al. (2007). Because Pm  1, this has a double structure. Viscosity acts only in an Ekman layer of thickness dv ¼ √ (v/2Ω) ≈ 0.1 m; magnetic diffusion acts in a layer whose thickness is comparable with  1=2 the EM skin depth, d  ¼ 12 jojm0 s  10 km. We therefore call this a “magnetic diffusion layer” (MDL), even though Coriolis and Lorentz forces affect its structure too. In the MDL, Eq. 25c is replaced by e ¼ v  BN  ∇  b  v  BN  1r  @ r b:

ð26Þ

As d/dv  1, the Ekman layer is only a tiny part of the MDL. Ekman-Hartmann theory simplifies by discarding the Ekman part, setting v ¼ 0 and abandoning the no-slip boundary conditions. The structure of the MDL still depends on rotation, and on the Elsasser number, Λ, defined in Eq. 20e. The boundary layers play a vital role in linking ζ in the main bulk of the FOC to b b on the CMB and e b on the ICB. M M They are therefore essential in determining bI and eI . At the CMB, ℬ ≈ 0.5 mT and Λ ≈ 0.07. The prevailing magnetic field, BN, therefore has very little effect on the boundary layer, which is controlled almost entirely by Coriolis forces and

Core-Mantle Coupling

97

magnetic diffusion. At the ICB, where ℬ may be even an order of magnitude greater than at the CMB, Λ > 1, and Lorentz forces are too significant to ignore in the boundary layer. Further details are given in RA12. The torque exerted by the waves on the mantle is proportional to the electrical conductivity, b sðv, y, fÞ, of the mantle, which we assume is nonzero only in the layer ro < r < r1 ¼ ro + d at the base of the mantle. The conductance of this layer is b ðy, fÞ ¼ S

ð r1

b sðr, y, fÞ dr:

ð27aÞ

ro

b < 109 S; we take It is commonly assumed that 107 S < S b ¼ 108 S below. S b either by leaking Electric currents, bj, flow in the mantle, V, from the core or by electromagnetic induction, through the time dependence of the EM field on the CMB. We shall be interested in the penetration of the fields b and e of the waves, at frequencies o of order 5  108 s1. The resulting magnetic and electric fields, b b and be, in the mantle depend on o and on L, the horizontal length scale imposed by b and e on the CMB. Associated with o is the skin depth of the mantle: 

1 db ¼ jojm0 b so 2

1=2

:

ð27bÞ

Starting with Glatzmaier and Roberts (1995), theoreticians have usually simplified EM theory in the mantle by adopting the “thin-layer approximation” (TLA). This originated from the modeling of laboratory MHD experiments (see, e.g., Müller and Bühler 2001). It is easily applied and therefore popular, although the conditions for its validity are seldom mentioned or questioned. The TLA demands that d  db ð LÞ. The horizontal part, beH , of the electric field be is then independent of r, so that bjH ¼ b s beH ,

and

b 5 ¼ Sb b eH , J

ð27c, dÞ

b is the total horizontal current carried by the layer. where J It may be helpful to visualize the TLA as a mathematical b held fixed. Then J b is a surface limit, d ! 0, b s ! 1 with S current responsible for a discontinuity in the magnetic field. If  ˇ  ˇ b ¼ ∇ w is the potential field above the conducting layer, ˇ

b ðy, fÞ  1r bðr o , y, fÞ ¼ m0 J b ðr o , y, fÞ  b b ðy, fÞbeH ðr o , y, fÞ  1r : ¼ m0 S

ð27eÞ

The potential field does not affect the torque, ΓM, on the mantle, although it does contribute to the torque that each Taylor cylinder exerts on the others.

We contrast two proposed conductivity distributions. Buffett et al. (2002) inferred from their studies of nutational resonances that d is only 200 m and that b s ¼ 3  105 S m1 , which is comparable to the core conductivity s, and gives db ¼ 10 km. The TLA should therefore be excellent in most applications. In contrast, the laboratory experiments of Ohta et al. (2008) suggest b s ¼ 100 S m1 and d ¼ 3  105 m; see also Yoshino (2010). This gives db ¼ 2  106 m , so it is doubtful if the TLA can be validly applied.The similar b ¼ 3=6  107 S are insufficonductances of the models S cient to justify the use of the TLA. This completes our critique of the basics of torsional wave coupling to the mantle. Some of its consequences are unexpected; but most are not. Even dimensional reasoning leads to   bM ¼ G b M z  bz , G z 0   b M ¼ O r 4 Sℬ b 2  4  1027 Nms, where G 0 o o

ð28a, bÞ

 2 for ℬ ¼ 0.5 mT. Here ℬ2o is an average of BNr over the CMB, and z is defined by: zð t Þ ¼

1 s2 AbðsÞ

ð ro 0

s2 AbðsÞzðs, tÞ ds

ð28cÞ

Perhaps unexpectedly, the boundary layer on the CMB  b , b M by a factor of S= S þ S described above reduces G 0 where S is the conductance of the boundary layer, defined by 1 S ¼ ð1 þ iBÞsd ¼ ð1 þ iBÞðs=2m0 jojÞ1=2 , 2

ð28dÞ

and B ¼ sgn(o). This gives |S| ≈ 3  109 S, which is b Ignoring this factor, the magnetic intercomparable with S. action of mantle and FOC is governed by     b M z  bz , Cd t z ¼ G b M z  bz , b tbz ¼ G Cd 0 0

ð28e, f Þ

where C ¼ 9.08  1036 kg m2 is the polar moment of inertia of the solidly rotating FOC. These equations provide an estimate of the e-folding time,  bt , taken  M by mantle conduction to kill ζ b C þ Cb G b  64 years. This is greater and bz : bt ¼ CC= 0 than the time taken by the waves to cross the core, which is tA ¼ ro/VA ≈ 5.6 years, for ℬ ¼ 2mT. If we take dbz  3  1012 s1 , as indicated by the LOD data of the first  section above, Eq. 28e, f suggest that z  bz  Cb þ C bz=C is about 2  1011 s1, so that   r o z  bz  7  105 m s1 , which is less than, but compabM  rable with, the inferred core surface flow. It also gives G z

C

98

Core-Mantle Coupling

8  1016 Nm. This increases to the target torque if we take b ¼ 1:2  109 S, but that reduces bt to 5.3 years, which is less S than tA. This highlights a difficulty that might be called the “magnetic coupling paradox,” and quantified by a quality factor:  1 b A P ¼ bt =tA ¼ ðℬ=ℬo Þ2 m0 SV :

ð29Þ

b that may be narrow or nonexisThere is a window for S, b M is large enough to explain variations in tent, in which G z LOD by torsional waves, but simultaneously is small enough to ensure that P > 1 so that the waves are not over damped by mantle conduction. According to the admittedly imprecise, order of magnitude estimates made here, the window is nonexistent. For the target torque to be reached or exceeded, b b G≳1:2  109 S, but P > 1 requires G≲1:1  109 S. See also Dumberry and Mound (2008). So far the existence of the SIC has been ignored, almost totally. We have however recognized that, for s < ri, two Taylor cylinders exist, C N ðsÞ and C S ðsÞ, in which ζN(s) and ζS(s) may be expected to differ. For simplicity, we assume e z, it is necessary to link here that they are equal. To evaluate G e b to b across a boundary layer strongly influenced by Λ. An analysis of the boundary layer leads to (see RA12)   bef ¼ m0 e sde j sBNr z  ez

on r ¼ r i ,

ð30aÞ

z

where

eM G 0 ð30b, cÞ

e M  2  1029 Nms . This cf. Eq. 28a, b. For ℬi ¼ 5 mT, G 0 large torque acts on the SIC whose moment of inertia is less b The coupling time, et , is therefore very much than 103 C. less than bt . Equations analogous to Eq. 28a, b give (for e s ¼ s) e jG e j 4 days: et  C= 0 M

z

core flow interacting with bumps on the CMB. In analyzing b T , it became clear that exchange of angular momentum with G z

the SIC is significant. Order of magnitude arguments showed b T , which is produced b T is G that potentially the largest part of G z

a,z

by the gravitational field of density anomalies in the mantle and possibly SIC, including bumps on their surfaces. This part b T is therefore intimately related to the gravitational torque of G z

where ez is the angular velocity of the SIC, and de ¼ 1 1=2  d  10 km . This leads to an expression si 2 jojm0 e M e for G , in which the main part that couples to the TC is   eM ¼ G e M z  bz , G z 0   ¼ O r 4i e sde ℬ2i ;

Synthesis In this review we have analyzed the various ways in which Earth’s core is coupled to the mantle and have presented estimates of the amplitudes of these couplings in order to show which may plausibly explain the available LOD data. In our first section, we provide observational evidence for core-mantle coupling. We show that Earth’s rotation rate has a roughly semi-decadal time variability, such that the LOD fluctuates at the ms level. To explain these LOD fluctuations, an internal coupling must exist between the mantle and the core that provides torques of order 1018 Nm, which we named “the target torque.” In the later text, we develop estimates of the strength of the viscous, topographic, gravitational, and electromagnetic torques. Only the viscous torque, GVz , appears to be too weak to explain the LOD signal. This is true even when we allow for the enhanced coupling that turbulence can provide. b T , is created by The topographic torque on the mantle, G

ð30dÞ

This is the time taken for a mismatch between ζ and ez to be wiped out by magnetic torques. Clearly the coupling between TC and SIC is substantial at frequencies of order oA. This supports the opinion, advanced several times in this review, that on semi-decadal time scales, the TC is effectively locked to the SIC in its rotation about Oz.

b GþT , there is an b G . When the two are treated together as G G z a,z GþT e equal but opposite torque, G on the SIC. Gravitational a,z oscillations (Buffett 1996) occur when the system is perturbed from a state of minimum gravitational energy in which b GþT ¼ G e GþT ¼ 0. An oscillation period of tG ¼ 4.1 years G a,z

a,z

was derived. If, as seems likely, strong magnetic coupling exists between the tangent cylinder (TC) and the SIC, the gravitational oscillation period increases to tG TC ¼ 8:9 years. T T b b The remaining part of G is G , and is produced by core z

c,z

convection. Its importance is uncertain. From what is known today, GTc,z may be 0 Nm or may exceed the target torque (cf. Kuang and Bloxham 1997; Hide 1998; Jault and Le Mouël 1999; Kuang and Chao 2001). A simple model of torsional waves traversing FOC can explain oscillations of period, t ≈ 6 years, but there is a paradox: The target torque cannot be attained by magnetic coupling between the waves and the mantle unless a dimensionless “paradox parameter,” II, defined in Eq. 29, is large enough. If this parameter is too large, however, the waves are damped out before they can cross the core. Whether the core can evade the paradox seems uncertain. The topographic, gravitational and magnetic torques all have significant uncertainties in their amplitudes, but the target torque falls within these uncertainties, i.e., conceivable any of them could explain the semi-decadal LOD signals. The coupling processes may be convolved. The recent model of

Core-Mantle Coupling

Buffett et al. (2009) allows for this, but argues that the gravitational torque dominates. In contrast, Gillet et al. (2010) infer that torsional oscillations in the FOC can explain the LOD observations without strong gravitational coupling. Improvements in data and modeling of Earth’s rotation (e.g., Gross 2009), the geomagnetic field (e.g., Hulot et al. 2002; Jackson 2003), core seismology (e.g., Dai and Song 2008), and the time-variations in the gravity field (e.g., Velicogna and Wahr 2006; Dumberry 2010) will all prove important in testing these core-mantle coupling arguments. This review has focussed on explaining variations in LOD owing to internal coupling between the mantle and core. This coupling produces changes primarily in the axial angular rotation rate, Ωz, on semi-decadal time scales. Detailed measurements now exist of variations in Earth’s full rotation vector on many time scales (e.g., Gross 2007), with the different directional components providing information on different geophysical phenomena (e.g., Mathews et al. 2002). Furthermore, rotation vector and magnetic field measurements now exist for other planets (e.g., Margot et al. 2007; Uno et al. 2009), and will improve in quality in the coming decades. Such measurements will allow the development of models of deep interior structure and dynamics in planetary bodies throughout the solar system (e.g., Tyler 2008; Noir et al. 2009; Goldreich and Mitchell 2010).

Cross-References ▶ Core Dynamo ▶ Earth’s Structure, Lower Mantle ▶ Energy Budget of the Earth ▶ Geomagnetic Field, Theory Acknowledgments We thank Richard Gross, Richard Holme, and Andrew Jackson for sharing their insights and their data. We are also grateful to Bruce Buffett and the referee (Mathieu Dumberry) for giving helpful advice.

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Crustal Reflectivity (Oceanic) and Magma Chamber Tanaka S (2010) Constraints on the core-mantle boundary topography from P4KP-PcP differential travel times. J Geophys Res 115: B04310. https://doi.org/10.1029/2009JB006563 Taylor JB (1963) The magnetohydrodynamics of a rotating fluid and the Earth’s dynamo problem. Proc R Soc Lond A274:274–283 Tyler RH (2008) Strong ocean tidal flow and heating on moons of the outer planets. Nature 456:770–773 Uno H, Johnson CL, Anderson BJ, Korth H, Solomon SC (2009) Modeling Mercury’s internal magnetic field with smooth inversions. Earth Planet Sci Lett 285:328–339 Velicogna I, Wahr J (2006) Acceleration of Greenland ice mass loss in spring 2004. Nature 443:329–331 Vočadlo I, Alfè D, Price GD, Gillan MJ (2000) First principles calculation of the diffusivity of FeS at experimentally accessible conditions. Phys Earth Planet Inter 120:145–152 Wahr J, de Vries D (1989) The possibility of lateral structure inside the core and its implications for nutation and Earth tide observations. Geophys J Int 99:511–519 Wahr J, Swenson S, Velicogna I (2006) Accuracy of GRACE mass estimates. Geophys Res Lett 33:L06401. https://doi.org/10.1029/ 2005GL025305 Wicht J, Christensen UR (2010) Torsional oscillations in dynamo simulations. Geophys J Int 181:1367–1380 Yoshida S, Sumita I, Kumazawa M (1996) Growth model of the inner core coupled with outer core dynamics and the resulting elastic anisotropy. J Geophys Res 101:28085–28103 Yoshino T (2010) Laboratory electrical conductivity measurement of mantle minerals. Surv Geophys 31:163–206. https://doi.org/10. 1007/s10712-009-9084-0

Crustal Reflectivity (Oceanic) and Magma Chamber Satish C. Singh Laboratoire de Géoscience Marines, Institut de Physique du Globe de Paris, Paris, France

Synonyms

101

Introduction Over 70% of the earth’s crust is formed by the cooling and crystallization of melt at ocean spreading centers, which represent over 55,000 km of chains of volcanoes in the middle of the oceans, called mid-ocean ridges. At ocean spreading centers, the oceanic plate separates causing the mantle to move upward, reducing the pressure and causing the melting of the mantle. Since the newly formed melt is lighter than the surrounding mantle material, it moves upward toward the surface of the earth. Part of the melt is erupted on the seafloor as lava, which cools very rapidly forming a cap of solid extrusive layer, also known as Layer 2A (Fig. 1). As there is a significant amount of water present at mid-ocean ridges, the water circulates deep in the crust. Therefore, the melt stays mainly in the middle of the crust and erupts along thin dikes. When these dikes are cooled and crystallized, they form a thick intrusive layer or Layer 2B. Below the dikes, the melt could reside for a long period, forming a steady state melt lens, called axial melt lens or axial magma chamber (AMC). The magma cools and crystallizes in this melt lens, forming a crystalline lower crust. The melt lens forms the lower limit for further penetration of water, and therefore, partial melt is generally present beneath the melt lens down to the crust– mantle boundary. Sometimes hot melt ascending from the mantle may get injected in this partially molten region. Based on this basic process, the oceanic crust is divided into three layers, lava (extrusive), dikes (intrusive), and gabbroic crust. The relative thicknesses of these layers depend on the spreading rate, which can vary from a few millimeters up to 200 mm per year. Based on the spreading rate, mid-ocean ridges are divided into four groups: fast, intermediate, slow, and ultraslow. East Pacific Rise (EPR) is a fast spreading center, where the spreading rate varies from 70 to 180 mm per year (Fig. 2). Juan de Fuca Ridge and Valu Fa Ridge are intermediate

Melt lens; Spreading center and ridge 0 Lava (Layer 2A)

Definition 2

Melt Lens Depth (km)

Axial magma chamber (Melt lens) is a thin melt lens observed at ocean spreading centers. Layer 2A or Lava is the top layer of oceanic igneous crust. Layer 2B is a dike sequence and lies above the axial melt lens. Layer 3 (gabbro) forms the lower oceanic crust. Moho is a boundary between the crust and mantle. Pg is a seismic ray that travels through the crust. PmP is a reflection from the crust–mantle boundary. Pn is a ray that travels in the upper mantle. Tomography is a technique to image the velocity structure of the earth.

Dike (Layer 2B)

4 Gabbro (Layer 3) 6

8

Moho

Mantle Peridotite

10

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 1 Classical model of the oceanic crust. Layers 2A and 2B form the upper crust whereas the gabbro layer corresponds to the lower crust

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102

Crustal Reflectivity (Oceanic) and Magma Chamber

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 2 Major spreading centers on the earth. JFR, Juan de Fuca Ridge; VFR, Valu Fa Ridge; CIR, Central Indian Ridge. The rectangles indicate the positions of data/results shown in this entry: 1, Lucky Strike Segment at Mid-

Atlantic Ridge (Figs. 5, 7, 9, and 12); 2, 9° N East Pacific Rise (Figs. 8, 11, and 14); 3, 14° S East Pacific Rise (Figs. 6, 10, and 13); and 4, Wharton basin (Figs. 15 and 16)

spreading centers, where the spreading rate varies from 55 to 70 mm per year. Mid-Atlantic Ridge and Central Indian Ridge are slow spreading centers with spreading rates of 20–55 mm per year. South-West Indian Ridge and Gakkel Ridge are ultraslow spreading centers with spreading rates of less than 20 mm per year. The melt production depends on the spreading rate and, therefore, the crustal thickness also varies from 3.5 to 8 km, with an average thickness of 6 km.

they are placed 5–20 km apart on the seafloor, and hence provide a very large-scale (5–10 km) velocity structure of the crust. Since the compressional (P) waves travel fast, they arrive first on OBS, and hence mainly P-wave velocities are estimated using OBS data. However, sometimes secondary (S) arrivals, converted phases Ps, are recorded on these OBS and provide information on S-wave velocity. Seismic reflection method is most commonly used to obtain crustal reflectivity. A streamer containing pressure sensors (hydrophones) is towed behind a vessel recording seismic energy generated by an array of air guns as the vessel moves at 4.5–5 knots (Fig. 3). The length of the streamer varies from a few hundred meters up to 15 km (Singh et al. 2009), and hydrophones are generally spaced at 12.5 m. For crustal imaging the air gun shot spacing is ~50 m. Depending upon the target depth, the recording length varies from 10 to 20 s (20–60 km depth). This technique is routinely used for oil and gas exploration, and has become a standard technique for crustal studies. There are standard tools to analyze these data. Since the source receiver distance is small as compared to the depth of the target, the data are initially presented as a function of time, and then are depth-converted using velocities obtained from the OBS study or from other sources.

Methods There are two main methods to study the oceanic crustal structures: seismic refraction or tomography and reflection methods. In seismic refraction method, a set of ocean bottom seismometers (OBS) are placed on the seafloor and an array of air guns is used to generate acoustic energy that travels through water column into the crust and mantle and is recorded on OBS (Fig. 3). There are three types of waves that arrive on an OBS: waves that travel in the crust (Pg) and the mantle (Pn) and waves reflected from the crust–mantle boundary, Moho (PmP). The travel times of arrivals of energy on these OBS are used to determine the velocity structure of the crust and upper mantle. Since OBS are small in numbers,

Crustal Reflectivity (Oceanic) and Magma Chamber

103

Vessel Sea surface

Streamer Airgun source Reflection

OBS

Seafloor

Pg Crust

P

Pm

Moho Mantle Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 3 Schematic diagram showing the seismic reflection and refraction survey to study the oceanic crust. Red star indicates the air gun source.

Pn OBS, ocean bottom seismometers; Pg, crustal arrival; Pn, mantle arrival; and PmP, reflection from the crust–mantle boundary. Red ellipse indicates melt lens

Crustal Structure at Fast Spreading Centers

2

4

6

8

0 Layer 2A

Depth (km)

Based on seismic refraction and reflection studies at fast spreading centers and on ophiolites from Oman, the crust at fast spreading centers is considered as a layer cake and is divided into three distinct layers: Layer 2A (pillow lava), Layer 2B (dike sequence), and Layer 3 (gabbro) (Fig. 4). The velocity in Layer 2A varies from 2.5 to 4.5 km/s, that in Layer 2B from 4.5 to 6 km/s, and in gabbro from 6 to 7 km/s. On the ridge axis, a magma chamber is generally present between the dike and the gabbro layer. The structure of Layer 2A is inferred either using the velocity from OBS study or from the reflection method. The boundary between Layer 2A and 2B is not sharp but there is a high velocity gradient where the velocity increases from 2.5 to 4–4.5 km/s in a thin transition zone, which leads to three arrivals (triplication) from Layer 2A (Fig. 5). However, since the first arrival tomography is performed on OBS data, spaced at 5–10 km distance, one can only get a smooth velocity of the crust. Therefore, a velocity contour of 4.5 km/s is generally used at the base of Layer 2A (Fig. 4, dashed curve). In seismic reflection data, the triplication from the Layer 2A/2B boundary lies in a narrow offset range at intermediate offsets, depending on the water depth and the thickness of Layer 2A (Fig. 5) (Harding et al. 1993; Seher et al. 2010a). Since the cusp of triplication has strong amplitude, it can be used to image the base of Layer 2A, and leads to a very nice image of Layer 2A. Since the energy lies in a limited offset range, the twoway travel time for these images varies depending on the velocity used for stacking. This becomes particularly important when the thickness map needs to be prepared from 3D seismic reflection data or compared along long profiles. In order to objectively compare Layer 2A

Velocity (km/s) 0

2

Layer 2B

4

Layer 3

6

Moho Mantle

8

10

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 4 Average one-dimensional velocity of oceanic crustal layers and mantle (solid brown line). Thin dashed line indicates velocity determined using tomographic type techniques

thickness, a constant velocity that provides the best image of Layer 2A is used to stack Layer 2A arrival and the two-way travel time is converted into depth using the tomographic velocity obtained using either OBS data or streamer tomography (Seher et al. 2010a). Plot Triplication Figure 6 shows the image of Layer 2A obtained along the southern East Pacific Rise (Kent et al. 1994). The thickness of the layer varies from ~200 m on the ridge axis and increases to 600 m off-axis (Kent et al. 1994; Harding et al. 1993). The thickening of Layer 2A off-axis could be associated with the

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Crustal Reflectivity (Oceanic) and Magma Chamber

1

Offset (km) 2 3

3.0 4

2 Reflection Data

Time (s)

Time (s)

Seafloor Layer 2A 4.0

AMC

3 2B 2A

4

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 5 Seismic reflection from the Lucky Strike Segment of MidAtlantic Ridge showing triplication (dashed curve) from Layer 2A/2B boundary. (Modified from Seher et al. 2010a)

accumulation of lavas away from the ridge axis. On slow spreading ridges, Layer 2A could be up to 1 km thick (Singh et al. 2006a; Seher et al. 2010a). Instead of thickening of Layer 2A away from the spreading center, its thickness decreases, which could be due to thinning by stretching (faulting) of the crust (Fig. 7). In fact, the thinnest (300 m) Layer 2A at the Lucky Strike segment is observed near the Median bounding faults (Seher et al. 2010a). There is still debate about the causes of Layer 2A reflection: it could be due to pillow lava and dike boundary as shown in Fig. 4 or due to alteration front associated with hydrothermal circulation or pressure collapse boundary (Christeson et al. 2007). Below Layer 2A, the velocity increases from 4.5 up to 6 km/s, corresponding to dike sequence, below which a magma chamber reflection might be present. The thickness of the dike sequence is ~1.5 km on a fast spreading center (Vera et al. 1990) and up to 2 km on a slow spreading center (Seher et al. 2010b).

Axial Magma Chamber (Melt Lens) On a fast spreading center, the axial magma chamber (AMC) reflector marks the boundary between the upper crust (lava and dikes) and the lower crust (gabbro). It is observed along a significant part of the fast spreading axis at 1.5–2 km below the seafloor. The width of the AMC varies from a few hundred meters to 4.5 km (Kent et al. 1993). The largest melt lens is observed beneath 9° N overlapping spreading center at the

AMC WIDTH: 650 M 5.0

W

approx. 4.0 km

E

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 6 Seismic reflection image of Layer 2A and axial magma chamber at 14° S East Pacific Rise (Position 3 in Fig. 2). (From Kent et al. 1994). The Layer 2A thickness varies from 200 m on the ridge axis to 400 m off-axis

East Pacific Rise (Kent et al. 2000; Singh et al. 2006a) (Fig. 8). Recently, Canales et al. (2009) have imaged an offaxis melt lens. A 3D seismic reflection study of the 9° N EPR suggests that there might be an extensive presence of melt sills off-axis (Carton et al. 2009). The AMC have also been observed beneath intermediate spreading centers such as Juan de Fuca Ridge (Canales et al. 2006) and Valu Fa Ridge (Collier and Sinha 1990). They are generally observed about 3 km below the seafloor. It has been difficult to image AMC reflection on slow spreading ridges, which has been due to strong scattering on the seafloor and complex 3D bathymetry. Using 3D seismic reflection technique, Singh et al. (2006b) discovered AMC beneath the Lucky Strike segment of the Mid-Atlantic Ridge (Fig. 9). The AMC is about 3 km wide and 7 km long at ~3 km below the seafloor. The thickness of the melt lens is difficult to determine. Forward modeling of seismic reflection data from 9° N at EPR suggests that it could be 50–100 m thick (Kent et al. 1993). Seismic full waveform inversion of data from two locations at the EPR suggests that it should be 50–60 m thick (Collier and Singh 1997; Singh et al. 1998, 1999). Using full waveform of seismic reflection data from 14° S at EPR, Singh et al. (1999) found that a 60 m thick melt lens is overlain by a 60 m thick solid roof and underlain by a 200 m thick solid floor. Above the roof layer, they find a 200 km thick low velocity layer, which they associate with the presence of hydrothermal circulation (Singh et al. 1999) (Fig. 10). The roof layer could be the transition zone between the hot melt (1,200 °C) below and the hydrothermal layer (400 °C) above. The presence of a solid floor suggests that magma cools and crystallizes in the melt lens and forms the solid floor.

Crustal Reflectivity (Oceanic) and Magma Chamber Lucky Strike Volcano F1

2000

Depth (m)

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 7 Seismic reflection of Layer 2A beneath the Lucky Strike volcano at slow spreading Mid-Atlantic Ridge (Position 1 in Fig. 2). F1 indicates an active fault. Layer 2A is thick beneath the ridges and thin beneath the fault

105

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4000

5000

6000

7000 8000 Across-axis distance (m)

9000

10000

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 8 Seismic reflection image of axial magma chamber (AMC) and Moho reflection at 9° N overlapping spreading center at the East Pacific Rise (Position 2 in Fig. 2). (Modified from Singh et al. 2006a)

Using 3D seismic reflection data, we can image the base of the melt lens. Here the melt lens is about 4.5 km wide and could be up to 250 m thick (as compared to 50–100 m observed elsewhere). This could be due to its very large size (4.5 km) and the presence of low velocity underneath it (Bazin et al. 2003). Singh et al. (2006a) suggest that up to 40% melt could be present between 1.5 and 6 km depth, making it the largest observed melt lens in the crust observed on the earth so far (Fig. 11).

Using full waveform and partial stacking technique of seismic reflection data, Singh et al. (1998) show that the melt along 14° S EPR consists of 2–4 km long pure melt zones at 15–20 km interval along the ridge axis (Fig. 10). They associate pure melt region where the shear wave velocity is nearly zero and partial melt zone where the shear wave velocity is nonzero (Taylor and Singh 2002). They suggest that a pure melt region corresponds to fresh supply of magma from the mantle and a partial melt region corresponds to

106

Crustal Reflectivity (Oceanic) and Magma Chamber

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 9 Axial magma chamber reflection image from the Lucky Strike Segment of the MidAtlantic Ridge (Position 1 in Fig. 2)

Dyke 0.6

0.6

Layer

0.8

S-wave

1.0

P-wave

AMC

0.8

S-wave

Solid proof Mush

Melt

15–20 km

2–4 km

Depth (km)

Depth (km)

Hydrothermal

P-wave

AMC

1.0

Solid Floor 1.2

1.2 0

2 4 6 Velocity (km/s)

0

2 4 6 Velocity (km/s)

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 10 Full waveform inversion results at two different locations along the 14° S EPR (Position 3 in Fig. 2). Inversion results show the presence of 60 m AMC, 60 m thick roof, and 200 m thick floor of the AMC. It also shows a hydrothermal layer above the roof of the AMC. The result on the

left panel corresponds to a partial melt region (less than 50% of liquid melt) whereas that on the right corresponds to a pure melt (more than 80% of liquid) region. The depth is from the seafloor. (Modified from Singh et al. 1999)

cooled and crystallized state of melt lens. They also find that a pure melt region is associated with hydrothermal activities on the seafloor, linking the supply of magma from the mantle with hydrothermal circulation on the seafloor. In order for the

melt lens to be in steady state, it would require a supply of magma from the mantle every 30 years as it will take 50 years to completely solidify a 50 m thick melt lens (Singh et al. 1999).

Crustal Reflectivity (Oceanic) and Magma Chamber

107 Along-axis distance (km) –20

–10

0

10

20

Depth (km bss)

0

–4

C –8 Moho

Low velocity anomaly

–12

2

3

4

5

6

7

8

Velocity (km/s)

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 11 Seismic reflection image showing the top and bottom of the axial magma chamber at 9° N EPR (Position 2 in Fig. 2)

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 12 Tomographic results from the Lucky Strike Segment of the Mid-Atlantic Ridge (Position 1 in Fig. 2) showing melt lens (red line), low velocity in the lower crust, and Moho across the ridge axis. The inverse triangle is the center of the ridge axis. (Modified from Seher et al. 2010c)

Oceanic Moho Lower Crust So far, no other crustal melt lenses have been observed beneath the axial melt lens. Therefore, only tomographic methods using OBS data are used to determine the velocity structure of the lower crust. Using two-ship expanding spread profile, Vera et al. (1990) and Harding et al. (1989) found that the velocity in the lower crust (5.5–6 km/s) on-axis was lower than that off-axis (6.8–7 km/s), which suggested the presence of partial melt in the lower crust. Using 3D tomography technique, Toomey et al. (1990) and Dunn et al. (2000) showed the presence of low velocity in the lower crust, suggesting the presence of partial melt. Singh et al. (2006b) found a large anomaly in twoway travel time between the Moho reflection and AMC and suggest that up to 40% of melt might be present in the lower crust (Fig. 8). A low velocity anomaly has also been observed beneath the Lucky Strike segment of the Mid-Atlantic Ridge beneath the melt lens (Seher et al. 2010c; Singh et al. 2006b). These results suggest that partial melt is present in the lower crust beneath the melt lens (Fig. 12). Based on the study of Oman ophiolite, Kelemen et al. (1997) and Boudier et al. (1996) suggested that the lower crust is formed by injection of melt sills in the lower crust instead of cooling and crystallization of magma in the upper crustal melt sill. However, no melt lenses have been imaged in the lower crust so far, even using 3D seismic reflection technique (Singh et al. 2006a), which suggest that the melt in the lower crust must be in small pockets, not in large melt sills.

The Moho is a boundary between the crust and mantle, where the velocity changes from 6.8–7.0 km/s to 7.8–8.1 km/s. For a sharp boundary, a clear Moho reflection is observed. However, the Moho is not just a simple boundary. Kent et al. (1994) identified three types of Moho reflection: (1) Impulsive Moho where a clear single reflection is observed (Fig. 13), (2) Diffuse Moho where reflection from Moho is patchy, and (3) Shingled Moho where reflection is shingled. Moho reflections are generally observed away from the spreading center, not beneath the melt lens. This is because the upper crustal melt lens and associated lower crustal melt would attenuate seismic energy and hence, it would be difficult to image Moho using conventional seismic reflection techniques. Secondly, it was accepted that Moho is formed away from the ridge axis. However, using 3D seismic reflection technique, Singh et al. (2006a) have imaged Moho beneath the wide axial melt lens (Fig. 14), suggesting the Moho is formed at zero age. Moho reflections are observed ~2 s two-way travel time below the seafloor. The average crustal velocity is about 6 km/s, and hence the average crustal thickness is about 6 km. However, Singh et al. (2011) have observed Moho reflection 1.3 s below the seafloor offshore NW Sumatra, where the crust was formed at the fast spreading Wharton ridge about 55 Ma ago. If we assume an average velocity of 6.0 km/s, which means at the crust there is only 3.5 km thick, it would be the thinnest crust ever observed in a fast spreading environment (Fig. 15). Since there is a positive velocity contrast at the crust– mantle boundary, large amplitude critical angle reflections

108

Shingled Moho Reflection

5.5

6.0 Two way travel time

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 13 Different types of reflection for Moho boundary observed at Southern East Pacific Rise (Position 3 in Fig. 2) (Kent et al. 1994). Top: Shingled Moho, Middle: Diffuse Moho, Bottom: Impulsive Moho

Crustal Reflectivity (Oceanic) and Magma Chamber

Diffuse Moho Reflection

5.5

6.0 Impulsive Moho Reflection

5.5

6.0

3000

4000

Time (ms)

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 14 Seismic reflection image of Moho beneath the melt lens at the 9° N East Pacific Rise (Position 2 in Fig. 2)

approx. 2.5 km

5000

6000

N 16000 Al

on

g–

E

W S

12000 Ax

is

(m

)

16000

8000

xis

12000

ro

Ac

(PmP) are generated that arrive at large offsets (15–30 km), and are recorded on OBS. These data are then used to map the Moho structure. However, there are two problems in using arrival times of PmP arrivals. The velocity gradient in the lower crust is very small or close to zero, and therefore rays that sample the lower crust do not turn in the lower crust making it difficult to estimate velocity in this zone. Since the arrival time of PmP arrival depends on the velocity in

)

(m

A ss–

the crust, particularly the lower 3 km of the crust, it is difficult to estimate both the velocity in the lower crust and the Moho structure. Since the velocity in the lower crust can be approximated to a large extent ~6.8 km/s, these data provide a reasonable constrain on the Moho structure, and are routinely used. Secondly, the OBS spacing is generally 5–20 km and the lateral resolution of the Moho structure is very poor.

Crustal Reflectivity (Oceanic) and Magma Chamber

Crustal Thickness Once we determine the velocity in the crust and Moho structure, we can estimate the crustal thickness. So far, most of the crustal thickness estimations are based on wide-angle seismic studies, and there are significant uncertainties in crustal thickness estimations for the reason explained above. However, combined analyses of seismic reflection, refraction, and gravity data suggest that wide-angle estimation of the crustal thickness can be very reliable. The crust formed at fast spreading centers is generally uniform and 6 km thick (White et al. 1992; Eittreim et al. 1994) but some variations (5–8 km) have been observed recently (Canales et al. 2003; Singh et al. 2006a; Barth and Mutter 1996). There is no significant crustal thickness variation across the fracture zone in a fast spreading environment, where it is also 5.5–6 km thick (Van Avendonk et al. 2001). The crust formed at slow spreading centers is generally thick at the center of the segment (6–7 km) and thin at segment ends (4–5 km) (Barclay et al. 1998; Seher et al. 2010b). The crust beneath fracture zones in a slow spreading environment is generally thin (3–4.5 km) (Detrick et al. 1993). The thinnest crust (3.5–4.5 km) is formed at ultraslow spreading centers (Jokat and Schmidt-Aursch 2007). These observations

109

suggest that the thickness of the crust depends on the spreading rate and melt distribution in the crust. On the other hand, thicker crusts are found beneath large igneous provinces, across hotspot tracks or at the interaction of plume and ridges, which are believed to be formed by higher mantle temperatures due to the presence of a plume (e.g., Parkin and White 2008; Watts et al. 1985; Charvis et al. 1999; Grevemeyer et al. 2001). For example, the crust beneath Iceland could be up to 30 km thick, and beneath La Reunion Island it is 13 km (Charvis et al. 1999). However, anomalously thin crust has been reported. A 5 km thick crust is observed near the South American trench (Grevemeyer et al. 2007) and in the area of IODP Hole 1256 (Hallenborg et al. 2003) for crusts formed at the EPR about 20 Ma. Rodger et al. (2006) found an ultrathin crust (4 km) that was formed near the central MAR about 85 Ma ago, which they associate to be due to a reduction in the spreading rate from slow to ultraslow. The crustal study at ODP holes 504B also found thin crust (5 km), which Collins et al. (1989) associate to be due to depleted mantle. Singh et al. (2011) have observed ultrathin crust (3.5–4.5 km) in the Indian Ocean (Fig. 15) that was formed at fast spreading Wharton center 55 Ma ago, which they associate to be due to the interaction of the Kerguelen plume with the Wharton

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 15 Seismic reflection image of extremely thin crust in the Indian Ocean (Position 4 in Fig. 2). (Modified from Singh et al. 2011)

C

110

Crustal Reflectivity (Oceanic) and Magma Chamber

Spreading center due to channeling of the cold lithosphere around a plume and its interaction with the spreading center.

NW 25000

SE 23000

5

Upper Mantle

10

Depth (km)

The velocity in the upper mantle is generally determined using rays that turn below the Moho, which are called Pn arrivals. The velocity in the mantle beneath ridge axis is ~7.6–7.8 km/s and is 8–8.1 km/s away from the ridge axis (Vera et al. 1990; Harding et al. 1989). An extensive 3D tomography and undershoot experiment were carried out at 9° N EPR. Using these data, Dunn et al. (2000, 2001) obtained a symmetric low velocity anomaly below the Moho in the mantle. Using the same data and some new data Toomey et al. (2007) found asymmetric low velocity anomaly at 10 km away from the ridge axis. There are serious problems with these results. First, the crustal thickness in this area varies from 6 to 8 km. Secondly, the velocity anomaly depends on the background velocity used during the inversion. Dunn et al. (2000) use a starting mantle velocity of 8.2 km/s whereas one year later Dunn et al. (2001) use 7.6 km/s. Although the crustal thickness is up to 8 km (Canales et al. 2003), Dunn et al. (2001) interpret velocity at 7 km below the seafloor; the part of the anomaly could be due to crustal variations. Toomey et al. (2007) show that the ray only penetrates down to 8 km below the seafloor but interpret the velocity at 9 km below the seafloor. These conflicting results and the inaccuracy in inversion led Singh and Macdonald (2009) to suggest that these results are not reliable. The only robust solution we have is that the velocity in the mantle is ~7.6–7.8 km/s below the ridge axis and 8–8.1 km/s away from the ridge axis (Vera et al. 1990; Harding et al. 1989). However, it is possible that melt sills get trapped in the mantle as shown by Nedimovic et al. (2005) and may lead to a low velocity anomaly in the mantle.

CMP number 24000

15

30

25

30

10 km

V.E. = 1

Crustal Reflectivity (Oceanic) and Magma Chamber, Fig. 16 Seismic reflection of deep penetrating faults in the mantle in the Indian Ocean (Position 4 in Fig. 2)

5–8 km below the Moho, related to bending of the subducting plate and the associated serpentinization, has been reported for the Central American trench (Ranero et al. 2003). Faults have also been imaged in the mantle offshore Sumatra down to 35 km depths (Fig. 16) suggesting that a significant part of the oceanic lithosphere is involved during oceanic earthquakes.

Summary Faults in the Oceanic Crust and Upper Mantle Along with magmatic process, tectonic processes also play an important role in shaping the oceanic crust. Singh et al. (2006b) have shown that median valley bounding faults could be imaged down to 3 km, close to the melt lens. They also found extensive faults above the melt lens. It is possible that some of these faults penetrate down to the crust–mantle boundary (Dusunur et al. 2009). In a subduction zone environment, there are two forces that act on the incoming oceanic plate: plate bending leading to normal faulting and compressive forces leading to thrust faulting. Water can enter into the mantle leading to serpentinization of the oceanic mantle. Normal faulting down to

The reflectivity of the oceanic crust and axial magma chambers at ocean spreading centers are described in detail using real seismic images. The oceanic crust consists of three distinct layers: Layer A is 200–1,000 m thick and consists mainly of pillow lavas. Layer B is 1–2 km thick and consists mainly of cooled basalts in dikes. Layer 3 is 2–3 km thick and is formed by the cooling and crystallization of magma in the melt lens. Axial melt lens has been imaged on fast and intermediate spreading centers and recently on a slow spreading center (Singh et al. 2006b). Melt lenses are 50–60 m and have a roof and a floor. Between the melt and Moho, lower crust, partial melt is generally present. The oceanic crust is about 6 km thick, which could vary from 3.5 to 8 km. The

Crustal Reflectivity (Oceanic) and Magma Chamber

P-wave velocity in the mantle lie between 7.6 and 8.1 km/s, but the detailed nature of the oceanic mantle is poorly constrained and requires further investigations. Faults due to earthquakes have been observed down to 35 km depth.

Cross-References ▶ Continental Crustal Structure ▶ Deep Seismic Reflection and Refraction Profiling ▶ Lithosphere, Oceanic ▶ Ocean Bottom Seismics ▶ Ocean, Spreading Center ▶ Seismic Data Acquisition and Processing ▶ Seismic Structure at Mid-Ocean Ridges ▶ Single and Multichannel Seismics

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112 Parkin CJ, White RS (2008) Influence of the Iceland mantle plume on the oceanic crustal generation in the North Atlantic. Geophys J Int 173:168–188 Ranero CR, Reston TJ, Belykh I, Gnibidenko H (1997) Reflective oceanic crust formed at a fast spreading centre in the Pacific. Geology 25:499–502 Ranero CR, Phipps Morgan J, McIntosh K, Reichert C (2003) Bendingrelated faulting and mantle serpentinization at the Middle America trench. Nature 425:367–373 Rodger M, Watts AB, Greenroyd CJ, Peirce C, Hobbs RW (2006) Evidence for unusually thin oceanic crust and strong mantle beneath the Amazon Fan. Geology 34:1081–1084 Seher T, Crawford W, Singh SC, Cannat M (2010a) Seismic layer 2A variations in the Lucky Strike segment at the Mid-Atlantic Ridge from reflection measurements. J Geophys Res 115:B07107. https:// doi.org/10.1029/2009JB006783 Seher T, Singh SC, Crawford W, Escartin J (2010b) Upper crustal velocity structure beneath the central Lucky Strike Segment from seismic refraction measurements. Geochem Geophys Geosyst 11:5. https://doi.org/10.1029/2009GC002894 Seher T, Crawford W, Singh SC, Cannat M, Combier V, Dusunur D, Canales J-P (2010c) Crustal velocity structure of the Lucky Strike segment of the Mid-Atlantic Ridge (37°N) from seismic refraction measurements. J Geophys Res 115:B03103. https://doi.org/10.1029/ 2009JB006650 Singh SC, Macdonald K (2009) Mantle skewness and ridge segmentation. Nature 458:E11–E12 Singh SC, Kent GM, Collier JS, Harding AJ, Orcutt JA (1998) Melt to mush variations in crustal magma properties along the ridge crest at the southern East Pacific Rise. Nature 394:874–878 Singh SC, Collier JS, Kent GMJS, Harding AJ, Orcutt JA (1999) Seismic evidence for a hydrothermal layer above the solid roof of axial magma chamber at the southern East Pacific Rise. Geology 27:219–222 Singh SC, Harding A, Kent G, Sinha MC, Combier V, Hobbs R, Barton P, White R, Tong V, Pye J, Orcutt JA (2006a) Seismic reflection images of Moho underlying melt sills at the East Pacific Rise. Nature 442:287–290 Singh SC, Crawford W, Carton H, Seher T, Combier V, Cannat M, Canales J, Dusunur D, Escartin J, Miranda M (2006b) Discovery of a magma chamber and faults beneath a hydrothermal field at the MidAtlantic Ridge. Nature 442:1029–1033 Singh SC, Midenet S, Djajadihardja Y (2009) Sesimic evidence of the locked and unlocked Sumatra subduction zone. Eos 90:471–472 Singh SC, Carton H, Chauhan A et al (2011) Extremely thin crust in the Indian Ocean possibly resulting from Plume-Ridge interaction. Geophys J Int. https://doi.org/10.1111/j.1365-246X.2010.04823.x Taylor M, Singh SC (2002) Composition and microstructure of magma bodies from effective medium theory. Geophys J Int 149:15–21 Toomey DR, Purdy GM, Solomon SC, Wilcock WSD (1990) The threedimensional seismic velocity structure of the East Pacific Rise near latitude 9°30’ N. Nature 347:639–645 Toomey DR, Jousselin D, Dunn RA, Wilcock WSD, Detrick RS (2007) Skew of mantle upwelling beneath the East Pacific Rise governs segmentation. Nature 446:409–414 Van Avendonk HJA, Harding AJ, Orcutt JA, McClain JS (2001) Contrast in crustal structure across the Clipperton transform fault from travel time tomography. J Geophys Res 106:10961–10981 Vera EE, Mutter JC, Buhl P, Orcutt JA, Harding AJ, Kappus ME, Detrick RS, Brocher TM (1990) The structure of 0 to 0.2 m.y. old oceanic crust at 9° N on the East Pacific Rise from expanding spread profiles. J Geophys Res 95:15529–15556 Watts AB, ten Brink US, Buhl P, Brocher TM (1985) A multi-channel seismic study of lithospheric flexure across the Hawaiian Emperor seamount chain. Nature 315:105–111 White RS, McKenzie D, O’Nions RK (1992) Oceanic crustal thickness from seismic measurements and rare earth element inversions. J Geophys Res 97:19683–19715

Curie Temperature

Curie Temperature Vincenzo Pasquale Department of Earth, Environment and Life Sciences, University of Genoa, Genoa, Italy

Synonyms Curie point

Definition A rock containing magnetic minerals loses its permanent magnetism when heated up to a critical temperature, referred to as Curie temperature or Curie point, Tc. On the atomic level, below Tc, the magnetic moments are aligned in their respective domains, and even a weak external magnetic field results in a magnetization. As the temperature increases to Tc and above, fluctuations due to the increase in thermal energy destroy that alignment, and the rock becomes paramagnetic.

Magnetic Minerals and Curie Temperature The most important magnetic rock-forming minerals are oxides of iron and titanium, and their compositions can be represented in the FeO-TiO2-Fe2O3 ternary diagram. Among the three major solid-solution series identified in this system, the titanomagnetite series, Fe3-xTixO4 (with 0  x  1), plays a major role in controlling rock magnetism (Stacey 1992). Above 600 °C there is a continuous solid solution in the twocomponent series magnetite (Fe3O4) and ulvöspinel (Fe2TiO4), which upon cooling is restricted toward the end-members. Magnetite, the most abundant and strongly magnetic mineral, is of ferrimagnetic type, i.e., neighboring magnetic moments are aligned antiparallel, as in antiferromagnetism, but unequal numbers or strengths, thus giving a net magnetization. There are many other magnetic minerals, but they are rare (iron sulfide), unstable (maghemite), or having a weak spontaneous magnetization due to the canting of its equal and nearly opposite atomic moments (canted antiferromagnetism), as hematite occurring in sedimentary rocks often in solid solution with ilmenite. Variation of Curie temperature with titanium substitution in the titanomagnetite series is approximately given by Tc ¼ 580 (1–1.26 x). In pure magnetite, Tc is 575–585 °C, but titaniferous inclusions can reduce Tc, which approximates room temperature for x ¼ 0.77 (Hunt et al. 1995). Saturation magnetization is also a function of temperature, and it disappears above Tc; at room temperature it decreases from 90 to 92 A m2 kg1 for x ¼ 0 to about zero for x ¼ 0.8. Because Ti4+

Curie Temperature

has no unpaired spins, the saturation magnetization decreases with increasing x (Alva-Valdivia and López-Loera 2011). The cell dimensions increase with increasing x. As a result of the increased cell dimension, there is a decrease in Curie temperature. Tc undergoes a small increase with pressure. At the boundary between the crust and the mantle (or Moho), the Tc increase should be no more than a few degrees of the experimental values at normal pressure.

Geomagnetic Implications There is a connection between the anomalies of geomagnetic field and the temperature-depth distribution, since rock magnetization is strongly affected by temperature variations. The rocks lose their magnetization at a depth where the temperature is greater than Tc and, consequently, their ability to generate detectable magnetic anomalies disappears. By transforming magnetic anomaly data into the Fourier domain and analyzing their spectra, it is possible to infer the depth of the magnetic layer bottom (MLBD), i.e., where magnetic rocks are replaced with nonmagnetic material. Its comparison with information about thermal state of the Earth’s crust can provide interpretation for both the magnetic source distribution and the Curie temperature. Chiozzi et al. (2005) gave an example of MLBD determination and its relation with Moho depth, thermal structure, and geothermal flow in Central and Southern Europe (Table 1). MLBD corresponds to the Moho in the Variscan units, and the expected temperature is close to Tc of magnetite. Beneath the Alpine units, the magnetic layer bottom is shallower than the Moho and corresponds to temperature, as inferred from the geothermal flow, of 550 °C, indicating the presence of Ti. In the Ligurian basin, the temperature at the MLBD minimum depth is of 570 °C, hence compatible with the Curie point of magnetite. However, this depth is slightly larger than the Moho. Like observed in some oceanic regions (Counil et al. 1989), this might mean that the uppermost part of the mantle has magnetization.

113 Curie Temperature, Table 1 Moho depth, depth and temperature of magnetic layer bottom, and geothermal flow of main tectonic units in Central and Southern Europe

Tectonic unit Variscan units (Central Europe, CorsicaSardinia block) Alpine units (Alps, Apennines, Molasse and Po basins) Ligurian basin

Moho depth (km) 29–33

Magnetic layer bottom Depth Temperature (km) (°C) 29–33 540–580

Geothermal flow (mW m2) 60–70

25–50

22–28

500–600

50–80

17–24

20–24

570–650

80–100

Cross-References ▶ Continental Crustal Structure ▶ Geomagnetic Field, Global Pattern ▶ Geomagnetic Field, Theory ▶ Heat Flow, Continental ▶ Magnetic Domains ▶ Remanent Magnetism

Bibliography Alva-Valdivia LM, López-Loera H (2011) A review of iron oxide transformations, rock magnetism and interpretation of magnetic anomalies: El Morro mine (Brazil), a case study. Geofis Int 50:341–362 Chiozzi P, Matsushima J, Okubo Y, Pasquale V, Verdoya M (2005) Curie-point depth from spectral analysis of magnetic data in Central-Southern Europe. Phys Earth Planet Inter 152:267–276 Counil JL, Achache J, Galdeano A (1989) Long-wavelength magnetic anomalies in the Caribbean: Plate boundaries and allochthonous continental block. J Geophys Res 94:7419–7431 Hunt CP, Moskowitz BM, Banerjee SK (1995) Magnetic properties of rocks and minerals. In: Ahrens TJ (ed) Rock physics and phase relations: a handbook of physical constants. American Geophysical Union, Washington, pp 189–204 Stacey FD (1992) Physics of the earth, 3rd edn. Brookfield Press, Brisbane

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Deep Scientific Drilling Ulrich Harms1 and Harold Tobin2 1 Scientific Drilling, GFZ German Research Centre for Geosciences, Potsdam, Germany 2 Department of Earth and Space Sciences, University of Washington, Seattle, WA, USA

Definition Scientific Drilling

Deep Scientific Drilling

Applying drilling methods developed in the hydrocarbon and mineral exploration industry to dig holes and retrieve samples of rock and fluid for scientific, not-for-profit purposes. Drilling for scientific research goals beyond near-surface geological and shallow groundwater exploration to depths greater than several hundred meters subsurface, through land, sea ice, or below the sea/lake floor.

Introduction Geoscientific insight into the deep Earth is mostly based on observations from the surface, indirect evidence through deep geophysical investigations or, increasingly, modeling. However, several critical Earth system processes and records of environmental and paleoclimate change are difficult to observe from ground level only. Hence, despite the great progress achieved in solid Earth science of the past decades, truly ground-truthed knowledge of the dynamics of Earth’s crust is limited. Today, direct access to the interior of the Earth’s crust is still confined almost exclusively to sedimentary basins where hydrocarbon resources are recovered through deep industry-financed wells, if typically shallow exploration and mining for groundwater and minerals are neglected. Deep © Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

drilling for mainly scientific reasons has so far remained a relatively rare exception because the very high cost of drilling is a challenging barrier in the design of research projects. Nevertheless, this threshold can be overcome if societal interests are pressing, if political and financial support is gained, and typically if international long-term cooperation is achieved. A small number of international Earth science programs devoted to scientific drilling have been established with widely agreed-upon strategic scientific goals, international co-funding, and a necessary minimum of infrastructure and operational capabilities. While the overarching research goals are almost identical, the drilling programs are separated, based on largely technological grounds, into oceanic (Integrated Ocean Discovery Program 2011) and continental (International Continental Scientific Drilling Program, Horsfield et al. 2014).

Goals of Scientific Drilling The international programs for scientific drilling address topics of broad international interest which can contribute significantly to solving important Earth science themes of high societal relevance (Fig. 1). The research themes of these programs can be summarized in three overarching branches: • The evolution and dynamics of Earth’s crust and lithosphere, especially in subduction zones and orogenic belts, with special emphasis on the physicochemical background of geological hazards such as earthquakes, volcanic eruptions, landslides, and meteorite impacts. • Environmental, climate, and paleoceanographic changes as recorded in geological archives of sediments in the oceans, in lakes, in ice shields and in other depositional environments, including the resulting biological consequences, as well as the exploration of the widely unknown microbial biosphere living at depth.

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Solar radiation Climate dynamics Global environments Volcanic systems Evolution of life

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Deep Scientific Drilling, Fig. 1 Sketch of research themes addressed by major international scientific drilling programs

• Basic science of unconventional resources such as methane hydrates or geothermal energy, and of novel utilization of the deep underground environment such as CO2sequestration.

This chapter addresses mainly the first topic on lithospheric dynamics, hazard, and evolution of the Earth because most deep scientific drilling projects are dealing to a large extent with questions about internal forcing, whereas climate, environmental, and subsurface biosphere research primarily employs shallow drilling techniques.

Drilling, Sampling, and Monitoring Techniques A standard in most deep drilling operations (e.g., hydrocarbon exploration) is the rotary drilling technique. A cutting bit is lowered into the ground on a steel tube rotated through a rotary drive in a drilling derrick. The propelled drill string consists of connected pipe elements through which a drilling fluid is pumped down the well. The drill mud, usually water with clay minerals and other added solids to provide viscosity, cools the bit and flushes out chips of the destroyed volume of rock (drill cuttings) to the surface through the annulus between the borehole wall and the drill string. The weighton-bit is controlled through adjusted variable lifting of the string which is pulled down and kept in tension by heavy

weight sections of drill pipe installed right above the drill bit. The drilling progress is supported by extending the string in the derrick with additional sections, or stands, of pipe. When drilling from a barge, ship or floating platform, the annulus extends only to the sea/lake floor, so mud and cuttings do not return to the drill rig. In this setting, drilling can be performed with water in place of drilling mud, and cutting spilling out on sea or lake bottom around the well. However, if pressure control and mud return is required, an outer second pipe, a so-called “riser,” is put in place, so the mud and cuttings can be pumped back to the deck. Furthermore, in waves or swell, a heave compensation device is required to ensure constant weight-on-bit. Drilling through tidally lifted ice necessitates defrosting of the pipe in the ice section. Figure 2 summarizes basic drilling methods and important elements used in scientific drilling. In normal rotary drilling technique, coring is performed with a hollow core bit that leaves a central column of rock that slides into the pipe while drilling progresses. After a few meters of coring, the whole assembly has to be pulled back out of the hole (called “pipe tripping”) to get the core to surface. In scientific drilling projects, by contrast, continuous coring is often desirable. To avoid time-consuming pipe tripping runs, wireline coring techniques are applied. An inner core barrel is lowered through the drill string and latches in above the core bit, collecting and holding the drilled-out rock column, and is finally retrieved by a winch-drawn steel cable, obviating the need for a complete pipe round trip with each

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Deep Scientific Drilling, Fig. 2 Principle technical components and styles of drilling operations in different environments

core. The actual formation-cutting method varies depending on the type of rock or sediment present. Typically, thin-kerfed diamond core bits with high-rotation speed are used for hard rock drilling, roller cone abrasion bits are used for softer sedimentary rock, and nonrotating sharp edged hollow metal pistons of several meters length are hydraulically shot (forced) into soft sediments to collect cores and advance the borehole (Fig. 3). After research wells are drilled, in situ measurements and monitoring of active processes in rocks and fluids allow for unprecedented data acquisition undisturbed by surface processes. A classic example is seismometer recordings in boreholes whose resolution and noise floor is improved by orders of magnitudes in comparison to surface measurements (Abercrombie 1997). Sealed long-term borehole observatories have been installed in many places on land and at sea that are designed to record changes in temperature, pressure, strain, tilt, acceleration, and hydrologic parameters, and to sample fluids or microbiology (e.g., Fischer et al. 2008; Kastner et al. 2006; Prevedel and Kück 2006; Araki et al. 2017).

Dynamics of Continental Crust An unprecedented and still standing depth record in ultradeep scientific drilling was achieved in 1989 in the former Soviet Union when the Kola Superdeep Borehole reached 12,261 m

after 25 years of drilling (Kozlovsky 1987; Fuchs et al. 1990). Temperatures of less than 200 °C, low tectonic stress, and special lightweight aluminum tools were key to this success. The well intersected Paleoproterozoic to Archean metavolcano-sedimentary sequences and plutonic rocks in the Kola Peninsula of the northeastern Baltic Shield. The main scientific targets were the origin and extent of ore deposits as well as the principal structure and composition of the deeper crust. Important findings include – beside the falsification of a hypothesized basaltic layer in an onion-skin-like deep crust – the discovery of high permeability and circulating fluids throughout the crust. The German Continental Deep Drilling Program (KTB) aimed at principal physical and chemical processes and the geological evolution of the mid-European continental crust in an amalgamated collision zone (Emmermann and Lauterjung 1997). After 4 years of drilling of high-grade metamorphic rocks, a final depth of 9101 m was reached in 1994 at formation temperatures of ~270 °C. A Paleozoic suture zone cut by the borehole was found to be strongly deformed by postorogenic Mesozoic fault zones in an antiformal stack. Down to 9 km, faults contain saline, gas-rich free Ca-Na-Cl-fluids, the geothermal gradient increases from about 21 to 28 °C/km, and fluids percolate freely along extensive fracture pathways. A detailed correlation of geophysical data with petrophysical properties at depth allowed for the calibration of surface geophysics to in situ conditions. Furthermore, for the first

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Deep Scientific Drilling, Fig. 3 World map with important scientific deep drilling sites

time, the state of stress and the rheological behavior of an ultra deep crustal profile were analyzed in-situ. Determinations of the minimal and maximum horizontal principal stress between 3 and 9 km depth yielded a uniform stress direction except for a major fault at 7 km depth (Brudy et al. 1997). In conjunction with the deep main well, a 4000-m deep pilot hole just 200 m away allowed for the installation of a depth laboratory, for example, fluid level (Schulze et al. 2000) and induced seismicity experiments (Baisch et al. 2002). Ultrahigh pressure metamorphic (UHPM) rocks form the 1000-km-long Qinling-Dabie-Sulu Belt in China, which was formed in a Triassic continental plate collision zone. From 2002 to 2005, a 5158-m deep core hole of the Chinese Continental Scientific Drilling project served to retrieve a vertical sample profile of the protracted multiphase deep (>150 km) subduction and rapid exhumation history of UHPM rocks (e.g., Liu et al. 2007; Zhang et al. 2006). The complex metamorphic path and its timing could be determined specifically in high-pressure minerals which survived late stage fluid-enhanced retrogression and overprint encased in zircons and other refractory accessories (Fig. 3).

Volcanic Systems Volcanism is one of the most spectacular expressions of the dynamics of tectonic processes and the gigantic scale of

recycling working in the Earth. The scientific investigation of volcanic processes is also crucial for societal security and thus an important mission for science. Since most of the history of volcanism is encapsulated in rocks that are overlain by sediments, younger lava flows or beneath the sea, drilling is in many cases a critical tool for the investigation of the different types of magmatism reaching surface. Mid-ocean Ridge Magmatism As defined by Conference Participants (1972), ophiolites represent a suite of oceanic crust plus upper mantle. Beneath a sediment cover lava flows and pillow basalts overlay sheeted dike complexes, gabbroic rocks and, as part of the upper mantle, serpentinized peridotites and similar ultramafic rocks. Several holes drilled by the Deep Sea Drilling Program 1969–1985 and the Ocean Drilling Program 1985–1996 confirmed this structure but at the same time provided a much more complex picture. After several meters of lava flows were encountered in the early years of these programs, deeper parts of the oceanic plutonic rock succession were drilled in composite sections in tectonic windows and contributed much to the understanding of the ocean crust and mantle. Several drillholes in the Mid-Atlantic Ridge provided insight into slow-spreading modes with emplacement of ultramafic rocks. A 1508-m-deep hole (735B) penetrated a long section of gabbroic rocks of the slow-spreading Southwest Indian Ridge off Madagascar (Dick et al. 2000).

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The first lava-dike transition was drilled in 6 Ma age crust of the Nazca Plate off Ecuador (Hole 504B) down to 2111 m below seafloor (Alt et al. 1996). And the first complete section of the upper oceanic crust was recovered by the Integrated Ocean Drilling Program at Site 1256 in 2005 and 2006 when the lava-sheeted dyke (1061 m) and the sheeted dyke-gabbro (1407 m depth) transitions of the Cocos Plate off Costa Rica were penetrated. The hole was deepened to 1521.6 m in 2011. The 15 Ma crust was formed in the East Pacific Rise by superfast spreading producing a thick succession (750 m) of lava flows, a relatively thin sheeted dyke complex and an upper gabbro section of still unknown extent (Alt et al. 2007). As the drilled fractionated gabbros (1407–1521.6 m) are not the residues of the overlying volcanics, the full sequence of plutonic ocean crust and the Moho transition to the upper mantle remains to be detected in future ocean drilling (Ildefonse et al. 2007; Teagle et al. 2011).

160 °C, and no gas kicks appeared due to enhanced permeability and 9 years of convection and groundwater circulation after the last eruptions. At 1996 m depth, rocks were cored that could be identified as feeder dyke material (Nakada et al. 2005). In the next stage of scientific drilling into volcanic features, calderas will be explored because they form through collapse during super-eruptions – the most explosive volcanism on Earth. An outstanding example of an actively inflating caldera is the Campi Flegrei volcanic field in Naples, Italy (Troise et al. 2007). A shallow crustal magma chamber, strong geothermal activity with enormous release of volcanic gases, and recent episodes of unrest call for drilling as a tool to investigate the driving mechanisms. Similarly to the Unzen drilling, a deviated well is planned to be drilled from land under the Bay of Naples towards the center of seismic unrest in the heart of the structure at 3.5 km depth.

Active Volcano Drilling and Subduction-Induced Volcanism The most common and typically most dangerous types of volcanoes on Earth are aligned along the subduction-related arc systems. They are typically fed by highly viscous andesitic to rhyolithic magmas whose degassing behavior is critical for their eruption mode, either effusive or explosive. While erosion and tectonics provide detailed insight into solidified and cooled volcanic structures, little is known about the physical and chemical conditions or permeability close to and in feeder fissures and magma conduits of active volcanoes. Despite challenging engineering and safety issues, some attempts have been made to drill into active volcanism. Still notable because of temperatures of above 1400 °C is the shallow drilling of the Kilauea Iki lava lake on Hawaii in the 1970s (Hardee et al. 1981). Furthermore, in Eastern California, the recent, rhyolithic Inyo Lava Domes were sampled to understand degassing mechanisms (Eichelberger et al. 1986). A series of devastating eruptions with frequently occurring pyroclastic flows and vulcanian explosions in the early 1990s at Mount Unzen Volcano in southeastern Japan was accompanied by volcano-tectonic earthquakes and isolated tremor allowing for a geophysical pinpointing of the ascent path and surface appearance of magma. The identified conduit locale was within depth reachable by drilling, and justified drilling to sample in situ conditions after eruptions ended. Within the framework of the International Continental Scientific Drilling Program, the Unzen Volcano Drilling Project hit 2004 the conduit through an almost 2000 m long but 1500 m deep well (Sakuma et al. 2008). Deviated drilling steered by downhole motors from the northern flank of Unzen was utilized to meet environmental and safety restrictions. Large-diameter casings were cemented in place to allow mud circulation for cooling of high-temperatures which were modeled to reach up to 600 °C. However, maximum temperatures were as low as

Mantle Plume Volcanism Hot spots cut across lithospheric oceanic and continental plates and build up giant volcanic edifices in Large Igneous Provinces (LIP) and Ocean Island Volcanoes. Their build up requires vast melt extraction from the deep mantle and provides insight into mantle compositions, transport mechanisms, and the evolution of volcanic structures over space and time. Several boreholes of the Ocean Drilling Program opened windows into the Ontong-Java Plateau of the Western Pacific (Fitton et al. 2004) and the Kerguelen Plateau in the Southern Indian Ocean (Frey et al. 2000) to confirm their LIP origin. In the Hawaii Scientific Drilling Project, the Mauna Kea volcano was cored with 95% recovery to a total depth of 3520 m (3508 m below sea level) to sample the evolution of plume melts over about 700,000 years as the volcanoes have drifted with the Pacific Plate over the Hawaii Plume (Stolper et al. 2009). The formation temperatures increase only from about 10 to 50 °C in the borehole (19 °C/km gradient below 2000 m) due to deep circulation of seawater in a complex hydrological regime including freshwater inflow as deep as 3000 m. The circa 850 m subaerial and 2440 m submarine layered lava and hyaloclastite accumulations represent almost 400 lithological units such as individual lava flows. They show geochemical and isotopic heterogeneity with tholeiitic versus alkaline magma transitions. This bimodal distribution of source components indicates a radial zoning within the melting region with a hot inner zone which does not contain entrained mantle material and must be based in the lowermost mantle region (Bryce et al. 2005). A further drilling target are supercritical fluids at 4–5 km depth in the Iceland Deep Drilling Project (Fridleifsson and Elders 2007). Through coupled geothermal industry research and scientific investigation of in situ fluid-magma exchange, the first well at Krafla, Northern Iceland, hit rhyolithic magma

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at about 2000 m depth before the well was cased for testing (Elders et al. 2011). The Yellowstone-Snake River HOTSPOT drilling project documented the volcanic history of this mantle plume through three boreholes (1821, 1912, and 1958 m deep) that cut through basaltic and rhyolitic series (Shervais et al. 2013). Scientific drilling into volcanic rocks of the ocean crust provided the most important test and confirmation of the seafloor-spreading hypothesis. Samples retrieved from thickened ocean crust confirmed another type of seafloor volcanism forming Large Igneous Provinces which contribute large magma volumes to the continental crust as well. These LIPs seem to represent transient plume heads while the plume tails persist in Ocean Island Volcanoes. Hawaii served as pivotal example to drill deep into a single volcano to sample different parts of the plume and the volcanic history. Furthermore, recent penetrations into active volcanoes demonstrated the technical feasibility of such operations at very high temperatures. This access to deep volcanic processes is opening novel possibilities for monitoring.

Impact Craters Almost 180 craters on Earth are currently known that have been formed by astrophysical chance when celestial bodies collided with Earth. The effects on life and the environmental consequences of the destructive forces as well as the input of extraterrestrial material, including organic compounds, is an overarching question in Earth systems history. Together with geophysical illumination, deep drilling into impact structures provides crucial data to study cratering, informing models of the impactors’ size, impact angle, the resulting energy release through melting, evaporation, ejection, and, most importantly, provides data for charting the aftermaths of such dramatic events. The best studied example to date is the Chicxulub crater (Dressler et al. 2003) in which the approximately 10-km-large impactor penetrated the whole crust and caused upper mantle rebound and an exemplary drastic biotic turnover 65.5 Ma ago (Schulte et al. 2010). The impact ejected a spherule-rich ash layer all over the whole globe and released vast amounts of carbonate and sulfate gases previously bound in sedimentary rocks at the site of impact. The resulting darkness, cooling, geochemical cycle perturbations and sudden change of environment interrupted the food chain and led to the major extinction event at that time. Since the year 2002, four main impact craters have been studied by scientific drilling: the 200-km-wide CretaceousPaleogene Chicxulub Crater in Mexico was drilled on land and offshore (Hecht et al. 2004; Gulick et al. 2019), the 1 Ma old and 12 km-wide Bosumtwi Crater in Ghana (Ferrière et al. 2008), the late Eocene, 60 km Chesapeake Bay Crater in the Eastern United States (Gohn et al. 2008), and the 14 km Lake

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Elgygytgyn Crater in Chukotka, Sibiria (Brigham-Grette and Melles 2009). One of the important results is an improved understanding of the cratering process. Giant km-sized megablocks of target rocks were drilled in the lower sections of the huge Chesapeake and Chicxulub craters at 800–1700 m depth. In Chicxulub, uplifted basement collapsed outward and formed a peak ring that was first covered in brecciated impact melt and coarse-grained suevite rocks followed by well-sorted suevites. Resurge deposits and reflected rimwave tsunami layers sedimented atop within about a day. This impact rock layer is about 130 m thick and shows the extremely turbulent event sequence with, e.g., charcoal from impact induced wildfires in the tsunami layer (Gulick et al. 2019).

Active Fault Zones Since the late 1990s, researchers have undertaken a wide range of ambitious projects aimed at drilling into active faults in a range of tectonic settings, both on land and at sea. These projects have been motivated by the recognition that fault zone process are poorly understood from surface data, and that a combination of (a) samples of fault materials from present-day active systems in the subsurface and (b) access through boreholes to in situ measurements and monitoring of ambient conditions in active faults, can provide key new observations for learning about these processes. Our understanding of the mechanics and dynamics of plate boundary faulting is severely limited by a lack of information on ambient conditions and mechanical properties of active faults at depth. A major goal in earthquake mechanics research is thus direct in situ sampling and instrumentation by drilling into the seismogenic zone of active faults, including both inter- and intraplate fault systems in many tectonic settings. Understanding of the complex physics of tectonic faulting, earthquakes, and the generation of tsunami in the Earth’s crust is one of the grand challenges of the geosciences in general and seismology in particular (Lay 2009). Because earthquakes take place deep below the Earth’s surface, direct observation of processes acting in the faults themselves has generally not been possible; what we know about earthquake processes is derived from seismologic observations, geodetic deformation, and studies of exhumed rocks containing products of slip at depth long ago. Researchers have learned and inferred a great deal from these studies, but they are inherently limited. Much of fault zone drilling is aimed at understanding earthquake slip, but in fact fault physics encompasses a spectrum of slip and slip rates ranging from purely aseismic behavior, such as steady creep, through events such as slow earthquakes and episodic creep, to true seismicity. It is not yet clear how similar or distinct the physics of these modes of fault slip may be (Ide et al. 2007).

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Hence, some of the key questions addressed through fault zone drilling projects include the following: What are the deformation mechanisms of fault slip in both aseismic and seismic conditions? Why is slip sometimes seismic, sometimes aseismic, and sometimes in an intermediate state, and what governs the transitions among these processes? How do earthquakes nucleate, and how does rupture grow and stop? Are there precursory phenomena in the fault or near field that indicate a preparatory phase to earthquake or slip nucleation? If so, can a prediction or early warning strategy based on borehole observation be developed? What are the processes governing tsunamigenic slip? What is the role of fluid content and fluid pressure in modulating faulting processes, especially during rapid seismic slip? How does permeability of the fault zone evolve interseismically and coseismically? What is the stress tensor in and around a fault zone throughout the earthquake cycle or other slip? What controls localization of slip vs. distributed strain in faulting? Pursuing these questions, a number of major national and international efforts have been carried out or planned in recent years. These prominently include the San Andreas Fault Observatory at Depth (SAFOD), the Taiwan Chelungpu Fault Drilling Project (TCDP), and the Gulf of Corinth drilling on land, the Nankai Trough Seismogenic Zone Experiment (NanTroSEIZE), the Japan Trench Fast Drilling Program (JFAST), and Hikurangi subduction drilling at sea. Fault zone drilling projects are different in many respects from most other scientific drilling efforts. Rather than the complete stratigraphic intervals or sampling of broader rock volumes targeted by most other deep drilling, fault zone targets include discontinuities and small anomalous intervals. The targets of interest are often those locations of poorest drilling conditions, such as a highly fractured damage and gouge zone with potential for excess pore pressure above hydrostatic condition, anomalous stress state, weak rock, and other drilling hazards. The scientific approach commonly focuses on obtaining very extensive downhole measurements, logs, cores, and seismic structure in and around these fault zone environments. Measurements at the time of drilling as well as long-term observations are emphasized. Nojima Fault Projects, Japan Soon after the Kobe earthquake in 1995 (M6.9), several research teams drilled multiple boreholes into and across the shallow portion of the Nojima fault (the locus of that event) from ~750 m to ~1800 m depth (Ito et al. 1999). Detailed compositional and structural analyses identified a narrow fault core of gouge and surrounding damage zone, with asymmetric structure in the footwall and hanging wall (Boullier et al. 2001; Fujimoto et al. 2001; Ohtani et al. 2001; Tanaka et al. 2001). Shear wave splitting studies of aftershocks showed a rapid evolution of fast direction after the

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earthquake, interpreted as evidence of fracture healing during a 12 monthly post-seismic period (Tadokoro and Ando 2002). SAFOD, California The San Andreas Fault Observatory at Depth borehole was drilled from 2004 through 2007 into the San Andreas fault in central California at the northern end of the locked Parkfield segment and the creeping section of the fault zone (Williams et al. 2010). A vertical pilot hole was drilled in 2002 to circa 2 km depth (Hickman et al. 2004), and the main hole was later drilled in several phases in a “dogleg” deviated 55° across the steeply dipping San Andreas fault zone to a vertical depth of 2.6 km (Fig. 4; Hickman et al. 2007). The initial main borehole was logged and cased; repeated casing diameter logs over time showed that two intervals of casing several meters wide were actively deforming, interpreted as a manifestation of creep in two discrete fault zones (Hickman et al. 2007). In 2007, the hole was reentered and sidetracks were drilled to obtain core samples (Williams et al. 2010). A core interval of approximately 40 m was obtained from one of the two identified major fault zones and additional cores were collected from adjacent wall rock intervals. Borehole seismometers installed at SAFOD detected earthquakes as small as M2 within a few hundred meters of the source, allowing unprecedented highfrequency detection of source processes (Ellsworth et al. 2007). A notable result from SAFOD has been the absence of evidence for either a thermal anomaly from frictional heating around the fault core or an excess pore fluid pressure substantially above hydrostatic anywhere in or around the faults, as well as geochemical evidence that the fault zone is a permeability barrier to cross-fault flow (Hickman et al. 2007; Wiersberg and Erzinger 2007), simultaneously adding evidence for absolute fault weakness in the San Andreas and opening new questions about the source of that weakness. Taiwan Chelungpu Drilling Project, Taiwan The devastating 1999 M7.6 Chi-Chi earthquake in Taiwan was a thrust event in a collisional tectonic setting that exhibited very large slip (>10 m), including large surface displacements, on the Chelungpu fault. The principal objective of drilling was to sample the main slip surface or zone, if one could be identified, at a relatively short post-seismic interval. The fault zone was drilled in 2004 and 2005 in an area of high-velocity large slip during the main shock, and crossed the fault zone and several hundred meters of the footwall in two adjacent holes (Ma et al. 2006). The main slip zone and several subsidiary (or older) fault cores were identified in the borehole at ~1100 m depth (Hung et al. 2007). Studies of the fault gouge particle size, thickness, and composition have yielded quantitative estimates for the work done in forming gouge and damage (fracture energy), contributing to the understanding of energy budgets during earthquakes (Ma et al. 2006).

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Wenchuan Earthquake Fault Scientific Drilling Project (WFSD), Sichuan, China The devastating Wenchuan earthquake (M7.9) of May 12, 2008, caused at least 68,000 deaths. It occurred on a reverse fault in crystalline basement rocks, and slip propagated to the near surface. Only 178 days after the earthquake, the Wenchuan Fault drilling program was begun in order to access the co-seismic slip zone for sampling and in situ observations, in six boreholes along the Longmenshan Fault system. The boreholes elucidated a complex structure of the fault system, thermal pressurization as fault-weakening mechanism, an extremely low frictional coefficient for faulting coincident with graphite occurrence and a relationship between deep fluid appearances and seismicity (Xu and Li 2019). Geophysical Observatory in the Eastern Sea of Marmara, GONAF A prime example of earthquake-endangered megacities is Istanbul, Turkey, due to its close proximity to the extremely active North Anatolian Fault Zone that routes compartmentalized just a few kilometers south of the city center through the Marmara Sea. In contrast to the majority of other fault zone drilling projects the GONAF boreholes did not penetrate the fault itself but were utilized to deploy extremely sensitive

borehole seismometers and strainmeters down to 300 m together with surface stations as a permanent observatory of deformation around the Sea of Marmara (Bohnhoff et al. 2017). The acquired high-resolution data from depth have meanwhile been used to elucidate the coeval release of slow strain and enhanced local seismic moment (Martinez-Garzon et al. 2019). These accomplishments are leading to an ongoing expansion of the GONAF and related networks for monitoring of the whole Marmara region and beyond. Deep Scientific Drilling to Study Reservoir-Triggered Earthquakes in Koyna, India A key societal challenge in Earth science remains predicting earthquakes. There is an earthquake-prone zone in Western India where such forecasts are possible as artificial water reservoir-triggered earthquakes occur repeatedly after monsoon rains since 1962. While the region is seismically stable, rooted in Precambrian basement overlain by about 1200 m of Deccan Trap basalt flows, a secluded zone of recurrent seismicity stretches North-South of two man-made water reservoirs. Periodic earthquakes at depths between 2 and 7 km offer a unique opportunity to install an observatory to monitor mechanical and physicochemical properties of rocks and fluids in the near-field of these events (Gupta and Nayak 2011).

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The 3-km-deep pilot well KFD1 drilled in 2017 transects 1247 m of basalt and 1767 m of granitic-gneissic basement rocks to shed light on deformation processes. Borehole geophysical logs analysis revealed anomalous zones with stressinduced shear wave anisotropy indicating the orientation of maximum horizontal compressive stress related to the active subsurface fault damage zone (Goswami et al. 2019, 2020). For the next phase, it is planned to drill a main well towards the proper fault zone down to about 6 km depth to install a borehole laboratory at depth for direct and permanent monitoring of an intraplate seismic zone. Subduction Zone Fault Drilling at Sea For well over 30 years, drilling efforts have targeted active large offset thrusts and decollement faults in subduction zone settings, but these have mostly been in areas shallower than the depth of seismogenic strain accumulation and release. Beginning in the 1980s with the Barbados Accretionary Wedge at the Atlantic Caribbean plate boundary, drilling was aimed at understanding the faulting conditions, ambient pore fluid pressure, and mechanics in a very low angle (~2 degree dip) décollement thrust fault at the base of an accretionary wedge (Moore 1989). Drilling, downhole logging, and long-term borehole observatory measurements there demonstrated that the fault zone was a mechanically very weak interval with high permeability, high excess pore fluid pressure, and patchily distributed areas of anomalously high porosity (Moore et al. 1998; Fisher 2005). Drilling at Costa Rica off the Nicoya peninsula also elucidated the structure of a shallow decollement zone. Structural studies and geochemical tracers of hydrological flow paths suggest that two strongly decoupled systems exist above and below the ~20m-thick main fault zone, which acts simultaneously as a lowpermeability barrier to cross fault flow and a highpermeability channel for along-fault flow (Tobin et al. 2001). Installation of a sealed borehole monitoring system there documented an apparently weakly overpressured décollement zone that exhibits temporal pressure excursions tied to apparent aseismic transients recorded geodetically on land (Davis and Villinger 2006). Similarly, instrumented boreholes at the Nankai Trough off Shikoku Island in Japan show pressure transients associated temporally and spatially with swarms of very low frequency earthquakes, suggesting a long-distance (tens of km) poroelastic connection between the earthquake events and the boreholes (Davis et al. 2006). The same data exhibit long-term pressure trends interpreted as a response to gradual interseismic strain accumulation. Drilling to understand megathrust earthquake processes and fault zone properties has been a major focus of both the Integrated Ocean Drilling Program and the International Ocean Discovery Program (IODP), and was a primary justification for the construction of the riser drilling vessel Chikyu. Major fault drilling programs have been undertaken by IODP

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at the Nankai Trough, Sumatra, Japan Trench, Costa Rica, and Hikurangi (New Zealand) subduction zones. The spate of extremely large earthquakes that began with the Sumatra 2004 M9.2 and continued with the 2010 Maule M8.8, the Tohoku 2011 M9.1, and others has led to renewed urgency to investigate these events. Scientific ocean drilling has contributed to a new view of how fault locking, slip, and friction work. IODP Expedition 362 drilling at Sumatra showed that diagenesis of well-cemented sediments led to unexpected conditions of frictional instability to the trench for the 2004 event (Hüpers et al. 2017), facilitating large slip. Nankai drilling on Expedition 316 in 2007 revealed evidence of frictional heating of fault zone rocks during past slip, indicative of rupture reaching shallower levels of the megathrust than was thought possible theretofore (Sakaguchi et al. 2011). Beginning in 2007, IODP embarked on its most ambitious fault zone drilling project to date, to drill a transect of boreholes at the Nankai Trough, including one ultradeep site targeting the plate boundary fault at 5 km below the sea floor. The Nankai Trough Seismogenic Zone Experiment (NanTroSEIZE) project represents the first attempt to drill, sample, and instrument the seismogenic portion of a plateboundary fault or megathrust within a subduction zone, where the 1944 M8.1 Tonankai earthquake slip zone is located, and where more recently discovered very low frequency earthquakes, seismic tremor, and transient slow slip have all been detected (e.g., Sugioka et al. 2012). The fundamental goal of NanTroSEIZE science plan (Tobin and Kinoshita 2006) has been to access, sample, and then instrument the faults in aseismic and seismogenic regions of the megathrust system. Over the period from 2007 through 2019, 12 Chikyu expeditions have targeted this zone, reaching a maximum depth of 3262 m below the sea floor, a record in scientific ocean drilling. Key results have included the delineation of the fault zones, and the establishment of the first three real-time cabled borehole observatories in a subduction zone system which have detected many transient slip events (Araki et al. 2017; Tobin et al. 2019). The long-range objective of drilling into and instrumenting the plate interface fault zone at 5–7 km depth remains as a future objective. Japan Trench Fast Drilling Project (JFAST) The unprecedented discovery of very large (40–80 m) fault slip to the trench region during the 2011 Tohoku M9.1 earthquake and tsunami led to “rapid-response” drilling being planned and executed within 13 months of the earthquake. JFAST was carried out as IODP Expeditions 343 and 343 T (Chester et al. 2013). The primary objective was to identify and sample the fault zone that slipped near its upper end at the trench, and measure the residual thermal anomaly from the frictional heating of fault slip. A challenging aspect of the drilling was the extreme water depth of nearly 7000 m at the sea floor. A borehole of 844 m was drilled, the slipped fault

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material recovered, and its frictional properties documented (Ujiie et al. 2013). Subsequently, a thermistor string was sealed into the borehole for several months, and detected the frictional heat signature of the earthquake, measured as 0.3 °C above the background. From these data, Fulton et al. (2013) concluded that slip had occurred under extremely low effective friction, with a coefficient of only 0.08. These are the first data to quantify fault slip conditions for large earthquakes.

Unconventional Resources: Geoenergy Among the unexplored geological resources requiring drilling for basic scientific research are gas hydrates, in which methane and other gaseous hydrocarbons combine with water to form crystalline, ice-like structures (clathrates). Two widespread environments of their occurrence have been known for a long time: in sediments of the continental slopes where they simulate bottom reflectors in seismic images, and below permafrost in polar regions (Kvenvolden 1988). Little is known about the global abundance and distribution of gas hydrates, and even less about the physical and chemical parameters that affect the formation and stability of gas hydrates. They bear a high potential of generating continental slope instabilities causing giant submarine landslides and contributing significantly to the atmospheric greenhouse effect when released into atmosphere (Dickens 2003). At sea, the best studied, drilled areas are in the Eastern Pacific off Vancouver Island and at Blake Ridge in the western Atlantic Ocean off Georgia (Tréhu et al. 2006). Continental gas hydrate accumulations were drilled in the Mackenzie River delta in northern Canada (Dallimore and Collett 2005). Two observation wells and one central production well drilled to 1200 m depth were used for an extensive field program including coring, downhole logging, cross-hole experiments, and production testing. All three wells show massive thermogenic methane gas hydrate accumulations between 890 and 1100 m depth with the highest concentration at the base of the gas hydrate stability zone (Takahashi et al. 2005). Similarly, in marine sediments, the hydrates are concentrated in coarsegrained, high-permeability layers.

The Iceland Deep Drilling Project, IDDP Global decarbonization plans require enhanced regenerative energy production. Accordingly one of the emerging research objectives in Earth science is geothermal energy. Much effort is invested in identifying (and tapping) hot water and steam and heat exchangers at depth or creating such systems artificially. Although such system can be tapped for up to some tens of years, the utilization of high enthalpy supercritical

Deep Scientific Drilling

magmatic fluids or the exploitation of magmatic heat and accompanying crystallization energy promises ten times higher energy output. In an attempt to test the theory of supercritical fluids (22 MPa and 374 °C for pure water) utilization, the IDDP project drilled in 2009 at Krafla in Iceland, a well in basaltic rocks that was intruded at 2100 m depth by rhyolitic magma (Elders et al. 2011). Drilling was stopped due to challenging 900 °C conditions but the well was tested for directly utilizing magmatic heat and inspired planning for a magma test site (Eichelberger 2019). The second IDDP well drilled on the Reykjanes Peninsula in SW Iceland to about 4500 m depth reaching about 535 °C undisturbed formation temperature (436 °C measured borehole temperature) and 34 MPa fluid pressure. The drill cores retrieved from different depths confirm increasing hydrothermal alteration of basaltic dike rocks up to pyroxene hornfels facies (Fridleifsson et al. 2018). Although cold-water injection was seismically monitored and fluid inflow zones could be characterized, a test confirming energy gain from supercritical magmatic fluids remains pending.

Summary This chapter summarizes the most significant, but not all, deep scientific drilling projects that have been carried out in recent years by international research teams. Key goals of these projects encompass topics of high societal relevance such as a better understanding of the physics driving geological disasters and a wise utilization of resources. Various several kilometer deep drilling ventures allowed for unprecedented sampling and in situ monitoring of dynamic processes governing formation and restructuring of the Earth’s crust such as collision events, volcanism, faulting, or impact cratering. Despite grand engineering challenges, deep Earth access techniques are now available for novel burgeoning scientific exploration. Scientific drilling promises to be an integral tool of research generating new observations for the time to come.

Cross-References ▶ Earthquakes and Crustal Deformation ▶ Geophysical Well Logging ▶ Impact Craters on Earth ▶ KTB Depth Laboratory: A Window into the Upper Crust ▶ Lithosphere, Continental ▶ Lithosphere, Oceanic ▶ Mantle Plumes ▶ Seismic Instrumentation ▶ Slow Earthquake ▶ Vertical Seismic Profiling

Deep Scientific Drilling

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D Deep Seismic Reflection and Refraction Profiling Kabir Roy Chowdhury Department of Earth Sciences, Utrecht University, Utrecht, The Netherlands

Synonyms Active source seismology; Controlled source seismology; Deep seismic sounding; Explosion seismology; Wide-angle reflection/refraction profiling

Definition Deep Seismic Reflection and Refraction Profiling Classically, multichannel recording, along a measurement line, of (mostly) seismic P-waves, artificially generated using large energy sources, after these have traveled deep thru the earth’s crust (and upper mantle). Later developments include multicomponent recording enabling analysis of shear waves. Deep reflection profiling is mostly done using vibrators (on land) or air guns (in water) at near-vertical distances (8–12 km) to image the structure of the crust and upper mantle. Wide-angle reflection/refraction profiling uses large explosions and recording distances (200–300 km), primarily to obtain velocity information down to the upper mantle. Notational Notes Below, all capitals (e.g., DSRRP) will be used for acronyms and italicized phrases within double quotes (e.g., “Seismic Noise”) will refer to articles elsewhere in this volume.

Introduction Vibrations caused by earthquakes are known to travel great distances and provide valuable information about the internal structure of the earth. Explosions and other artificial sources could also be used to generate such waves. Man-made

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earthquake signals have several advantages of their own, e.g., their location – both in space and time – is precisely known. Further, due to relatively shorter distances of observation, higher frequencies are better preserved and allow a better resolution of the subsurface structure. Development of marine seismics has extended the coverage of this technique to offshore regions too. As a science, deep seismics is positioned between exploration seismics and seismological studies. With the shared theory of wave propagation in (visco)elastic media, similar instrumentation, and computer-intensive processing/ interpretation techniques, there is increasing overlap between these two approaches. Improvements in data acquisition and processing have extended the depth range of exploration seismics from basin level downwards and now cover the entire continental crust down to Moho and the upper mantle region immediately below it (50–100 km). Such studies are called deep seismic sounding, long-range reflection-refraction, wideangle crustal-seismics, etc. depending upon their focus. The acronym DSRRP, derived from the title, will denote studies using seismics to study the (continental) crust, including active and passive continental margins, and the upper mantle of the earth, using man-made and natural sources. There are several good resources about the general theory of wave propagation in elastic media (e.g., Aki and Richards 2002) and exploration seismics (e.g., Sheriff and Geldart 1995 and ▶ “Seismic Data Acquisition and Processing”); special topics pertaining to the latter in the context of DSRRP will be elaborated whenever needed.

Basic Principles Figure 1 illustrates the principle of the seismic experiment using a simplified earth model. Here, a horizontal boundary separates the top layer of velocity (actually wave speed) v1 and thickness d from a half-space of velocity v2, with v1 < v2, mostly valid within the earth. Wave energy, generated at the source, travels through the media and is recorded by many (usually hundreds of) receivers on the surface as seismic traces, their collection being a seismic record. Plot of a record – amplitudes as a function of recording distance versus travel time – shows distinct spatiotemporal alignments, representing different wave types (phases), three of which are sketched in the figure. Depending upon their travel path, the phases show up in the record as first or later arrivals and carry information about some properties of the medium. The amplitudes of these phases – and their variation with the recording distance (offset) – yield additional information about material properties of the medium, e.g., density, elasticity, etc. The subject matter of this article forms an extension of exploration seismics. Sheriff and Geldart (1995) and Liner

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(2004) provide good coverage of its various aspects, e.g., theory and data acquisition, processing, and interpretation. More detailed treatment of these topics is provided by Aki and Richards (2002; theory of wave propagation), Vermeer (2002; data acquisition), and Yilmaz (2001; data processing). DSSRP-specific modifications/extensions to the standard field and processing procedures will be described later. Based on Fig. 1, a brief overview of some key concepts from seismics – relevant to DSRRP – follows. Linear arrivals represent the travel times of both direct and refracted arrivals, if v1 and v2 are constant, the slopes (dt/dx) being inversely proportional to the corresponding velocities. Using these, and the intercept time (tint, Fig. 1) of the refracted arrival, the layer thickness d can be calculated. Critical distance is the minimum recording distance for observing the refracted wave. According to Snell’s law, only when  the energy is incident on the boundary at the critical angle ¼ sin 1 vv12 will it be refracted along the boundary, will propagate with the velocity v2 of the lower medium, and will keep sending energy back into the upper medium according to Huygens’ principle. Crossover distance denotes the distance, beyond which the faster refracted phase overtakes the direct wave (v2 > v1) and becomes the first arrival; the latter can also be used to determine the thickness d. In a layered earth with velocities increasing with depth, refraction arrivals from deeper boundaries become the first arrival after their respective crossover distances. Historical note: A. Mohorovicic used the overtaking of the direct P-phase from an earthquake by a faster (refracted) one, which he called Pn, to infer the thickness of the earth’s crust and the presence of a higher-velocity medium underneath (IGCP-559 2010a); the boundary defining the base of the crust has since been named MOHO in his honor. Reflections represent that part of wave energy incident upon a layer boundary, say at the point “V” in Fig. 1, which will be transmitted back into the upper medium, according to Snell’s law. The reflected arrival is hyperbolic in shape, with its apex at time t0, corresponding to the vertical two-way time (¼ 2d/v1) from the source down to the boundary and back. Reflection amplitude is sensitive to the (contrast of) material properties across the boundary and the incidence angle and provides valuable additional information regarding the media. Supercritical incidence occurs, as the recording distance increases, with the incidence angle changing in Fig. 1 from (near) vertical (e.g., point “V”) through the critical incidence (e.g., point “C”) to supercritical (e.g., point “W”). Note that the reflection hyperbola becomes asymptotic to the linear arrival belonging to the layer above but never crosses it. This relationship also holds in a multilayered situation with increasing velocities. Wide-angle incidence implying large recording distances (compared to the target depth) sees a sharp increase of

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T i m e

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Deep Seismic Reflection and Refraction Profiling, Fig. 1 Schematics of a multichannel seismic experiment to image the earth (layer of wave speed V1 overlying a half-space of wave speed V2). Energy travels from the source through the medium and is recorded by the receivers (filled triangles), planted in the ground, as seismic traces. Dash-dots represent energy directly traveling from source to the

receivers along the surface, continuous lines show reflections from the layer boundary, and dashes indicate energy critically refracted along the boundary. The upper half uses the same patterns to show the travel times of these three phases. Letters V, C, and W indicate points representing (near) vertical, critical, and wide-angle incidence of energy on the boundary

reflection coefficient and enables imaging of crustal boundaries with small contrasts. The quantity tx  t0 (Fig. 1), called normal moveout (NMO), can be measured well in this region and helps velocity estimation. The seismic signal window gets compressed though at large offsets due to the geometry (kinematics) of the travel time curves (Fig. 1, top). Modern DSRRP represents an integration of the nearvertical and wide-angle modes of seismic investigation. While near-vertical multichannel reflection seismic (NMRS) provides powerful tools for structural imaging developed for the exploration industry, wide-angle reflection-refraction (WARR) is valuable for constraining the velocities of the crustal units and thus provides the geological framework.

waves – both vertically and especially horizontally – are much larger. Such sources including quarry blasts using several kilotons of TNT – sometimes under water – were recorded over long distances, often across international boundaries (Steinhart and Meyer 1961; György 1972; Kaila et al. 1978). Calibrating and monitoring of nuclear explosions necessitated several long-distance (mostly) refraction profiles, with nuclear explosions as sources (see ▶ “Traveltime Tomography Using Controlled-Source Seismic Data”). Later, very long-range crustal seismic measurements were carried out in the Soviet Union using dedicated peaceful nuclear explosions (Pavlenkova and Pavlenkova 2006). Although chemical explosions continue to be used for WARR studies, powerful hydraulically driven mechanical vibrators on land and large compressed air sources (air guns) under water are preferred nowadays for NMRS; these have sufficient energy to return signals back to the surface from Moho and upper mantle. Receivers used in DSRRP are similar to those used in seismology and in exploration seismics but need some special characteristics. Long measurement profiles imply deployments over a large distance/area, often necessitating stand-alone capabilities, especially for power supply and storage capacity; see Mereu and Kovach (1970), in which mention is also made of an early German effort (MARS-66) using a portable four-channel analog tape recorder. Modern DSRRP receivers digitize in real time and can often be

Data Acquisition Seismic field measurements need three components, viz., man-made source to produce seismic energy (usually at the earth surface or within water), which then propagates within the earth, receivers (usually also at the surface) to pick up a part of this energy returning back after reflection/refraction/ diffraction (scattering in general) from within, and recorder to store the received signals (nowadays after digitization); see ▶ “Seismic Data Acquisition and Processing” for details. Sources used in DSRRP have to be stronger than those in exploration, as the distances traveled by the seismic

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programmed to record at certain pre-arranged time windows (see Texan in the references). For near-vertical land/marine deployments, standard industry equipment is used; these are frequently cable-free and use radio communication for data transfer. Time-of-shot recording was the weakest link in DSRRP – especially for WARR deployment, covering distances of several hundred kilometers. Early experiments used short-wave radio link or the parallel recording of a continually broadcasting radio transmitter. Availability of GPS-based time signals and high-quality internal clocks has mitigated this problem.

Field Layouts Receiver arrays were utilized early on for long-range refraction-seismic studies for their directivity vis-a-vis possible nuclear test sites – some becoming permanent installations, e.g., NORSAR (Norway), GBA (India), and WKA (Australia). Recently, the Program for Array Seismic Studies of the Continental Lithosphere (see PASSCAL in the references) has made hundreds of identical instruments available, enabling temporary deployment in several countries. Using these, with fixed and portable deployments recording both active and passive sources, a 15-year observation has been completed in the USA. See Fig. 2 and USARRAY and EarthScope in the references. Multichannel near-vertical land/marine studies use standard industry equipment and deployments (see also ▶ “Seismic Data Acquisition and Processing”), the main difference being a much longer recording time (15–30 s or longer). When using vibratory sources on land, long sweep and listening times are needed too. If the continental crust is covered by a shallow water layer (e.g., British Isles), marine data acquisition offers twin advantages of speed and better signal-to-noise ratio, enabling better imaging. DSS-DSRRP onshore-offshore recorded earlier from near-vertical outward till at least the pn-phase (refraction below Moho) was recorded as the first arrival (Gamburtsev 1952). Outside the Soviet Union, this technique was first extensively used in India starting 1972, to study the tectonic framework of the subcontinent (Reddy et al. 1999) and was later adapted for sub-basalt (Deccan Trap) exploration. Many DSRRP investigations nowadays combine the two modes – an industry standard common midpoint profile (NMRS) is frequently interspersed with a few widely spaced explosions, observed at large distances (WARR). Variations include two-ship marine recording and simultaneous onshoreoffshore measurements, useful for investigating continental/ accretionary margins.

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Special Processing and Interpretation Tools Processing of DSRRP data followed initially the standard industry scheme. Soon, however, the special needs of deep seismic data were realized. Consequently, schemes using modules newly developed by the industry/academia were established, and older data were frequently reprocessed, often resulting in improved images and interpretation (Cook and Vasudevan 2006). Line drawings were used initially to prepare DSRRP data for migration and interpretation. NMRS signals are generally weak and laterally discontinuous and embedded in a noisy background. Lines were therefore drawn on the paper section to form more or less continuous alignments – taken to represent boundaries for interpretation. The process was strongly subjective. This approach paralleled interpreting WARR data with long refraction segments, which neglected signals with limited lateral continuity. Coherency filter was proposed to mitigate this. Kong et al. (1985) formulated a procedure to automatically identify signals present in the seismic section. Using the concept of phase coherency, it uses a few user-defined parameters, to yield repeatable results, and can detect weak but laterally continuous signals in the presence of incoherent noise – although it does not preserve (relative) amplitudes. Some form of coherency filtering is nowadays commonly used for processing NMRS data. Noise reduction versus amplitude preservation is, however, an important issue because modern processing techniques can use amplitude information to advantage. An example of amplitude-preserving noise reduction is provided by Kumar et al. (2011), wherein curvelets are used to suppress incoherent seismic noise. Statistical processing of DSRRP data, for objective highgrading using a coherency criterion, was proposed by Hansen et al. (1988), based upon statistical hypothesis testing. It provided some estimate of the robustness of the results, albeit at the cost of additional computation time. Vasudevan and Cook (1998) introduced the concept of skeletonization to delineate regions of the deep crust based upon their seismic signature; van der Baan (2000) included local signal statistics for high grading the signals. Attempts have also been made to treat the entire deep-reflection wave field as backscattering from a random medium (Hurich 1996; Hurich and Kocurko 2000; Pullammanappallil et al. 1997) and analyze it to extract parameters describing the medium (see below). Vertical vis-a-vis horizontal tectonics used preferentially could lead to different interpretations of the same data. DSS profiles in György (1972) and later literature (e.g., Kaila et al. 1979) frequently contain intra-crustal normal faults. With

Deep Seismic Reflection and Refraction Profiling Deep Seismic Reflection and Refraction Profiling, Fig. 2 Plot of EarthScope stations – permanent and temporary – including those of USARRAY. For details, see legend in the inset. Full resolution original at http://www.earthscope. org ! Research ! Maps ! EarthScope Overall Maps (Archive) ! June 2015. (Figure courtesy of www. earthscope.org)

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the paradigm shift associated with the formulation of plate tectonics, some of these may need to be revisited. Gibbs (1986) illustrated this by using a DSS section sans the earlier interpretive lines for an alternate interpretation.

Main Results NMRS recordings from 12 to 15 s contain coherent reflected energy from down to 35–45 km, depending upon the presence/absence of sedimentary cover. Crystalline crust appears to be much more reflective than assumed earlier, although there are transparent zones too. In general, the mature continental middle crust is less reflective than the lower crust – probably indicating differences in their rheology (brittle vs. ductile). Intra-crustal dipping reflective zones – of both thrust and normal fault types – are encountered frequently. At times, these cut through the Moho into the upper mantle as frozen evidence of paleo-subduction. Moho appears frequently as the termination of diffuse reflectivity, the boundary itself occasionally showing sharp displacement (sign of strength). WARR recordings can be modeled with longer boundaries separating tectonic/velocity blocks at a regional scale. These also provide occasional evidence of sharp offsets in crustal boundaries including Moho; the latter seems, in some cases, to be doubled. The role of deeper structures – especially deep faults – in controlling the evolution of shallower geology, e.g., deposition, deformation, fluid movement, etc., is being increasingly appreciated. The knowledge gained is of economic significance – for understanding systems associated with economic accumulation of hydrocarbons and minerals and to help steer search for them. Early DSS(RP) in Eastern Europe has been nicely summarized in György (1972), from which Fig. 3 is taken. Deep Seismic Reflection and Refraction Profiling, Fig. 3 DSRRP data recorded in Ukraine in the 1960s. Both refracted and reflected energy is clearly visible at a distance of about 233 km and is correlatable trace to trace with a geophone spacing of 100 m. The first arrival, here marked PM, is the head wave from Moho (Pn), and the strong later phase, marked PM refl , is the wide-angle reflection from the base of the crust. (Figure courtesy of Geophysical Transactions from György (1972, p. 50))

Deep Seismic Reflection and Refraction Profiling

It shows a part of a long-offset (≈233 km) DSS profile recorded in Ukraine in the 1960s, as part of extensive DSS surveys in Eastern Europe, which had established the observability of deep refracted and reflected phases in the wideangle range. High apparent velocity of the refracted Pnphase (first arrival at large offsets) and its relationship with (later) reflected phase help identify the base of the crust as their origin. Some figures in the above reference also show near-vertical reflections from the Moho. György (1972, pp. 44–68) includes on page 66 a contour map of the Moho depth below Ukraine, the detail of which, although debatable, is impressive, especially considering its vintage. It is based on an astounding 6000 line km of DSS profiling with a dense network, following the methodology described in Gamburtsev (1952). The contours indicate depths between 22.5 and 55 km, with rapid lateral variations at places. The latter were interpreted as deep faults displacing the Moho. (Inter)national consortia in DSRRP started in 1975, when an industry crew measured a reflection profile in Texas, using a vibratory source, with a recording time of 15 s (Finlayson 2010b). A large amount of data has since then been collected/analyzed in several national academic industrial collaborations – Table 1 provides a partial overview; see also IGCP-559 (2010c) for more details. Recently, some international DSRRP experiments have studied specific geological problems, e.g., Himalaya and Tibetan Plateau (Zhao et al. 1993), active continental margin in Central Andes (ANCORP Working Group 2003), and structures in East Antarctica from SEAL geotransect (Kanao et al. 2011). Such transects are often multidisciplinary in character, e.g., Palomeras et al. (2011). International biennial symposia have been organized (roughly) every 2 years since 1984 to showcase the data and discuss the results from DSRRP surveys. The publications resulting from these meetings provide a historical record of

Deep Seismic Reflection and Refraction Profiling

the progress of DSRRP – with respect to both technological advancement and scientific knowledge: Barazangi and Brown (1986a, b), Matthews and Smith (1987), Leven et al. (1990), Meissner et al. (1991), Clowes and Green (1994), White et al. (1996), Klemperer and Mooney (1998a, b), Carbonell et al. (2000), Thybo (2002), Davey and Jones (2004), Snyder et al. (2006), Heikkinen et al. (2011), Rawlinson and Goleby (2012), Santosh et al. (2014), Carbonell et al. (2016), Rawlinson et al. (2017), and Malinowski et al. (2019). Some additional information is also available in IGCP-559 (2010b). (Re)processing, synthesis, and interpretation of the vast amount of DSRRP data – near-vertical and wide-angle – are not easy. The data quality depends upon the geological settings and the data acquisition technique and parameters used. Uniform processing of this dataset of mixed quality/vintage is necessary though for regional syntheses and interpretation, to understand the internal architecture of the continental crust (e.g., Phinney and Roy Chowdhury 1989; Cook and Vasudevan 2006). Results from DSRRP contain several surprising features – some of which are still being interpreted – and have yielded new insights into the processes that shape the continental crust. Below are some of these highlights; the acronyms referring to the consortia/projects are explained in Table 1. Imaging deeper with multichannel exploration seismics began in 1975. The COCORP consortium – the acronym is reported to have been coined past midnight at a bar in Mexico – pioneered the use of industry standard sources (vibrators), recording layout (NMRS), and processing for deep seismic profiling on land and obtained useful signals from depths of 40–50 km, by using 4–5 vibrators simultaneously and extending the recording time (Oliver et al. 1976). Later, similar studies confirmed that the crust underlying the basement possesses variable reflectivity, including some transparent regions. Moho, the base of the crust, often showed up on such images as the termination of a zone of diffuse reflectivity, and not as a long and sharp boundary inferred from earlier (refraction) studies. One of the early surprises of the COCORP lines was the discovery of a mid-crustal zone of strong reflectivity below southern Georgia, USA; the surrency bright spot was reconfirmed during a later survey (Pratt et al. 1993). Such zones have since been reported in other surveys too, e.g., the Quebrada Blanca Bright Spot (QBBS) in Andean subduction zone (ANCORP Working Group 2003) and below the Dnieper-Donets paleorift (Pylypenko et al. 2011). The possible causes of this strong reflectivity remain controversial (see below). Besides WARR recordings of quarry blasts, Germany had an early start in near-vertical recordings of deep reflections and statistical evaluation of their amplitudes (e.g., Dohr 1957; Dohr and Fuchs 1967). More recently, their DEKORP program has included investigations across active collisional

133 Deep Seismic Reflection and Refraction Profiling, Table 1 Some (inter)national DSRRP efforts Acronym COCORP

Location USA

BIRPS

UK

Period 1975 onwards 1981–1998

DEKORP

Germany

1983–1997

LITHOPROBE ECORS

Canada France

1984–2003 1983–1991

MONA LISA IBERSEIS

UK Spain

1993 2001

SEAL ... INDEPTH

E. antarctica

2002–2004

China

ANCORP

Chile

1992 onwards 1996

KRISP

Kenya

1985–1994

BEST

Russia

2002

BABEL

Baltic Sea

1989

EAGLE

Ethiopia

2003

SINOPROBE ...

China

2008–2012

Remarks Pioneered nearvertical imaging Marine seismic imaging Deep drilling (KTB) Multidisciplinary IFREMER (marine) On/Off -shore Spanish universities and institutes Japan USA and other countries Germany and other countries European and US universities Denmark and Poland European groups, On/Off -shore European and US universities Multidisciplinary

zones, e.g., Alps (Gebrande et al. 2006) and the Andes (ANCORP Working Group 2003). For details, see DEKORP in the references. Marine seismic imaging of the continental crust was seized upon to (partly) alleviate the unfavorable signal-tonoise ratio (SNR) for deep seismic data acquisition on land, e.g., noise from traffic, industry, etc. – although marine seismics has its noise sources too. Phinney (1986) used data from a 48-channel marine survey (recorded by USGS during 1973–1979) over the Long Island Platform. The original stack sections of lines 36 and 23, reprocessed to 12 s, showed clear evidence of a rich crustal architecture, with half grabens, wedges, and other tectonic features indicating both compressional and extensional phases of a Wilson cycle. Existence of known – and expected – hydrocarbon-bearing structures had attracted marine seismic exploration activity in the 1970s and 1980s to the waters around the British Isles. The latter, surrounded by North Sea and the northeast margin of the Atlantic Ocean, are a part of the European continental shelf. Starting 1981, this situation was utilized to great advantage by the BIRPS consortium – essentially by extending the marine seismic exploration recording time to 15 s. The

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DEKORP played an important role in the site selection phase of the German superdeep drilling program, KTB, which was set up to drill ≈10,000 m down through an ancient geodynamic suture – see Emmermann and Lauterjung (1997) for an overview of KTB. Later, besides providing a reference point for the seismics at the drill site, the KTB program has yielded direct evidence of shear zones, anisotropy, and substantial amounts of fluids deep in the crust; the analysis of the latter has provided new insights into their origin and role in controlling/influencing geodynamic processes. The Russian super-deep drilling program in the Kola Peninsula (Smythe et al. 1994) reached the record depth of 12,200 m and also provided valuable ground truth regarding the macrostructure of the mature continental crust and the origin of crustal reflectivity. Upper-mantle reflectivity-frozen subduction has also undergone a paradigm shift as a result of DSRRP investigations. Reflections from upper mantle were already reported by György (1972). DSRRP has provided vivid proof that reflective zones may extend into the upper mantle and has helped understand their structure. Dipping reflectivity in this region has provided information regarding paleotectonics and may – in some cases (see Fig. 4) – be evidence for paleo-subduction (see also BABEL Working Group 1990; Morgan et al. 1994). Rheology of crust, Moho, and upper mantle depends upon their composition and in situ physical conditions. Intra-crustal faulting, affecting even the Moho, was already inferred by György (1972) and Kanasewich et al. (1969). DSS data from the Indian shield (Kaila et al. 1979) was used by Roy Chowdhury and Hargraves (1981) to infer constraints about the thermo-tectonic evolution of the early earth. With

Outer Isles Thrust Shot point numbers APPROX DEPTH (km)

0

3000

Moine Thrust

2000

1000

100 East

West

PZ SHELF

OIT

0

MT(B)

LEWISIAN BASEMENT MT(A) 5

15 LOWER CRUST MOHO

30

10

FT

TWO-WAY TIME (S)

very first profile, MOIST (recorded by the preceding BURPS group), contained strong reflections from the lower crust, Moho, and upper mantle (see Fig. 4), which could be connected to surface geology on land (Smythe et al. 1982), and even prompted correlation of tectonic evolution across the Atlantic (Brewer and Smythe 1984). Later BIRPS profiles, e.g., WINCH (Brewer et al. 1983), provided evidence for shallower/younger tectonics being controlled by older/deeper crustal structures. The DRUM profile extended the recording to 30 s. The density of coverage and the quality of data allowed Flack and Warner (1990) to map deep reflections in 3D and enabled Chadwick and Pharaoh (1998, p. 268) to produce a contour map of Moho beneath the UK, which may be compared with a similar map below Ukraine (György 1972, p. 66) mentioned earlier. Integrated transects, using additional geophysical tools, e.g., magnetotellurics, electromagnetics, and geochronology, characterized the LITHOPROBE program in Canada. It took advantage of the geology to investigate both ancient processes, e.g., assembly of continents and modern crustal dynamics of active subduction, detachment, and imbrication. For example, the Kapuskasing Uplift, one of the few lower crustal exposures on earth surface, was imaged in the KSZ transect (Percival et al. 1989), whereas the transects SNORCLE and SC examined younger mountain building processes (e.g., Clowes et al. 1983). Relating to the ground truth, the strength of exploration seismics, is the Achilles’ heel of DSRRP (and seismology). Superdeep drill holes provide the only opportunities of directly correlating observations to the rock properties; the German DEKORP program was able to utilize this in a symbiotic manner.

25 km MOIST NORTHERN SCOTLAND

Flannan Thrust Deep Seismic Reflection and Refraction Profiling, Fig. 4 Annotated line drawing interpretation of the MOIST profile data, showing coherent reflections from throughout the crustal column. Moho shows up as a more or less continuous boundary. Several dipping thrust-like features (e.g., Outer Isles Thrust) can be seen at all depth

ranges, which also include transparent zones. One dipping reflective zone (Flannan Thrust) is seen to penetrate through the Moho into the upper mantle. (Figure from Finlayson (2010a); see Brewer and Smythe (1984) for details)

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Deep Seismic Reflection and Refraction Profiling, Fig. 5 Unmigrated seismic section from URSEIS profile across southern Urals. (Figure from Diaconescu et al. (1998), with permission from Geology). Moho (older than 1 billion years) is bright on the left and is offset sharply in the middle of the figure (Makarovo fault zone) by ≈5 km. Migrated image of the boxed upper part (not shown) contains laterally continuous stratification over this fault zone

improved acquisition/processing, such features are now regularly reported from different parts of the world. (Near) vertical offset in the Moho, inferred at many places from teleseismic, gravimetric, and other geophysical observations, is now often imaged in NMRS. This has important bearings regarding the nature of Moho. This first-order seismic boundary is often thought to be a surface that re-equilibrates after tectonic events above (e.g., thrusting) or below (e.g., underplating). DSRRP images contain counter examples too, showing that Moho topography can survive later orogenic cycles, e.g., BABEL Working Group (1990) and Diaconescu et al. (1998). The reason for this behavior is not well understood. The thermal- and the stress- regimes in the lower crust and upper mantle determine the interaction between the two during the formation of Moho as the boundary layer. Chadwick and Pharaoh (1998) interpret a DSRRP line by associating increased reflectivity of the Moho there to its being a detachment surface resulting from low-angle shear (Fig. 6). Local doubling of Moho is seen especially in some WARR data – its evolutionary mechanism remains unclear though. WARR had taken a back seat with the increasing use of NMRS. With progress on some of the processing issues, it has made a comeback following the adage: structures from reflection and velocities from refraction. It is, sometimes, a part of onshore/offshore measurements and uses three-component receivers to study crustal anisotropy. Mooney and Brocher (1987) provide a review of earlier coincident WARR studies. Another example is BABEL Working Group (1993). Standalone WARR remains useful at places with difficult logistics; see the KRISP experiment in the Kenyan Rift Valley (Khan et al. 1999). Deep seismic investigations using nuclear explosions: During 1965–1988, the then USSR carried out a series of

D

very long-range seismic experiments using peaceful nuclear explosions (PNEs), supplemented with chemical explosions placed every 100–150 km (Benz et al. 1992). Employing three-component receivers, typically 10 km apart, and recording out to 1550–3000 km (Pavlenkova and Pavlenkova 2006), these datasets provide an invaluable basis for current and future research; QUARTZ, the profile studied most, used 400 three-component receivers, 3 PNEs, and 51 chemical explosions (Morozov et al. undated IRIS communication; see references); analysis of its data has generated new ideas about wave propagation in the lower crust and upper mantle. Analysis/modeling of WARR data typically starts by picking travel times of first and later arrivals that are interesting, the choice being decided by the data quality and the geological aim. Usually, some prominent (mid) crustal reflected/refracted arrivals are identified and picked, along with arrivals from Moho, and possibly deeper ones. Methods of deriving crustal models from this data include ray tracing (Zelt and Smith 1992), tomography (Hole 1992), computation of synthetics, etc. (see also ▶ “Traveltime Tomography Using Controlled-Source Seismic Data”). Frequently, a preliminary velocity model is iteratively fine-tuned to obtain a desired level of fit to the observed travel times. There are several schemes for ray tracing; see ▶ “Seismic Ray Theory” and Zelt (1995) for overviews. Of late, amplitude information has also been incorporated in such schemes. Maguire et al. (2006) contains examples of the different aspects of the procedure in the context of the EAGLE experiment (Fig. 7). Different modeling/inversion schemes yield different results from same input. The DOBREfraction‘99 Working Group (2003) provides an example of this. Pavlenkova (2011) has interpreted PNE arrival times to infer upper mantle rheology and suggests layering along the travel paths with boundaries at 100 and 200 km (fluids?).

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Deep Seismic Reflection and Refraction Profiling

Deep Seismic Reflection and Refraction Profiling, Fig. 6 Schematic representation of low-angle shear near Moho (c) with parts of BIRPS seismic sections GRID (a) and NEC (b). (For details, see Chadwick and Pharaoh (1998); figure courtesy of Tectonophysics)

Analyzing amplitude of DSRRP signals is a crucial step, to differentiate between competing models of the crust. Improvements in the data quality, and careful processing, allow attempts to quantify the material properties that influence the strength of the recorded signals. Properties that are of primary interest are reflection coefficients (RC) across various boundaries, their variation with the angle of incidence (AVO), and the Q(uality) factor along the path (see ▶ “Seismic Data Acquisition and Processing” for definitions). These properties are especially important while considering the probable cause(s) of the bright spots, e.g., fluids, intrusions, layering, etc. Warner (1990) reported RC values of about 0.1 for lower crustal layers and about 0.15 for Moho – both derived from the WAM profile (Peddy et al. 1989). The polarity (sign) of RC is important in differentiating between likely causes of bright spots sometimes seen in DSRRP images. However,

even using polarity-preserving processing, it is often only possible to determine relative values of RC. ANCORP Working Group (2003) discusses the issues involved for a couple of strong reflectors – Nazca and QBBS – while reporting RC > 0.2 for the latter. Combining NMRS and WARR data, Makovsky and Klemperer (1999) report RC values between 0.3 and 0.4 for the NBS bright spot in the Tibetan middle crust. Obtaining estimates of Q along the travel path of NMRS signals (relatively high-frequency body waves) is also difficult – the effects due to scattering, conversion, etc. (apparent-Q) cannot be readily separated from the intrinsic attenuation (see also ▶ “Seismic Viscoelastic Attenuation”). Combining the two, Hobbs (1990) obtained a value of 500  200 for effective-Q for the lower crust below the WAM profile. Morozov et al. (1998) used the data from the QUARTZ profile mentioned above to obtain a 2D

Deep Seismic Reflection and Refraction Profiling

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Deep Seismic Reflection and Refraction Profiling, Fig. 7 Velocity modeling by ray tracing for the EAGLE project (in the Ethiopian rift). Note the variable coverage of the ray tracing (above) and the smooth result, with a few long segments, in the final P-wave velocity model

(below) – both these characteristics are typical for WARR data analysis. (For details, see Maguire et al. (2006); figures courtesy of Geol. Soc. London, with permission from the author)

Q-model for the upper mantle below this PNE profile down to a depth of 500 km.

Seismix-2016: “Seismic imaging at the cross-roads: Active, passive, exploration and solid Earth”

Some Recent Developments DSSRP has traditionally positioned itself between basin-level seismics (using artificial sources) and deep-earth seismology (using natural sources). Recent methodological advances in the above fields have blurred this distinction. This is exhibited by the themes of the biannual meetings: Seismix-2010: “Seismic imaging of continents and their margins: New research at the confluence of active and passive seismology” Seismix-2012: “Advances in seismic imaging of crust and mantle: Preface”

These may be contrasted with seismic probing of continents and their margins, which, with small variations, was used for almost the first three decades. Note: most submissions at the latest biannual meeting (Seismix-2018) have not (yet) been published but are available at Seismix-2018-abs (2018). These abstracts (56 talks and 48 posters) further show the maturing of DSRRP – both regarding methodology and inclusion of other geophysical data for a better (geodynamic) interpretation. Improvements in data acquisition are reported in Seismix-2018-abs (2018, pp. 40, 41) using the newly developed DAS (distributed acoustic sensing) technology. Both of these use the data for VSP (vertical seismic profiling) in a mineral exploration setting. Use of OBS (ocean bottom

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seismometers) is reported on page 45 of the same abstract collection to improve lower crustal velocity models and also better calibrate pressure measurements from air gun sources. Noise as a seismic source is now an exciting field of research and is referred to as seismic interferometry or daylight imaging. The basic idea is that noise contains useful information. Noise recorded at two locations can, for example, be used to obtain relevant properties of the intervening medium. See Curtis et al. (2006), Wapenaar and Snieder (2007), and Snieder and Wapenaar (2010) for further details and also ▶ “Seismic, Ambient Noise Correlation” and ▶ “Seismic Noise.” Rawlinson and Goleby (2012) mentioned the use of noise for crustal imaging in their introductory article for Seismix-2010. Ito et al. (2012) use autocorrelation functions of ambient noise to obtain seismic images in a subduction zone. An interesting case of using autonomous recording nodes to record noise in between the active recording to obtain better Vs models is reported by Behm et al. (2019). Inversion – Full/constrained/joint, i.e., the ability to reproduce each wiggle of the seismic observations is the ultimate goal for its interpretation, in terms of the viscoelastic properties of the causative medium (see ▶ “Seismic, Waveform Modeling and Tomography”); DSRRP benefits from research in this field. For a quick introduction to this topic, including underlying problems, see Virieux and Operto (2009), Virieux et al. (2017), and Brittan and Jones (2019). True full waveform inversion (FWI) of seismic remains an unattained goal, but some progress is reported by, e.g., Górszczyk et al. (2019), which use both OBS and standard multichannel marine reflection data to optimize the crustal velocity model by a joint full waveform inversion. Rawlinson et al. (2016) use a detailed crustal starting model from ambient noise seismics for a constrained inversion in teleseismic tomography. Multidisciplinary studies have been largely not discussed in this entry, which has focused on P-wave measurements. Availability of high-quality multi-geophysical data does enable extraction of additional information though. Shear waves, for example, can provide valuable constraints regarding lithology and composition – see Eccles et al. (2011) for an example. Nowadays, data from other geophysical surveys, e.g., gravity, magnetic, magnetotelluric, geodetic, etc., are frequently available near the DSRRP transects; these can be used to improve both structural and petrophysical interpretation. See Dong et al. (2013) reporting results from the SinoProbe Programme (2008–2012) covering deep lithospheric exploration of China. Integrating diverse geophysical datasets, however, needs special attention during interpretation, to obtain a model which satisfies all data in some optimal fashion. This approach may be further subdivided into joint assessment and true joint/constrained inversion.

Deep Seismic Reflection and Refraction Profiling

Palomeras et al. (2011) falls in the first category, so does Yegorova and Pavlenkova (2014), who report inverting for density along the PNE profiles, with their starting model taken from seismic data. Roberts et al. (2012) is an example of attempting to constrain the final model for a multidisciplinary dataset (seismic refraction, MT, and gravity) using computer emulation. The road ahead will see DSSRP results being increasingly applied to geodynamic problems related to the continental crust/lithosphere (e.g., Seismix-2018-abs 2018, pp. 20, 75). Better acquisition/processing will result in crustal images/properties with a higher resolution, enabling more nuanced conclusions regarding their evolutionary history. Inroads are already being made into more fundamental questions like earthquake processes (Górszczyk et al. 2019). True joint inversion of disparate datasets will remain a challenging problem – not in the least due to the relative scaling for the different datasets. Syracuse et al. (2017) report results from a joint inversion using P- and S-arrival times and dispersion data from Rayleigh waves from USARRAY deployment and Bouguer gravity anomalies. Several important issues related to FWI of different datasets are mentioned by Seismix-2018-abs (2018, p. 58), e.g., optimal acquisition design, large-scale inversions involving hundreds of millions of parameters, problem of nonlinearity, and optimizing highand low-frequency contents of the result.

Research Problems The massive amount of DSRRP data collected during the past half century has produced many new insights but has also brought up some (yet unsolved) problems. Origin of crustal reflectivity, clearly visible in DSRRP images of mature continental crust, cannot be explained uniquely. Surface exposures of basement rocks mostly show steep dips due to earth’s surface being stress-free and explain transparent zones at shallow depths. Starting at intermediate depths, these structures are expected to become subhorizontal and seismically imageable and do reveal layering including strongly reflective zones. Smithson and Brown (1977, Fig. 5) had already proposed a complex crustal model with a threefold subdivision based on geo-scientific data. Processes and materials to explain such low- and high-angle deformation, sometimes even affecting Moho and upper mantle, remain a challenge. Shear zones, decollements, imbrications, laminae, metamorphism (facies changes, mylonitization), sill-like intrusions, and fluids (water, brine, melt, magma) have all been proposed as candidates for crustal (and Moho) reflectivity. Both Kola (Smythe et al. 1994) and KTB (Emmermann and Lauterjung 1997) super-deep drill holes have identified such conditions/structures at depth; extrapolations would point to

Deep Seismic Reflection and Refraction Profiling

the presence of crustal fluids in quantities more than that earlier expected, both in bound and free form. See also Meissner et al. (2006), ▶ “Continental Crustal Structure” and ▶ “Crustal Reflectivity (Oceanic) and Magma Chamber”. Bright spots – zones of very strong reflectivity – present an extra challenge. Pratt et al. (1993) investigated the surrency bright spot (Georgia, USA), first assumed to be fluid related due to its flat nature and concluded an (ultra)mafic body instead. Makovsky and Klemperer (1999) inferred 10% (volume) of free aqueous fluids as the cause for the Tibetan bright spot. Figure 8 is an interesting example of two reflective zones, probably with different, but related, origins. Simancas et al. (2003) reported a mid-crustal highly reflective body below SW-Iberia from NMRS observations; later modeling of dense WARR data suggested the presence of high-velocity material. Palomeras et al. (2011) used additional data (heat flow, gravity, etc.) to further study this zone and inferred the presence of sill-like lenses of mantle material. Origin and nature of (continental) Moho are also not well understood, probably because this boundary – geophysically defined as a first-order transition of P-wave velocity from ≈6.8 to ≈8.2 km/s – does not always have the same evolutionary history. However, DSRRP has replaced the earlier model for (seismological) Moho, consisting of long refraction segments – more or less flat – by a (seismic) boundary with a complex and variable structure. It seems, at places, to be the equilibrium surface established under the deviatoric stress regime after a tectono-thermal event involving the lower crust and upper mantle. At others, it Deep Seismic Reflection and Refraction Profiling, Fig. 8 Migrated seismic section from the ANCORP line 2 in the Central Andean subduction zone. (Reproduced from http://wwwapp1.gfz-potsdam.de/www/pb3/ dekorp/an-fig/Amline2.gif, with permission from the author). The strongly reflective zone on upper right is the Quebrada Blanca Bright Spot (QBBS), and the reflective zone dipping to the right is the Nazca reflector. Superimposed red dots show seismicity in this area of active subduction. (For further details, see ANCORP Working Group (2003))

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seems to be underplated or overlain by sill-like intrusions, from a later magmatic episode. Elsewhere, Moho is only identifiable as the terminal zone of a diffused lower-crustal reflectivity. At many places, it seems to exhibit enough strength, to retain its earlier structure through later tectonic events; examples include frozen subduction (Fig. 4) and offset Moho (Fig. 5). Role of (multiple) scattering in lower crust has attracted attention lately – both to explain observations and to relate to geological evidence. Surface outcrops of Moho (read: lower crustal and upper mantle rocks) are extremely rare in the continental setting, e.g., Kapuskasing Uplift (Eastern Canada), Musgrave Range (Central Australia), and IvreaVerbano Zone (IVZ, Northern Italy). While the first two have been studied by DSRRP (LITHOPROBE transect KSZ and Central Australian Basin transect, respectively), IVZ – although sans seismics – has been extensively studied in the past by geologists, e.g., Zingg (1990), and geophysicists, e.g., Berckhemer (1969). Recently, statistical analyses of detailed geological maps of several exposed (lower) crustal rocks have yielded the tantalizing possibility of a self-similar (fractal) description of their shapes in terms of typical horizontal and vertical scale lengths. For the IVZ, a 2D von Karman distribution of the structure with bimodal petrology has been derived (Holliger and Levander 1992). Holliger et al. (1993) computed synthetics for such a simulation of IVZ and were able to qualitatively reproduce lower-crustal reflectivity observed in NMRS. At large distances (WARR configuration), the

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synthetics from the random medium contained laterally correlatable events, which could be erroneously used for velocity analysis, migration, etc., a possibility already suggested earlier (e.g., Levander and Gibson 1991; Emmerich et al. 1993). Another explanation offered for increased reflectivity in the lower crust, including transition zones, is lamellar structures with associated wave propagation effects (amplitude, anisotropy); see Meissner et al. (2006) and ▶ “Magnetic Anisotropy.” Both these possibilities, random heterogeneity and lamination, bring up the role of multiple scattering in the DSRRP wave field and question the propriety of using conventional tools from basin exploration for processing such data. Douma and Roy Chowdhury (2001) used synthetics to show that multiple scattering has a limited effect for a 1D bimodal model of IVZ but also mentioned the need for 2D full-wave numerical simulations. In the upper mantle too, multiple scattering seems to play an important role. Menke and Chen (1984) had invoked this to explain long-range propagation of Pn-phase from earthquakes. More recently, DSRRP data from PNE profiles, e.g., QUARTZ, has shown surprisingly strong propagation of high-frequency (5 Hz) Pn-phase to distances of 3,000 km! Estimating descriptive parameters of a possibly random medium in lower crust and the upper mantle can be done by modeling or direct estimation. To explain the long-distance Pn-phase in the QUARTZ data, Ryberg et al. (1995) modeled an upper mantle zone of horizontally stretched, randomly

Deep Seismic Reflection and Refraction Profiling, Fig. 9 Longrange seismic lines used for modeling by Nielsen and Thybo (2006). Early Rise used chemical explosives; the other lines used PNEs. B and

Deep Seismic Reflection and Refraction Profiling

distributed velocity anisotropy. Nielsen and Thybo (2006) modeled a larger dataset (Fig. 9) and inferred random heterogeneity for both lower crust and upper mantle. Following some earlier work (Hurich 1996; Pullammanappallil et al. 1997; Hurich 2003; Carpentier and Roy Chowdhury 2007), Carpentier et al. (2011) have recently analyzed the data from Line 48 of the LITHOPROBE transect AG statistically. Assuming a 2D von Karman medium, they estimated the horizontal scale length of the medium directly from the seismic wave field (Fig. 10), which also shows their interpretation. The comparison with an earlier line drawing-based interpretation (above) illustrates the similarities and differences between the two approaches.

Summary DSRRP has, over half a century, produced quality images of the continental crust and its margins, revealing their complex structure. These – including some unexpected results, e.g., frozen subduction – have contributed significantly to our ideas about the processes, current and ancient, involved in their evolution. As the deep structures, mostly inaccessible, play an important role in the development of the shallower geology, understanding these (deep faults) also helps in the optimal exploration of economic resources, e.g., hydrocarbons, ore deposits, etc., and in the study of natural hazards associated with volcanism and earthquakes.

ML mark the locations of NMRS lines BABEL and MONA LISA, respectively. (Figure courtesy of Tectonophysics)

Deep Seismic Reflection and Refraction Profiling

141 SOA

2000 CP

1000

NVZ GB

3000

FEGB

4000 LOP

5000

6000

7000

8000

9000

CDP

LRP NEM

NRSZ

OP

OP

OP

D

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Deep Seismic Reflection and Refraction Profiling, Fig. 10 Above: Tectonic interpretation of LITHOPROBE line AG-48 (AbitibiGrenville, Canada, taken from Calvert et al. (1995); figure courtesy of

Nature). Below: interpretation by Carpentier et al. (2011) overlain on their estimation of the horizontal scale length (ax) of the medium from the seismic data. (Figure courtesy of Tectonophysics)

Cross-References

Bibliography

▶ Continental Crustal Structure ▶ Continental Rifts ▶ Crustal Reflectivity (Oceanic) and Magma Chamber ▶ Earth’s Structure, Upper Mantle ▶ Lithosphere, Continental ▶ Magnetic Anisotropy ▶ Plate Tectonics, Precambrian ▶ Seismic Data Acquisition and Processing ▶ Seismic Imaging, Overview ▶ Seismic Instrumentation ▶ Seismic Monitoring of Nuclear Explosions ▶ Seismic Noise ▶ Seismic Ray Theory ▶ Seismic Tomography ▶ Seismic Viscoelastic Attenuation ▶ Seismic Waves, Scattering ▶ Seismic, Ambient Noise Correlation ▶ Seismic, Waveform Modeling and Tomography ▶ Traveltime Tomography Using Controlled-Source Seismic Data

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Differential Rotation of the Earth’s Inner Core Xiaodong Song Department of Geology, University of Illinois at UrbanaChampaign, Urbana, IL, USA

Definition Differential rotation of the inner core refers to the difference between the rotation of the Earth’s solid inner core and the daily rotation of the Earth’s crust and the mantle.

Background Driven by gravity, the Earth differentiated into iron core and rocky mantle at the early stage of its formation. The central core is liquid because of high temperature; but at the center of the Earth, an inner core is formed and gradually grows as the Earth cools, and the liquid iron freezes under tremendous pressure (three million times the atmospheric pressure). The size of the inner core is slightly smaller than that of the moon. Separated from the solid mantle by the fluid outer core, the inner core is thus free to rotate independently under an external torque. The early idea of inner core rotation came from studies of the geodynamo, which generated the Earth’s magneticfield. Gubbins (1981) first suggested that electromagnetic forces between the electrically conducting inner core and the

Differential Rotation of the Earth’s Inner Core

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magnetic field generated in the fluid outer core would cause a differential rotation of the inner core. In a computer simulation of a three-dimensional, self-consistent geodynamo, Glatzmaier and Roberts (1995) predicted that the inner core is driven to rotate by electromagnetic coupling at a few degrees per year faster than the mantle and the crust.

Seismological Observations Song and Richards (1996) reported first evidence for differential rotation of inner core from seismic observations. The Differential Rotation of the Earth’s Inner Core, Fig. 1 Ray paths of PKP waves and example of waveform doublet used to detect temporal change of travel times through the inner core, (a) Ray paths of three branches of PKP waves turning in the solid inner core (DF), the bottom of the fluid outer core (BC), and the midouter core (AB). (b) Highly similar waveforms recorded at one station in Alaska from a waveform doublet from South Sandwich Islands. The two events are 10 years apart, one in 1993 and the other in 2003. (c) Superimposed and enlarged PKP waveforms from the box in (b). The waves through the outer core (BC and AB) are aligned, but the wave through the inner core (DF) shows a small time shift (about 0.1 s). (From Zhang et al. 2005)

basic idea of their method is simple. They chose a fixed monitoring station and compare seismic waves that travel through the inner core from earthquakes at the same region but a few years apart. This is like the twinkling of a star, which is caused by disturbance of the light as it passes through the atmosphere (an analogy used by coauthor Paul Richards). The seismic waves used are short-period (about 1 s) longitudinal waves (P waves) that have propagated through the fluid core (generically called PKP waves) (Fig. 1). At the distance range of about 146–154°, there are three possible paths, i.e., PKP (DF) waves that traverse the inner core, PKP(BC) waves that turn at the bottom of the outer core, and PKP(AB) waves that

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turn at the middle of the outer core. The basic data used in detecting the inner core rotation is the time difference between the PKP(BC) and PKP(DF) arrivals, abbreviated as the BC-DF time. Because these two waves travel very closely together in the crust and mantle and in most of the fluid core, the differential time BC-DF is relatively insensitive to uncertainty in source location and to three-dimensional heterogeneities of the crust and mantle along the ray paths. To remove influence of small difference in earthquake location and epicentral distance, the differential time residual is formed by subtracting the observed BC-DF time from the predicted difference for a standard earth model. In addition, residuals of the differential times between PKP(AB) and PKP(BC), which pass through the fluid core only, are used to examine possible systematic biases in source locations. Song and Richards found that BC-DF residuals along certain paths have changed systematically with time over a few decades. In particular, the seismic waves that emanate from earthquakes in South Sandwich Islands in the South Atlantic travel through the inner core and reach College, Alaska seismic station. They have sped up systematically by about 0.3 s over 28 year-year time interval from 1967 to 1995. They interpreted the temporal change as evidence of an inner core rotation. The estimate of the rotation rate depends on the underlying aspherical structure of the inner core that results the temporal change from a rotation. The inner core is known to be very anisotropic (i.e., seismic waves traveling through the inner core along different directions have different speeds) (Morelli et al. 1986; Woodhouse et al. 1986; see also review by Song 1997; Tromp 2001). The anisotropy is roughly axissymmetric with the P-wave speed faster by some 3% along the Earth’s rotation axis than along the equatorial plane. However, the anisotropy changes significantly laterally and with depth. The observed temporal change in BC-DF residuals was first interpreted as a change of the orientation of the fast axis of the inner core anisotropy, which yielded a rate of 1.1° per year (Song and Richards 1996), but it was later and preferably interpreted as a shift of lateral velocity gradient in the inner core caused by the inner core rotation (Creager 1997; Song 2000), which yielded a rate ranging from 0.2–0.3 to 0.3–1.1° per year. Subsequent studies provided further support and most estimates of the rotation rate are a few tenths of a degree per year faster than the rotation of the Earth (a super-rotation) (see reviews by Tromp 2001; Song 2003; and Souriau et al. 2003). The methods used include BC-DF times for additional paths (Ovchinnikov et al. 1998; Song and Li 2000; Xu and Song 2003), inner core scattering waves (Vidale et al. 2000), so-called earthquake waveform doublets (Li and Richards 2003; Zhang et al. 2005; see below), and normal modes (Laske and Masters 2003). However, whether the inner core

Differential Rotation of the Earth’s Inner Core

rotation is real has also been hotly debated because of potential biases such as earthquake mislocation (e.g., Poupinet et al. 2000), failure to detect the motion (Souriau 1998), and discrepancy in the inferred rotation rate (e.g., Creager 1997; Vidale et al. 2000; Laske and Masters 2003). The most important source of errors is systematic event mislocation. Because global stations used to locate the earthquakes are not exactly the same in different years, the temporal variation could potentially be an artifact of the changes of the station distributions (Poupinet et al. 2000; Souriau et al. 2003). To resolve the debate, the two sides (Dr. Poupinet and the author) worked directly in 2004–2007. A thorough presentation of the collaborative work was published (Sun et al. 2006). The basic conclusion is that earthquake mislocation is too small to explain the observed time shifts in the data, and that inner core rotation is still the best explanation. Perhaps the strongest support comes from studies of earthquake waveform doublets (Zhang et al. 2005). A waveform doublet is a pair of earthquakes occurring at different times but at essentially the same spatial position, as evidenced by their highly similar waveforms at each station recording both events (Poupinet et al. 1984). Zhang et al. (2005) observed that the waves that traveled through the area outside the inner core, in the crust, the mantle, and the outer core are all the same. Only when they travel through the inner core at different times are they different (Fig. 1). The changes are two folds: first, and most prominently, the PKP(DF) waves (passing through the inner core) move faster in a systematic pattern over time, by about one tenth of a second per decade; second, the PKP(DF) waves themselves change in shape over time, an independent signature for motion of the inner core. The fact that the wave shapes are similar when they emanate from the source region allows them to measure precisely the small time shifts and to pin down precisely where the changes took place. The temporal change of BC-DF differential times for the South Sandwich Islands to College, Alaska path is constrained to be 0.0092 s/year with standard deviation of 0.0004 s/year. The best estimate of the rotation rate is 0.3–0.5° per year faster than the Earth’s mantle. Subsequent studies of earthquake doublets suggest that the waves bouncing off the inner core boundary also shows temporal variation (Wen 2006; Cao et al. 2007; Song and Dai 2008).

Concluding Remarks The discovery of the inner core super-rotation has attracted attention of the academia and popular media. It has implications for understanding the geodynamo and angular momentum transfer in the interior of the Earth (Buffett and Glatzmaier 2000). The origin of the Earth’s magnetic field has been regarded as one of the great unsolved problems in

Differential Rotation of the Earth’s Inner Core

modern physics. The observation of the inner core rotation provides a unique observational constraint on the geodynamo at the center of the earth. The existence of the differential rotation of the inner core has now generally, although not universally, been accepted. However, major questions remain. What is the acceptable range of the rotation rate? Is the inner core rotation variable? Does the rotation change direction? Does the inner core oscillate within a certain range? The inner core possesses large hemispheric scale variations (Tanaka and Hamaguchi 1997; Niu and Wen 2001). This is difficult to reconcile with an inner core of a constant rotation, which may be expected to average out the lateral varying structure of the inner core over geological time as the inner core grows from the crystallization of liquid iron. If the rotation is variable or oscillating, what is the time scale? What is the driving force? How does it relate to the geodynamo processes? Recently, Lindner et al. (2010) proposed a nonparametric modeling method, which is able to separate mantle structure, inner core structure, and inner core motion. The study suggests an average rate of inner-core rotation of about 0.39° per year and that the rotation has accelerated from about 0.24° to about 0.56° per year within the last 55 years. The speed of the differential rotation, averaging about 10 km per year at the inner core equator, is by 50,000 times the fastest motion of the tectonic plates (20 cm per year) at the Earth’s surface. The minimum torque acting on the inner core is estimated 1.19  1016 N m, which could easily result from the imbalance of much larger electromagnetic and gravitational torques. With the continuing accumulation of high quality seismic data, I believe we’ll reach a new understanding on the inner core motion within next 50 years.

Cross-References ▶ Body Waves ▶ Core Dynamo ▶ Core-Mantle Coupling ▶ Earth Rotation ▶ Earth, Density Distribution ▶ Earth’s Structure, Core ▶ Earth’s Structure, Global ▶ Earthquakes, Location Techniques ▶ Geodynamics ▶ Geomagnetic Field, Global Pattern ▶ Geomagnetic Field, Theory ▶ Magnetic Anisotropy ▶ Mantle Convection ▶ Propagation of Elastic Waves: Fundamentals ▶ Radioactivity in Earth’s Core ▶ Seismic Properties of Rocks

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Bibliography Buffett BA, Glatzmaier GA (2000) Gravitational braking of inner-core rotation in geodynamo simulations. Geophys Res Lett 27:3125–3128 Cao AM, Masson Y, Romanowicz B (2007) Short wave-length topography on the inner-core boundary. Proc Natl Acad Sci U S A 104(1):31–35 Creager KC (1997) Inner core rotation rate from small-scale heterogeneity and time-varying travel times. Science 278:1284–1288 Glatzmaier GA, Roberts PH (1995) A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle. Phys Earth Planet Inter 91:63–75 Gubbins D (1981) Rotation of the inner core. J Geophys Res 86:11695–11699 Laske G, Masters G (2003) The Earth’s free oscillations and the differential rotation of the inner core. In: Dehant V et al (eds) Earth’s core: dynamics, structure, rotation. Geodynamics series, vol 31. American Geophysical Union, Washington, DC, pp 5–21 Li A, Richards PG (2003) Using earthquake doublets to study inner core rotation and seismicity catalog precision. Geochem Geophys Geosyst 4(9):1072. https://doi.org/10.1029/2002GC000379 Lindner D, Song X, Ma P, Christensen DH (2010) Inner core rotation and its variability from nonparametric modeling. J Geophys Res 115: B04307. https://doi.org/10.1029/2009JB006294 Morelli A, Dziewonski AM, Woodhouse JH (1986) Anisotropy of the inner core inferred from PKIKP travel times. Geophys Res Lett 13:1545–1548 Niu FL, Wen LX (2001) Hemispherical variations in seismic velocity at the top of the Earth’s inner core. Nature 410:1081–1084 Ovchinnikov VM, Adushkin VV, An VA (1998) On the velocity of differential rotation of the Earth’s inner core. Dokl Akad Nauk 362:683–686 Poupinet G, Ellsworth WL, Frechet J (1984) Monitoring velocity variations in the crust using earthquake doublets: an application to the Calaveras Fault, California. J Geophys Res 89:5719–5731 Poupinet G, Souriau A, Coutant O (2000) The existence of an inner core super-rotation questioned by teleseismic doublets. Phys Earth Planet Inter 118:77–88 Song XD (1997) Anisotropy of the earth’s inner core. Rev Geophys 35:297–313 Song XD (2000) Joint inversion for inner core rotation, inner core anisotropy, and mantle heterogeneity. J Geophys Res 105:7931–7943 Song XD (2003) Three-dimensional structure and differential rotation of the inner core. In: Dehant VM, Creager KC, Zatman S, Karato S (eds) Earth core: dynamics, structure, rotation. Geodynamics series, vol 31. American Geophysical Union, Washington, DC, pp 45–63 Song XD, Dai W (2008) Topography of Earth’s inner core boundary from high-quality waveform doublets. Geophys J Int 175:386–399. https://doi.org/10.1111/j.1365-246X.2008.03909.x Song XD, Li AY (2000) Support for differential inner core superrotation from earthquakes in Alaska recorded at South Pole station. J Geophys Res 105:623–630 Song XD, Richards PG (1996) Seismological evidence for differential rotation of the Earth’s inner core. Nature 382:221–224 Souriau A (1998) New seismological constraints on differential rotation of the inner core from Novaya Zemlya events recorded at DRV, Antarctica. Geophys J Int 134:F1–F5 Souriau A, Garcia R, Poupinet G (2003) The seismological picture of the inner core: structure and rotation. Compt Rendus Geosci 335:51–63 Sun XL, Poupinet G, Song XD (2006) Examination of systematic mislocation of South Sandwich Islands earthquakes using station pairs: implications for inner core rotation. J Geophys Res 111:B11305. https://doi.org/10.1029/2005JB004175

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148 Tanaka S, Hamaguchi H (1997) Degree one heterogeneity and hemispherical variation of anisotropy in the inner core from PKP(BC)PKP(DF) times. J Geophys Res 102:2925–2938 Tromp J (2001) Inner-core anisotropy and rotation. Annu Rev Earth Planet Sci 29:47–69 Vidale JE, Dodge DA, Earle PS (2000) Slow differential rotation of the Earth’s inner core indicated by temporal changes in scattering. Nature 405:445–448 Wen L (2006) Localized temporal change of the Earth’s inner core boundary. Science 314:967–970

Differential Rotation of the Earth’s Inner Core Woodhouse JH, Giardini D, Li X-D (1986) Evidence for inner core anisotropy from free oscillations. Geophys Res Lett 13:1549–1552 Xu XX, Song XD (2003) Evidence for inner core super-rotation from time-dependent differential PKP travel times observed at Beijing Seismic Network. Geophys J Int 152:509–514 Zhang J, Song XD, Li YC, Richards PG, Sun XL, Waldhauser F (2005) Inner core differential motion confirmed by earthquake waveform doublets. Science 309:1357–1360

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Earth Rotation Harald Schuh1 and Sigrid Böhm2 1 Department 1 “Geodesy” at Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany 2 Research Division Higher Geodesy, Department of Geodesy and Geoinformation, TU Wien, Vienna, Austria

Definition Earth Earth rotation

Solid Earth including oceans and atmosphere. Temporal variation of the orientation and the rotation speed of the Earth.

Introduction The rotation of the Earth or Earth rotation, respectively, specifies the spatiotemporal change of the Earth rotation vector. The direction of the Earth rotation vector corresponds to the instantaneous rotation axis of the Earth and its absolute value equals the rotation speed. The Earth’s rotation is not uniform and is given in terms of Earth orientation parameters (EOP): precession and nutation are long-term and periodic changes of the direction of the Earth rotation vector with respect to a space-fixed reference system. Polar motion is the variation of the direction of the Earth rotation vector with respect to an Earth-fixed reference system (qv ▶ “Geodesy, Networks, and Reference Systems”). Changes in the Earth rotation speed are expressed as deviations of Universal Time 1 (UT1) from the uniform atomic time (Universal Time Coordinated, UTC) dUT1 ¼ UT1–UTC or as variations in the length of day (LOD). The subgroup of polar motion and dUT1 or LOD is called Earth rotation parameters (ERP). Fundamental information on Earth rotation theory and observation and about the relations of Earth rotation variations with geo© Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

physical processes is given in the seminal works of Munk and MacDonald (1960), Lambeck (1980), and Moritz and Mueller (1987). While the exist`ence of precession was already known to the Greek astronomer Hipparchus in the second century before Christ, nutation was not discovered before the eighteenth century by James Bradley. Observations of polar motion were taken for the first time by the Bonn astronomer Friedrich Küstner at the end of the nineteenth century by measuring latitude variations. From the 1970s to the twentieth century onward, space geodetic techniques like Very Long Baseline Interferometry (VLBI) (qv ▶ “Very Long Baseline Interferometry”), Satellite Laser Ranging (SLR), Lunar Laser Ranging (LLR), Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), and the Global Navigation Satellite Systems (GNSS) like the Global Positioning System (GPS) (qv ▶ “GPS, Data Acquisition, and Analysis”) have been employed in addition to astronomical methods. Approximately since the year 2000, the latter are no longer routinely applied. Nowadays, the achievable accuracies of the measurements are better than 2  104 arcsec or less than 0.6 cm, if projected to the Earth’s surface. EOP can be determined from space geodetic observations within the respective parameter estimation process. The transformation between Earth-fixed and space-fixed system and thus the EOP are thereby components of the technique’s observation equation and can be solved for as unknowns. The most precise technique to observe polar motion is GNSS, whereas precession/nutation and dUT1 can be measured directly only by VLBI. Due to the correlation of nutation and dUT1 with the orbital elements of the satellites, satellite techniques are only sensitive to the time derivation of those parameters, i.e., nutation rates and LOD. Promising new devices for the observation of high-frequency variations of the instantaneous Earth rotation vector are large ring laser gyroscopes. These instruments can access absolute rotation by measuring the beat frequency of two laser beams rotating in opposite direction, the Sagnac frequency (Schreiber et al. 2004).

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Information about the long-term behavior of the EOP is obtained from historical records of lunar and solar eclipses and from the analysis of sedimentary deposition. Evidence for a secular increase in LOD can be found for instance in paleontological studies of coral growth rate. The observed EOP show a wide spectrum of variations. Its interpretation allows for drawing valuable conclusions about the structure of the Earth and the dynamical features of the Earth’s interior, the atmosphere, hydrosphere, and cryosphere. Even anthropogenic influences on Earth rotation such as the impact of mass transfer due to an increased CO2 emission are subject of scientific investigations (de Viron et al. 2002). Medium and long-period as well as long-term mass displacements affect Earth rotation (cf. e.g., Gross (2007) or Seitz and Schuh (2010) for a comprehensive compilation and further references). According to the conservation of total angular momentum of the Earth in short terms, the rotation of the solid Earth undergoes variations, which are mirror image to changes in atmosphere and oceans. Any mass variation in one or more components of the system Earth that changes the Earth inertia tensor leads to a corresponding variation in the Earth rotation. One can also think of a mass movement which does not change the inertia tensor, because once a mass element moved away it is replaced by another, like in a circular ocean current. Such a mass transport causes a motion relative to the considered reference frame, which affects Earth rotation as well. In which way changes of the inertia tensor and relative motions can be related to variations in Earth rotation is shown in the successive section.

This is one form of Euler’s dynamical equations for rigid body rotation referred to a body-fixed coordinate system. The angular momentum H can be expressed as the product of the tensor of inertia I and the angular velocity vector v. The inertia tensor is a symmetric matrix containing the moments of inertia and the products of inertia of a rotating body and thus characterizing the mass distribution in the body. Since single particles do not move with respect to the body-fixed system, this tensor is invariant in the case of a rigid body. For a nonrigid (deformable) body, the inertia tensor becomes time variable, and the particles can move with respect to the body frame, thus allowing for relative motions which introduce relative angular momentum. The angular momentum H of a rotating deformable body is then written as H ¼ Iv þ h

ð3Þ

with h denoting relative angular momentum. The first summand is often referred to as mass or matter term, and the second is called motion term. As the Earth is a nonrigid body, the equation of rotational motion 2 has to be extended considering the above stated differences to the motion of a rigid body, leading to L¼

d ðIv þ hÞ þ v  ðIv þ hÞ dt

These are the Euler-Liouville equations or simply Liouville equations. The deviations from uniform, rigid rotation are formulated as follows for the rotation vector v: o1 ¼ Om1 , o2 ¼ Om2 , o3 ¼ Oð1 þ m3 Þ

Mathematical Formulation of Earth Rotation The dynamical equation of motion of a rotating body with respect to a space-fixed system is given by L¼

dH dt

ð1Þ

relating the torque L acting on the body to the temporal change of its angular momentum H. This is the basic relation for the development of (Newtonian) precession/nutation theories, since it describes the motion of the rotating body in a space-fixed reference system. To characterize polar motion and changes of the rotation rate of a rotating body, Eq. 1 has to be referred to body-fixed axes, rotating with the angular velocity v: L¼

dH þvH dt

ð2Þ

ð4Þ

ð5Þ

where Ω is the mean angular velocity of the Earth. The mi are small dimensionless quantities describing the excursions of the rotation vector from its uniform rotation due to time-variable changes in the mass distribution of the Earth system and relative particle motion. Assuming the body-fixed axes to be principal axes of inertia and the body to be rotationally symmetric, these mass changes can be taken into consideration by composing the tensor of inertia from constant and time-variable parts: I 11 ¼ A þ DI 11 I ij ¼ DI ij ,

I 22 ¼ A þ DI 22 i 6¼ j

I 33 ¼ C þ DI 33

ð6Þ

where the constant parts are the polar moment of inertia C, and the equatorial moment of inertia A of the undeformed Earth and the time-variable components are the quantities ΔIij. If second-order terms are neglected and the relations 5 and 6 are introduced, the linearized equations of motion can be rewritten as

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1 m_ ¼ c1 , sE 2 1 m2 þ m_ 1 ¼ c2 , sE m3 ¼ c3 þ const:

m1 

ð7Þ


 with sE ¼ CA A O being the Euler frequency, which would be the frequency of the resonance of a rigid Earth, corresponding to a circular motion of the rotation axis with a period of approximately 305 days. If the Earth is considered to be an elastic or at least deformable body, the Euler frequency is replaced by the complex Chandler frequency  2p i sC ¼ T C 1 þ 2Q , with TC ~ 433 days denoting the observed period of the Chandler wobble and Q ¼ 30. . .200 being a dimensionless dissipation factor (see next section for a brief discussion of the Chandler wobble). The ci are called excitation functions. They contain the changes of the tensor of inertia and the relative angular momenta:  2  1 O DI 13 þ ODI_23 þ Oh1 þ h_2  L2 O ð C  AÞ  2  1 O DI 23  ODI_13 þ Oh2  h_1 þ L1 ð8Þ c2 ¼ 2 O ð C  AÞ  ðt 1 c3 ¼ 2 O2 DI 33 þ Oh3  O L3 dt O C 0 c1 ¼

2

The first two equations of 7 for m1 and m2 express polar motion, while the third equation for m3 describes the magnitude of the rotation speed and hence LOD variations. Provided that all the time-variable parts of the excitation functions (changes of the inertia tensor ΔIij, relative angular momenta hi, and external torques Li) are introduced as known quantities from models or observations, the Euler-Liouville equations can be solved for v. Earth rotation variations can be thus calculated or predicted, respectively, from observed or modeled changes in the components of the system Earth, such as atmosphere or oceans. In general, relative motions (relative angular momenta) contribute more to LOD variations while the major effect on polar motion comes from alterations of the inertia tensor. The excitation functions are in practice mostly replaced by so-called angular momentum functions, introduced by Barnes et al. (1983). Elaborate explications of the basic Earth rotation formalism can be found in the fundamental books mentioned in the introduction.

Definition and Observation of Earth Orientation Parameters The elements which are used to perform the transformation between a space-fixed and an Earth-fixed reference system are

commonly referred to as EOP. The realizations of such reference systems by assigning coordinates to selected celestial objects or terrestrial sites, respectively, are called reference frames. The definition and maintenance of the reference frames is one major task of the International Earth Rotation and Reference Systems Service (IERS). Conventional reference frames are the space-fixed International Celestial Reference Frame (ICRF) and the Earth-fixed International Terrestrial Reference Frame (ITRF). The current version of the ICRF is its third realization, the ICRF3 (Charlot 2018). The ITRF2014 (Altamimi et al. 2016) has been adopted as the official terrestrial reference frame and will be followed by the ITRF2020 which is currently under construction. The definition of the EOP actually depends on the kind of the applied transformation method. The IERS recommends the transformation according to the IAU (International Astronomical Union) 2000 and 2006 Resolutions (Petit and Luzum 2010). The transformation procedure is defined from the Geocentric Celestial Reference System (GCRS) to the International Terrestrial Reference System (ITRS) or vice versa. The ICRS is not a geocentric system – its origin is located in the barycenter of the solar system. To transfer from a barycentric system to a geocentric system, effects, which are not directly related to Earth rotation, like aberration and parallaxes have to be taken into account. The orientation of the ICRS however corresponds to the orientation of the GCRS. The transition from the GCRS to the ITRS is described as a sequence of time-dependent rotation matrices: ½ITRS ¼ W ðtÞ  RðtÞ  QðtÞ  ½GCRS

ð9Þ

W(t) (with “W” for wobble) designates the polar motion matrix. It contains the coordinates xp and yp of the reference pole CIP (Celestial Intermediate Pole) in the Earth-fixed system and the angle s0 , which provides the position of the TIO (Terrestrial Intermediate Origin) on the equator of the CIP. The TIO and the Celestial Intermediate Origin (CIO) realize an instantaneous prime meridian in the respective system. These terms are part of the transformation concept using the non-rotating origin (NRO), which replaced the older transformation scheme with ecliptic (plane of the Earth’s orbit) and equator. The rotation between TIO and CIO is performed with R(t) using a quantity named Earth Rotation Angle (ERA). The ERA is directly related to UT1, which is relevant for Earth rotation research. Precession and nutation are represented by the matrix Q(t). It comprises rotations around the angles X and Y, the coordinates of the CIP in the celestial system and around the angle s, which locates the CIO on the equator of the CIP. In case of using the transformation according to IAU 2000 resolutions, the five quantities {xp, yp, dUT1, X, Y} therefore represent the EOP. If

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the older transformation concept based on ecliptic and equator is applied, X and Y are replaced by Δε and Δc, nutation in obliquity and longitude. The before cited Celestial Intermediate Pole is the reference pole measurements of space geodetic techniques are related to. The CIP thus defines the observed axis. This is a pure convention, realized by an accordingly adapted precession-nutation theory. The direction toward the CIP does not correspond to any physically defined axis, like the rotation axis, the figure axis, or the angular momentum axis; nevertheless, it is possible to mathematically connect it to each of those axes. Casually, it is often said that Earth rotation observations represent the motion of the Earth rotation axis. Regarding the measurements of space geodetic techniques, this is actually not entirely correct, since they are not sensitive to the instantaneous Earth rotation vector but to the complete rotation matrix only. The CIP defines an intermediate pole, separating the motion of the pole of the ITRS in the GCRS into a celestial part and a terrestrial part. The celestial part (precession and nutation, {X, Y}) comprises all motions with periods greater than 2 days, as seen from space. This is equivalent to frequencies between 0.5 and +0.5 cycles per sidereal day (cpsd). With the minus sign labeling retrograde motions (opposite to the sense of Earth rotation) and the plus sign labeling prograde motions (in the sense of Earth rotation), all motions outside of the retrograde diurnal band in the Earth-fixed system, i.e., frequencies below 1.5 and above 0.5 cpsd, are allocated to the terrestrial part (polar motion, {xp, yp}). The celestial motions, precession and nutation, are long-term and periodic changes of the direction of the Earth rotation axis or actually of the CIP axis, with respect to a space-fixed reference system, which is realized, for example, by the positions of extragalactic radio sources observed by VLBI. Due to precession, the Earth axis moves with respect to the space-fixed system on a cone with an aperture of 23.5°, corresponding to the angle between the Earth’s equator plane and the ecliptic. The revolution period amounts to approximately 25,800 years. This time interval is also called Platonic year. The tidal forces of moon and sun are responsible for this steady motion. Since the Earth is not a sphere but can be characterized as an oblate spheroid (qv ▶ “Geodesy, Figure of the Earth”) and its rotation axis is inclined by 23.5° with respect to the ecliptic normal, the gravitational torques force the equatorial plane into the ecliptic. Because of its rotation, the Earth acts like a gyroscope and swerves by moving on the abovementioned cone, whose surface can be calculated very precisely. The smaller periodic change in the direction of the Earth rotation axis that is superimposed to precession is called nutation. This comprises motions of the Earth rotation axis with respect to the space-fixed system with periods from a few days to 18.6 years, caused by gravitational influences of sun, moon, and the planets of our solar system. Precession and

Earth Rotation

nutation can be modeled and predicted precisely using timedependent harmonic series expansions. The arguments of the individual harmonic terms are thereby calculated from combinations of five fundamental astronomical arguments. The currently most precise precession-nutation model adopted by IAU Resolutions (2000, 2006) is IAU 2006/2000A. In this model, effects of planets, ocean tides, mantle anelasticity, and electromagnetic coupling mechanisms between core and mantle as well as between inner and outer core are considered (Mathews et al. 2002). As for the planets, their direct effect on Earth as well as the indirect effect of the sun acting on a planet, which also causes subtle nutation terms, is taken into account. Remaining parts of the axis motion with respect to the space-fixed system that are not covered by the precessionnutation model can be measured by means of VLBI and are provided by the IERS as so-called celestial pole offsets. These residuals originate from still deficiently modeled and unpredictable effects, like the free core nutation (FCN). The FCN is a proper mode of the Earth, caused by a misalignment of the rotation axes of mantle and core. Whereas first theoretical estimations indicated a period of about 460 days for this mode, VLBI measurements show that the FCN period is more likely to be around 430 days with highly variable amplitude. The terrestrial part of the change of the direction of the Earth axis or rather the CIP axis is designated polar motion (Fig. 1). Polar motion has an order of magnitude of several meters and is expressed in a two-dimensional coordinate system by the pole coordinates xp and yp. According to the definition of the IERS and its precedent organizations, the x-axis is oriented in the direction of the Greenwich meridian, and the y-axis is oriented positively toward 90° west longitude. Already in 1765, the Swiss mathematician Leonhard Euler calculated a circular motion of the pole (of an Earth at that time assumed to be rigid) with a period of around 304 days. Today, we know that polar motion is mainly composed from an annual variation and the Chandler oscillation or Chandler wobble, named after Seth Carlo Chandler who first detected a period of approximately 14 months when analyzing polar motion observations at the end of the nineteenth century. The Chandler wobble is another proper motion of the Earth with strongly varying amplitude. More precisely, this nearly circular motion is a damped oscillation, which would have vanished due to friction in the Earth’s interior after a few decades if there was not a constantly revolving excitation. Although the Chandler oscillation is known for more than 100 years, its underlying excitation mechanism is still under investigation. Today, there is broad consensus that the necessary excitation energy emerges from irregular processes in the atmosphere-ocean system (Gross 2000). A radical change in the phase of the Chandler wobble around 1925 could also not yet fully be explained. Variations of the Earth’s magnetic field are nowadays quoted as potential causes for the phase variations. The interference of the

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2020 2020

2015

2010 2010 2000 2005 1990 2000 1980

Time (yr)

1995

1970

1990

1960

1985

1950

1980

1940 1930

1975

1920 1970 1910 1965 1900 1960 -10 0

x p (m)

10

0 10 y (m) p

Earth Rotation, Fig. 1 Polar motion from the combined EOP series C04 14 of the IERS (1962–2019, daily resolution) and position of the mean pole from 1900 to 2017

Chandler wobble and the annual motion leads to a beat-like rising and ebbing of the polar motion amplitude to a maximum of 9 m with a beat period of about 6.3 years. The polar motion spectrum below 1 year is dominated by irregular variations appearing in 3 and 5 months intervals. The most important short-period variations are due to the ocean tides caused by sun and moon with essentially diurnal and semidiurnal periods (Karbon et al. 2019), albeit the total effect is only about 1/100 of the Chandler wobble. Analysis of longterm polar motion shows a distinct variation of approximately 11 years. It is usually associated with processes in the Earth’s interior but could also be connected to a corresponding cycle of the solar activity. In addition there are other, decadal variations with periods around 30 years and between 70 and

80 years. These are assumed to be caused by geodynamic couplings between Earth’s core and mantle. Recently, Adhikari and Ivins (2016) identified terrestrial water storage variability as the source of decadal-like changes in observed polar motion during their study period 2003–2015. Secular polar motion, i.e., the long-term linear trend in the position of the pole, amounts to about 10.5 cm/year toward Labrador, Canada (e.g., Schuh et al. 2001 or Adhikari et al. 2018). This effect is supposed to be predominantly due to melting of the polar ice masses and postglacial rebound, but also long-term mass movement due to mantle convection is suspected to play a role. For an extensive review on precession, nutation, and wobble of the Earth, refer to Dehant and Mathews (2015). LOD is used to express the speed of Earth rotation alternatively to the difference dUT1 between UT1, which is directly connected to the Earth rotation speed, and the regular atomic time UTC. The parameter, which is usually quoted to characterize changes in LOD, is actually ΔLOD, the so-called excess LOD which represents the deviation of the effective LOD from the nominal value of 86,400 s (Fig. 2). Variations of LOD can be assigned to different period ranges. In the short-period range with periods from around 1 day and half a day, the strongest influence emerges from the ocean tides caused by sun and moon (Karbon et al. 2018). Hydrodynamic models or altimetry (qv ▶ “Geoid Determination, Theory and Principles”) provide variations of ocean heights and velocities of the water masses, from which global changes of angular momentum of the world oceans are calculated. These are subsequently used to estimate the oceanic influence on the Earth rotation parameters. In this way high-frequency ERP variations can be predicted from tide and circulation models, and they can already be observed by the high-precision space geodetic techniques mentioned above. In the range of a few weeks to months, periods of 14 and 28 days, due to solid Earth tides (qv ▶ “Earth Tides”), are dominant. In addition, there are other strong variations between 40 and 90 days basically excited by zonal winds. Seasonal variations due to changes in the angular momentum of the atmosphere show semiannual and annual periods. The variation of LOD with a period of approximately 1 year is predominantly due to annually changing wind patterns. A definite amplification of the annual variation every 4–6 years is associated with largescale climate signals related to the El Niño phenomenon by which an ocean circulation with a very characteristic pattern in the Southern Pacific is connected to variations of the meteorological parameters. The existence of decadal fluctuations of LOD at the ms level suggests an internal coupling between the Earth’s core and mantle exhibiting torques on the order of 1018 Nm. Four coupling mechanisms and associated torques are commonly identified, the viscous, topographic, gravitational, and electromagnetic torques. Except for the viscous torque which seems too weak, any of the other

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Earth Rotation, Fig. 2 Excess length of day from the combined EOP series C04 14 of the IERS (1962–2019, daily resolution)

torques, would be capable of explaining these LOD variations (qv ▶ “Core-Mantle Coupling”). Because of tidal friction and long-term mass variations, a secular prolongation of the day by about 1.8 ms in 100 years is observed as well (Stephenson et al. 2016). This is, of course, an average long-term trend deduced from eclipse data over the last 2,700 years. Besides the secular trend, there is also evidence of LOD fluctuations on a timescale of centuries and some indication of an around 1500 years oscillation.

IERS International Earth Rotation and Reference Systems Service The IERS is an international service for the determination of EOP and their dissemination to interested users. The definition and realization of reference systems for geodesy and astronomy can be regarded as its superior task. According to the terms of reference (Dick and Thaller 2018), the primary objectives of the IERS are to serve the astronomical, geodetic, and geophysical communities by providing the following: • The International Celestial Reference System (ICRS) and its realization, the International Celestial Reference Frame (ICRF). • The International Terrestrial Reference System (ITRS) and its realization, the International Terrestrial Reference Frame (ITRF).

• EOP required to study Earth orientation variations and to transform between the ICRF and the ITRF. • Geophysical data to interpret time/space variations in the ICRF, ITRF, or EOP, and to model such variations. • Standards, constants, and models (i.e., conventions) encouraging international adherence. In addition, the IERS collects, archives, and distributes several products such as reference frames, monthly Earth orientation data, daily rapid service estimates of near realtime Earth orientation data, and their predictions. The IERS also announces the differences between astronomical and civil time for time distribution by radio stations and leap seconds which – if necessary to keep the differences dUT1 smaller than 0.9 s – are added at midnight of July 31 or December 31. Further products are related to global geophysical fluids such as mass and angular momentum distribution, annual reports and technical notes on conventions and other topics, and long-term Earth orientation information. The IERS began operation on January 1, 1988, as common institution of the IAU and the International Union of Geodesy and Geophysics (IUGG) and replaced thereby the former International Polar Motion Service (IPMS) and the Earth rotation section of the Bureau International de l’Heure (BIH). The service consists, among other parts of a Central Bureau, combination centers of the space geodetic techniques and several product centers, for example, for the collection of data about geophysical influences on Earth rotation (Global Geophysical Fluids Center, GGFC). These are atmospheric

Earth Rotation

and oceanic variations, hydrodynamic effects like groundwater variations, and processes in the Earth’s interior which lead to changes in the rotational behavior of the Earth. The Central Bureau is located at the BKG (Bundesamt für Kartographie und Geodäsie) in Frankfurt am Main, Germany. Apart from regular publication of EOP, the IERS also issues guidelines (IERS Conventions, Petit and Luzum (2010)), which contain models and standards recommended for the data processing of space geodetic techniques.

Summary Earth rotation is conventionally described by the EOP, which represent the link between a space-fixed (celestial) and an Earth-fixed (terrestrial) coordinate system. Those EOP expressing the temporal variations of the orientation of the Earth correspond to precession/nutation (changes of the direction of Earth rotation axis with respect to a space-fixed reference frame) and polar motion (wobbling of the Earth with respect to its axis). Small variations of the speed of Earth rotation are expressed by UT1 minus the uniform atomic time (UTC) or as variations in the LOD. A precise knowledge of the Earth’s attitude is needed for all positioning and navigation tasks on Earth and in space. It also gives fundamental information about the interactions between the various geophysical components of system Earth and allows deriving conclusions about phenomena of celestial mechanics. The EOP are nowadays monitored by space geodetic techniques and assembled and published by the IERS.

Cross-References ▶ Core-Mantle Coupling ▶ Earth Tides ▶ Geodesy, Figure of the Earth ▶ Geodesy, Networks, and Reference Systems ▶ Geoid Determination, Theory and Principles ▶ GPS, Data Acquisition, and Analysis ▶ Very Long Baseline Interferometry

Bibliography Adhikari S, Ivins ER (2016) Climate-driven polar motion: 2003–2015. Sci Adv 2(4):e1501693. https://doi.org/10.1126/sciadv.1501693 Adhikari S, Caron L, Steinberger B, Reager JT, Kjeldsen KK, Marzeion B, Larour E, Ivins ER (2018) What drives 20th century polar motion? Earth Planet Sci Lett 502:126–132. https://doi.org/10. 1016/j.epsl.2018.08.059 Altamimi Z, Rebischung P, Métivier L, Collilieux X (2016) ITRF2014: a new release of the International Terrestrial Reference Frame modeling non-linear station motions. J Geophys Res 121:6109–6131. https://doi.org/10.1002/2016JB013098

155 Barnes RTH, Hide R, White AA, Wilson CA (1983) Atmospheric angular momentum functions, length-of-day changes and polar motion. Proc R Soc Lond A387:31–73 Charlot P (2018) The third realization of the International Celestial Reference Frame. Presentation given at the IAU General Assembly 2018, Vienna. https://www.iau.org/static/science/scientific_bodies/ divisions/a/2018/Charlot.pdf. Accessed May 2019 de Viron O, Dehant V, Goosse H, Crucifix M, Participating CMIP Modeling Groups (2002) Effect of global warming on the lengthof-day. Geophys Res Lett 29(7):1146. https://doi.org/10.1029/2001 GL013672 Dehant V, Mathews PM (2015) Precession, nutation and wobble of the earth. Cambridge University Press, Cambridge. https://doi.org/10. 1017/CBO9781316136133 Dick WR, Thaller D (eds) (2018) IERS annual report 2017. International Earth Rotation and Reference Systems Service, Central Bureau. Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main Gross RS (2000) The excitation of the Chandler wobble. Geophys Res Lett 27(15):2329–2332 Gross RS (2007) Earth rotation variations – long period. In: Herring TA (ed) Physical geodesy. Treatise on geophysics, vol 3. Elsevier, Amsterdam IAU Resolutions (2000) http://www.iau.org/static/resolutions/IAU2000 _French.pdf IAU Resolutions (2006) http://www.iau.org/static/resolutions/IAU2006 _French.pdf Karbon M, Balidakis K, Belda S, Nilsson T, Hagedoorn J, Schuh H (2018) Long-term evaluation of ocean tidal variation models of polar motion and UT1. Pure Appl Geophys 175(5):1611–1629. Springer International Publishing AG, part of Springer Nature. https://doi.org/ 10.1007/s00024-018-1866-1 Karbon M, Balidakis K, Belda S, Nilsson T, Hagedoorn J, Schuh H (2019) Long-term evaluation of ocean tidal variation models of polar motion and UT1. In: Braitenberg C, Rossi G, Geodynamics and Earth Tides Editor (eds) Geodynamics and Earth tides observations from global to micro scale. Pageoph topical volumes. Birkhäuser, Basel, pp 17–35, eBook ISBN 978-3-319-96277-1, Softcover ISBN 978-3319-96276-4, Buchreihen ISSN 2504-3625. https://doi.org/10.1007/ 978-3-319-96277-1 Lambeck K (1980) The Earth’s variable rotation, geophysical causes and consequences. Cambridge University Press, Cambridge Mathews PM, Herring TA, Buffett BA (2002) Modeling of nutation and precession: new nutation series for nonrigid Earth, and insights into the Earth’s interior. J Geophys Res 107(B4). https://doi.org/10.1029/ 2001JB000390 Moritz H, Mueller II (1987) Earth rotation: theory and observation. Ungar, New York Munk WH, MacDonald GJF (1960) The rotation of the Earth: a geophysical discussion. Cambridge University Press, Cambridge Petit G, Luzum B (eds) (2010) IERS conventions (2010). IERS technical note 36. Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main Schreiber U, Velikoseltsev A, Rothacher M, Klügel T, Stedman G, Wiltshire D (2004) Direct measurement of diurnal polar motion by ring laser gyroscopes. J Geophys Res 109(B6). https://doi.org/10. 1029/2003JB002803 Schuh H, Nagel S, Seitz T (2001) Linear drift and periodic variations observed in long time series of polar motion. J Geod 74:701–710 Seitz F, Schuh H (2010) Earth rotation. In: Xu G (ed) Sciences of geodesy. Springer, Berlin Stephenson FR, Morrison LV, Hohenkerk CY (2016) Measurement of the Earth’s rotation: 720 BC to AD 2015. Proc R Soc A 472: 20160404. https://doi.org/10.1098/rspa.2016.0404

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Earth Tides

Earth Tides

a Ω Moon

John M. Wahr (Deceased)

Definition

b The Earth tide is the deformation of the solid Earth caused by the gravitational attraction of the Sun and moon. The most striking gravitational effects of the Sun and moon are the orbital motion of the Earth’s center of mass. The moon causes the Earth to orbit the Earth–moon center of mass, and the Sun causes that center of mass to orbit the Sun.

The Tidal Force But the Sun and moon also cause points within the Earth to be displaced relative to each other: they cause the Earth to deform. For example, the side of the Earth nearest the moon is attracted toward the moon more than is the center of the Earth. And the side farthest from the moon is attracted less than the center of the Earth. Consequently, both the far and near sides of the Earth are pulled radially outward away from the center. The regions of the Earth that are at right angles to the Earth– moon vector are pulled radially inward. This deformation pattern is illustrated in Fig. 1. The total gravitational acceleration vectors due to the moon are shown in Fig. 1a. The vectors all point toward the moon, but are of unequal length and direction, although the differences are greatly exaggerated in the figure. The orbital acceleration is, to a high degree of accuracy, equal to the acceleration vector at the Earth’s center of mass. Subtracting that vector from the other vectors in Fig. 1a results in the pattern of vectors shown in Fig. 1b. These residual vectors represent that portion of the lunar gravitational force tending to deform the Earth. The residual acceleration field multiplied by the local material density is defined as the lunar tidal force, and the deformation it induces is called the lunar tide. Note from Fig. 1b that because of the Earth’s diurnal rotation the tidal force at a fixed point in the Earth varies through two complete cycles in 1 day; a fixed point rotates through one outward bulge, then through an inward bulge, and then into the other outward bulge, all within a half a rotation period (i.e., within 12 h). This semidiurnal time dependence is split into many periodic terms with frequencies closely spaced about 2 cycles/day, due to the time variability of the orbital motion of the moon. Furthermore, because the moon is not always in the plane of the Earth’s equator, there is also significant variability at frequencies

Ω Moon

c

B Ω A

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Earth Tides, Fig. 1 All three panels show the Earth as seen from above the north pole. The Earth’s diurnal rotation is represented by W. The arrows in (a) illustrate the magnitude and direction of the gravitational acceleration toward the moon. The differences in the lengths and directions of the arrows are greatly exaggerated. The lunar tidal acceleration is defined by subtracting the acceleration vector at the Earth’s center of mass from the other acceleration vectors. The resulting vectors are shown in (b). The tidal force (the local density times the acceleration) deforms the Earth into the elliptical shape shown greatly exaggerated in (c). Points A and B are used in the text to illustrate tidal strain and tilt

closely spaced about 1 cycle/day (imagine, e.g., that the Earth–moon vector is inclined at 45° to the Earth’s rotation axis) and about 0 cycles/day (imagine that the moon is directly above the north pole). The solar tidal force and the solar tide are defined in a similar manner, and can also be decomposed into semidiurnal, diurnal, and long period terms. For a quantitative description, it is useful to work with the tidal potential, denoted here as VT and defined so that its gradient is the tidal force per unit mass. Define a nonrotating coordinate system with origin at the center of the Earth. The total lunar gravitational potential at the point r inside the Earth is (assuming the moon to be a point mass) V¼

GM jr  R j

ð1Þ

where M is the mass of the moon, G is Newton’s gravitational constant, and R is the position vector of the moon. Let r, θ, f

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be the radial distance, colatitude, and eastward longitude of the Earth-fixed point, r, so that f + Ωt (where Ω ¼ 1 cycle/day is the Earth’s angular velocity of rotation) is the azimuthal angle of r in nonrotating, inertial space. The factor 1/|r – R| can be expanded as a sum of complex spherical harmonics, Y m n ðy, fÞ, so that "

1 n h i GM X X r n m m V ¼ Re d Y ðy, fÞeim Ot R n¼0 m¼0 R n n

# ð2Þ

where Re denotes the real part (included in Eq. 2 so that the sum over m includes only m  0), R is the radial coordinate of the moon, and d m n are coefficients that depend on the angular position of the moon relative to nonrotating inertial space. The coordinate R and the moon’s angular position depend on time, due to the moon’s orbital motion. The n ¼ 0 term in Eq. 2 is a spatial constant and so is physically meaningless. It can be ignored. The n ¼ 1 term represents a spatially constant force, equal to the lunar force at the Earth’s center of mass. Consequently, the n ¼ 1 term is removed from Eq. 2 when defining the tidal potential VT. Since r  a where a is the Earth’s radius, and since a/R ≈ 1/ 60 for the moon (and is much smaller for the Sun), the factor (r/R)n in Eq. 2 causes the contributions to Eq. 2 to fall off rapidly with increasing n. For most purposes it is sufficient to keep only the n ¼ 2 terms in VT, SO that VT has the approximate form "

2 h iX r V T ¼ Re cm Y m ðy, fÞeim Ot a m¼0 2 2

# ð3Þ

  2 m where cm are complex, time-varying 2 ¼ GM =Rða=r Þ d 2 coefficients that depend on the orbital coordinates of the moon. The dominant time dependence in Eq. 3 comes from the eimΩt term. This term results in semidiurnal, diurnal, or long period tides, depending on whether m is 2, 1, or 0, respectively. The imΩi time-dependent cm time dependence into 2 in Eq. 3 split the e terms with frequencies closely spaced about mΩ.

The Earth’s Response The tidal force causes ocean tides as well as Earth tides. The observed ocean tide is actually the difference between the response of the ocean and the response of the solid Earth. Ocean tides are complicated and difficult to model, for a number of reasons that need not concern us here. Open ocean tides are typically less than 1 m in height, although

ocean tides at some coastal locations can be as large as several meters peak-to-peak. Earth tides are much easier to understand and model. The tidal force shown in Fig. 1b tends to deform the solid Earth into the elliptical shape shown greatly exaggerated in Fig. 1c. Tidal displacements of the solid Earth are typically several tens of centimeters. But, unlike ocean tides, Earth tides cannot be observed without sensitive instruments. The reason is that Earth tides cause both the ground and the observer to be displaced by the same amount. One way to detect Earth tides is to use a gravimeter. There are three contributions to the observed tidal variation in the gravitational acceleration: (1) the direct attraction of the Sun and moon, (2) the change in the Earth’s gravity field due to tidal deformation within the Earth, and (3) the change in the gravitational acceleration at the gravimeter caused by the radial tidal displacement of the Earth’s surface beneath the gravimeter (if the surface moves outward, gravity decreases, and vice versa). All three contributions are roughly of the same order. The total tidal variation of the observed gravitational acceleration is on the order of 100 mgals or more (1 gal ¼ 1 cm/s2). The unperturbed gravitational acceleration at the Earth’s surface is about 103 gals. So, tides affect gravity at about the 107 level. This is roughly the relative importance of tides on any physical observable, and can be traced to the fact that tidal displacements (a few tens of centimeters) are about 107 of the Earth’s radius (about 6  108 cm). A simple, idealized gravimeter is a mass hanging from a spring. The amount the spring is stretched at equilibrium is proportional to the local gravitational acceleration. So by continually monitoring the length of the spring, tidal variations in gravity can be determined. Although real gravimeters are more complicated than this, the basic idea is the same. Most instruments, in fact, do use springs. To illustrate the effects of tides on gravity, Fig. 2 shows 1 month of hourly gravitational acceleration data as observed with a spring-type instrument at Walferdange, Luxembourg (data courtesy of Olivier Francis). The tides are the dominant signal in the data. The semidiurnal and diurnal tides are especially evident. The longer period (2 weeks and 1 month) amplitude modulations evident in the figure are caused by beating between frequencies separated by 1 cycle/ 13.7 days and between frequencies separated by 1 cycle/ 27.6 days. Long period gravity tides have small amplitudes at the latitude of Walferdange, and are hard to distinguish in Fig. 2. One problem common to all relative gravimeters is calibration. For spring-type instruments, the proportionality

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Earth Tides Gravity measured at Walferdange, luxembourge

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Earth Tides, Fig. 2 One month of hourly gravity data, taken with a spring-type gravimeter at Walferdange, Luxembourg. (Data courtesy of Olivier Francis). The tides are the dominant signal and are described further in the text

constant between the length of the spring and gravity depends on the spring constant, which can never be precisely known. This, for example, could cause errors in the vertical scale in Fig. 2. Another method of detecting Earth tides is to use a strainmeter. This is an instrument that continually monitors the distance between two points on the Earth’s surface, separated by anywhere from a few centimeters to a kilometer. The fact that Earth tides perturb the distance between points can be seen by considering points A and B in Fig. 1c. The distance between those points has been changed by the tidal deformation, because A has moved radially outward more than B. Also, not evident in the figure, A and B have been displaced horizontally by different amounts. Typical tidal variations in the distance between two points are on the order of 107 times the unperturbed distance. A third way to detect tidal deformation is with tiltmeters. These instruments measure the tilt of the Earth’s surface with respect to the direction of gravity. Tidal tilt of the surface can be understood from Fig. 1c by noting that the tidal bulge has affected the slope of the line between points A and B. The direction of gravity also tips during the deformation, but by a different amount. It is this difference that is observed. Typically, observed tidal tilts are a few tenths of a microradian.

Tidal deformation can also be observed using space geodetic techniques, including satellite laser ranging (SLR), very-long-baseline-interferometry (VLBI), and GPS positioning. These techniques are sensitive to displacements of the ground beneath the observing stations. In fact, one of the primary geophysical objectives of these techniques is to detect tectonic motion of the stations. The effects of Earth tides are evident in the data, but they cannot generally be determined with as much relative accuracy as can the tidal effects on surface gravity, strain, or tilt. Satellite orbit tracking can also be used to detect tidal variations in the Earth’s gravity field. Those variations affect the orbit of a satellite, and so can show up in the satellite tracking data. In fact, for lower-flying satellites the ranging data are usually more sensitive to these tidal gravity variations than they are to tidal displacements of the ground stations. These various observable tidal effects can be conveniently described, mathematically, by using dimensionless parameters known as Love numbers, defined as follows. The deformation of a spherically symmetric, nonrotating Earth in response to a Y m n ðy, lÞ applied external potential can be described with that same Y m n ðy, lÞ angular dependence. Mathematically, this is because the Y m n separate spherically symmetric differential equations. The change in the Earth’s gravitational potential at r ¼ a (the unperturbed surface of the Earth) and the radial displacement at r ¼ a are both proportional to Y m n , and the horizontal displacement vector at r ¼ a is proportional to ∇ Y m n : The proportionality constants depend on n but are independent of m. Thus, the same constants are pertinent for all three Y m 2 terms in Eq. 3. Let F(θ, l), Ur(θ, l), Uθ(θ, l), and Ul(θ, l) denote tidal effects at the Earth’s surface (r ¼ a) on, respectively, the Earth’s gravitational potential, and the radial, southward, and eastward displacements of the point (θ, l). Then, F, Ur, Uθ, and Ul for our assumed spherical Earth have the form Fðy, lÞ ¼ kV aT ðy, lÞ h Ur ðy, lÞ ¼ V aT ðy, lÞ g ð4Þ l Uy ðy, lÞ ¼ @ y V aT ðy, lÞ g l Ul ðy, lÞ ¼ @ V a ðy, lÞ g sin y l T where k, h, and l are the dimensionless Love numbers, g is the unperturbed gravitational acceleration at r ¼ a, and " # 1 X a m m im Ot V T ðy, lÞ ¼ Re c2 Y 2 ðy, lÞe ð5Þ m¼0

is the tidal potential (Eq. 3) evaluated at r ¼ a.

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159

Tidal variations in gravity, strain, tilt, surface displacements, and satellite orbital motion can all be parameterized in terms of k, h, and l. As one example, consider the gravitational acceleration at a fixed point on the Earth’s surface. There are three contributions to this acceleration, as discussed above. First, the direct gravitational acceleration from the moon (or Sun) in the inward radial direction is @ r V T ðr ¼ aÞ ¼  2a V aT (using the r2 radial dependence of Vt shown in Eq. 3). Second, F (the tidal change in the Earth’s gravitational potential at r ¼ a) has an angular dependence 3 Ym 2 and so, outside of the Earth, has radial dependence r . Thus, the effect of F on the radially inward gravitational acceleration is @ r Fðr ¼ aÞ ¼ 3a F ¼ 3a kV aT Third, in the absence of tides, the surface gravitational acceleration, g, varies with radius as r2. So, if the surface point is displaced radially by Ur, the resulting perturbation in the gravitational acceleration at the displaced surface point is U r @ r g ¼  2a gU r ¼  2a hV aT : Adding these three contributions together gives the total tidal effect on the observed acceleration as 2 Dgðy, lÞ ¼ d V aT ðy, lÞ a

ð6Þ

where d ¼ 1  32 k þ h is called the gravimetric factor. Similar exercises show that tidal tilt is described by the diminishing factor γ ¼ 1 + k–h, and that tidal strain, although more complicated than either tilt or gravity, depends on the Love numbers h and l. The numerical values of the Love numbers depend on the Earth’s internal properties. It is possible to learn about some of those properties by comparing tidal observations with predictions based on theoretical results for the Love numbers. For example, tidal observations have been used to place constraints on the Earth’s anelastic properties. The effects of anelasticity are frequency- and stress-dependent, but are not well understood. Because tides sample the Earth in unique frequency and stress regimes (at lower frequencies and larger deep-earth stresses than seismology, e.g.,), tidal studies have proven useful as complements to other types of anelastic observations. The use of tides to study the Earth’s internal structure can be difficult, for several reasons. One problem is instrument calibration, as described above for gravimeters. Any error in the calibration maps directly into a frequency independent, multiplicative error in the estimated Love numbers. But there are also a number of other complicating effects, most related to the fact that the Earth is not really spherically symmetric and nonrotating. Some of those effects are potentially useful. For example, the Earth does rotate, and because of that rotation the Earth’s internal properties are closer to being elliptically, rather than spherically,

symmetric. In this case, the results of Eq. 4 are still approximately valid, but the Love numbers k, h, and l are notably dependent on frequency near 1 cycle/day. This diurnal frequency dependence is particularly sensitive to the shape of the core-mantle boundary, and so diurnal tidal results can be used to help constrain that shape. Another omission in the theory described above is the ocean. Tides in the ocean cause time-varying pressure loads on the surface of the solid Earth with the same frequencies as the Earth tides. These loads cause the Earth to deform, and this “load tide” (the tidal deformation that would occur in the absence of oceans is called the “body tide”) affects all Earth tide observations to some extent. Tidal gravity observations, for example, can be perturbed by up to 10% near coasts, and typically a few percent in the interior of continents. Tidal tilt and strain can be perturbed by several hundred percent near coasts. It is often difficult to model and remove the load tide well enough to use the remaining body tide to learn about the Earth’s deep interior. There are uncertainties in ocean tide models. Furthermore, the Earth’s response to the ocean tides, particularly the contributions to tilt and strain near coasts, can be sensitive to the local material properties that may or may not be adequately known. On the other hand, because of these possible uncertainties, people have sometimes been able to use tidal observations to help constrain the nearby ocean tide or the underlying material properties. In fact, for tilt and strain local effects can be important on both the load tide and the body tide. By comparing observed tidal amplitudes from instruments in an array of tiltmeters or strainmeters, geophysicists can learn about the local geology and underlying structure. There have even been attempts to look for time-dependent variations in tidal amplitudes near active earthquake faults that might be caused by sudden changes in local material properties preceding an earthquake. A related problem is that tilt and strain amplitudes are also affected by the local topography and by the shape and size of any cavity the instrument is placed in. (Tiltmeters and strainmeters are often placed in boreholes or tunnels to minimize the effects of the surface environment.) These effects are rarely interesting, and they cannot be observationally separated from the effects of local geology. Instead, they must be modeled separately and removed from the data.

Summary The effects of Earth tides can be detected in several types of geophysical measurements. These tidal observations can be inverted to learn about large-scale material properties of the Earth’s deep interior and about local geological structure.

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In practice, a useful inversion is often difficult due to the large number of possible geophysical interpretations, and due to uncertainties associated with instrument calibration, topographic and cavity effects, and ocean tidal loading of the solid Earth.

Cross-References ▶ Earth Rotation ▶ Earth, Density Distribution ▶ Geodesy, Figure of the Earth ▶ Geodesy, Networks, and Reference Systems ▶ Geodesy, Physical ▶ Geodynamics ▶ GPS, Tectonic Geodesy ▶ Gravity Method, Satellite

Bibliography Agnew DC (2007) Earth tides. In: Schubert G, Herring T (eds) Treatise on geophysics, vol 3. Elsevier, Oxford, pp 163–195 Baker TF (1984) Tidal deformations of the earth. Science Progress (Oxford) 69:197–233 Harrison JC (1984) Earth tides (Benchmark papers in geology). Hutchinson Ross, Pennsylvania Melchior P (1983) The tides of the planet earth. Pergamon, Oxford Wilhelm H, Zurn W, Wenzel HG (1997) Tidal phenomena. Springer, Berlin

Earth, Density Distribution Frank D. Stacey1 and Paul M. Davis2 1 Division of Exploration and Mining, CSIRO, Kenmore, Australia 2 Earth and Space Sciences, UCLA, Los Angeles, CA, USA

Definition Except for iron meteorites, the Earth is the densest body in the solar system. As has been recognized for more than two centuries, it is, on average, about twice as dense as common surface rocks. But many of the early speculations on its internal structure were bizarre, ignoring the most basic physics and the simplest density information, and prompting a remark in one of the early papers on the subject with a sound scientific base (Oldham 1906): “Many theories of the earth have been propounded ... till geologists have turned in despair from the subject, and become inclined to confine their attention to the outermost crust of the earth, leaving the center as a playground for mathematicians.” Seismological studies,

Earth, Density Distribution

for which Oldham was one of the pioneers, have changed that by giving us a clear picture of the interior. But debates on many details and improvements in the methods of investigating them continue.

The Mean Density The first well-informed estimate of the Earth’s mean density was by Isaac Newton, who appealed to four pieces of evidence: the Earth’s surface gravity, the period of the Moon’s orbit, the size of Jupiter, and the orbital periods of its satellites. It was evident to him that the density of Jupiter is a quarter of the density of the Earth. Arguing that the density of Jupiter would not be lower than that of water (it is now known to be 1327 kg m3), and recognizing that the densest Earth materials would have sunk to the deep interior, he inferred that the mean Earth density is between 5000 and 6000 kg m3, a remarkable coincidence with the now established value r ¼ 5513:4ð6Þ kg m3 . More precise estimates awaited the determination of the gravitational constant, G. Any astronomical or geophysical measurement of gravity, g, or, equivalently the acceleration of a body in a gravitational field, depends on the product of G and a mass. In the approximation of a spherical, nonrotating Earth, mass M, and radius R, as Newton showed, the surface gravity is g ¼ GM/R2 (the departure from this simple equation caused by rotation and ellipticity is about 0.5%). Since g and R have been reasonably well known since Newton’s time, a determination of G is equivalent to a determination of M, or r, and was referred to as such by early investigators. Satellite measurements have given the most precise value of the product GM ¼ 3.986004415(8) 1014 m3 s2, where M includes the mass of the atmosphere, 8.84  107 of the total. The first attempts to measure G were by M. Bouguer, who, in the 1730s, was conducting an astrogeodetic survey in the extreme topography of the Andes and noted a plumb line deflection by Mount Chimborazo, in Ecuador. He also compared the gravity on a plateau, at Quito, with observations on the nearby flood plain of the Esmeralda River, the first use of what we now know as the Bouguer method of gravity interpretation. In both cases Bouguer recognized that he was thwarted by heterogeneity of the geological structures and that his estimates of r were unsatisfactory, but in 1774 his plumb line deflection method was repeated more successfully by N. Maskelyne on the slopes of Mt. Schiehallion in Scotland. Maskelyne’s estimate was revised several times by others, using different assessments of the density of the mountain. This introduced a measure of uncertainty, so that geophysical measurements were generally disregarded for about 50 years after 1798, when H. Cavendish made a series of measurements of the gravitational force between laboratory masses, using a torsion balance developed for the

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purpose by J. Michell. However, Cavendish’s result was also a subject of doubt. Although the measurements were excellent, the final result was clouded by an arithmetic error, recognized in 1842 by F. Baily, leading to a revival of interest in geophysical methods. The most significant of these was by G. B. Airy, who, in 1856, measured the variation of gravity with depth in a coal mine near Newcastle, in northeast England. Although limited by inadequate measurements of rock density, Airy’s experiment was well conceived. It did not require an absolute measurement of gravity but only the differences in its value at different levels in the mine, as indicated by the fractional differences between the periods of pendulums operated by observers who exchanged electrical signals to indicate their timing. The linked variations in gravity and density, recognized by Airy, are central to the development of earth models. The Airy method was used by others for the following 50 years, but with mixed results, and by 1900 the superiority of torsion balance measurements was clear. Now, it appears that conventional balance experiments, pioneered in the late 1800s by P. von Jolly and J. H. Poynting, are better still and the current best estimate of G derives primarily from them: G ¼ 6.6743(7)  1011 m3 kg1 s2. The uncertainty (1 standard deviation), 1 part in 104, is far larger than the uncertainties in the other fundamental physical constants and translates to a similar uncertainty in M, and therefore r, through the very precise value of (GM).

The Moment of Inertia Constraint The understanding of the Earth’s density structure made little progress, beyond Newton’s inference that denser materials would have sunk to the center, until the late 1800s. By then astronomical measurements of precession, combined with astrogeodetic measurements of ellipticity, had given an estimate of the moment of inertia of the Earth, C. This is expressed as a coefficient, f relating C to the mass, M, and equatorial radius, a C ¼ f Ma2

ð1Þ

For a uniform sphere f ¼ 0.4 and a lower value is a measure of the concentration of mass toward the center. Using recent data the value for the Earth is f ¼ 0.330968(2), with the uncertainty in parenthesis. This number represents a quite strong central concentration of mass and prompted speculation on the Earth’s deep composition. The cosmic abundance of iron, apparent from meteorites and the solar spectrum, invited the supposition that the Earth has an iron core. This idea was pursued by E. Wiechert who, in 1897, showed that both M and f could be matched by a model with an iron core and rocky mantle, with both materials approximating their familiar low pressure densities, if the core radius was about

0.78a. But in 1906 R. D. Oldham reported seismological evidence for a smaller core, with a radius that he estimated to be about 0.4a (it is now known to be 0.546a), requiring higher densities for both the core and the mantle. Two developments were needed: recognition of the increase in density by the compression of deep burial and the introduction of a transition zone of increasing intrinsic density within the mantle. Both required more detailed seismological data than were available to Wiechert and Oldham.

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Self-Compression Williamson and Adams (1923) recognized that information about self-compression in a homogeneous layer was available from seismic wave velocities. Using the velocities of both compressional and shear waves V P ¼ f½K S þ ð4=3Þm=rg1=2 ; V S ¼ fm=rg1=2

ð2Þ

they separated the adiabatic bulk modulus, KS, from the rigidity modulus, m, by writing F ¼ V P 2  ð4=3ÞV S 2 ¼ K S =r ¼ ð@P=@rÞS

ð3Þ

That is, the seismological parameter F gives directly the variation of pressure, P, with density, r, in a homogeneous region with an adiabatic temperature gradient. Since dP/dz. ¼ rg, g being gravity at depth z, dr=dz ¼ rg=F

ð4Þ

This is the Williamson–Adams equation for the variation of density with depth in a homogeneous region. An additional term, αrt, was added by F. Birch to account for a temperature gradient differing from the adiabat t ¼ dT=dz  ðdT=dzÞAdiabatic

ð5Þ

where α is the volume expansion coefficient. However, this term is very small except at the limited depths of steep temperature gradients, the lithosphere, and the base of the mantle (D00 ). In the outer core, it is certainly negligible. Another reason for imperfection of the Williamson– Adams equation is that, for composite materials, such as mantle rocks that are comprised of several minerals with different properties, there is a frequency dependence of elasticity. The unrelaxed elastic moduli that apply to rapid cycles of low stress, as in seismic waves, are slightly higher than the relaxed moduli applying to strong static compression. Averaged over the lower mantle, the difference is about 0.5% (Stacey and Davis 2008, Table 18.1). This means that dr/dz is 0.5% greater than by the Williamson–Adams equation.

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Interpreted as an error in the inferred temperature gradient, it amounts to ~ 100 K over the depth of the lower mantle, but earth models are not so accurate that this is regarded as serious. When self-compression, as represented by the Williamson–Adams equation, is allowed for a simple twolayer model with the observed core size and a rocky mantle cannot match the moment of inertia. It is necessary to bring some mass up higher into the mantle. This was the vital next step to realistic earth models, introduced by K. E. Bullen and B. Gutenberg in the 1930s: the mantle transition zone. By then, seismological information was sufficiently detailed to indicate the depth range of the transition zone and the earth models that followed were quite close to our present understanding. As we now know, the transition zone is a combination of transitions at several levels in the depth range of 220–660 km, where mantle minerals are converted to denser phases. The transition at 660 km is conventionally regarded as the boundary between the upper and lower mantles. The lower mantle, which has two-thirds of the mantle volume and three-fourths of its mass, appears to be more or less homogeneous over almost the whole depth range to the core, at 2900 km depth, and conforms well to the Williamson–Adams equation with a density 18% higher than for the low pressure forms of the minerals. The upper mantle is more complicated. Bullen (1975) reviewed this stage in the development of our understanding of the Earth’s density.

Free Oscillation Data In the 1960s, observations of the Earth’s modes of free oscillation introduced a new constraint on the density structure. Whereas body wave seismology gives the ratio KS/r throughout the Earth but provides no formal separation of KS and r, the spheroidal modes involve oscillations in the radial distribution of mass with consequent gravitational as well as elastic restoring forces. The equations of motion and the mode frequencies involve not just the ratios KS/r and m/r but also Gr. With G a known constant, the Gr term allows an assessment of absolute density independent of elasticity. This is illustrated by considering the frequencies of two modes for the simple case of a uniform solid sphere. The spheroidal mode 0S2 is an alternation between prolate and oblate deformations and, since there is radial motion, the Gr term has an important control on the frequency, f which is given by
 f 2 ð0 S2 Þ ¼ ð2=3pÞGr þ 5=3p2 m=rR2

ð6Þ

whereas the lowest toroidal mode, 0T2, is an alternating twist between two hemispheres, with no radial motion and a frequency given by


 f 2 ð0 T2 Þ ¼ 7=4p2 m=rR2

ð7Þ

Gr ¼ ð3p=2Þ f 2 ð0 S2 Þ  ð10p=7Þ f 2 ð0 T2 Þ

ð8Þ

so that

The frequencies of more than 500 modes have been identified (Masters and Widmer 1995), providing a direct measure of global average properties, including the radial density profile. Fine details, such as the sharpness of boundaries, still require analysis of body wave data, but the broad scale structure is derived from the mode frequencies. The Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981) was produced in this way. It signaled a turning point in seismological studies, with less attention to global average properties and increasing emphasis on lateral heterogeneities, boundary structure, and anisotropy. A powerful new use of free mode data to clarify the radial density structure was presented by Masters and Gubbins (2003), who selected a combination of modes sensitive to a restricted depth range, rather than inverting the whole data set. By concentrating on a region close to the inner core boundary, they obtained an estimate of the density difference between the inner and outer cores, 820 kg m3, compared with the PREM difference, 597 kg m3. The revision amounts to 1.8% of the density at that level, compared with the estimated 0.5% precision of the new technique, and may be interpreted as a measure of the revisions to PREM that may be expected in an updated model. From the splitting of mode frequencies, free oscillation data allow identification of lateral density variations independently of elasticity. This is referred to in a later paragraph on lateral heterogeneity.

A Density Gradient Anomaly An interesting feature of PREM is the decrease in density with depth in the lithosphere and asthenosphere, over the depth range 24.4–220 km. The obvious interpretation is a steep temperature gradient, but this presents difficulty. The inverted density gradient is slight, little more than 0.1 kg m3 per kilometer of depth, but the difficulty would arise even with zero gradients. For a uniform layer, density is a function only of pressure and temperature, so we can write dr=dz ¼ ð@p=@PÞT dP=dz þ ð@p@T ÞP dT=dz ¼ 0:109 kgm3 =km and substitute the physical properties of mantle material in this depth range to obtain dT/dz ¼ 8.6 K/km. (For dr/dz ¼ 0 this would be 7.9 K/km.) Over the depth range of 195.6 km,

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the anomalous gradient gives a temperature range of 1680 K and since the temperature at 24.4 km (the averaged crustmantle depth of PREM) cannot be taken as less than about 700 K, the implied temperature at 220 km is close to 2400 K. This is about 600 K too high; the temperature at 660 km depth required to meet the (P,T) conditions of the phase transition there is about 1900 K (Boehler 2000) and the 220 km temperature must be lower than that. The discrepancy can be understood if PREM underestimates the effect of anisotropy in the asthenosphere, causing the inverted density gradient as a modelling artifact.

Extrapolation to Zero Pressure and Inferences About Composition The density profile of PREM, with minor equation of state adjustments, is plotted in Fig. 1, together with equation of state extrapolations to zero pressure and a temperature of 300 K. The extrapolations use equations of state fitted to each of the ranges, assuming that the materials can be decompressed and cooled without phase changes. Thus, the broken line in the figure represents the densities that the high pressure phases would have under ambient conditions. For each of the inner and outer cores and lower mantle, the extrapolated densities are uniform, within the uncertainties of the calculation, and so indicate homogeneity. However, this is not true for the upper mantle, where the model indicates heterogeneity additional to the boundaries that mark recognized phase changes. The density estimates provide the basic data needed for a discussion of composition. Two long-standing debates that

14,000 ρ

Density (kg m−3)

12,000 10,000 8,000 ρ0 6,000

ρ

4,000

ρ0

2,000 0

1,000

2,000

3,000

4,000

5,000

6,000

Radius (km)

Earth, Density Distribution, Fig. 1 Density profile of the Earth based on PREM (solid line) and the profile of r0,0, the density obtained by extrapolating to zero pressure and a temperature of 300 K (broken line)

hang on this are the composition of the core, that is, the light elements that must be dissolved in iron-nickel alloy to reduce its density to that of the core, and the possibility of a significant compositional difference between the upper and lower mantles. As Birch (1952) recognized, there is no plausible alternative to iron as the major core constituent, but the outer core is about 10% less dense than pure iron and Poirier (1994) reviewed the arguments for rival candidate light solutes. At core pressures iron occurs as a hexagonal close-packed (ε) phase, for which the extrapolated density would be r0,0 ¼ 8352 23 kg m3 (Stacey and Davis 2004), compared with the core density, extrapolated to the same conditions and also in the ε phase, 7488 kg m3 for the outer core and 7993 kg m3 for the inner core. The effects on the iron density of each of the candidate light elements are known from metallurgical literature; although none of the observations were made on ε iron, they are taken as a sufficient guide to their potential contributions to the core density deficit. There are undoubtedly many elements dissolved in the core, but the favored important ones, with the mass percentages required in the outer core if each were the only light ingredient, are H (1.4%), C (10.6%), O (12.7%), Si (17.7%), and S (18.2%). The core densities can, in principle, be explained by any combination of these elements, but geochemical arguments restrict the range. The mass percentages of each, in the inner and outer core, respectively, favored by Stacey and Davis (2008, Table 2.5) are H (0.07%, 0.08%), C (0.45%, 0.50%), O (0.11%, 5.34%), Si (nil, nil), and S (8.02%, 8.44%). There are two main arguments bearing on the similarity of the compositions of the upper and lower mantles, the seismological evidence for penetration of the 660-km boundary by subducting slabs, and the densities of high pressure phases of mantle minerals. The perovskite phase of (Mg,Fe)SiO3 is believed to be the dominant lower mantle mineral. It is produced by high pressure laboratory experiments and survives in a metastable state at P ¼ 0 provided it is not heated. With no Fe the density at ambient conditions is 4107 kg m3. The second most important lower mantle mineral is almost certainly ferro-periclase (magnesiowustite), (Mg, Fe)O, which may account for as much as 20% of the composition. It is less dense than the perovskite (3585 kg m3 for MgO) but more readily accepts substitution of Fe for Mg and would be 5956 kg m3 for FeO. The magnesium silicate perovskite can accept Al ions, but neither of these minerals readily accepts Ca, so a calcium perovskite is believed to occur also, but, unlike the magnesium perovskite and periclase, it does not survive decompression and is not as well studied. It is evident that, with a modest iron content, a mixture of these minerals can match the lower mantle density, r0,0 ¼ 4144 kg m3, and that, on the basis of density, there is no evidence for a compositional difference between the upper and lower mantles.

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Lateral Density Variations

Bibliography

Convection is ubiquitous in the Earth and this means that lateral density variations, with hot, dilated materials rising, and replaced by denser, cool materials. Heterogeneity of the mantle inferred from body wave tomography is generally consistent with the convective pattern, acknowledging that relics of earlier convective patterns are still apparent, but elastic moduli are more sensitive to temperature than is density and a thermodynamic analysis is needed for interpretation (Stacey and Davis 2008, Sect. 19.7). The splitting of mode frequencies presents evidence of lateral density variations that does not rely on such interpretation, but, in spite of several detailed studies, the difficulties are such that available results must be regarded as preliminary (Ishii and Tromp 1999; Masters et al. 2000; Romanowicz 2001). The general conclusion is that, in addition to temperature variations, compositional heterogeneity of the deep mantle must be invoked to explain the observations, in confirmation of a thermodynamic assessment (Forte and Mitrovica 2001; Stacey and Davis 2008, p. 288) of results of body wave tomography (Robertson and Woodhouse 1996; Su and Dziewonski 1997). With respect to the outer core, observable lateral heterogeneity must be discounted as quite implausible, in view of the speed of convective motion apparent from the geomagnetic secular variation. Compositional layering, with a stable lowdensity layer at the top of the core has been suggested (Braginsky 1999), but is not generally supported.

Birch F (1952) Elasticity and constitution of the Earth’s interior. J Geophys Res 57:227–286 Boehler R (2000) High pressure experiments and the phase diagram of lower mantle and core materials. Rev Geophys 38:221–245 Braginsky SI (1999) Dynamics of the stably stratified ocean at the top of the core. Phys Earth Planet Inter 111:21–34 Bullen KE (1975) The Earth’s density. Chapman and Hall, London Dziewonski AM, Anderson DL (1981) Preliminary reference earth model. Phys Earth Planet Inter 25:297–356 Forte AM, Mitrovica JX (2001) Deep-mantle high viscosity flow and thermochemical structure inferred from seismic and geodynamic data. Nature 410:1049–1056 Ishii M, Tromp J (1999) Normal-mode and free-air gravity constraints on lateral variations in velocity and density of Earth’s mantle. Science 285:1231–1236 Masters G, Gubbins D (2003) On the resolution of density within the earth. Phys Earth Planet Inter 140:159–167 Masters G, Widmer R (1995) Free oscillations: frequencies and attenuation. In: Ahrens TJ (ed) A handbook of physical constants 1: global earth physics. American Geophysical Union, Washington, DC, pp 104–125 Masters G, Laske G, Gilbert F (2000) Matrix autoregressive analysis of free oscillation coupling and splitting. Geophys J Int 143:478–489 Oldham RD (1906) The constitution of the interior of the earth as revealed by earthquakes. Q J Geol Soc Lond 62:456–475 Poirier J-P (1994) Light elements in the Earth’s core: a critical review. Phys Earth Planet Inter 85:319–337 Robertson GS, Woodhouse JH (1996) Ratio of relative S to P heterogeneity in the lower mantle. J Geophys Res 101:20041–20052 Romanowicz B (2001) Can we resolve 3D density heterogeneity in the lower mantle? Geophys Res Lett 28(6):1107–1110 Stacey FD, Davis PM (2004) High pressure equations of state with applications to the lower mantle and core. Phys Earth Planet Inter 142:137–184 Stacey FD, Davis PM (2008) Physics of the earth, 4th edn. Cambridge University Press, Cambridge Su WJ, Dziewonski AM (1997) Simultaneous inversion for 3D variations in shear and bulk velocity in the mantle. Phys Earth Planet Inter 100:135–156 Williamson ED, Adams LH (1923) Density distribution in the earth. J Wash Acad Sci 13:413–428

Summary The Earth’s density structure is now known in sufficient detail to provide a sound basis for discussions of internal composition. High pressure equations of state (Stacey and Davis 2004), the modern version of finite strain theory, provide reliable extrapolations of density to zero pressure, as well as distinguishing internal heterogeneities caused by temperature variations from compositional and miner-alogical differences. Although fine details in the deep interior are not all resolvable, it is evident that the bulk of the Earth does not have very strong lateral heterogeneities, such as are obvious in the crust. Nevertheless, the heterogeneities that are seen are indications of the dynamics of the Earth, making them prime targets for continuing investigation.

Earth’s Structure, Core Lianxing Wen Department of Geosciences, State University of New York at Stony Brook, Stony Brook, NY, USA

Definition Cross-References ▶ Geodesy, Physical ▶ Gravity Measurements, Absolute ▶ Gravity, Global Models ▶ Seismic, Velocity, and Density Relationships

Structure of Earth’s core is referred to as the properties of velocity, density, attenuation, anisotropy, composition, and surface feature of Earth’s core and how these properties change with depth, geographic location, and time. Attenuation is a measure of material’s ability to absorb energy as seismic waves pass through it.

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Anisotropy is a material property that is directionally dependent. Mushy zone is a layer of mixing solid inner core and liquid outer core materials. Overall Structure The Earth’s core occupies the center portion of Earth with a radius of 3480 km. It consists of a liquid outer core and a solid inner core. Earth’s core is composed of Fe/Ni and some minor light elements, such as S, C, O, and Si. Those light elements are preferentially enriched in the outer core after the continuing solidification of outer core material to the inner core. Seismic compressional velocity and density exhibit jumps at the inner core boundary at a radius of about 1221 km. The inner core is anisotropic in both velocity and attenuation, with seismic waves exhibiting higher velocities and higher attenuation as they propagate along the polar paths than along the equatorial paths. In first order, the inner core exhibits an eastwest hemispheric difference of various seismic properties, a

change of anisotropy with depth, an irregular surface, some localized mushy zones at the top, and temporal change of its surface in a time scale of days or months (Fig. 1). Radial and Lateral Variations With possible exception at the bottom and the top, the outer core is homogeneous due to vigorous mixing of the liquid iron/ nickel. The inner core exhibits complex lateral and radial variations of structure, with the most prominent feature being the hemispheric difference in seismic velocity, attenuation, and anisotropy, between the “eastern” hemisphere (defined between longitudes from 40°E to 180°E) and the “western hemisphere” (defined between longitudes from 180oW to 40°E). The eastern hemisphere exhibits a higher velocity, higher attenuation, weaker anisotropy, and a larger transitional depth from top isotropy to deep anisotropy. The top 235 km of the eastern hemisphere also exhibits a flat velocity gradient with depth. At depth, the innermost 300 km of the inner core may have different forms of anisotropy from the rest of the inner core.

Inner Core 0.5% faster High attenuation (Q=250)

–0.3% slower Low attenuation (Q=600)

Mushy zone

West

East

? Crust Mantle

Topography

Weak anisotropy (~0.7%)

Strong anisotropy (~4%) Innermost inner core

Outer Core Inner Core

Earth’s Structure, Core, Fig. 1 Cartoon summary of east-west hemispheric differences in seismic properties (velocity, attenuation, and anisotropy), surface topography, and mushy zone in the Earth’s inner core

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Irregular Surface The inner core surface is irregular, despite that its global feature is not yet clear. In the regions that have been studied by seismology, most regions exhibit flat topography. But in some localized regions, the inner core surface exhibits at least two scales of topography: a height variation of 14 km changing within a lateral distance of no more than 6 km and a height variation of 4–8 km with a lateral length scale of 2–4 km. Mushy Zone at the Top A localized 4–8 km thick mushy zone is discovered across the inner core boundary beneath southwest Okhotsk Sea, with seismic properties intermediate between those of the inner and outer core and of a mushy zone. The discovered localized mushy zone is surrounded by a sharp inner core boundary nearby. Temporal Changes In some localized regions, mostly beneath Africa, South America, and central Pacific, the inner core surface changes its radius by several kilometers in a time scale of days or months. The temporal change of inner core surface appears episodic, rapidly migrating, and alternatively enlarged and shrunk. Possible Physical Mechanisms The proposed physical mechanisms for explaining the inner core structure include: (1) For the inner core anisotropy: solidification texturing due to the growth of inner core material and preferential alignment of anisotropic Fe crystals caused by inner core convection, magnetic force, and preferential growth of the inner core in the equator (2) For the east-west hemispheric features: a mantle-induced inner core solidification and a self-sustained translational inner core with the eastern hemisphere melting and the western hemisphere solidifying (3) For the temporal change of inner core surface and inner core topography: small-scale variation of temperature and/or composition near the inner core boundary and deformation of inner core surface by small-scale force near the inner core boundary (4) For the mushy zone: the current outer core composition being close to eutectic in most regions resulting in a sharp inner core boundary, but deviation in some localized regions generating a mushy zone

Earth’s Structure, Global

▶ Earth’s Structure, Global ▶ Magnetic Anisotropy

Bibliography Alboussière T, Deguen R, Melzani M (2010) Melting-induced stratification above the Earth’s inner core due to convective translation. Nature 466:744–747. https://doi.org/10.1038/nature09257 Aubert J, Amit H, Hulot G et al (2008) Thermochemical flows couple the Earth’s inner core growth to mantle heterogeneity. Nature 454:758–761. https://doi.org/10.1038/nature07109 Dai Z, Wang W, Wen L (2012) Irregular topography at the Earth’s inner core boundary. Proc Natl Acad Sci U S A 109:7654–7658. https:// doi.org/10.1073/pnas.1116342109 Ishii M, Dziewonski AM (2002) The innermost inner core of the Earth: evidence for a change in anisotropic behavior at the radius of about 300 km. Proc Natl Acad Sci 99:14026–14030 Morelli A, Dziewonski AM, Woodhouse JH (1986) Anisotropy of the inner core inferred from PKIKP travel times. Geophys Res Lett 13:1545–1548 Niu F, Wen L (2001) Hemispherical variations in seismic velocity at the top of the Earth’s inner-core. Nature 410:1081–1084 Tanaka S, Hamaguchi H (1997) Degree one heterogeneity and hemispherical variation of anisotropy in the inner core from PKP(BC) – PKP(DF) times. J Geophys Res Solid Earth 102:2925–2938 Tian D, Wen L (2017) Seismological evidence for a localized mushy zone at the Earth’s inner core boundary. Nat Commun 8:165. https:// doi.org/10.1038/s41467-017-00229-9 Wen L (2006) Localized temporal change of the Earth’s inner core boundary. Science 314:967–970. https://doi.org/10.1126/science.11 31692 Yao J, Tian D, Sun L, Wen L (2019) Temporal change of seismic Earth’s inner core phases: inner core differential rotation or temporal change of inner core surface? J Geophys Res Solid Earth 124:6720. https:// doi.org/10.1029/2019JB017532

Earth’s Structure, Global Jean-Paul Montagner Seismological Laboratory, Institut de Physique du Globe, University Paris-Diderot, UMR CNRS/7154, Paris, France

Definition Structure of the Earth Crust Mantle

Cross-References ▶ Differential Rotation of the Earth’s Inner Core ▶ Earth, Density Distribution

Core

The manner in which the Earth is constructed. It is characterized through physical and chemical parameters. The outer layer of the solid Earth that lies above the Mohorovicic discontinuity. A thick layer of rock below the crust down to the core-mantle boundary CMB at 2,900 km depth. The central part of the Earth below CMB. It is composed of the fluid outer core (2,900

Earth’s Structure, Global

Anelasticity

Anisotropy

Tomography

down to 5,100 km) and the solid inner core below. Property of a substance in which there is no definite (linear) relation between stress and strain. Property of a substance being directionally dependent, as opposed to isotropy (identical properties in all directions). Imaging technique of 3D objects, developed in medical sciences, astrophysics, geophysics, etc. It is based on the use of large density of penetrating waves illuminating the object. The images are displayed as 2D cross sections (in Greek, “tomos” means slice) of the object.

Introduction The Earth is the largest in size and mass of the Solar System’s four terrestrial planets. Of these four planets, Earth also has the highest density, the highest surface gravity, the strongest magnetic field, and fastest rotation. It is also the third planet from the Sun and the densest and fifth largest of the eight planets in the Solar System. Earth is a terrestrial planet, meaning that it is a rocky body, rather than a gas giant like Jupiter. Earth is currently the only place where life is known to exist and the only terrestrial planet with active plate tectonics. Earth Sciences in general, and Geophysics in particular, are built on the foundations of physics, mathematics, geology, and astronomy and are closely connected to other disciplines such as chemistry and biology. Its roots therefore go far back in history, but the science has blossomed only in the last century with the impressive increase in our ability to measure the properties of the Earth and the processes in play inside the Earth, on and above its surface. Geophysics includes the study of the Earth’s solid layers, fluid envelopes, and near-space environment. However, in this contribution, the subject is narrowed to the solid Earth. The interior structure of the Earth, like that of the other terrestrial planets, is divided into layers by their physical (rheological) or chemical properties. Scientific understanding of Earth’s internal structure is based on geological, geochemical, and geophysical observations at the surface of the Earth including topography and bathymetry, observations of rock in outcrops, and samples brought to the surface from greater depths, primarily by volcanic activity. Geophysics can make use of measurements of different physical fields such as the gravity, magnetic fields, and the fields associated with the propagation of seismic waves passing through the Earth. The Earth’s gravity field can be used to calculate its mass, its moment of inertia. By adding some additional

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physical constraints, the density variations within the Earth can be calculated. The magnetic field of internal origin provides fundamental information on the deepest layer of the Earth, the core (see contributions by Foss, “Magnetic Data Enhancements and Depth Estimation” and Arora, “Magnetic Methods, Principles”). This interdisciplinary observational approach is complemented by laboratory experiments with minerals at pressures and temperatures characteristic of the Earth’s deep interior. The technological advances of the last century in laboratory and field instrumentation, computing, and satellite-based remote sensing are largely responsible for the explosive growth of geophysics. These advances enabled geophysicists to make more and more accurate measurements, to collect and to analyze enormous amounts of data, and to model more and more complex systems. This new view of how the Earth works enabled a fundamental understanding of structure of the Earth as well as earthquakes, volcanoes, and mountain building, indeed all of geological processes and objects.

1D Structure of the Earth The global structure of Earth can be defined in two different ways, either by considering its physical properties or its chemical properties. Geophysics is the only Earth science discipline enabling to image the invisible deep inner structure of our planet by a direct measurement of its physical properties. To first order, the structure of the earth is dominated by its radial structure (spherically symmetric model). During the first part of the twentieth century, geophysicists determined the onion-like fine structure of the Earth. Seismic waves enable to investigate the elastic and to some extent anelastic properties of the whole Earth. Characterizing the Interior of the Earth with Earthquake Waves Global seismology started in 1889 when Ernst von Rebeur-Paschwitz in Potsdam (Germany) associated a disturbance recorded on a tiltmeter with the occurrence of a remote Japanese earthquake. Since then, seismologists modeled the propagation of seismic waves from teleseismic events by using the ray theory considering seismic waves as virtual particles following rays between earthquake focus and receivers, and their travel times are sensitive to the structure of the Earth. During the first decades of the twentieth century (see Dziewonski and Romanowicz (2015) for details), the layering of Earth was indirectly inferred using the travel times of refracted and reflected seismic waves generated by earthquakes. The changes in seismic velocity of P-waves (compression waves) and S-waves (shear waves) between

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different layers causes refraction owing to Snell-Descartes’s law. Reflections are caused by a large increase in seismic velocity and are similar to light reflecting from a mirror. Seismologists were able to find and characterize the main seismic discontinuities (depth and velocity jump) within the different layers of the Earth (see Vinnik, this issue). First of all, the Mohorovicic discontinuity between crust and mantle in the depth range 10–60 km then the core-mantle boundary at 2,900 km depth by Oldham (1906). The core does not allow shear waves to pass through it, showing that its outer part is fluid. Finally, Inge Lehmann (1936) demonstrated the existence of a solid inner core. Mantle itself can be divided into upper mantle and lower mantle by a discontinuity at 660 km depth. In the upper mantle, the finding of a discontinuity around 410 km depth defines the “upper transition zone” between 410 and 660 km depth. At the base of the mantle, about 200 km above the core-mantle boundary, a discontinuity was discovered, identifying the still mysterious D”-layer. There are some other candidates for entering the club of seismic discontinuities, the Hales discontinuity at 80 km depth, the Gutenberg discontinuity at the base of continental and oceanic lithosphere, the Lehmann discontinuity at 220 km, and also at 520 km and around 900 km depths, but their global nature is still debated. The layer between 660 km and 900 km is commonly named the “lower transition zone.” Most seismic discontinuities are interpreted as a mineralogical phase transition between minerals of mantle silicates (olivineperovskite system), but some of them might be a boundary between different chemical compositions. From its elastic properties, the different layers are presented in Table 1 and Fig. 1. The Earth can be divided into the outer silicate solid crust, the upper mantle (including lithosphere, asthenosphere, and upper transition zone), the lower mantle (with the lower transition zone and the

Earth’s Structure, Global, Table 1 The 1D structure of the Earth Layer (Kilometers) 0–80 0–35 35–80 35–2,890 35–660 80–220 410–660 660–2,890 660–900 2,740–2,890 2,890–5,150 5,150–6,360

Lithosphere (locally varies between 5 and 200 km) Crust (locally varies between 5 and 70 km) Uppermost part of mantle Mantle Upper mantle Asthenosphere Upper transition zone Lower mantle Lower transition zone D”-layer Outer core Inner core

D”-layer at its base), the liquid outer core much less viscous than the mantle, and the solid inner core. From 1D Models to 3D Tomographic Models Many reference models (spherically symmetric models) have been developed during the last century, but the most popular ones are the PREM (Preliminary Reference Earth Model by Dziewonski and Anderson 1981) (Fig. 2) and IASP91 (Kennett and Engdahl 1991). PREM introduced for the first time an anisotropic layer (with the specific case of radial anisotropy with a vertical symmetry axis) in the uppermost 220 km of the mantle. Montagner and Kennett (1996) by trying to reconcile body wave and normal mode data showed that a significant amount of radial anisotropy is also necessary in the mantle transition zones between 410 and 900 km depth. However, important deviations between observed and theoretical travel times of seismic waves with respect to 1D-reference models cannot be explained by these simple 1D models. These differences, though relatively small (10%) (see Babuska and Cara (1991), Anderson (2007), and Mainprice (2015) for reviews). Consequently, the petrological models which are assemblages of different minerals are less anisotropic than pure olivine. The amount of anisotropy is largely dependent on the percentage of these different minerals and on the mechanisms aligning the crystallographic axes according to CPO. For example, the anisotropy of the pyrolytic model, mainly composed of olivine and orthopyroxene (Ringwood 1975), will depend on the relative orientation of the crystallographic axes of different constituents (Christensen and Lundquist 1982). For the sake of mathematical simplicity, it was usually assumed that the propagating elastic medium is isotropic, in spite of the large amplitude of anisotropy in several depth

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Earth’s Structure, Global, Fig. 4 Tomographic images of slabs (Top, van der Hilst et al. 1997) and plumes (Bottom, French and Romanowicz 2015)

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Earth’s Structure, Global, Fig. 5 Competing models for explaining the origin of hotspots (Left, Courtillot et al. 2003; Anderson 2001). (From Foulger et al. 2005)

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ranges. And only 1D reference models of attenuation are incorporated in seismic modeling. Global tomographic models improved over the years not only by an increase in the number of data but also by more general parameterizations. They are now including anisotropy (radial anisotropy in Nataf et al. (1994); general slight anisotropy in Montagner and Tanimoto (1991)) and anelasticity (Romanowicz 1995). Whereas isotropic heterogeneities enable to map hot and cold regions within the Earth, seismic anisotropy gives access to the convective flow. Gaboret et al. (2003) and Becker et al. (2003, 2008) demonstrated that mantle circulation inferred from geodynamic models is in good agreement with radial and azimuthal anisotropy distributions (Fig. 6). Seismic anisotropy has many other applications such as the determination of the base of continental roots (Montagner 1994; Gaherty and Jordan 1995; Silver 1996; Gung et al. 2003; Debayle et al. 2005) and the investigation of small-scale convection at the base of the lithosphere and of convection boundary layers within the Earth, in the transition zones (Trampert and van Heijst 2002), D”-layer (see Lay and Garnero et al.), and even within the inner core (see Souriau and Calvet 2015 in structure of the inner core).

Sc

Louisville Reunion Afar

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Low-velocity

Relative velocities

Relative velocities

Because of its strong dependence on temperature, partial melting, and water content, mapping anelastic attenuation in the Earth has the potential to provide valuable information on Earth’s three-dimensional (3D) structure and dynamics, in complement to what can be learned from mapping elastic isotropic velocities and anisotropic parameters. A significant challenge is to separate the effects of anelastic (or intrinsic) attenuation from those of elastic focusing and scattering (or extrinsic) due to propagation in elastic 3D structure. Seismic anisotropy and anelasticity provide new ways to investigate geodynamic processes within the different layers of the Earth.

Outlook: New Challenges So far, global 3D-tomographic models are well resolved for very large spatial scales (>1,000 km). And the global flow circulation in most of the mantle is dominated by the “degree 2 pattern.” However, smaller wavelengths are not so well resolved even though many geological objects with a lateral

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Earth’s Structure, Global, Fig. 6 Numerical modeling of flow circulation superimposed to 3D tomographic model (Gaboret et al. 2003)

extent of the order of 100 km, such as slabs or mantle plumes, play a key role in global geodynamics. New innovative numerical techniques for forward modeling (such as Spectral Element Method) and for inverse problems (adjoint tomography of Tarantola 1986; Tromp et al. 2005) are now available and make it possible to use the full waveforms of seismograms. The first images of different volcanic plumes including Hawaiian plume (Montelli et al. 2004) were first obtained by incorporating finite frequency effects (see ▶ “Seismic Tomography”). Therefore, a new revolution in global tomography is underway (see Fig. 4b for images of plumes). Impressive progress through global tomography, geochemistry, and mineral physics has been made during the last 30 years in our understanding of global geodynamics, demonstrating how active and turbulent is our planet. All layers are heterogeneous, interact, and exchange matter. The geosciences community must improve the lateral resolution and the quality of the 3D images and incorporate on a routine basis anisotropy and anelasticity. There is a real need for a 3D seismic reference Earth model in agreement with geological and mineralogical data and fluid dynamics modeling. This progress brings about the expansion of new instruments such as very dense seismic networks on land (e.g., US array, Hi-Net, Euro-Array, etc.) and on the sea floor, by exploring other planets and by incorporating new data

such as seismic noise (Shapiro et al. 2005; Nishida et al. 2009) implementing more powerful numerical techniques. Geophysics only provides some pieces for our puzzling global geodynamics, and a multidisciplinary effort is necessary to fully understand the spatiotemporal evolution of our planet. The exploration of other telluric planets is necessary for having other examples of solutions chosen by nature, and the seismometer within InSight mission successfully deployed in January 2019 on planet Mars will provide such answers.

Cross-References ▶ Body Waves ▶ Core-Mantle Coupling ▶ Differential Rotation of the Earth’s Inner Core ▶ Earth’s Structure, Core ▶ Earth’s Structure, Lower Mantle ▶ Earth’s Structure, Upper Mantle ▶ Free Oscillations of the Earth ▶ Geodynamics ▶ Inverse Theory, Linear ▶ Lithosphere, Continental ▶ Magnetic Anisotropy ▶ Mantle Convection ▶ Mantle D00 Layer

Earth’s Structure, Global

▶ Mantle Plumes ▶ Plate-Driving Forces ▶ Propagation of Elastic Waves: Fundamentals ▶ Seismic Discontinuities in the Transition Zone ▶ Seismic Imaging, Overview ▶ Seismic Instrumentation ▶ Seismic Ray Theory ▶ Seismic Structure at Mid-Ocean Ridges ▶ Seismic Tomography ▶ Seismic, Migration ▶ Subduction Zones ▶ Surface Waves

Bibliography Aki A, Richards PG (2002) Quantitative seismology, 2nd edn. University Science Books, Sausalito Anderson DL (2007) New theory of the Earth. Cambridge University Press, Cambridge Babuska V, Cara M (1991) Seismic anisotropy in the Earth. Kluwer, Dordrecht Dahlen FA, Tromp J (1998) Theoretical global seismology. Princeton University Press, Princeton Gutenberg B, Richter CF (1954) Seismicity of the Earth. Princeton University Press, Princeton Lay T, Wallace TC (1995) Modern global seismology. Academic, San Diego Nolet G (2008) A breviary of seismic tomography. Cambridge University Press, Cambridge, UK Poirier J-P (1991) Introduction to the physics of the earth interior. Cambridge University Press, Cambridge, UK Richter CF (1958) Elementary seismology. Freeman, San Francisco Treatise on Geophysics (2015) Gerald Schubert (Editor-in-Chief), Elsevier B.V. Turcotte DL, Schubert G (2002) Geodynamics, 2nd edn. Cambridge Univeristy Press, Cambridge

Further Reading Anderson DL (2001) Top-down tectonics? Science 293:2016–2018 Becker TW, Kellogg JB, Ekström G, O’Connell RJ (2003) Comparison of azimuthal seismic anisotropy from surface waves and finite-strain from global mantle-circulation models. Geophys J Int 155:696–714 Bercovici D, Karato S-I (2003) Whole-mantle convection and the transition-zone water filter. Nature 425:39–44 Birch F (1952) Elasticity and constitution of the Earth’s interior. J Geophys Res 57:227–286 Courtillot V, Davaille A, Besse J, Stock J (2003) Three distinct types of hotspots in the Earth’s mantle. Earth Planet Sci Lett 205:295–308 Debayle E, Kennett BLN, Priestley K (2005) Global azimuthal anisotropy and the unique plate-motion déformation of Australia. Nature 433:509–512 Dziewonski AM (1984) Mapping the lower mantle: determination of lateral heterogeneity in P velocity up to degree and order 6. J Geophys Res 89:5929–5952 Dziewonski AM, Anderson DL (1981) Preliminary reference earth model. Phys Earth Planet Inter 25:297–356 Dziewonski AM, Hager BH, O’Connell R (1977) Large-scale heterogeneities in the lower mantle. J Geophys Res 82:239–255 Dziewonski AM, Romanowicz B (2015) Overview. In: Romanowicz B, Dziewonski A (eds) Treatise on geophysics, vol 1. Elsevier B.V., pp 1–28

175 Foulger GR, Natland JH, Presnall DC, Anderson DL (2005) Plates, plumes, and paradigms. Geological society of America special, vol 388, 881pp French S, Romanowicz B (2015) Broad plumes rooted at the base of the Earth’s mantle beneath major hotspots. Nature 525:95–101 Fukao Y, Widiyantoro S, Obayashi M (2001) Stagnant slabs in the upper and lower mantle transition region. Rev Geophys 39:291–323 Gaboret C, Forte AM, Montagner J-P (2003) The unique dynamics of the Pacific hemisphere mantle and its signature on seismic anisotropy. Earth Planet Sci Lett 208:219–233 Gaherty JB, Jordan TH (1995) Lehmann discontinuity as the base of an anisotropic layer beneath continents. Science 268:1468–1471 Gung Y, Panning M, Romanowicz B (2003) Global anisotropy and the thickness of continents. Nature 422:707–711 Holmes A (1928) Radioactivity and Earth movements. Trans Geol Soc Glasgow 18:559–606 Karato S-I, Jung H, Katayama I, Skemer P (2008) Geodynamic significance of seismic anisotropy of the upper mantle: new insights from laboratory studies. Annu Rev Earth Planet Sci 36:59–95 Kennett BLN, Engdahl ER (1991) Travel times for global earthquake location and phase identification. Geophys J Int 122:429–465 Lehmann I (1936) Publications du Bureau Central Séismologique International, série A, Travaux Scientifiques 14: 87–115 Mainprice D (2015) Seismic anisotropy of the deep Earth from a mineral and rock physics perspective. In: Schubert G, Bercovici D (eds) Treatise on geophysics, vol 2. Elsevier, pp 487–538 Masters G, Jordan TH, Silver PG, Gilbert F (1982) Aspherical earth structure from fundamental spheroïdal mode data. Nature 298:609–613 Montagner J-P (1994) What can seismology tell us about mantle convection? Rev Geophys 32:115–137 Montagner J-P, Kennett BLN (1996) How to reconcile body wave and normal-mode reference earth models? Geophys J Int 125:229–248 Montagner J-P, Tanimoto T (1991) Global upper mantle tomography of seismic velocities and anisotropies. J Geophys Res 96:20337–20351 Montelli R, Nolet G, Dahlen F, Masters G, Engdahl E, Hung S (2004) Finite-frequency tomography reveals a variety of plumes in the mantle. Science 303:338–343 Morgan WJ (1971) Convection plumes in the lower mantle. Nature 230:42–43 Murakami M, Hirose K, Kawamura K, Sata N, Ohishi Y (2004) Postperovskite phase transition in MgSiO3. Science 304:5672 Nataf H-C, Nakanishi I, Anderson DL (1994) Anisotropy and shear velocity heterogeneity in the upper mantle. Geophys Res Lett 11:109–112 Nishida K, Montagner J-P, Kawakatsu H (2009) Global surface wave tomography using seismic hum. Science 326(5949):112. https://doi. org/10.1126/science1176389 Oldham RD (1906) The constitution of the Earth. Q J Geol Soc Lond 62:456–475 Pekeris C (1935) Thermal convection in the interior of the Earth. Geophys J 3:343–367 Romanowicz B (1995) A global tomographic model of shear attenuation in the upper mantle. J Geophys Res 100:12375–12394 Romanowicz B, Mitchell BJ (2015) Deep earth structure – Q of the earth from crust to core. In: Romanowicz B, Dziewonski A (eds) Treatise on geophysics, vol 1. Elsevier, pp 789–828 Sengupta MK, Toksoz MN (1976) Three dimensional model of seismic velocity variation in the Earth’s mantle. Geophys Res Lett 3:84–86 Shapiro NM, Campillo M, Stehly L, Ritzwoller MH (2005) Highresolution surface-wave tomography from ambient seismic noise. Science 307:1615–1618 Silver PG (1996) Seismic anisotropy beneath the continents: probing the depths of geology. Annu Rev Earth Planet Sci 24:385–432

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Souriau A, Calvet M (2015) The Earth’s cores. In: Romanowicz B, Dziewonski A (eds) Treatise on geophysics, vol 1, Elsevier, pp 725–758 Tanimoto T, Anderson DL (1985) Lateral heterogeneity and azimuthal anisotropy of the upper mantle: love and Rayleigh waves. 100–250s. J Geophys Res 90:1842–1858 Thurber C, Ritsema J (2015) Theory and observations – seismic tomography and inverse methods. In: Romanowicz B, Dziewonski A (eds) Treatise on geophysics, vol 1. Elsevier, pp 307–338 Trampert J, van Heijst HJ (2002) Global azimuthal anisotropy in the transition zone. Science 296:1297–1299 Tromp J, Tape C, Liu Q (2005) Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophys J Int 160:195–216 van der Hilst RD, Widiyantoro S, Engdahl R (1997) Nature 386:578–584 Vinnik L, Kosarev GL, Makeyeva LI (1984) Anisotropy of the lithosphere from the observations of SKS and SKKS phases. Proc Acad Sci USSR 278:1335–1339 Wilson JT (1963) Evidence from islands on the spreading of ocean. Nature 197:536–538 Woodhouse JH, Dziewonski AM (1984) Mapping the upper mantle; three-dimensional modeling of Earth structure by inversion of seismic waveforms. J Geophys Res 89:5953–5986

velocity province (LLSVP) Ultra low velocity zone (ULVZ)

beneath the Atlantic Ocean and Africa. They are continental-sized in lateral dimension, and up to and over 1000 km high. A ULVZ is a thin (tens of kilometers thick) and isolated (100–1000 km laterally) structure with very low seismic wave speeds.

Introduction Earth’s interior has three fundamentally distinct divisions: (1) the relatively buoyant and thin outer crust, which averages roughly 6–7 km in thickness beneath oceans and 30–40 km in continents; (2) the mantle, which extends to nearly half way to Earth’s center; and (3) the dominantly iron (Fe) core, which has a fluid outer part and a solid inner part. Figure 1 highlights these basic shells, along with a number of additional significant mantle subdivisions. The mantle is divided into an upper and lower part, with the boundary near 660 km depth. The

Earth’s Structure, Lower Mantle Edward J. Garnero1, Allen K. McNamara2 and James A. Tyburczy1 1 School of Earth and Space Exploration, Arizona State University, Tempe, AZ, USA 2 Department of Earth and Environmental Sciences, Michigan State University, East Lansing, MI, USA

Synonyms Deep Earth; Deep mantle

Definition Seismic Tomography

Forward Modeling Phase Transition D00

Large low shear

An inverse method that utilizes seismic wave travel times and/or waveforms to estimate Earth’s seismic velocity and density structure, either regionally or globally. The procedure involving the generation of models that predicts observations, which is frequently used in deep Earth seismology. A crystallographic rearrangement of atoms in a mineral that occurs at high pressure. The depth shell of Earth occupying the lowermost 200–300 km of Earth’s lower mantle. (Pronounced “D-double-prime”) Two LLSVPs exist at the base of the mantle, one beneath the Pacific Ocean, and one

Earth’s Structure, Lower Mantle, Fig. 1 Major depth shells of Earth, along with a number of subdivisions. TZ, transition zone (the upper mantle between 410 and 660 km depth); CMB, core-mantle boundary; ICB, Inner core boundary

Earth’s Structure, Lower Mantle

upper mantle has another distinct discontinuity near 410 km depth. As shown in Fig. 2, the dominant mineral in the upper mantle is olivine, a magnesium-iron silicate mineral with composition (Mg,Fe)2[SiO4], that undergoes phase transitions to higher density mineral forms (or polymorphs) near 410, 520, and 660 km depths. The region between 410 and 660 km is referred to as the mantle transition zone. Below 660 km, mantle silicates undergo another phase transformation, compacting into the perovskite structure (recently given the mineral name bridgmanite) with composition (Mg,Fe) [SiO3], plus magnesium-iron oxides, (Mg,Fe)O (known as ferropericlase and also referred to as magnesiowüstite in earlier literature). In the lowermost several hundred kilometers of the mantle, a region referred to as D00 , perovskite has been shown to undergo an additional phase transition into the “post-perovskite” structure (Murakami et al. 2004), and depending on the temperature, post-perovskite might backtransform into perovskite at greater depth, remaining as perovskite down to the CMB (see The D00 Layer encyclopedia entry). The lower mantle depth shell accounts for nearly 56% of Earth’s volume, and extends from 660 km depth down to the core-mantle boundary (CMB) at 2891 km depth (and thus includes the D00 layer). The major discontinuities throughout Earth’s mantle are thus well-explained by phase transitions in the olivine-perovskite system. While uncertainties are present in our understanding of the exact compositional makeup of Earth’s mantle, it is generally believed that bridgmanite is the dominant mineral structure of the lower mantle, possibly accounting for ~80% of the lower Earth’s Structure, Lower Mantle, Fig. 2 (a) Variations in seismic properties of the Earth’s mantle with depth, from the surface to the core-mantle boundary, showing the major discontinuities in P-wave velocity, S-wave velocity, and density. (b) Mineralogical makeup of the mantle. Cpx, clinopyroxene; Opx, orthopyroxene

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mantle, and hence is the most abundant mineral on Earth. This perovskite mineral structure is the same as that occurring in well-known electronic and piezoelectric materials, such as barium titanate. A fundamental difference between the silicate minerals of the upper mantle and lower mantle perovskite is that every silicon (Si) atom in perovskite is surrounded by six oxygen (O) atoms (silicon is octahedrally coordinated), whereas for upper mantle silicates the silicon is bonded to four O atoms (tetrahedral coordination). Thus a primary effect of the higher pressure in the lower mantle is that O atoms occupy a smaller volume, thus more can be organized around (coordinated with) Si. This more compact rearrangement of atoms in lower mantle phases means that they have higher density than upper mantle materials, as well as differences in other properties such as the material’s stiffness and resistance to shearing. These changes affect the velocities of seismic waves; thus, at depths where phase transitions occur, there are jumps (i.e., discontinuous increases) in seismic compressional and shear wave speeds (referred to as P-wave and S-wave velocities, or VP and VS, respectively). Velocity and density depth distributions in Earth’s mantle are presented in Fig. 2; these values represent an estimation of the globally averaged properties. The exact depth of a phase transition in any given mineral group depends upon temperature and pressure (and chemical composition). Earlier conceptions of the lower mantle (Birch 1952) postulated that the lower mantle had a constant composition, and hence the temperature and velocity variations could be described by simple adiabatic compression.

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However, in light of more recent studies, it is interesting to consider the fact that lateral temperature variations are likely in the lower mantle, because the mantle is convecting, and hence phase boundaries are not expected to be perfectly flat (e.g., Lebedev et al. 2002). Rather, they should have relief that directly relates to the temperature field (or to perturbations in chemical composition, since minerals may transform at different pressures if there are variations or impurities in the mineral composition). The Earth presumably was significantly hotter in the distant geologic past (from heat generated during formation and differentiation of the mantle and core), and therefore we expect phase boundaries in the earliest Earth to have been at different depths than today.

Tools for Studying the Lower Mantle Earth’s mass and moment of inertia, determined from orbital and rotational dynamics, provide important information about the planet’s internal density distribution. These data, combined with assumptions about starting materials for Earth’s formation from analyses of meteorites, have long indicated a predominantly iron core with a silicate rock mantle (Carlson 2003). Seismic studies early in the twentieth century confirmed the existence of the core (outer and inner), as well as the fact that the outer core is liquid (from the absence of shear waves on the opposite side of the planet following an earthquake, because shear waves cannot propagate in fluids). A number of Earth science disciplines contribute to our body of knowledge of the lower mantle (and the deep Earth in general). For example, the field of mineral physics includes high pressure laboratory experiments aimed at simulating the pressures and temperatures within the deep Earth. Predictions of the melting temperature of iron at the conditions of the boundary between the inner and outer core (at radius of ~1220 km), as well as measurements delineating the temperature of the olivine to bridgmanite plus ferropericlase phase transition at 660 km depth, provide two important temperature tie-in points that help to constrain Earth’s geotherm; temperature must be extrapolated from these points toward the CMB, where a very large temperature jump occurs. Based on such studies, the top of the lower mantle at 660 km depth is at a pressure of about 23 GPa (1 Pa ¼ 1 N m2 and 105 Pa corresponds to approximately 1 atmosphere of pressure) and a temperature of about 1800 200 K (e.g., Ito and Katsura 1989; Frost 2008). The CMB is at a pressure of about 130 GPa; the temperature, while less constrained, has been estimated to be about 4000 500 K (Boehler 1996; Van der Hilst et al. 2007) but recently lower values have been proposed (Nomura et al. 2014). The temperature jump from the CMB to the outermost core may be as much as 1000–1500 K. A number of notable uncertainties are present that currently preclude precise knowledge of lower mantle

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temperature, including the amount of heat-producing radiogenic materials, the vigor of mantle convection, the heat flux from the fluid core into the mantle, and important thermal properties of bridgmanite at lower mantle pressure and temperature conditions (e.g., thermal conductivity, heat capacity, and the coefficient of thermal expansion). However, using a diamond anvil cell high pressure-device to achieve pressures as great as those of the lower mantle and even the core, researchers have recently refined our understanding about two important characteristics of bridgmanite. As mentioned above, bridgmanite (br) can transform into a higher pressure (and density) structure termed “post-perovskite” (ppv) at D00 pressures (around 120 GPa). Thus, if the systematics of this phase transition can be well defined from laboratory experiments or theory, then seismic mapping of the br!ppv transition depth (as well as that of possible ppv!br backtransformation) adds an additional temperature tie-in point for the geotherm (Hernlund et al. 2005). Another discovery describes a change with increased pressure in the electronic spin state of iron in magnesiowüstite and perovskite (a change from a high spin configuration with a maximum number of unpaired d-shell electrons to a low spin configuration in which d-shell electron pairing is maximized). Experiments indicate that this will occur between the depths of 1000 and 2200 km (Lin et al. 2013; Badro 2014). Current research predicts that this spin transition may result in a softer material, which can thus be more compressed and increase the density of the deeper mantle, possibly affecting convective dynamics as well as seismic velocities, element partitioning, and oxidation state (Shim et al. 2017; Liu et al. 2018). The vigor of mantle convection plays a central role in the cooling of Earth. The most commonly used parameter to determine if convection is occurring and to describe the strength of convection is the Rayleigh number, Ra: Ra ¼

agrDTD3 , k

where α is the coefficient of thermal expansion, g is the acceleration due to gravity, r is density, ΔT is the nonadiabatic temperature change across the convecting layer (in this case, the whole mantle) of thickness D,  is the viscosity, and k is the thermal diffusivity. Ra in the lower mantle is estimated to be on the order of 107; values greater than ~103 indicate that convection is occurring and values as high as 107 indicate convection with motions on the order of cm per year, which is considered quite vigorous when considered over geologic timescales. Furthermore, factors such as the temperature dependence of viscosity can strongly affect the style of convection, causing warmer regions to undergo significantly more strain than cooler regions. Thus, numerical convection calculations incorporate our best understanding of Earth’s properties and provide important information

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about the dynamics and evolution of the mantle (and planet as a whole). Erupted materials offer an opportunity to study mantle chemistry. Specifically, some hotspot volcanoes are thought to be caused by mantle plumes that originate in the lower mantle. However, it is possible that not all hotspots tap the lower mantle, and instead have different depths of plume source origin (Courtillot et al. 2003). It is widely accepted that lavas from hot spot volcanoes are enriched in incompatible elements and often have a higher 3He/4He ratio than lavas erupted at midocean ridges. This has led to a view that the lower mantle contains a distinct reservoir (or multiple reservoirs) that has undergone only limed stirring with the upper mantle (Hofmann 1997). Subsequent and ongoing work continues to challenge and extend that perspective (Jackson et al. 2017; Stracke et al. 2019). We address geodynamic issues in the “Three-Dimensional Structure and Dynamics” section below. Seismic imaging is the primary tool that gives us the ability to map lateral variations of mantle properties from the upper mantle to the CMB. Seismic tomography provides an image of the full three-dimensional heterogeneity field, albeit at relatively long horizontal wavelengths, on the order of 1000 km and greater (e.g., Kustowski et al. 2008; Ritsema et al. 2011; French and Romanowicz 2015). Variations in seismic velocity as well as density (e.g., Trampert et al. 2004; Ishii and Tromp 2004; Koelemeijer et al. 2017) have been mapped, and have important implications on convective flow in the interior. Information about the small-scale heterogeneity field has been provided by a number of forward modeling seismic studies. For example, some studies have suggested that small-scale scatterers (~1 km) exist throughout the lower mantle (Hedlin and Shearer 2000). Other studies find evidence for discrete reflectors in some locations, particularly beneath subduction zones (e.g., Rost et al. 2008; Frost et al. 2018). The exact nature of such small-scale heterogeneity is unknown, but may indeed relate to an incompletely mixed mantle, or heterogeneities entrained from the top (dense things falling) or the bottom (buoyant things rising).

Three-Dimensional Structure and Dynamics For decades it has been known that Earth’s mantle contains significant lateral variations in seismic properties, which are due to variations in temperature and/or chemical composition compared to the local average properties. Results from seismic tomography depict VP, VS, and r changes throughout the mantle. Variations are neither expected nor detected in the fluid outer core; the extremely low viscosity and relatively fast convection promote homogenization (Stevenson 1987). Seismically detected mantle heterogeneity is the strongest in the outermost few hundred kilometers of the mantle (i.e., near

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the planet’s surface), due to the strong variations in temperature and composition associated with plate tectonic and mantle convection processes (Masters et al. 2000). The next most seismically heterogeneous depth shell is the D00 region at the base of the mantle (see Encyclopedia entry on The D00 layer). Figure 3 shows seismic velocity variations in the lower mantle. D00 silicate rock is juxtaposed against the fluid iron outer core, which represents roughly a 75% increase in density. Thus the mantle side of the CMB can accommodate longlived stable structures over a wide range of densities (between the mantle and core density); it is possible that cumulates denser than average lowest mantle silicates can survive entrainment by the convective currents in the overlying mantle (e.g., McNamara and Zhong 2005; Garnero and McNamara 2008; Garnero et al. 2016). However, other possibilities are viable (Deschamps et al. 2007; Schuberth et al. 2009; Davies et al. 2015). The lower mantle above D00 displays variability in seismic properties, but at a much weaker level than that imaged at the top and bottom of the mantle. In some regions it is plausible that low amplitudes of mid-mantle heterogeneity may reflect our limited ability to confidently image the mantle there (due to a lack of seismic information because earthquakes and seismic recorders are not uniformly distributed on Earth). Nonetheless, all seismic tomography studies to date agree that heterogeneity is greater at the top and bottom of the mantle relative to the middle of the mantle. Plate tectonics involve the creation of plates at mid-ocean ridges, and destruction or consumption of plates at subduction zones, where the cold and dense oceanic crust and lithosphere (compared to the surrounding mantle) descend into the interior. Tomographic analyses have shown that lineations in high velocity wave speeds beneath subduction zones (e.g., Grand et al. 1997; Li et al. 2008; Obayashi et al. 2013) are consistent with cold material subducting deep into the lower mantle. While the penetration depth of subducting slabs is not adequately constrained beneath all subduction zones, it is widely accepted that many slabs descend through the 660 km discontinuity, in some cases reaching to the CMB (such as beneath the Caribbean). It is important to understand the large-scale long-term flux of material from Earth’s surface to great depths in the interior because of the importance for long-term (geological time) effects on surface volatiles, the atmosphere, and global climate. The amount of mass flux between the upper mantle down into the lower mantle depends on several poorly known quantities such as the viscosity of the descending slab and mantle, the viscosity increase at the 660 km discontinuity, a possible viscosity increase in the upper lower mantle near 1200 km depth (Rudolph et al. 2015), as well as the degree of viscous coupling between the oceanic lithosphere and underlying asthenosphere (before and after subduction). Timeintegrated mass transfer into (and out of) the lower mantle

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Earth’s Structure, Lower Mantle, Fig. 3 Tomographically derived shear velocity heterogeneity in the lower mantle (French and Romanowicz 2015) is shown for four positions of the globe. Heterogeneity has been isosurfaced (red) in the lower half of Earth’s mantle at the 0.75% level. The two large low shear velocity provinces (LLSVPs) are present

plays a role in long-term mixing and lower mantle residence time of deep mantle minerals. For example, weak viscous coupling between slabs and the surrounding mantle predicts lower flux rates into (and out of) the lower mantle, and hence less mixing between the upper and lower mantle. A strong coupling between the slab and mantle predicts the opposite. Uncertainties in current data and predictions result in difficulties in distinguishing between these possibilities. An analogous dependency between mixing and convective parameters exists for the bottom of the mantle, where recent work argues for large low shear velocity provinces (referred to as LLSVPs) being compositionally distinct from the surrounding mantle (Cottaar and Lekic 2016; Garnero et al. 2016; Deschamps et al. 2019; Roy et al. 2019). Convective currents associated with subduction sweep basal dense material to regions away from downwellings, forming thermochemical “piles” (Fig. 4) (McNamara and Zhong 2005; McNamara et al. 2010). The contrast in properties (e.g., density, viscosity, temperature) between piles and surrounding mantle will determine the morphology and vertical extent of the pile material, as well as the degree of entrainment (and

hence its evolutionary behavior). The two nearly antipodal large low shear velocity provinces, or “LLSVPs” (Cottaar and Lekic 2016; McNamara 2019), one beneath the central Pacific and the other beneath the southern Atlantic and Africa (Fig. 3), are accompanied (in certain areas) by low velocity conduits that are associated with mantle plumes (e.g., Montelli et al. 2004), which appears to correlate with the surface locations of hotspots and reconstructed origination locations of large igneous provinces (Burke et al. 2008; Torsvik et al. 2010). Thus the present-day large-scale structure and circulation pattern of the lower mantle represent a time-integrated effect of anomalous temperature and/or chemistry in Earth’s lowermost and uppermost mantle boundary layers (D00 and the lithosphere, respectively). At smaller scale lengths, seismic studies have found evidence for seismic wave scattering. The imaged size of heterogeneities in the lower mantle ranges from kilometer to tens of kilometers in scale (e.g., Vidale and Hedlin 1998; Wen and Helmberger 1998; Frost et al. 2017). This observation indicates chemical anomalies at a spectrum of physical scales and a mantle that is not fully mixed. Indeed, analyses of magmas

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Earth’s Structure, Lower Mantle, Fig. 4 Numerical geodynamics calculations for a model mantle consisting of three distinct chemistries are shown for the whole mantle, from Earth’s surface to the core-mantle boundary. Panel (a) displays the chemical field, showing the bulk of the upper and lower mantle in blue, as one chemistry type, dense lower mantle piles (pink) as a second distinct chemistry, and thin ultra-low

velocity zones at the core mantle boundary (red) that accrue near the pile margins. (b) The temperature field associated with (a) is shown. Red and blue are hotter and colder regions, respectively. The dense piles can be stable for billions of years, though a low level of entrainment is constant, as evidenced by the white streaks throughout the mantle in panel (McNamara et al. 2010)

from hot spot volcanoes have yielded characteristic isotopic signatures distinct from those of mid-ocean ridge basalts, arguing for unique, possibly isolated deep mantle reservoirs that feed mantle plumes. Irregular convective patterns (e.g., see Fig. 4) are consistent with such heterogeneities spreading through the lower mantle as well as being entrained into mantle plumes, but not necessarily in a systematic fashion (since convection patterns appear multi-scaled and may be complex). Alternatively, scattering of seismic energy can occur from small pockets of partially molten mantle material. This scenario has been motivated by the detection of thin (a few kilometers) laterally discontinuous layers right at the CMB, called ultra-low velocity zones (ULVZs). ULVZs have been argued to contain some degree of partial melt, and hence offer a plausible source of scatters composed of melt pockets (whether or not they are of a distinct chemical composition). Subduction of oceanic crust into the deep mantle also

represents a viable source of lower mantle heterogeneity (e.g., Li et al. 2014; Mulyukova et al. 2015).

Future Progress As seismic sensors are deployed in new regions across Earth’s surface, new geographical regions will be sampled with greater certainty, and thus our understanding of global processes will greatly improve. As sensor deployments become denser (with smaller distances between sensors), Earth’s deep structure can be imaged with a much greater resolution. Thus, small-scale structures and processes can be inferred with greater certainty and tied to estimations of global processes. The seismic work only maps present-day structure, which must be put in a dynamical and evolutionary framework (e.g., Hernlund and McNamara 2015) with the predictions

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from geodynamical convection calculations, which in turn must first be informed by an understanding of the material properties. The latter comes from the field of mineral physics, both in laboratory experiments at high temperature and pressure and with computer simulations of material behavior at the atomic scale. The time evolution of Earth is also informed by geochemical analyses of erupted materials and meteorites. Thus, future work that advances our understanding of the structure, dynamics, and evolution of Earth’s interior will be multidisciplinary in nature.

Summary Earth’s lower mantle represents the largest volume of any depth shell on Earth. It is unique in that its chemistry, structure, dynamics, and evolution represent a time-integrated effect of the chemistry and dynamics of the surface (lithosphere, asthenosphere) and lowermost mantle (D00 , ULVZ, LLSVP, and CMB) boundary layers. The degree to which the lower mantle is recycled into the upper mantle depends upon many poorly known convective parameters, such as the viscosity and viscous coupling of descending slabs, as well as material properties, such as the nature and origin of plausibly dense deep mantle chemically distinct piles. Future work of improved seismic imaging coupled with continued advancements in other deep Earth disciplines, such as geodynamics, mineral physics, and geochemistry, will greatly help to reduce uncertainties in our understanding of Earth’s evolutionary pathway, and present-day structure and dynamical state.

Cross-References ▶ Core-Mantle Coupling ▶ Earth’s Structure, Core ▶ Earth’s Structure, Upper Mantle ▶ Mantle Convection ▶ Mantle D00 Layer ▶ Mantle Viscosity

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Earth’s Structure, Lower Mantle Burke K, Steinberger B, Torsvikc TH, Smethurstc MA (2008) Plume generation zones at the margins of large low shear velocity provinces on the core–mantle boundary. Earth Planet Sci Lett 265:49–60 Carlson RW (2003) In: Carlson RW (ed) Treatise on geochemistry, volume 2: Geochemistry of the mantle and core. Elsevier, 586pp Cottaar S, Lekic V (2016) Morphology of seismically slow lower-mantle structures. Geophys J Int 207:1122–1136. https://doi.org/10.1093/ gji/ggw324 Courtillot V, Davaille A, Baesse J, Stock J (2003) Three distinct types of hotspots in the Earth’s mantle. Earth Planet Sci Lett 205:295–308 Davies DR, Goes S, Sambridge M (2015) On the relationship between volcanic hotspot locations, the reconstructed eruption sites of large igneous provinces and deep mantle seismic structure. Earth Planet Sci Lett 411:121–130 Deschamps F, Trampert J, Tackley PJ (2007) Thermo-chemical structure of the lower mantle: seismological evidence and consequences for geodynamics. In: Yuen DA et al (eds) Superplume: beyond plate tectonics. Springer, Dordrecht, pp 293–320 Deschamps F, Konishi K, Fuji N, Cobden L (2019) Radial thermochemical structure beneath Western and Northern Pacific from seismic waveform inversion. Earth Planet Sci Lett 520:153–163. https:// doi.org/10.1016/j.epsl.2019.05.040 French SW, Romanowicz B (2015) Broad plumes rooted at the base of the Earth’s mantle beneath major hotspots. Nature 525:95–99 Frost DJ (2008) The upper mantle and transition zone. Elements 4:171–176 Frost DA, Rost S, Garnero EJ, Li M (2017) Seismic evidence for Earth’s crusty deep mantle. Earth Planet Sci Lett 470:54–63 Frost DA, Garnero EJ, Rost S (2018) Dynamical links between smalland large-scale mantle heterogeneity: seismological evidence. Earth Planet Sci Lett 482:135–146 Garnero EJ, McNamara AK (2008) Structure and dynamics of Earth’s lower mantle. Science 320:626–628 Garnero EJ, McNamara AK, Shim S-H (2016) Continent-sized anomalous zones with low seismic velocity at the base of Earth’s mantle. Nat Geosci 9(7):481–489. https://doi.org/10.1038/ngeo2733 Grand SP, van der Hilst RD, Widiyantoro S (1997) Global seismic tomography: a snapshot of convection in the Earth. GSAToday 7:1–7 Hedlin MAH, Shearer PM (2000) An analysis of large scale variations in small-scale mantle heterogeneity using Global Seismic Network recordings of precursors to PKP. J Geophys Res 105:13,655–13,673 Hernlund JW, McNamara AK (2015) The core-mantle boundary region. In: Schubert G (editor-in-chief) Treatise on geophysics, 2nd edn, vol 7. Elsevier, Oxford, pp 461–519 Hernlund JW, Thomas C, Tackley PJ (2005) A doubling of the postperovskite phase boundary and structure of the Earth’s lowermost mantle. Nature 434:882–886 Hofmann AW (1997) Mantle geochemistry: the message from oceanic volcanism. Nature 385:219–229 Ishii M, Tromp J (2004) Constraining large-scale mantle heterogeneity using mantle and inner-core sensitive normal modes. Phys Earth Planet Inter 146:113–124 Ito E, Katsura T (1989) A temperature profile of the mantle transition zone. Geophys Res Lett 16:425–428 Jackson MG, Konter JG, Becker TW (2017) Primordial helium entrained by the hottest mantle plumes. Nature 42:340–343. https://doi.org/10. 1038/nature21023 Koelemeijer P, Deuss A, Ritsema J (2017) Density structure of Earth’s lowermost mantle from Stoneley mode splitting observations. Nat Commun 8:15241. https://doi.org/10.1038/ncomms15241 Kustowski B, Ekström G, Dziewonski AM (2008) Anisotropic shearwave velocity structure of the Earth’s mantle: a global model.

Earth’s Structure, Upper Mantle J Geophys Res Solid Earth (1978–2012) 113(B6):B06306. https:// doi.org/10.1029/2007JB005169 Lebedev S, Chevrot S, Van der Hilst RD (2002) Seismic evidence for olivine phase changes at the 410-and 660-kilometer discontinuities. Science 296:1300–1302 Li C, van der Hilst RD, Engdahl ER, Burdick S (2008) A new global model for P wave speed variations in Earth’s mantle. Geochem Geophys Geosyst 9(5):Q05018 Li M, McNamara AK, Garnero EJ (2014) Chemical complexity of hotspots caused by cycling oceanic crust through mantle reservoirs. Nat Geosci 7:366–370 Lin J-F, Speziale S, Mao Z, Marquardt H (2013) Effects of electronic spin transitions of iron in lower mantle minerals: implications for deep mantle geophysics and geochemistry. Rev Geophys 51:244–275. https://doi.org/10.1002/rog.20010 Liu J, Dorfman SM, Zhu F, Li J, Wang Y, Zhang D, Xiao Y, Bi W, Alp EE (2018) Valence and spin states of iron are invisible in Earth’s lower mantle. Nat Commun 9:1284. https://doi.org/10.1038/s41467-01803671-5 Masters G, Laske G, Bolton H, Dziewonski AM (2000) The relative behavior of shear velocity, bulk sound speed, and compressional velocity in the mantle: implications for chemical and thermal structure. In: Karato S-I, Forte AM, Liebermann RC, Masters G, Stixrude L (eds) Earth’s deep interior: mineral physics and tomography from the atomic to the global scale. AGU, Washington, DC, pp 63–87 McNamara AK (2019) A review of large low shear velocity provinces and ultra low velocity zones. Tectonophysics 760:199–220 McNamara AK, Zhong S (2005) Thermochemical structures beneath Africa and the Pacific Ocean. Nature 437:1136–1139 McNamara AK, Garnero EJ, Rost S (2010) Tracking deep mantle reservoirs with ultra low velocity zones. Earth Planet Sci Lett 299:1–9 Montelli R, Nolet G, Dahlen F, Masters G, Engdahl E, Hung S (2004) Finite-frequency tomography reveals a variety of plumes in the mantle. Science 303:338–343 Mulyukova E, Steinberger B, Dabrowski M, Sobolev SV (2015) Survival of LLSVPs for billions of years in a vigorously convecting mantle: replenishment and destruction of chemical anomaly. J Geophys Res Solid Earth 120:3824–3847. https://doi.org/10.1002/2014JB011688 Murakami M, Hirose K, Kawamura K, Sata N, Ohishi Y (2004) Postperovskite phase transition in MgSiO3. Science 304(5672) Nomura R, Hirose K, Uesugi K, Ohishi Y, Tsuchiyama A, Miyake A, Ueno Y (2014) Low core-mantle boundary temperature inferred from the solidus of pyrolite. Science 343:522–525 Obayashi M, Yoshimitsu J, Nolet G et al (2013) Finite frequency whole mantle P wave tomography: improvement of subducted slab images. Geophys Res Lett 40:5652–5657 Ritsema J, Deuss A, van Heijst HJ, Woodhouse JH (2011) S40RTS: a degree-40 shear-velocity model for the mantle from new Rayleigh wave dispersion, teleseismic traveltime and normal-mode splitting function measurements. Geophys J Int 184:1223–1236 Rost S, Garnero EJ, Williams Q (2008) Seismic array detection of subducted oceanic crust in the lower mantle. J Geophys Res 113: B06303. https://doi.org/10.1029/2007JB005263 Roy SK, Takeuchi N, Srinagesh D, Kumar MR, Kawakatsu H (2019) Topography of the western Pacific LLSVP constrained by S-wave multipathing. Geophys J Int 218(1):190–199. https://doi.org/10. 1093/gji/ggz149 Rudolph ML, Lekic V, Lithgow-Bertelloni C (2015) Viscosity jump in Earth’s mid-mantle. Science 350(6266):1349–1352 Schuberth BSA, Bunge H-P, Ritsema J (2009) Tomographic filtering of high-resolution mantle circulation models: can seismic heterogeneity

183 be explained by temperature alone? Geochem Geophys Geosyst 10: Q05W03 Shim S-H, Grocholski B, Ye Y, Alp EE, Xu S, Morgan D, Meng Y, Prakapenka VB (2017) Stability of ferrous-iron-rich bridgmanite under reducing midmantle conditions. Proc Natl Acad Sci 114(25):6468–6473. https://doi.org/10.1073/pnas.1614036114 Stevenson DJ (1987) Limits on lateral density and velocity variations in the Earth’s outer core. Geophys J R Astron Soc 88:311–319 Stracke A, Genske F, Berndt J, Koornneef J (2019) Ubiquitous ultradepleted domains in Earth’s mantle. Nat Geosci 12:851–855. https:// doi.org/10.1038/s41561-019-0446-z Torsvik TH, Burke K, Steinberger B, Webb SJ, Ashwal LD (2010) Diamonds sampled by plumes from the core-mantle boundary. Nature 466:352–355 Trampert J, Deschamps F, Resovsky J, Yuen DA (2004) Probabilistic tomography maps chemical heterogeneities throughout the mantle. Science 306:853–856 Van der Hilst RD, de Hoop MV, Wang P, Shim S-H, Ma P, Tenorio L (2007) Seismostratigraphy and thermal structure of Earth’s coremantle boundary region. Science 315:1813–1817 Vidale JE, Hedlin MAH (1998) Evidence for partial melt at the coremantle boundary north of Tonga from the strong scattering of seismic waves. Nature 391:682–685 Wen L, Helmberger DV (1998) Ultra-low velocity zones near the coremantle boundary from broadband PKP precursors. Science 279:1701–1703

Earth’s Structure, Upper Mantle Guust Nolet IRD Geoazur and Université Côte d’Azur, Sophia Antipolis, France

Definition The upper mantle is defined as that part of the mantle between the crust and the phase transition of γ–olivine to perovskite. The total mass of the upper mantle is 1.06  1024 kg, about a quarter of the total mass of the mantle. Its volume, 2.95  1011 km3, is a third of the total volume of the mantle. Until the discovery of a major transition near 660-km depth, the upper mantle was better known as Bullen’s “layer B,” extending to 400-km depth. A sharp gradient or discontinuity in seismic velocity causes a bend in the travel time curves of P-wave arrivals near 20°, such that the slowness drops from more than 12 s/deg. to 10 s/deg. This phenomenon was observed as early as 1931 by Jeffreys, who correctly adopted a suggestion by J.D. Bernal that it represents a high-pressure modification of olivine. The modern notion of the upper mantle actually extends into Bullen’s next layer, “layer C,” which has now been abandoned as a physically meaningful subdivision.

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The depths to the top and the bottom of the upper mantle cannot be given precisely because they depend on the region. The Mohorovic̆ić discontinuity marks the transition from the crust to the upper mantle, with a density jump of the order of 10%. It is found at very shallow depth near ocean ridges where oceanic crust is formed but extends to more than 70-km depth beneath the Himalayas and the Andes. The “transition zone” between upper and lower mantle is marked by the transition of olivine (Mg2SiO4) to its polymorph β-olivine (Wadsleyite) near 410-km depth to γ-olivine near 520-km depth and finally to perovskite (MgSiO3), which occurs near a depth of 660 km, but with variations of 20 km depending on temperature and composition and possibly with variations more than twice that large for narrow areas inside hot plumes and cold slabs. A typical temperature estimate for the phase transition gives a temperature of about 1600 °C at 660-km depth. The upper mantle is one to two orders of magnitude less viscous than the lower mantle, and its electrical conductivity is much lower, with the possible exception of regions of partial melt. The upper mantle itself is subdivided into several depth regions, characterized by their differences in density, mechanical properties, and/or seismic velocity: Lithosphere It is a strong layer with a viscosity of the order of 1021Pa s and a thickness typically about 80–150 km, though it may exceed 200 km beneath old cratons, representing the rigid “plates.” The temperature gradient is strongly superadiabatic. The geotherm in the lithosphere connects the low temperature at the bottom of the crust, 600–900 °C under continents but as low as 250 °C in the oceans, to the (approximately adiabatic) temperature in the asthenosphere (1200–1300 °C). Gutenberg discontinuity The transition between lithosphere and asthenosphere, historically known as the Gutenberg discontinuity, is defined as the depth where shear velocity starts to decrease with depth. The most likely cause is that the weakening effect of temperature exceeds the effect of pressure at this depth. However, where it is sharp the rock composition, melt and/or the presence of water must also play a role. Sharp transitions may be mapped using receiver functions, in which case the boundary is often denoted as the lithosphere–asthenosphere boundary or LAB. Byrnes et al. (2015) discuss the various options for this LAB. Asthenosphere It is a mechanically weak layer with low viscosity of the order of 1019 Pa s. This layer is present under the oceans and the younger part of continents, but its existence beneath the oldest cratons is uncertain. It may contain a small degree of partial melt. Lehmann discontinuity It is a positive velocity jump named after Inge Lehmann who first observed this feature beneath

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Europe and North America near 150-km depth. It is generally interpreted as a sharp boundary representing the bottom of the asthenosphere, but not globally observed (Gu et al. 2001). “410-km” discontinuity It is a globally observed jump over a narrow depth interval in seismic velocity, marking the transition from the α ! β phase of olivine near 400-km depth. This transition occurs at a temperature of about 1400–1450 °C. “520-km” discontinuity Predicted by the phase diagram for olivine, which transforms to the γ phase under certain temperature conditions, this gradient in seismic velocities is occasionally observed. Reflections or refractions from other depths have been observed – such as the “Hales discontinuity” within the lithosphere – and are occasionally attributed to a global physical or chemical transition, but a lack of consistency between such observations makes it more likely that the observations should be explained as wave energy returned from threedimensional structure rather than from a discontinuity that represents an omnipresent layering. Early regionalized refraction studies of the upper mantle already showed a marked variability in P and S velocity with depth. Surface wave studies, which are the preferred tool to investigate layers of low velocity, indicate minimum S velocities as low as 4.05 km/s under the Western United States, 4.09 km/s under the East African Rift, and 4.14 km/s under the Pacific Ocean, but body waves show a minimum S velocity as high as 4.63 km/s under the North American craton. The Lehmann discontinuity is observed in few regional models, but it is present in the Preliminary Reference Earth Model (PREM) (Dziewonski and Anderson 1981), which shows a major velocity increase in P velocity of 7.1% at 220 km. By way of compensation, PREM has only a minor increase of 2.6% from 8.90 to 9.13 km/s at the “410-km” discontinuity. This, however, disagrees with all regional observations and reveals a major shortcoming of the reference model PREM. In fact, the variability in the velocity jump at 410 km is not very pronounced in the early refraction models. Typically, the P velocity in such models jumps about 5%. An average over seven regional models gives a P velocity jump from 8.78 0.06 km/s to 9.26 0.03 km/s, where the uncertainty represents one standard deviation. The jump in P velocity at 660 km over the same set of models is roughly 4% but more variable than that at 410 km: from 10.22 0.20 to 10.72 0.10 km/s. Near 660-km depth model PREM is in agreement with regional studies. Though the regional models based on refracted waves provide the most accurate estimates of absolute velocities, the velocity–depth relationships represent horizontal averages

Earth’s Structure, Upper Mantle

over thousands of km. We know that velocity variations exist over length scales much smaller than that. New tomographic techniques are able to delineate such variations and allow us to image them.

Tomographic Techniques Knowledge of the three-dimensional upper mantle structure comes from tomographic techniques, notably: • Transmission tomography of body wave delay times for both P and S velocity structure. • Phase or group delay tomography using surface waves, primarily sensitive to S velocity. • Imaging of waves reflected or converted at discontinuities (“receiver functions”), sensitive to the depth location of such discontinuities. • Waveform modelling is able to combine the information of the previous three wave types in one inversion, though at the expense of very large computation times. • Inversion of the splitting of normal mode frequencies by lateral heterogeneity, for long wavelength heterogeneity in both P and S velocity. For an extensive overview of transmission tomographic techniques, see ▶ “Seismic Tomography”. Each of these techniques has seen important progress in recent times. Ray theory is being replaced with first-order scattering theory that allows for the modelling of the frequency-dependent dispersion in delay times (Sigloch et al. 2008) or in receiver functions (Deng and Zhou 2015). Single station (or “single channel”) stacking of converted or reflected phases is being replaced by multichannel techniques, in particular Kirchhofftype stacking along diffraction hyperboles much as one does to migrate seismic sections in exploration seismics (see Bostock 2009). Seismic tomography allows the mapping of “anomalies” or deviations from a regular layering of the upper mantle. The most remarkable anomalies are listed here: Subduction zones and the associated low velocity structure in the mantle wedge can be very clearly imaged because of increased seismicity when actively subducting. The observed velocity anomalies are consistent with average (cold) temperature anomalies of several hundred degrees. Oceanic ridges are mapped best with surface waves, which leaves the exact depth extent as well as the temperature anomaly somewhat uncertain. Cratonic roots were originally detected through the large negative delay times they cause for S waves, consistent

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with a cold lithosphere (as much as 400° colder than oceanic lithosphere), but the depth extent is mapped using teleseismic surface waves. In the tectosphere model of Jordan (1981), the high density induced by the cooling is compensated by a chemical composition that is depleted in basaltic components. Upper mantle plumes are rather narrow anomalies in the upper mantle but can be imaged adequately using body wave delays as well as surface wave observations from field deployments of portable seismographs. The velocity anomalies are consistent with temperature anomalies of 100–300 °C inferred from petrological studies. Topography of the “410”- and “660”-km discontinuity is globally imaged using underside reflections of waves such as SS. This involves some geographic averaging, but topographic anomalies of 20 km or more are observed. These anomalies are all visible in the variations of isotropic velocity variations. There is strong evidence, notably from the birefringence of SKS waves, that the seismic velocity in the shallow part of the upper mantle depends locally on the propagation direction of the wave; under favorable circumstances, this could in principle allow the mapping of mantle flow, since the crystal structure is assumed to align itself to the flow direction. However, at present there is little agreement between different tomographic models of anisotropy, partly because the SKS information is insensitive to depth, while the crustal correction, which is badly known, is in itself as large as the signal from anisotropy in surface waves (Ferreira et al. 2010).

Global Models Schaeffer and Lebedev (2013) developed a detailed tomographic model of shear velocity variations in the upper mantle and compare this with five other models. At a length scale of thousands of km, all models are in agreement, but large differences still exist at shorter scales (500–1000 km). Very low velocities beneath oceanic ridges extend to about 100-km depth. The roots beneath the old cratons do not seem to extend far beyond 200-km depth. Hosseini et al. (2018) collect 36 global models, not all independent. Figure 1 shows the upper mantle P-wave perturbations averaged over 12 of these models. Figure 2 shows the average S-wave perturbations averaged over 17 models. The ability to resolve depth is superior for S-wave models since these are partly based on surface wave information. The averages displayed in Figs. 1 and 2 illustrate the global features that are present in all tomographic models: old continental shields (Canada, Scandinavia, Western

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Earth’s Structure, Upper Mantle, Fig. 1 An average over 12 tomographic models of P-wave velocity perturbations, plotted at 4 different depths in the upper mantle using the web-based toolbox of Hosseini et al. (2018)

Australia, West and South Africa) are characterized by high P and S velocities down to depths of about 250 km. The ridge in the North Atlantic interacts with the Iceland mantle plume and is still clearly visible at 250-km depth that shows low P and S anomalies. The oceans are generally characterized by low velocities throughout much of the upper mantle, with the exception of the South Atlantic and the westernmost Pacific. Plume-like anomalies, with a high degree of vertical continuity in the upper mantle, are visible in the P velocity model beneath Hawaii, the Society Islands (Tahiti), and the East-African rift area (Afar). For a proper imaging of plumes, tomographic techniques that correct for the effects of diffraction around small or narrow anomalies are more suitable. Figure 3 shows models PRI-05 for P and S velocity in the upper mantle from Montelli et al. (2006), which clearly show a number of narrow upper mantle plumes. Underside reflections from the upper mantle discontinuities allow the depth variations to the “410-” and “660-km” discontinuity to be mapped globally. This has most recently been done by Houser et al. (2008) using precursors to SS waves (Fig. 4).

Regional Models Regional studies are able to offer a better resolution than the global efforts, certainly when using data from large and dense seismic arrays such as Skippy (Australia), USArray (USA), or HiNet (Japan), and other, usually temporary, deployments of broadband seismographs. Efforts to image the upper mantle beneath them have led to very precise images of subduction zone structure in particular. Figure 5 shows the slab subducting in the Fiji–Tonga region. Though this slab image is continuous down to 600-km depth, slab geometry in the upper mantle is very variable, whereas some slabs remain for at least some time lodged in the transition zone others have no trouble sinking through the phase transition barrier at 660-km depth, with a negative Clapeyron slope and an increase of as much as two orders of magnitude in viscosity (Fukao et al. 2001). A very different fate of subducting slabs is shown for the now inactive Farallon subduction beneath western North America, where the former Farallon plate is slowly disintegrating, letting mantle rock flow through major ruptures such as the “slab gap” in Fig. 6 (Sigloch et al. 2008).

Earth’s Structure, Upper Mantle

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Earth’s Structure, Upper Mantle, Fig. 2 An average over 17 tomographic models of S-wave velocity perturbations, plotted at 6 different depths in the upper mantle using the web-based toolbox of Hosseini et al. (2018)

Outlook

Cross-References

It is now clear that the simple, layered models of the upper mantle are inadequate to explain many of the dynamic processes we observe at the Earth’s surface. The resolving power of tomographic techniques is still growing because of theoretical improvements and the densification of seismic networks. The coming decades will undoubtedly reduce the high degree of nonuniqueness that still plagues tomographic images and allows for more reliable estimates of temperature and anisotropy. Floating seismic robots (see ▶ “Floating Seismographs (MERMAIDS)”) are opening up the oceans for dense seismic observations.

▶ Body Waves ▶ Earth’s Structure, Global ▶ Floating Seismographs (MERMAIDS) ▶ Geodynamics ▶ Gravity Field of the Earth ▶ Lithosphere, Continental ▶ Lithosphere, Oceanic: Thermal Structure ▶ Mantle Plumes ▶ Plate-Driving Forces ▶ Seismic Tomography ▶ Seismic, Receiver Function Technique

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Earth’s Structure, Upper Mantle, Fig. 3 Models PRI-P05 (P velocity) and PRI-S05 (S velocity anomaly) at a depth of 350 km (left) and 600 km (right). The color scale ranges from 1.5% (red) to

+1.5% (blue) in P velocity and 3% in S velocity. Upper mantle plumes show as narrow quasi-circular slow anomalies in red. (From Montelli et al. (2006) with permission from the AGU)

410 Topography

660 Topography

Earth’s Structure, Upper Mantle, Fig. 5 A west-to-east cross section through the Tonga subduction zone crossing the trench at 19S, 173 W shows the subducting slab as a continuous feature down to 600-km depth (blue positive anomaly) and coinciding with the deep seismicity (circles). The negative velocity anomaly above the slab represents most likely the effect of dehydration. Active volcanism on Fiji and Tonga is indicated by black triangles. Also indicated are the Central and Eastern Lau Spreading Centers. (Source: Zhao et al. 1997)

-17 -14 -11 -8 -5 -2 2 5 8 11 14 17

kilometers Earth’s Structure, Upper Mantle, Fig. 4 The depth variations to the “410-” and “660-km” discontinuities, with respect to 410- and 650-km depth, respectively. (From Houser et al. (2008), with permission from the AGU)

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189 Houser C, Masters G, Flanagan M, Shearer P (2008) Determination and analysis of long-wavelength transition zone structure using precursors. Geophys J Int 174:178–194 Jordan T (1981) Continents as a chemical boundary layer. Philos Trans R Soc Lond 301A:359–373 Montelli R, Nolet G, Dahlen F, Masters G (2006) A catalogue of deep mantle plumes: new results from finite-frequency tomography. Geochem Geophys Geosys 7:Q11007 Schaeffer A, Lebedev S (2013) Global shear speed structure of the upper mantle and transition zone. Geophys J Int 194:417–449 Sigloch K, McQuarrie N, Nolet G (2008) Two-stage subduction history under North America inferred from multiple-frequency tomography. Nat Geosci 1:458–462 Zhao D, Xu Y, Wiens DA, Dorman L, Hildebrand J, Webb, S (1997) Depth extent of the Lau back-arc spreading center and its relation to subduction processes. Science 278:254–257

Earth’s Structure, Upper Mantle, Fig. 6 A view from the northeast at the subducted Farallon slab beneath western North America shows major ruptures that allow mantle rock to flow around it, such as the “slab gap” indicated by the broken line. The slab itself is delineated by a surface with an anomaly of +0.4% in P velocity (high velocity anomalies associated with the lithosphere are not mapped). The location of Yellowstone, another suspected passage of hot mantle rock toward the surface, is indicated by the arrow. Numbers along the axes indicate depth in km, latitude, and longitude (negative for west). Color shading indicates depth. The plate boundaries between the North America, Pacific, and Juan de Fuca plates are indicated as purple lines at the surface. (Figure courtesy Karen Sigloch)

▶ Seismic, Waveform Modeling and Tomography ▶ Subduction Zones ▶ Surface Waves

Earthquake Lights John S. Derr1, France St-Laurent2, Friedemann T. Freund3 and Robert Thériault4 1 Tijeras, NM, USA 2 LaSalle, QC, Canada 3 Code SCR, NASA Ames Research Center/San Jose State University, Moffett Field, CA, USA 4 Québec Ministry of Energy and Natural Resources, Québec, QC, Canada

Synonyms Luminous phenomena associated with earthquakes

Bibliography Definition Bostock M (2009) Teleseismic body-wave scattering and receiver-side structure. In: Romanowicz B, Dziewonski A (eds) Seismology and structure of the Earth. Elsevier, Amsterdam, pp 219–246 Byrnes J, Hooft E, Toomey D, Villagomez D, Geist D, Solomon S (2015) An upper mantle seismic discontinuity beneath the Galápagos archipelago and its implications for studies of the lithosphere asthenosphere boundary. Geochem Geophys Geosys 16:1–19 Deng K, Zhou Y (2015) Wave diffraction and resolution of mantle transition zone discontinuities in receiver function imaging. Geophys J Int 201:2008–2025 Dziewonski A, Anderson D (1981) Preliminary reference Earth model. Phys Earth Planet Inter 25:297–356 Ferreira A, Woodhouse J, Visser K, Trampert J (2010) On the robustness of global radially anisotropic surface wave tomography. J Geophys Res 115:B04313 Fukao Y, Widiyantoro S, Obayashi M (2001) Stagnant slabs in the upper and lower mantle transition region. Rev Geophys 39:291–323 Gu Y, Dziewonski A, Ekström G (2001) Preferential detection of the Lehmann discontinuity beneath continents. Geophys Res Lett 28:4655–4658 Hosseini K, Matthews K, Sigloch K, Shepard G, Domeier M, Tseknistrenko M (2018) Submachine: web-based tools for exploring seismic tomography and other models of Earths deep interior. Geochem Geophys Geosys 19:1464–1483

Earthquake lights (EQLs) are anomalous luminosities associated with and presumably caused by the accumulation of stress, most characteristically before or during a seismic event or during an aftershock sequence. There is some disagreement over whether the term “earthquake lights” should also be applied to similar luminous phenomena observed in earthquake-prone areas or along faults without immediately associated seismic activity. In these cases the EQLs may be due to locally high stress levels in the Earth’s crust that wax and wane without catastrophic rock failure or by earthquakes which are too distant or late relative to the observed EQL.

Classification of Earthquake Lights The earliest known report of earthquake lights dates to at least 373 BCE when the Greek cities Helice and Buris were destroyed by an earthquake accompanied by “immense

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columns of fire” (Seneca). Ancient references will always be questionable, especially when the surviving accounts are written later, but such references indicate that the people at that time were aware that lights might accompany earthquakes. Many sightings from various countries around the world have since been reported. They have been summarized in a number of papers (see, e.g., Thériault et al. 2014; St-Laurent et al. 2006; Derr 1973). As a result of extensive observations associated with the Saguenay earthquake sequence from November 1988 to January 1989, St-Laurent (2000) examined and extended earlier EQL classifications (e.g. Montandon 1948). Six types of luminous phenomena were documented: 1. “Seismic lightning” may appear to be similar to ordinary lightning. The latter appears to be an almost instantaneous flash lasting typically 30 ms, but seismic lightning is an electric discharge lasting typically 150 ms, bursting out of the ground, rising upward into the atmosphere, and illuminating large areas of the sky without thunder. “Seismic lightning” can be mistaken for “sheet” or “heat” lightning at night which is too distant for thunder to be heard. 2. Luminous bands in the atmosphere at indeterminate height, sometimes horizontally or vertically and sometimes in a bundle, similar in appearance to some polar aurorae although, at much lower altitude, possibly even extending from the ground level upward. 3. Globular incandescences (moving or static), sometimes attached to luminous bands, called “orbs,” “globes,” or “meteors.” They have the appearance of ball lightning, e.g., of luminous spheres floating in midair, sometimes coalescing, lasting for up to a few minutes. Earthquake Lights, Fig. 1 Earthquake lights from Mt. Kimyo, Matsushiro area, Japan, 26 September 1966, 0325 (JST). Luminosity lasted 96 s (Derr 1973; Yasui 1968). (© Seismological Society of America)

Earthquake Lights

4. Fire tongues, small “flames” flickering or creeping along or near the ground, or like ignis fatuus. 5. Seismic “flames” seen emerging from the ground like an evanescent gas flame but very rarely causing visible burns. 6. Coronal and point-discharge-like lights. A possible seventh type might be “luminous clouds” like those filmed shortly before the M 7.9 Wenchuan, China, earthquake of 12 May 2008 (YouTube 2008). These are phenomena in the high atmosphere, similar to “fire rainbows,” possibly linked to processes at the upper edge of the atmosphere and/or to ionospheric perturbations.

Examples of EQL While reliable pictures of EQL are scarce, one example of type 1 comes from the Matsushiro earthquake swarm from 1965 to 1967, Fig. 1 (Yasui 1968, 1971; discussed in Derr 1973). Video of type 1 can be found in Heraud and Lira (2011). An example of type 3 was photographed at Tagish Lake, Yukon Territory, Canada, Fig. 2 (Jasek 1998), around 1 July, probably 1972 or 1973. While it is not possible to say that this picture is definitely EQL, these orbs are typical of the most-frequently observed EQL, as described by Montandon (1948). The only known photo of “flames,” type 5, was taken during the aftershock sequence to the Vrancea, Romania, earthquake of 4 March 1977, discussed in St-Laurent (2000). A sketch of type 6 comes from the account by Joseph A. Dallaire during the Saguenay M 5.9 mainshock of

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Earthquake Lights, Fig. 2 Earthquake lights from Tagish Lake, Yukon-Alaska border region, around 1 July, probably 1972 or 1973 (exact date unknown). Estimated size: 1 m diameter. Closest orbs slowly drifted up the mountain to join the more distant ones (Jasek 1998). (Photo credit: Jim Conacher, used with permission)

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Earthquake Lights, Fig. 3 Drawing by artist from witness’ description of a bright, fast-moving bluish-white light, preceded and accompanied by crackling noise emitted by the trees. Laterrière, Québec, 19 km N

of the epicenter, observed a few seconds before the M 5.9 mainshock of 25 November 1988. Not to scale (St-Laurent 2000). (© Seismological Society of America)

25 November 1988, Fig. 3 (presented in St-Laurent 2000). These examples are notable for the differences they exhibit: illumination of a broad area of the sky, discrete glowing plasmas, evanescent “flames,” and an electric discharge bursting through the surface of the Earth. They are representative of a number of other sightings, although the last one, the rapidly moving electric discharge, is so far unique in the detail of its description.

Example of Type 3 EQL Schmidt and Mack (1912; Translation by Steven Soter, personal communication) give a number of eye-witness descriptions of type 3 EQL. The most detailed was by Friedrich Konzelmann, foreman in an Ebingen factory: On the evening of 16 November [1911] between 10 and 10:30, I went home with my wife. My house is about 200 m from the city

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Earthquake Lights

Earthquake Lights, Fig. 4 Earthquake lights over Ebingen, Germany, 16 November 1911, ~10 pm. Observed and drawn by Friedrich Konzelmann (Schmidt and Mack 1912) on the national road to Strasbourg. As we were about 30 m from our house, I heard a distant noise like thunder from Sigmaringen, followed by a faint vibration. As I glanced in the direction from which the noise came, I suddenly noticed at some distance a bright flash from the ground, which then at a considerable height turned into a ball of light, about the size of 20 suns. This ball maintained its brightness about 2–3 seconds and then divided itself like lightning in the direction of Ebingen. In my opinion the earthquake came from southeast to northwest. The earthquake began with the lighting of the fireball. After the rolling went past, I looked backward to the city and saw above the Katzenbuckel a similar but somewhat smaller fireball. The whole surroundings were brightly illuminated. (Fig. 4)

Source of EQL and Associated Electromagnetic (EM) Phenomena The leading theory of how these various light phenomena are generated, based on extensive laboratory experiments, attributes them to effects by mobile positive hole charge carriers, which become activated when rocks are subjected to high levels of stress (Freund 2002, 2010, 2019). Positive holes exist in common rocks in a dormant state as peroxy links, O3Si/OO\SiO3, many of which are located at grain boundaries or may straddle grain boundaries. When stresses are applied, grains are moving relative to each other causing the peroxy links to break. They thereby activate electrons and positive holes, which immediately begin to recombine, returning to the inactive peroxy state. The difference between the rate of activation and rate of recombination determines how many of the positive hole charge carriers are able to flow out of the stressed rock volume into

the surrounding less stressed or unstressed rocks at speeds up to about 100 m/s (360 km/h or 220 miles per hour). Those positive holes constitute an electric current with attendant magnetic field variations and electromagnetic (EM) emissions. As positive holes arrive at the Earth surface, they create electric microfields powerful enough to field-ionize air molecules and to produce positive airborne ions, corona discharges, and EQL. They also recombine at the surface, returning to the peroxy state and leading to a distinct IR emission, spectroscopically identified in laboratory experiments and known from satellite infrared images as thermal infrared (TIR) anomalies. How many peroxy a given rock contains represents an important factor. Gabbro and diabase often form vertical sheets of magmatic rocks called dykes, which have deep roots in the Earth’s crust, even the upper mantle. They are emplaced during periods of extensional tectonics and are believed to be particularly rich in peroxy. When such rocks are stressed very rapidly, within milliseconds, for instance, by the passage of an earthquake wave, the number density of activated positive holes can instantly reach such high values that the electrical properties of the rocks change dramatically and outburst of electric charges can occur at the speed of light (Freund 2019). The positive hole theory accounts not only for EQL but also for other pre-earthquake phenomena such as: 1. Air ionization at the ground-to-air interface with either exclusively positive airborne ions or corona discharges producing positive plus negative ions that tend to rise through the atmosphere. They are suspected to be linked

Earthquake Precursors and Prediction

2.

3. 4.

5.

6.

7.

to widely reported anomalous pre-earthquake fog/haze/ cloud formation. Changes in the electrical conductivity of the soil and geoelectric potentials, both as a function of lateral distances and as a function of depth. Geomagnetic anomalies in the Earth’s crust due to stressactivated telluric currents that flow through the rock column. Ionospheric perturbations in response to air ionization processes in the low atmosphere and/or low to ultralow electromagnetic emission from the Earth. Ultralow and extremely low-frequency (ULF/ELF) and radio-frequency (RF) emissions. Such signals are being used successfully to forecast earthquakes 2–3 weeks in advance in the Peru subduction zone (Heraud and Lira 2011; Heraud et al. 2017). Anomalous thermal infrared (TIR) emissions from around future epicentral areas, occurring particularly from topographic highs. Unusual animal behavior thought to be triggered by air ionization at the ground-to-air interface, in particular by the formation of positive airborne ions or by broadband ultralow frequency (ULF) electromagnetic (EM) emissions from within the Earth’s crust, due to stress-activated fluctuating telluric currents, which interfere with physiologically important EM emissions from living organisms such as the alpha brain waves (Freund and Stolc 2013).

Conclusion At present there is no consensus among seismologists on the phenomenology and physical mechanism of EQL and associated EM phenomena. However, if the positive hole theory is correct, the entire suite of pre-earthquake phenomena, including EQL, may become useful in recognizing the build-up of tectonic stresses in the Earth’s crust and, hence, for forecasting earthquakes.

Cross-References ▶ Earthquake Precursors and Prediction ▶ Electrical Properties of Rocks ▶ Electrical Resistivity Surveys and Data Interpretation ▶ Geoelectromagnetism ▶ Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity

Bibliography Derr J (1973) Earthquake lights: a review of observations and present theories. Bull Seismol Soc Am 63:2177–2187 Freund F (2002) Charge generation and propagation in rocks. J Geodyn 33:545–572

193 Freund F (2010) Toward a unified theory of pre-earthquake signals. Acta Geophys 58:719–766 Freund F (2019) Co-seismic earthquake lights: the underlying mechanism. Pure Appl Geophys. https://doi.org/10.1007/s00024-01902142-2 Freund F, Stolc V (2013) Nature of pre-earthquake phenomena and their effects on living organisms. Animals 3:513–531 Heraud J, Lira J (2011) Co-seismic luminescence in Lima, 150 km from the epicenter of the Pisco, Peru earthquake of 15 August 2007. Nat Hazards Earth Syst Sci 11:1025–1036. EQL Video at: https://www. youtube.com/watch?v¼f14pQakxXjc. Last accessed 1 February 2020 Heraud J, Centa V, Bleier T (2017) Images of the benioff zone in the lithosphere using electromagnetic energy released from stress in tectonic plates. 32nd URSI GASS, Montreal, 19–26 August 2017 Jasek M (1998) Tagish Lake UFO photo report. http://www.ufobc.ca/ yukon/tagish.htm. Last accessed 1 February 2020 Montandon F (1948) Lueurs et malaises d’origine séismique. Geogr Helv 3(1):157–178 Schmidt A, Mack K (1912) Das Süddeutsches erdbeben vom 16 November 1911. Abschnitt VII: Lichterscheinung. Württembergische Jahrbücher fur Statistik und Landskunde, Jahrg. part 1, 131–139 Seneca, Natural Questions, VI, 26.2–4 St-Laurent F (2000) The Saguenay, Québec, earthquake lights of November 1988–January 1989. Seismol Res Lett 71:160–174 St-Laurent F, Derr J, Freund F (2006) Earthquake lights and the stressactivation of positive hole charge carriers in rocks. Phys Chem Earth 31:305–312 Thériault R, St-Laurent F, Freund F, Derr J (2014) Prevalence of earthquake lights associated with rift environments. Seismol Res Lett 85(1):159–178 Yasui Y (1968) A study of the luminous phenomena accompanied with earthquake (part 1). Mem Kakioka Magn Obs 13:25–61 Yasui Y (1971) A study of the luminous phenomena accompanied with earthquake (part 2). Mem Kakioka Magn Obs 14:67–68 YouTube (2008) http://www.youtube.com/watch?v¼KKMTSDzU1Z4. Last accessed 1 February 2020

Earthquake Precursors and Prediction Toshiyasu Nagao1, Masashi Kamogawa2 and Seiya Uyeda3 1 Institute of Oceanic Research and Development, Tokai University, Shizuoka, Japan 2 Division for Earthquake Prediction Research, Global Center for Asian and Regional Research, University of Shizuoka, Shizuoka, Japan 3 Japan Academy, Tokyo, Japan

Definitions and Introduction Earthquake (EQ hereafter) prediction or forecast is to specify the source location, magnitude M, and occurrence time of EQ with certain accuracy before its occurrence. The terms prediction and forecast are often used with different nuances, the latter allowing for more stochastic nature. We will mainly use the former term in this entry. EQ prediction is often roughly classified by the length of concerned time into long term

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(101.5–101 years), intermediate term (101–100.5 years), and short term ( n, though it has been so in all actual studies of earthquakes. It is now well understood that the matrix A can be ill-conditioned which implies that the system (4) admits more than one solution, equally well fitting the observations. The constraints (5) are introduced for the purpose of reducing the set of permissible (feasible) solutions. When a “best-fitting” solution is obtained, other solutions almost satisfying the equations are also of great interest. This is because the data used in geophysical applications often contains noise and the models used are themselves approximations to reality, so that the “best-fitting” solution may not represent actuality. In addition, though the different solutions may be very close together in data-space, it is possible that they may be somewhat different in model-space, that is, they may physically represent rather different solutions. Thus, more than one solution must be found and the set of (almost) equally well-fitting solutions examined to find features common to the set of solutions, as was done by Das and Kostrov (1990), Henry et al. (2000), and others studies using this method. Das and Kostrov (1990, 1994) also suggested additional tests to determine the robustness of such features. Only features clearly required by the solutions should be interpreted in a geophysical context, as was done in the specific examples discussed below. For the system of equations (4) together with the constraints (5) to comprise a complete mathematical problem, it is necessary to formulate the exact form of what the “best-fit” to observations means. We include here only constraint (5a) in the mathematical formulation for the sake of simplicity, the inclusion (5b) and (5c) being simple, and is discussed, for example, by Das and Kostrov (1990, 1994). We have to minimize the vector of residuals: r ¼ b  Ax

ð6Þ

For this purpose, some |norm of the vector r must be adopted. One may choose to minimize the ‘1, the ‘2 or the

‘1 norm, all three being equivalent in the sense that they tend to zero simultaneously. Hartzell and Heaton (1983) used the ‘2 norm to solve the optimization problem with the positivity constraint, employing computer programs in Lawson and Hanson (1974). Das and Kostrov (1990, 1994) have used the linear programming method (Press et al. 1986) to solve the system (4) and minimize the ‘1 norm subject to the condition (5a). The ‘1 norm is considered more robust (Tarantola 1987), and Hartzell et al. (1991) have investigated the effect of minimizing different norms on the solution and have shown that the most robust features are properly obtained in the solutions, whichever norm is used. Das and Kostrov (1990, 1994) have checked that when the ‘1 norm is minimized, the ‘2 and the ‘1 norms for the solution are also small. In reality, the actual scales at which the rupture propagation takes place in the Earth is microscopic whereas the inverse problem must be solved using finite grids. Therefore, it is important to consider how well the inverse solution obtained gives a picture of the actual solution. To do this, it is necessary to decide how large the cells in the inversion should be. Clearly, the smaller the cells, the better the approximation to the integral equation being solved (Eq. 1) but the larger the number of unknowns. On the other hand, though the approximation to the integral equation improves by going to smaller and smaller cells, the condition number of the matrix A increases (due to columns becoming very similar) making the problem more and more unstable. As pointed out by Backus and Gilbert (1967), since the amount of data is finite, the problem of finding a continuous function from it is indeterminate. Das and Suhadolc (1996) and Das et al. (1996) have suggested that synthetic tests using artificially created data should be used to best estimate the spatial and temporal grid sizes and that the spatial and temporal grid sizes should be consistent. Henry et al. (2000) have also compared the Green functions obtained using various grid sizes to aid in the grid size selection. Sarao et al. (1998) have studied the effect of non-uniform seismic station distribution around the earthquake epicenter.

Constraints from Laboratory Experiments As mentioned earlier, advanced technological developments has led to major improvements in laboratory experiments on rupture propagation. One important result is the demonstration that ruptures can not only exceed the shear wave speed of the medium but can even reach its compressional wave speed value. Though super-shear ruptures

Earthquake Rupture: The Inverse Problem

speed had been found in laboratory experiments as far back as the 1970s (Wu et al. 1972), since these experiments were conducted on plastic polymer, under very low normal stresses, this was considered unrealistic for real earthquakes – both the material and the low normal stress – and the results were ignored. In the late 1990s to the early 2000s, a group at Caltech, led by Rosakis, measured earthquake speeds in the laboratory, not only exceeding the shear wave speed (Rosakis et al. 1999; Xia et al. 2004) but even reaching the compressional wave speed (Xia et al. 2005). The validity of such experiments was confirmed by Mello et al. (2010), who compared velocity and acceleration records from the 2002 Mw 7.9 Denali, Alaska, earthquake at a station close to the fault with those found in the laboratory experiments. Though these experiments were carried out on man-made material (Homalite) under normal temperature and pressure, they revolutionized our way of thinking. To mimic more realistic conditions, a group led by Schubnel at the Ècole Normale Supérieure (ENS) in Paris has carried out experiments on rock samples (Westerly granite) and obtained super-shear rupture speeds (Passelègue et al. 2013). The rupture front position was obtained by analysis of acoustic high-frequency recordings on a multistation array. This is clearly very close to the situation in seismology, where the rupture details are obtained by seismogram (time-series) analysis, as discussed earlier. In the real Earth, earthquake ruptures propagate through material at higher temperatures and pressures than those in these experiments, causing lower crustal, intermediate, and deep earthquakes. The same group at ENS has started looking at this problem in the laboratory (Schubnel et al. 2013) by simulating the nucleation of acoustic emission events that resemble deep earthquakes. These events are caused by an instantaneous phase transition from olivine to spinel, which would occur at the same depth and result in large stress releases observed for some deep earthquakes. Now this group has installed laboratory equipment which can reach pressures of 4GPa and temperatures of 1100 °C. For the larger samples, this would simulate conditions in the Earth approximately at 15–20 km depth. On smaller size samples, this would cover the 20–100 km depth range. Thus, in future we will be able to better simulate ruptures at the depths where earthquakes actually occur, thereby improving our understanding of such earthquakes. From the practical point of view, the idea that only crustal earthquakes cause damage to the built environment is not true. The 1970 Mw 7.8 Peru earthquake occurred at a depth of 45 km (International Seismological Centre Bulletin), and killed more than 66,000 people and completely devastated the city of Huarez, Peru. It remains till today the worst natural disaster in the

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western and southern hemispheres of the Earth. Thus, study of sub-crustal earthquakes also has important societal implications.

Discussion of Results of Inversion Finally, we discuss some examples of what we have learned regarding the mechanics of the earthquake source from the solution of such inverse problems. The first great earthquake which for which the slip history and distribution was obtained was the great 1985 Mw 8.0 Michoacan, Mexico, earthquake (Mendoza and Hartzell 1988). It was found that the main slip took place in two patches on the fault and that aftershocks occurred in the regions of low fault slip or regions of transition from high to low slip, that is, where stresses were increased due to the earthquake. Some studies since then have been summarized by Das and Henry (2003), and many earthquakes have taught us important, and often unexpected, features of earthquake mechanics. Super-Shear Wave Rupture Speeds One very important outcome of the study of the inverse problem of earthquake source mechanics has been the result that actual earthquakes can have rupture speeds exceeding the shear wave speed and can even approach the compressional wave speed of the medium. Robinson et al. (2006a) found this when studying the >400 km long strike-slip 2001 Mw 7.8 Kunlun, Tibet, earthquake, which propagated for nearly 100 km at this very high speed. The fault geometry was shown to be a controlling factor for the changes in rupture speed (Robinson et al. 2006a), with the longest, straight portion of the fault having the fastest rupture speed (Fig. 2). The off-fault regions of the super-shear portions of this earthquake had great damage (Bhat et al. 2007), and this again has obvious important implications for earthquake hazard mitigation. By inspecting all long strike-slip faults in the world, and noting their long straight portions, Robinson et al. (2010) made a list of all faults likely to have sustained super-shear rupture and called them “earthquake fault superhighways.” Das (2015) compiled a list of earthquakes known to rupture at super-shear speeds and discussed its importance for seismic hazard assessment and for earthquake engineering applications. Table 1 gives an updated list of crustal earthquakes known to have reached these speeds, and Table 2 the list of earthquake fault superhighways. Earthquake Rupture Complexity Another result that has come from the study of inverse problems is the complexity of earthquake ruptures. Forward modeling had shown that such complexity can exist (see,

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Earthquake Rupture: The Inverse Problem, Fig. 2 The 14 November 2001 Mw 7.8 Kunlun, Tibet, earthquake. (a) (top) Tectonic setting and topography. The star indicates the epicenter. (bottom) Segmentation of the Kunlun fault is shown by vertical blue dashed lines, together with the CMT solution (red) obtained by Robinson et al. (2006a). The Harvard CMT solution for the 8 November 1997 Manyi earthquake is shown in

Earthquake Rupture: The Inverse Problem

blue and the focal mechanisms of two earlier earthquakes in the region with magnitudes >7 are shown in black. The faulting extent in the 2001 Kunlun earthquake is shown in red. (b) Relocated aftershocks for the 6-month period following the earthquake are shown by gray dots, together with their 90% confidence error ellipses. The two faults on which rupture occurred are marked (in red) as MKF (Main Kunlun

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e.g., Das and Kostrov 1988). Detailed analyses of body wave seismograms have now shown that such complexity does exist in real earthquakes (Das and Henry 2003). For example, Henry et al. (2000) showed that the 1998 Mw 8.1 Antarctic Plate earthquake jumped over a barrier of width as large as 70–100 km, with a time delay of about 40s, and then

continued propagating for about another 60 km, this second earthquake itself having a Mw 7.6–7.8 (Fig. 3). Both this complexity and the pattern of aftershocks is controlled by preexisting fracture zones on the ocean floor. The great 2001 Mw 8.4 Peru earthquake (Fig. 4) showed how a barrier on the fault could stall the rupture process (Robinson et al.

Earthquake Rupture: The Inverse Problem, Table 1 Strike-slip earthquakes known to have reached super-shear rupture speeds Year 1979

Mw 6.5

Location Imperial Valley, California

Super-shear segment length 35 km

Data used Strong ground motion

1999 1999 2001 2002 2012 2013 2018

7.6 7.2 7.8 7.9 8.6 7.5 7.5

Izmit, Turkey Duzce, Turkey Kunlun, Tibet Denali, Alaska N. Sumatra Craig, Alaska Palu, Indonesia

45 km 40 km >400 km 340 km 200, 400, 400 km 100 km 30 km

” ” Teleseismic ” ” ” ”

Reference Archuleta (1984); Spudich and Cranswick (1984) Bouchon et al. (2000) Bouchon et al. (2001) Robinson et al. (2006a) Walker and Shearer (2009) Wang et al. (2012) Yue et al. (2013) Socquet et al. (2019)

Earthquake Rupture: The Inverse Problem, Table 2 Earthquake fault superhighways

1 2 3

Fault system and location Red River, Vietnam/China San Andreas, California Sagaing, Burma

Total (km) 1000 1050 1000

Segment lengthsa 280, 230, 290 160, 230 700

Affected population (millions)b 25.7 13.1 9.1

4

Great Sumatra

1600

6.7

5 6 7

1000 1100 1600

8

Dead Sea, Jordan/Israel Chaman/Herat, Luzon, Philippines Pakistan/ Afghanistan Kunlun, Tibet

100, 160, 220, 200 100, 125 170, 320, 210 130

9 10 11

Altyn Tagh, Tibet Bulnay, Mongolia Denali, Alaska

1200 300 1400

1600

270, 130, 180, 100 100, 100, 150 100, 200 130

5.2 2.5 2.1

Size and dates of past earthquakesc 7.7 (1733), 8.0 (1833) 7.9 (1857), 7.9 (1906) N.D.(1839), 7.3 (1930), 7.3 (1930), 7.6 (1931), 7.5 (1936), 7.4 (1946) 7.7 (1892), 7.6 (1909), 7.5 (1933), 7.4 (1943), 7.6 (1943) N.D. (1068), N.D. (1170), N.D. (1202) N.D. (1892) 7.8 (1990)

0.15

7.5 (1937), 7.8 (2001)

0.062 0.020 Negligible

7.6 (1932) 7.8,8.2 (1905) 7.8 (2002)

a

Lengths of straight segments, identified as superhighways, listed from south to north Current population, in millions, within 50 km of the superhighways, this being the region expected to be most damaged by earthquake rupture along the superhighways c Magnitude of old earthquakes are surface wave magnitude or moment-magnitude, as available; N.D. if unknown b

ä Earthquake Rupture: The Inverse Problem, Fig. 2 (continued) Fault) and SF (Secondary Fault). The star shows the rupture initiation region. Available CMT solutions are shown in green. Schematic graphs showing the final slip on the two faults are shown as blue lines, with the

rupture speeds in the three phases of rupture on the MKF indicated at the top. The central portion is where rupture reached the compressional wave speed of ≈ 6 km/s. (Figure based on Robinson et al. (2006a))

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2006b). Based on marine geophysical data from the portion of the Nazca plate that was subducting in this region, this barrier was identified as being due to bathymetry (such as seamounts located on the inner corner of a fracture zone) related to a subducting fracture zone. Robinson and Cheung (2010) found that barriers limited the faulting extent of the 2003 Mw 8.3 Tokachi-Oki, Japan, earthquake and that the

regions of high slip (“asperities”) aligned with other seamounts on the subducting plate about to enter the trench (Fig. 5). The fact that all the earthquakes mentioned in this section are oceanic is simply due to the fact that many such earthquakes have occurred in the last three decades, and does not imply that earthquakes on land do not have complexity!

Earthquake Rupture: The Inverse Problem, Fig. 3 (a) Tectonic setting of the 25 March 1998 Mw 8.1 Antarctic Plate earthquake. Marine gravity anomalies are from an updated version of Sandwell and Smith (1997), illuminated from the east, with contours every 20mGal. The relocated aftershocks for a one-year period following the earthquake are shown as white diamonds, the main shock epicenter being the black star. Background seismicity for the period January 1, 1964, to July 31, 1997, is shown as white circles. Selected linear gravity features are identified by white lines and are labeled F1–F6. F1a is the southward continuation of F1. (b) (Top) Principal features of the main shock rupture process. Arrows show location and directivity of the two sub-events and are labeled with start and end times of rupture segments, obtained by Henry et al. (2000). Focal mechanisms are shown in green for the initiation, the first sub-event plotted at the centroid location obtained in

the study of Henry et al. (2000), and for the second sub-event. The open triangle shows the Harvard CMT centroid location. Epicenters shown in (a) which fall in the region of this map are shown in gray. Linear gravity features F1a, F3a, are F4–F6 are shown as solid gray lines. T1a and T3a (dashed lines) are the locations of tectonic features inferred by Henry et al. (2000). (Bottom) Final moment distribution for the preferred solution of Henry et al. (2000) shown as black line graph, with the baseline of the graph being the physical location of the fault. The peak moment density is 1.25  1019 N m km1. Regions of the fault with > 1 vertical advection dominates, while the conductive component prevails for Pez n ¼ k, is known as strictly overdetermined. 2. Case when n > m ¼ k, is known as strictly underdetermined. 3. Case when m ¼ n ¼ k, is known as evenly determined. 4. Case when k < min(m,n) is known as underdetermined. Here, k is the rank of matrix A. In any real problem the observed data is, in general, inadequate and inaccurate. As a result the resulting inverse problem is inconsistent, underdetermined, and most of the time unstable. Under such adverse conditions when the exact solution does not exist, the basic task in solving the matrix Eq. 5 pertains to obtaining a solution that meets the desired objectives. Backus (1970, 1996), Backus and Gilbert (1967, 1968, 1970), Jackson (1972, 1979), and others studied these problems and developed methodologies to obtain a unique solution. It is inevitable that in such cases there would be a trade-off between the propagation of data error

to parameter estimation and the parameter resolution. The inevitable nonuniqueness is removed by incorporating the available a priori information about the unknown parameter vector x. The various methods differ in handling the a priori knowledge about data error and model and in controlling the trade-off between error propagation and parameter resolution. To be able to solve Eq. 5 one designs a matrix Ag, termed as generalized inverse, such that x ¼ Ag b:

ð6Þ

It is obvious that matrix Ag will not possess all the properties of the exact matrix inverse, A1, which can be defined only for non-singular matrix. For various cases mentioned above, the generalized inverse will possess only a subset of these properties. One of the important properties of exact inverse is AA1 ¼ A1 A ¼ I. In case of a matrix A of order m  n, the generalized inverse matrix, Ag, will lead to the product matrices, S ¼ AAg of order m  m and R ¼ AgA of order n  n and these, in general, will not be identity matrices. The former product matrix S is termed as data resolution matrix or information density matrix, while the latter product matrix R is known as parameter resolution matrix. These product matrices play an important role in the evaluation of quality of solution. Norms of the deviation of matrices S and R from respective Identity matrices provide quantitative measures of quality of information contained in data vector and of the degree of parameter resolution, respectively. The matrix Ag can be synthesized in several ways for a given type of coefficient matrix A. The important approaches widely employed to solve the Eq. 5 lead to 1. 2. 3. 4. 5.

Least square inverse Minimum norm inverse Damped least square (DLS) inverse Rao–Mitra generalized inverse Moore–Penrose generalized inverse based on singular value decomposition (SVD)

All these approaches are briefly presented below followed by Backus–Gilbert method which also provides a solution of the linear inverse problem stated as Eq. 1.

Generalized Inverses Least Square Inverse The least square (LS) solution of a strictly overdetermined problem was first proposed by Gauss in 1795. The LS solution exists even for inconsistent system of equations. It is obtained by minimizing the L2 norm of misfit error vector, that is,

Inverse Theory, Linear

9 8 Minimize eT e ¼ ðb  AxÞT ðb  AxÞ > > > > > >  T 1 T > > > > > > A A b Solution : x ¼ A > > > > > >   > > 1 T > > g T > > Inverse : A ¼ A A A = < g Information density matrix : S ¼ AA > > > >  T 1 T > > > > > > ¼ A A A A ¼ 6 I m > > > > > > > g > > Parameter resolution matrix : R ¼ A A > > > > >  T 1 T ; : ¼ A A A A ¼ In

817

Here, the parameter l is a control parameter that represents relative weighting given to the two objective functions. The expressions of matrices S and R suggest that neither the data quality nor resolution of parameters is optimal. ð7Þ

The expressions of matrices S and R reveal that the data quality is not optimal while all the parameters are resolved exactly. Minimum Norm Inverse The minimum norm (MN) solution of a strictly underdetermined problem is obtained through constrained minimization of the L2 norm of the unknown vector x. 9 8 Minimize xT x > > > > > > > > > > Subject to the constraint b  Ax ¼ 0 > > > > > >   > > 1 > > T T > > Solution : x ¼ A AA b > > > > > > > >   1 = < g T T Inverse : A ¼ A AA > Information density matrix : S ¼ AAg > > > > > > >  T 1 > > > > T > > ¼ AA AA ¼ I > > m > > > > > > g > > Parameter resolution matrix : R ¼ A A > > > > > >   1 ; : T T ¼A A A 6¼ I n

ð8Þ

The expressions of matrices S and R suggest that the data quality is optimal while all the parameters are not resolved exactly and there exists linear dependence amongst some components of the parameter vector. Damped Least Square Inverse The DLS solution of an overdetermined problem when rank k < min(m,n) was concurrently developed by Marquardt, Phillips, Tikhanov, and Twomey. It is obtained by minimizing the two objective functions defining norms of error e and of unknown vector x, subject to the constraint of x satisfying the matrix Eq. 5. 9 8 > Minimize eT e ¼ ðb  AxÞT ðb  AxÞ and xT x > > > > > > > > > > > Subject to the constraint b  Ax ¼ 0 > > > > > >   > > 2 1 T T > > > > A þ l I A b Solution : x ¼ A > > > > > >   > > 1 = < T T 2 ¼ A AA þ m I b   1 > > > > Inverse : Ag ¼ AT A þ l2 I AT > > > > > > > >   1 > > T T 2 > > or A AA þ m I > > > > > > > > g > Information density matrix : S ¼ AA 6¼ I m > > > > > > > ; : Parameterre solution matrix : R ¼ Ag A 6¼ I n

Weighted Damped Least Square Inverse To incorporate any available information about the error characteristics of data and/or about smoothness characteristics of parameter vector, the two objective functions optimized in case of DLS inverse, can be redefined in terms of weighting matrices We and Wm. We is the data error covariance matrix and Wm is the matrix defining smoothness characteristics of the desired model. Wm can be constructed from the low order differences of parameter vector components. 9 8 Minimize eT W e ¼ ðb  AxÞT > > > > > > > > T > > W ð b  Ax Þ and x W x > > e m > > > > > > > > Subject to the constraint b  Ax ¼ 0 > > > > > >   > > 1 > > 2 T T > > W A þ l W A W b or Solution : x ¼ A > > e m e > > = <   2 1 T 1 T 1 1 ð10Þ x ¼ W m A AW m A þ l W e b > > > >   > > 1 > > > Inverse : Ag ¼ AT W e A þ l2 W m AT W e or > > > > > > >   > > 1 > > 2 1 T g 1 1 T > > A ¼ W A AW A þ l W > > m e m > > > > > > g > Information density matrix : S ¼ AA ¼ 6 I > > m > > > ; : Parameter resolution matrix : R ¼ Ag A 6¼ I n All the above inverses can be considered as special cases of the generalized inverse defined by Rao and Mitra 1971) which is discussed next. Rao–Mitra Inverse Rao and Mitra (1971) put forth a definition of generalized inverse, Ag, as any matrix that provides a solution of matrix Eq. 5 as x ¼ Ag b:

ð11Þ

It is evident that this inverse satisfies the relation AAg A ¼ A:

ð12Þ

This condition leads to two more properties AAg is idempotent, Ag A is idempotent: ð9Þ It may be mentioned that idempotent matrices are those for which any power of the matrix is equal to itself. Rao and Bhimasankaram (1992) presented a suit of lgorithms for evaluating the generalized inverse using Gaussian elimination like steps based on elementary row nd

I

818

Inverse Theory, Linear

column operations that reduce the coefficient matrix to a Hermite canonical form. An earlier definition of generalized inverse, A+, given by Moore (1920) and rediscovered by Penrose (1955), vas based on SVD of matrix A. It satisfies the conditions 9 8 AAþ A ¼ biA > > > > > > þ þ = < A AA ¼ Aþ > AAþ is idempotent > > > > > ; : A þ A is idempotent

ð13Þ

f ðxÞ ¼ Gðx, yÞgðyÞdy, a < x < b

ð18Þ

c

For Eqs. 1 and 18 to coexist, it is essential that the functions K(x,y) and G(x,y) satisfy the relations

ð14Þ

Here U and V are the modal matrices of orders m  m and n  n, respectively and Λ is an m  n matrix whose leading k diagonal elements correspond to the nonzero singular values and all other elements are zero. The MPI is given by Aþ ¼ V Lþ U T

Continuous Inverse Problem The solution of Equation 1 can be represented as ðd

Moore–Penrose inverse (MPI) is always unique while Rao–Mitra inverse (RMI) is, in general, not. In fact, MPI is one member of the set of all possible RMIs. To elaborate this point, it may be stated that for a given SVD of matrix A, A ¼ ULVT :

problem expressible as a Fredholm’s equation of first kind like Eq. 1. The main feature of this methodology is the possible detailed study of the trade-off between parameter resolution and error propagation. The Backus–Gilbert method (BGM) is briefly described here both for continuous and discrete cases.

ðb

Gðx, yÞK ðx, y0 Þdx ¼ dðy  y0 Þ,

ð19Þ

Gðx, yÞK ðx0 , yÞdy ¼ dðx  x0 Þ:

ð20Þ

a

and ðd

ð15Þ c

Here Λ+ is the n  m matrix whose first k diagonal elements correspond to the inverse of nonzero singular values and all other elements are zero. In contrast, the RMI is given by Ag ¼ V Lg U T :

ð16Þ

Here matrix Λg is an n  m matrix whose leading k  k submatrix block is diagonal with nonzero diagonal elements corresponding to the inverse of nonzero singular values and all the other submatrices are arbitrary. This freedom of having arbitrary elements outside the main k  k block leads to the nonuniqueness of MRI. This, in turn, leads to nonunique solutions of matrix Eq. 5, which can be derived from a given solution, x(1), of Eq. 5 as x ¼ xð1Þ þ ðAg A  I Þj

The function δ(r–r0 ) is the transcendental Dirac-delta function. Backus and Gilbert brought into focus the infinitely undetermined nature of the real continuous linear inverse problems represented by Eq. 1. They emphasized that from the finite observations, it would never be possible to exactly determine the infinite unknowns constituting the sought model property function f(x). They circumvented the problem by seeking not the exact solution but its approximation as a linear superposition of the observed data values. Let the finite observation set comprise m values gi ¼ g(yi) with c  y1  y2  . . .  ym–1  ym  d. Further let the unknown function f(x) need be estimated at the n argument values a  x1  x2  .. . .  xn–1  xn < b. Now Eqs. 1 and 18 can be rewritten as

ð17Þ

ðb g i ¼ gð y i Þ ¼

j being an arbitrary vector. One possible vector x(1) can be x + the solution obtained using MPI.

KiðxÞf ðxÞdx,

81 < i < m

ð21Þ

ðd   f j ¼ f x j ¼ G j ðyÞgðyÞdy,

81 < j < n

ð22Þ

a

and

Backus–Gilbert Method Backus and Gilbert (1967, 1968, 1970) studied in detail the problem of synthesizing solutions to a linear inverse

c

Inverse Theory, Linear

819

Expressing Eqs. 25, 27, and 28 in matrix form, we get

where   K i ðxÞ ¼ K ðx, yi Þ and G j ðyÞ ¼ G x j , y :

9 8 T b > > =

> ; : T u c¼1

ð23Þ

Backus and Gilbert sought an approximation of Eq. 22 as m   X   f j ffi fbj ¼ f x j ¼ c i x j gi

ð24Þ

with

1

ðb Mij ¼ K i ðxÞK j ðxÞdx

Substituting for gi from Eq. 21, we get

a

ð m X   b ci x j K i ðxÞf ðxÞdx fj¼ b

1

ðb

    ai ¼ 2 K i ðxÞJ x, x j D x, x j dx

a

a

ðb

  b ¼ D2 x, x j dx

For Ki, and f being square integrable, the order of integration and summation can be interchanged and the equation can be rewritten as fbj ¼

ðb "X m a

a

ui ¼ K i ðxÞdx a

Using Lagrange’s method of undetermined multipliers, the solution of Eq. 29 is found to be

where m   X   A x, x j ¼ ci x j K i ðxÞ,

c¼

ð26Þ

1

is termed by Backus and Gilbert as averaging kernel and is also known in literature as resolving kernel or scanning function. From Eq. 25, it is evident that for fj to be exactly estimated, the averaging kernel should be the Dirac-delta function δ(x– xj). This equality is, in general, not achievable. In order to make A(x,xj) as close to δ(x–xj) as possible, the coefficients c0i s are chosen to satisfy a judiciously chosen δ-ness criterion. One such criterion can be ðb        2 q j ¼ q x j ¼ J x, x j A x, x j  D x, x j dx:

ð27Þ

a

Several pairs of functions J,D have been used by different workers. Backus and Gilbert used J(x, xj) ¼ 12(x–xj)2 and D(x,xj) ¼ δ(x–xj). They interpreted qj as the half-width or spread of the resolving kernel A. Hence, c0i s are so chosen that these minimize qj subject to the condition that A is unimodular, that is, ðb





A x, x j dx ¼ 1 a

I

a

ðb

# ðb     ci x j K i ðxÞ f ðxÞdx ¼ A x, x j f ðxÞdx ð25Þ

1

ð29Þ

M 1 u uT M 1 u

ð30Þ

Substituting these c-values in Eq. 24, one gets the estimate of unknown physical property function f at argument value xj. All values f1, f2, . . . fn are obtained by repeating the above steps for each x1, x2,. . .,xn. The quality of solution can be evaluated by analyzing the characteristics of the averaging kernel A(x,xj), which ideally should be the Dirac-delta function. Hence, the deviation of the center of A(x,xj) from xj and the finite range of x over which A(x,xj) is nonzero may be taken as pointers to the quality of estimation. The discussion till now was confined to error-free data. However, if the data is erroneous and its error covariance matrix is E, the procedure need to be modified to get a solution. In such cases, the superposition coefficient vector c is obtained by minimizing besides qj given by Eq. 27 and the error propagation measure ej given by e j ¼ cT Ec

ð31Þ

The two objective functions are combined into one as r j ¼ cT Nc,

ð28Þ where

820

Inverse Theory, Linear

N ¼ lM þ ð1  lÞE, 0  l  1:

ð32Þ

The procedure described above can be followed after replacing matrix M by N and the resulting estimate of coefficient vector will then be c¼

N 1 u : uT N 1 u

ð33Þ

Equations 21 and 22 make it obvious that for l ¼ 1, spread gets minimized but error propagation is uncontrolled while for l ¼ 0, the spread is unlimited and error propagation is minimized. Discrete Inverse Problem The implementation of BGM for the discrete case represented by Eq. 5 was given by Gupta (1998) where it is shown that, for error-free case, if the unimodularity constraint is not applied the BG estimate is same as the minimum norm estimate. However, when this constraint is applied the BG estimate has an additional correction term. The solution of matrix Eq. 5 is synthesized as x ¼ Ag b:

ð34Þ

To synthesize Ag some norm of (Ag A–I) is minimized. In particular, for L2 norm, we can minimize   uT ðfAg A  I Þ AT AgT  I u:

ð35Þ

This will lead to the minimum norm solution xmin. When the unimodularity condition, asking for each row vector of the resolution matrix R ¼ AgA ¼ AT(AAT)A to be of unit length, is imposed as constraint, the solution xBG is found to be given as xBG ¼ xmin þ

 u  Ru  T u mmin uT Ru

Summary This article presents salient aspects of linear inverse problems. The LS formulations and the Backus–Gilbert formulation are discussed in brief. The information density and parameter resolution matrices and their importance are discussed. Finally, the methods for finding the optimal multi-objective optimization parameter l are briefly mentioned. For more details on SVD one can refer to the article by A. Manglik, while for global optimization techniques for nonlinear inverse problems one can refer to the articles by M. K. Sen and by W. Sandham published in this series.

Cross-References

Here uT ¼ (11 . . . 1). Minimization yields  1 Ag ¼ AT AAT :

each other, one gets an L-shaped curve. This curve reveals that for large value of l, error propagation is uncontrolled and the spread is optimized while for small value of l, error propagation is optimized and the spread is uncontrolled. The tip of the L-curve provides the optimal value of l. In case of DLS method, the same exercise can be carried out to obtain the optimal value of l. Lot of research has been put in to devise the strategies to find optimal l values. Most of these works analyze the dependence of solution on l through eigen analysis. For details, one can refer Calvetti et al. (2000), Hansen (1992), Reichel and Sadok (2008), and Rezghi and Hosseini (2009).

ð36Þ

Trade-Off Between Resolution and Error Propagation In Backus–Gilbert (BG) method Eq. 32 demonstrates the trade-off between the two objective functions. For different values of l one gets different solutions. Hence, it is expected that some value of l will provide optimal trade-off solution. BG demonstrated in their paper of 1970 that when the values of objective functions qj and rj for each l are plotted against

▶ Inverse Theory, Artificial Neural Networks ▶ Inverse Theory, Global Optimization ▶ Inverse Theory, Singular Value Decomposition

Bibliography Aki K, Richards PG (2002) Quantitative seismology. University Science Book, Sausalito Backus GE (1970) Inference from inadequate and inaccurate data. Proc Natl Acad Sci 65:1–7, 281–287; 67:282–289 Backus GE (1996) Trimming and procrastination as inversion techniques. Phys Earth Planet Inter 98:101–142 Backus GE, Gilbert JF (1967) Numerical applications of a formalism for geophysical inverse problems. Geophys J R Astron Soc 13:247–273 Backus G, Gilbert JF (1968) Numerical applications of a formalism for geophysical inverse problems. Geophys J R Astron Soc 16:169–205 Backus G, Gilbert JF (1970) Uniqueness in the inversion of inaccurate gross earth data. Philos Trans R Soc Lond A 266:123–192 Bjorck A (1996) Numerical methods for least squares problems. SIAM, Philadelphia Borcea L (2002) Electrical impedance tomography – topical review. Inverse Prob 18(6):R99–R136 Calvetti D, Morigi S, Reichel L, Sgallari F (2000) Tikhonov regularization and the L-curve for large discrete ill-posed problems. J Comput Appl Math 123:423–446

Inverse Theory, Monte Carlo Method Ghosh DP (1971a) The application of linear filter theory to the direct interpretation of geoelectrical resistivity sounding measurements. Geophys Prospect 19:192–217 Ghosh DP (1971b) Inverse filter coefficients for the computation of resistivity standard curves for a horizontally stratified earth. Geophys Prospect 19:769–775 Gupta PK (1998) Chapter 7: The Backus Gilbert method. In: Indira NK, Gupta PK (eds) Inverse methods: general principles and applications in earth system sciences. Narosa Publishing House, New Delhi, pp 60–68 Gupta PK, Niwas S, Gaur VK (1996) Straightforward inversion scheme (SIS) for one-dimensional magnetotelluric data. Proc Indian Acad Sci (EPS) 105(4):413–429 Gupta PK, Niwas S, Gaur VK (1997) Straightforward inversion of vertical electrical sounding data. Geophysics 62(3):775–785 Habashy TM, Abubakar A (2004) A general framework for constraint minimization for the inversion of electromagnetic measurements. Prog Electromagn Res 46:265–312 Hadamard J (1902) Sur les problemes aux derives partielles et leur signification physique. Princeton Univ Bull 13:49–52. Reprinted in his Oeuvres, Vol. III, Centre Nat. Recherche Sci., Paris, 1968, 1099–1105 Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34:561–580 Indira NK, Gupta PK (eds) (1998) Inverse methods: general principles and applications to earth system sciences. Narosa, New Delhi Jackson DD (1972) Interpretation of inaccurate, insufficient, and inconsistent data. Geophys J R Astron Soc 28:97–109 Jackson DD (1979) The use of a-priori data to resolve nonuniqueness in linear inversion. Geophys J R Astron Soc 57:137–158 Kunetz G (1972) Processing and interpretation of magnetotelluric soundings. Geophysics 37:1005–1021 Lawson CL, Hanson RJ (1995) Solving least squares problems. SIAM, Philadelphia Lines LR, Treitel S (1984) Tutorial: a review of least-squares inversion and its application to geophysical problems. Geophys Prospect 32:159–186 Mallat S (2009) A wavelet tour of signal processing: the sparse way. Elsevier, Amsterdam Menke W (1984) Geophysical data analysis: discrete inverse theory. Academic, Orlando Moore EH (1920) On the reciprocal of the general algebraic matrices. Bull Am Math Soc 26:394–395 Oldenburg DW (1976) Calculation of Fourier transform by the BackusGilbert method. Geophys J R Astron Soc 44(2):413–431 Oldenburg DW, Li Y (2004) Inversion for applied geophysics – a tutorial, presented at EMI workshop held at Hyderabad Penrose R (1955) A generalized inverse for matrices. Proc Camb Philos Soc 51:406–413 Rao AR, Bhimasankaram P (1992) Linear algebra. Tata McGraw-Hill, New Delhi Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York Rawlinson N, Pozgay S, Fishwick S (2010) Seismic tomography: a window into deep Earth – review. Phys Earth Planet Inter 178:101–135 Reichel L, Sadok H (2008) A new L-curve for ill-posed problems. J Comput Appl Math 219:493–508 Rezghi M, Hosseini SM (2009) A new variant of L-curve for Tikhonov regularization. J Comput Appl Math 231:914–924 Rojas M, Sorensen DC (2002) A trust region approach to the regularization of large scale discrete forms of ill-posed problems. SIAM J Sci Comput 23(6):1843–1861 Saltelli A, Ratto M, Tarantola S, Campolongo F (2006) Sensitivity analysis practices: strategies for model based inferences – review. Reliab Eng Saf 91:1109–1125

821 Scales JA, Smith ML, Treitel S (2001) Introductory geophysical inverse theory. Smizdat, Golden, White River Junction Sneider R, Trampert J (1999) Inverse problems in geophysics. In: Wirgin A (ed) Wavefield inversion. Springer, New York, pp 119–190 Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. SIAM, Philadelphia Tikhanov AN, Arsenin VY (1977) Solutions ill-posed problems. Wiley, New York Treitel S, Lines L (2001) Past, present, and future of geophysical inversion – a new millennium analysis. Geophysics 66(1):21–24 Twomey S (1977) Introduction to the mathematics of inversion in remote sensing and indirect measurements. Elsevier Scientific, Amsterdam Zhdanov MS (2002) Geophysical inverse theory and regularization problems. Elsevier, New York

Inverse Theory, Monte Carlo Method Malcolm Sambridge1 and Kerry Gallagher2 1 Seismology and Mathematical Geophysics, Research School of Earth Sciences, The Australian National University, Canberra, ACT, Australia 2 UMR 6118 – Géosciences Rennes, Geosciences, Université de Rennes 1, Rennes Cedex, France

Definition Monte Carlo method. A computational technique making use of random numbers to solve problems that are either probabilistic or deterministic in nature. Named after the famous Casino in Monaco. Monte Carlo inversion method. A method for sampling a parameter space of variables representing unknowns, governed by probabilistic rules. Markov chain Monte Carlo (McMC). A probabilistic method for generating vectors or parameter variables whose values follow a prescribed density function.

Introduction Because geophysical observations are made at (or very near) the Earth’s surface, all knowledge of the Earth’s interior is based on indirect inference. There always exists an inverse problem where models of physical properties are sought at depth that are only indirectly constrained by the available observations made at the surface. Geophysicists have been dealing with such problems for many years and in doing so have made substantial contributions to the understanding of inverse problems. Pioneering work on linear inverse problems in the 1960s arose out of the need to understand how to use new surface observables from seismology to constrain radial variations in geophysical properties at depth within the Earth. Data were

I

822

few in number and attention was focused on the mathematical structure of the inverse problem and the ways in which reliable information could be recovered. This resulted in a series of important papers beginning with Backus and Gilbert (1967, 1968, 1970). Since that time the geosciences, like many other fields, have moved into a data-rich environment with increasing availability of computational power. Considerable progress has been made over 30 years utilizing the class of linear (typically least squares) parameter estimation algorithms, which are common to many areas of the physical sciences (Aster et al. 2005). In many of the inverse problems encountered the dependence of data on models is nonlinear and this must be taken into account for meaningful solutions. Often this is achieved by performing a local linearization and using Inverse Theory, Linear. As the mathematical relationship between data and unknowns becomes complex then linearized methods fail because they depend heavily on having a starting model for the iterative process which must be close enough to the solution for convergence. Over the last 30 years there has been considerable progress in the solution of highly nonlinear inverse problems involving a limited number of parameters so that thorough exploration can be made of the character of models. Many algorithms have been devised, most of which make use of random numbers to make decisions, that is, in how to generate a set of values of the unknowns whose predictions can be compared to the available data. The original description of Monte Carlo methods by Hammersley and Handscomb (1964) is “the branch of experimental mathematics which is concerned with experiments on random numbers.” By this definition all inversion techniques that make use of random numbers are Monte Carlo methods. Many of the Inverse Theory, Global Optimization inversion methods fall within the class. The particular approach known as Markov chain Monte Carlo is the primary focus of the present entry.

Inverse Theory, Monte Carlo Method

oscillatory field corresponding to a pair of tuning parameters, as well as the nature of the data itself result in a complicated 2-D fitness function. The set of unknowns that gives the best fit (i.e., smallest misfit) to data corresponds to the global maximum of the multimodal function and to find it one must employ Inverse Theory, Global Optimization techniques such as model space search. In this example the global maximum (at the red central peak) was efficiently found with the neighborhood algorithm of Sambridge (1999), which utilizes ideas from the field of computational geometry. Optimization techniques based on local Inverse Theory, Linear would only be suitable once a trial solution is found within the vicinity of the global peak (shaded red). Adaptive Monte Carlo-based direct search approaches like genetic algorithms, simulated annealing, and the neighborhood algorithm (see “▶ Inverse Theory, Global Optimization”) are able to solve this (two unknown parameters) problem relatively easily due to their ability to detect the variation of the fit and concentrate sampling where there is most benefit. This example demonstrates the complexity of inverse problems in cases where the data are highly oscillatory waveforms, a common situation in fields such as acoustics and seismology, where the dimension of the problem is often much higher. For the 2-D example in Fig. 1, the objective is to tune a particular instrument for maximum sensitivity, and it is appropriate to seek a global maximum. More generally in inverse problems the fitness landscape would represent the difference between observations and predictions made by a mathematical model. In this case simply finding the best-fit solution is inadequate. One needs to characterize the uncertainty in the

Nonlinearity and Multimodal Fitness Functions Figure 1 shows a fitness surface from an inverse problem that arises in the analysis of infrasound array data. (Infrasound arrays are used by the United Nations Comprehensive Test Ban Treaty Organization to monitor international adherence to the nuclear test ban treaty.) The height of the surface represents the degree of agreement between two oscillatory fields. There are just two unknowns in this case, which represent tuning parameters in the infrasound array. The object of the exercise is to best tune the array for sensitivity to incoming atmospheric signals, which means finding the point on the surface where the fitness is maximum. We see a curved rim of local maxima from background to foreground, with a broader valley of low fit to the left of the central maximum (red). The physics of the forward problem, that is, calculating the

Inverse Theory, Monte Carlo Method, Fig. 1 A multimodal data fit surface arising from the mismatch between two oscillatory fields in the infrasound inversion problem (Kennett et al. 2003)

Inverse Theory, Monte Carlo Method

823

solution, for example, assess how noise in the data lead to errors and trade-offs in the estimated model. Linearized techniques (see “▶ Inverse Theory, Linear”) could be used, but all uncertainty estimates are then based on the assumption of local linearity and do not truly reflect the global nature of the data constraint. Another issue that often arises in inverse problems is that of nonuniqueness (see “▶ Inverse Theory, Singular Value Decomposition”). In this case it is not possible to fully constrain the unknowns from the data. The model is unbounded and so best data fit solutions do not exist, and extra assumptions or independent information must be introduced to achieve a single optimal solution. In linearized inversion some form of regularization is used. An example is damping a solution back to some reference set of values (or model) (Aster et al. 2005). It is well known that in this case the details of the solution depend on the nature of regularization used. In addition, uncertainty estimates produced by linearized theory often reflect the choice of regularization. Typically, the least well-constrained components of the solution require the most regularization and resulting uncertainty estimates are severe underestimates of the real errors (Aster et al. 2005 for an example) potential leading to overconfidence in the results.

Bayesian Inference An alternative approach to inversion is Bayesian inference. Many textbooks and review papers are available. Discussions within a geophysical context can be found in Tarantola and Valette (1982), Duijndam (1988a, b), Sambridge and Mosegaard (2002), and Mosegaard and Sambridge (2002). In Bayesian inference, all information on the unknowns is represented in terms of probability density functions (PDF). Within this framework it is accepted that all inference is relative. What one learns from the data gets added to what is known prior to collecting the data and represented in terms of an a posteriori PDF. The most commonly used form of Bayes’ rule is given below

whose density is distributed according to this function. This is termed sampling the posterior PDF. The posterior PDF is the product of the likelihood and the prior PDF. Only the former contains the data vector, d. The likelihood increases as the model fits the data better relative to the noise in the data. The form of the likelihood depends on the statistical character of the data errors. A simple example is a multidimensional Gaussian function characterized by a mean and a covariance matrix, both of which are usually known or assumed. The prior PDF represents information known about the model before collecting the data represented in a probabilistic manner, and may take a variety of forms. Again a multidimensional Gaussian is the most simple, but rarely is real information in this convenient form. Prior PDFs can be the most controversial component of Bayesian inference as there is always a degree of subjectivity in any choice, and the only way to represent no prior information is to not have a prior PDF. Comparisons of Bayesian and alternate approaches can be found in Malinverno and Parker (2005). We see then that instead of seeking a single optimal solution, in a Bayesian framework many samples are sought. Assessment of the constraints placed on the model is achieved by examining collective properties. Typically, this is done by plotting the distribution of samples as a function of one or more subsets of unknowns, calculating credible intervals to represent uncertainty and covariance matrices to examine the trade-offs between parameters. The main task to be carried out is then to generate random samples that follow the multidimensional posterior PDF p(m|d) arising from the inverse problem. McMC methods are practical tools for dealing with complicated probability distributions. Used correctly they result in (quasi)-independent samples whose density follows any target PDF. They have been the subject of much research in fields from Theoretical Physics to Computational statistics. For summaries, see Smith (1991), Smith and Roberts (1993), and Bernardo and Smith (1994). Below we describe the McMC method briefly and provide a simple illustrative example.

Markov Chain Monte Carlo pðmjd Þ ¼ kpðdjmÞpðmÞ

ð1Þ

where p(m|d) is the PDF of the model vector, m, containing the unknowns, given the data vector, d, containing the data; p(d|m) is the likelihood function measuring the probability of the data, d, being observed given the model m; p(m) is the a priori PDF on the model (which is known or assumed about m before the data are collected), and k is a constant of proportionality. In a Bayesian framework, all information on the unknown variables in the model is represented by the posterior PDF, p(m|d) and one usually sets about trying to generate an ensemble of candidate solutions to the inverse problem

Fixed Dimension Approach McMC can be regarded as a combination of random Monte Carlo sampling and a Markov chain random walk strategy around the model space. The aim is to produce an ensemble of models from a probability distribution, that is, the posterior PDF, using only function evaluations. The basic approach was developed from the work of Metropolis et al. (1953), placed in a Bayesian framework by Hastings (1970), and a useful overview is given in Gilks et al. (1996). The practical applications lagged behind theoretical developments as a consequence of the need for many simulations. However,

I

824

Inverse Theory, Monte Carlo Method

the increase in computing power over the last 15 years or so has led to a rapid increase in use of this methodology in geophysics and other fields of Earth Sciences (e.g., Mosegaard and Tarantola 1995; Malinverno 2002; Sambridge et al. 2006; Gallagher et al. 2009). The algorithm is as follows: first we choose an initial model from the prior distribution, and calculate its likelihood. Then we generate a new model by making a random perturbation (Monte Carlo) to the current model. This new model is known as the proposed model and depends only on the values of the current model (Markov chain). The final stage is to decide whether we replace the current model with the proposed model, or stay at the current model and repeat the whole process. This important step is determined from the acceptance criterion, which is defined below

pðm0 Þpðdjm0 Þqðmjm0 Þ a ¼ min 1, pðmÞpðdjmÞqðm0 jmÞ

ð2Þ

where m0 and m are the proposed and current models, respectively, q(a|b) is the probability of proposing model a, given a current model b, and the other distributions are as defined earlier. The decision to accept or reject a proposed model is made by comparing the value of α (which is always between 0 and 1) to a uniform (between 0 and 1) random number, u. If u < α then we replace the current model with the proposed model, if not we discard the proposed model and stay at the current model. We then continue the sampling process (perturb the new current model and so on) for many iterations. The choice of the proposal function is not critical to the correctness of the sampler, but does affect the efficiency, performance, and convergence. A typical choice might be a normal distribution, centered on the current model, and then we need to tune the performance through the scale parameter of this distribution (e.g., the variance). If we choose too small a scale parameter, the proposed model will be very similar to the current model, their likelihoods will be similar, and we will almost always accept the proposed model. If we choose too large a scale parameter, the proposed model will tend to be very different to the current model, and lead to large changes in the likelihood, which are more likely to be rejected. In practice, both situations mean that we tend to move slowly around the model space. The proposal functions need then to be tuned for particular problems to achieve a reasonable balance between accepting and rejecting the proposed models. A reasonable rate of acceptance is around 30–40%. Generally, we can choose proposal functions that are symmetrical, so that q(a|b) ¼ q(b|a), so these terms cancel out in the acceptance criterion. Also, if we choose uniform prior distributions, then the prior terms also cancel. The acceptance criterion then reduces to the original Metropolis et al. (1953) algorithm. After an initial period sampling (known as the burn-in), the current model from each iteration is taken as representing a

sample from the posterior distribution (the sampling chain is then stationary). If the model space has N dimensions, and we are interested in the distribution on one of the parameters, mi, for example, then formally we need to solve the following integral ð   pðmi Þ ¼ p mi , m j dm j ;

ð3Þ

j ¼ 1, . . . , i  1, i þ 1, . . . , N that is we need to integrate out the variation in all parameters except mi. This is known as marginalizing and p(mi) is the marginal probability distribution of mi. Using the McMC samples, we can just plot all value of mi as a histogram as the sampling effectively deals with the integration. Also, it is straightforward to calculate estimates of the expected (or average) value for any parameter. Formally, we have the expected value for parameter mi defined as ð Eðmi Þ ¼ pðmi Þ mi dmi

ð4Þ

Using the McMC samples, we simply average over all the samples accepted for that parameter, that is, Eðmi Þ ¼

Na 1 X mj N a j¼1 i

ð5Þ

where Na is the number of samples accepted (post-burn-in) for model parameter mi. Transdimensional Approach A major issue concerning most inverse problems, and the approaches used to solve them, is how best to balance the twin desires of fitting the observations and avoiding introduction of unjustified complexity in the resulting models. Green (1995) introduced a transdimensional form of McMC (referred to as Reversible Jump), in which the inversion procedure involves the inference of the model complexity (see also birth–death McMC, Geyer and Møller 1994). For finite dimension models (with a fixed number of unknowns) this then typically becomes a question of determining the dimension of the model. If we are dealing with two models with dimensions k and k0 , then acceptance criterion can be written as

pðk0 Þpðm0 jk0 Þpðdjm0 , k0 Þqðmjm0 Þ a ¼ min 1, pðkÞpðmjkÞpðdjm, kÞqðm0 jmÞ

ð6Þ

Here we separate the prior on the number of dimension, p(k), from the model parameter prior, p(m|k). The proposal function q() becomes more complex as we now want to

Inverse Theory, Monte Carlo Method

825

propose models with different dimensions. Moreover, we need to allow for the transformation from one model to another to ensure that theoretical probability requirements are maintained. In dealing with a situation where we are simply increasing or decreasing the number of parameters, we can write (

 0 ) pðk0 Þpðm0 jk0 Þpðdjm0 , k0 Þg uk a ¼ min 1, jJj pðkÞpðmjkÞpðdjm, kÞgðuk Þ

ð7Þ

0

Here uk and uk are vectors of random numbers of length r0 and r, respectively, and used to transform from one model to another, such that r + k ¼ r0 + k0 , and g(.) is the probability distribution used to generate these random numbers. The term

|J| is the Jacobian, and allows for the transformation between the two models, that is, j J j¼

ð8Þ

The last equation for α is actually a general form for the acceptance criterion, although for fixed-dimensional problems the Jacobian is generally 1, and the proposal functions are of the form as described earlier. The details of the reversible jump acceptance criterion are discussed in more detail by Green (2003), Malinverno (2002), and Sambridge et al. (2006) and examples of the implementation algorithms for variable dimension problems are given in Jasra et al. (2006), Bodin and Sambridge (2009), Charvin et al. (2009), and Hopcroft et al. (2009). One important characteristic of the

I

1.01

2.5 1.25

 0 @ m0 , uk @ ðm, uk Þ

2 1.006 1

m1

σ = 0.01

1.15

1.5 m0

m1

m0

0.5 1.05

0 σ = 0.001 m1

0.95

0

a

1000

2000 3000 Iterations

4000

1.002 1

0.996

−0.5 −1 5000

0.992

1

2

b

1.008

3 Iterations

4

5 ⫻ 105

600

400 1.004

m1

200

1

d

0 0.6

0.8

1 m0

1.2

1.4

0.996

1 m1

1.004

1.008

600

0.996

400 0.992 0.6

c

200 0.8

1 m0

1.2

1.4 0 0.992

e Inverse Theory, Monte Carlo Method, Fig. 2 (a) Initial 5,000 iterations for sampling of two parameters for the linear regression problem. (b) Post-burn-in iterations for parameter m1. (c) The green points show the post-burn-in sampling, and the contours are the log likelihood function. The blue cross is the best solution (equivalent to the analytical

maximum likelihood or least squares solution). (d) Marginal distribution for parameter m0. The histogram is constructed from the post-burn-in accepted samples, and the curve is the analytical solution for the marginal distribution. The two vertical bars mark the upper and lower bounds on the 95% credible interval. (e) As (d) but for parameter m1

826

Inverse Theory, Monte Carlo Method

Bayesian transdimensional formulation is that it is naturally parsimonious. For two models that fit the data equally well, it will tend to favor simpler models over complex ones as a consequence of the posterior probability effectively being penalized through the addition of more terms to the prior distribution.

Again, we can see that the sampler has managed to distribute itself across the distribution according to the posterior probability and these histograms are good representations of the marginal distributions.

Summary A Simple Example To demonstrate McMC in action, we choose a simple two parameter linear regression problem, that is, yi ¼ m0 þ m1 xi þ ei ,

i ¼ 1, . . . , N

ð9Þ

where m0 and m1 are the model parameters, y is an observed/ measured value, and ε is the data error. In a Bayesian formulation, this problem has an analytical solution for a uniform prior on the model parameters, assuming the data error is known (Lee 1989, p. 180). We chose m0 ¼ m1 ¼ 1 and generated 100 synthetic data (y) for random values of x between 0 and 100, and added noise (ε ¼ 0.5). We used Gaussian proposal distributions, with different scale parameters (sm0 ¼ 0:01, and sm1 ¼ 0:001). In Fig. 2 we show the sampling for the two parameters, starting from a randomly selected model. Figure 2a shows that the sampler has not reached the stationary state until at least 1,500–2,000 iterations. Figure 2b shows the sampling for parameter m1 for later iterations which is clearly stationary. The sampling resembles a white noise spectrum about the mean (or expected) value, lacking any internal structure as a function of iteration. It is also clear that the sampler manages to move toward the upper and lower extreme values of the parameter range (determined by the posterior PDF and the proposal function scale parameter). These are diagnostic (but qualitative) characteristics of stationarity. Figure 2c shows the 2-D distribution of samples of the post-burn-in accepted samples (here we thinned the chain taking every 100th sample), compared to the log likelihood function (which is proportional to the log of the posterior distribution as we use uniform priors). The density of the sampling increases around the high likelihood values, but there are still some samples from the lower likelihood regions. Figure 2d, e shows the marginal distributions for the two parameters as a frequency histogram, and also the analytical solutions (scaled to the same peak height). These are constructed simply by taking all the accepted values for a given model parameter, as the sampler effectively integrates out the other parameters. Also shown are the 95% credible interval ranges for each parameter. These are constructed by sorting all the samples for a given parameter in ascending order, and finding the indices for the credible values such that 2.5% of the samples are less than the lower credible value, and 2.5% of the values are greater than the upper credible value.

Monte Carlo sampling, relying on random numbers, has been used in geophysics for over 40 years, although the increase in computing power has seen a commensurate increase in applications in the last 15 years or so. This approach avoids the use of gradients, is robust to local minima, and so is suitable for nonlinear inverse problems which often have complex misfit (or fitness) surfaces in high dimensions. McMC, particularly when used in a Bayesian formulation, provides a means of sampling a model space according to the (unknown) posterior distribution for the model parameters. Transdimensional (or reversible jump) Markov chain Monte Carlo generalizes this approach to allow models of different dimensions to be considered, and provides a means of choosing between models of differing complexity. Quantifying the posterior distribution with McMC is then a solution to the inverse problem and various types of inference can be made from this distribution (e.g., expected values, marginal distributions, credible intervals) to characterize the model space.

Cross-References ▶ Inverse Theory, Artificial Neural Networks ▶ Inverse Theory, Global Optimization ▶ Inverse Theory, Linear ▶ Inverse Theory, Singular Value Decomposition Acknowledgments We would like to thank FAST (French-Australia Science and Technology exchange program) for their support during the preparation of this entry. This project is supported by the Commonwealth of Australia under the International Science Linkages program.

Bibliography Aster R, Borchers R, Thurber CH (2005) Parameter estimation and inverse problems. International Geophysics Series, vol 90. Elsevier, Amsterdam Backus GE, Gilbert JF (1967) Numerical applications of a formalism for geophysical inverse problems. Geophys J R Astron Soc 13:247–276 Backus GE, Gilbert JF (1968) The resolving power of gross Earth data. Geophys J R Astron Soc 16:169–205 Backus GE, Gilbert JF (1970) Uniqueness in the inversion of inaccurate gross Earth data. Philos Trans R Soc Lond A 266:123–192 Bernardo JM, Smith AFM (1994) Bayesian theory. Wiley, Chichester Bodin T, Sambridge M (2009) Seismic tomography with the reversible jump algorithm. Geophys J Int 178:1411–1436

Inverse Theory, Singular Value Decomposition Charvin K, Gallagher K, Hampson G, Labourdette R (2009) A Bayesian approach to infer environmental parameters from stratigraphic data 1: methodology. Basin Res 21:5–25 Duijndam AJW (1988a) Bayesian estimation in seismic inversion part I: principles. Geophys Prospect 36:878–898 Duijndam AJW (1988b) Bayesian estimation in seismic inversion part II: uncertainty analysis. Geophys Prospect 36:899–918 Gallagher K, Charvin K, Nielsen S, Sambridge M, Stephenson J (2009) Markov chain Monte Carlo (McMC) sampling methods to determine optimal models, model resolution and model choice for Earth science problems. Mar Pet Geol 26:525–535 Geyer CJ, Møller J (1994) Simulation procedures and likelihood inference for spatial point processes. Scand J Stat 21:369–373 Gilks WR, Richardson S, Spiegalhalter DJ (1996) Markov chain Monte Carlo in practice. Chapman & Hall, London Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82:711–732 Green PJ (2003) Chapter 6: Trans-dimensional McMC. In: Green PJ, Hjort N, Richardson S (eds) Highly structured stochastic systems. Oxford statistical sciences series. Oxford University Press, Oxford, pp 179–196 Hammersley JM, Handscomb DC (1964) Monte Carlo methods. Chapman & Hall, London Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109 Hopcroft P, Gallagher K, Pain CC (2009) A Bayesian partition modelling approach to resolve spatial variability in climate records from borehole temperature inversion. Geophys J Int 178:651–666 Jasra A, Stephens DA, Gallagher K, Holmes CC (2006) Analysis of geochronological data with measurement error using Bayesian mixtures. Math Geol 38:269–300 Kennett BLN, Brown DJ, Sambridge M, Tarlowski C (2003) Signal parameter estimation for sparse arrays. Bull Seismol Soc Am 93:1765–1772 Lee PM (1989) Bayesian statistics: an introduction. Edward Arnold, New York/Toronto Malinverno A (2002) Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem. Geophys J Int 151:675–688 Malinverno A, Parker RL (2005) Two ways to quantify uncertainty in geophysical inverse problems. Geophysics 71:15–27 Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equations of state calculations by fast computing machines. J Chem Phys 21:1087–1092 Mosegaard K, Sambridge M (2002) Monte Carlo analysis of inverse problems. Inverse Probl 18:R29–R54 Mosegaard K, Tarantola A (1995) Monte Carlo sampling of solutions to inverse problems. J Geophys Res 100:12431–12447 Sambridge M (1999) Geophysical inversion with a neighbourhood algorithm – I. Searching a parameter space. Geophys J Int 138:479–494 Sambridge M, Mosegaard K (2002) Monte Carlo methods in geophysical inverse problems. Rev Geophys 40:3.1–3.29 Sambridge M, Gallagher K, Jackson A, Rickwood P (2006) Transdimensional inverse problems, model comparison and the evidence. Geophys J Int 167:528–542 Smith AFM (1991) Bayesian computational methods. Philos Trans R Soc Lond A 337:369–386 Smith AFM, Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J R Stat Soc Ser B 55:3–23 Tarantola A, Valette B (1982) Inverse problems ¼ quest for information. J Geophys 50:159–170

827

Inverse Theory, Singular Value Decomposition Ajay Manglik National Geophysical Research Institute (Council of Scientific and Industrial Research), Hyderabad, India

Definition Condition number

Ratio of the largest singular value to the smallest singular value of a matrix. For a singular matrix, it is infinite, and it is very large for an ill-conditioned matrix.

I

Description Linear geophysical inverse theory (Inverse Theory, Linear) deals with solving a system of linear equations, represented in matrix form as Ax ¼ b,

ð1Þ

where A is a m  n Jacobean matrix, x a n  1 column vector of perturbations in unknown model parameters with respect to a reference model, and b the corresponding m  1 column vector of perturbations in observed responses. For a full rank square matrix (m ¼ n ¼ p : the rank of matrix A), it is easy to find a unique inverse A1 and estimate the model parameters as x ¼ A1b. However, geophysical inverse problems (Tarantola 1987) mostly deal with inversion of rectangular matrix A wherein m is either greater than n (overdetermined system representing more number of observations than the unknown parameters, also known as least squares solution) (Lawson and Hanson 1974; Lines and Treitel 1984) or less than n (underdetermined system representing less number of observations compared to the unknown parameters, also called minimum norm solution). For many geophysical problems, the rank p of the matrix is less than m and n. In such situations, linear geophysical inverse theory aims at finding out a pseudo-inverse A of A such that b x ¼ A b,

ð2Þ

where b x represents column vector of the estimated model parameters. Moore (1920) and Penrose (1955), using a spectral decomposition form of matrix A, showed that there exists a unique pseudo-inverse A of the matrix A such that (1) AAA ¼ A, (2) AAA ¼ A,

828

Inverse Theory, Singular Value Decomposition

(3) (AA)T ¼ AA, and (4) (AA)T ¼ AA, where superscript T denotes the transpose of matrix. Further details can be found in Inverse Theory, Linear. Singular value decomposition (SVD) is a powerful tool to invert matrices that are either singular or numerically very close to singular (ill-conditioned system). Standard matrix equation solvers can fail in such situations. The advantage of SVD is that one need not a priori know the nature of the problem, that is, whether it is overdetermined, under-determined, or partially determined. SVD is very general in the sense that it can be applied to any arbitrary size matrix. The technique was discovered independently by Beltrami in 1873 and Jordan in 1874 as a tool to solve square matrices and was extended to rectangular matrices by Eckart and Young in the 1930s (Stewart 1993). Its use as a computational tool dates back to the 1960s (Golub and Kahan 1965; Golub and Reinsch 1970). Golub and van Loan (1996) demonstrated its usefulness and feasibility in a wide variety of applications. Following SVD, a matrix A of size m  n is factorized in the form A ¼ UL VT ,

ð3Þ

where U is a m  m unitary matrix, Λ is a m  n diagonal matrix, and V is a n  n unitary matrix. The columns of V form a set of orthonormal vectors called parameter eigenvectors, and the columns of U form a set of orthonormal basis vectors called data eigenvectors. For a special case of m ¼ n, the matrix Λ has the form 2

0



0

b ¼ Ub0 ;

s2 ⋮



0 7 7 7, ⋱ ⋮5

0

0



s1  s2   sn :

x0 ¼ L1 b0 :

For a real matrix, the columns of matrices U and V are orthogonal, that is, UTU ¼ I and VTV ¼ I, where I is the identity matrix and T is transpose of matrix. For complex matrices transpose is replaced by complex conjugate. Inverse of A The expression for the inverse of A can be obtained by first substituting Eq. 3 into Eq. 1 to get ð6Þ

ð9Þ

From Eq. 7 and Eq. 9, and orthogonality property of data and parameter eigenvectors, we get x ¼ VL1 UT b,

ð10Þ

or the inverse of A can be written as A1 ¼ VL1 UT :

ð11Þ

Rank-Deficient Matrix and Its Inverse The advantages of above spectral decomposition can be seen in the case of A being a singular matrix. Let us assume that only p of n singular values of A are non-zero. In this case, the data and parameter eigenvectors can be split into two subspaces representing non-zero eigen-space and null-space, respectively, that is, 

     0 ; U ¼ Up , U0 ; V ¼ Vp , V0 , 0

ð12Þ

then 

ð5Þ

ð8Þ

Since Λ is a diagonal matrix, we can easily get

sn

where s1, , sn are called the singular values of matrix A. The number of non-zero singular values is equal to the rank of A. In the SVD representation, the singular values are arranged in the order of decreasing amplitude, that is,

ð7Þ

Lx0 ¼ b0 :

Lp L¼ 0 ð4Þ

x ¼ Vx0 ,

and multiplying Eq. 6 by UT, which gives

3

s1 6 0 6 L¼6 4⋮

ULVT x ¼ b,

and then applying the following orthogonal transformation

A ¼ Up , U0

  Lp

0

0

0



Vp V0

T ,

ð13Þ

which gives A ¼ Up Lp VTp :

ð14Þ

This indicates that matrix A can be reconstructed only from non-zero singular values and corresponding vector spaces. An inverse of a rank-deficient matrix can be written as: T A ¼ Vp L1 p Up ,

ð15Þ

by substituting Eq. 12 into Eq. 11. Ill-Condition Matrix Equation 15 assumes that the singular values of the system are either zero or positive scalars. If some of the singular values

Inverse Theory, Singular Value Decomposition

829

are very small, then it is easy to see that the inverse of such values can be very large and, in the case of error in data, can lead to unstable solutions. Stability of the matrix inverse can be defined in terms of ratio of largest to smallest singular value, also called the condition number ¼

s1 : smin

Adx ¼ «:

ð17Þ

dx0 ¼ L1 «:

ð18Þ

In terms of singular value decomposition, it amounts to boosting the small singular values of the matrix ATA. This can be shown by expressing Eq. 18 in terms of singular values decomposition of the matrix A, which gives 

s A ¼ V diag UT , s2 þ l2

ð19Þ

where diag(. . .) represents the elements of a diagonal matrix. Here, it can be seen that adding l to small singular values stabilizes the inverse. Sensitivity to Errors in Data Field observations normally contain noise superimposed on the actual response. If we assume a noise vector ε, then we observe b + ε instead of b. This leads to an error δx in the estimated model parameters. Therefore, for noisy data Eq. 1 may be written as: Aðx þ dxÞ ¼ b þ «:

ð20Þ

ð22Þ

This shows that in the case of small singular values, small error in data yields large error in estimated model parameters. This problem of instability of estimated model parameters can be overcome by either truncating the small singular values or boosting these by adding a Marquardt damping factor, as discussed in the previous section. Such treatment nevertheless results in loss of finer details of the model parameters. Resolution Matrices In linear inverse theory, the concepts of parameter resolution and data information density analyses are used to analyze the resolution of estimated model parameters and the regions of data contributing maximum to these model estimates. These resolution matrices can be expressed in terms of data and parameter eigenvectors. Parameter resolution matrix for a rank-deficient system can be obtained by substituting Eqs. 1, 14, and 15 into Eq. 2, which gives

Marquardt (1963) suggested that an ill-conditioned matrix ATA may be stabilized by adding an arbitrary damping factor l to its diagonal elements, that is,  1 A ¼ AT A þ l2 I AT :

ð21Þ

Expressing matrix A in spectral form and applying orthogonal transformation δx0 ¼ Vδx and ε0 ¼ Uε, Eq. 21 gives

ð16Þ

The matrix is ill-conditioned if the condition number is very large. In such situations, the system can be stabilized by specifying a cutoff threshold 0 for the condition number and ignoring all singular values that lead to the condition number larger than this threshold. Let there be p non-zero singular values arranged in decreasing order of their amplitudes of which only the first q singular values have the condition number smaller than 0. Then (q + 1, , p) singular values can be reset to zero thus reducing the rank of the matrix to q. This can be understood as ignoring finer details of the parameter space, which are associated with small singular values. Geophysical inverse problems normally fall under this category. For least squares inverse problems, where an inverse of A is given by the expression  1 A ¼ AT A AT ,

Substituting from Eq. 1, we get

ℜ ¼ Vp VTp :

ð23Þ

For a well-resolved system, ℜ is an identity matrix I. For a rank-deficient system, this matrix provides information about the equivalence of various model parameters. Similarly, data information density matrix can be written as ℑ ¼ Up UTp ,

ð24Þ

by substituting Eqs. 2, 14, and 15 into Eq. 1. Jupp and Vozoff (1975) provided an elegant description of the concepts of stability of solution and resolution of model parameters for geophysical data contaminated by noise, and Vozoff and Jupp (1975) showed its application to direct current (DC) electrical resistivity and magnetotelluric (MT) data interpretation. Example The following example demonstrates the use of SVD for an overdetermined system and ill-conditioned matrix 2

3:0 6 1:01 6 A¼6 4 0:5

1:0 2:9

0:3

0:31

2:0

3 2:9999 0:9999 7 7 7, 0:5 5 0:29

ð25Þ

I

830

Inverse Theory, Singular Value Decomposition

2

which has rank p ¼ 3. It is decomposed into U, Λ, and V with three non-zero singular values as 5.2573, 2.5613, 0.0078, and 2

0:7700

6 6 0:5380 6 U ¼6 6 0:3285 4 2

0:6311

0:0932

0:6086

0:2595

0:4809

0:0988

0:0056

0:5796

0:4047

6 V ¼6 4 0:5740 0:5784

0:8188 0:4071

0:0115

0:7073

0:5275

6 A ¼ 4 0:0952 8:6133

0:1905 23:5143

0:0013

3 0:5 7 0:0013 5:

ð30Þ

0:5003

It can be seen that stabilizing of the inverse of the matrix comes at the cost of compromising on the resolution of model parameters. Here, first and third parameters are coupled when the rank of the matrix is reduced.

7 0:0019 7 5:

Summary

0:7069

SVD is a powerful tool to solve geophysical linear inverse problems which often deal with rank-deficient singular matrices. The advantage of SVD is that one need not a priori know the nature of the problem, i.e., whether it is overdetermined, under-determined, or partially determined. It is very general in the sense that it can be applied to any arbitrary size matrix.

First three columns of U form matrix Up whereas Vp is equal to V. Following Eq. 15, the pseudo-inverse of A is obtained as 8:2480 23:4520

0:0013 1:0

3

ð26Þ

2

0:5

3

7 0:5224 7 7 7, 0:4604 0:6699 7 5 0:8438

0:4997 6 R4 0:0013

41:7228

76:3959

3

7 0:3010 0:1915 5, 41:6213 76:3367

Cross-References ▶ Inverse Theory, Linear

ð27Þ with the resolution matrix R as the identity matrix I. In this example, third singular value is small having the condition number of 672.89. In order to stabilize the solution, we can reset this singular value to zero (new rank ¼ 2). Thus, the data and parameter eigenvectors become 2

0:7700

6 6 0:5380 6 Up ¼ 6 6 0:3285 4 0:0988

0:6311

3

7 0:6086 7 7 7 and 0:4809 7 5

0:0056 2 3 0:5796 0:4047 6 7 7 Vp ¼ 6 4 0:5740 0:8188 5, 0:5784 0:4071

ð28Þ

yielding the pseudo-inverse as 2

0:1846 0:0368 6  A ¼ 4 0:1177 0:2533 0:1850 0:0375 and the resolution matrix R as

3 0:0398 0:0100 7 0:1896 0:0126 5, ð29Þ 0:0403 0:0100

Acknowledgments The example is computed using Matlab. CSIR-NGRI contribution number NGRI/Lib/2019/Pub-81 under the project MLP6404-28(AM).

Bibliography Golub GH, Kahan W (1965) Calculating the singular values and pseudoinverse of a matrix. J Soc Indus Appl Math Series B 2:205–224 Golub GH, Reinsch C (1970) Singular value decomposition and least squares solutions. Numer Math 14:403–420 Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. Johns Hopkins, Baltimore Jupp DLB, Vozoff K (1975) Stable iterative methods for the inversion of geophysical data. Geophys J R Astron Soc 42:957–976 Lawson CL, Hanson RJ (1974) Solving least squares problems. Prentice-Hall, Englewood Cliffs Lines LR, Treitel S (1984) A review of least-squares inversion and its application to geophysical problems. Geophys Prospect 32:159–186 Marquardt DW (1963) An algorithm for least-squares estimation of nonlinear parameters. SIAM J Appl Math 11:431–441 Moore EH (1920) On the reciprocal of the general algebraic matrices. Bull Am Math Soc 26:394–395 Penrose R (1955) A generalized inverse for matrices. Math Proc Camb Philos Soc 51:406–413 Stewart GW (1993) On the early history of the singular value decomposition. SIAM Rev 35:551–566 Tarantola A (1987) Inverse problem theory: methods for data fitting and model parameter estimation. Elsevier Science, Amsterdam Vozoff K, Jupp DLB (1975) Joint inversion of geophysical data. Geophys J R Astron Soc 42:977–991

Isostasy

Isostasy A. B. Watts Department of Earth Sciences, University of Oxford, Oxford, UK

Synonyms Isostacy – (Dutton 1882).

Definition Isostasy – a principle or general law (Heiskanen 1931). Isostasy considers there is a certain surface within the Earth, known as the depth of compensation, on which the vertical stresses due to an overlying column of rock are equal (isos ¼ Greek isoB “equal,” stasis ¼ Greek stάsιB “a standing still”). Isostasy implies a state of hydrostatic equilibrium such that the Earth’s crust and mantle float on their substrate and light regions have a greater elevation than dense regions. Isostatic equilibrium – an idealized state that the outermost layers of the Earth tend toward following their disturbance by the addition or removal of loads associated, for example, with the waxing and waning of ice sheets, the growth and decay of volcanoes and deposition, sliding and slumping of sediments. Local isostasy – a hypothesis that the outermost layers of the Earth are weak such that the deformation caused by loading and unloading is localized, intense, and may involve faulting. Regional (or flexural) isostasy – a hypothesis that the outermost layers of the Earth have an intrinsic strength such that they resist the deformation associated with loading and unloading and bend (or flex) over a broad area rather than break. Thermal isostasy – a hypothesis that the outermost layers of the Earth form a cooling thermal boundary layer and that regional differences in topography are controlled by differences in the temperature structure such that hot regions have a greater elevation than cold regions.

Historical Background Isostasy is a principle that has played a major role in our understanding of the structure, morphology, and tectonic evolution of Earth’s outermost layers. The term was first coined by Dutton (1889), but there is evidence (Delaney 1940; Vai

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2006) that the equilibrium state of the Earth’s crust was being considered much earlier by, for example, the Italian scientists Leonardo da Vinci (1452–1519) who pondered the occurrence of marine fossils at high elevations in mountain ranges and by Luigi Fernando Marseli (1658–1730) who was puzzled about the juxtaposition of ocean deeps and mountains. J. F. W. Herschel (1792–1871) was one of the first to link subsidence in one area to uplift in another by some sort of subcrustal mantle flow and both C. Lyell (1797–1875) and C. Babbage (1790–1871) through observations at the Temple of Serapis appreciated that the same area of the crust could experience both subsidence and uplift. The break through that led to the formulation of the principle of isostasy came following George Everest’s pioneering geodetic work in South Africa and India. Airy (1855) and then Pratt (1855) used Everest’s deflection of the vertical data in northern India to address the question of how the Himalaya mountains were supported at depth. Fisher (1881), who favored Airy’s hypothesis that the Himalaya were supported by a deep crustal “root,” concluded that the Earth’s crust was in a state of hydrostatic equilibrium and floated on its fluid substrate, much like an iceberg on seawater. Today, we tend to think of isostasy as a restraining force rather than a driving force like ridge push or trench pull. Isostasy is a process that acts on the topography created, for example, at plate boundaries by continental rifting, breakup, and mountain building so as to reshape it and redistribute the stresses that have built up. The existence of regions of large topographic relief, unusually thick and thin crust, and large-amplitude gravity and geoid anomalies are all pointers that isostatic equilibrium does not prevail everywhere. The main challenge in isostatic studies has been to quantify the degree to which hydrostatic equilibrium of Earth’s crust is achieved and the time and spatial scales over which it operates. A number of isostatic models have been proposed, three of which are illustrated in Fig. 1. Airy and Pratt are local models in which changes in the Earth’s topography are supported by either change in the thickness of a uniform density crust or by lateral change in density of the crust and/or mantle. Both models imply that the crust and mantle respond to loads (e.g., volcanoes) or unloads (e.g., erosion) locally such that neighboring regions are not involved in their support or removal. While the Airy and Pratt models are a potential source of stress in the crust (Bott and Dean 1972; Bott and Kusznir 1979) and may drive extension and compression they imply an intrinsically weak crust that responds to loads and unloads locally and in some cases by faulting. Vening Meinesz, on the other hand, is a regional model in which loads and unloads are supported by flexure of a thin elastic plate and stresses accumulate by bending.

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It is important to remind ourselves that the models in Fig. 1 are highly idealized and only represent the state that the Earth’s outermost layers would tend toward in the absence of disturbing forces. These disturbing forces include many of the great cycles of geology such as the waxing and waning of ice-sheets, the growth and decay of volcanoes, and sedimentation and erosion. Each process is potentially a disturbing force that could act to delay or even prevent isostatic equilibrium from being achieved.

There is now a considerable body of work that indicates Earth is a dynamic planet that is continuously being deformed. Isostasy, on the other hand, is a static concept that refers to a crust and mantle that has already moved and is now in equilibrium. It therefore appears at odds with current geodynamic views. But isostatic studies of the deformation that follows, for example, megathrust earthquakes, lateglacial rebound, volcano emplacement, and sediment accumulation provide “snapshots” of the way that the crust and

Isostasy, Fig. 1 The main types of isostatic models. Each model implies a state of hydrostatic equilibrium such that the Earth’s outermost layers are in a state of flotation on their more fluid substrate. Names (e.g., Airy-Heiskanen) refer to the workers who first conceived and quantified the models. Vertical (lithostatic) stresses are constant on the depth of compensation in the Airy-Heiskanen (Airy 1855; Heiskanen 1931) and Pratt-Hayford (Pratt 1855; Hayford 1909) models and vary laterally in the Vening-Meinesz model (Vening Meinesz 1931). (a) Airy model: isostatic equilibrium is achieved by variations in the thickness of a

uniform low-density crust that overlies a high-density mantle. (b) Pratt model: isostatic equilibrium is achieved by lateral changes in the density. (c) Vening-Meinesz model: topography is considered as a surface load, and isostatic equilibrium is achieved by flexural downwarping and upwarping of the crust and mantle over a broad area. The load is supported by the strength of the lithosphere (which is determined by its effective elastic thickness, Te) and the buoyancy of the underlying asthenosphere

Isostasy

mantle respond to loads and unloads over certain temporal and spatial scales (Watts 2007). Together, these “snapshots” provide information on whole lithosphere behavior and the clues as to how the crust and mantle would actually deform in response to continually applied loads and unloads. A goal of isostatic studies has been to determine the vertical motion history of the Earth’s crust and mantle on short, through intermediate to long time-scales. Therefore, isostasy is central to studies of global change, especially those that are aimed at determining the relative role of tectonics and sea-level change in controlling the position of the shoreline today and in the geological past. Most significantly, isostatic studies provide information on how the lithosphere responds to the load and unload shifts that occur during climate-driven glacial and inter-glacial cycles. We are presently in an inter-glacial, for example, yet the isostatic effects of the Last Glacial Maxima (LGM) are still clearly visible today in tide gauge and satellite Global Positioning System (GPS) data. Isostasy is a key to understanding the lithosphere on which we live and how it interacts with the asthenosphere below and the cryosphere, hydrosphere, and atmosphere above.

Concepts and Applications The isostatic models illustrated in Fig. 1 imply different gravity anomalies and crust and mantle structures. Airy and Pratt, for example, predict that free-air gravity anomalies will generally be small over volcanoes because they are underlain by either a thick, uniform-density, crustal “root” or a constant thickness low-density crust, the negative gravitational effect of which will compete with the positive effect of the volcano. Vening Meinesz, on the other hand, predicts a shallower depth to Moho beneath a volcano than Airy or Pratt and, hence, the free-air gravity anomaly will be generally large over a volcano because its gravity effect will now dominate over that of the compensation. Figure 2 shows the topography, free-air, and isostatic gravity anomalies in the region of northern India where the early geodesists carried out their classical work. The isostatic anomaly has been calculated by subtracting (and thereby removing) the positive gravity effect of the topography and the negative gravity effect of the Airy compensation from the observed free-air gravity anomaly. The Root Mean Square (RMS) of the free-air gravity anomaly variation about its mean is 87.8 mGal while the RMS of the Airy isostatic gravity anomaly is 45.9 mGal. These RMS values show that the freeair gravity anomaly is substantially reduced after a correction for isostasy and so we may conclude that equilibrium prevails to a significant degree in the northern India region. Despite this success, Fig. 2a shows that significant departures remain in the Airy isostatic anomaly map. The most

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striking are the positive-negative anomaly “couples” that correlate with the edge of the Tibetan Plateau, particularly, its boundary with the Ganges and Tarim foreland basins. The negative anomalies are usually on the basin side while the positive anomalies are on the plateau side, which implies that the Moho is deeper than expected for Airy on the basin side and shallower on the plateau side. Such a Moho geometry can be explained if we assume that the load of the Himalaya and Kunlun Mountains is supported by the strength of the underlying crust and mantle which flexes over a broad region that extends beyond the mountains themselves. The Vening Meinesz model takes into account these ideas of loads and bending and Fig. 2c shows that when an elastic plate flexure model is used to compute the compensation rather than an Airy model then the RMS of the isostatic anomalies are reduced even further. A useful parameter in the Vening Meinesz model is the effective elastic thickness, Te, since it is a proxy for the long-term strength of the lithosphere. Figure 2c shows that there is a well-developed minimum in the RMS of the isostatic anomaly for Te ~ 30 km, suggesting this as the best fit Te estimate for the region (see also Crosby 2007 who found a similar best fit Te for the northern India region). This is, of course, only an average and so there will be areas that are stronger and areas that are weaker than this. Indeed, Braitenberg et al. (2003) and Jordan and Watts (2005) have already shown, using inverse methods, that Te is 30 km over the Ganges and Tarim foreland basins. Another way to test isostasy on a regional scale is to compare compilations of the seismically constrained crustal structure directly to the predictions of isostatic models. Figure 3 shows, for example, a 18,000 km long transect of the Indian, Atlantic, and Pacific Oceans along which observed and calculated depths to Moho are compared. The observed Moho is based on CRUST 2.0 (Bassin et al. 2000), a 2°  2° grid of P-wave velocity and crustal layer thickness data derived from active source seismic data. The calculated Moho is based on an Airy model with the same density and crustal thickness as used in Fig. 2. The figure shows that the calculated Airy Moho tracks the observed seismic Moho well, especially so at rifted margins and beneath some continents. The main departures are in the central Atlantic Ocean where the observed Moho depth is shallower than predicted by Airy and beneath the Eastern US Appalachians the Gulf of Mexico and the Persian Gulf where the observed Moho depth is deeper. The shallower depths in the can be explained by our assumption of a relatively high subcrustal oceanic mantle density (3330 kg m3) which if reduced (e.g., to 3180 kg m3) beneath a mid-oceanic ridge would result in a shallower Moho depth and better agreement with the seismic Moho depths. The idea of a relatively low-density subcrustal mantle beneath the ridge suggests a Pratt rather than an Airy

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Isostasy, Fig. 2 Topography, free-air, and isostatic gravity anomalies in the region of the Tibetan plateau. (a) Upper panel: Topography based on GEBCO (2003). The map shows the plateau and the flanking Tarim and Ganges foreland basins, the major faults (ATF ¼ Altyn Tagh Fault, MBT ¼ Main Boundary Thrust), and the north-south profile (thick red line) plotted in (b). Middle panel: Free-air gravity anomaly based on the EGM2008 2.5  2.5 min grid (Pavlis et al. 2008). Lower Panel: Airy isostatic anomaly calculated by subtracting the gravity effect of the topography and its compensating “root” from the observed free-air gravity anomaly. Calculations assume a density of water, crust, and mantle of 1030, 2800, and 3330 kg m3, respectively, and a zero

elevation crustal thickness of 31.2 km. (b) Predicted crustal structure based on the Airy and flexure models and free-air and isostatic gravity anomaly profiles of the Tibetan plateau at longitude 85°E. (c) Root Mean Square (RMS) of the isostatic anomaly as a function of elastic thickness, Te, which is a proxy for the long-term strength of the lithosphere. The Moho depth for Te ¼ 0 km corresponds to the predictions of an Airy model which implies a plate with no strength. The minimum RMS and hence the isostatic model that best explains free-air gravity anomaly data in the vicinity of the Tibetan plateau is for a flexure model with Te ¼ 30 km

model of isostatic compensation. Indeed, the Pratt model has been assumed in thermal cooling models (Sclater and Francheteau 1970) for the variation of seafloor depth and heat flow with age and the model may well apply to other cases of thermal isostasy, for example. at mid-plate oceanic swells (Nakiboglu and Lambeck 1985) and fracture zones (Sibuet and Veyrat-Peinet 1980).

The deeper depths than predicted by Airy in Fig. 3 correlate with thick sediment accumulations in the Mississippi River Delta and the Zagros foreland basin. Here, sediment loads due to rifting and thermal contraction and thrusting and folding providing the accommodation space have flexed the Moho downwards over a broad area such that Moho is deeper than would be expected for an Airy model. The deeper depths

Isostasy

Isostasy, Fig. 3 Comparison of the crustal structure based on seismic refraction data to the predicted crustal structure assuming an Airy model along a 18,000 km long great circle profile that extends from the Pacific Ocean in the west across the Atlantic Ocean to the Indian ocean in the

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east. The seismic refraction data is based on CRUST2.0 http://igppweb. ucsd.edu/~gabi/rem.html and the Airy model is based on the same parameters as assumed in Fig. 2

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Isostasy, Fig. 4 Comparison of the observed crustal structure derived from seismic refraction data (Contreras-Reyes et al. 2010) to the predictions of an Airy and flexure model of isostasy along a NNE-SSW trending profile of the Louisville Ridge seamount chain at latitude 27.5° S, southwest Pacific Ocean. Black filled triangles indicate the location of the ocean bottom seismometers that were used to constrain the seismic structure of the crust. Thick solid red lines show the depth to the seismically constrained Moho. Thin solid black line separating the purple and pink filled regions show the 6.0 km/s P wave iso-velocity contour. Purple fill ¼ volcanoclastic material. Pink fill ¼ volcano edifice. Blue fill ¼ deformed oceanic crust. (a) Airy model based on the same parameters as assumed in Fig. 2. (b) Flexure model with Te ¼ 10 km

beneath the Eastern US Appalachians are puzzling and may reflect a downward flexure related to thinner lithosphere, as defined by teleseismic studies (Rychert et al. 2005). The role of flexure in contributing to crustal structure is well illustrated in Fig. 4 which shows the results of a recent seismic reflection and refraction survey of the Louisville

Ridge seamount chain near its intersection with the TongaKermadec trench (Contreras-Reyes et al. 2010). The survey utilized a large-volume air gun source array and 39 ocean bottom seismometer receivers spaced ~15 km apart. The geometry was such that full ray coverage of the seamount and the underlying oceanic crust and uppermost ~10 km of

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the mantle was achieved. Figure 4a shows that the seismic Moho (which is resolved here to 100 m) cannot be explained by an Airy model which predicts too deep a Moho beneath the seamount chain and too shallow a Moho in flanking regions. The best fit between observed and calculated Moho depths (Fig. 4b) is for a flexure model of isostasy with Te ¼ 10 km. Such a model explains well the observed depth to Moho beneath both the seamount edifice and flanking moat regions.

Current Investigations One of the most important results to have emerged from isostatic studies over the past 40 years has been to show that a link exists between the Te derived from flexural modelling and the long-term strength of the lithosphere. Data from experimental rock mechanics, for example, suggest that the strength of oceanic lithosphere is limited by brittle deformation in its uppermost part and by ductile flow in its lowermost part (Goetze and Evans 1979). This strength “profile” implies that loads applied to the surface (or base) of the lithosphere will be supported partly by the brittle and ductile strength of the lithosphere and partly by a central “core” which deforms elastically. It has been shown that oceanic Te reflects the thickness of this “core” which will be small in regions of young crust, large loads, and high curvature of bending and high in regions of old crust, small loads, and low curvature (Watts and Burov 2003). Oceanic Te is in the range 0–40 km and since the oceanic crust is only ~8 km thick, we may conclude that not only the crust but also the mantle is involved in supporting long-term geological loads. Figure 5 shows that Te at seamounts and ocean islands and deep-sea trench and outer rise systems increases with the age of the lithosphere at the time of loading, being small at volcano and trench loads that form on young seafloor (e.g., on-ridge) and large at loads on old seafloor (e.g., off-ridge). Thus, as oceanic lithosphere cools, subsides, and increases in age following its creation at a spreading center, it becomes more rigid in the way that it responds to loads. Interestingly, most seamount and oceanic island values fall within the 300–600 °C isotherm envelope based on the cooling plate model of Parsons and Sclater (1977), while most trench and outer rise values require a higher controlling isotherm in the region 450–800 °C. This difference in the range of controlling isotherm can be explained by difference in the load age, which is small (essentially present day) at deep-sea trenchouter rises and large (~1.5 to 100 Myr) at seamounts and oceanic islands. Indeed, a load age dependency explains much of the scatter in plots of Te vs. age of the lithosphere at the time of loading. Other possible contributors are uncertainties in load, infill and mantle densities, thermal

Isostasy

perturbations due to hot and cold spots, and yielding in regions of large loads and high curvature. Most of the Te estimates plotted in Fig. 5 are based on a forward modelling approach in which the gravity anomaly, surfaces of flexure, or vertical motion history are calculated and compared to observations. The best fit Te is then determined by a “trial and error” comparison of the observed and calculated. While forward modelling is a satisfactory way to estimate Te, the number of sites where information on both load and plate age are available is limited. Technological improvements in satellite radar altimetry and shipboard multibeam (swath) bathymetry data acquisition systems have led to the development of new analysis techniques and an increase in Te estimates. Watts et al. (2006), for example, used satellite-derived gravity data together with a reciprocal admittance technique to predict bathymetry for different values of Te. By comparing observed and calculated bathymetry, these workers obtained 9758 individual estimates of Te, 291 of which were at sites where both load and plate age were known. They found a greater than expected scatter in global plots of Te vs. age of the lithosphere at the time of loading with no clear indication of a controlling isotherm. Kalnins and Watts (2009) used satellite-derived gravity and shipboard bathymetry data and a sliding window admittance technique and found the controlling isotherm varied regionally. This may help explain some of the scatter in global plots of Te against load and plate age. Bry and White (2007) used satellite-derived gravity and shipboard bathymetry data and an inversion technique to determine Te along a number of fixed length profiles of deep-sea trench outer rise systems and found no evidence of a controlling isotherm. Hunter and Watts (2016), however, used a similar circum-Pacific data set and inverse modelling approach and found a strong dependence of Te and plate age. They attributed the difference with Bry and White (2007) to a combination of factors which include: the manner of the long wavelength gravity field removal; the method used to calculate gravity anomalies; the constraint placed on bending moment; and the location of the trench axis. One problem with spectral methods is the influence of window size on the recovered Te (Crosby 2007). Windows need to be large enough to resolve the largest Te and small enough to avoid “contamination” from more than one type of tectonic setting (e.g., on-ridge, off-ridge). The physical meaning of Te and its relationship to load and plate age is less clear in the continents than it is in the oceans. Unlike oceanic lithosphere, continental lithosphere has a multilayered rheology (Burov and Diament 1995; Meissner and Strehlau 1982) and there may be more than one brittle and ductile layer that supports an individual load or unload. In this situation, Te is not a particular depth in the lithosphere but a measure of its integrated strength. Continental Te will depend on both the curvature of flexure (which depends in turn on the

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Isostasy, Fig. 5 Plot of Te vs. age of the oceanic lithosphere at the time of loading. (a) Seamounts and oceanic islands and (b) Deepsea trench and outer rise systems. The data is based on Table 6.1 in Watts (2001) with additional values from Freedman and Parsons (1986), McQueen and Lambeck (1989), and 15 other studies that have been carried out since 2000. Solid and dashed lines show the calculated depth of the 300 °C and 600 °C isotherms based on the cooling plate models of Parsons and Sclater (1977) and McKenzie et al. (2005), respectively

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size of the load or unload) and how well individual competent layers are coupled, being low for high curvature and poorly coupled systems and high for low curvature and well-coupled systems (Watts and Burov 2003). As in the oceans, continental Te has been estimated using both forward and spectral modelling techniques. Most forward modelling estimates are based on fitting either the gravity anomaly or the surfaces of flexure (as inferred, for example, from basement depth data) at rift-type, strike-slip,

intra-cratonic, and foreland basins to model predictions. Sometimes, both gravity anomaly and depth to basement data are available and this yields most reliable estimates (Haddad and Watts 1999). Continental Te estimates range from 5–100 km and no simple relationship has yet been found between Te and load and plate age (Watts 2001). The data is suggestive, however, of some age dependence. Archaen and Proterozoic cratons, for example, are generally associated with high Te values (>70 km) while Phanerozoic

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and younger mountain belts generally have lower values (25–60 km) (Pérez-Gussinyé and Watts 2005). Young rifted margins (e.g., South China Sea) appear to have lower Te values than old margins (e.g., Amazon) (Watts et al. 2009) and this has been interpreted as the result of heating during continental break-up, which weakens the lithosphere, and cooling following rifting that strengthens it (Burov and Poliakov 2001). Plume-influenced rifts (e.g., Afar, North Atlantic) are associated with the lowest Te values (Ebinger et al. 1989; Watts and Fairhead 1997) but it is not clear how much strength the weakened rifted lithosphere in such tectonic settings is capable of regaining following a rifting event. According to the Wilson Cycle, ocean basins open and close and compressional structures in mountain belts develop on or close to the site of extensional rifted margins. Isostatic studies of the foreland basins suggest that they may inherit the Te structure of the underlying rifted margin (Lin and Watts 2002; Ali and Watts 2009). The foreland basin may be either narrow or wide, depending on whether it formed on a young or old margin. The very high (>70 km) Te values that are associated with some foreland basins (e.g., central Andes) may be explained by the fact that the fold and thrust belts, the main driving load for foreland basin subsidence, will sometimes over-ride the rifted margin altogether and “telescope” onto the more rigid cratonic interior, thereby inheriting its value. As in the oceans, spectral approaches have provided new insights into continental Te, especially its spatial and temporal variation. The first studies (e.g., Dorman and Lewis 1970; Banks et al. 1977; McNutt and Parker 1978) were based on a Bouguer admittance, Fourier transform periodogram method, which generally yielded low values. Forsyth (1985), however, pointed out that the low values might be a result of subsurface (i.e., buried) loads that were uncorrelated with surface topographic loads. Subsurface loads are difficult to define and so he suggested using the Bouguer coherence, rather than the Bouguer admittance, because it was less sensitive to the ratio of surface to subsurface loading. Forsyth’s method has been widely applied (e.g., Zuber et al. 1989) and as Bechtel et al. (1990) showed for North America, it yields low as well as high values, especially in cratonic regions. Subsequent studies have incorporated new techniques, including maximum entropy estimators (Lowry and Smith 1995) and multitapers (McKenzie and Fairhead 1997). McKenzie and Fairhead (1997) recommend using the free-air admittance rather than the Bouguer coherence because the surface topography, unlike buried loads, is a known load, the gravity effect of which should not really be removed from the free-air gravity anomaly. Forsyth’s Bouguer coherence technique they argued maybe biased by the effects of erosion and his method will

Isostasy

yield over-estimates of Te rather than true values. PérezGussinyé and Watts (2005) used both Bouguer coherence and free-air admittance methods and found the highest values in Europe (>70 km) over the Archaen and Early Proterozoic ( 70 km (Fluck et al. 2003). The consequences of such strength variations are profound, especially as they impact the computation of the isostatic gravity anomaly (Kirby 2019), the patterns of glacial isostatic rebound (Whitehouse et al. 2006), the solid earth tide

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Isostasy, Fig. 6 The Te structure of Europe as derived from the coherence between the observed Bouguer gravity anomaly and topography as a function of wavelength. Modified from (Pérez-Gussinyé and Watts 2005). (a) Te and its relationship to the terrane structure. Thin white lines show major sutures. I ¼ Iapetus, R ¼ Rheic, A ¼ Alpine. T ¼ Thor. S ¼ Sorgenfrei-Tornquist. The tectonic provinces are Ko, Kola; Ka, Karelia; and Svf, Svecofennian. Thin dashed lines show the Caledonian (Ca), Variscan (Va), and Alpine (Al) deformation fronts. Peri-G. ¼ Peri-Gondwanaland. Av ¼ Avalonia. EEC ¼ East European Craton. Black filled circles show the location of the coherence and RMS plots in (b). (b) Examples of the observed and predicted Bouguer coherence and RMS between the observed (solid line) and calculated (dashed line) coherence for a 400  400 km and 1000  1000 km analysis windows centered on Belgium and northern Italy. Note that northern Italy appears as a region of low Te in both windows, but the high Te region of Belgium only has a clear RMS minima in the larger window

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(Mantovani et al. 2005), and the terrane structure of continental lithosphere (Pérez-Gussinyé et al. 2007).

Current Controversies and Gaps in Knowledge Isostatic studies have never been far from controversy as is evidenced by the vigorous debates on local vs. regional models that took place at the turn of the twentieth century. Central to these debates was the applicability of isostatic models to particular geological features and certain “schools of thought” emerged among geologists, geophysicists, and geodesists. There were even debates either side of the Atlantic when an Airy model was preferred by Europeans and a Pratt model by North Americans. Today, there is now general agreement on the relative role of local and regional models of isostasy. Debate is focused instead on the flexure model, particularly Te, what it means and what it tells us about the long-term strength of the lithosphere, the time-scales of isostatic adjustment, and the geological processes that shape the top and base of the lithosphere. The Long-Term Strength of the Lithosphere In the oceans, there is widespread agreement on oceanic Te and what it indicates about the long-term strength of the lithosphere. While there is scatter in plots of oceanic Te vs. plate age, most of it can be attributed to viscoelastic stress relaxation (Watts and Zhong 2000), curvature and yielding (McNutt and Menard 1982), and spatial variations in the controlling isotherms that determine Te (Kalnins and Watts 2009). There has not, however, been the same agreement concerning continental Te estimates and their relationship to age. The controversy came to a head in the so-called crème brûlée vs. jelly sandwich debate (Burov and Watts 2006). Proponents of the crème brûlée model follow McKenzie and Fairhead (1997) and Jackson (2002) who believe that because Te is small and similar in thickness to the seismogenic layer, the strength of the lithosphere resides mainly in the uppermost part of the crust. In their view, the lower crust and uppermost mantle are weak which explains why deep earthquakes are rare. Proponents of the jelly sandwich model, however, follow Meissner (1986) who argue that the strength resides mainly in the upper crust and the uppermost mantle. In his view, only the lower crust is weak. This model is compatible with the results of seismic reflection profiling which show a layered lower continental crust and with the results of flexure studies which show a wide range of Te values. Low Te is indicative of thin competent layers while high Te values reflect thick competent layers. Very high Te values (>70 km) are also permissible, depending on composition, and occur when the upper and lower crust and/or the lower crust and mantle are coupled. Deep earthquakes are rare, not because the mantle is

Isostasy

too weak, but because it is too strong and the regional tectonic stresses required to initiate faulting and earthquakes are not large enough. The debate has led to a reexamination of the methodologies used to estimate continental Te, the physical meaning of Te and the geological implications of a weak or strong mantle. Pérez-Gussinyé et al. (2004), for example, showed that it is necessary when comparing observed and calculated free-air admittances to first window them using the same multitaper estimator while Crosby (2007) showed that in order to recover Te in cratonic shield regions, large window sizes (up to 1500 km) are needed. Otherwise, Te maybe underestimated. Watts and Burov (2003) pointed out that while Te may well be low and similar to the seismogenic layer thickness in some tectonic settings (e.g., Basin and Range, western USA), the two parameters are not the same. Te reflects the integrated strength of the lithosphere while the seismogenic layer is the thickness of the uppermost part of the crust that responds to stresses by faulting and earthquakes. Finally, Handy and Brun (2004) argued that a strong uppermost mantle is required in order to explain the integrity of sinking slabs in subduction zones and the deep structure of rifted margins and orogens. The continental Te debate has proved a useful discussion point in isostatic studies. It highlights the importance of Te while at the same time illustrates gaps in our knowledge about the Earth’s lithosphere, especially regarding its long-term rheology. Other gaps are: Time-Scales of Isostatic Adjustment Isostasy varies temporally, as well as spatially. Exactly how slow or fast isostasy operates is of fundamental importance to our understanding of the Earth’s vertical motion history as well as to topics such as sea-level change. For example, Bloom (1967) pointed out that the height of a wave-cut notch above present day sea-level depends not only on the amplitude of sea-level change but how quickly isostatic equilibrium is achieved. If isostasy is fast, then a sea-level rise will have time to load the crust and upper mantle and so the height of the notch will be low due to regional subsidence. If, on the other hand, isostasy is slow then there will be time and a notch will be cut at a higher elevation. Several attempts have been made to estimate the time scales of isostatic adjustment associated with load shifts on the top or base of the Earth’s crust. Watts (1978), for example, compared the Te derived from oceanic flexure studies to the thermal and seismic thickness of the lithosphere and concluded that on loading there must be a rapid reduction in the thickness of the mechanical layer that supports a load as the lithosphere relaxes from its short-term seismic to its long-term elastic thickness. Beaumont (1979) used an elastic-plastic model based on the creep laws of olivine to argue that while oceanic lithosphere was essentially elastic on long-time scales, continental lithosphere would continue to relax on

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assumes an asthenosphere viscosity of 1021 Pa s and a lithosphere viscosity that increases from 1021 Pa s to 1035 Pa s, depending on thermal age. The figure shows a rapid decrease in the thickness of the lithosphere that supports a load and hence a decrease in its strength. This weakening results in an increase in subsidence and a narrowing of the zone of subsidence with increase in load age. Eventually, the strength reduction slows and the thickness that supports a load approaches a steady state value. The continents display a wider range of load and unload ages than the oceans and elastic thickness estimates have now been published for short-term (a few s to a few hundred s) earthquake loading (2–100 s) through intermediate (~10 ka) glacial isostatic rebound to long-term (>1 Ma) flexure. There are regions (e.g., western and eastern USA) where elastic layer thickness has been estimated for two or more timescales. These data suggest that the relaxation observed in oceanic lithosphere may also characterize continental lithosphere, albeit over a much wider range of relaxation time-scales. For example, the Hebgen Lake (Nishimura and Thatcher 2003), Lake Bonneville (Nakiboglu and Lambeck 1983),

time-scales of up to several tens of Myr. Willett et al. (1985) used a viscoelastic plate that overlies an inviscid substrate to model continental Te data and showed that on loading there would be a weakening of the lithosphere that reduced the thickness of the mechanical layer by as much as a factor of 3–8, depending on the initial thermal structure of the lithosphere that was assumed. And finally, Watts and Zhong (2000) and Watts et al. (2013) have used the actual temperature structure of the oceanic lithosphere constrained from seafloor depth and heat flow data to calculate a viscosity profile and then examined how the equivalent Te would vary following volcano loading. They showed that because viscosity increases from low values in the lower lithosphere to high values in the upper lithosphere, the lithosphere relaxes from the bottom-up such that stresses migrate upwards and the reduction in the thickness of the mechanical layer that supports a surface or subsurface load is initially fast and then slows with increase in the load age. Figure 7 compares the Te for a number of seamounts and ocean islands in the world’s ocean basins to predicted curves based on a multilayered viscoelastic model. The model

Age of Load

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ans

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Oce

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s S) ent tin ern U n Co ast .E (e.g ???

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Isostasy, Fig. 7 Plot of elastic layer thickness vs. age of load for oceanic and continental lithosphere. Modified from Watts (2007). The thick solid blue line and thin dashed blue lines show the predicted change in elastic layer thickness for young, intermediate, and old oceanic lithosphere according to Watts and Zhong (2000). Filled blue circles show the data from individual seamount and ocean island loads which range in age from ~1–100 Myr. The data indicate that by ~10 kyr oceanic loads have undergone most of their relaxation. The thick dashed brown lines show the predicted change in elastic thickness for continental lithosphere. Filled grey rectangles are based on earthquake loading (postseismic and co-seismic), filled blue rectangles are based glacial isostatic

Flexural loading/unloading Oceans Continents

adjustment and filled beige and yellow rectangles are based on continental flexure studies (see Watts 2007 for references). Open symbols show elastic layer thickness estimates from the same tectonic region (circles ¼ western USA, squares ¼ eastern USA) based on the data in Nakiboglu and Lambeck (1983), Di Donato et al. (2000), Lowry et al. (2000), Nishimura and Thatcher (2003), and Pazzaglia and Gardner (1994). The data suggest two modes of relaxation that correspond to “fast” and “slow” isostasy. Interestingly, the relaxation time separating these modes varies between oceans and continents with western USA relaxing much quicker (~10 yr) than the oceans and eastern USA much slower (~70 Myr)

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and western Cordillera (Lowry et al. 2000) loads which reflect ages of 0–120 s, 10 ka, and ~ 60 Ma are associated with elastic layer thicknesses of 38, 30, and 5 km, respectively. Te therefore decreases with increase in load age. The decrease is similar in form to the ocean loading cases modelled by Watts and Zhong (2000) and Watts et al. (2013), but the viscosity of the asthenosphere required is less and ~ 3  1018 Pa s. This viscosity results in a rapid initial and a slow final weakening of the lithosphere and a transition between the two weakening regimes at load ages of ~30 a. Interestingly, the transition age appears to vary regionally such that loads on tectonic western North America relax quicker than those emplaced on the more stable parts of eastern North America (Watts et al. 2013). Isostasy and Landscape Evolution The behavior of the lithosphere and how it interacts with the cryosphere, hydrosphere, and atmosphere above and the asthenosphere below is a fundamental question that needs to be addressed if we are to fully understand geological processes and construct realistic models. We know, for example, from flexure studies that the lithosphere behaves as a low-pass filter in the way that it responds to loads and unloads, suppressing the short-wavelength deformation and passing the long-wavelengths. The physical properties of the lithosphere therefore smooth out the effects of surface and subsurface processes such as those involved in landscape evolution Isostasy, Fig. 8 Simple model for flexure of the lithosphere due to river incision and excavation. The model assumes that the lithosphere responds to loads (and unloads) initially as a thick elastic plate (Te ¼ 80 km), then as a viscous material and finally as a thin, essentially elastic, plate (Te ¼ 10 km). The response shows an initial long-wavelength, low-amplitude uplift a transient short-wavelength high-amplitude uplift and subsidence and a final short-wavelength high-amplitude uplift and subsidence that together reflects the thinning of the lithosphere from its initial shortterm thickness to its long-term elastic thickness. Thick solid red line shows the final topography after it has been modified by slope diffusion (assumed erosion time ¼ 50 ka, diffusion constant ¼ 50 m2 a1)

and mantle convection, making it difficult to see the full effect of these processes. It is being increasingly recognized that isostasy plays a major role in the development of landscapes and seascapes, both in the constructive forces such as tectonics that build them and the destructive forces such as erosion and largescale slope failures that destroy them. Examples include the flexures that follow mechanical unloading on normal faults (ten Brink and Stern 1992), fault scarp retreat (Gilchrist and Summerfield 1990), and glacial erosion (Pelletier 2004; Stern et al. 2005). Because flexure involves rock uplift and subsidence it is capable of modifying the landscape, especially river knickpoints, base levels and pre-existing drainage basins. This is particularly well seen in the rim uplifts that flank rifts (Tucker and Slingerland 1994) where flexure is a major factor in controlling the drainage divide that separates scarp and dip streams. Other examples have been described from the Cotswold Hills, southern England (Lane et al. 2007) where rim uplifts flank regions of fluvial incision and excavation. Figure 8 illustrates the isostatic adjustment that would occur following fluvial incision and excavation of a 150 m deep, 60 km wide, region of relatively low-density sediments. The calculations assume that the lithosphere behaves as a viscoelastic plate, similar to that implied by the Te data in Fig. 7 and used by Watts and Zhong (2000) to calculate the time-dependant flexure that follows seamount loading. The

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main model features are an initial thick-plate elastic response, a viscous time-dependent response, and a final thin-plate, essentially elastic, response. The figure shows that lithospheric weakening due, for example, to load-induced stress relaxation has the capacity to create topography that locally either enhances or subdues the more regional deformation associated with the initial short-term response. Karner (1984) showed that topography is also created during lithospheric strengthening when, for example, the oceanic lithosphere is thermally rejuvenated (and weakened) at midplate hotspot swells and subsequently cools. The topography in this case subdued the original swell. Incision and excavation, together with refilling, are common features of the fluvial response to glacial and inter-glacial cycles. Isostasy will therefore have a major role to play in the deformations that result. For example, incision due to meltwater charged rivers is a feature of glacial periods when sealevel was low while valley filling and delta construction are features of inter-glacials when sea-level was high. Repeated episodes of incision and valley fill will therefore re-surface a landscape, creating in the process isostatic rim uplift and subsidence, as seems to have occurred in the region of the Mississippi River where it debauches in to the Gulf of Mexico (Blum et al. 2008), One consequence of this resurfacing is that apparent sea-level may vary along-strike the northern Gulf of Mexico rifted margin such that the Mississippi River valley in Texas is presently subsiding relative to the adjacent Alabama coast. The Earth’s lithosphere is deformed not only by surface geological loads and unloads but also by subsurface loads, some of which originate within the crust while others have their source in the subcrustal mantle. Like surface loads, subsurface loads are associated with mass excesses and deficiencies and create flexures of the Moho and other crustal layers that are manifest in topography and gravity anomaly data. For example, subsurface loads due to sinking downgoing slabs, intra-crustal thrusting, obduction, and ophiolite emplacement cause flexures that contribute to both topography and gravity anomalies (e.g., Royden and Karner 1984; Lin and Watts 2002; Ali and Watts 2009). Long-Wavelength Gravity and Topography Anomalies, Isostasy, and Mantle Dynamics It has been known since the pioneering studies of Kaula (1967) that Earth’s gravity field spectrum is rich and has significant power at short as well as long wavelengths. Isostatic loads and unloads, however, are compensated so that their associated gravity anomaly will be reduced and may approach zero at the longest wavelengths. This is clearly seen in the theoretical free-air admittance which shows that Airy, Pratt, and flexure models have a distinct “roll-over” that approaches zero at the longest wavelengths because of isostasy (Watts 2001). The existence in oceanic regions of

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observed admittances in the range 5–20 mGal km1 at wavelengths of 1000–4000 km (Crosby 2007; Watts 2007; Wieczorek 2007) is therefore of much geodynamic interest. Watts (2007) has shown that the long-wavelength admittance remains approximately constant at ~18 mGal km1 in the central Pacific Ocean as the analysis window size is increased from 1000 to 4000 km. He argued that the long-wavelength admittance cannot be attributed to isostasy and so must reflect dynamic processes in the subcrustal mantle. As McKenzie (2010) has shown, continents, like the oceans, are characterized by a high admittance at long wavelengths. The higher values he obtained (35–50 mGal km1) are expected since topography displaces air in the continents rather than water thereby increasing the contribution of the gravity effect of the topography. The observation of high, positive, admittances at long wavelengths in both oceans and continents is interesting, especially as it suggests a degree of correlation between gravity and topography at these wavelengths. A correlation between long wavelength gravity and topography was noted by Kaula (1967) and as McKenzie (1977) points out, makes a strong argument for some form of mantle dynamics. Density-driven convection, for example, has the potential to contribute to the free-air gravity anomaly and the surface topography in both upwelling and downwelling regions. Long-wavelength topography that cannot be attributed to isostasy has been dubbed a “depth anomaly” by Menard (1973) and “dynamic topography” by (Hager et al. 1985) and convection maybe a significant contributor to the depth of mid-plate swells (Watts 1976), old (Huang and Zhong 2005) and young (Marks et al. 1990; Crosby and McKenzie 2009) ocean floor, the subsidence and uplift history of foreland-type (Liu and Nummedal 2004), rift-type (Muller et al. 2000), and some intra-cratonic sedimentary basins (Coakley and Gurnis 1995). An outstanding question is how best to separate the effects of isostasy from gravity anomaly and topography data and isolate the planform of motions in Earth’s mantle. Some progress on the problem has now been made by better defining the wavelength band of flexural isostasy. Figure 9 shows, for example, a plot of the power spectral density (in mGal2) of Earth’s free-air gravity anomaly as a function of spherical harmonic degree and equivalent wavelength. The figure shows the power spectra increases at short wavelengths (1100 km). The rise at short wavelengths follows the predictions of a model in which topography is uncompensated. The flattening at intermediate wavelengths suggests some degree of compensation of topography, although not in the form as predicted by the Airy model of isostasy. The observed spectra are clearly higher than the calculated spectra in this wavelength band and suggest instead a flexure model of

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Isostasy

Isostasy, Fig. 9 Comparison of the power spectrum of Earth’s free-air gravity anomaly field (filled black circles) to calculated spectra based on the gravity effect of uncompensated topography (Red filled circles) and the gravity effect of the topography and its Airy compensation (Green filled circles) and flexural compensation (Light blue filled circles). From Watts and Moore (2017). The observed spectra have been calculated from the spherical harmonic coefficients that describe the EGM2008 gravity field model (Pavlis et al. 2008) and the uncompensated and isostatic spectra have been calculated using equations in Kaula (1967)

and Watts and Moore (2017) and the Earth2014 topography model of Hirt and Rexer (2015). Grey band ¼ twice the minimum Root Mean Square (RMS) difference between the observed spectra and the calculated spectra based on a flexure model of isostasy, used to define a region of permissibility for the best fit elastic thickness, Te. HDB ¼ High Degree Band. LDB ¼ Low Degree Band. Inset shows the RMS difference between observed spectra and calculated spectra of the gravity effect of the topography and its flexural compensation as a function of Te. The best fit global Te is 34.0  4.0 km

isostasy with a Te of 34.0  4.0 km. This range of Te is interesting, especially as it is a global mean and indicates that there are regions of Earth with lower and higher values. The rise at long wavelengths (>1100 km) cannot be explained by the gravity effect of the flexure model of isostasy, or any other isostatic model, and therefore defines for us the limits of isostasy and the role played by non-isostatic processes such as those associated with mantle convection in contributing to Earth’s gravity field spectra. Improved gravity anomaly, topography and seismic field data, localized spectral analysis techniques, and better geodynamical models that include a lithosphere offer the most promise of better quantifying the contribution of isostasy to Earth’s gravity and topography fields and crustal structure in the future.

the crust to float, in hydrostatic equilibrium, on its denser substrate such that light areas stand at a greater elevation than dense areas. Today, we consider isostasy as a highly idealized state that the crust and upper mantle approaches, but rarely achieves. Nevertheless, the comparison of observed gravity anomalies to the predictions of local and regional isostatic models have led to a new understanding about the thermal and mechanical properties of the lithosphere. We now know, for example, that the lithosphere responds to long-term (i.e., >105 years) loading and unloading in a similar manner as would a strong elastic plate that overlies a weak fluid substrate. In the oceans, the thickness of the elastic layer that supports a volcano load increases with the age of the lithosphere at the time of loading. In the continents, the relation between elastic thickness and age is not as clear. However, there is evidence that older Archaen and Early Proterozoic cratons have high values of the elastic thickness (i.e., >70 km) while younger Middle/late Proterozoic and Phanerozoic orogenic belts and rifts have lower values. Data from experimental rock mechanics suggest that the elastic

Conclusions Isostasy is a principle that is central to the Earth Sciences. It was first formulated in the 1880s to describe the tendency of

Isostasy

thickness derived from isostatic studies is a proxy for the long-term strength of the lithosphere. The elastic thickness differs from the seismogenic layer thickness, which is the thickness of the uppermost part of the lithosphere that responds to stresses by faulting and earthquakes and the seismic “lid” thickness, which is the mechanical thickness that supports short-term loads and unloads. Isostatic studies suggest that the time-scales of isostatic adjustment varies regionally such that tectonically active areas respond to loads and unloads much quicker than cratonic areas. The behavior of the lithosphere on seismic to geologic time-scales is important to take into account when modelling geological processes, especially those linking dynamic topography and mantle convection.

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Isostasy Variscan crust, J. Geological Soc. London Special Publication No. 24, 1–10 Meissner R, Strehlau J (1982) Limits of stresses in continental crusts and their relation to the depth-frequency distribution of shallow earthquakes. Tectonics 1(1):73–89 Menard HW (1973) Depth anomalies and the bobbing motion of drifting islands. J Geophys Res 78:5128–5137 Muller RD, Lim VSL, Isern AR (2000) Late tertiary tectonic subsidence on the northeast Australian passive margin: response to dynamic topography? Mar Geol 162:337–352 Nakiboglu SM, Lambeck K (1983) A reevaluation of the isostatic rebound of Lake Bonneville. J Geophys Res 88:439–447 Nakiboglu SM, Lambeck K (1985) Comments on thermal isostasy. J Geodynamics 2:51–65 Nishimura T, Thatcher W (2003) Rheology of the lithosphere inferred from postseismic uplift following the 1959 Hebgen Lake earthquake. J Geophys Res 108. https://doi.org/10.1029/2002JB002191 Parsons BE, Sclater JG (1977) An analysis of the variation of ocean floor bathymetry and heat flow with age. J Geophys Res 82:803–827 Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM 2008. EGU General Assembly 2008 Pazzaglia FJ, Gardner TW (1994) Late Cenozoic flexural deformation of the middle U.S. Atlantic passive margin. J Geophys Res 99:12,143–12,157 Pelletier JD (2004) Estimate of three-dimensional flexural-isostatic response to unloading: rock uplift due to late Cenozoic glacial erosion in the western United States. Geology 32:161–164 Pérez-Gussinyé M, Watts AB (2005) The long-term strength of Europe and its implications for plate-forming processes. Nature 436. https:// doi.org/10.1038/nature03854 Pérez-Gussinyé M, Lowry AR, Watts AB, Velicogna I (2004) On the recovery of effective elastic thickness using spectral methods: examples from synthetic data and from the Fennoscandia shield. J Geophys Res 109. https://doi.org/10.1029/2003JB002788 Pérez-Gussinyé M, Lowry AR, Watts AB (2007) Effective elastic thickness of South America and its implications for intracontinental deformation. Geochem Geophys Geosyst 8. https://doi.org/10.1029/ 2006GC001511 Pérez-Gussinyé M, Metois M, Fernandez M, Verges J, Fullea J, Lowry AR (2009) Effective elastic thickness of Africa and its relationship to other proxies for lithospheric structure and surface tectonics. Earth Planet Sci Lett 287:152–167 Pratt JH (1855) On the attraction of the Himalaya mountains, and of the elevated regions beyond them, upon the plumb line in India. Phil Trans R Soc A 145:53–100 Royden L, Karner GD (1984) Flexure of the lithosphere beneath Apennine and Carpathian foredeep basins: evidence for an insufficient topographic load. Am Assoc Pet Geol 68:704–712 Rychert CA, Fischer KM, Rondenay S (2005) A sharp lithosphereasthenosphere boundary imaged beneath eastern North America. Nature 436. https://doi.org/10.1038/nature03904 Sacek V, Ussami N (2009) Reappraisal of the effective elastic thickness for the sub-Andes using 3-D finite element flexural modelling, gravity and geological constraints. Geophys J Int 179:778–786 Sclater JG, Francheteau J (1970) The implications of terrestrial heat flow observations on current tectonic and geochemical models of the crust and upper mantle of the earth. Geophys J R Astron Soc 20:509–542 Sibuet J-C, Veyrat-Peinet B (1980) Gravimetric model of the Atlantic equatorial fracture zones. J Geophys Res 85:943–954 Simons FJ, Zuber MT, Korenaga J (2000) Isostatic response of the Australian lithosphere: estimation of effective elastic thickness and anisotropy using multitaper spectral analysis. J Geophys Res 105:19,163–119,184

Isostasy, Thermal Stark CP, Stewart J, Ebinger CJ (2003) Wavelet transform mapping of the effective elastic thickness and plate loading: validation using synthetic data and application to the study of the south African tectonics. J Geophys Res 108. https://doi.org/10.1029/ 2001JB000609 Stern TA, Baxter AK, Baxter PJ (2005) Isostatic rebound due to glacial erosion within the Transantartic Mountains. Geology 33:221–224 Tassara A, Swain C, Hackney R, Kirby J (2006) Elastic thickness structure of South America estimated using wavelets and satellitederived gravity data. Earth Planet Sci Letts 253:17–36 ten Brink U, Stern T (1992) Rift flank uplifts and hinterland basins: comparison of the Transantarctic Mountains with the great escarpment of southern Africa. J Geophys Res 97:569–585 Tucker GE, Slingerland RL (1994) Erosional dynamics, flexural isostasy, and long-lived escarpments: a numerical modeling study. J Geophys Res 99:12,229–212,243 Vai GB (2006) Isostasy in Luigi Ferdinando Marsili’s manuscripts, in The origins of geology in Italy, edited. Geol Soc, Amer Vening Meinesz FA (1931) Une nouvelle methode pour la réduction isostatique régionale de l’intensité de la pesanteur. Bull Géod 29:33–51 Watts AB (1976) Gravity and bathymetry in the Central Pacific Ocean. J Geophys Res 81:1533–1553 Watts AB (1978) An analysis of isostasy in the world’s oceans: 1. Hawaiian-emperor seamount chain. J Geophys Res 83:5,989–6,004 Watts AB (2001) Isostasy and flexure of the lithosphere, 458 pp. Cambridge University Press, Cambridge Watts AB (2007) An overview, in Treatise of Geophysics. In: Watts AB (ed) Crust and lithosphere dynamics, vol 6. Elsevier, pp 1–48 Watts AB, Burov EB (2003) Lithospheric strength and its relationship to the elastic and seismogenic thickness. Earth Planet Sci Letts 213:113–131 Watts AB, Fairhead JD (1997) Gravity anomalies and magmatism at the British Isles continental margin. J Geol Soc Lond 154:523–529 Watts AB, Moore JDP (2017) Flexural isostasy: constraints from gravity and topography power spectra. J Geophys Res 122. https://doi.org/ 10.1002/2017JB014571 Watts AB, Zhong S (2000) Observations of flexure and the rheology of oceanic lithosphere. Geophys J Int 142:855–875. https://doi.org/10. 1046/j.1365-246x.2000.00189.x Watts AB, Sandwell DT, Smith WHF, Wessel P (2006) Global gravity, bathymetry, and the distribution of submarine volcanism through space and time. J Geophys Res 111. https://doi.org/10.1029/ 2005JB004083 Watts AB, Rodger M, Peirce C, Greenroyd CJ, Hobbs RW (2009) Seismic structure, gravity anomalies, and flexure of the Amazon continental margin, NE Brazil. J Geophys Res 114. https://doi.org/10. 1029/2008JB006259 Watts AB, Zhong SJ, Hunter J (2013) The behavior of the lithosphere on seismic to geologic timescales. Annu Rev Earth Planet Sci 41. https:// doi.org/10.1146/annurev-earth-042711-105457 Whitehouse P, Latychev K, Milne GA, Mitrovica JX, Kendall R (2006) Impact of 3-D earth structure on Fennoscandian glacial isostatic adjustment: implications for space-geodetic estimates of presentday crustal deformations. Geophys Res Lett 33. https://doi.org/10. 1029/2006GL026568 Wieczorek MA (2007) Gravity and topography of the terrestrial planets, in Treatise of Geophysics. Volume 10: Planets and Moons, edited by T. Spohn, pp. 165–206, Elsevier Willett SD, Chapman DS, Neugebauer HJ (1985) A thermo-mechanical model of continental lithosphere. Nature 314:520–523 Zuber MT, Bechtel TD, Forsyth DW (1989) Effective elastic thickness of the lithosphere and mechanisms of isostatic compensation in Australia. J Geophys Res 94:13,919–13,930

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Isostasy, Thermal Derrick Hasterok1 and David S. Chapman2 1 Earth Sciences Department, University of Adelaide, Adelaide, SA, Australia 2 Department of Geology and Geophysics, The University of Utah, Salt Lake City, UT, USA

Definition Isostasy

Compositional isostasy Thermal isostasy

Isostatic equilibrium

(Greek: isos “equal,” stasis “standstill”) term used in geology to refer to the condition of gravitational equilibrium such that the rigid outer part of Earth’s crust and uppermost mantle “float” at an elevation which depends on its density. This concept explains how different topographic heights can exist at Earth’s surface. The part of isostasy traceable to density variations arising from compositional (rock type and mineralogy) differences. The part of isostasy traceable to density variations arising from temperature differences and thermal expansion within the rock column. The condition whereby a certain area of crust and upper mantle reaches the state of isostasy such that a hydrostatic equilibrium exists at depth below a level of compensation. The region is said to be in isostatic equilibrium.

Introduction The condition in Earth whereby variations in crustal thickness and density determine the elevation of Earth’s solid surface is known as isostasy and has been appreciated for a century and a half (Pratt 1855; Airy 1855; Watts 2001, for general review). Local isostasy requires a hydrostatic equilibrium condition at a depth below a compensation level, implying that the integral of density over a rock column is constant. When density differences arise from differences in rock composition (rock type, mineralogy), the isostatic condition is referred to as compositional isostasy. It is compositional isostasy, and in particular the density contrast across the crust–mantle boundary, that dominates isostasy discussions in textbooks. If, on the other hand, density differences arise from different thermal states of the lithosphere and thermal expansion of rock, the condition is properly called thermal isostasy. This entry is

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devoted to thermal isostasy, computation of thermal isostasy effects, and examples of thermal isostasy in oceanic regions and on the continents.

Thermal Isostasy Rock density is influenced by temperature through thermal expansion. If rock composition is identical in two vertical columns through the lithosphere, the warmer column has a lower density than the cooler lithosphere. Applying the isostatic condition of hydrostatic equilibrium at some depth of compensation leads to a prediction of elevation changes for the two regions. Consider two regions of different lithospheric thermal states. Temperature differences between the two regions represented by a regional geotherm T(z) and a reference geotherm Tref(z), respectively, multiplied by a thermal expansion coefficient αV and integrated over depth predict an elevation change, ΔεT, given by

0

½T ðzÞ  T ref ðzÞdz:

ð1Þ

The maximum depth of integration, zmax, is the depth at which the colder geotherm converges to a mantle adiabat assumed to be identical for both lithospheric regions. Most studies use a coefficient of thermal expansion of 3.0  105 K1 within the crust, consistent with the values for major crustal forming rocks including granite and gabbro. Within the mantle, a value of thermal expansion of 3.2  105 K1 is used.

Oceanic Thermal Isostasy The clearest example of thermal isostasy occurs in oceanic regions as hot, near-molten lithosphere at a spreading ridge 0

2 0.5 5

25 Depth [km]

Isostasy, Thermal, Fig. 1 Thermal evolution of oceanic lithosphere after formation at a spreading oceanic ridge system. (a) Temperaturedepth profiles as a function of lithosphere age in millions of years. (b) Subsidence of the seafloor as a function of seafloor age. Solid dots are data in 2 My age bins. Gray shading represents 1 standard deviation about the mean bathymetry.

25 50

50 t = 150 Ma

75 oceanic geotherms 100 0 a

500

4 5 6

1000 1500

Temperature [°C]

Sediment Corrected Bathymetry

3 Bathymetry [km]

DeT ¼ aV

ð zmax

cools over tens of millions of years leading to thermal contraction, increasing rock density, and subsidence of the seafloor by about 3,000 m. It is important to note that lateral compositional variations are small in oceanic regions. Oceanic crust about 7 km thick is formed at a spreading center and its thickness and composition are modified only in minor ways over its maximum 165 My existence. The thermal state of oceanic lithosphere, on the other hand, varies from a near molten state at the ridge to extremely cold lithosphere after up to 165 My of cooling. These contrasting thermal states have consequences for the rigidity of the lithosphere but in particular for its density and therefore elevation. Almost since the discovery of plate tectonics and seafloor spreading, thermal isostasy has been invoked to explain the bathymetry of the oceans as a function of age. Figure 1a illustrates the time evolution of an oceanic geotherm for a cooling plate model. The initial condition is 1,450 °C (Stein and Stein 1993); other cooler values lead to similar subsidence curves if thermal expansion values are adjusted accordingly. The ocean lithosphere initially cools rapidly as shown by the large difference between the 0.5 My geotherm and the 5 and 25 My geotherms (Fig. 1a). At longer times, the cooling slows and the geotherms reach near steady state shown by the 150 My geotherm. The average temperature change in the lithosphere between zero age at the ridge and about 150 My seafloor is 725 K. Bathymetry is proportional to the integrated cooling of the oceanic lithosphere (Eq. 1), additionally incorporating the changing load of water as the seafloor subsides. This subsidence (Fig. 1b) is explained very well by a simple onedimensional (1-D) model of lithospheric cooling. A midocean ridge, or zero age lithosphere, has on average a depth of 2.7 km, a consequence of both its thermal state, crustal thickness, and composition. But as the lithosphere ages and cools, temperature changes but crustal thickness and composition do not. Cooling over the first 9 My causes a subsidence of 1 km to a depth of 3.7 km. At a lithospheric age of 36 My, the solid surface has subsided a further kilometer to a water depth of

0 b

50

100 Age [Ma]

150

Isostasy, Thermal

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4.7 km. As the lithosphere reaches its thermal equilibrium, the bathymetry also approaches its equilibrium value of 5.5 km.

Continental Thermal Isostasy On continents, thermal isostasy has been used successfully to examine the evolution of regions that mimic oceanic spreading, such as continental rifts and elements of provinces with extensive volcanism or back-arc regions (McKenzie 1978; Jarvis and McKenzie 1980; Brott et al. 1981; Lachenbruch and Morgan 1990; Hyndman et al. 2005). However, direct thermal effects on continental elevation are difficult to discern because of the potential masking effect of compositional variations in lithospheric density and crustal thickness. The elevation effect due to compositional variation can be estimated by a simple isostatic calculation using continental extremes. A mountainous region with a crustal thickness of 50 km and a density of 2,800 kg m3 (granodiorite) would have an elevation 5 km higher than a rift province with a crustal thickness of 25 km and a density of 2,900 kg m3 (gabbro). Both columns assume a similar mantle density of 3,340 kg m3. The potentially large compositional effect can easily mask the effect of thermal isostasy. Therefore, in order to isolate the effect of thermal isostasy in the continents, it is necessary first to remove compositional isostatic effects. Normalizing Compositional Elevation Compositional variations involving both crustal thickness and density are removed with a simple isostatic correction to

the observed elevation (Han and Chapman 1995). The adjustment normalizes any crustal column to an arbitrary crustal standard. In refining this method, Hasterok and Chapman (2007a) used a standard crustal thickness h0 c of 39 km and a standard crustal density r0 c of 2,850 kg m3. These values represent the average crustal thickness and density of North American provinces. A lithospheric mantle density of 3,340 kg m3 is used based on the xenolith derived estimate by Griffin et al. (1999) for Proterozoic lithosphere. Consider a crustal column with a crustal thickness, hc, and crustal density, rc, at an elevation of εobs (Fig. 2a). If the density and thickness of the observed crustal column are adjusted to match the standard density and thickness, the elevation would change by an amount Δεc (Han and Chapman 1995) given by   r0 r DeC ¼ h0c 1  c  hc 1  c rm rm

ð2Þ

where rm is the density of the mantle. If the province surface is below sea level (e.g., continental shelf), an additional term involving the water depth (where εobs is the bathymetry) and seawater density, rw, is required and the elevation adjustment becomes   r0 r e r DeC ¼ h0c 1  c  hc 1  c  obs w : rm rm rm

ð3Þ

By adding the elevation adjustment to the observed elevation, one arrives at the final adjusted elevation, εadj, given by

Δε

εobs

εobs < 0

Δεw εobs

ρc

hc‘

‘

ρc

hc

ρc

2800

0

hc

ρw

0

2900

-2

MSL

Standard Column

2

εobs ≥ 0

Density [kg m-3]

2

3000

DOC a

Isostasy, Thermal, Fig. 2 Compositional isostasy adjustments to continental elevation. (a) Cartoon illustrating parameters used in compositional correction for elevation. MSL is mean sea level and DOC is the depth of compensation. The observed crustal columns (left for regions above sea level and right for continental margins below sea level) are adjusted to an arbitrary standard crust (center) of thickness 39 km and

-2

ρm 2700 b

-4

ρm

20

30

40

50

Crustal Thickness [km]

density 2,850 kg m3. (b) Nomogram contouring the compositional elevation adjustment for observed crustal thickness and average crustal density. Contour interval is 0.5 km. Dashed lines for crustal thickness of 39 km and density of 2,850 kg m3 represent the standard crustal column. Zero line is locus of values for which the elevation adjustment is zero.

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Isostasy, Thermal

eadj ¼ eobs þ DeC :

ð4Þ

Three important physical parameters must be estimated in order to make the compositional adjustment: εobs, hc, and rc. The observed elevation is obtained from GTOPO30 for elevations above sea level and from bathymetric maps for elevations below sea level. Crustal thickness is obtained from 1-D whole crustal VP models. Crustal density is estimated by using empirical velocity–density (VP-r) relationships. The lateral dimension of each province analyzed varies from
70 km in some provinces in the Western USA to >500 km in shield provinces. See Hasterok and Chapman (2007a, b) for details. The magnitudes of compositional elevation adjustments made using Eqs. 2 and 3 are shown in Fig. 2b. The zero contour is the locus of crustal thickness and density combinations for which no crustal adjustment is made and therefore passes through the standard crust of thickness 39 km and density 2,850 kg m3. Over the extreme range of crustal thicknesses (20–60 km) and densities (2,700–3,000 kg m3), compositional elevation adjustments can be as much as 4 km. Regions that are thinner and/or more dense than the standard crust receive positive elevation adjustments, whereas regions that are thicker and/or less dense than the standard crust receive negative elevation adjustments. Most adjustments in Fig. 2b fall in the range +/1 km. From this nomogram we can also determine the uncertainty in elevation adjustment introduced by variations in crustal thickness and density. For example, to achieve an uncertainty of 250 m requires an uncertainty in crustal thickness less than 3 km and in density less than 43 kg m3. Continental Thermal State The thermal state of continental lithosphere is described by a set of geotherms. Whereas oceanic geotherms may be identified in terms of age, continental geotherms do not correlate well with rock age. Instead, the most important controlling parameter for continental geotherms is surface heat flow (Pollack and Chapman 1977b; Chapman and Pollack 1977; Blackwell 1971; Lachenbruch and Sass 1977; Rao et al. 1982). There exist a number of geotherm models of the continental crust (Blackwell 1971; Lachenbruch and Sass 1977; Chapman 1986; Chapman and Furlong 1992). Many other models are based on locally derived estimates of thermal conductivity and/or heat production. Although locally and xenolith-derived models may be best in the region for which they are developed, they may be poor estimates when extended to other regions. Therefore, we feel it is best to use a consistent geothermal model based on the fewest parameters necessary to demonstrate the usefulness of this method. We chose the method of Chapman (1986) and Chapman and Furlong (1992) to compute our geotherms because they yield a set of geotherms that are well documented, commonly

used, and yield temperatures which fall between other warm and cool models. The properties of the geotherm model include a lithospheric structure divided into an upper and lower crust and mantle lithosphere. Thermal conductivity within the crust is determined using a pressure-/temperature-dependent relationship of Chapman (1986). Thermal conductivities within the upper and lower crust are initially set as 3.0 and 2.65 W m1 K1, respectively, corresponding generally to felsic and mafic crystalline rocks at STP conditions. Mantle lithospheric thermal conductivities include a radiation term at temperatures above 500 K. Surface heat production is determined assuming a 40% radiogenic contribution to surface heat flow with a characteristic depth of 8 km (Pollack and Chapman 1977a). This characteristic depth is also used as the decay length of an exponentially decreasing function used to compute heat generation within the upper crust. Heat production of the lower crust is set as 0.4 mW m3 consistent with many studies of exposed granulite terranes. Mantle heat production is assumed to be 0.02 mW m3, a lower value justified by Chapman (1986) and Chapman and Furlong (1992). Temperatures within the asthenosphere are computed using an adiabat with a potential temperature of 1,300 °C and increase with depth of 0.3 ° C km1. Steady state, 1-D, conductive continental geotherms computed for surface heat flow values between 40 and 100 mW m2 are shown in Fig. 3a. Details of the geotherm calculation and boundary conditions used are given by Chapman (1986) and further discussed in Hasterok and Chapman (2007a). The thermal structure in some regions differs from the geotherm models used in this study as a result of nonsteady state processes and differences in thermophysical properties. However, we have chosen to keep our thermal models as simple as possible, interpreting the more difficult and uncertain parameters, that is, heat production and thermal conductivity, as residuals from a reference thermal isostatic model. Revealing Thermal Isostasy on Continents Because the thermal state of continents is not clearly related to the age of the lithosphere as it is in the oceans, one cannot confirm continental thermal isostasy by simply plotting elevation of continental regions against age of the province. Other approaches must be explored. One such approach examined elevation as a function of heat flow for several tectonic provinces around the globe (Han and Chapman 1995; Nagihara et al. 1996). Hasterok and Chapman (2007a, b) followed the same approach in much greater detail and made an assessment of the uncertainties related to removing compositional isostatic effects. A thermal isostatic curve for the continents, derived from the geotherm family in Fig. 3a and Eq. 1, is shown in Fig. 3b. We use a reference geotherm corresponding to a surface heat flow of 47 mW m2 and assign a lithosphere having this

Isostasy, Thermal

851

70 60 50

100

q0 = 40 mW m-2

200 0

Continental Geotherms 500

1000

Temperature [°C]

2 1 0 -1

1500 b

3

Observed

Adjusted Elevation [km]

100

50

150

a

3 Observed Elevation [km]

Depth [km]

0

40

60

80

Heat Flow [mW m ]

2 1 0 Misfit < 0.8

-1

100 -2

Adjusted

c

40

60

80

100 -2

Heat Flow [mW m ]

Isostasy, Thermal, Fig. 3 Thermal isostasy on continents. (a) Generalized family of continental geotherms parametric in surface heat flow values from 40 to 100 mW m2. Geotherms are truncated by the 1,300 ° C mantle adiabat. (b) Observed elevation versus heat flow for individual tectonic provinces of North America. Symbols denote dominant style of tectonic province: circle (shield), star (basin), square (collision), diamond (volcanic), triangle (extension). Error bars are one standard

deviation of the mean for heat flow. The geotherms in (a) together with Eq. 1 are used to calculate a theoretical thermal isostasy relationship shown as a bold line. (c) Elevation adjusted for compositional effects versus heat flow for the same tectonic provinces in (b). Elevation error bars are predicted from a Monte Carlo analysis of uncertainties in the seismic velocity to density conversion. Data trend confirms expected 3,000 m of thermal isostasy effect.

thermal state an elevation of 0 km. Although the zero elevation reference heat flow at 47 mW m2 is assumed here, the actual zero elevation could easily be set at another value. This initial assumption for the zero-elevation reference heat flow is refined further by Hasterok and Chapman (2007b). Within the range of surface heat flow for continental tectonic provinces (40–90 mW m2), the predicted elevation range resulting from thermal isostasy is approximately 3 km. This prediction is similar to the oceanic bathymetry difference between hot ridges and cold abyssal plains. Figure 3b also demonstrates how much continental thermal isostasy is obscured by compositional effects. There is little correlation between observed elevation of 36 North American tectonic provinces and the continental thermal isostatic curve. Much of this scatter is the result of variations in crustal density and thickness between the different tectonic provinces. Compositional elevation adjustments for the 36 tectonic provinces of North America range from 763  221 m in the Middle Rocky Mountains to 2,207  119 m in the Gulf of California (Hasterok and Chapman 2007b). Nearly all of the provinces have compositional elevation adjustments between 1 and +1 km. The results suggest that the compositional contribution to the elevation of North America accounts for
3 km of observed elevation variation. Patterns are evident in variations of elevation adjustment as a function of crustal thickness and density (Fig. 3c). The magnitude and range of elevation adjustments correlate with the type of tectonic regime that characterizes each North American province. Precambrian cratons have compositional adjustments between 0.5 and +0.5 km. Cenozoic rifts have elevation adjustments >0.5 km, and adjustments for volcanic provinces range from just under 0–1.0 km. Collisional orogens have a

very large range of elevation adjustments relative to the other tectonic environments, ranging from 0.75 to 1.0 km. The elevation adjustment is added to the observed elevation to determine the compositionally adjusted elevation (Fig. 3c). Although the data shown in Fig. 3c exhibit much scatter, two trends are clear. First, the adjusted elevation difference between hot and cold provinces is
3 km, a magnitude roughly equal to the elevation range observed in the oceans between hot spreading centers and cold abyssal plains. Second, whereas observed elevations do not define a clear trend with heat flow, the adjusted province elevations are less scattered and define a pattern that more closely resembles our predicted thermal isostatic curve (black line). The considerable scatter in Fig. 3c may be caused by imperfect compositional adjustment (seismic velocity characterization, velocity to density transform), by regions not being in isostatic equilibrium, or by surface heat flow not being perfectly representative of the thermal state of the lithosphere as described by the geotherm model used. The North American Cordillera The elevation of the North American Cordillera provides an excellent case study for thermal isostasy. Conventional wisdom suggests that many continental mountain belts, the result of continent–continent collision, are elevated because the collision process has thickened the continental crust. Airy isostasy considerations predict terrain elevations of 1,000–2,000 m as a result of thickened crust. But Hyndman (2010) point out that modern seismic data reveal that many of these collision provinces, in particular the North American Cordillera, do not have crustal roots. The Canadian Cordillera, for example, has a crustal thickness of only 35 km in comparison to the Canadian Shield crustal thickness of 40 km. But the elevation of the

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qs ¼ qr þ qm

ð5Þ

Figure 5a illustrates several geotherms expected for a set of regions with identical surface heat flow but differing upper crustal heat production. The integrated lithospheric temperatures is much lower than expected by the reference geotherm family, instead isostatically equivalent to a typical region with low surface heat flow (Fig. 5b). In Fig. 3, a typical partitioning of surface heat flow between a radiogenic upper crustal component (40%) and a basal component (60%) is assumed. If this model is reconsidered to allow for variations in partitioning fraction, the surface heat flow and adjusted elevation plot (Fig. 5c) can be used to identify regions of anomalous radioactivity or sublithospheric heat flow. Hasterok and Gard (2016) used thermal isostasy in this manner to estimate the heat production variations within the Australian continent (Fig. 5c).

0

400

800

1200

0 Heat Flow

Cordillera

Craton 40

Moho

Depth [km]

Xenoliths

80

120

160 Tomography

a

200 4 3

Elevation [km]

Australia Not all variations in heat flow between provinces can be easily attributed to increases in temperature. Proterozoic Australia, for instance, has typical heat flow values >80 mW m2, which is more consistent with active tectonic zones than shields or cratons. Numerous studies have attributed these high heat flow anomalies to very high crustal radioactivity (e.g., McLaren et al. 2003). Surface heat flow, qs, can be decomposed into a radiogenic heat flow, qr, and a sublithospheric heat flow, qm,

Temperature [ºC]

Mantle adiabat

thinner crust Cordillera is more than 1,000 m higher than the Canadian Shield, exactly opposite to the Airy compositional isostasy prediction. The solution to this apparent conundrum is found in thermal isostasy (Hyndman 2010). Hyndman et al. (2005) and Currie and Hyndman (2006) have shown that high upper mantle temperatures characterize most subduction back-arc regions (see also Lewis et al. 1992, 2003). And that many mountain belts also occur in current or recent back-arc locations. The temperature-depth field for the Cordillera, therefore, is much hotter than for a cratonic region (Fig. 4a). The Cordillera geotherm approaches a mantle adiabat at a depth of about 60 km, defining the lithospheric thickness for the Cordillera, while the Craton geotherm does not approach the mantle adiabat until a depth of more than 200 km. If Eq. 1 is applied to these two regions, the estimated difference in elevation, once compositional effects have been removed should be about 1,600 m. Figure 4b (from Hyndman 2010) shows exactly that effect. Elevations for both back-arc mountain belts and cold stable areas corrected for compositional density effects are plotted against local crustal thickness. Each group of data exhibit the expected trend of increasing elevation with increasing crustal thickness, but the Cordillera elevations are consistently 1,600 m higher than stable area elevations for the same crustal thickness.

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Isostasy, Thermal, Fig. 4 Thermal isostasy case study for North America Cordillera (after Hyndman 2010). (a) Temperature-depth plot for the northern Cordillera in comparison to the adjacent craton. Shaded regions in the mantle are general results from tomography; solid dots are temperature-depth loci from xenolith studies (see Hyndman 2010 for details). (b) Elevation adjusted for composition (density) versus crustal thickness (Table 3 of Hasterok and Chapman 2007b) for the hot Cordillera and cold stable areas of North America. The
1.6 km elevation difference can be explained by thermal isostasy using Eq. 1

The decomposition method can also aide interpretation of Australian non-thermal isostatic variations. Some regions sit outside the bounds of allowable heat flow–adjusted elevation pairs which indicate a violation of the basic model (Fig. 5c). The three provinces that sit outside this region in Australia include the Pilbara Craton, Yilgarn Craton, and the AlbanyFraser belt (a reworked portion of the Yilgarn Craton). The

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Isostasy, Thermal, Fig. 5 Decomposition of surface heat flow using thermal isostasy. (a) A demonstration of radiogenic effect on geotherms with the same surface heat flow and differing upper crustal heat production. (b) The differences in geotherms lead to elevation anomalies with respect to the reference thermal isostatic curve. (c) Case study of Australian heat flow and elevation (after Hasterok and Gard 2016) used to estimate radiogenic and mantle heat flow contributions to the surface heat flow. The sub-horizontal contours are lines of constant mantle heat flow whereas the sub-vertical contours are lines of constant radiogenic heat flow. The individual geological provinces separate into three

distinct sets, one with typical ranges of radioactivity and mantle heat flow when compared with North America, a second set of provinces with anomalously low elevation/high surface heat flow consistent with high radiogenic heat flow, and a third set outside the thermal bounds allowable by the model–a consequence of low-density Archean lithospheric roots. The error bars on the individual provinces have been removed to emphasize the background contours. The geotherms used to compute these models have been updated, but the thermal elevation, in Eq. 1, and the process of elevation adjustment are the same.

lithospheric mantle beneath these regions are often assumed to be depleted – compositionally buoyant – Archean lithosphere, violating the thermal isostatic model that only includes compositional adjustment for crustal buoyancy.

methods to adjust elevations for compositional effects. When those adjustments are made, there is a clear trend of increasing elevation of a continental province with surface heat flow – a confirmation of thermal isostasy. The magnitude of elevation differences between hot and cold continental regions is also about 3,000 m, comparable to thermal isostasy effects in marine areas. An excellent example of thermal isostasy effects is drawn from the North American Cordillera where Airy isostasy is unable to explain the elevation of the Cordillera terrains relative to stable shield areas. Instead, the hot upper mantle associated with current or recent back-arc activity, conveniently explains the 1,600 m of elevation difference that is observed.

Thermal Isostasy on Terrestrial Planets Venus and Mars also have convecting mantles beneath rigid lithosphere and presumably have significant variations in heat flow across their surfaces. Indeed, volcanism on Mars and Venus may be plume-related, but the cooling of the lithosphere following volcanic periods may result in significant temporal changes in topography. Cooling may be responsible for anomalies in ancient coastline elevations observed on Mars today (Ruiz et al. 2004).

Cross-References Summary Thermal isostasy, that part of isostasy traceable to density differences being caused by temperature differences and thermal expansion of rock, is an important but underappreciated condition in crustal geophysics. The condition/process of thermal isostasy received much attention in oceanic regions with the discovery of plate tectonics and seafloor spreading. It was quickly recognized that systematic cooling of the lithosphere formed at a spreading ridge could explain the 3,000 m of seafloor subsidence away from a ridge. On continents, thermal isostasy effects are obscured by compositional isostasy, principally crustal thickness, which produces over 3,000 m of elevation effects. Recent analyses, however, have developed

▶ Heat Flow, Continental ▶ Lithosphere, Continental: Thermal Structure ▶ Lithosphere, Oceanic: Thermal Structure ▶ Radiogenic Heat Production in the Continental Crust

Bibliography Airy G (1855) On the computation of the effect of the attraction of mountain-masses, as disturbing the apparent astronomical latitude of stations in geodetic surveys. Philos Trans R Soc Lond 145:101–103 Blackwell D (1971) The thermal structure of the continental crust. In: Heacock J (ed) The structure and physical properties of the earth’s crust. Geophysical monograph series, vol 14. AGU, Washington, DC, pp 169–184

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854 Brott C, Blackwell D, Ziagos J (1981) Thermal and tectonic implications of heat flow of the western Snake River Plain, Idaho. J Geophys Res 86:11709–11734 Chapman D (1986) Thermal gradients in the continental crust. In: Dawson J, Carswell D, Hall J, Wedepohl K (eds) The nature of the lower continental crust, vol 24. Geological Society Special Publication, Chapman, London, pp 63–70 Chapman D, Furlong K (1992) Thermal state of the continental lower crust. In: Fountain D, Arculus R, Kay R (eds) Continental lower crust geotectonics, vol 23. Elsevier, New York, pp 179–199 Chapman D, Pollack H (1977) Regional geotherms and lithospheric thickness. Geology 5:265–268 Currie R, Hyndman R (2006) The thermal structure of subduction backarcs. J Geophys Res 111:B08404. https://doi.org/10.1029/2005JB004024 Griffin W, O’Reilly S, Ryan C (1999) The composition and origin of subcontinental lithospheric mantle. In: Fei Y, Bertka C, Mysen B (eds) Mantle petrology: field observations and high pressure experimentation: a tribute to Francis R. (Joe) Boyd, vol 6. Special Publication of the Geochemical Society, Griffin, Lancaster, PA, pp 13–45 Han U, Chapman D (1995) Thermal isostasy: elevation changes of geologic provinces. J Geol Soc Korea 31:106–115 Hasterok D, Chapman D (2007a) Continental thermal isostasy: 1. Methods and sensitivity. J Geophys Res 112:B06414. https://doi. org/10.1029/2006JB004663 Hasterok D, Chapman D (2007b) Continental thermal isostasy: 2. Application to North America. J Geophys Res 112:B06415. https://doi.org/ 10.1029/2006JB004664 Hasterok D, Gard M (2016) Utilizing thermal isostasy to estimate sublithospheric heat flow and anomalous crustal radioactivity. Earth Planet Sci Lett 450:197–207. https://doi.org/10.1016/j.epsl.2016.06.037 Hyndman R (2010) The consequences of Canadian Cordillera thermal regime in recent tectonics and elevation: a review. Can J Earth Sci 47:621–632. https://doi.org/10.1139/E10-016 Hyndman R, Currie C, Mazzotti S (2005) Subduction zone backarcs, mobile belts, and orogenic heat. Geol Soc Am Today 15:4–10. https://doi.org/10.1130/1052-5173(2005)0152.0.CO;2 Jarvis G, McKenzie D (1980) Sedimentary basin formation with finite extension rates. Earth Planet Sci Lett 48:42–52 Lachenbruch A, Morgan P (1990) Continental extension, magmatism, and elevation; formal relations and rules of thumb. Tectonophysics 174:39–62

Isostasy, Thermal Lachenbruch A, Sass J (1977) Heat flow in the United States and the thermal regime of the crust. In: Heacock J (ed) The Earth’s crust. Its nature and physical properties, geophysical monograph series, vol 20. AUG, Washington, DC, pp 625–675 Lewis T, Bentkowski W, Hyndman R (1992) Crustal temperatures near the Lithoprobe Southern Cordillera Canada transect. Can J Earth Sci 29:1197–1214 Lewis T, Hyndman R, Flueck P (2003) Heat flow, heat generation, and crustal temperatures in the northern Canadian Cordillera: thermal control of tectonics. J Geophys Res 108:2316–2334. https://doi.org/ 10.1029/2002JB002090 McKenzie D (1978) Some remarks on the development of sedimentary basins. Earth Planet Sci Lett 40:25–32 McLaren S, Sandiford M, Hand M, Neumann N, Wyborn L, Bastrakova I (2003) The hot south continent: heat flow and heat production in Australian Proterozoic terranes. In: Hillis RR, Müller RD (eds) Evolution and dynamics of the Australian plate. Geological Society of American Special Paper, vol 22. Geological Society of America, Boulder, pp 151–161. https://doi.org/10.1130/0-8137-2372-8.157 Nagihara S, Lister C, Sclater J (1996) Reheating of old oceanic lithosphere: deductions from observations. Earth Planet Sci Lett 139:91–104 Pollack H, Chapman D (1977a) Mantle heat flow. Earth Planet Sci Lett 34:174–184 Pollack H, Chapman D (1977b) On the regional variation of heat flow, geotherms, and lithospheric thickness. Tectonophysics 38:279–296 Pratt J (1855) On the attraction of the Himalaya mountains, and the elevated regions beyond them, upon the plumb-line in India. Philos Trans R Soc Lond 145:53–100 Rao R, Rao G, Reddy G (1982) Age dependence of continental heat flow – fantasy and facts. Earth Planet Sci Lett 59:288–302 Ruiz J, Fairén AG, Dohm JM, Tejero R (2004) Thermal isostasy and deformation of possible paleoshorelines on Mars. Planet Space Sci 52:1297–1301. https://doi.org/10.1016/j.pss.2004.06.003 Stein C, Stein S (1993) Constraints on Pacific midplate swells from global depth–age and heat flow – age models. In: Pringle M (ed) The Mesozoic Pacific. Geology, tectonics, and volcanism, geophysical monograph series, vol 77. AGU, Washington, DC, pp 53–76 Watts A (2001) Isostasy and the flexure of the lithosphere. Cambridge University Press, Cambridge

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KTB Depth Laboratory: A Window into the Upper Crust Ulrich Harms and Jochem Kück Scientific Drilling, GFZ German Research Centre for Geosciences, Potsdam, Germany

Definition Depth observatory or depth laboratory is an up to several kilometer-deep borehole or underground facility enabling observations of geophysical, geological, mineralogical, geochemical, and biological phenomena in the deep Earth. It is equipped with various sensors allowing for long-term data acquisition at crustal depth level. KTB Kontinentales Tiefbohrprogramm der Bundesrepublik Deutschland was the Continental Deep Drilling Program of the Federal Republic of Germany with two, 4,000 m and 9,101-mdeep, boreholes drilled during the years 1987–1994. Borehole monitoring is either the systematic direct collection of physical, chemical, rock mechanical and biological data from formations penetrated by boreholes thus covering various geological timescales or the indirect observation of information from greater depth, e.g., through elastic (seismic) waves. Installations of downhole instruments and subsequent data transmission to data centers for further processing are either permanent, e.g., cemented in place, or retrievable at the end of the observation period. Scientific drilling encompasses an Earth science research program that utilizes drilling machines to produce a deep well, retrieve samples and acquire data from depth to gain novel scientific insights.

Introduction Long-lasting and successful international scientific drilling projects are not only striving to provide direct access to the © Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

deep underground for geoscientific research purposes but also to monitor in situ physical, chemical, and rock mechanical properties over prolonged time periods to understand the dynamics and processes inside the Earth. One of the paramount examples of the utilization of very deep research boreholes is the two 4- and 9.1-km-deep wells of the Continental Deep Drilling Program of Germany, in short KTB (Fig. 1). These wells have been drilled not only to understand the complex geological terrane structure of Central Europe and its related physical and thermoelastic properties but also to utilize them as windows into the deep underground for monitoring purposes. After 25 years of operating the KTB Depth Laboratory (KTB-DL) with still ongoing monitoring and testing activities, an assessment of possibilities and challenges for utilizing boreholes for Earth science is timely.

Scientific Background The KTB project is based on a German Earth science community effort that started in 1978. Embedded in the framework of the International Lithosphere Program, research was focused on the continental crust through deep geophysical sounding and direct access through deep drilling. The German Federal Ministry for Research and Technology granted funding for KTB in 1982 with the objective to create a window into the upper crust of Central Europe. Research started with a 2-year preparatory phase including comparative geoscientific studies at five locations in West Germany and project studies for a deep well drilling program. Data gained and geological hypothesis advanced in this phase then resulted in a decision for a more detailed site survey at two locations, namely, the Upper Palatine (Oberpfalz) and the Black Forest (Schwarzwald). Studies at both locations, including drilling of several shallow boreholes, were executed and compiled from 1984 to 1986 and finally discussed in 1986, leading to a decision for the Upper Palatine site near the town of Windischeschenbach, Eastern Bavaria, SE Germany.

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KTB Depth Laboratory: A Window into the Upper Crust

KTB Depth Laboratory: A Window into the Upper Crust, Fig. 1 Aerial view of the KTB-DL site with the KTB-VB pilot hole (upper left corner), KTB-HB main hole below the drill rig (foreground),

the former field lab in the main building behind the rig. The logging installations are within the orange circles

The first operational phase lasted from 1987 to 1990 with drilling of the 4,000.1-m-deep KTB-VB pilot hole, followed by a 1-year long phase to conduct measurements, testing, and evaluation, resulting in research concepts and a detailed engineering and management plan for the KTB-HB main borehole. Following this multiphase approach, the ultradeep KTB-HB (9,101 m) was then drilled from 1990 to 1994. Drilling, coring, and downhole logging of the deep KTBHB (Figs. 2 and 3) required extensive and detailed technical planning and key technology developments. The two most outstanding advances were a fully automated drill pipe handling system reducing trip times about half and the vertical drilling systems providing an almost vertical borehole path to drastically minimize drill string friction. Both technologies, pipe handler and steering systems, have since become standards in deep drilling (Engeser 1996). Ultradeep borehole measuring necessitated a wire line downhole logging system capable of safely handling depths beyond 8 km. This was achieved by installing a cable-friendly custom-made friction winch between the drum winch and the borehole and by running deep logs in tapered cable mode: a 1–2 km long, lightweight, HT temperature-resistant cable at

bottom and the long, heavy, and strong cable above (Kück et al. 2020). The finalization phase in 1995 comprised the dismantling of most components of the drilling facilities, except for the logging equipment and the 80-m-high drill rig itself. Data evaluation, documentation, publication, and administrative phaseout completed this phase. In the following years, the two fully accessible, cased, and cemented deep wells were prepared as a deep crustal laboratory (KTB-DL) with concrete pads around the wells and a complete logging infrastructure including office and workshop space (Fig. 1). From 1996 on, the two KTB wells with logging and testing infrastructure were operated by the GFZ German Research Centre for Geosciences as depth laboratory. During the first 10 years, more than 250 wire line operations were performed at the KTB-DL, resulting in more than 1,000 km of logging runs. The establishment of a KTB-DL was an indispensable part of the overall KTB concept from the very beginning, because only long-term observations (months to years) allow for investigations of temporally varying processes under natural in situ conditions years after all drilling related disturbances have decayed. Such a borehole lab enables carrying out even

KTB Depth Laboratory: A Window into the Upper Crust

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KTB Depth Laboratory: A Window into the Upper Crust, Fig. 2 Overview of the main activities in the KTB-DL since its start in 1996; blue ¼ scientific experiments, green ¼ technical activities,

insert: drilling progress chart of the KTB main hole showing delays during technical challenges below 6760 m depth

very long experiments at far lower costs compared to the very high daily costs of an active drilling. The key scientific objectives of the KTB as defined already in 1983 (Emmermann and Lauterjung 1997) were:

involved. The overarching goal was to conduct basic geoscientific research on the physical and chemical conditions in the deeper continental crust and the acting processes aiming to provide new insight into the structure, the dynamics, and the evolution of an intracontinental crustal area. The different research fields directed their efforts to the verification of fundamental concepts in Earth sciences. One key example was the transition from brittle to ductile deformation in a temperature range from 250 °C to 300 °C where a fundamental change in rock parameters was expected to occur (Brudy et al. 1997). Others such as the proof of several kilometer-wide networks of permeable fractures in the deeper crust and the related temperature profile and heat flow are shown in Emmermann and Lauterjung (1997) and detailed references therein.

1. To improve the understanding of the geological structure and evolution of the Central European Earth’s crust 2. To study, sample, and monitor the geophysical structures and heterogeneities at depth and to compare them with results from surface investigations 3. To shed new light on the fluid system at depth, its pathways, and its influence on geological processes such as metamorphism 4. To determine the upper crustal temperatures, the geothermal gradient, and the crustal heat flow 5. To obtain data on stress and strain in the upper crust and better understand earthquake nucleation and propagation 6. To establish a “deep crustal laboratory” and allow for longterm observations at depth These targets were complemented by a large number of specific science goals as defined during the course of this long-lasting project by the various Earth science disciplines

KTB Data Infrastructure The utilization of the KTB boreholes as deep laboratory is based on the complex digital data record available for the site. Cores, cuttings, and fluids were regularly sampled in short intervals and are mostly still available to interested science parties.

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KTB Depth Laboratory: A Window into the Upper Crust

KTB Depth Laboratory: A Window into the Upper Crust, Fig. 3 Drilling and casing scheme of the KTB boreholes showing the current KTB Depth Laboratory layout

Sample materials were investigated in great detail for a wealth of critical parameters in mineralogy, geochemistry, petrophysics, and geomechanics. These data and complementing downhole logging data, borehole tests, and surface geophysics including deep geophysical sounding and their joint

interpretation have been published and are digitally available for download. The data set was mainly acquired during the drilling phase in the KTB field laboratory (Emmermann and Lauterjung 1997). KTB samples and data served and still serve to publish a large number of scientific papers. This wealth of

KTB Depth Laboratory: A Window into the Upper Crust

information available makes the cuboid around KTB with its size of about 10 km depth times several kilometers lateral extension one of the best explored crystalline basement blocks on Earth. The complete digital data sets including metadata and the related links to all published KTB reports are available from Kück et al. (2020).

KTB Depth Laboratory Utilization The KTB-DL layout with two deep wells at just 200 m distance is shown in detail in Fig. 3, while the supporting surface infrastructure surrounding of the boreholes (downhole logging tools, winches, cables, workshops, offices, etc.) is indicated in Fig. 1. The main well is still accessible down to 6,700 m with c. 200 mm diameter instruments (Tmax ¼ 190 °C and pmax ¼ 66 MPa), while the pilot well is still fully accessible down to 3,970 m with 90 mm diameter devices (at Tmax/pmax of 115 °C/20 MPa). This setup allowed for experiments and tests over more than two decades. Over the first 5 years, more than 200 weeks of experiments were performed in the wells, while over the last 5 years (2013–2018), about 60 weeks of operations happened on site. In summary all post-drilling activities at the KTB-DL can be ordered into four categories: Geoscientific Experiments During the first KTB-DL phase from 1996 to 2006, the KTB holes were intensely used with an uninterrupted sequence of scientific experiments that were planned ahead already during the KTB drilling time. Experiments carried out investigated eight main research areas: deep temperature gradient, gravity field (deepest gravity meter survey), tidal effects (frac breathing at 4 km) (Schulze et al. 2000), deep magnetic survey through steel casing, seismic imaging in crystalline rocks/ deepest VSP/MSP (Rabbel et al. 2004), seismology (seismometer at 4 km for almost 2 years), rheology of the Earth’s crust/fracture propagation from brittle to ductile below 9 km (Bohnhoff et al. 2004), and fluid transport. The duration of these experiments varied between a few weeks and 4 years. More than 200 scientists and technicians from 22 institutes and companies worked at the KTB-DL and the surrounding area during this time. The longest experiment lasting from 2002 to 2005 was a pumping and injection test, each part one and a half year long with the aim to investigate the fracture network intersected by the pilot hole at c. 3,995 m and the natural rock fluids therein. An amount of 23,000 m3 was produced from the well, and later almost 85,000 m3 of freshwater was injected into the crystalline rock (Fig. 2). Key results such as the far-reaching fluid pathways at depth with high fracture permeabilities have been published in a special volume of Geofluids (Erzinger and Stober 2005) and by Kümpel (2006).

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Test and Calibration of New and Existing Instruments, Methods, and Installations Since 2006 the boreholes are used mainly by GFZ and other research institutes for verification and testing of new and experimental instruments and methods as well as for technical works such as tool testing and equipment preparation before and after campaigns elsewhere. In the framework of GFZ’s and other research institutes’ downhole tool development programs and the ongoing utilizations of geophysical borehole sondes, the wells are regularly used for performance tests and to calibrate instruments, e.g., XRF-density/elements sonde, memory logging-while-tripping sondes, borehole heat exchange reactor, and active seismic prediction sonde. Future experiments will target at fracture-fluid systems in deeper regions (>6,000 m) and the deep biosphere in the formation fluid filled and since many years undisturbed 4-km-deep pilot hole with an open hole temperature at 110 °C. Commercial Utilization of the KTB Wells Aside from the academic utilization of the KTB-DL, the holes are also used by a number of companies to test or verify new downhole methods, tools, cables, and winches under in situ conditions (temperature, pressure, in-hole cable length). Logging service companies frequently use the deep well to season new logging cables (reduce cable torque) before using them in actual logging jobs. Outreach and Education Based on the public outreach facility installed during drilling, a vibrant environmental and geoscientific education center with an on-site museum, the “GEO-Zentrum an der KTB,” has been established at the KTB site since 1998. It is running continuously with communal and local state support as well as European Union funding. The number of visitors as of 2018 is in the range of 25,000 per year. The center serves as KTB museum with the KTB drill rig as landmark and focuses in addition on Earth and environmental education for teachers and school classes. It is also used for academic education such as for courses in the framework of the International Continental Scientific Drilling Program (ICDP). The access to the KTB wells with conditions of utilization has been detailed by Harms and Kück (2016) at http://www. gfz-potsdam.de/en/section/geomechanics-and-rheology/infra structure/ktb-deep-crustal-lab/.

Long-Term Utilization of a Deep Borehole Site The majority of scientific drilling projects worldwide as well as commercial drillings have strict legal obligations concerning well completion and abandonment. This commonly requires closing the borehole safely by cutting casings, cementing the well, and other procedures to ensure hydraulic and gas sealing of

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intersected reservoirs. For long-term use of boreholes, it is therefore necessary that a company or institute has to take responsibility for safety and final borehole closure and to commit to its enduring mining law responsibility. In the case of the KTB-DL the GFZ German Research Centre for Geosciences is legally responsible. A crucial lifetime determining factor of borehole observatories is the long-term accessibility of the borehole, both logistically at surface and inside the well due to borehole wall instability, together with the availability of funds to work over collapsed hole sections. Only few deep drill holes stay stable over many years, and even a steel casing is in danger to be damaged and deformed by failing rock in a strong triaxial stress field. In general, the deeper the hole is, the higher is the risk of collapse which has to be expected. Unsurpassed until today is the Kola Superdeep SG-3 in NW Russia with more than 12 km depth which had an observatory phase after the end of the drilling operations since 1994. It offered 8,500 m of the mainly cased well (Popov et al. 1999) also for international utilization. Despite highly qualified and experienced experts, the difficult and uncertain conditions after end of the Soviet Union led to an insignificant utilization and withdrawal of governmental support. The facilities were abandoned in the year 2008, and the infrastructure deteriorated soon after. Fortunately, other borehole laboratories have derived from scientific drilling projects in various geological environments because the drilled rocks proved to be stable and bearing no special risks. Organizations like geological surveys, research institutes, or universities committed to remain responsible, to maintain sites, and to offer access to wells for protracted periods. Examples are the 2,500-m-deep Outokumpu borehole laboratory OKU-1 in Finland since 2006, the 2,500-mdeep COSC-1 underground lab near Åre in Sweden since 2014, and the 3,000-m-deep KFD1 hole of the Koyna drilling project in India since 2017, all supported by the ICDP. More depth laboratories can be expected in the coming years because long-term monitoring has become an inherent component of deep scientific drilling projects.

Summary The KTB-DL after 25 years in operation continues to offer deep access to the upper crystalline crust. The infrastructure available and the stable conditions of the vertical 9191-mdeep KTB main hole (accessible to 6,700 m) and the 4000-mdeep KTB pilot hole (open hole section from 3,850 to 3,970 m) at only 200 m lateral distance invite for both ultradeep geoscientific experiments and for technical use as test bed. The KTB-DL was used in the first 10 years after drilling primarily for further scientific investigations of fluids and seismicity while tool development and testing purposes prevail in the experiments in the recent past. Multi-purpose

KTB Depth Laboratory: A Window into the Upper Crust

utilization including science, commercial use and education and outreach as well as stable funding conditions have ensured 25 years of lifetime of the KTB wells. New depth laboratories and observatories have emerged in the past 10 years and more can be expected from upcoming deep scientific drilling projects.

Cross-References ▶ Borehole Seismic Networks and Arrays ▶ Deep Scientific Drilling ▶ Seismic Monitoring of Nuclear Explosions ▶ TOPO-EUROPE: From the Deep Earth to the Surface of Continental Europe and Its Margins

Bibliography Bohnhoff M, Baisch S, Harjes H-P (2004) Fault mechanisms of induced seismicity at the superdeep German Continental Deep Drilling Program (KTB) borehole and their relation to fault structure and stress field. J Geophys Res 109:B02309. https://doi.org/10.1029/ 2003JB002528 Brudy M, Zoback MD, Fuchs K, Rummel F, Baumgärtner J (1997) Estimation of the complete stress tensor to 8 km depth in the KTB scientific drill holes: implication for crustal strength granites. J Geophys Res 102:18453–18475 Emmermann R, Lauterjung J (1997) The German Continental Deep Drilling Program KTB: overview and major results. J Geophys Res 102:B818,179–B818,201. https://doi.org/10.1029/96JB03945 Engeser B (ed) (1996) KTB report 95-3. Das Kontinentale Tiefbohrprogramm der Bundesrepublik Deutschland, KTB. Bohrtechnische Dokumentation, the continental deep drilling project of the Federal Republic of Germany. Documentation of the used drilling technology, Niedersächsisches Landesamt für Bodenforschung. ISBN 978-3-928-55916-4 Erzinger J, Stober I (2005) Introduction to special issue: long-term fluid production in the KTB pilot hole, Germany. Geofluids 5:1–66. https://doi.org/10.1111/j.1468-8123.2004.00107.x Harms U, Kück J (2016) Superdeep tests and experiments at 9.1 km and 4 km. J Large Scale Res Facil 2:A75. https://doi.org/10.17815/jlsrf-2-132 Kück J, Conze R, Harms U (2020) Data from the German Continental Deep Drilling Project (KTB, Kontinentale Tiefbohrung). GFZ Data Services. https://doi.org/10.5880/GFZ.KTB.top Kümpel H-J, Erzinger J, Shapiro S (2006) Two massive hydraulic tests completed in deep KTB pilot hole. Sci Drill 3:40–44. https://doi.org/ 10.2204/iodp.sd.3.05.2006 Popov Y, Pevzner S, Pimenova V, Romushkevicha R (1999) New geothermal data from the Kola superdeep well SG-3. Tectonophysics 306:345–366. https://doi.org/10.1016/S0040-1951(99)00065-7 Rabbel W, Beilecke T, Bohlen T, Fischer D, Frank A, Hasenclever J, Borm G (2004) Superdeep vertical seismic profiling at the KTB deep drill hole (Germany): seismic close-up view of a major thrust zone down to 8.5 km depth. J Geophys Res 109(Issue B9). https://doi.org/ 10.1029/2004JB002975 Schulze K, Kümpel H-J, Huenges E (2000) In-situ petrohydraulic parameters from tidal and barometric analysis of fluid level variations in deep wells: some results from KTB. In: Stober I, Bucher K (eds) Hydrogeology of crystalline rocks. Water science and technology library, vol 34. Kluwer, Rotterdam, pp 79–104. https://doi.org/10. 1007/978-94-017-1816-5_4

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Legal Continental Shelf: Geology, Geophysics, and Tectonics Elana Geddis1, Ray Wood2 and Vaughan Stagpoole3 1 Harbour Chambers, Wellington, New Zealand 2 CRP-OCS Ltd, Haumoana, New Zealand 3 GNS Science, Lower Hutt, New Zealand

Synonyms Continental margin; Continental shelf; Extended continental margin; Extended continental shelf

Definition The legal continental shelf comprises the submerged prolongation of the land mass of a State beyond 200 nautical miles (M) from the territorial sea baselines, consisting of the seabed and subsoil of the shelf, the slope, and the rise. Formulae to determine the outer limits of the legal continental shelf are in Article 76 of the United Nations Convention on the Law of the Sea (UNCLOS). Application of the Article 76 definition frequently turns on an understanding of the geology, geophysical structure, and tectonic evolution of the sea floor.

Introduction A State has exclusive rights over the seabed resources of its legal continental shelf as the natural prolongation of its land territory. The assertion of those rights is an inherent entitlement, not a claim. In 1982 UNCLOS imposed a legal framework on the scientific concept of the continental shelf, creating a clear definition of the outer limits of the legal continental shelf © Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

and a process for international verification of those limits. These are contained in Article 76 of UNCLOS and Annex II of the Final Act of the Third United Nations Conference on the Law of the Sea (the Statement of Understanding). Under Article 76 the continental shelf is defined as extending to the outer edge of the continental margin, subject to two constraints. Article 76 was drafted to deal with the vast majority of continental margins, but it does not adequately deal with margins characterized by a narrow and steep morphological shelf and slope and an extraordinarily extensive rise. The Statement of Understanding extends the principles of Article 76 to those margins. Where the outer limits of the continental shelf extend beyond 200 M from the coast, a State must make a submission to the UN Commission on the Limits of the Continental Shelf (CLCS) to verify that the outer limits have been calculated correctly. The recommendations issued by the CLCS in response to States’ submissions address the real world complexities of continental margins and continue to refine the understanding of the legal continental shelf and the scientific concepts that underpin it.

Defining the Outer Edge of the Continental Margin and the Limits of the Continental Shelf Article 76 defines the continental margin to include the shelf, the slope, and the rise as the “submerged prolongation of the land mass” distinct from the “deep ocean floor.” The outer edge of the continental margin is determined through the application of two formulae, both calculated from the foot of the continental slope (Fig. 1). The outer edge of the margin lies at points 60 M from the foot of the slope or where the thickness of sediment is 1% of the distance from the foot of the slope. In the absence of evidence to the contrary, the foot of the slope is identified at the point of maximum change in seafloor gradient.

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Legal Continental Shelf: Geology, Geophysics, and Tectonics, Fig. 1 Cartoon showing a hypothetical continental margin, summarizing the formulae and constraints described in Article 76 that determine the outer limits of the continental shelf. (Modified from Kapoor and Kerr 1986)

Article 76 also constrains the legal continental shelf to no more than 350 M from the coast or 100 M from the 2500 m isobath, whichever is greatest. On margins with expansive, relatively shallow slopes the 2,500 m + 100 M constraint can lie seaward of the 350 M line. Significantly, the 2,500 m + 100 M constraint applies to “submarine elevations that are natural components of the margin” but does not apply to “submarine ridges” (limiting these to 350 M). The distinction between “submarine elevations” and “submarine ridges” remains both conceptually and analytically difficult. Although the CLCS Technical Guidelines (1999) reveal general principles that are considered, each feature must be assessed individually in terms of its morphological, geological, and tectonic continuity (Brekke and Symonds 2011). Applying the Article 76 definition requires four steps (Fig. 2): (1) the outer edge of the continental margin is identified using the two formulae; (2) the two constraint lines are determined; (3) the most seaward constraint line is applied; and (4) the resulting line composed of formulae and/or constraint lines is connected by lines up to 60 M long (Wood et al. 2011). These steps are described in the CLCS Technical Guidelines (1999). Margins with a narrow morphological shelf, steep slope, and a very extensive rise may meet the criteria under the

Statement of Understanding. The Statement of Understanding has four elements that identify continental margins to which it applies. For States that meet its criteria, the Statement of Understanding, in conjunction with the CLCS Technical Guidelines (1999), defines the outer limits of the continental shelf at points where the sediment thickness is at least 1 km.

Prolongation of the Land Mass The concept of the submerged (or natural) prolongation of the land mass is central to the definition of the legal continental shelf. Establishing where prolongation occurs and the nature of that prolongation are of primary importance because they determine (a) whether a legal continental shelf exists and (b) how far it extends beyond 200 M (Wood et al. 2011). Natural prolongation has three aspects (Wood et al. 2011): • Morphologic – the seafloor shape is a continuation of the land mass morphology. • Geologic – the rocks beneath the seafloor are the same as, or related to, those of the land mass. • Tectonic – the rocks beneath the seafloor share their history with those of the land mass.

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Legal Continental Shelf: Geology, Geophysics, and Tectonics, Fig. 2 The four steps for establishing the outer limits of the continental shelf. (From Wood et al. 2011)

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The transition from land mass to deep ocean floor is the result of the composition and density of the rocks beneath the seafloor, the geological processes that form and shape them, and the tectonic forces that act on them (Wood et al. 2011). The morphological transition is relatively simple on many margins, with a shelf dipping gently to a depth of several 100 m, a steeper slope deepening to a depth of several 1,000 m, and a gently dipping rise lying between the slope and the flat deep ocean floor (Fig. 1). Other margins have a complex morphology with ridges, seamounts, and canyons, and the transition with the deep ocean floor can be harder to recognize (Fig. 3).

Role of Geology and Tectonics The Article 76 definition divides the Earth into land territory, the continental margin, and the deep ocean floor with its oceanic ridges. Applying that definition to a particular margin often requires complex scientific analysis. On simple margins the shelf, the slope, and the rise may be easily identified from the seafloor morphology. The primacy of seafloor morphology is reinforced by both the CLCS Technical Guidelines (1999) and subsequent CLCS recommendations. But frequently geology, geophysics, and tectonics will be important in interpreting that morphology to:

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Legal Continental Shelf: Geology, Geophysics, and Tectonics, Fig. 3 (a) Foot of slope positions along Reykjanes Ridge determined from analysis of (b) multibeam data and regional tectonics. (From CLCS 2016)

• Distinguish between areas that are part of the continental margin and those that are part of the deep ocean floor • Distinguish between “submarine ridges” and “submarine elevations that are natural components of the margin” • Clarify the location of the foot of the continental slope The role of geology and tectonics in the application of Article 76 is demonstrated by examples from the recommendations on the submissions by Iceland, Ascension Island (the United Kingdom), Bouvetøya and Shaka Ridge (Norway), Ogasawara Composite High (Japan), and the Kerguelan Plateau (Australia). Iceland is the product of voluminous volcanism related to the interaction of a mantle hotspot with a seafloor spreading ridge. Because its origin is similar to that of the volcanism that forms the deep ocean floor, geologically it is difficult to identify the boundary of the continental margin. In its recommendations the CLCS concluded that hotspot-ridge interaction had significantly changed the seafloor spreading process and the morphology of parts of the Reykjanes Ridge (CLCS 2016). It therefore considered that the region defined by the Iceland hotspot interaction with the seafloor spreading process is part of the continental margin of Iceland for the purposes of Article 76. Even given this criterion for identifying the outer limits of the continental margin, finding those limits is not

straightforward. Figure 3a shows the foot of slope points along the Reykjanes Ridge that were accepted by the CLCS. It is apparent that their choice is not based entirely on morphology but also on identification of the V-shaped seafloor region formed by hotspot activity (Fig. 3b). Significantly, the CLCS could not agree on whether the continental margin is a “submarine ridge” or a “submarine elevation” for the purposes of Article 76. A similar argument was used by the United Kingdom for prolongation from Ascension Island. The United Kingdom submitted that Ascension Island is an integral component of the Mid-Atlantic Ridge and its natural prolongation therefore extends along the ridge. The CLCS agreed that islands surmounting ridges rising from the deep ocean floor are entitled to a legal continental shelf. But it determined that the rugged seafloor surrounding the Ascension Island volcanic edifice is part of the deep ocean floor and there is no morphological feature in the vicinity of Ascension Island that could be characterized as continental margin extending beyond 200 M (CLCS 2010). In summary, Iceland was able to demonstrate that the volcanic processes that formed the land mass affected areas of the surrounding seafloor so that they are morphologically and geologically distinct from the adjacent deep ocean floor. Iceland could therefore show the continental margin extended beyond 200 M. Ascension Island, on the other hand, could not

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identify a geological distinction between the volcanic edifice on which the island sits and the Mid-Atlantic Ridge. Similar to Ascension Island, Bouvetøya is a volcanic island on the Southwest Indian Ridge in the South Atlantic, a seafloor spreading ridge. Bouvetøya and the elevated area of seafloor extending northeast from it to the eastern end of Shaka Ridge are interpreted to have formed by hotspot related volcanism. On the basis of seafloor morphology and the geochemistry of the basement rocks, the CLCS agreed that there was natural prolongation of Bouvetøya to the eastern end of Shaka Ridge (CLCS 2019). In addition, the CLCS relied on geochemical analysis to conclude that the Shaka Ridge is a “submarine elevation” that forms a natural part of the Bouvetøya margin, distinct from the oceanic basalts of the deep ocean floor of the Southwest Indian Ridge. Geology and tectonics also played an interesting role in determining the outer limits of Japan’s continental shelf around the Ogasawara Composite High, on the eastern flank of the IzuOgasawara Arc southeast of the islands of Japan. The Ogasawara Composite High is composed of several east-west trending seamounts, and straddles an active subduction trench and volcanic arc. The presence of an active subduction trench could be considered a fundamental obstacle to establishing prolongation. But the CLCS identified an elevated morphological saddle across the subduction trench, which it considered sufficient to demonstrate prolongation to the high (CLCS 2012). This was reinforced by its acceptance that, based on the presence of thrust faulting, the western part of the high had been accreted to the Izu-Ogasawara Arc, and thus is a “submarine elevation” forming a natural part of the Japanese margin. The eastern part of the high, by contrast, had not been accreted and was concluded by the CLCS to be a “submarine ridge.” In Australia’s continental shelf submission on the Kerguelan Plateau region, geology was used to distinguish between elements of the plateau region which are “submarine elevations” and those which are “submarine ridges.” Heard and MacDonald Islands are embedded within the late Cretaceous magmatic crust of the Central Kerguelan Plateau. These basement rocks and those of the morphologically connected Southern Kerguelan Plateau show affinity with continental crust based on geochemical evidence. The CLCS therefore agreed that the Central and Southern Kerguelan Plateau regions are “submarine elevations” and natural components of the continental margin of Heard and MacDonald Islands (CLCS 2008). However, the Commission decided that there is insufficient data to establish the geological origin of Williams Ridge, a ridge extending southeast from the Central Kerguelan Plateau. Therefore, even though it is morphologically part of the margin, the CLCS considered it is a “submarine ridge,” not a natural component of the continental margin.

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Summary The definition of the legal continental shelf in Article 76 enables States to define the outer limits of the continental shelf with geographic precision. The elements of the definition, and the discipline imposed by the requirement to submit outer limits to the CLCS for review, have required States to study the geology, geophysical structure, and tectonic evolution of the continental margin in new detail. This is particularly important for complex margins where the shelf, slope, and the rise cannot be easily identified from the seafloor morphology, and there is uncertainty in distinguishing the transition from land mass to deep ocean floor. The examples from Iceland, Ascension Island, Bouvetøya and Shaka Ridge, the Ogasawara Composite High, and the Kerguelan Plateau show the value of tectonic and geological evidence when applying the Article 76 definition. Although seafloor morphology remains the primary evidence, there is in many cases a proven need to consider a range of geological, geochemical, and geophysical evidence to determine the outer limits of the legal continental shelf.

Bibliography Brekke H, Symonds P (2011) Submarine ridges and elevations of Article 76 in light of published summaries of recommendations of the Commission on the Limits of the Continental Shelf. J Ocean Dev Int Law 42:289–306 Commission on the Limits of the Continental Shelf (1999) Scientific and technical guidelines of the Commission on the Limits of the Continental Shelf (CLCS/11). https://www.un.org/Depts/los/clcs_new/ commission_documents.htm. Accessed 31 July 2019 Commission on the Limits of the Continental Shelf (2008) Recommendations of the Commission on the Limits of the Continental Shelf (CLCS) in regard to the submission made by Australia on 15 November 2004. https://www.un.org/Depts/los/clcs_new/submissions_files/ submission_aus.htm. Accessed 31 July 2019 Commission on the Limits of the Continental Shelf (2010) Summary of recommendations of the Commission on the Limits of the Continental Shelf in regard to the submission made by the United Kingdom of Great Britain and Northern Ireland in respect of Ascension Island on 9 May 2008. https://www.un.org/Depts/los/clcs_new/submissions_ files/submission_gbr.htm. Accessed 31 July 2019 Commission on the Limits of the Continental Shelf (2012) Summary of recommendations of the Commission on the Limits of the Continental Shelf in regard to the submission made by Japan on 12 November 2008. https://www.un.org/Depts/los/clcs_new/submissions_files/sub mission_jpn.htm. Accessed 31 July 2019 Commission on the Limits of the Continental Shelf (2016) Summary of recommendations of the Commission on the Limits of the Continental Shelf in regard to the submission made by Iceland in the Ægir Basin area and in the western and southern parts of Reykjanes Ridge on 29 April 2009. https://www.un.org/Depts/los/clcs_new/submis sions_files/submission_isl_27_2009.htm. Accessed 31 July 2019 Commission on the Limits of the Continental Shelf (2019) Summary of recommendations of the Commission on the Limits of the Continental Shelf in regard to the submission made by Norway in respect of Bouvetøya and Dronning Maud Land on 4 May 2009. https://www.

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un.org/Depts/los/clcs_new/submissions_files/submission_nor_30_ 2009.htm. Accessed 31 July 2019 Kapoor DC, Kerr AJ (1986) A guide to maritime boundary delimitation. Carswell, Toronto, p 123 Wood R, Henrys S, Stagpoole V, Davy B, Wright I (2011) Legal continental shelf. In: Encyclopedia of solid Earth geophysics, 1st edn. Springer, Heidelberg

Lithosphere, Continental David E. James Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, DC, USA

Definition and Introduction The continental lithosphere consists of the continental crust and, typically, some nonconvecting part of the underlying upper mantle (Fig. 1). In plate tectonics terms, the continental lithosphere is part of the rigid outer rind of the Earth, which is segmented into several major plates. The cold lithosphere lies atop a hotter, more mobile (low strength) asthenosphere. In this context, oceanic lithosphere for the most part obeys comparatively simple thermal models, with the boundary

Ocean basin (125 Million years)

Mobile belt (1–2 Billion years) Kimberlite

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Stable craton (3 Billion years) Eruptions

Cru Ma st ntle Ancient keel

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e phit Gra nd mo Dia

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Lithosphere, Continental, Fig. 1 Schematic cross section showing transition from oceanic lithosphere to younger continental lithosphere and into the thick, cratonic lithosphere (tectosphere) beneath the ancient continental nucleii. Melts rising from depth ascent through relatively thin oceanic and young continental lithosphere to produce volcanoes at the surface, whereas thick tectosphere acts as a barrier to melts percolating from depth, requiring evolved, volatile-rich kimberlitic magmas to bore a conduit through the mantle to the surface

between lithosphere and asthenosphere represented by an isotherm marking the transition in mantle peridotite between elastic and ductile behavior (but see ▶ “Lithosphere, Oceanic” for the growing list of complications). By contrast, the continental lithosphere is heterogeneous and its structure highly variable. That contrast reflects the fact that oceanic lithosphere is formed in much the same way worldwide and that it is continually recycled into the interior of the Earth via the plate tectonic “conveyer belt.” Continental lithosphere, on the other hand, is generally too buoyant to be subducted, although an increasing number of instances have been reported in which parts of continental lithospheric mantle show evidence of gravitational instability, forming “drips” where lithosphere delaminates and sinks into the deeper mantle (Zandt et al. 2004). The detachment of continental mantle from the crust in such circumstances may be an important factor in intraplate deformation and volcanism (Carlson et al. 2005). Thus, while no known oceanic lithosphere is older than about 200 my, the continental lithosphere has evolved over billions of years, one consequence of which is that it is seen today as the resulting collage of many superimposed tectonic events, including rifting, subduction, continental collision, accretion, and hot spot magmatism, that recur repeatedly over geologic time. The study of continental lithosphere has been revolutionized over the past two decades with the advent of portable

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broadband seismology. Passive array experiments (see ▶ “Seismological Networks”) have made it possible to obtain high-resolution images of deep lithospheric structures beneath every continent (James 2007). The greatest concentration of high-density broadband deployments has been in western North America, Europe, and Asia, with large-scale experiments also in Africa, Central and South America, and Australia. Within the USA, the Transportable Array of the national EarthScope project has already resulted (at the time of this writing) in uniformly high-resolution seismic imaging of the mantle beneath the western half of the nation. The lithosphere so revealed consists of a collage of overlapping structures produced by ongoing subduction, trench migration, ridge override, regional extension, mantle upwellings and volcanism, and lithospheric delamination. In some tectonically active regions, a thin crust is the only lithosphere remaining intact (Carlson et al. 2005). This complex and disrupted lithosphere is in marked contrast to that of stable cratonic regions, including the continental nuclei of northcentral North America, western Australia, and southern Africa. The cratons of southern Africa are perhaps the best studied of the Archean cratons of the world, where results from large-scale passive array experiments show that lithospheric roots as deep as 250 km or more underlie the undisturbed parts of the cratons (James and Fouch 2002; Ritsema and van Heijst 2000). Despite remarkable progress in imaging the continental lithosphere over the past 2 decades, a precise definition of continental lithosphere remains elusive. No single definition of continental lithosphere can be made to fit all circumstances. Particularly controversial is the definition of what constitutes the base of the lithosphere. Many definitions of continental lithosphere appear in the literature – seismological, mechanical, rheological, thermal, and compositional. Depending on one’s view, the continental lithosphere may be a mechanical boundary layer, a thermal boundary layer, or a chemical boundary layer. In the remainder of this entry, we shall explore these various concepts in some detail. Our goal here is to describe the continental lithosphere in terms of global Earth structure, plate tectonics, and the long-term evolution of the continents. By this view, the continental lithosphere is envisaged to be a long-lived plate tectonic unit, consisting both of continental crust and, typically, a very significant mantle “keel” that is attached to and translates with the continent. In this sense, it is seen as a stable component of the continent and does not participate in the convective processes of the deeper dynamic mantle.

Mechanical Models A simple expression of the lithosphere is elastic thickness. While rarely used to describe continental lithospheric in a

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geologic or structural sense, elastic thickness is a useful concept in that it models the mechanical response of the lithosphere as an elastic plate overlying a weak (asthenospheric) substrate. As discussed in detail elsewhere in this volume (see ▶ “Lithosphere, Mechanical Properties”), the elastic thickness of the lithosphere may be determined through topographic response to surface loads (such as volcanoes), the relationship between topography and Bouguer gravity anomalies, or the shape of depositional surfaces in basins. The thickness of elastic lithosphere so determined is rarely as much as 100 km and in areas of extension or hotspot activity may be less even than crustal thickness. The elastic thickness of the continents is chiefly controlled by the thermal state of the lithosphere (see ▶ “Lithosphere, Mechanical Properties”). As Forsyth (1989) points out, however, the apparent elastic thickness is also affected by the state of stress and the rheological stratification in the crust and mantle, so mechanical thickness may not bear a simple relationship to a particular isotherm. One important aspect of the mechanical structure of the continental lithosphere as it relates to lithospheric evolution is the large decrease in strength of crustal rocks in the deep crust. The steep geothermal gradient in the crust produces increasing ductility of crustal rocks with depth. The brittle zone in tectonic areas typically extends only to about 15 km, below which the rocks deform plastically. The topmost part of the underlying mantle, however, is relatively much stronger as temperatures there are still far from the melting point of mantle rocks. Thus, it is widely posited that there is a zone of weak lower crust sandwiched between strong upper crust and strong upper mantle, creating a possible zone of crust– mantle decoupling (see ▶ “Lithosphere, Mechanical Properties”). While the flexural lithosphere is a useful mechanical concept, it fails as a plate tectonic description of lithosphere. In many tectonic regions, the flexural lithosphere is not even as thick as the continental crust. As low-density sialic crust is not easily subducted into the mantle, it must persist as the primary long-lived component of continental lithosphere. From the perspective of plate tectonics, it may be argued that the full definition of the lithosphere should account both for the longterm structural coherence of continental plates and the impact of asthenospheric processes that affect either the formation or the removal of lithospheric mantle beneath the crust.

Thermal Models A convenient definition for thermal lithosphere, or the thermal boundary layer, is the outer layer of the Earth in which heat transfer is dominated by conduction (see ▶ “Heat Flow, Continental” and ▶ “Lithosphere, Continental: Thermal Structure”). Thus, the stronger lithosphere acts as a barrier

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to thermally induced buoyancy forces that drive convective heat transfer in the underlying asthenosphere (Morgan 1984). In the case of the oceanic lithosphere, this conductively cooled layer of the Earth is thought to have finite thickness (i.e., the lithosphere ceases to undergo thermal contraction as it ages beyond about 80 Ma, implying that heat is supplied to the lithosphere from the underlying asthenosphere) (see ▶ “Lithosphere, Oceanic” and ▶ “Lithosphere, Oceanic: Thermal Structure”). For the continents, the situation is far more complex. The continental lithosphere does not undergo simple monotonic cooling, but may be subjected to repeated episodes of thermal or tectonic disturbances. Thus, superimposed upon the geological age of the lithosphere is the thermal or tectonic age of the lithosphere, factors that have led to significant compositional heterogeneity in both crust and lithospheric mantle. We consider first the thickness of the thermal boundary layer beneath the stable cratonic cores of the continents. Here, the lithosphere is both at its thickest and its strongest. If the effective viscosity of the asthenosphere beneath the cratonic lithosphere is taken to be about 1021 Pa s, based upon glacial unloading and gravity/topography correlations (see ▶ “Mantle Viscosity”), it implies a temperature of transition from conductive to convective heat transfer of about 1300–1400 °C for typical mantle compositions (Morgan 1984). If crustal heat production is taken into account, the thickness of the thermal lithosphere of the stable continental cratons as calculated from surface heat flow may range from about 90 to 220 km (Rudnick et al. 1998). The lithospheric thickness obtained by thermal modeling depends upon other factors, the most important of which is the composition, including volatile content, of the subcrustal mantle. If the continental lithospheric mantle consists of low-density peridotite depleted of its low-melting basaltic fraction and devolatilized over time due to metamorphic and magmatic events (see discussion below), the continental mantle will be less dense and substantially more refractory than the rest of the mantle. The chemical boundary layer represented by this depleted subcontinental mantle peridotite may stabilize the continental mantle root against thermal disruption to depths below the thermal boundary layer (Jordan 1978, 1981). Tectono-thermal events, such as those that are widespread across the western USA, will have the effect of thinning the existing lithosphere. Regions where prior thermal disturbances have occurred, therefore, may be more prone to reactivation, whereas stable cratons, particularly those buffered by bordering Proterozoic mobile belts as in southern Africa, should be comparatively less vulnerable to thermal perturbations from the underlying asthenosphere (Pollack 1986). Tectonic reactivation seems to be particularly true of continental rifting, where repeated episodes apparently recur along the same long-lived zones of weakness. The greatly

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expanded scope of tomographic studies in recent years has revealed a number of localities in which various forms of lithospheric delaminations, or “drips” appear to be occurring, including a number in the western USA (e.g., West et al. 2009; Zandt et al. 2004). Such processes suggest that lithospheric mantle beneath tectonically active areas of the continent may be thinned or removed and later regenerated. Indeed, a wide range of epeirogenic (vertical) movements within continental interiors have been interpreted to be due to thermal disturbances that produce lithosphere heating and thinning, followed by conductive relaxation and asymptotic thickening of the lithospheric thermal boundary layer.

Seismological/Compositional Models: Tectosphere It is well known that average seismic velocities, especially shear velocity, are much higher under cratons than under oceanic or tectonically active areas. This was first shown by surface wave dispersion measurements and later by measurements of vertically travelling ScS phases. Sipkin and Jordan (1980) showed that ScS one-way travel-time anomalies associated with the upper mantle beneath stable continental interiors could be as large as 3.5 s (fast) relative to the upper mantle beneath the western Pacific. Moreover, the highvelocity continental paths are also associated with high Qs, suggesting that the anomalies are due to thermal and/or compositional variations in the upper mantle (Jordan 1981). Since the early studies of Sipkin and Jordan, numerous other studies based on three-dimensional inversion of travel-time anomalies for velocity structure have been carried out regionally across the globe (see summaries in Carlson et al. 2005; James 2007). The results nearly all show that regional velocity contrasts, even between provinces entirely within the continents, extend to at least 200–250 km (James 2007). The notion of a long-lived thermochemical boundary layer in the upper mantle beneath vast regions of Precambrian crust raised many difficult questions as to how they formed and how they survived over geologic time and to a significant extent these questions remain only partially answered. There is general agreement, however, that the longevity and stability of ancient cratonic roots must be due, at least in part, to a combination of factors, primarily compositional buoyancy and strength and secondarily buffering of cratons by marginal Proterozoic mobile belts. The concept of a stabilizing chemical boundary layer is at the heart of the temperaturedepletion compensation hypothesis laid out by Jordan in an important series of papers on the continental tectosphere (see Jordan (1981) for references) and discussed in some detail below. The term tectosphere was introduced by Jordan in an attempt to circumvent some of the difficulties associated with the conflicting definitions of lithosphere alluded to above. It is

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a particularly convenient concept for understanding those seismological and petrological/geochemical aspects of deep continental structure that appear to indicate continent-ocean heterogeneity to depths of at least 200 km, and perhaps considerably more. There is a large body of knowledge about the composition of deep cratonic mantle from the study of xenoliths that have been carried to the surface in explosive volcanic pipes. The development of accurate mineralogical geothermometers and geobarometers, which allow the equilibrium depth and temperature of xenolith source regions to be determined, made it possible to “map” xenolith compositions as functions of position in the mantle. A comprehensive summary of results from the geochemical study of mantle xenoliths can be found in Carlson et al. (2005). Most of the ultramafic nodules associated with older cratonic areas are highly depleted in their basaltic melt fraction to a depth of at least 200 km and are approximately the same age as the overlying crust (Carlson et al. 2005; James et al. 2004). Depleted mantle xenoliths, which are by far the most common, are characterized by significantly lower density and higher seismic velocities than the more typical fertile suboceanic mantle (Boyd and McCallister 1976). This fact led Jordan (1981) to

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postulate a thick chemical boundary layer for the continental tectosphere. The effect of removing a basaltic partial melt from a fertile mantle peridotite (such as the hypothetical pyrolite of suboceanic mantle) is to reduce the density of the residue and to increase seismic velocities. Jordan compared densities of depleted peridotite nodules with densities of a hypothetical pyrolite composition and shows that average depleted peridotite of subcontinental mantle has a normative density about 1.3% less than that of pyrolite. This reduction in density will approximately compensate for the density contrast in a homogeneous mantle that would result from the temperature contrast of about
400° between a typical shield geotherm and the average oceanic adiabat. The model then, is based on the supposition that the lower density of the subcontinental mantle due to basalt depletion is gravitationally balanced by the density decrease in the oceanic mantle caused by a much higher geotherm, hence the expression temperature-depletion compensation (Fig. 2). The thickness of the chemical boundary layer is generally taken to be about 150–200 km. The tectosphere model remains controversial. The mechanisms by which a chemical boundary layer can be built up to form a deep mantle keel beneath ancient continental crust are

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Lithosphere, Continental, Fig. 2 Schematic diagram (adapted from Jordan, e.g., (Jordan 1988) with reference to previous work) showing the principal components of the isopycnic (“equal density”) hypothesis. The cold, strong lithosphere (A) extends to at least 250 km beneath the ancient cratons and migrates with the craton during plate motions. At comparable depths beneath the oceans (B), the mantle is hot, weak, and mobile. The depleted mantle peridotite that makes in the cratonic keel is

depleted in heavier constituents (e.g., Fe) and is therefore intrinsically less dense than more fertile oceanic mantle at the same temperature and pressure. At the same time, the sub-cratonic mantle is colder than the suboceanic mantle so the densities are approximately equal (the “temperature-depletion compensation” model). The chemical buoyancy of the cratonic root, along with its anhydrous strength, helps stabilize it against convective disruption over geologic time

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poorly understood, although there are many modes by which magma can be extracted from continental mantle over time (Carlson et al. 2005). The most difficult aspect of deep mantle roots to model, however, has been their remarkable dynamical stability over billions of years of a chemical boundary layer whose thickness varies laterally by 150 km or more (Shapiro et al. 1999). Convective instability should act to disperse and thin the zone of depleted mantle, and the fact that this does not occur appears to be due largely to the anhydrous, highly viscous, nature of the depleted mantle keel. One aspect of lithospheric formation that may be important in some cases involves underplating of continental lithosphere by oceanic lithosphere. There is evidence, for example, that among the eclogites brought up in kimberlite pipes in Africa some may be relicts of basaltic oceanic crust emplaced beneath the continental lithosphere in Precambrian time (MacGregor and Manton 1986). The data on which this conclusion is based come primarily from measurements of oxygen isotopic ratios combined with trace element and radiogenic isotopic ratios. The oxygen isotopic ratios (low @O18), in particular, are difficult to explain as originating anywhere other than in oceanic crust. Similar evidence for underplating of cratonic mantle by oceanic lithosphere has also been found in sulfide inclusions in eclogitic diamonds from southern Africa, where episodic diamond formation has been linked to the accretion of oceanic lithosphere onto early cratonic nuclei, with subsequent stabilization of the tectospheric mantle (Shirey et al. 2001).

Lithosphere–Asthenosphere Boundary A fundamental concept of plate tectonic theory is that the cold, rigid lithospheric plates of the Earth’s outer rind are decoupled from the hot, ductile underlying mantle of the convecting Earth at the lithosphere–asthenosphere boundary (LAB). The detailed nature of this global boundary, despite many years of intense study, is still very poorly known. Yet understanding the seismic and rheological character of the decoupling zone between lithosphere and asthenosphere is a critical element in understanding the dynamical interaction between the lithospheric plates moving across the surface of the globe and the convecting asthenospheric mantle beneath. While seismic studies of the LAB beneath the ocean basins show that in many cases – although by no means all – the LAB can plausibly be related to an isotherm or rheological interface, the LAB beneath continents is far more problematic, particularly beneath the stable continental interior, where the LAB has been remarkably difficult even to detect, and its specific nature remains largely unknown. Much of the information gleaned over the past 2 decades on the velocity structure and thickness of the continental lithosphere comes from seismic tomography, which images

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3-D variations in velocity distribution in the upper mantle. Where continental lithospheric mantle is present, it is seen as a higher velocity layer overlying a lower velocity asthenosphere. On the other hand, tomographic imaging, while an invaluable tool for 3-D mapping of velocity structure, provides little or no information as to the sharpness of the lithosphere– asthenosphere boundary or the velocity contrast across it. This is a significant limitation, as knowledge of the fine structure of the LAB is crucial to sorting out the competing effects of varying geothermal gradients and the presence or absence of hydrous/carbonate fluids or partial melts for decoupling the continental lithosphere from the mobile mantle beneath. Direct approaches to determining velocity structure at the base of the lithosphere generally involve investigating the small seismic signals produced either by the conversion or the reflection of teleseismic body waves incident on the underside of the lithosphere–asthenosphere discontinuity (e.g., Rychert et al. 2005; Sacks and Snoke 1977). A converted phase is generated when some fraction of the energy of a compressional/shear wave that is obliquely incident on a velocity discontinuity in the Earth is converted to a shear/compressional wave. One widely used method, receiver function analysis (see ▶ “Seismic, Receiver Function Technique”), can effectively isolate the small signals on seismic records that represent conversions from the LAB. In principle, the waveform and amplitude of converted phases can provide detailed information about the structure (sharpness) and the velocity contrast – including the sign of the velocity contrast – across a discontinuity. Similarly, underside reflections from the LAB, which are seen on seismic records as small precursors to the large amplitude SS or PP surface reflections from distant earthquakes can be used to determine the sharpness and velocity contrast across the LAB. Unlike conversions, however, underside reflections sample substantial areas of the LAB (typically ~ 1,000 km in the case of SS phases) and, therefore, measure average regional, rather than local, structure. Unlike most discontinuities in the Earth, the LAB represents a reversal in velocity with depth, the effect of which is to produce a converted or reflected signal with opposite polarity to that produced by a velocity increase. Results of receiver function studies of converted phases from the LAB have been decidedly mixed. An important study published in 2005 (Rychert et al. 2005) revealed a sharp LAB at a depth of about 100 km beneath northeastern USA (Fig. 3). The authors suggested on the basis of the magnitude of the negative shearwave velocity contrast across the LAB (~5.10%) and the sharpness of the velocity transition (~ 10 km or less) that the LAB cannot be produced by thermal or compositional contrast alone, but must represent a transition from solid (meltfree) lithosphere to a porous asthenosphere containing at least a few percent of partial melt or fluid phase. Thus, the LAB

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Lithosphere, Continental, Fig. 3 A schematic 3-D rendering of LAB conversion points beneath six permanent seismic stations (red triangles) in northeastern USA (From Rychert et al. 2005. With permission). Inset map at top shows location of 3-D rendering. Topography is shown as shaded relief on the upper 3-D panel. The lower surface represents the base of the lithosphere, which ranges in depth from 90 km (orange) to 110 km (pink). Blue circles on the LAB discontinuity surface indicate the conversion points of the Ps phases. Black lines connect piercing points to the station at which the conversion is observed (Rychert et al. 2005)

L could correspond to the solidus, or point of incipient melting, in the mantle beneath the continent, or it could mark the depth at which an impermeable barrier (the lithosphere) blocks the upward migration of fluids or partial melts from deeper in the mantle. In either case, the LAB in the Rychert et al. study represents a remarkably sharp rheological boundary, implying an abrupt transition from rigid lithosphere to ductile asthenosphere. Subsequent studies employing similar methodologies to those of Rychert et al., however, have proven difficult to interpret (e.g., Rychert and Shearer 2009), in that LAB depths so measured, particularly those beneath the stable continental interior, are incompatible either with tomographic images or with expected lithospheric thickness based on xenolith data or heat flow analysis (Pollack 1986; Romanowicz 2009). Thus, while the lithosphere–asthenosphere boundary beneath continents may be sharp and readily identified in some areas, notably tectonic regions or those of younger age, it would appear that beneath large parts of the stable continents the LAB may simply be too diffuse and/or heterogeneous to be identified as a single coherent discontinuity.

Summary and Conclusions Lithosphere, as it is understood in plate tectonic phraseology, is a term that was developed almost entirely in the context of

oceanic plates. There, the discontinuity separating oceanic lithosphere from asthenosphere is widely recognized to be a rheological boundary between a strong melt-free plate and a weak, probably partially molten asthenosphere. In many important respects, we now know that this easily visualized oceanic model – universally displayed in textbooks both elementary and advanced – is a cartoonish oversimplification when applied to the continental lithosphere. Although there do appear to be regions where the continental lithosphere is almost as well defined as it is beneath ocean basins, the distinction between lithosphere and asthenosphere at depth beneath the continents, specifically the boundary between the two, is still shrouded in uncertainty. The failure of our ability seismologically to determine where lithosphere ends and asthenosphere begins has major implications for plate tectonic models in a dynamical earth. If we do not understand the coupling (or lack thereof) between lithosphere and asthenosphere, and the depth at which that occurs, then the task of sorting out the geodynamical drivers for plate tectonics becomes an exceedingly difficult exercise. While seismology ultimately has the ability to ferret out the truth of how continental plates are coupled or decoupled from the mobile mantle beneath, we are not quite at that point in the science. Nonetheless, the study of the continental lithosphere, particularly the lithospheric mantle, is entering a period of unprecedented expansion. In the USA, the EarthScope program’s Transportable Array of 400 broadband seismic systems has

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migrated to the country’s midsection, with nearly 1,000 stations occupied at the time of this writing. Similar large-scale programs are well underway in Japan, China and Europe, giving hope that within the next several years the continental lithosphere and the lithosphere–asthenosphere boundary will have been mapped at high resolution over much of the globe’s surface.

Cross-References ▶ Heat Flow, Continental ▶ Lithosphere, Continental: Thermal Structure ▶ Lithosphere, Mechanical Properties ▶ Lithosphere, Oceanic ▶ Lithosphere, Oceanic: Thermal Structure ▶ Mantle Viscosity ▶ Seismic, Receiver Function Technique ▶ Seismological Networks

Lithosphere, Continental: Thermal Structure Rychert CA, Shearer P (2009) A global view of the lithosphereasthenosphere boundary. Science 324:495–498 Rychert CA, Fischer KM, Rondenay S (2005) Scattered wave imaging of a sharp lithosphere-asthenosphere boundary beneath eastern North America. Nature 436:542–545 Sacks IS, Snoke JA (1977) The use of converted phases to infer the depth of the lithosphere-asthenosphere boundary beneath South America. J Geophys Res 82:2011–2017 Shapiro SS, Hager BH, Jordan TH (1999) Stability and dynamics of the continental tectosphere. In: van der Hilst RD, McDonough WF (eds) Composition, deep structure and evolution of continents. Elsevier, Amsterdam, pp 115–133 Shirey SB, Carlson RW, Richardson SH, Menzies AH, Gurney JJ, Pearson DG, Harris JW, Wiechert U (2001) Emplacement of eclogite components into the lithospheric mantle during craton formation. Geophys Res Lett 28(13):2509–2512 Sipkin SA, Jordan TH (1980) Multiple ScS travel times in the Western Pacific: implications for mantle heterogeneity. J Geophys Res 85:853–861 West JD, Fouch MJ, Roth JB, Elkins-Tanton LT (2009) Vertical mantle flow associated with a lithospheric drip beneath the Great Basin. Nat Geosci 2:439–444 Zandt G, Gilbert HJ, Owens TJ, Ducea M, Saleeby J, Jones CH (2004) Active foundering of a continental arc root beneath the southern Sierra Nevada in California. Nature 431:41–46

Bibliography Boyd FR, McCallister RH (1976) Densities of fertile and sterile garnet peridotites. Geophys Res Lett 3(9):509–512 Carlson RW, Pearson DG, James DE (2005) Physical, chemical and chronological characteristics of continental mantle. Rev Geophys 43, RG1001, 1–24 Forsyth DW (1989) Lithosphere: mechanical properties. In: James D (ed) The encyclopedia of solid earth geophysics. Van Nostrand Rhinehold, New York, pp 655–660 James DE (2007) Crust and lithospheric structure – natural source portable array studies of the continental lithosphere. In: Dziewonski AM, Romanowicz B (eds) Treatise on geophysics. Elsevier, New York, pp 479–531 James DE, Fouch MJ (2002) Formation and evolution of Archaean cratons: insights from Southern Africa. In: Ebinger C, Fowler CMR, Hawkesworth CJ (eds) The early earth: physical, chemical and biological development. Geological Society, London, pp 1–26 James DE, Boyd FR, Schutt D, Bell DR, Carlson RW (2004) Xenolith constraints on seismic velocities in the upper mantle beneath southern Africa. G-cubed 5:1–32. https://doi.org/10.1029/2003GC00055 1(Q01002) Jordan TH (1978) Composition and structure of the continental tectosphere. Nature 274:544–548 Jordan TH (1981) Continents as a chemical boundary layer. In: The origin and evolution of the Earth’s continental crust. Trans Roy Soc London Ser A 301(1461):359–373 MacGregor ID, Manton WI (1986) Roberts Victor eclogites: ancient oceanic crust. J Geophys Res 91:14063–14079 Morgan P (1984) The thermal structure and thermal evolution of the continental lithosphere. Phys Chem Earth 16:107–193 Pollack HN (1986) Cratonization and thermal evolution of the mantle. Earth Planet Sci Lett 80:175–182 Ritsema J, van Heijst H (2000) New seismic model of the upper mantle beneath Africa. Geology (Boulder) 28(1):63–66 Romanowicz B (2009) The thickness of tectonic plates. Science 324:474–476 Rudnick RL, McDonough WF, O’Connell RJ (1998) Thermal structure, thickness and composition of continental lithosphere. Chem Geol 145:395–411

Lithosphere, Continental: Thermal Structure Claude Jaupart1 and Jean-Claude Mareschal2 1 Université de Paris Institut de Physique du Globe, Paris, France 2 Centre GEOTOP-UQAM, University of Québec, Montréal, QC, Canada

Definition Thermal conduction is the dominant form of heat transport across the lithosphere. The thermal structure of the continental lithosphere and its thickness are determined by the distribution of heat-producing elements in the crust and lithospheric mantle and the boundary condition at its base. Vertical seismic velocity profiles, geothermobarometry on mantle xenoliths, and sedimentary records of subsidence provide additional constraints on lithospheric thickness and temperature. The base of the lithosphere can be defined as the intersection of the conductive temperature profile with the mantle isentrope.

Introduction The lithosphere is the superficial shell or boundary layer that lies at the top of the convecting mantle and experiences little internal deformation. The concept of lithosphere has evolved

Lithosphere, Continental: Thermal Structure

over the years, leading to various definitions and estimates of its thickness. The base of the lithosphere was initially defined by a seismic low-velocity zone associated with partial melting. In the oceans, it was also determined by the variations of seafloor depth and surface heat flux as a function of age. It was clear from the beginning that all these definitions are related to the thermal structure. For instance, partial melting requires temperatures above the solidus of mantle rocks. These definitions were introduced, and tested, in the oceans with varying degrees of success but could not be readily extended to the continents. Over large continental areas, no seismic lowvelocity zone can be detected, and, for a long time, the meager scattershot heat flux data set allowed conflicting interpretations. In the oceans, the lithosphere is made out of homogeneous starting material that is well characterized. It is affected by few thermal perturbations and tectonic events during its short residence time at Earth’s surface. Its thermal structure and thickness are essentially functions of only one variable, age, and evolve in simple ways that can be studied with several independent geophysical techniques. In comparison, the continental lithosphere is heterogeneous and thick. Not only was it formed a long time ago by processes that are still debated today, but it has been affected by a succession of perturbations and tectonic events which have left their imprints. Its thermal structure is sensitive to the large amounts of heat-producing elements in the crust and cannot be characterized by age only. Because the continental lithosphere is thick, its thermal relaxation time is very large, which offers opportunities to study mechanisms that are no longer active today. Because of its old age, it preserves structural and chemical records of ancient geological processes. From a purely thermal standpoint, it plays an important role in regulating the Earth’s heat loss and in storing large quantities of radioactive elements that are no longer available to power mantle convection.

The Thermal Boundary Layer of Mantle Convection The lithosphere can be defined from a purely thermal standpoint. There are two basic mechanisms of heat transport within the Earth, conduction and convection. Convection refers to the transport of energy by matter that is set in motion by buoyancy forces. Without mass transport, energy can be transferred by conduction in association with a temperature gradient. These mechanisms are important in different parts of the Earth. At shallow depths beneath the surface, the cold upper layer does not deform easily on geological timescales, so that conduction is the dominant heat transport mechanism. The lithosphere may be defined from this perspective, but this definition does not account for heat that is supplied by convection from below. One must add a relatively thin convective

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boundary layer at the base of the lithosphere, which connects the rigid and purely conductive upper region to the wellmixed mantle below. In this basal boundary layer, heat transport by conduction is not negligible, as discussed below in more detail. Thermal Structure In the continental lithosphere, heat is in part supplied from below by convection and in part generated by the decay of radioactive elements. In steady state and assuming that heat transport occurs in the vertical direction only, the heat balance equation is: 0¼


dq þH dz

ð1Þ

where z is the depth, q is the vertical heat flux, and H is the rate of heat production. In the upper part of the thermal boundary layer, there is no convection, and the heat flux is given by the Fourier law of heat conduction: q ¼
l

dT dz

ð2Þ

where T is temperature and l thermal conductivity. The temperature gradient can be measured at Earth’s surface in deep boreholes and determined in the lithospheric root below the crust using the thermodynamic equilibrium conditions between mineral assemblages found in the sample from the mantle, mantle xenoliths found in kimberlite pipes. Typical values in stable continents are 15 K km
1 and 5 K km
1, respectively (Fig. 1). The difference between these two values is due to radiogenic heat production in the continental crust. Below the lithosphere, the temperature gradient is dictated by the convective regime and is expected to be close to an isentrope. By definition, entropy is conserved when going up or down an isentrope, such that: dT S agT S ¼ dz Cp

ð3Þ

where α is coefficient of thermal expansion, g is acceleration of gravity, TS is temperature, and Cp is specific heat at constant pressure. For the Earth’s mantle, the isentropic gradient is typically 0.5 K km
1, which is much less than the conductive gradient in the lithospheric mantle. One identifies the isentrope by its potential temperature, i.e., the value of its temperature at atmospheric pressure. Three different depths may be defined in the thermal boundary layer (Fig. 2). The shallowest boundary, h1, corresponds to the base of the rigid upper part and of what we shall call the thermal lithosphere. The deepest boundary, at depth h3, corresponds to the top of the well-mixed mantle and lies

L

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along the mantle isentrope. With no knowledge of boundary layer characteristics and heat transport mechanisms, one cannot determine h1. An intermediate depth, h2, is obtained by downward extrapolation of the conductive geotherm to the mantle isentrope. h3 is obtained from seismic velocity anomalies, as it is such that temperatures do not deviate significantly from those beneath oceans or from the average mantle velocity profile. These thickness determinations say nothing about h1. Yet, it is h1 which defines the mechanically coherent unit (the “plate”) which moves at Earth’s surface and sets the thermal relaxation time which follows tectonic and magmatic perturbations. Uncertainty on this thickness has severe consequences because the diffusive relaxation time is / h2/k, where k is thermal diffusivity. In steady state, Q0, the heat flux at Earth’s surface, is obtained by integrating the heat balance Eq. (1) and can be broken down in three components:

0 1

Pressure (GPa)

2 3 4 5 6 7 0

500

1000

1500

Temperature (°C) Lithosphere, Continental: Thermal Structure, Fig. 1 Pressure and temperature estimates from studies of mineral assemblages in xenoliths from the lithospheric mantle beneath the Kirkland Lake kimberlite pipe, Abitibi subprovince, Canadian Shield. Data are taken from Vicker (1997). The best fit (dashed line) indicates a temperature gradient of 5 K km
1. Note that it intersects Earth’s surface at a temperature of about 400 °C, showing that the temperature gradient is larger than 5 K km
1 in the crust

0

T0

Tb

Q

b

h1 h2

Isentropic temperature profile

Q=

Tm

Q0 ¼ Qcrust þ Qlith þ Qb

ð4Þ

where Qcrust and Qlith stand for the contributions of heat sources in the crust and in the lithospheric mantle and Qb is the heat flux at the base of the lithosphere. To describe heat transport mechanisms and the lithosphere structure, one must

T Conductive boundary layer

Convective boundary layer

h3 Well-mixed mantle

z Lithosphere, Continental: Thermal Structure, Fig. 2 Left: Schematic vertical temperature profile through the continental lithosphere, illustrating three different thicknesses and temperatures. Thickness h1 corresponds to the rigid upper part of the thermal boundary layer. h2 is determined by downward continuation of the temperature profile in the rigid upper part to the isentrope that approximates the temperature profile

in the well-mixed convective mantle. h3 denotes the base of the thermal boundary layer, such that temperature is along the well-mixed isentrope. Right: three different regions with different heat transport characteristics. Heat is transported by conduction only above depth h1 and by convection only below depth h3. Between these two depths, a convective boundary layer is such that both heat transport mechanisms are important

Lithosphere, Continental: Thermal Structure

875

Tb
T0 T
T0 ¼l m h1 h2

ð5Þ

Well-established convection theory leads to a closure equation relating Qb to the temperature difference across the convective boundary layer, (Tm – Tb) (Davaille and Jaupart 1993; Solomatov 1995). In this case, therefore, measurement of the surface heat flux Q0 allows determination of the basal heat flux and of the temperature difference (Tm – Tb). In turn, this leads to the values of thicknesses h1 and h2 as well as of basal temperature Tb. In the oceans, therefore, the values of all the variables can be deduced from surface measurements. In the continents, the problem is more complicated because the surface heat flux includes the crustal contribution Qcrust which may vary by large amounts. This crustal component is typically about 30 mW m
2 out of a total of 42 mW m
2 in Archean (i.e., older than 2.5Gy) cratons. In such conditions, surface heat flux measurements do not provide a direct measure of the heat flux at the base of the lithosphere, which must be inferred from other data with methods that will be discussed below. Qcrust depends on the composition of the continental crust, which itself reflects the past history of magma emplacement and tectonic deformation/accretion. Observations show that it varies by large amounts even in a single geological province. It cannot be calculated from first physical principles and must be determined on a case-by-case basis using a host of different measurements. The lithosphere owes its physical properties to partial melting and melt extraction processes which leave a solid residue that is both less dense and mechanically stronger than the starting mantle material. Mechanical strength is due to both the lower temperature in the lithosphere than in the convective mantle and dehydration that occurred during partial melting (Pollack 1986). Melting may be achieved by two different mechanisms operating in different settings. One mechanism is decompression in an upwelling and has been

Heat Flux at the Base of the Continental Crust and at the Base of the Lithosphere Constraints on the thermal structure of the continental lithosphere may be obtained with several geophysical methods. To obtain continuous temperature profiles, however, one can only use surface heat flux data and downward continue 0

0

100 5 200

10

1000

Depth (km)

Q0 ¼ Qb ¼ l

documented in considerable detail in mid-ocean ridges (McKenzie and Bickle 1988). Decompression melting begins at a depth which depends on mantle temperature and water content (e.g., 60–80 km for the oceans today). Continents are not linear features and were probably not generated by linear mantle upwellings such as present-day mid-ocean ridges. Their formation has been linked to large mantle plumes or subduction zones (Carlson et al. 2005). For mantle plumes, the mechanism is basically the same as that beneath midocean ridges, and the depth where melting starts dictates the thickness of the buoyant solid residue. In a dry mantle, high potential temperatures are required for the generation of thick lithospheric roots (Herzberg 1983; Fig. 3). For a thickness of 250 km, for example, the mantle potential temperature must be about 1800 °C. On the other hand, if the Archean mantle was wet, melting occurred at shallower depths and at lower temperatures (Grove and Parman 2004). In this case, a second event is required to account for the large thickness of continental lithosphere, involving large-scale compression and/or thrusting. For both types of models, the lithosphere thickness provides a strong constraint on the mechanism of continent formation.

Pressure (GPa)

introduce three temperatures, T0 at the upper boundary, which, for all practical purposes, may be taken as fixed and equal to 0 °C, Tb at the base of the lithosphere, and Tm along an isentrope in the well-mixed convective interior (Fig. 2). One can reasonably assume that the mantle potential temperature is the same beneath continents and oceans and hence that it can be deduced from the composition of mid-ocean ridge basalts (Kinzler and Grove 1992). Today, this temperature is about 1325 °C. Tb, the temperature at the base of the thermal lithosphere (the rigid lid), cannot be determined from geophysical measurements. In the oceans, one may ignore Qcrust and Qlith, and, for seafloor older than 100My, the surface heat flux is equal to the basal heat flux. In this case:

300

1400 1800 Temperature (°C)

400 2200

Lithosphere, Continental: Thermal Structure, Fig. 3 Solidus temperature of dry mantle peridotite as a function of pressure and depth, from Herzberg (1983). Also shown are two isentropes that intersect the solidus at depths of 60 and 250 km

L

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shallow temperature measurements. To this aim, one needs to specify the heat flux at the base of the lithosphere. A first step is to determine the amount of heat produced in the continental crust or, alternatively, the heat flux at the Moho discontinuity. Uranium and thorium are the main heat-producing elements in rocks. They are located mostly in accessory minerals and grain boundaries, which depend weakly on the bulk chemical composition. Thus, their concentrations are not related to physical properties such as density and seismic velocity and cannot be retrieved from large-scale geophysical studies. In addition, they vary by large amounts at all scales, from that of a petrological thin section to that of a whole massif. Within an apparently homogeneous pluton, they can change in both vertical and horizontal directions due to fluid migration and late-stage alteration. In the Bohus granite, Sweden, for example, thorium concentrations vary by a factor of 5 over horizontal distances as small as a few tens of meters (Landstrom et al. 1980). In a geological province, radio-element concentrations cannot be estimated using data from other provinces and must be measured in all the major rock types present. They depend on lithology and are high in granites and metasediments, so that a geological map gives a rough idea of the spatial distribution of heat production. With the very wide ranges that exist for each rock type, however, the standard deviation of the heat production distribution is large and often larger than the mean. As a consequence, a single thermal model for a province would gloss over the important lateral temperature variations that occur due to the heterogeneous crustal structure. For a reliable thermal calculation in a specific area, one cannot use the local heat production values of a few rock types found on the outcrop because temperatures are sensitive to heat that is generated over rather large volumes. Average heat production values must be determined on a scale that is intermediate between the dimensions of individual plutons and the size of the province. Vertical variations of heat production can be determined in deep boreholes or from exposed crustal sections. On a vertical scale of ≈ 10 km, data from the deep boreholes at Kola, in the Russian part of the Baltic Shield (Kremenentsky et al. 1989), and the KTB, Germany (Clauser et al. 1987), show no systematic change with depth. At KTB, heat production between 8 and 9 km depth is the same as between 1 and 2 km and higher than above 1 km. At Kola, heat generation in the Archean rocks between 8 and 12 km is higher (1.47 mW m
3) than in the shallower Proterozoic section (0.4 mW m
3). Over a crustal thickness scale, heat production is lower in mid-crustal assemblages than in the upper crust. This vertical variation is not monotonous and cannot be described by a simple function valid everywhere, as shown by exposed crustal sections such as the Vredefort in South Africa (Nicolaysen et al. 1981), the Cordilleran core complexes of

Lithosphere, Continental: Thermal Structure Lithosphere, Continental: Thermal Structure, Table 1 Element concentration and heat production in the continental crust. (From Rudnick and Gao (2014)). Element concentrations are given in ppm Upper crust Middle crust Lower crust Total crust

U 2.7 1.3 0.2 1.3

Th 10.5 6.5 1.2 5.6

K 23,300 19,200 5,080 15,080

mW m
3 1.65 1.00 0.19 0.89

Lithosphere, Continental: Thermal Structure, Table 2 Estimates of bulk continental crust heat production from heat flux data. (From Jaupart and Mareschal (2014)) Age group Archean Proterozoic Phanerozoic Total continents a

Aa 0.56–0.73 0.73–0.90 0.95–1.21 0.79–0.99

Qbcrust 23–30 30–37 37–47 32–40

Range of heat production in mW m
3 Range of the crustal heat flow component in mW m
2

b

Arizona (Ketcham 1996), or the Pikwitonei-Sachigo and Kapuskasing-Wawa areas of Canada (Fountain et al. 1987; Ashwal et al. 1987). One robust feature is the depletion of lower crustal assemblages in comparison to upper and middle crustal ones (Table 1). Table 1 lists estimates derived from global geochemical models of the crust and a very large number of measurements. The crust is split into three different crustal horizons corresponding to average regional seismic models. The model crust is 40 km thick with middle crust in the 12–23 km depth range and is such that Qcrust ¼ 36 mW m
2. Another method relies on heat flux data and will be explained below (Table 2). With this method, one may also evaluate heat production variations between provinces. The two independent methods lead to comparable results and show that crustal heat generation typically accounts for more than half of the surface heat flux. Heat Flux and Heat Production With the large variations in the concentrations of heatproducing elements that have been documented in crustal rocks, it is not surprising that, in continents, the surface heat flux changes by large amounts over small horizontal distances of a few tens of kilometers. The largest anomalies occur in association with enriched granitic plutons. These plutons are usually of small size, and the local value of the heat flux is not representative of the average crust that lies below the measurement site. In such conditions, studies of the thermal structure of continents require large data sets. The challenge is to obtain average values of heat flux and heat production that are representative of the whole crust in

877

QM ¼ Q0
Qcrust ¼ Qlith þ Qb

ð6Þ

which depends on heat production in the lithospheric mantle and on the heat flux at the base of the lithosphere. Both components come from depths larger than the crustal thickness, so that lateral variations, if they exist, get smoothed out by horizontal diffusion. For an average lithospheric thickness of 250 km, variations of the basal heat flux Qb over wavelengths smaller than 500 km are not detectable in the surface heat flux (Mareschal and Jaupart 2004). A similar statement can be made for the contribution of heat sources in the lithospheric mantle, Qlith. Thus, on scales that are smaller than 500 km, one must consider that the Moho heat flux is

Heat f lux (mW m-2)

Individual measurements

Local averages (≈ 250 km)

Heat f lux (mW m-2)

an area. We discuss the scales over which this can be done in a reliable manner. There is only a weak relationship between local values of heat flux and heat production in a geological province (Fig. 4). For a representative thermal model, one must determine an average heat flux value over an area that is sufficiently large for the smoothing of small-scale variations due to isolated anomalous massifs. Figure 4 also shows data for ten ≈ 250 km  250 km windows distributed in North America. These are located in the same area as the individual heat flux values discussed previously and show that a relationship between heat flux and heat production begins to emerge at this scale. Even at this scale, the relationship is not defined tightly due to insufficient sampling of heat production. A single heat flux value is sensitive to crustal heat production over large horizontal distances and records the contribution of many different geological units. In contrast, the heat production values correspond to a small number of rocks found at shallow depth in boreholes used for heat flux determinations. The limited sampling achieved at a scale of 250 km does not account for the full range of rock types and heat production rates in the crust. For representative heat production averages, one needs to work at a larger scale. Figure 4 shows data for five major geological provinces in North America with typical dimensions of ≈ 500 km  500 km. These provinces cover the geological history of the stable continent and represent different types of continental crust. Furthermore, they have been extensively deformed and eroded, so that rocks from a large range of depths can be found at the surface. The metasedimentary-plutonic belts of the western Superior Province, Canadian Shield, for example, include rocks from all metamorphic grades up to granulite facies. The province-wide averages exhibit a remarkable linear relationship, with a heat flux intercept of 33 mW m
2 corresponding to crust with zero surface heat production. The data for the ten ≈ 250 km  250 km windows lie close to this global linear relationship (Fig. 4). The Moho heat flux is equal to:

Heat f lux (mW m-2)

Lithosphere, Continental: Thermal Structure

Province-wide averages (≈ 500 km)

Heat production (μW m-3) Lithosphere, Continental: Thermal Structure, Fig. 4 Relationship between heat flux and radiogenic heat production in North America at different scales, from Levy et al. (2010). Top panel: for individual measurements, there is clearly no relationship between the two. Middle panel: at an intermediate scale of about 200 km, a relationship between the average values of heat flux and heat production begins to emerge. The dashed line corresponds to the best-fit linear relationship defined by province-wide averages. Bottom panel: it shows a strong correlation between the average values of heat flux and heat production for the five major geological provinces of North America. The best-fit linear relationship has a slope of 9 km and an intercept of 33 mW m
2. Symbols are as follows: App, Appalachian Province (400My); Gre, Grenville Province (1100My); THO, Trans-Hudson Orogen (1800My); Sup, Superior Province (>2500My); Slave, Slave Province (>2500My)

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uniform and that variations of the surface heat flux come solely from changes of crustal heat production. The strong correlation that is observed between the province-wide average surface heat flux and heat production indicates that variations of the Moho heat flux are small throughout the whole North American continent (Fig. 4). If there were large variations of the Moho heat flux, they would need to be compensated by opposite variations of the average lower crustal heat production. No physical mechanism can explain how such independent variables can be linked to one another. Variations of the Moho heat flux are therefore less than the intrinsic uncertainty on the heat flux measurements and the magnitude of departures from the heat flux-heat production relationship, which is about 2–3 mW m
2 (Mareschal and Jaupart 2004). The absolute value of the Moho heat flux is not specified by this analysis, however, and this issue is discussed in the next section. The Moho Heat Flux Direct determination of the heat flux through the lithospheric mantle can be made using xenolith samples brought from lithospheric depths by powerful kimberlitic eruptions. Coexisting mineral assemblages and mineral compositions allow estimates of pressure and temperature and of the geothermal gradient. Together with a thermal conductivity estimate, they lead to values of the heat flux beneath the crust. For the Kirkland Lake pipe, within the Superior Province, Canada (Fig. 1), Rudnick and Nyblade (1999) obtained a best-fit

Moho heat flux estimate of ≈ 18 mW m
2 within a total range of 17–25 mW m
2. Russell and Kopylova (1999) derived a best fitting value of 15 mW m
2, within a range between 12 and 24 mW m
2, for the Jericho kimberlite area of the Archean Slave Province, also in Canada. Figure 5 shows data from several kimberlite pipes in the Slave Craton (2008). There are significant temperature differences between the different data sets. Surprisingly, the youngest eruptions, which provide recent samples of the lithospheric mantle, yield the highest temperatures. A similar result is obtained in the Kaapvaal Craton, South Africa (Michaut et al. 2007). If these temperature variations are indeed due to age and reflect the time evolution of lithospheric temperatures, they are not consistent with secular cooling of the Earth’s mantle. The pipes do not overlap so that the data may in fact record lateral variations of temperature in the lithospheric mantle, which can only be explained by changes of heat production. Nevertheless, these data lead to similar values of the temperature gradient and heat flux. Bounds on the Moho heat flux can be obtained by taking advantage of the variations of heat flux and crustal structure in a geological province. Combined with heat production data for the various rock types involved, one can isolate the variable crustal component and the uniform Moho heat flux. In Canada, Moho heat flux values of 10–15 mW m
2 have thus been derived for the Grenville Province, the Trans-Hudson Orogen (THO), for the Abitibi belt of the Superior Province (Jaupart and Mareschal 2015). Crustal thickness and

0 553 My (Gahcho Kue) 173 My (Jericho)

Depth (km)

50

55 My

100

150

200 250 400

600

800

1000

1200

600

800

1000

1200

1400

Temperature (°C) Lithosphere, Continental: Thermal Structure, Fig. 5 Relationship between temperature and depth in the lithospheric mantle of the Slave Province, Canada, from Canil (2008). The Slave Province is small and has been pierced repeatedly by kimberlite pipes in the same area. Left panel: data from the two oldest pipes. Right panel: data for two young

pipes compared to those for the oldest pipe in the area. On average, temperature estimates from the two young pipes are larger than those of the older pipes by about 150 K. Such differences may indicate lateral temperature variations in the lithospheric mantle

Lithosphere, Continental: Thermal Structure

50 ns hia lac pa Ap

150

ce

vin

Pro

Depth (km)

100

r erio

Lithospheric Geotherms Using information on heat flux values at the surface and at the Moho, one may determine geotherms through the lithosphere. The calculation requires values for thermal conductivity that are discussed in the Appendix. Fig. 6 shows two such geotherms for two provinces in North America, which are consistent with both seismic and heat flux data (Levy et al. 2010). These two geotherms intersect the well-mixed mantle isentrope at different depths, indicating large lateral thickness variations for the continental lithosphere. The maximum thickness is reached beneath the Archean Superior Province and is estimated to be 280  30 km.

0

Sup

composition can also be constrained by combining seismic refraction, gravity, and heat flux. Such methods lead to values of QM between 7 and 15 mW m
2. Another method relies on Pn velocities from seismic refraction surveys (Perry et al. 2006). For given values of the surface heat flux, the Moho temperature depends on the amount of crustal heat production and hence on the Moho heat flux. One can therefore obtain a relationship between Moho temperature and seismic velocity with the Moho heat flux as control parameter. One can also derive independently a relationship between temperature and seismic velocity using laboratory measurements of elastic properties and seismic velocities, with the bulk mantle composition as control parameter. The two relationships are only consistent with one another over a restricted range of composition and Moho heat flux. For depleted mantle compositions appropriate to the Superior Province, a good fit between predicted and observed Pn velocity values is obtained if the Moho heat flux is within a range of 12–25 mW m
2 (Perry et al. 2006). The three different and independent methods that we have described have all been applied to the Abitibi belt in the Canadian Shield and can be combined to tighten the final range to 12–15 mW m
2 for that subprovince. Lower and upper bounds on the Moho heat flux have been derived using other arguments. Upper bounds on the mantle heat flux are obviously provided by the lowest heat flux measured. Values of 20–23 mW m
2 have been reported within the Canadian Shield (Mareschal et al. 2000). Regional values as low as 18 mW m
2 have also been reported for the Baltic or Siberian Shield. One can refine this estimate to 18 mW m
2 using lower bounds on crustal heat production. The methods that we have described here rely on different data and hence are associated with different sources of uncertainty. That such completely independent methods converge to similar results allows some confidence in the final range obtained. Values lower than 12 mW m
2 are not consistent with the xenolith data, whereas values higher than 18 mW m
2 can be excluded because of the heat flux data. This upper bound appears to be valid for all the shields (Jaupart and Mareschal 2015).

879

200

250

300

0

1000 500 Temperature (°C)

1500

Lithosphere, Continental: Thermal Structure, Fig. 6 Two steadystate lithospheric geotherms beneath the Appalachian and Superior provinces of North America, from Levy et al. (2010). These two geotherms were deduced from measurements of the surface heat flux and heat production and were constrained by requiring consistency with seismic data. The dashed line is the isentrope for a potential temperature of 1350 °C

Thermal Transients Because the continental lithosphere is thick, it is characterized by a long thermal relaxation time. For a thickness of 250 km and a thermal diffusivity k equal to 10
6 m2 s
1, the characteristic diffusion time t ¼ h2/k is ≈ 2 Gy. This is much longer than that of the oceanic lithosphere, which is only about 200 My (for a thickness of 80 km). We shall now discuss two types of thermal transients in thick lithosphere. Thermal relaxation of the lithosphere following a deep thermal perturbation can lead to subsidence, which has been recorded in intracratonic sedimentary basins located away from active plate boundaries. Subsidence histories for two such basins of North America, the Michigan and Williston basins, are shown in Fig. 7 and emphasize their long durations exceeding 200 My. On the basis of models for heat transport and cooling in the vertical direction only, the thickness of the North American lithosphere is found to about 115 km beneath the Michigan basin and 270 km beneath the Williston (Haxby et al. 1976; Ahern and Mrkvicka 1984). There is no obvious explanation for such a large thickness difference in the middle of the North American continent, which suggests that the subsidence models may be oversimplified. These models

L

880

Lithosphere, Continental: Thermal Structure

1 Illinois

te

0.8

m

ra pe

tu r e

2

xe

d

Williston 0.6

Fi

1

τ2/τ1

Sediment thickness (km)

0

Fi

0.4

xe

d

a he

t fl

ux

0.2 3 0 500

400

300 Age (Ma)

200

100

0

Lithosphere, Continental: Thermal Structure, Fig. 7 Subsidence histories for the Michigan and Williston basins, North America

rely on 1-D calculations and hence are only valid if the thermal anomaly stretches over a horizontal distance that is much larger than the lithosphere thickness. The Michigan and Williston basins have been attributed in large part to plate flexure driven by a deep load (Nunn and Sleep 1984; Ahern and Mrkvicka 1984). This load is related to the initial thermal anomaly and has a radius of about 120 km beneath the Michigan basin (Haxby et al. 1976), which is smaller than the thickness of the lithosphere beneath the North American craton. In these conditions, the assumption of purely vertical heat transfer is not tenable. Kaminski and Jaupart (2000) have reevaluated thermal models for the continents by taking into account lateral heat transport within thick lithosphere. They considered a lithosphere of thickness h with a cylindrical thermal anomaly of radius a between depths h and b. At the surface (z ¼ 0), temperature is fixed at To. At steady state, the base of the lithosphere is at T ¼ To + ΔT. Assuming, for example, that the thermally perturbed region is initially at a uniform temperature To + ΔT, temperature can be written as the sum of the equilibrium temperature and a dimensionless perturbation θ: T ðr, z, tÞ ¼ T o þ DT

h

i z þ y ðr, z, tÞ , h

ð7Þ

where r is the radial distance. The temperature perturbation may be expressed in terms of a vertical component θZ, which corresponds to the 1-D solution, and a radial component which can be calculated using Bessel functions. For thermal anomalies with planforms that are not circular, lateral heat transfer is dominated by the smallest horizontal dimension. The acceleration of cooling due to lateral heat loss can be measured simply by the time to achieve 90% of the final subsidence, noted t2, scaled to that for an infinitely wide initial perturbation, t1 (corresponding to the 1-D calculation) (Fig. 8). Results are sensitive to the boundary condition at the base of the lithosphere and show that the impact of lateral heat

0.5

1

1.5 a/h

2

2.5

3

Lithosphere, Continental: Thermal Structure, Fig. 8 Time to achieve 90% subsidence in thick lithosphere as a function of the radius of the deep lithospheric thermal anomaly, a (scaled to the lithosphere thickness h). The subsidence time has been scaled to the value obtained with a 1-D thermal model for purely vertical heat transport. Results have been obtained for two different boundary conditions at the base of the lithosphere. (Adapted from Kaminski and Jaupart (2000))

transport is significant for a/h < 1, corresponding to a < ≈ 250 km in the North American craton. This is the case of all the major intracratonic basins. In comparison to the oceans, thermal relaxation in the continents is lengthened by the larger lithosphere thickness and shortened by lateral heat transport. Different subsidence histories and durations are generated by the same basic process operating over different lateral distances. Secular Transients We have shown above that continental temperatures depend strongly on radiogenic heat production. Thus, continents cool down continuously due to the rundown of radioactivity with time. One should also add that conditions at the base of the lithosphere also evolve with time due to secular cooling of the whole Earth. This is ill-constrained and we focus our attention to the effects of decaying heat sources. For the Appalachian geotherm of Fig. 6, for example, crustal heat production contributes about 300 °C. For average Th/U and K/U ratios, radiogenic heat production decreases by a factor of 2 in about 2.7 Gy. We thus deduce that continental temperatures decrease at a rate of about 50 K Gy
1, which is comparable to the bulk mantle cooling rate (Abbott et al. 1994). This simple argument relies on a quasi-steady-state approximation, such that ancient geotherms are calculated using the steadystate heat equation together with past values of heat production. For thick lithosphere, this approximation is not valid and may lead to significant errors. In thick continental lithosphere, the timescale for diffusive heat transport is comparable to the half-lives of uranium, thorium, and potassium. As noted above, the characteristic time for heat diffusion equal to h2/k is 2 Gy for 250-km-thick

Lithosphere, Continental: Thermal Structure

0

0

881

Temperature (°C)

Temperature (°C) 1000 500

1500

00

400

800

1200

1400

1 50

Pressure (GPa)

100

tr ar cul Se i ans

3 4 5 6

.5 G

e, 0 stat

ady Ste

200

1 te, -sta ady

yr

Ste

0G

150

, ent

Depth (km)

2

yr

r Gy

250 Lithosphere, Continental: Thermal Structure, Fig. 9 Present-day vertical temperature profile for transient thermal model with decaying heat sources (thick line) compared with two steady-state geotherms (solid line for present-day heat production; dashed line, for heat production rate values 1.5 Gy ago) for the same set of parameters (h ¼ 250 km, crustal heat production Ac ¼ 1.0 mW m
3, lithospheric mantle heat production Am ¼ 0.02 mW m
3, basal heat flux Qb ¼ 10 mW m
2). Surface heat flow is of 49 mW.m
2 for the transient solution, 46 mW m
2 for the present-day steady-state profile, 62 mW m
2 for the 1.5 Ga steady-state profile

lithosphere, which is close to the decay time of radiogenic heat production (≈2.7 Gy). Thus, heat diffusion is not efficient enough to keep up with the decaying heat sources, and lithospheric temperatures are not in equilibrium with the instantaneous rate of heat production. In these conditions, the vertical temperature profile exhibits significant curvature and may be hotter than a steady-state profile by as much as 150 K depending on the absolute value of the heat generation rate (Fig. 9). Another consequence is that forcing a steady-state temperature profile through xenolith (P-T) data leads to an overestimate of the mantle heat flux. For illustration purposes, Fig. 9 shows the instantaneous vertical temperature profile with decaying sources and a steady-state geotherm derived for the same present-day values of crust and mantle heat production. The difference between the two geotherms increases with depth because of increasingly inefficient transport of heat to the Earth’s surface. For comparison, we have also calculated a steady-state geotherm corresponding to heat production values 1.5 Gy ago. The transient geotherm falls between the steady-state geotherms calculated for heat production rates at 0 and 1.5 Ga, showing that it is sensitive to heat generation in the past. At the base of the lithosphere, all geotherms are parallel because of the constant heat flux bottom boundary condition. Secular transient behavior is not important at shallow depths,

7 8 Lithosphere, Continental: Thermal Structure, Fig. 10 Xenolith P-T data from the Kimberley, Jagersfontein, Premier and Frank Smith kimberlite pipes, Kaapvaal Craton, South Africa (Rudnick, personal communication, 2003). Thick line: vertical temperature profile for a transient thermal model with decaying heat sources and for a lithosphere thickness of 225 km. Parameter values are crustal heat production Ac ¼ 0.8 mW m
3, mantle heat production Am ¼ 0,025 mW.m
3, and basal heat flux Qb ¼ 12 mW m
2. Note that the geotherm terminates above several deep data points, due to the starting assumption on lithosphere thickness

due to the efficiency of diffusive heat transport over small distances. In the crust, differences between the transient and present-day steady-state profiles are very small and can be neglected for all practical purposes. Between 50 and 150 km depth, heat flux is about 13 mW m
2 for the present-day steady-state model and 16 mW m
2 for the transient one. Maximum departure from thermal equilibrium is achieved at the base of the lithosphere. The magnitude of transient effects essentially depends on mantle heat production as well as on lithosphere thickness. Large values of heat production in the lithosphere do not introduce large transients at shallow depths. In contrast, even small values of heat production lead to significant effects in a thick lithosphere. The secular transient geotherm is compared to xenolith (P-T) data from Kaapvaal Craton, South Africa, for h ¼ 225 km in Fig. 10. For this solution, the surface heat flux is 47 mW m
2, which is close to measurements in this region, and heat production in the lithospheric mantle is 0.025 mW m
3, which lies in the lower range of values proposed by Rudnick and Nyblade (1999). The basal heat flux is 12 mW m
2. The Kaapvaal xenolith (P-T) data suggest a change of slope in the geotherm at a pressure of about 5 GPa (Bell et al. 2003). According to some authors, this marks the transition between lithosphere and asthenosphere, with the deepest data points belonging to the convecting mantle below

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Lithosphere, Continental: Thermal Structure

the lithosphere. Rudnick and Nyblade (1999) and Bell et al. (2003) have argued that all data points belong to unperturbed lithosphere. Significant curvature of the secular transient geotherm allows a good fit to all the data and does not suggest a break in the geotherm. Due to the intrinsic decay of radioactive heat sources, the lithospheric mantle therefore undergoes secular cooling even when thermal conditions at the base of the lithosphere remain steady. Predicted cooling rates are in the range of 50–150 K Gy
1. Such transient behavior also has implications for heat flow studies. Assuming that the decay of radiogenic heat production can be represented by a single exponential, such that Am(t) ¼ Aoh exp(α–t), the contribution of lithospheric heat production to the surface heat flux is: pffiffiffiffiffi tan at DQlith ðtÞ ¼ Ao h pffiffiffiffiffi exp ð
at Þ ð8Þ at This is compared to the instantaneous amount of heat produced in the lithosphere: DQlith,i ðtÞ ¼ Ao h exp ð
at Þ

ð9Þ

Ratio Qlith/Qlith,i increases as a function of lithosphere thickness and is 1.5 for h ¼ 300 km, for example. This shows that surface heat flux measurements record some time-average of the deep lithospheric heat production.

Summary The continental lithosphere is much thicker than its oceanic counterpart, and its thermal structure depends strongly on radiogenic heat production in both the crust and in the mantle. Because of these two characteristics, its behavior depends on its geological history and is not a simple function of age. Beneath Archean shields, it can be as thick as 300 km.

Cross-References ▶ Deep Seismic Reflection and Refraction Profiling ▶ Energy Budget of the Earth ▶ Heat Flow Determinations, Continental ▶ Heat Flow, Continental ▶ Heat Flow, Seafloor: Methods and Observations ▶ Lithosphere, Mechanical Properties ▶ Lithosphere, Oceanic: Thermal Structure ▶ Mantle Convection ▶ Radiogenic Heat Production of Rocks ▶ Sedimentary Basins ▶ Seismic Velocity and Temperature Relationships ▶ Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity

Appendix: Thermal Conductivity For an isotropic medium, Fourier’s law is: q ¼
l∇T

ð10Þ

where l is the thermal conductivity measured in W m
1 K
1. More generally, if the medium is not isotropic, the thermal conductivity must be defined as a second-order tensor. Thermal conductivity can be broken into lattice and radiative components, which have different behaviors as temperature and pressure change. At low temperatures, heat transport is in large part affected by lattice vibrations. Lattice conductivity decreases with increasing temperature. At larger temperatures, electromagnetic waves transport energy in continuous media. Radiation is rapidly attenuated but the medium re-emits energy if the temperature is sufficiently high. In a temperature gradient, the net energy flux will be / T3 ∇T, which has the same form as the flux for lattice conduction. One can therefore define a radiative component of conductivity,lR, such that the radiative heat flux is: qR ¼
lR ∇T

ð11Þ

The gray body law states that: lR ¼

16 n2 3 sT 3 ϵ

ð12Þ

where n is the refractive index, ϵ is the opacity, and s is the Stefan constant of blackbody radiation (s ¼ 5.67  10
8 W m
2 K
4). Lattice Conductivity For individual crystals, theoretical arguments suggest that, below the Debye temperature, the temperature dependence of lattice conductivity takes the following form: rffiffiffiffiffiffiffiffi 298 lðT Þ ¼ l298 T

ð13Þ

For polycrystalline assemblages, this equation is no longer valid and one relies on empirical fits to the data. Reasonable agreement can be obtained with a relationship of the type: lð T Þ ¼ A þ

B 350 þ T

ð14Þ

where T is in oC and where constants A and B depend on the rock type (Table 3). The effect of pressure on conductivity can be treated independently of temperature, so that one can write:

Lithosphere, Continental: Thermal Structure

883

Lithosphere, Continental: Thermal Structure, Table 3 Constants for calculating the thermal conductivity of different rock types. (From Clauser and Huenges (1995)) Rock type Metamorphic rocks Felsic rocks Mafic rocks Ultramafic rocks

T range (°C) 0–1200 0–1400 50–1100 20–1400

A 0.75 0.64 1.18 0.73

B 705 807 474 1293

Lithosphere, Continental: Thermal Structure, Table 4 Pressure dependence of lattice conductivity Material Mafic granulites (Finland) Olivine a

βa  10
2 GPa
1 2.4  0.8 3.2

Reference Kukkonen et al. (1999) Xu et al. (2004)

Coefficient in Eq. 15 for thermal conductivity

lðT, PÞ ¼ lo ðT Þð1 þ bPÞ

ð15Þ

where β is a constant coefficient (Table 4). The effect of pressure increases conductivity by about 20% between the surface and 200 km. For olivine, several independent laboratory measurements suggest that: lR ¼ 0:368  10
9 T 3

ð16Þ

This equation leads to 0.13 < lR < 1.8 W m
1 K
1 for 700 < T < 1700 K. It is only valid in a single crystal if the mean free path of photons is independent of temperature. For mantle rocks, one must account for scattering and for the effect of interfaces in a mineral assemblage. Such complications led Marton et al. (2005) to use a constant radiative conductivity component lR ¼ 1 W m
1 K
1 for temperatures higher than 700 K.

Bibliography Abbott D, Burgess L, Longhi J (1994) An empirical thermal history of the Earth’s upper mantle. J Geophys Res 99:13835–13850 Ahern JL, Mrkvicka SR (1984) A mechanical and thermal model for the evolution of the Williston basin. Tectonics 3:79–102 Ashwal LD, Morgan P, Kelley SA, Percival J (1987) Heat production in an Archean crustal profile and implications for heat flow and mobilization of heat producing elements. Earth Planet Sci Lett 85:439–450 Bell DR, Schmitz MD, Janney PE (2003) Mesozoic thermal evolution of the southern African mantle lithosphere. Lithos 71:273–287 Canil D (2008) Canada’s craton: a bottoms-up view. GSAToday 18:4–10 Carlson RW, Pearson DG, James DE (2005) Physical, chemical, and chronological characteristics of continental mantle. Rev Geophys 43:2995–3007 Clauser C, Huenges E (1995) Thermal conductivity of rocks and minerals. In: Ahrens TJ (ed) A handbook of physical constants: rock physics and phase relations. AGU, Washington, DC, pp 105–126

Clauser C, Gieses P, Huenges E, Kohl T, Lehmann H, Rybach L, Safanda J, Wilhelm H, Windlow K, Zoth G (1987) The thermal regime of the crystalline continental crust: implications from the KTB. J Geophys Res 102:18,417–18,441 Davaille A, Jaupart C (1993) Transient high-Rayleigh-number thermal convection with large viscosity variations. J Fluid Mech 253: 141–166 Fountain DM, Salisbury MH, Furlong KP (1987) Heat production and thermal conductivity of rocks from the Pikwitonei-Sachigo continental cross section, Central Manitoba: implications for the thermal structure of Archean crust. Can J Earth Sci 24:1583–1594 Grove TL, Parman SW (2004) Thermal evolution of the earth as recorded by komatiites. Earth Planet Sci Lett 219:173–187 Haxby WF, Turcotte DL, Bird JM (1976) Thermal and mechanical evolution of the Michigan basin. Tectonophys 36:57–75 Herzberg CT (1983) Solidus and liquidus temperatures and mineralogies for anhydrous garnet lherzolite to 15 GPa. Phys Earth Planet Inter 32:193–202 Jaupart C, Mareschal JC (2014) Constraints on crustal heat production from heat flow data. In: Rudnick RL (ed) Treatise on geochemistry, 2nd ed, The crust, vol 4. Elsevier, New York, pp 65–84 Jaupart C, Mareschal JC (2015) Heat flow and thermal structure of the lithosphere. In: Watts AB (ed) Treatise on geophysics, 2nd ed The lithosphere, vol 6. Elsevier, New York, pp 218–253 Kaminski E, Jaupart C (2000) Lithosphere structure beneath the phanerozoic intracratonic basins of North America. Earth Planet Sci Lett 178:139–149 Ketcham RA (1996) Distribution of heat-producing elements in the upper and middle crust of southern and west Central Arizona: evidence from the core complexes. J Geophys Res 101: 13,611–13,632 Kinzler RJ, Grove TL (1992) Primary magmas of Mid-Ocean ridge basalts 2. Applications. J Geophys Res Solid Earth 97(5):6907–6926 Kremenentsky AA, Milanovsky SY, Ovchinnikov LN (1989) A heat generation model for the continental crust based on deep drilling in the Baltic shield. Tectonophys 159:231–246 Kukkonen IT, Jokinen J, Seipold U (1999) Temperature and pressure dependencies of thermal transport properties of rocks: implications for uncertainties in thermal lithosphere models and new laboratory measurements of high-grade rocks in the central Fennoscandian shield. In: Surveys in geophysics, vol 20. Kluwer, Dordrecht, pp 33–59 Landstrom O, Larson SA, Lind G, Malmqvist D (1980) Geothermal investigations in the Bohus granite area in southwestern Sweden. Tectonophys. 64:131–162 Levy F, Jaupart C, Mareschal JC, Bienfait G, Limare A (2010) Low heat flux and large variations of lithospheric thickness in the Canadian shield. J Geophys Res Solid Earth 115:408 Mareschal JC, Jaupart C (2004) Variations of surface heat flow and lithospheric thermal structure beneath the North American craton. Earth Planet Sci Lett 223:65–77 Mareschal JC, Poirier A, Rolandone F, Bienfait G, Gariépy C, Lapointe R, Jaupart C (2000) Low mantle heat flow at the edge of the North American continent, Voisey Bay, Labrador. Geophys Res Lett 27:823–826 Marton FC, Shankland TJ, Rubie DC, Xu Y (2005) Effects of variable thermal conductivity on the mineralogy of subducting slabs and implications for mechanisms of deep earthquakes. Phys Earth Planet Inter 149:53–64 McKenzie D, Bickle MJ (1988) The volume and composition of melt generated by extension of the lithosphere. J Petrol 29:625–679 Michaut C, Jaupart C, Bell DR (2007) Transient geotherms in Archean continental lithosphere: new constraints on thickness and heat

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884 production of the subcontinental lithospheric mantle. J Geophys Res Solid Earth 112(11):04408 Nicolaysen LO, Hart RJ, Gale NH (1981) The Vredefort radioelement profile extended to supracrustal strata at Carletonville, with implications for continental heat flow. J Geophys Res 86:10,653–10,661 Nunn JA, Sleep NH (1984) Thermal contraction and flexure of intracratonal basins: a three-dimensional study of the Michigan basin. Geophys J R Astron Soc 79:587–635 Perry HKC, Jaupart C, Mareschal JC, Shapiro NM (2006) Upper mantle velocity-temperature conversion and composition determined from seismic refraction and heat flow. J Geophys Res Solid Earth 111(10):07301 Pollack HN (1986) Cratonization and thermal evolution of the mantle. Earth Planet Sci Lett 80:175–182 Rudnick RL, Gao S (2014) Composition of the continental crust. In: Rudnick RL (ed) Treatise on geochemistry (2nd ed), The crust, vol 4. Elsevier, New York, pp 1–51 Rudnick RL, Nyblade AA (1999) The thickness and heat production of Archean lithosphere: constraints from xenolith thermobarometry and surface heat flow. In: Fei Y, Bertka CM, Mysen BO (eds) Mantle petrology: field observations and high pressure experimentation: a tribute to Francis R. (Joe) Boyd. The Geochemical Society, Houston, pp 3–12 Russell JK, Kopylova MG (1999) A steady-state conductive geotherm for the north central slave, Canada: inversion of petrological data from the Jericho kimberlite pipe. J Geophys Res 104:7089–7101 Solomatov VS (1995) Scaling of temperature- and stress-dependent viscosity convection. Phys Fluids 7:266–274 Vicker PA (1997) Garnet peridotite xenoliths from kimberlite near Kirkland Lake, Canada. In: Unpublished MS thesis. University of Toronto, p 127 Xu Y, Shankland TJ, Linhardt S, Rubie DC, Langenhorst F, Klasinski K (2004) Thermal diffusivity and conductivity of olivine, wadsleyite and ringwoodite to 20 GPa and 1373 K. Phys Earth Planet Inter 143:321–336

Lithosphere, Mechanical Properties Evgueni Burov (Deceased)

Synonyms Geosphere

Definition Lithosphere (mechanical). This is the rigid (“litho” ¼ stone) outer layer of the Earth that remains mechanically strong over geological time spans. Mechanical lithosphere includes rigid layers of crust and outermost mantle capable to maintain high differential tectonic stresses, from 10 MPa to 1 GPa. Mechanical lithosphere is 1.5–2 times thinner than “seismic,” “thermal,” or “chemical” lithosphere.

Lithosphere, Mechanical Properties

Mechanical properties of the lithosphere. This term refers to the integrated strength of lithospheric plates, their rheological structure and parameters, and mechanical behavior in response to various tectonic loads.

Introduction Mechanical properties of the lithosphere are of primary importance for local and global geodynamics. In particular, compared to the convective mantle, high long-term mechanical strength makes the lithosphere a unique stress/strain guiding and accumulating envelope with lasting mechanical memory. High strength prohibits internal heat advection, so thermal conduction is main heat transfer mechanism in the lithosphere, in contrast to the convective mantle. High strength also stabilizes vertical lithological structure of lithosphere making it a stagnant layer. In contrast to viscous mantle, long-term rheology of the lithosphere is strongly influenced not only by its ductile but equally elastic and brittle properties. It is probably the nonviscous properties of the lithosphere that shape it in the characteristic plate tectonics patterns. The term lithosphere has been introduced in the second half of the nineteenth century, while the notion of mechanical lithosphere appeared in early twentieth century, in conjunction with that of seismic lithosphere (see ▶ “Earth’s Structure, Global”), after formulation of the continental drift theory by Wegener and first interpretations of regional isostasy by J. Barrel and VeningMeinesz (Watts 2001; see entry ▶ “Isostasy”). The fact that the lithosphere has finite measurable strength has been demonstrated from observations and models of regional isostatic compensation of large topographic loads. Before that, the lithosphere was considered either as a very strong solid layer (Pratt’s model) or, in turn, a weak fractured layer (Airy’s model). Postglacial rebound studies of early twentieth century have contributed to the definition of the mechanical lithosphere as the uppermost layer of the solid earth characterized by slow viscoelastic relaxation, in contrast to the underlying, relatively low-viscosity asthenosphere. The long-term mechanical base of the lithosphere, hm, is limited by the depth to isotherm 500–600 °C in oceans and 700–800 °C in continents, compared to almost twice as deep 1,330 °C isotherm delimiting the thermal lithosphere (see entries ▶ “Lithosphere, Oceanic: Thermal Structure” and ▶ “Lithosphere, Continental: Thermal Structure”). As suggested on the basis of recent mantle–lithosphere interaction models (e.g., Schmeling et al. 2008), it is the elastic and plastic properties of the lithosphere that essentially determine the geometry of lithospheric plates and the mechanisms of formation of constructive, destructive, and transform plate boundaries at global scale. At smaller scale, mechanical properties of the lithosphere control formation and evolution of major geological structures such as rifts, passive margins, foreland basins, mountain ranges,

Lithosphere, Mechanical Properties

plateau or strike-slip faults. They also control short-term processes such as seismicity (Watts and Burov 2003).

Mechanical Properties at Different Timescales Mechanical properties of the lithosphere are timescale dependent (e.g., Watts 2001). At seismic timescales (1 Myr), the thickness of the elastic core is reduced to values smaller than hm, and the stresses are slowly relaxed at timescales on the order of several Myr. The long-term properties of the lithosphere were first assessed from geodynamic-scale observations, first of all, regional isostasy. According to these data, the lithosphere exhibits a large spectrum of long-term behavior from quasi-elastic to brittle and viscous. Observation of crustal and lithosphere scale faults and distributed seismicity shows that some domains within the lithosphere deform in brittle-plastic regime over long time spans, while relaxation of deformation below, for example, oceanic volcanic islands, points to long-term viscoelastic strength. However, the long-term properties of the lithosphere are more generally studied indirectly using extrapolations from rock mechanics data.

Sources of Information on the Mechanical Properties of the Lithosphere Flexural studies and experimental rock mechanics data are major sources of quantitative information on long-term behavior and strength of the lithosphere. In structured viscous-elastic-plastic media, all rheological properties are interrelated, and various time- and scale-dependent factors may cause variations in the effective elastic, brittle, and ductile deformation. Consequently, observations of long-term deformation are of primary importance for assessment of the effective mechanical properties of the lithosphere. These and other sources of information are summarized below: 1. Observations of long-term (t > 1,000 year) lithosphere response to tectonic loading, in the order of importance: • Observations of regional isostatic compensation: gravity-flexural studies providing estimates for the equivalent elastic thickness of the lithosphere, Te

885

• Vertical motions due to postglacial rebound, lake, and volcanic island loading • Observations of lithosphere folding (folding wavelength is a function of plate strength) • Field observations of ductile and brittle deformation in the outcrops • Interpretations of deformational microstructures and paleo-stresses • Seismic tomography (e.g., evidence for more or less strong slabs at depth) • Borehole stress measurements 2. Intermediate timescale observations (t > 3–10 year): • Slow-rate rock mechanics experiments (strain rates 10
9 to 10
4 s
1, for ductile properties) • Geodetic (GPS-INSAR) data over >5 year time spans (strains, viscoelastic properties) • Inter-seismic deformation; slow earthquake data 3. Observations of deformation in response to short-term loading (t < 1 year): • Short-term rock mechanics experiments (elastic and brittle properties) • Distribution of intraplate seismicity (brittle properties) • Tidal deformation (viscoelastic properties) • Post-seismic relaxation (viscoelastic properties) • Geodetic (GPS-INSAR) data (strains, viscoelastic properties) • Attenuation of S waves (proxy to low-viscosity zones) • Magnetotelluric sounding (reduced electrical resistivity is proxy to low-viscosity zones) 4. Physical considerations and self-consistent thermomechanical models: • Estimates of the minimal integrated strength of the lithosphere required for lifetime stability of geological structures, subduction or transmission of tectonic stresses, and forces over large spatial scales, including horizontal pressure gradients caused by lateral variations in lithospheric density structure and topography (gravity potential energy theory). For example, lithosphere must be strong enough to transmit ridge push and slab pull forces on the order of 1011–1013 N per unit length. • Lithosphere scale numerical thermo-mechanical models of tectonic processes integrating multidisciplinary data, which allows for testing the validity of data and hypotheses on lithosphere rheology. Observations of long-term deformation provide key parameters such as the integrated strength of the lithosphere. These parameters are needed to constrain rock mechanics data obtained at laboratory conditions, because they are too far from geological conditions: short timescales (t < 5 years), small spatial  scales (l ~ 0.1 m), high strain rates _e > 10
9 s
1 , small strains (ε < 10), high temperatures, simple deformation, largely mono-phase samples. Rock

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mechanics data allow only for assessment of general form of rheology laws, their sensitivities, and relative strengths of different kinds of rocks. Their extrapolation to geodynamic scales (t > 106 years, l ~ 1,000 km, e_ < 10
14 s
1 , ε > 100, cold temperatures, aggregate rocks) needs re-parameterization and validation using real-scale observations and models. Interpretation of short-term data (seismic, satellite geodesy, INSAR) is not straightforward. In particular, interpretation of intraplate seismicity (see entry ▶ “Seismicity, Intraplate”) and post-seismic relaxation data is questioned due to the lack of evidence that mechanisms of this shortterm deformation can be physically linked to those of longterm deformation.

Observations of Flexural Behavior and Effective Long-Term Strength of the Lithosphere Observations of regional isostatic compensation (e.g., Watts 2001; see entry “▶ Isostasy”) have shown that the lithosphere has substantial long-term elastic rigidity that allows for transferring and maintaining intraplate stress levels (10 MPa–1 GPa) over geodynamic time spans (>several Myrs). Studies of gravity anomalies observed over mountain ranges and subduction zones have demonstrated that lithospheric plates bend like thin elastic plates of finite stiffness in response to tectonic, topography, and sedimentary loads. With improvement of geophysical measurement techniques during the second half of twentieth century, multiple studies (specifically, forward flexural models) have produced robust estimates of flexural rigidity, D, and equivalent elastic thickness, Te, of lithospheric plates. These data arguably present a major source of information on the long-term mechanical properties of lithosphere. Flexural studies reveal strong variations of lithosphere strength, from near zero at ocean ridges to 100–150 km thick quasi-elastic cores detected within old and cold cratons (Burov and Diament 1995; Kirby and Swain

2009). In flexural models of regional isostatic compensation (Watts 2001), D is varied until the model-predicted basement or Moho topography provides optimal fit to observations. Gravity data (see entry ▶ “Gravity Anomalies, Interpretation”) are used when basement topography is hidden (e.g., by sediments) or not representative of flexure (e.g., modified by erosion). The most robust gravity models are forward models. Inverse models are widespread due to the ability to cover large zones, but their results are more sensible to errors and should be cross-checked with forward models. For example, if not properly formulated, gravity admittance techniques may generate spurious results in continents (e.g., Kirby and Swain 2009). D provides a direct measure for the integrated long-term strength of the lithosphere and is linked to the equivalent elastic thickness of the lithosphere, Te: D ¼ ETe3(12(1
v2))
1, where E and v are Young’s modulus and Poisson’s ratio, respectively. Plate bending or flexure is characterized by its vertical deflection, w(x) and local radius of curvature, Rx(x) or 2 2 curvature, K ðxÞ ¼
R
1 x ¼ @ w=@x (Fig. 1). The flexural equation, when expressed using bending moment Mx(x) is rheology independent and is valid for all, elastic and inelastic plates: Mx ðxÞ

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 1 0 @2 @x2

C B   B ET 3e @ 2 wðxÞC C þ @ Fx @wðxÞ B   B12 1
v2 @x2 C @x @x A @ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |{z}

x = x0

K ðxÞ

DðxÞ

þDrgwðxÞ ¼ rc ghðxÞ þ pðxÞ where Fx is horizontal fiber force, Δr is the density contrast between surface and subsurface material (i.e., between mantle density rm and topography/sediment/water density rc), h(x) is topography elevation, and p(x) is additional surface or subsurface load. The elasticity is used just as simplest

elastic plate flexure model (oceans) –V0

rheological interpretation

x = xb exx

Δs w

x=0

x –Zn

Te

exx z Elastic layer

sxx

e

ittl

Br

Du

sxx

ctil

Brittle seismogenic layer

–M0

ð1Þ

Zn

Ts Te

e

K Z

Ductile layer

Lithosphere, Mechanical Properties, Fig. 1 Classical flexural model of oceanic lithosphere (left). Right: brittle-elastic-ductile yield stress envelope (YSE) and interpretation of the equivalent elastic thickness Te of the lithosphere. εxx(z) is flexural strain, sxx(z) is flexural stress,

K(x) is local plate curvature, Δs is differential stress, Zn(x) is neutral plain, and Ts is brittle seismogenic layer. w(x) is vertical plate deflection. V0 and M0 are boundary cutting force and flexural moment, respectively

Lithosphere, Mechanical Properties

887

rheological interpretation of bending strength: elastic bending stress is linear function of curvature and depth: sxx(x, z) ≈ (0.5Te
z)EK(1
v2). Te is thus effective parameter and should not be automatically related to any real layer within the lithosphere. For inelastic plates, Te and D have a sense of “condensed” plate strength and are direct proxies for the long-term integrated strength, B, of the lithosphere (Watts 2001). For example, for a single-layer oceanic plate (Te ¼ Te_ocean, Fig. 1): hðm

s f ðx, z, t, e_ Þ dz

B¼ 0

0

while T e

Te

ocean

ocean

B  
1 B @sxxf B ¼ B12 @y B @

Rheological Properties of Lithosphere According to Rock Mechanics Data

The long-term mechanical behavior of rocks is represented by extended Maxwell solid, in which total strain increment equals a sum of elastic, viscous (ductile), and plastic (brittle) increments while the elastic, viscous, and plastic stresses are mutually equal. The weakest rheological term thus defines the effective behavior of this solid. Goetze and Evans (1979) used Maxwell solid and rock mechanics data to introduce the yield stress envelope (YSE) of the lithosphere (Figs 1 and 2). This approach consists in predicting, for a representative background strain rate and depth–pressure–temperature profile, 1 13 the maximal yield strength Δsmax(z) as function of depth, z. If M x ð xÞ elastic differential stress estimate Δse(z) < Δsmax(z), the zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl }|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl { 0h 1C ðm deformation is elastic and differential stress Δs(z) ¼ Δse(z). C C f @ s ðz
Z n ÞdzAC ; If Δse(z)  Δsmax(z), then Δs(z) ¼ Δsmax(z) and the deforxx C A mation is brittle or ductile depending on z. 0

< hm ð2Þ

where sxxf is brittle-elastic-ductile bending stress (Burov and Diament 1995); Te is usually much smaller than hm. Mx and, hence, D can be obtained from depth integration of sxxf . D and Te may spatially vary due to their dependence on local bending that leads to localized plate weakening (called plastic or ductile hinging) in the areas of utmost flexure, for example, near subduction zones or below mountains and islands.

–2000 0

–1000

OCEANS Δσ [Mpa] 0

1000

Elastic Properties The elastic behavior is described by linear Hooke’s law: sij ¼ leii dij þ 2Geij

where l and G are Lame’s constants. Repeating indexes mean summation, δ is Kronecker’s operator. For most rocks l ≈ G ¼ 30 GPa (Turcotte and Schubert 2002).

2000 –2000

CONTINENTS Δσ [Mpa] –1000 0 1000

Moho OLIVINE

–20

2000

QUARTZ

craton

DIORITE

JS

Moho

h min m

–40

ð3Þ

h min m

OLIVINE

Depth [km]

–60 h max m

–80

JS

–100

25 Ma

50 Ma

50 Ma

–120

75 Ma

–140 –160

150 Ma

h max m

500 Ma

125 Ma

750 Ma

150 Ma

1000 Ma

175 Ma

COMPRESSION

250 Ma

100 Ma

TENSION

CB

2000 Ma

COMPRESSION

TENSION

–180

Lithosphere, Mechanical Properties, Fig. 2 Commonly inferred (brittle(Byerlee)-elastic-ductile) rheological yield stress envelopes (YSEs) as function of thermotectonic age for oceans and continents. In mantle, maximal strength can be limited by Peierls or GBS law instead of

Byerlee’s law. For continents, variations in crustal composition and fluid content result in various Jelly Sandwich, Jelly Sandwich, and more rare Crème Brûlée. (After Burov (2007))

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Lithosphere, Mechanical Properties

 e_ d ¼ Aa
m f w Dsn exp
H ðRT Þ
1

Brittle-Plastic Properties Brittle resistance, t, is a linear function of normal stress, sn and, that is, of pressure (Byerlee 1978): t ¼ 0:85sn ,

sn  200 MPa

t ¼ 50 MPa þ 0:6sn ,

1, 700 MPa > sn > 200 MPa

ð4Þ

Byerlee’s law is equivalent of Mohr-Coulomb (CoulombNavier) plastic failure criterion (e.g., Burov 2007): t ¼ C0 þ tan ðfÞsn

ð5Þ

where C0 is cohesive strength (1026 Pa s in cold mantle near Moho depth. Several ductile mechanisms such as diffusion creep, grain boundary sliding (GBS), pressure solution, and cataclastic flow may play important role at appropriate conditions, but the leading part belongs to the dislocation creep (Kohlstedt et al. 1995).  e_ d ¼ Af w Dsn exp
H ðRT Þ
1 for Ds < 200 MPa ðPower lawÞ   2 e_ ¼ Af w exp
H 1
Ds=sp =RT d

ð7Þ

for Ds > 200 MPa ðHarper
Dorn lawÞ where e_ d is shear strain rate, A is material constant, n is power law constant, fw is water fugacity factor, Δs is differential stress, R is universal gas constant, H ¼ Q + PV is creep activation enthalpy, Q is activation energy (100–600 kJ/mol), P is pressure, V is activation volume, T is temperature in K. sp is Peierls-like stress (sp ~ several GPa). For tectonically relevant Δs/sp ratios (1 GPa). Since Peierls plasticity includes strong water-weakening (H ¼ Q + p(V
βΔVw)), where ΔVw is molar volume change due to incorporation of hydroxyl ions in the main rock and β is experimental parameter, it may play an important role in localization of deformation in subduction zones. This role may be shared with GBS creep, which might be responsible for aseismic localization of deformation in the lithosphere mantle.

Mechanical Properties of Oceanic Lithosphere Versus Continental Lithosphere Due to its temperature dependence, ductile strength limits the depth to the base of mechanical lithosphere, hm, to that of the isotherm 500–600 °C in oceans and 700–800 °C in continents (in continents, higher pressure for given temperature increases yield strength via the H term in Equation 7). Oceanic YSEs (0 < Te < 50–70 km) derived from (Eqs. 3, 4, 5, 6, 7, and 8) predict brittle seismogenic behavior in the upper parts of lithosphere, where depth of brittle-ductile transition, BDT, varies from few km near ridges to 40 km near subduction zones (Figs. 2 and 3). The competent domains below the elastic core are dominated by aseismic ductile creep, where important ductile strength is preserved down to the depths of 80–100 km near subduction zones. Continental YSEs (0–5 < Te < 150 km) reflect strong rheological stratification between the upper, intermediate, lower crust and mantle lithosphere (Figs. 2 and 3). There might be several BDT depths, typically at 15–25 and 30–45 km (Watts and Burov 2003; Burov 2007, 2010). The depth to the mechanical bottom of strong ductile lithosphere may vary from 30 to 200 km. Te estimates (Fig. 3) reveal important difference between the mechanical properties of oceanic and continental lithosphere. In oceans, Te (500–700 Ma) are in stationary thermal regime; the age dependence of Te is expectedly small, but Te scatter is still strong revealing influence of factors other than thermal age. Using rock mechanics data (Fig. 4), it can be shown that continental Te should be strongly controlled by the crustal thickness and mechanical state of crust-mantle interface (Burov and Diament 1995). When the lower crust is mechanically strong, crust and mantle are mechanically coupled yielding a single strong layer with high Te:

ductile layer

curvature K(left). After Burov (2010). Homogeneous horizontal stresses will shift Te and Ts up or down in opposite directions, but will not change the character of their dependence on plate curvature

T e  h1 þ h 2 . . . ¼

X

hi

ð9Þ

n

where hi are thicknesses of crustal and mantle competent layers. Mechanical decoupling between crust and mantle leads to structural weakening, that is, dramatically smaller Te (Burov and Diament 1995): X ffi  1=3 rffiffiffiffiffiffiffiffiffiffiffi T e  h31 þ h32 . . . ¼ 3 h3i  max ðhi Þ
60 km

90 km

0

Te = 20 km

Tm < 450°C

40 km

«Indi-Asia»

JS1

stable subduction

SLA B

SLA B

Depth (km)

0

1800

0

Tm < 650°C

40 km 50 km

JS2

«Alpes 1»

1800

«double crust», short-lived subduction, pure shear

Te = 25 km B

40 km

Te = 20 km

Tm = 600°C

SLA

SLA

B

Depth (km)

0

60 km

JS3 600

0

«Alpes 2»

1800

subduction + underplating

Distance (km)

40 km

Tm = 600°C

CB 0

RT-instability + pure shear

«Carpathes» 1800

Distance (km)

Lithosphere, Mechanical Properties, Fig. 5 Numerical thermomechanical models of long-term geological processes (here, continental collision) allow for validation of inferred rheology profiles (Jelly Sandwich, JS1–JS3, and Crème Brûlée, CB) by testing the compatibility of model-predicted and observed tectonic evolution. Models JS2, JS3, and CB have similar Te (20–25 km) but different mantle strength, leading to

strong differences in resulting collision styles. The models show that strong mantle lithosphere is needed to drive continental subduction while strong crust cannot play same role (its low density prohibits subduction if there is no mantle drag). Tm – temperature at Moho. Colors: orange, yellow – crust;blue, azure – mantle;purple – oceanic slab. (Modified from Burov (2010))

mechanical models of geological processes show that predicted evolution is highly sensitive to rheology choices (Fig. 5). It has been demonstrated that in most cases mantle lithosphere has to be rheologically strong to insure stability of geological structures and the observed styles of deformation. In particular, stability of mountain ranges, continental subduction (e.g., Fig. 5, Burov 2010) and most observed rifting modes (Buck 1991; Bassi 1995) need initially strong mantle lithosphere with mechanical thickness of the mantle part more than 30–50 km. Collision models (Fig. 5) quantify relationships between the collision styles and deep lithosphere structure and rheology. They show that tectonic evolution depends not only on the integrated strength (Te) but as much on strength distribution with depth, that is, on which particular lithological layer (upper crust, lower crust, or mantle lithosphere) provides main contribution to Te. Rift models show that narrow rifts are associated with initially strong mantle lithosphere, while metamorphic core complexes “need” warm lithospheres with negligible mantle strength. Wavelengths of boudinage and deca-kilometric upper crustal fault spacing observed in rift zones also appear to be a direct function of the ductile properties of the underlying lower

crust. In compressional settings, observed folding wavelengths provide constraints on plate rheology since these wavelengths (50–1,000 km) are proportional to 5–10 times thickness of competent layers in the lithosphere and correlate with Te values (e.g., Burov 2007).

Links Between Different Timescales: Burger’s Rheology Model The abundance of short-timescale observations such as earthquake distributions or post-seismic relaxation explains attempts to interpret these data in terms of longterm rheology. Yet, these attempts are typically based on misinterpretation of Goetze’s YSEs that are valid only for geodynamic strain rates. Earthquakes and relaxation occur at locally and temporarily high strain rates that are not representative of long-term rates. Hence, the fact that continental lithosphere below Moho depth is mainly aseismic cannot be interpreted as a sign of weak ductile behavior (Fig. 4). It rather indicates that either mantle stress levels are insufficient to induce brittle sliding, or that seismogenic Byerlee’s law does not operate at high

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pressure (>36–70 km depth), or that the frictional behavior is strain-rate dependent. At seismic strain rates (1016 times higher than geodynamic strain rates), the entire lithosphere acts as a brittle-elastic body and no ductile flow can occur. Strictly speaking, Te and maximal seismogenic depth Ts should rather anti-correlate (Watts and Burov 2003, Fig. 4). However, Ts cannot be used to constrain Te without knowing intraplate stress level. The fact that Ts distributions in oceans and continents are similar, while their rheological and mineralogical structures are different, adds to the argument that seismicity is primarily related to stress levels. Post-seismic relaxation data provide controversial results yielding effective viscosities of deforming domains about 1–2 orders of magnitude smaller than postglacial rebound data that provide minimal viscosity (1019 to 5  1019 Pa s) of the Earth’s weakest layer – asthenosphere. Since these estimates are based on inversion of surface deformation, they strongly depend on initial assumptions on lithosphere structure and properties, while it is impossible to determine which layer in the lithosphere relaxes post-seismic deformation. Nevertheless, whatever is this layer, the estimated viscosities appear too low for long-term properties. To explain this controversy, one can consider Burger’s model of solids. According to this model, lithospheric behavior is described by two independent serially connected terms, one of which is Kelvin solid responsible for the primary creep (seismic, post-seismic) and the second one is Maxwell solid responsible for secondary longterm creep (geodynamic timescale). The viscosity of the first term is independent of that of the second term. In other words, the physics of deformation mechanisms activated at seismic-scale strain rates is different from that of the mechanisms acting at geodynamic strain rates.

Summary Lithosphere has important long-term ductile, elastic, and brittle-plastic strength and is capable of maintaining (not relaxing) differential stresses at geological timescales. These properties can be accessed mainly from estimates of the equivalent elastic thickness, Te, combined with rock mechanics data, validated by analytical and numerical thermo-mechanical models of geological and geodynamic processes. In oceans, Te fits in the mechanical core of the lithosphere, but in continents, it generally does not represent any particular layer due to strong rheological stratification. The brittle-plastic properties of the lithosphere are governed by Byerlee’s law to 30–40 km depth and likely by Peierls plasticity and/or GBS creep below. The ductile properties are dominated by dislocation power-law creep, which is strongly nonlinear and rock-type dependent. Highest integrated strength (Te ~ 100–150 km) is detected in cratons (Cratonic Jelly Sandwich rheology) where strong crust is mechanically coupled with strong mantle. In warmer

Lithosphere, Mechanical Properties

continental lithospheres, mechanical decoupling between crust and mantle leads to structural weakening resulting in ~50% reduction of Te (Jelly Sandwich rheology). In most cases, mantle lithosphere provides main contribution to the integrated plate strength. The cases where mantle is weaker than crust (Crème-Brûlée rheology) refer to hot lithospheres such as metamorphic core complexes. There is probably no exploitable link between short-term deformation such as earthquake data or post-seismic relaxation and long-term properties of the lithosphere.

Cross-References ▶ Earth’s Structure, Global ▶ Gravity Anomalies, Interpretation ▶ Isostasy ▶ Lithosphere, Continental ▶ Lithosphere, Continental: Thermal Structure ▶ Lithosphere, Oceanic: Thermal Structure ▶ Seismicity, Intraplate

Bibliography Bassi G (1995) Relative importance of strain rate and rheology for the mode of continental extension. Geophys J Int 122:195–210 Buck WR (1991) Modes of continental extension. J Geophys Res 96:20161–20178 Burov E (2010) The equivalent elastic thickness (Te), seismicity and the long-term rheology of continental lithosphere: time to burn-out “crème brûlée”? Insights from large-scale geodynamic modeling. Tectonophysics 484:4–26 Burov EB (2007) Plate rheology and mechanics. In: Schubert G (ed) Treatise on geophysics, vol. 6: crust and lithosphere dynamics (vol ed: Watts AB), vol 99, no 152. Elsevier, Amsterdam, 611 pp. TOGP00102. ISBN:978-0-444-51928-3 Burov EB, Diament M (1995) The effective elastic thickness (Te) of continental lithosphere: what does it really mean? J Geophys Res 100:3895–3904 Byerlee JD (1978) Friction of rocks. Pure Appl Geophys 116:615–626 Goetze C, Evans B (1979) Stress and temperature in the bending lithosphere as constrained by experimental rock mechanics. Geophys J R Astron Soc 59:463–478 Kameyama M, Yuen D, Karato S-I (1999) Thermal-mechanical effects of low-temperature plasticity (the Peierls mechanism) on the deformation of a viscoelastic shear zone. Earth Planet Sci Lett 168:159–172 Kirby JF, Swain CJ (2009) A reassessment of spectral Te estimation in continental interiors: the case of North America. J Geophys Res 114: B08401. https://doi.org/10.1029/ 2009JB006356 Kohlstedt DL, Evans B, Mackwell SJ (1995) Strength of the lithosphere: constraints imposed by laboratory experiments. J Geophys Res 100:17587–17602 McNutt MK, Menard HW (1982) Constraints on yield strength in the oceanic lithosphere derived from observations of flexure. Geophys J R Astron Soc 71:363–395 Schmeling H, Babeyko AY, Enns A, Faccenna C, Funiciello F, Gerya T, Golabek CJ, Grigull S, Kaus BJP, Morra G, Schmalholz SM, van Hunen J (2008) A benchmark comparison of spontaneous subduction models – toward a free surface. Phys Earth Planet Inter 171:198–223

Lithosphere, Oceanic Turcotte DL, Schubert G (2002) Geodynamics. Cambridge University Press, Cambridge, p 456 Watts AB (2001) Isostasy and flexure of the lithosphere. Cambridge University Press, Cambridge, p 458 Watts AB, Burov E (2003) Lithospheric strength and its relationship to the elastic and seismogenic layer thickness. Earth Planet Sci Lett 213:113–131

Lithosphere, Oceanic James S. McClain Department of Earth and Planetary Sciences, University of California, Davis, Davis, CA, USA

Definition The term “lithosphere” is applied to a variety of concepts and observations, and the actual definitions for the term often depend on the type of observations made. Most generally, lithosphere refers to a strong and cool outer carapace for the Earth. It lies above a weaker, hotter, and more mobile asthenosphere. The oceanic lithosphere comprises a crustal component and a mantle component, where the latter is significantly thicker, but models for the former are better constrained. The boundary between the lithosphere and asthenosphere (LAB) is the subject of a great deal of study and controversy and remains unresolved. Models for the oceanic lithosphere are intimately connected to the ideas about the nature of LAB, and to the theory of seafloor spreading. Because the crust and upper mantle must undergo cooling as they spread away from the mid-ocean and back-arc spreading centers, it is anticipated that temperature-dependent properties of the LAB deepen as the oceanic plates age. Thus, virtually all models for the oceanic lithosphere include the constraint that it must thicken with age. Modern models allow for a maximum thickness for older lithosphere, because cooling diminishes at ages approaching 100 Ma. Constraints for thickening of the oceanic lithosphere, and the nature of the lithosphere/asthenosphere boundary, are supplied by observations of age-dependent changes in seafloor depth and heat flow, by observations of a seafloor deformation, and by a variety of seismic and magnetotelluric observations.

Introduction The term “lithosphere” is applied to a variety of concepts and observations, and actual definitions for the term often depend directly or indirectly on the type of observations made. Most generally, lithosphere refers to a relatively strong and cool outer carapace for the Earth. The concept was initially put forth by Barrell (1914), who considered the need for a strong

893

outer layer of the Earth to sustain tectonic loads over geologic timescales. More broadly, the lithosphere is a cold outer layer that is physically, compositionally, and/or rheologically distinct from the warmer, next deepest part of the Earth’ upper mantle, known as the “asthenosphere.” Often, lithosphere is considered a primarily conductive boundary layer, and the asthenosphere is considered an adiabatically convective layer. Furthermore, most models lead to the conclusion that the older lithosphere is denser than the underlying asthenosphere, thus leading to a density instability that contributes to the formation of subduction zones. However, in detail, these ideas remain controversial, and full acceptance of any one model or definition remains elusive. Ideas about the nature of the lithosphere are closely linked to understanding the nature of the lithosphere-asthenosphere boundary, or LAB (see Fig. 1). For example, as we shall see below, the LAB is often observed because the lithosphere displays higher upper mantle seismic velocities (known as the seismic “lid”), while the asthenosphere is associated with the seismic “low-velocity zone” (LVZ) that is often observed, particularly for shear waves. The oceanic lithosphere shares many of the above concepts with its continental counterpart, but there are significant differences between the two. The term “lithospheric plate” is sometimes used to describe that part of the Earth that moves as a contiguous unit coherent plate. However, the vast majority of models for the oceanic lithosphere are intimately connected to the theory of seafloor spreading. Because the crust and upper mantle must undergo cooling as they spread away from the mid-ocean ridges and back-arc spreading centers, it is anticipated that temperature-dependent properties of the mantle and crust, including the rheology, physical properties (such as seismic velocity), and compositional changes, should deepen as the oceanic plates age. Thus, virtually, all models for the oceanic lithosphere include the constraint that it must thicken with age. Also, as we shall see, all modern models allow for a maximum thickness for older lithosphere, as cooling seems to slow or cease at ages approaching 100 Ma. This means that the uppermost young asthenosphere also must move along with the plate. Constraints for thickening of the oceanic lithosphere, the nature of the LAB, and the difference between the lithosphere and asthenosphere are supplied by changes in seafloor depth and heat flow, by observations of a seafloor deformation, and by a variety of seismic and magnetotelluric observations. We look in detail at each of these below. The nature of the LAB remains controversial. Most models argue for a boundary between the completely solid lithosphere and partially melted asthenosphere. Others argue for a hydration boundary, with a lithosphere depleted in water, while the asthenosphere retains its water (e.g., Karato and Haemyeong 1998). Others argue for a boundary that includes variation in the crystal fabric of the mantle, or solid-solid transitions.

L

894 0

Crustal Lithosphere

MOHO

-20 -40

Mantle Lithosphere

-60

Depth (km)

Lithosphere, Oceanic, Fig. 1 Cartoon defining different levels of the oceanic upper mantle and crust, as used in this entry. Density is only approximate, based on the PREM model of Dziewonski and Anderson (1981). The lower boundary of the asthenosphere or low velocity zone is poorly defined for the oceans

Lithosphere, Oceanic

Lid

-80

LAB

-100 -120 -140

Asthenosphere

Low Velocity and/or High Conductivity Zone

-160 -180 -200 -220 2.5

3

3.5

4

Psuedo Density

Different authors have used the term lithosphere to be restricted to the uppermost mantle, while others include the crust. For the purposes of this entry, we shall include the crust and uppermost mantle as components of the lithosphere. The inclusion of the crust is critical, because the crust plays an important compositional role in the creation of the lithosphere at the ridges, and because the crust plays a major role in the hydration of the lithosphere as it spreads away from the axes, and dewatering of the plate as it is subducted. For this entry, we will exclude oceanic sediments, and anomalous regions such as islands, continental margins, and arcs. We include that part of the lithosphere that is generated on seafloor spreading centers and at back-arc spreading centers.

The Mantle Component of the Lithosphere Despite the importance of the crust to lithospheric processes, it is the uppermost mantle that makes up the bulk of the observations constraining models for the lithosphere and LAB (see Fig. 1). In fact, a vast number of studies have been made of the nature of the mantle lithosphere, the mantle asthenosphere, and the boundary between them. An exhaustive review is beyond the scope of this entry, but we will try to discuss some of the key elements. Directly or indirectly, the LAB for many models depends on temperature. While a slight change in the temperature gradient in the Earth can result in a low velocity zone in the mantle, many studies suggest that the LAB is a fairly sharp transition, and thus it is a result of a more fundamental change in the mantle. Many workers argue for a boundary between a totally solid lithosphere and an asthenosphere consisting of a small but significant percentage of melt (e.g., Anderson and Spetzler 1970; Kennett and Furumura 2015; Eilon and Abers 2017). The effects of such melt may be enhanced by its

concentration at the LAB, or by the geometry of melt layering. However, partial melt models are not universally accepted (e.g., Karato and Park 2019). Others argue that the LAB may represent a hydration boundary (e.g., Hirth and Kohlstedt 1996; Karato and Haemyeong 1998). Other transitions have been proposed, including a change in the orientation of the anisotropy at the LAB, deformation-related grain size effects, and the presence of elastically accommodated grain boundary sliding (see Karato and Park 2019). Solid-state phase boundaries also may explain the transition. Most of these mechanisms for the LAB are temperature sensitive, and this suggests that the depth of the LAB must change as the lithosphere ages and cools. As we shall see below, most models include such age-related changes. Many models also imply an upper limit on the thickness of the lithosphere. To the extent that these models converge, the thickness of the lithosphere (or depth to the LAB) appears to lie between 90 and 135 km. Models Constrained by the Cooling of the Lithosphere as It Ages Measurements of heat flow and seafloor depth are directly related to the temperatures in the ocean lithosphere, and to the changes in temperature as it ages. The ocean forms an isothermal boundary layer (near 0 °C) at the seafloor, and temperatures in the lithosphere are controlled by thermal conduction and convection of heat to the seafloor from the mantle. With the recognition of the theory of seafloor spreading it was predicted that the seafloor should contract (deepen) and heat flow should decrease as it ages. Early models focused on conductively cooled models. Davis and Lister (1974) considered a simple half-space cooling model and calculated the depth as a function of age. They show that conductive cooling of the lithosphere and resulting

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contraction lead to an ocean depth anomaly (Δh) that is proportional to the square root of age and can be represented as, pffiffiffiffiffi 2 DhðtÞ ¼ pffiffiffi ae T m kt , p

ð1Þ

where age is given by t, the thermal diffusivity of the “lithosphere” is given by k, and the emplacement temperature of new crust and mantle (t ¼ 0) is given by Tm.The effective thermal expansion coefficient, αe, is adjusted for the replacement of rocks with lithosphere density (rl),by water (rw), or   rl ae ¼ ð2Þ a: rl
rw l In addition to increasing in depth, it was found that the conductive model predicts a decrease in measured heat flow (q) with 1 qðtÞ ¼ kT m pffiffiffiffiffi , kt

ð3Þ

where k is the thermal conductivity of the lithosphere. Davis and Lister (1974) and Parsons and Sclater (1977), and others, pointed out significant discrepancies between the above predictions and actual measurements made on the seafloor. The indication that seafloor depths were too deep and heat flow too low on the young seafloor was most prominent. It was suggested that hydrothermal circulation at the ridges essentially cooled the crust and upper mantle far more efficiently than conductive models, thus accounting for the observations. This was confirmed by the discovery of very high temperature venting (e.g., Corliss et al. 1979) that spectacularly confirmed the role of hydrothermal circulation in cooling the lithosphere, as well as accommodating chemical alteration of the oceanic crust (see Fig. 2a and b). In addition to the anomalous depths and heat flow in younger plate ages, studies revealed that older seafloor exhibits just the opposite effects, with seafloor depths being shallower and heat flow higher than the half-space predictions (see Fig. 2a and b). These observations suggest that cooling of older (>70 Ma) lithosphere is greatly reduced or eliminated. That is, the average seafloor depth approaches a maximum, while the heat flow approaches a minimum above older

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Lithosphere, Oceanic, Fig. 2 Different observables used to constrain models for the oceanic lithosphere, primarily its mantle component. (a) Observed heat flow data (in red) averaged in two million year bins (modified from Stein and Stein (1992) compared to predictions for a conductively cooled half space (in gray-dashed line). Heat flow is too low and scattered for ages less about 40 Ma, indicating nonconductive (hydrothermal circulation) cooling must be taking place. (b) Observed seafloor depth (in blue) compared to predictions for a conductively cooled half space (in gray-dashed). Depths are too shallow beyond 80 Ma, indicating temperatures have stabilized and heat is being supplied from below. (c) Plot of earthquake depths and effective elastic thickness on top of expected isotherms for a half space model

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lithosphere. A common explanation is that the lithosphere is a zone of conductive cooling, while the underlying asthenosphere is undergoing adiabatic convective heat flow, maintaining a limit to cooling, and thus yielding a maximum thickness to the lithosphere. Parsons and Sclater (1977) found that the model that best fit their large data set was a lithosphereasthenosphere boundary that reached a depth of 125 km under older lithosphere, with a temperature of 1350 °C. These early models have been greatly refined by better data sets and the inclusion of more constraints. In their widely cited paper, Stein and Stein (1992) found an improved fit for a larger data set, with an older lithosphere thickness of 95 km (+/
15 km), and a weakly constrained basal temperature of 1475 °C (+/
250 °C). Their model, labeled GDH1, has become a standard of comparison for more recent efforts to study the evolution of the lithosphere. Stein and Stein (1994) also considered in detail the role of hydrothermal circulation, both high temperature circulation at the spreading centers, and more diffuse and lower temperature circulation on seafloor that may persist to >60 Ma. Afonso et al. (2007) further refined models for the lithosphere by including gravity and geoid height measurements, as well as consideration of seismic velocities as constraints on the LAB and lithosphere. In their model, they include the buoyancy effects of the crust, and the effects of solid phase transitions in the mantle, and using a basal temperature of 1300 °C to find a “thermal lithosphere” thickness of 105 km. Very important is their finding that the density of the lithosphere may not be greater than the underlying asthenosphere, contrary to accepted models. Grose and Afonso (2013) consider the insulating effect of the crust, and radiative heat transport in their analysis of the temperature effects on lithosphere evolution. They find their models that include consideration of pressure-dependent thermal properties, hydrothermal circulation, and insulating oceanic crust achieve good fits to various observations. They find an asymptotic lithosphere thickness of 109 (+/
6) km if a radiative contribution is small, and 123 km if it is significant. They also find significantly lower heat flow values for young ocean lithosphere, when compared to GDH1. Korenaga and Korenaga (2016) argue that the presence of a relatively low density crust may add buoyancy to the lithosphere, not accounted for by many models. More recently, Richards et al. (2018) reexamine data after corrections for sediment and crustal thickness. They also include constraints from laboratory measurements on the temperature and pressure dependence of thermal parameters such as conductivity and thermal expansion coefficients. In their model, they predict the maximum lithosphere thickness of 135 km. Models Constrained by Rheological Observations One of the major concepts of the mantle lithosphere is that it behaves with a rheology that is distinct from that of the

Lithosphere, Oceanic

underlying asthenosphere. Specifically, it is widely considered that the lithosphere can be defined by elastic or brittle behavior while the asthenosphere has a viscoelastic or ductile rheology. These differences have been exploited in a number of studies and give distinct values for the long-term thickness of the “lithosphere.” It has long been known that vertical loads added to the Earth’s crust, such as volcanoes, mountain ranges, and glacial ice sheets, will cause the surface to subside. Similarly, removal of the load by erosion or melting ice will cause uplift. This leads to the concept of an isostatic compensation depth, at which the pressures exerted by the overlying loads are equal. However, if the lithosphere is strong and elastic, it can support and distribute that load over a region that is broader than that of the load applied. That is, there is less subsidence than would be normally expected. The resulting bending of the lithosphere under the load can be thought of as a response of an elastic plate overlying a viscoelastic asthenosphere. The wavelength and amplitude of bending are reflected in deviations in seafloor depth, and measurement of the seafloor can be used to determine an “effective plate thickness” (Watts 2001). This thickness is not necessarily the same as that of the lithosphere. The effective plate thickness as a function of age is shown in Fig. 2c (modified from Stein and Stein 1992). We see the equilibrium lithospheric thickness is far less than the estimates from overall seafloor depths and heat flow values discussed above, with maximum values reaching about 50 km, roughly corresponding to the 600 °C isotherm in the half-space cooling model. Similarly, workers have considered the depth of “intraplate” earthquakes (earthquakes occurring away from plate boundaries). The maximum depth of seismicity can be considered the boundary for brittle and nonbrittle behavior (e.g., Wiens and Stein 1983; Stein and Stein 1992), and that in turn could be considered a proxy for lithospheric thickness. As can be seen in Fig. 2C, the result yields a “lithospheric” thickness that also corresponds to the 600 °C isotherm for the half space cooling model, and a maximum thickness of about 60 km for a 100 Ma crust. However, Afonso and others (2007) argue that temperatures at the base of the seismogenic zone would be higher (700–800 °C) than those predicted by simpler models, but still much below shallower than any melting boundary. Models Constrained by Seismic and Magnetotelluric Observations The best information about the Earth’s interior structure comes from observations exploiting seismic wave propagation. This is true for the oceanic lithosphere and asthenosphere as well, although often in the context of high seismic velocity “lid” overlying a low velocity zone (LVZ). Thus, the characterization of the lithosphere is closely linked to the observations of the underlying LVZ, and the boundary

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between them, the LAB (see Fig. 1). Over the years, the major constraints on the upper mantle structure have resulted from observations of surface waves. With the development of better and more ocean bottom seismometers (OBS), body wave phases converted at the LAB have provided new, more detailed constraints. In addition, certain diffuse seismic phases, typically labeled Po and So, are believed to be scattered P and S waves that are trapped in the lid, and they can be used to further characterize the lithosphere. Surface Waves Studies

Seismic surface waves have long provided the firm evidence that a low velocity zone (LVZ) exists in the upper mantle between the Moho and the 400 km discontinuity (e.g., Forsyth 1975). Surface waves are primarily sensitive to shear wave velocities. Low velocity zones are frequently reported for shear waves and appear to be particularly welldefined under the oceans at a depth of 100–150 km. While not observed for all studies, the LVZ was present in models often enough to be included in for horizontally polarized shear waves (Sh) in the widely cited and used Preliminary Earth Reference Model (PREM, Dziewonski and Anderson 1981). Dziewonski and Anderson allow a negative velocity gradient with respect to depth below the Moho (Fig. 3). Vertically polarized shear waves (Sv) do not share this feature in PREM, requiring significant seismic anisotropy in the uppermost mantle. PREM includes this negative velocity gradient down to the Lehman discontinuity at 220 km, which is a

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Lithosphere, Oceanic, Fig. 3 Examples of shear wave velocities (km/s) and the log of electrical conductivity (S/m) for profiles extending through the crust and upper mantle. The Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981) is shown, for both vertically and horizontally polarized shear waves (Sv and Sh) are shown in blue. Results of Nishimura and Forsyth (1989) are shown in red for two different ages in the Pacific, and a conductivity profile for the NoMELT experiment is shown in Green (Evans et al. 2019)

relatively sharp increase in velocity. Thus, the PREM does not include a LAB or separate lithosphere and asthenosphere. In general, by virtue of their long wavelengths, surface wave models do not resolve details of the lithosphereasthenosphere boundary (LAB) or of the lithosphere itself. The presence of the LVZ obviously implies a high velocity lid, however, the nature of the LAB remains controversial. Several important features have been observed using surfaces waves. Perhaps most important has been the observation that the depth of the seismic LVZ increases with seafloor age (e.g., Forsyth 1975, 1977; Nishimura and Forsyth 1989) (see Figs. 3 and 4). This implies that, whatever the nature of the LAB, it must be at least partly controlled by temperature. The fact that it deepens with age, and that the lithosphere thickens with age, requires that the deeper parts of the older lithosphere initially were part of the young asthenosphere. Burgos et al. (2014) used an inversion data set comprising some 300,000 different dispersion measurements for group and phase velocities for surface Rayleigh and Love waves. They found regional variations in structure of the upper mantle, including age-related thickening of the lithosphere. In order to quantify those changes, they chose four different “proxies” for the depth of the LAB. They chose the maximum negative velocity gradient for Sv waves (the vertically polarized shear wave velocities) as one of their proxies. For another proxy, Burgos et al. (2014) used the azimuthal seismic anisotropy revealed by the shear wave inversions, and its relationship to present-day plate motion. Seismic

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Age (Ma) Lithosphere, Oceanic, Fig. 4 Lithosphere-Asthenosphere Boundary (LAB) depth for a variety of seismic studies using different techniques. We also include one magnetotelluric experiment result. Regardless of the technique, all of the studies are consistent with the concept of a lithosphere that thickens with age because of progressive cooling of the upper oceanic mantle. Most allow models where cooling is reduced or ceases in lithosphere older than 100 Ma. Burgos et al. (2014) used surface waves, and a variety of proxies to represent the boundary. Included here are results using the maximum shear wave negative velocity gradient (for vertically polarized shear waves) (Vs). Another of their proxies was the

transition from azimuthal anisotropy (C) (see text), which gives a slightly deeper LAB at older ages. Nishimura and Forsyth (1989) also use surface waves to define a LAB. Hannemann et al. (2017) used instrument receiver functions to determine LAB depth beneath the Atlantic. The Rychert and Shearer (2011) results were from shear body wave multiples (SS) reflecting from the LAB. We have only replotted their results from “normal” plate locations. Shito et al. (2015) modeled propagation of phases (Po and So) propagating in the lithosphere to constrain LAB depths. Evans et al. (2019) present an estimate of LAB depth from a magnetotelluric survey southeast of Hawaii

anisotropy both for compressional (P) and shear (S) waves has been widely observed in the oceans, particularly for the uppermost mantle (e.g., Raitt et al. 1971; Blackman and Kendall 2002; Lin et al. 2016). In the lithosphere, it is widely accepted that the anisotropy coincides with the preferred fabric of olivine in the mantle rocks. This preferred fabric is the result of strains related to plate motion at the time for formation; it is “frozen” in and preserves a record of plate motion during formation at the ridge axis. For the more fluid asthenosphere, they suggested that the anisotropy could change and reflect the present-day deformation, perhaps related to subsequent changes in present-day plate motion or to other smaller scale convection. Thus, they choose the depth where the anisotropy diverts away from present-day plate motion as the depth of the LAB. Burgos et al. (2014) find that both of the above proxies, as well as others, reveal a deepening of the LAB with age from 0 to 150 Ma. However, it is important to note that different proxies give different depths, again illustrating the problem of defining what is meant by the term “lithosphere” (see Fig. 4). For example, Burgos found a mature lithospheric thickness of about 110–120 km for the proxies shown in Fig. 4, but he also found using the maximum radial anisotropy as a proxy led to a lithospheric thickness of 70–80 km.

In addition to seismic velocities, a number of studies have shown that the seismic LVZ roughly correlates with a zone of high seismic attenuation (or low Q) (e.g., Anderson and Hart 1978; Eilon and Abers 2017). The lithosphere or lid forms a zone of low attenuation or high Q. Hammond and Humphreys (2000) argue that attenuation measurements argue for a partial melt in the asthenosphere, and thus in their model the LAB is the melt boundary. Studies Using Seismic Body Wave Phases

All of the above-discussed observations share in the fact that the depth resolution of resulting models is poor because of the long wavelengths required for deeply sampling surface waves. This characteristic tends to “smooth out” boundaries if they are present (see the excellent review of Karato and Park (2019) for more detail). Such smooth boundaries are consistent with changing varying variables such as pressure and temperature. For example, in the absence of any other type of transitionseismic, velocity gradient will depend on the ratio of the pressure gradient to the temperature gradient. Higher pressure gradients will drive the seismic velocity gradient positive, while higher temperature gradients will lead to a negative velocity gradient. The critical temperature gradient is about 6.5 °C/km (Birch 1952), and with a correction for the

Lithosphere, Oceanic

Earth’s curvature, the critical gradient is about 8.5 °C/km. These are quite reasonable geotherms for the mantle. In recent years, new advances in ocean instrumentation and data analysis techniques have improved our understanding of the oceanic upper mantle. New techniques allow studies of faint seismic body wave arrivals that may provide better vertical resolution than those provided by surface waves studies. Many recent studies have made use of the so-called “instrument response functions” to improve the resolution for models for the lithosphere, asthenosphere, and LAB. Such studies exploit the body waves traveling nearly vertically to a receiving seismometer or array of seismometers. As the body waves intersect a boundary, conversions between P waves and vertically polarized shear waves (Sv) may occur. In general, only a fraction of the incident wave energy will be converted. The size of these converted phases depends on the sharpness and velocity changes across the boundary. The resulting phases are denoted Ps (where a small fraction of the incident P wave converts to a shear wave at a target interface) and Sp (S to P conversions). For the former, the arriving P wave will be followed by a low amplitude shear wave for stations that overlie the targeted boundary. In the case of Sp, the arriving shear wave will be preceded by a lowamplitude P wave that formed at the boundary. Such converted phases can be difficult to discern from scattered energy, but with proper processing of the resulting seismograms, they can be detected and analyzed. For example, Kawakatsu et al. (2009) argue that such conversions observed in the western Pacific require a sharp LAB that cannot be explained by gradual temperature/pressure variations. They argue the data require a LAB to be less than 15 km thick and include a velocity reduction of 7–8%. The large change leads them to suggest that boundary’s effect is enhanced by partial melt that occupies horizontal layers. Such a sharp boundary probably precludes simple changes in temperature gradient as the cause of the lithosphereasthenosphere boundary. Hannemann et al. (2017) made a careful study of potential P to S conversions for a broad-band seismic array deployed in the Atlantic over 79 Ma lithosphere. Using a variety of techniques, they attempted to identify Ps conversions for the mantle, using data for both single stations, and stacked data from multiple stations. They made a tentative identification of the LAB for single-station data. If correct, it suggests the LAB is 70–80 km deep (see Fig. 4), and also argues for a relatively sharp transition. In addition to converted phases, a number of workers have examined precursors to shear wave arrivals such as those reflected from the core (ScS) or from the surface (SS) (e.g., Goes et al. 2013). Rychert and Shearer (2011) examine differences between SS phases reflecting from the Earth’s surface and those that may reflect from the LAB. Those

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differences will be subtle. However, they can be enhanced by stacking multiple seismograms from ray paths that reflect from the LAB at similar geographic locations and ages. They find that the LAB thickens with age as expected, but at depths corresponding to temperatures of 820–1020 °C (see Fig. 4). Rychert and Shearer (2011) allow for, but do not necessarily prove, the presence of a sharp boundary. They suggest the IAB may correspond to a hydration boundary, a melt boundary, or a permeability boundary. However, these temperatures are low for normal melting. Other Geophysical Studies

Additional constraints about mantle lithology are derived from the response of the Earth to electromagnetic induction caused by long wavelength electromagnetic waves that are vertically incident from sources such as fluctuations in the Earth’s magnetosphere or lightning strikes trapped by the ionosphere. In these studies, the asthenosphere correlates with a high conductivity zone (e.g., Evans et al. 2019) (see Fig. 3), while lithosphere would be represented by relatively high conductivities. In addition to the importance of seismic surface waves and body waves, observations have been made of unusual compressional (Po) and shear (So) phases that seem to be trapped in the lithosphere. These are typically characterized by a long, diffuse coda that follows the initial P and S waves propagating along the Moho. They tend to be of very high frequency compared to regular body wave phases. (Note there is some confusion of notation. Some workers specify these waves a Po and So, while some use notation as Pn and Sn. We use the former notation, and restrict the use of Pn and Sn for the first arriving mantle phases, that tend to be low amplitude.) Po and So are most prominent for over long travel paths, indicating that reverberation in a high Q medium, such as the lithosphere, is important. Sereno and Orcutt (1987) point out that reverberation in the water and sediment columns may also contribute to the complexity of the coda. However, Kennett and Furumura (2013) and Shito et al. (2013) argue for a propagation path best represented by scattering from three-dimensional heterogeneities. Kennett and Furumura (2015) model both Po and So phases using a stochastic model for the scale of heterogeneities. They find that scattering from “quasi-laminate” features with a horizontal characteristic scale of 10 km, and vertical characteristic scale of 0.5 km, provides the best fit. More work is needed in the study of these phases, but they do provide constraints of a lithospheric thickness. Using synthetic seismogram modeling, Shito et al. (2015) found that these trapped phases generated by earthquakes in the western Pacific demonstrate a systematic thickening (to 80 km.) of the lithosphere with age to at least 50 million years (see Fig. 4).

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The Crustal Component of the Oceanic Lithosphere

ship was towing the dredge bag, it was never certain where the dredge was when it collected the sample. Recently, more precisely located samples have become possible through the use of well-navigated submersibles and remotely operated vehicles (ROVs). The location of these vehicles, and any samples they recover, can be determined very precisely through acoustic techniques and the use of GPS in the ship above. In addition, a relatively new technique, rock coring or “wax coring,” has gained usage. With this tool, a weight is lowered rapidly from the ship, nearly straight down. The turnaround time is rapid compared to dredging or submersibles, so numerous samples can be collected. Because the corer is dropped straight down, rather than dragged, the location of the samples can be determined from the ship’s GPS position. The samples are small and usually volcanic glasses from the surface of the seafloor basalts, and with newer analytical tools and techniques, they can be analyzed for the detailed chemistry of the glass. Regardless of the sampling method, all of the samples are examined and described on the ship. When they are returned shore, they undergo extensive and higher precision analysis using a variety of techniques. Even from the earliest studies, basalts are the most commonly observed rock type, and they result from partial melting of ultramafic peridotites in the mantle. For rocks recovered that originated on the mid-ocean ridges, the vast majority are so-called MORBs or mid-ocean ridge basalts. MORBs are characterized by a composition that includes depletion in the incompatible elements (e.g., K, Ti, P, Ba, Pb, Th, U, Zr, Rb, and Sr) and light rare earth elements (atomic numbers between 57 and 71). These observations,

The top few kilometers of the oceanic lithosphere comprise the ocean crust. To a first approximation, the crust is viewed as the mafic member of the lithosphere and is generated by processes at the mid-ocean ridges (or at back-arc spreading centers). Three lines of evidence dominate our knowledge of the crust. These include direct sampling of seafloor rocks where they are exposed, seismic experiments that reveal the seismic velocity structure of the crust, and the study of crustal analogues exposed above sea level. These studies have converged on the so-called Penrose or “ophiolite” model (Fig. 5). While this model is highly successful in its predictions, the research in the last 20 years has led to the recognition, at least for some crust generated at ridges with slow to ultra-slow spreading rates, that the ophiolite models must be revised to include newly discovered crustal features. Samples of Crustal Rocks Oceanic crustal rocks are generally recovered from near the mid-ocean ridges before sediments have a chance to accumulate on the aging seafloor. The vast majority of these rocks are mafic basalts, however, other rocks such as gabbros and serpentinites are sometimes exposed at transform faults or on rifted mid-ocean ridges. From the Challenger expedition in the 1870s until the 1980s, most seafloor samples were collected by dredging, where a chain bag, with a steel jaw, is dragged behind a drifting or slowly moving ship. The problem with dredging, in addition to the uncertainty of success, is that because the

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coupled with a relatively low 87Sr/86Sr ratio, indicate that the mid-ocean ridge basalts originate from partial melting of a depleted upper mantle beneath the ridges. An additional constraining factor from seafloor basalts was that the detailed composition required that magma undergoes partial fractionation at shallow (i.e., lower crustal) pressures. This led to models that included a magma chamber in the lower crust. With the improved analytical instrumentation and techniques, the compositions of MORBs have been further parsed into N-MORB (normal-MORB) and E-MORB (enrichedMORB), and intermediate compositions (transitional or T-MORB). These observations generally are interpreted to indicate they originated in a heterogeneous, but still depleted, mantle source. Ophiolites, Ocean Drilling, and the Ocean Crust

Ophiolite sequences have been studied for over 100 years and have been the source of substantial controversy. Complete ophiolites (see Fig. 5) include a coherent stratigraphy extending from submarine lava flows into sheeted dikes, representing the feeders for the lava flows. Beneath the sheeted dikes are gabbros, often divided between isotropic gabbros (no preferred fabric) at the top, and layered (cumulate) gabbros at the base. Beneath the gabbros are ultramafic peridotites, varying between harzburgites, lherzolites, and dunites. In general, the peridotites are serpentinized to a greater or lesser extent, but it is unclear whether the serpentinization occurs during the original formation of the crust at the mid-ocean ridges or during emplacement and uplift onto land. Early on, ophiolites were thought to originate as oceanic crust and uppermost mantle, and to have been thrust up onto the continents by convergent processes (see, Moores 1982). Substantial debate centered on whether particular ophiolites originated in arc or back-arc settings, or whether some may have originated on midocean ridges. The former argument is primarily based on the detailed geochemistry of the ophiolite lavas, which seem to mimic those dredged from arcs. In contrast, some ophiolites are stratigraphically overlain by pelagic sediments, with no evidence for the volcaniclastic sediments observed in arc settings. Regardless of the origin of any particular ophiolite, it is clear that ophiolites provide an important analog for ocean crust generated on midocean ridges. The presence of sheeted dikes to deliver magma from a crustal magma chamber, and the presence of gabbros originating from the freezing of the magma chamber, leads to a model that is consistent with the petrologic models derived from seafloor sampling as described above. The Ocean Drilling Project, its predecessor Deep Sea Drilling Project, and successor the Integrated Ocean Drilling Project provide support for the “ophiolite” model. In particular, drilling results in the Costa Rican rift (Hole 504b) and off

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southern Mexico (Hole 1257) have penetrated through the seafloor lavas and into the sheeted dikes. For the latter site, gabbros intercalated with sheeted dikes have been reached at the base of the sheeted dike column. Seismic Studies of the Ocean Crust Except for studies of subareal ophiolites, rare exposures on the seafloor, and a few deep drilling efforts, seismic studies provide the only information about the oceanic crust below the seafloor. They provide information on structure and seismic velocities that lead to inferences about the petrology of the upper mantle and crust. After World War II, marine institutions began to exploit the seismic refraction technique to explore the Earth beneath the sea. These experiments usually yielded observations of compressional or P-wave seismic velocities, although converted shear waves were sometimes observed, and shear velocity structures could be inferred. In these early studies, theoretical and computational tools necessarily restricted models to laterally homogeneous structures, and a consistent four-layer model for the crust and overlying sediments was developed (e.g., Shor et al. 1969). The first layer was that of sediments, with velocities averaging about 2 km/s. Beneath that was “Layer 2,” and because the seafloor was known to reveal lavas in unsedimented areas, it was expected that layer 2 was the “volcanic layer.” The thickest part of the oceanic crust is “Layer 3,” with velocities of about 6.8–7.2 km/s and thicknesses on the order of 2–4 km. The lithology of layer 3, also known as the “oceanic layer,” has been controversial, with some models preferring a gabbroic composition, and others a serpentinite composition. The former model has gained general acceptance, partly because of the presence of gabbros in ophiolites (e.g., Christensen and Salisbury 1975; McClain 2003). Underlying layer 3 is layer 4, with compressional wave velocities of about 8 km/s, and corresponds with the upper mantle. The lithology of the mantle is generally taken to be peridotite, consistent with the above-mentioned models for partial melting of the mantle leading to the rise and formation of a mafic crust. Thus, the crust comprises layers 2 and 3, and the boundary between the crust and mantle, layers 3 and 4, is known as the Moho or Mohorovicic discontinuity. With better experiments, the original four-layered model was parsed into more finely divided structures. The earliest refinement was the observation of “Layer 2A.” This was the shallowest and lowest velocity portion of the upper crust, and the low seismic velocities were attributed to the high porosities of basalt flows. This interpretation is further supported by the observation that the velocities of layer 2A tend to increase to normal layer 2 values. That leads models where layer 2A thins over a few million years. This was interpreted as infilling of pores by metasomatic precipitation of minerals as the crust ages (e.g., Houtz and Ewing 1976). Such an explanation

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requires fluid circulation and chemical alteration in the crust that must continue over millions of years after the crust forms. In the 1970s, the development of new theoretical and computation tools allowed workers to move beyond the simple layered models to those that continuously vary, but usually vertically stratified models. As models grew more sophisticated, it became possible to correlate them with ophiolite structures, using both velocities and thicknesses (e.g., Christensen and Salisbury 1975; McClain 2003), and the so-called Penrose model or “ophiolite model” for the oceanic lithosphere has become the accepted (Fig. 5). For the crust, this led to models where Layer 2 was divided into an upper zone with relatively low velocities (layer 2A) discussed above, underlain by a high velocity gradient extending to layer 3. The high-gradient zone, sometimes labeled layer 2B, was attributed to a decrease in crustal porosity and is correlated with the transition to sheeted dikes of the ophiolite model. At mid-crustal levels, some experiments revealed a slight inflection in seismic velocities between the lower and upper crust (layer 2 and layer 3), while others did not (see Fig. 2). The origin of this slight transition was (and is) attributed either to a metamorphic boundary between lower and higher grade metasomatic changes in lithology, or to a transition from diabase dikes to the gabbros in the lower crust. Finally, the Moho is more gradational in these refined models, and the mantle velocities are well correlated with those expected for peridotites. The Midocean Ridges

The oceanic crust is generated by a complex interaction of magmatic, tectonic, and hydrothermal processes that take place at the midocean ridges where seafloor spreading occurs. The end result, at least on intermediate- to fast-spreading ridges, should be an ophiolite-like structure. The majority of seismic, petrological, and observational experiments over the last 2–3 decades have focused on the tectonic plate boundaries, with particular attention on the ridges. Interpretation of the data from these studies required models that allowed for two- or three-dimensional structure, rather than the laterally homogenous models from earlier. This required a breakthrough in seismic studies. Early ray-tracing techniques have given way to more robust tomographic inversion. In addition, the development of longer seismic hydrophone streamers and new processing and imaging algorithms now permits detection of subbasement (crustal and upper mantle) reflections using multichannel reflection experiments. One of the critical controversies about the structures beneath midocean ridges was the presence of a seismically resolvable axial magma chamber (AMC). As stated above, petrologic data seemed to require some sort of magma chamber in the lower crust, and the ophiolite model includes a thick gabbro layer. However, models resulting from seismic

Lithosphere, Oceanic

refraction surveys were equivocal, with some revealing crustal AMC (e.g., Orcutt et al. 1975). McClain et al. (1985) were able to provide convincing evidence for a narrow magma chamber on the East Pacific Rise at 13° north. The seismic velocity required for the chamber was 4.5 km/s, suggesting a partial melt rather than a pure magma. Further evidence was that the resolved AMC was found in an extensive tomographic experiment on the East Pacific Rise at 9o30’N (Toomey et al. 1990). For fast-spreading ridges, the question of magma chamber existence was largely resolved when a large multichannel experiment was conducted along the northern East Pacific Rise (e.g., Detrick et al. 1987). Using the long hydrophone streamers, researchers were able to resolve reflections from the base of Layer 2A, as well as the Moho. However, most striking was the presence of a reflection at midcrustal levels along most of the length of the ridge. Analysis of the reflection amplitude and the change in the sign of the reflection require that the reflections result from a sudden decrease in seismic velocity, and the explanation for this velocity inversion was the presence of an axial magma chamber. Furthermore, the large amplitudes of the reflections require a rather low velocity compared to models obtained from refraction experiments. This suggests that reflections result from pure or nearly pure melt sills. The discrepancy results because refraction experiments do not resolve the sills, but instead image an average structure made up of the sills and partial melt zones. Early models have suggested these sills form at the top of the magma chamber, and may be responsible for the isotropic gabbros observed in some ophiolites. The larger partial melt zones would form the cumulate gabbros that make up the bulk of the lower crust. Subsequent to the initial experiment of Detrick et al. (1987), similar magma reflections have been imaged on other segments of the fast or ultrafast spreading East Pacific Rise (e.g., Singh et al. 2006), as well as ridges with intermediate spreading rates (e.g., Canales et al. 2009). Most studies suggest that the crustal AMC is generally narrow. Hydrothermal Activity One of the most spectacular discoveries in the annals of earth science was the observation of vigorous hydrothermal circulation at spreading centers (e.g., Corliss et al. 1979). While the presence of hydrothermal circulation was not surprising, the exceptionally high temperatures, the deposition of hydrothermal minerals, and a vigorous chemosynthetic biological community were a major breakthrough in the study of midocean ridges. It is clear that the chemical, biological, and geological contribution of these vents is felt throughout the world’s oceans. Thus far, over 200 hydrothermal vent fields have been discovered, and as many as 1000 may lie on the world’s midocean ridges (Baker and German 2004).

Lithosphere, Oceanic

Early studies suggested that a substantial portion of the Earth’s heat was lost along the high temperature hydrothermal circulation on midocean ridges. More recent analyses indicate that more heat is lost through lower temperature circulation in older crust. However, it is clear that near-axis circulation is a major component of the crust forming process, and their discovery has led to a new appreciation for the interdisciplinary nature of midocean ridge science. High temperature venting is observed on all ridges, and at all spreading rates. Such systems require a heat source and a permeability structure that permits a focused flow of fluids to the surface. For fast-spreading ridges, heat sources are plentiful, with the presence of axial magma chambers and frequent eruptive events. Indeed, the narrowness of the axial magma chamber requires that circulation penetrates at least to the top of the chamber and/or into the gabbros at its flanks (e.g., Lister 1983). The geometry of the down-going limb of the seawater into the crust remains uncertain. It may be that it penetrates along-axis, perhaps along faults or fissures, or it may penetrate from the sides, also along small normal faults. For slower spreading ridges, the heat source is unclear. Certainly volcanic eruptions do occur, so a heat source is present at least some of the time. However, it may be that water also can penetrate to the deeper hot, but not molten, mantle rocks. The large normal faults often present on slow spreading ridges may accommodate this deeper penetration. Along-Axis Variations in Ridge Processes and SlowSpreading Ridges It has long been known that midocean ridges display a systematic variation in cross-sectional topography, varying from rifted ridges, with rift valleys of 1–2 km deep, and nonrifted ridges that display topographic highs. In general, rifting is associated with slower-spreading ridges, while nonrifted ridges exhibit rapid seafloor spreading. There are a number of exceptions to the rule. For the Reykjanes Ridge adjacent to the Iceland Hot Spot, the slow-spreading ridge is unrifted, and on the Galapagos Ridge the ridge segments closest to the hot spot also are unrifted. For ridges spreading at intermediate rates, both rifted and unrifted ridge segments are observed (e.g., the Gorda and southern Juan de Fuca Ridges) and are often separated by only a transform fault. These observations suggest that the presence or absence of a rift valley is more likely controlled by magma supply. On slow spreading ridges, where the isotherms are narrow, less melt may reach the surface unless excess temperatures and magma are supplied by a nearby mantle plume. For intermediate spreading ridges, local variations in magma supply, perhaps along-axis flow, may influence the formation of a rift valley. With the advent of multibeam bathymetric systems, it was found that ridges are segmented on a variety of scales, with offsets between segments also varying in character and

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length. Macdonald and Fox (1988) described a correlation between segment depths and cross-sectional geometry, and they suggested that these variations were linked to focused magma supply at segment centers and subsequent magma flow along axis to the segment ends. It is logical to assume that the magma supply would be tied to the resulting crustal structure and thickness, with thicker crust at the centers. The seismic reflection results of Detrick et al. (1987) seemed to support that linkage. However, it appears very little correlation or variation in thickness for spreading rates greater than 30 mm/year, and it appears that segmentation does not have a major influence on crustal thickness. For ridges that spread slowly, or very slowly, crustal thicknesses are far more variable. With the improvement of techniques, midocean ridge researchers have been able to overcome the complexities caused by topography on rifted ridges. For such ridges, typically spreading at slow rates, crustal thickness is closely tied to location along a segment, and it is likely that magma and mafic crust production are enhanced at segment centers. Tolstoy et al. (1993) correlated a thick crust in the shallow center of a rifted segment on the midAtlantic with gravity “bulls eyes,” or negative values centered at its center. The crust is thin at the segment ends, where the ridge intersects a transform. The observation is that thin crust exists at the intersections of rifted ridges and transforms, where we expect the lowest mantle temperatures, and a suppressed magma supply (e.g., Cannat 1996). The discovery of oceanic core complexes, with exposures of gabbro and/or peridotite on the seafloor, provided observational support models for “new” crustal processes at ridges with slow to ultra-slow spreading centers (Cann et al. 1997) (see Fig. 6). It was recognized that the ophiolite model may not suffice for such locations. Instead, one has to incorporate substantial rifting that dissects the crust, and exposes the lower crust and upper mantle. This rifting leads to a complex laterally heterogeneous structure in the oceanic crust and upper mantle (e.g., Canales et al. 2008). It appears that magma intrusions are not continuous beneath oceanic core complexes, and gabbros may not form a continuous layer as it does for the ophiolite model. During nonmagmatic rifting phases, we would argue that the mafic crustal thickness could be zero; i.e., the mantle peridotites exist at the seafloor. However, depending on the depth of serpentinization in the peridotites, with the consequent lowering of velocities, the seismic crust (velocities less than or equal to 7 km/s) may have a nonzero thickness (Fig. 6). The discovery of spectacular, serpentine-hosted hydrothermal circulation, at least one core complex, confirms the importance of rifting and serpentinization on the slow and ultra-slow spreading ridges (Kelley et al. 2001). It is clear that models for the oceanic crust must be refined to allow for slow to ultraslow spreading rates.

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904 Lithosphere, Oceanic, Fig. 6 Cartoon of rift-dominated processes on slow or ultraslow spreading ridges. Substantial normal faulting along a detachment fault exposes the deeper gabbros and peridotites, as volcanic/magmatic processes cannot “keep up” with the rifting. Rock units are the same as those in Fig. 5. Serpentinization of exposed or shallow peridotites will slow the seismic velocity, perhaps to crustal values. Thus, even where there are no crustal rocks, a thin “seismic crust” may be observed

Lithosphere, Oceanic

CORE COMPLEX exposed gabbro

RIFT VALLEY exposed peridotite

Detachment fault surface Extrusive basalts Sheeted dikes

Cumulate gabbro

Serpentinized periodotite

Isotropic gabbro

Cumulate ultramafics

Unserpentinized periodotite

Summary The oceanic lithosphere forms the relatively cool and strong surface layer under the oceans, and it has been the subject of numerous studies and speculation over many decades. It includes two components, the crust and uppermost mantle. However, a strict description and characterization of the lithosphere remains elusive and depends on the types of observations that are used to constrain models. Higher seismic velocities, low seismic attenuation, and low electrical conductivity characterize the mantle component of the lithosphere. It is underlain by an asthenosphere that is characterized by lower seismic velocities (the seismic low velocity zone), higher seismic attenuation (low Q), and higher electrical conductivity. The boundary between lithosphere and the asthenosphere deepens with age, i.e., the oceanic lithosphere thickens with age, a product of cooling of the plate as it spreads away from the ocean spreading center. That cooling continues for some distance but appears to decrease or cease when the lithosphere has aged on the order of 100 million years, after it has reached a thickness of 80–140 km. These maxima suggest that temperatures at the base of the older plate are maintained by convection in the asthenosphere. The oceanic crust represents the mafic component of the lithosphere and is generated by a complex interaction of magmatic/volcanic processes, tectonic processes (seafloor spreading), and hydrothermal processes. The “competition” between magmatic processes that build the crust and rifting processes that spread the ridges is age dependent. At intermediate to fast spreading ridges, the crust is best described by the so-called “Penrose” model with a consistent igneous stratigraphy (see Fig. 5). For slow to ultraslow spreading ridges, the

crust is laterally heterogeneous as rifting brings deeper layers of the crust to or near to the surface.

Cross-References ▶ Crustal Reflectivity (Oceanic) and Magma Chamber ▶ Isostasy ▶ Lithosphere, Continental ▶ Seismic Waves, Scattering

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Lithosphere, Oceanic: Thermal Structure Earl E. Davis1 and David S. Chapman2 1 Pacific Geoscience Centre, Geological Survey of Canada, Sidney, BC, Canada 2 Department of Geology and Geophysics, The University of Utah, Salt Lake City, UT, USA

Definition Lithosphere The outermost shell of the Earth that is sufficiently strong to support long-term geologic loads and transmit stress. These properties are attributed to low temperature and possibly to composition. Asthenosphere The mobile layer underlying the lithosphere where heat transfer is dominated by convection. Its properties are attributed to high temperature and trace quantities of water. Plate tectonics Lateral motion of large intact pieces of lithosphere (plates) that include both continental and oceanic regions. Mountain building, volcanoes, and earthquakes are the consequences of the relative motion between plates at their convergent, divergent, and transform boundaries.

Lithosphere, Oceanic: Thermal Structure

Boundary-layer cooling Conductive cooling of an initially hot semi-infinite medium, applied to the vertical cooling of aging oceanic lithosphere created at a seafloor spreading center. Plate cooling The description of the cooling of an initially hot layer of uniform thickness having a lower boundary maintained at a uniform temperature. Rayleigh number A dimensionless number reflecting the ratio of the forces driving and resisting convection. Nusselt number A dimensionless number reflecting the vigor of convection, equal to the ratio of the total heat transport relative to the transport that would take place by conduction alone across the same temperature differential.

Introduction The thermal structure of the oceanic lithosphere is of profound importance. One of the dominant mechanisms of heat loss from the Earth is the creation at seafloor spreading centers and subsequent cooling of oceanic lithosphere. Ocean lithosphere constitutes roughly 60% of the Earth’s surface, and the total heat flow through its surface comprises more than 80% of the global mantle heat flow (i.e., excluding the contribution from radiogenic heat production in continental crust; Jaupart et al. 2007). The thermal structure of the lithosphere plays key roles in constraining the rigidity and ultimate strength (seismic rupture potential) of the outermost layer of the Earth and influencing many geodynamic processes including the Earth’s most fundamental and defining one, plate tectonics. Driving forces for the motion of tectonic plates, and numerous consequent geological processes, are derived from the integrity and density of the oceanic lithosphere as controlled to a large degree by its thermal structure. And the volume of ocean basins, and hence the continental freeboard relative to sea level, is influenced directly by the thermal structure of oceanic lithosphere. “Lithosphere” has multiple meanings. It was first defined as the outer layer of the Earth, including the crust and upper mantle, having sufficient strength to support loads such as those imposed by mountains, volcanoes, and major river deltas (Barrell 1914). It was later identified as the high seismic velocity lid that overlies a deeper low-velocity zone in the mantle; its base was inferred to correspond to the closest approach of the geotherm to the mantle melting temperature (Gutenberg 1959). More recently and most fundamentally, it has become a geodynamic term that refers to the part of the mantle that forms relatively rigid plates, and within which

Lithosphere, Oceanic: Thermal Structure

heat is transported by conduction. Flexural strength, longterm immobility, seismic properties, and possibly composition all are properties that are used to define the lithosphere and its thickness, and all are tied to temperature. Beneath the lithosphere is the relatively mobile asthenosphere, where heat transport is dominated by vigorous (high Rayleigh number, high Nusselt number) convection. The oceanic lithosphere forms a thermomechanical upper boundary layer to a “conveyor-belt” convection system, in which the Earth’s interior heat is liberated through the seafloor as the layer moves laterally away from the locus of formation at a seafloor spreading center and is absorbed by the cooled layer when it sinks back into the asthenosphere at a subduction zone. Convection in the asthenosphere is to a large extent decoupled from the lithosphere. It is characterized by Rayleigh and Nusselt numbers of the order of 107 and 200, respectively, and is driven both by cooling at the top by heat lost through the lithosphere and by heating from the bottom and internally by deep and distributed sources derived from secular cooling and radiogenic heat production. Oceanic lithosphere includes a c. 7-km-thick basaltic oceanic crust, a product of partial melting of the upper mantle at a seafloor spreading center, and underlying upper mantle that has been cooled by an amount that gives the material its short- and long-term mechanical and elastic properties that contrast with those of the underlying mobile asthenosphere. Contrasting composition, resulting from the refractory process of producing the oceanic crust and from water extraction during asthenospheric ascent, may also play a part in defining the properties of the oceanic lithosphere. These various aspects are summarized schematically in Fig. 1. Previous reviews can be found in Davis (1989) and Jaupart and Mareschal (2007).

Key Observations Because the creation and cooling of ocean lithosphere is a transient process, observations used to constrain its thermal structure are most commonly placed in the context of lithospheric age and considered in the context of a model for transient cooling. Various observations are discussed in this section, and theoretical models are discussed in the next. Heat Flux Heat flux through the seafloor provides the most direct constraint on the thermal structure of the underlying lithosphere, although observational uncertainties are often large (Fig. 2a). Detailed studies of the thermal regime just below the seafloor commonly reveal a hydrologic regime that in young areas is dominated by high Rayleigh- and Nusselt-number pore-water convection (see discussions in Harris and Chapman 2004 and

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▶ Heat Flow, Seafloor: Methods and Observations entry, this volume), with high permeability in the uppermost few hundred meters of young igneous crust resulting from the volcanic and tectonic processes involved in crustal creation. Hydrothermal alteration of the crust slowly reduces the permeability, but the effects of convective heat transport are observed to persist for tens of millions of years (Von Herzen 2004). Heat passing advectively through the seafloor is virtually impossible to assess quantitatively and often goes unrecognized. Hence, wholesale compilations of seafloor heat flux observations are strongly biased (e.g., Stein and Stein 1992), particularly in young, sparsely sedimented areas where igneous crustal outcrops, serving as permeable ventilation points, are common. Where thick and extensively continuous accumulations of low-permeability sediment are present, advective ventilation ceases, and the heat flow becomes fully conductive and thus measurable. If circulation within the igneous crust persists, however, local heat flux variations can still be present, and large numbers of measurements must be made over a large area to obtain a meaningful average of the conductive seafloor heat flux at any given age. To avoid the biasing effects of advective heat loss, two approaches have been used. Values from global data sets have been selected to be far from areas of basement outcrop (relying on statistics for a reliable mean to emerge; e.g., Sclater et al. 1976; Hasterok et al. 2011). A more direct approach involves surveys carried out with large numbers of observation points positioned carefully in context of local and regional sediment/igneous hydrologic structure to rule out effects of ventilation and properly account for the effects of sub-sedimentary fluid flow that produces scatter (e.g., Lister et al. 1990; Davis et al. 1999; Fisher et al. 2003). From such studies, a reliable relationship between lithospheric heat flux and age has emerged, with heat flux decreasing proportionately with age–1/2 out to an age of roughly 100 Ma (Fig. 2b) and becoming fairly constant at greater ages at roughly 48 mW m
2. Seafloor Depth The most reliably measured consequence of lithospheric cooling is seafloor subsidence (Fig. 3a). Whereas heat flow determinations are affected by hydrothermal circulation in the upper oceanic crust, seafloor subsidence with increasing age away from a ridge results from lithospheric thermal contraction and hence reflects the integrated temperature state of the entire lithosphere. Bathymetry thus provides a robust constraint for understanding lithospheric thermal structure. As in the case of seafloor heat flux, seafloor depths must be considered with some care, with attention given to several potential perturbations. This vigilance requires checks on

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Lithosphere, Oceanic: Thermal Structure, Fig. 1 Schematic illustration of the creation and subsequent thermal evolution of oceanic lithosphere (a). Mechanical properties of the lithosphere are likely influenced primarily by temperature (brown shading) and possibly composition (light gray shading). Red arrows depict the migration of partial melt supplying crustal formation. Dark red arrows depict three modes of convection in the asthenosphere, including (1) upwelling beneath migrating lithosphere wherever it is thickening, (2) small-scale convection driven by cooling and associated instability below the base of the lithosphere, and (3) plumes driven by deep-seated thermal instabilities. Wherever temperatures are higher than the solidus (b), water will be absorbed by partial melt. Any migration of this melt will leave a dried residual asthenosphere with anomalous melting temperature and mechanical properties. Conductive cooling at great age may be limited to a compositionally established depth or arrested by regulated convection in the asthenosphere (as shown by the plate-cooling geotherm in (c)). Properties such as seismic velocity (d) and viscosity (e) are likely

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controlled by a combination of temperature, water content, and locally ponded partial melt. The zone of partial melt in the asthenosphere rising to supply material for crustal creation and lithospheric thickening has been discussed by Plank and Langmuir (1992) and others; the solidi (for dry and damp peridotite) are adopted from the summary of Asimow et al. (2004); the old ocean geotherm is computed using the temperaturedependent thermal conductivity structure summarized by Hasterok and Chapman (2011) including radiative heat transport (e.g., Hofmeister 1999) and an assumed radiogenic heat production rate of 0.5 mW m
3 in the oceanic crust; effects of sediment accumulation or crustal hydrothermal circulation are not included; velocity structure is adopted from surface wave tomographic results of Schaeffer and Lebedev (2013) with the localized low-velocity zone following results of Mehouachi and Singh (2018); viscosity, based on Mitrovica and Forte (2004), is highly schematic and includes an anomalous zone corresponding to the seismic low-velocity layer

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Lithosphere, Oceanic: Thermal Structure, Fig. 2 Oceanic heat flux. Data (from Hasterok et al. 2011) are plotted versus lithospheric age in (a) and versus age–1/2 in (b). Because raw heat flux data in young, sparsely sedimented areas are systematically biased by hydrothermal circulation, the data in these plots have been filtered to exclude sites where sediment thickness is less than 325 m and which are within 85 km of a known seamount. Data are drawn from all oceans. Solid dots show median heat

flux for 2 My age bins; gray shading is +/
1 standard deviation for the measurements in a bin. Open symbols represent high-quality data sets drawn from experiments where the environment of the heat flux sites is known from seismic imaging of the seafloor and other geophysical observations. The dashed and solid lines represent heat flux for halfspace and plate cooling models, respectively

isostatic compensation, sediment loading, crustal thickness variations, and sub-lithospheric density anomalies. Early data compilations were made with only modest regard to these factors (e.g., Davis and Lister 1974; Parsons and Sclater 1977); later studies, such as those of Schroeder (1984), Marty and Cazenave (1989), Johnson and Carlson (1992), Hillier and Watts (2005), and Crosby et al. (2006), have put more effort into “filtering” data to define depth/age transects that are relatively free of the effects of known or suspected major mantle convective upwellings and downwellings, crustal thickness anomalies, and thick sediment accumulations. Despite the various potential sources of “noise” (which, as in the case of heat flux data, add bias, in this case typically causing observed depths to be anomalously shallow), all compilations have shown a similar result, with seafloor depths increasing proportionately with age1/2 until 70–80 Ma (Fig. 3b). Depths at greater ages fall off of this trend and generally stabilize at about 5.6 km. Other systematic behavior has been gleaned from the compilations, most notably that the rates of subsidence vary both locally and regionally and that the rate of subsidence is related to the depth of the local ridge axis (Davis and Lister 1974; Marty and Cazenave 1989; Hillier and Watts 2005; Crosby et al. 2006). This behavior is believed to reflect variations in the temperature of the asthenosphere supplying material at and near spreading centers (Klein and Langmuir 1987).

Depth of the Lithosphere-Asthenosphere Boundary A third measureable indication of the thermal state of oceanic lithosphere is lithospheric thickness itself, identified principally by observations from seismology. Laboratory observations and field calibrations show that shear-wave velocity in mantle rocks is a strong function of temperature, with values decreasing rapidly (up to 0.1% K
1) with approach to the melting temperature (Gibb and Cooper 1998; Priestley and McKenzie 2006). Hence, shear-wave velocities determined from dispersive surface waves provide a useful constraint on temperature at depth. An even greater influence derives from the presence of partial melt, which is controlled by temperature and the presence of water. Partial melt also reduces compressional wave velocity and electrical resistivity. Surface wave data show a clear age-dependent structure for the uppermost mantle. A low-velocity zone is distributed widely in the oceans; it is thick and rises to the surface at midocean ridges and extends beneath a high-velocity lid (the “seismic lithosphere”) that thickens with age out to roughly 70 Ma (Ritzwoller et al. 2004; Schaeffer and Lebedev 2013). Overall it centered at roughly 100 km. The long wavelength of surface waves poses a severe limitation on the depth resolution of the structure defined. This limit is being overcome by use of receiver function analyses permitted by a small but growing number of long-term broadband

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Lithosphere, Oceanic: Thermal Structure, Fig. 3 Oceanic bathymetry. Data (from Hasterok et al. 2011) are plotted versus lithospheric age in (a) and versus age1/2 in (b). Bathymetry plotted is the position of the top of the igneous crust, 1-D isostatically adjusted for the effect of local

sediment loading. Solid dots are median values for 2 My bins of seafloor age; gray shading is +/
1 standard deviation for the measurements in a bin. Dashed and solid lines represent bathymetry for half-space and plate cooling models, respectively

seismometer deployments (e.g., Kawakatsu et al. 2009), by analysis of SS waveforms that contain a reflection from the base of the lithosphere (Rychert and Shearer 2011), and by active-source seismic reflection data where recording has been extended to several tens of seconds (e.g., Stern et al. 2015; Mehouachi and Singh 2018). Observations made using these methods, particularly the last, require a relatively sharp lower boundary to the lithosphere and a thin zone hosting partial melt beneath (although this inference has been debated; e.g., Olugboji et al. 2016; Niu and Green 2018). Magnetotelluric data also strongly suggest the presence of partial melt at the base of the lithosphere, distributed in a way that is sufficiently interconnected to cause anomalously high electrical conductivity with a bias in the direction of plate motion (e.g., Naif et al. 2013). Depths to the base of the lithosphere defined using these various techniques vary significantly, with a systematic difference between those estimated from shear-wave dispersion observations (gray symbols in Fig. 4) and those determined from seismic reflections and other techniques (black symbols). Both groups, however, suggest an age dependence of lithospheric thickness in younger settings and stabilization in older settings. All methods show great promise; additional studies will be needed to define just how consistent or variable the relationship between lithospheric thickness and age might be and to determine whether seismic reflections define the thickness of the lithosphere more accurately than surface wave dispersion.

Other Observational Constraints Additional constraints on lithospheric thermal structure are provided by observations of flexural rigidity, gravity, and geoid. Gravity data provide little information about the very long-wavelength signature of lithospheric cooling except to confirm that the topography is isostatic. At the shorter wavelengths of abyssal hills, seamounts, and swells, gravity is sensitive to the vertical mass distribution and can be used to discriminate whether topography is supported or unsupported by lithospheric strength and to constrain the depth of compensation of isostatic topography (e.g., Watts 2007; Crosby and McKenzie 2009). The geoid is sensitive to longwavelength density structure and provides a useful constraint on the thermally influenced density of oceanic lithosphere. A linear relationship between geoid height and age is seen over young seafloor, with a proportionality ranging between 0.1 and 0.14 m Myr
1, whereas the response is flat at ages greater than 80 Ma (e.g., Haxby and Turcotte 1978; Sandwell and Schubert 1980; De Laughter et al. 1999). Flexural rigidity (usually expressed as effective elastic thickness) and ultimate strength are strongly controlled by temperature. The first of these related properties is reflected by the long-term load-bearing capacity and is determined using the wavelength of flexure and the relationship between gravity and topography in the vicinity of large seamounts, trenches, fracture zones, and other crustal or tectonic loads. As in the case of the thickness of the high shear-wave-velocity lid, a clear age dependence has been revealed that shows that

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Lithosphere, Oceanic: Thermal Structure, Fig. 4 Oceanic lithosphere thickness measured as the depth to the seismic low-velocity zone. Data, primarily from the Pacific Basin, are plotted versus lithospheric age in (a) and versus age½ in (b). Symbols indicate types of analysis and uncertainties: gray symbols indicate estimates derived from surface wave dispersion (Leeds et al. 1974, Leeds 1975, crosses; Burgos et al. 2014, squares); black crosses are from receiver function analyses

(Kawakatsu et al. 2009 and Olugboji et al. 2016); black x’s are from SS precursor analysis (Rychert and Shearer 2011); inverted triangles are from active-source seismic reflection profiling (Mehouachi and Singh 2018); and the red diamond is from magnetotelluric data analysis (Naif et al. 2013). The dashed line shows the rate of thickening expected for half-space cooling

the effective elastic thickness is limited by temperature, although the scatter in thicknesses and inferred maximum temperatures is large, with the latter falling between 200 and 600 °C. This reflects in part the dependence of this property on the duration of loading. A limiting constraint on ultimate strength is provided by the occurrence of earthquakes, and when placed in a context of age, a relationship is also seen. The maximum depth of hypocenters increases with increasing age, and the limiting temperatures for seismogenic failure falls at roughly 600 °C (McKenzie et al. 2005). Again, the rate of loading that leads to seismogenic failure is an important factor (in addition to temperature) controlling this property. Settings in which deep earthquakes occur are normally complex (such as in subduction zones), however, and this restricts the quantitative utility of the depth distribution of earthquakes for defining the thermal structure of the lithosphere.

Boundary-Layer Cooling The simplest physical description of the thermal structure of oceanic lithosphere is provided by boundary-layer cooling theory (Parker and Oldenberg 1973; Davis and Lister 1974). Except very near the ridge axis, cooling is vertical and equivalent to cooling of an initially uniform-temperature half space. The lithosphere is defined simply as cooled asthenosphere with the boundary defined by an isotherm that deepens with age and thus distance from the ridge. Heat flux through the seafloor, Q, decreases with age, t, as Q ¼ l Ta (π k t)–1/2, and seafloor depth, h, increases as h ¼ 2 αeff Ta (k t / π) t1/2, where Ta ¼ the initial temperature of the asthenospheric material from which the lithospheric forms (relative to seawater at T ≈ 0 °C), k ¼ thermal diffusivity, αeff ¼ α r0 / (ra
rw) is the effective volumetric thermal expansivity adjusted for seawater loading, and ra and rw are the densities of the asthenosphere and seawater, respectively. The thermal properties are known to be sensitive to temperature (e.g., Hofmeister 1999); this dependence must be considered when estimating temperatures at depth (e.g., McKenzie et al. 2005), but it does not change the simple predicted dependence of subsidence and heat flux on age. The simplest application of boundary-layer cooling theory produces a heat flux singularity at the ridge axis; this is dealt with by applying a heat-balance boundary condition at the ridge axis (Davis and Lister 1974). Two important predictions of the theory are that the heat flux integrated from t ¼ 0 to t ¼ a is 2 Q(a), and that the heat flux is related to seafloor depth as Q ¼ (r cP / αeff) dh/dt,

Ways of Understanding Lithospheric Thermal Structure Because the lithosphere is not in an equilibrium state in most oceanic regions, the local thermal structure cannot be defined directly from seafloor heat flux, subsidence, or other observations. It must be estimated using transient cooling models, constrained by the observations described above along with estimates of material properties at depth.

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where cP is the specific heat at constant pressure and αeff is appropriate for 0.7 of Ta (Lister 1977). Wherever cooling follows boundary-layer behavior, this relationship can be used to obtain a robust estimate of the heat loss from the oceanic lithosphere without a need for relying on sparse, scattered, and potentially biased seafloor heat flux data. When plotted as functions of t–1/2 and t1/2 as they are in Figs. 2b and 3b and considered in light of the heat flux/ subsidence relationship given above, the heat flux and depth observations clearly demonstrate the efficacy of boundarylayer cooling theory for characterizing lithosphere younger than 70–80 Ma. The same is true for the geoid, which is predicted by boundary-layer cooling theory to decline at a rate of 0.16 m Myr
1 (Haxby and Turcotte 1978), and for the thickening of the lithosphere itself, which (as defined by seismic surface wave dispersion analysis) occurs at a rate of roughly 11 km Ma–1/2 (Fig. 4b). The cessation of simple subsidence, the flattening of the geoid, and the stabilization of the lithosphere/asthenosphere boundary depth beyond about 70 Ma, as well as the relatively constant heat flux beyond an age of roughly 100 Ma, all require something more complex than heat loss via transient cooling, however. Plate Cooling Models That ocean depths at great ages are relatively uniform and consistently shallower than those predicted by boundarylayer cooling provided the fundamental rationale for plate cooling models, which describe the lithosphere as a layer of constant thickness having a fixed lower boundary temperature (McKenzie 1967). This description was initially used out of mathematical convenience. There was little physical justification for the lithosphere to have an intrinsic thickness, or for constant heat flux to be supplied by asthenospheric convection beneath old lithosphere, until physical arguments were made for the latter by Parsons and McKenzie (1978) by way of an age-dependent onset of convection in the growing thermal boundary layer below the cooled, mechanically immobile lithosphere. Physical modeling of convection using material having a strongly temperature-dependent viscosity has shown how this phenomenon can serve to stop and possibly even reverse subsidence and later stop the decline in heat flux with age (Davaille and Jaupart 1994; Crosby et al. 2006). The key factor in this model is the growth of the thermal boundary layer beneath the conductively cooled plate. Once this layer reaches a sufficient thickness to become convectively unstable, convection begins to bring heat advectively from the large reservoir of the asthenosphere beneath. A balance is ultimately reached between the heat conducted through the lithosphere and that carried by convection in the asthenosphere. Alternate means suggested for maintaining an effectively steady-state lithosphere at great age include convection driven from below (plumes, distributed radiogenic heat, and

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secular cooling) and creation near ridge axes of a compositionally distinct layer (Oxburgh and Parmentier 1977; Morgan 1997). The first by itself is precluded as a general case by both topographic and heat flux observations which show no statistically significant influence of augmented heat flux at intermediate ages where the evolution of the lithosphere is well matched by boundary-layer cooling theory. Regulated heat flux by asthenospheric convection is also argued against by the contrast between heat flux at the base of the lithosphere beneath old oceanic and old continental regions, the latter being less than half of the former. The second seems unlikely in light of the observations of seismic reflections at the base of the lithosphere (e.g., Fig. 4), which suggest that there is a weak layer at the lithosphereasthenosphere boundary (depicted schematically in Fig. 1e). Any compositional differences resulting from the extraction of partial melt and water from the asthenosphere in the source zone for the oceanic crust (e.g., Plank and Langmuir 1992; Asimow and Langmuir 2003; see Fig. 1a) must be concentrated in a volume near the ridge axis and not carried laterally for lithospheric underplating at greater age. The effects of composition and water content must not to be discounted in terms of their influence on viscosity and melting temperature, however (e.g., Hirth and Kohlstedt 1996; Asimow and Langmuir 2003; Asimow et al. 2004). Preconditioning via partial melt of the asthenosphere from which the lithosphere is created may happen wherever asthenospheric ascent occurs to replace material lost to the thickening and laterally migrating lithosphere. Influence of Plumes Convective plumes in the asthenosphere, with defining characteristics of localized upwelling and anomalously high temperatures (possibly more than 200 K higher than typical asthenospheric temperatures; Putirka 2005), have been argued to thin or “reset” the thermal age of oceanic lithosphere (see review in Detrick et al. 1989), with the eventual collective contribution from plumes resulting in the departure from boundary-layer cooling behavior (Heestand and Crough 1981) and possibly in the supply of much of the heat flux to the base of plates (Malamud and Turcotte 1999), although these possibilities remain controversial. Locally the influence of plumes can certainly be large; surface volcanism from plumes is typically highly focused, but the associated topographic swells are large and extensive, often exceeding 1 km in amplitude and having a wavelength approaching 1000 km. A similarly broad thermal anomaly is implied for a source of the swells, but given the large thermal time constant of the lithosphere, seafloor heat flux provides an inherently poor constraint on the depth of the thermal source; attempts have been made to infer anomalies at depth with understandably mixed results (e.g., Von Herzen et al. 1989). A study of the geoid signature over the Hawaiian Swell by Moore and

Lithosphere, Oceanic: Thermal Structure

Schubert (1997) suggests that the lower part of the lithosphere is involved, but a study of the topography and gravity over swells by Crosby and McKenzie (2009) reveals an admittance (30 mGal km
1) that is consistent with the thermal density anomaly being below the base of the plate.

Driving Forces for Lithospheric Motion An element that is essential to a complete description of oceanic plates is the source of the force that drives their motion. Great attention has been given to the subject of plate driving forces ever since the formulation of the concept of plate tectonics (McKenzie 1969; Forsyth and Uyeda 1975; Runcorn 1980; Richardson 1992). Possible forces considered have been traction from large-scale convection at the base of the lithosphere (both oceanic and continental), negative buoyancy at subduction zones, and gravitational sliding away from mid-ocean ridges. Drive from basal traction forces is probably insignificant. Convection in the asthenosphere is generally too small in scale and directionally random, outside of the general flow required to maintain a mass balance between lithospheric creation at ridges and destruction at subduction zones which would generate a bias in the wrong direction. In addition, the low seismic shear velocity, high electrical conductivity, and possible presence of melt (Fig. 1) suggest that the lithosphere is mechanically decoupled from the asthenosphere. Such decoupling reduces any force transfer at the base of the asthenosphere, including near mid-ocean ridges where there could be a potential for “ridge push.” Most of the driving force for plate tectonics probably arises from the lithosphere itself, specifically the thermally derived density contrast between the lithosphere and asthenosphere. Wherever boundary-layer cooling governs the thickness of the lithosphere (i.e., where plate age is less than 70–80 Ma), gravitational forces associated with the thermally elevated lithosphere drive it down along its sloping lower boundary. This results in a body force that is constant with age (e.g., Lister 1975). The greatest role the lithosphere plays in moving itself is believed to occur at subduction zones, where cooled oceanic lithosphere sinks into the asthenosphere. The dominance of this negative buoyancy force is apparent in the global variations of plate velocities: plates bounded by subduction zones that are lengthy and consume old oceanic lithosphere, specifically the Pacific and Australian/Indian plates, move at the fastest rates. The negative buoyancy of subducting plates also provides the force for the genesis of mountain belts.

Summary A broad suite of observations gathered in context of powerful physical models have brought plate tectonics from a largely

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geometrical description of the history of the surface of the Earth to a robust theory that explains many geological processes. Among other things, this theory addresses the mechanisms by which plates form and are driven, the thermal structure of oceanic lithosphere that influences its mechanical and geodynamic behavior, and the dominant modes of heat loss from the Earth. In particular, the mathematics of halfspace cooling – applied to the physical model of seafloor spreading and constrained by carefully considered observations – leads to simple but profound approximations for the heat loss, bathymetry, and lithospheric thickness in young (< 70–80 Ma) oceanic regions of the Earth and suggests the following relationships for predicting heat flow through the seafloor (Q, in mW m
2), depth below sea level of the seafloor (d, in km), and depth below seafloor of the lithosphere-asthenosphere boundary (l, in km) as functions of age (t, in My): Q ¼ 500 t
1=2 d ¼ 2:6 þ 0:34 t1=2 l ¼ 11 t1=2 Departures from these relationships occur in regions of anomalous upper mantle temperatures and in regions of greater age, but with modifications for seafloor greater than 80 My to account for the limits of conductive cooling, these simple approximations explain heat flow, bathymetry (thermal subsidence), and lithosphere thickness for more than two thirds of the Earth’s surface. Uncertainties remain, however, and further work is required to resolve several lingering questions about lithospheric creation, cooling, and eventual stabilization. Among these are (1) to what degree is the mechanical behavior of oceanic lithosphere (particularly its resistance to thinning by convection of any form in the asthenosphere) established compositionally by the extraction of partial melt and water at ridge axes or in young regions; (2) to what degree does asthenospheric convection stimulated by top-down cooling regulate heat loss through and limit the growth of the lithosphere at great age; and (3) to what degree do plumes, the consequence of convective instabilities generated deep in the asthenosphere, contribute to heat flux at the base of plates, and do they convectively thin or merely conductively warm the plates beneath which they ascend? Addressing the first will improve constraints on plate drive generated by gravitational sliding forces, which are maximum for the case of a mechanical lithosphere being defined purely as a cooling boundary layer. Addressing the second and third will help resolve just what the primary regulating factor is that constrains the rate of heat loss from the Earth, such as the form and vigor of asthenospheric convection, the quotient of asthenospheric temperature and lithospheric thickness, or the rate of

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production and destruction of lithosphere at ridges and subduction zones. It is noteworthy that in addressing the primary topic of this chapter, the thermal structure of the oceanic lithosphere, we have had to lean heavily on indirect constraints. Even with the help of boundary-layer and plate cooling models, uncertainties of the parameters required for estimating deep lithospheric thermal structure from seafloor heat flux – specifically thermal conductivity, heat capacity, and heat production rate – lead to equivalent uncertainties in estimates of temperatures at depth. For this reason we have chosen to integrate other observations that constrain temperatures, as reflected in Figs. 1 and 4, and we urge readers to accept any estimates of deep temperatures with suitable caution.

Cross-References ▶ Heat Flow, Seafloor: Methods and Observations ▶ Lithosphere, Continental: Thermal Structure

Bibliography Asimow PD, Langmuir CH (2003) The importance of water to oceanic mantle melting regimes. Nature 421:815–820 Asimow PD, Dixon JE, Langmuir CH (2004) A hydrous melting and fractionation model for mid-ocean ridge basalts: application to the mid-Atlantic ridge near the Azores. Geochem Geophys Geosyst 5(1):24pp Barrell J (1914) The strength of the Earth’s crust. I. Geologic tests of the limits of strength. J Geol 22:28–48 Burgos G, Montagner J-P, Beucler E, Capdeville Y, Mocquet A, Drilleau M (2014) Oceanic lithosphere-asthenosphere boundary from surface wave dispersion data. J Geophys Res 119:1079–1093 Crosby AG, McKenzie D (2009) An analysis of young ocean depth, gravity and global residual topography. Geophys J Int 178:1,198–1,219 Crosby AG, McKenzie D, Sclater JG (2006) The relationship between depth, age, and gravity in the oceans. Geophys J Int 166:555–573 Davaille A, Jaupart C (1994) Onset of thermal convection in fluids with temperature-dependent viscosity: application to the oceanic mantle. J Geophys Res 99:19853–19866 Davis EE (1989) Thermal aging of oceanic lithosphere. In: Wright JA, Louden KE (eds) Handbook of seafloor heat flow. CRC Press, Boca Raton, pp 145–167 Davis EE, Lister CRB (1974) Fundamentals of ridge crest topography. Earth Planet Sci Lett 21:405–413 Davis EE, Chapman DS, Wang K, Villinger H, Fisher AT, Robinson SW, Grigel J, Pribnow D, Stein JS, Becker K (1999) Regional heat flow variations on the sedimented Juan de Fuca Ridge eastern flank: constraints on lithospheric cooling and lateral hydrothermal heat transport. J Geophys Res 104:17675–17688 De Laughter J, Stein S, Stein CA (1999) Extraction of a lithospheric cooling signal from oceanwide geoid data. Earth Planet Sci Lett 174:173–181 Detrick RS, White RS, Courtney RC, Von Herzen RP (1989) Heat flow on midplate swells. In: Wright JA, Louden KE (eds) Handbook of seafloor heat flow. CRC Press, Boca Raton, pp 169–190

Lithosphere, Oceanic: Thermal Structure Fisher AT, Stein CA, Harris RN, Wang K, Silver EA, Pfender M, Hutnak M, Cherkaoui A, Bodzin R, Villinger H (2003) Abrupt thermal transition reveals hydrothermal boundary and role of seamounts within the Cocos plate. Geophys Res Lett 30:1550. https:// doi.org/10.1029/2002GL016766 Forsyth D, Uyeda S (1975) On the relative importance of the driving forces of plate motion. Geophys J R Astron Soc 40:163–200 Gibb TT, Cooper RF (1998) Low-frequency shear wave attenuation in polycrystalline olivine: grain boundary diffusion and the physical significance of the Andrade model for viscoelastic rheology. J Geophys Res 103:27267–27279 Gutenberg B (1959) Physics of the Earth’s interior. Academic Press, New York Harris RN, Chapman DS (2004) Deep-seated oceanic heat flux, heat deficits, and hydrothermal circulation. In: Davis EE, Elderfield H (eds) Hydrogeology of the oceanic lithosphere. Cambridge University Press, Cambridge, pp 311–336 Hasterok D, Chapman DS (2011) Heat production and geotherms for the continental lithosphere. Earth Planet Sci Lett 307:59–70 Hasterok D, Chapman DS, Davis EE (2011) Oceanic heat flow: implications for global heat loss. Earth Planet Sci Lett 311:386–395 Haxby WF, Turcotte DL (1978) On isostatic geoid anomalies. J Geophys Res 83:5473–5478 Heestand RL, Crough TS (1981) The effect of hot spots on the oceanic age-depth relation. J Geophys Res 86:6107–6114 Hillier JK, Watts AB (2005) Relationship between depth and age in the North Pacific Ocean. J Geophys Res 110. https://doi.org/10.1029/ 2004JB003406 Hirth G, Kohlstedt DL (1996) Water in the oceanic upper mantle: implications for rheology, melt extraction and the evolution of the lithosphere. Earth Planet Sci Lett 144:93–108 Hofmeister A (1999) Mantle values of thermal conductivity geotherm from phonon lifetimes. Science 283:1699–1709 Jaupart C, Mareschal J-C (2007) Heat flow and thermal structure of the lithosphere. Treat Geophys 6:217–251 Jaupart C, Labrosse S, Mareschal J-C (2007) Temperatures, heat, and energy in the mantle of the Earth. Treat Geophys 7:253–303 Johnson HP, Carlson RL (1992) Variations of sea floor depth with age: a test of models based on drilling results. Geophys Res Lett 19:1971–1974 Kawakatsu H, Kumar P, Yasuko T, Shinohara M, Kanazawa T, Araki E, Suyehiro K (2009) Seismic evidence for sharp lithosphereasthenosphere boundaries of oceanic plates. Science 324:499–502 Klein EM, Langmuir CH (1987) Global correlations of ocean ridge basalt chemistry with axial depth and crustal thickness. J Geophys Res 92:8089–8115 Leeds AR (1975) Lithosphere thickness in the Western Pacific. Phys Earth Planet Inter 11:61–64 Leeds AR, Knopoff L, Kausel EG (1974) Variations of upper mantle structure under the Pacific Ocean. Science 186:141–143 Lister CRB (1975) Gravitational drive on oceanic plates caused by thermal contraction. Nature 257:663–665 Lister CRB (1977) Estimators for heat flow and deep rock properties based on boundary layer theory. Tectonophysics 41:157–171 Lister CRB, Sclater JG, Davis EE, Villinger H, Nagihara S (1990) Heat flow maintained in ocean basins of great age: investigations in the north-equatorial West Pacific. Geophys J Int 102:603–630 Malamud BD, Turcotte DL (1999) How many plumes are there? Earth Planet Sci Lett 174:113–124 Marty JC, Cazenave A (1989) Regional variations in subsidence rate of oceanic plates: a global analysis. Earth Planet Sci Lett 94:301–315 McKenzie D (1967) Some remarks on heat flow and gravity anomalies. J Geophys Res 72:6261–6273 McKenzie D (1969) Speculations on the consequences and causes of plate motions. Geophys J R Astron Soc 18:1–32

Lithospheric Magnetic Anomalies from Satellite Data McKenzie D, Jackson J, Priestley K (2005) Thermal structure of oceanic and continental lithosphere. Earth Planet Sci Lett 233:337–349 Mehouachi F, Singh SC (2018) Water-rich sublithospheric melt channel in the equatorial Atlantic Ocean. Nat Geosci 11:65–69 Mitrovica JX, Forte AM (2004) A new inference of mantle viscosity based upon joint inversion of convection and glacial isostatic adjustment data. Earth Planet Sci Lett 225:177–189 Moore WB, Schubert G (1997) Lithospheric thinning and chemical buoyancy beneath the Hawaiian Swell. Geophys Res Lett 24:1287–1290 Morgan JP (1997) The generation of a compositional lithosphere by midocean ridge melting and its effect on subsequent off-axis hotspot upwelling and melting. Earth Planet Sci Lett 146:213–232 Naif S, Key K, Constable S, Evans RL (2013) Melt-rich channel observed at the lithosphere-asthenosphere boundary. Nature 495:356–359 Niu Y, Green DH (2018) The petrological control on the lithosphereasthenosphere boundary (LAB) beneath ocean basins. Earth Sci Rev 185:301–307 Olugboji TM, Park J, Karato S, Shinohara M (2016) Nature of the seismic lithosphere-asthenosphere boundary within normal oceanic mantle from high-resolution receiver functions. Geochem Geophys Geosyst 17:1265–1282 Oxburgh ER, Parmentier EM (1977) Compositional and density stratification in oceanic lithosphere – causes and consequences. J Geol Soc 133:343–355 Parker RL, Oldenberg DW (1973) Thermal model of ocean ridges. Nature 242:137–139 Parsons B, McKenzie D (1978) Mantle convection and the thermal structure of the plates. J Geophys Res 83:4485–4496 Parsons B, Sclater JG (1977) An analysis of the variation of ocean floor bathymetry and heat flow with age. J Geophys Res 82:803–827 Plank T, Langmuir CH (1992) Effects of the melting regime on the composition of the oceanic crust. J Geophys Res 97:19749–19770 Priestley K, McKenzie D (2006) The thermal structure of the lithosphere from shear wave velocities. Earth Planet Sci Lett 244:285–301 Putirka KD (2005) Mantle potential temperatures at Hawaii, Iceland, and the mid-ocean ridge system, as inferred from olivine phenocrysts: evidence for thermally driven mantle plumes. Geochem Geophys Geosyst 6:Q05L08. https://doi.org/10.1029/2005GC000915 Richardson RM (1992) Ridge forces, absolute plate motions, and the intraplate stress field. J Geophys Res 97:11739–11748 Ritzwoller MH, Shapiro NM, Zong S-J (2004) Cooling history of the Pacific lithosphere. Earth Planet Sci Lett 226:69–84 Runcorn K (1980) Some comments on the mechanism of continental drift. In: Davies PA, Runcorn SK (eds) Mechanisms of continental drift and plate tectonics. Academic Press, London Rychert CA, Shearer PM (2011) Imaging the lithosphere-asthenosphere boundary beneath the Pacific using SS waveform modeling. J Geophys Res 116:B07307. https://doi.org/10.1029/2010JB008070 Sandwell D, Schubert G (1980) Geoid height versus age for symmetric spreading ridges. J Geophys Res 85:7235–7241 Schaeffer AJ, Lebedev S (2013) Global shear speed structure of the upper mantle and transition zone. Geophys J Int 194:417–449 Schroeder W (1984) The empirical age-depth relation and depth anomalies in the Pacific Ocean basin. J Geophys Res 89:9873–9883 Sclater JG, Crowe J, Anderson RN (1976) On the reliability of ocean heat flow averages. J Geophys Res 81:2997–3006 Stein CA, Stein S (1992) A model for the global variation in oceanic depth and heat flow with lithospheric age. Nature 359:123–129 Stern TA, Henrys SA, Okaya D, Louie JN, Savage MK, Lamb S, Sato H, Sutherland R, Iwasaki T (2015) A seismic reflection image for the base of a tectonic plate. Nature 518:85–88 Von Herzen RP (2004) Geothermal evidence for continuing hydrothermal circulation in older (>60 M.y.) oceanic crust. In: Davis EE,

915 Elderfield H (eds) Hydrogeology of the oceanic lithosphere. Cambridge University Press, Cambridge, pp 414–447 Von Herzen RP, Cordery MJ, Detrick RS, Fang C (1989) Heat flow and the thermal origin of hot spot swells: the Hawaiian swell revisited. J Geophys Res 94:13783–13799 Watts AB (2007) Crust and lithosphere dynamics: an overview. Treat Geophys 7:1–53

Lithospheric Magnetic Anomalies from Satellite Data Stavros Kotsiaros Planetary Magnetospheres Lab, NASA Goddard Space Flight Center, Greenbelt, MD, USA University of Maryland College Park, College Park, MD, USA Technical University of Denmark (DTU), Kongens Lyngby, Denmark

Synonyms Crustal (magnetic) field; Lithospheric (magnetic) field

L Definition Geomagnetic field is Earth’s magnetic field. Units are nT (nanoTesla). The geomagnetic field at the Earth’s surface has a magnitude of 25,000 nT to 65,000 nT. LEO satellites are spacecraft orbiting above Earth’s surface with an altitude of less than 2000 km. Typical altitudes of satellites measuring Earth’s magnetic field are between 300 km and 700 km. Magnetic observatories are land-based stations collecting measurements of Earth’s magnetic field. Magnetic observatories collect absolute measurements in contrast to variometer stations which measure magnetic field variations. Geomagnetic storms are perturbations in the Earth’s magnetic field caused by temporary disturbances of Earth’s magnetosphere usually caused by fluctuations in the stream of charged particles originating from the sun known as the solar wind. Space weather refers to the electromagnetic conditions in Earth’s space environment such as in the solar wind and within the magnetosphere and ionosphere.

Introduction The Earth possesses a rich magnetic field generated mainly by the motion of conducting metals, mostly molten iron and nickel, in our planet’s liquid (outer) core. The liquid core is found in a layer between ~2900 km and ~ 5150 km (Busse

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2011) deep inside Earth. This self-sustaining process which maintains Earth’s magnetic field is known as the geodynamo and produces a field with a strength of 25,000 nT near the equator and 65,000 nT near the poles making up about 95% of Earth’s total magnetic field. The remaining few percent is produced by magnetized material in the Earth’s lithosphere as well as by electric currents in the ionosphere at about 90 km to 1000 km altitude and magnetosphere, which extends to distances larger than several thousands of km, creating fields ranging from a few nT to thousands of nT at polar latitudes (Olsen and Stolle 2012). And lastly, a very small fraction is produced by induced electric currents in the Earth’s mantle, lithosphere, and ocean creating fields of a few nT. The Earth’s crust and the underlying uppermost part of the mantle make up the lithosphere which is about 100 km thick (Langel and Hinze 1998). The lithospheric magnetic field exhibits high regional variability because it depends on the local geological history, petrochemical composition, and physical properties of the magnetized rocks of the region. It can locally reach magnitudes of several hundreds or even thousands of nT, and since it is a small fraction of Earth’s total field, it is often referred to as the anomaly field and the associated magnetic signatures of the magnetized rocks are the lithospheric magnetic anomalies. Therefore, to determine the lithospheric field globally, detailed magnetic field observations with global coverage are needed. Low Earth Orbiting (LEO) satellites provide the most effective means of acquiring such observations and provide efficient coverage for the determination of Earth’s lithospheric field on a global scale. As an example, Fig. 1 shows the global distribution of magnetic ground observatories (red dots) collecting measurements of Earth’s magnetic field versus the measurements taken along a satellite orbit during 24 h (yellow ground track). It is clear that the present ground observatory network provides a sparse data distribution compared to the spatial coverage (even from only 24 h) of satellite observations. Continuous satellite data for several months or even years provide a dense net of magnetic field observations far superior to what is possible with observations close to Earth’s surface. However, due to orbital mechanics, the orbits of LEO satellites used in geomagnetism are usually inclined, that is, the angle between the orbit plane and the equatorial plane is not exactly 90° (it is usually ~90° 3° to 6°), which results in leaving a small cap around the two geographic poles unsampled (see polar views on Fig. 1). These unsampled regions are known as “polar gaps.”

The Lithospheric Field from Space An extensive review of measuring the Earth’s lithospheric field with satellites is given by Langel and Hinze (1998) from the Polar Orbiting Geophysical Observatories (POGO) launched between 1965 and 1971, until the Ørsted satellite

Lithospheric Magnetic Anomalies from Satellite Data

which was launched in 1999. POGO were the first satellites to provide global measurements of the Earth’s magnetic field whereas Ørsted marked the start of the International Decade of Geopotential research, an international effort to promote and coordinate a continuous monitoring of the geopotential (magnetic and gravity) field variability in the near-Earth environment. Two missions followed after the launch of Ørsted which provided the most detailed data for lithospheric field modeling and enhanced our knowledge of the global contour of Earth’s magnetized lithosphere. The first mission, CHAMP, was launched in 2003 and the second one, called Swarm, was launched in 2013. Ørsted, CHAMP, and Swarm have provided continuous measurements with an extended time-space coverage that helped improve our ability to characterize and understand the various sources that contribute to Earth’s magnetic field. Currently, Swarm is the only operational mission dedicated to measuring Earth’s magnetic field. It is a constellation of three identical satellites two of which are separated by ~150 km (at the equator and < 150 km in higher latitudes) in the east-west direction and orbit as a pair at lower altitudes (460 km) and a third flying at higher altitude (530 km) sampling different local times than the lower pair. The Swarm satellite mission provides a new dimension of detecting lithospheric anomalies from space. By using simultaneous (or near-simultaneous) measurements from the lower satellite pair, cross-track gradients (which approximate east-west gradients at nonpolar latitudes) of the magnetic field can be estimated. Gradients suppress the signals from external sources and specifically amplify the smallscale lithospheric anomalies. A full basis of magnetic space gradiometry, that is, the measurement of magnetic field gradients from satellites, and their advantages in the lithospheric field exploration is given by Kotsiaros and Olsen (2012). Specifically, east-west gradients contain valuable information on North-South oriented features of the lithospheric anomalies. Similarly, information on east-west features requires knowledge of the North-South gradient. North-South gradients can be approximated by along-track gradients (at nonpolar latitudes) which in turn can be estimated by finite differences along the satellite’s track. However, on the contrary to cross-track gradients, along-track gradients are not instantaneous since they are estimated from measurements from a single satellite taken at different time instances. Therefore, the spacing of the along-track finite differences is important for maximizing the signal to noise ratio. Kotsiaros et al. (2015) has shown the optimal estimation of along-track gradients and their added benefit for lithospheric anomaly detection using data from the CHAMP mission. Figure 2 shows how lithospheric anomalies are seen  by dB (a) vector data (the radial component Br) (b) across-track df , dB and (c) along-track dy gradients. While across-track gradients amplify north-south structures, along-track gradients amplify

Lithospheric Magnetic Anomalies from Satellite Data

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Lithospheric Magnetic Anomalies from Satellite Data, Fig. 1 Ground track of 24 h of a satellite orbiting at ~450 km altitude (yellow curves). The red dots indicate the locations of ground-based magnetic observatories. (Source of Earth’s background image: NASA)

L

east-west structures. Evidently, both gradients contain additional information and show anomalies with a higher level of detail compared to vector data. In general, both the magnitude and spatial scale of the lithospheric anomaly signals are attenuated at satellite altitude. The process of propagating the measured anomaly signals from satellite altitude down to the Earth’s surface is called downward continuation. Although downward continuation is a very useful tool for estimating the magnetic field at the Earth’s surface, systematic errors are also propagated and amplified together with the signal. This creates artifacts in the mapping of lithospheric anomalies.

Mathematical Background The magnetic field measured by a satellite is a superposition of contributions from various sources and their separation based on magnetic field measurements is challenging. The

lithospheric magnetic field signal is therefore entangled with the signals originating from the Earth’s core, ionosphere, and magnetosphere. The first attempt to separate the various magnetic field contributions was performed more than 100 years ago by Gauss (1877) (a revised translation from the German text is provided by Glassmeier and Tsurutani (2014)). Gauss introduced the concept of spherical harmonic analysis and the various sources can be separated based on the fact that each of their contributions has a specific spatial and temporal characteristic. For example, with Earth’s surface as reference, the magnetic field sources can be separated into “internal” originating in the Earth’s core and lithosphere and “external” originating in the ionosphere and magnetosphere. The magnetic field due  to internal sources varies as a function of ~ar , with a ¼ 6371.2 km (the mean Earth’s radius) and r > a the radial distance of the measurement location, whereas   the field due to external sources varies as a function of ~ ar .

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Lithospheric Magnetic Anomalies from Satellite Data

Lithospheric Magnetic Anomalies from Satellite Data, Fig. 2 Lithospheric anomalies at satellite altitude (450 km) detected by (a) vector data    dB (radial component, Br), (b) across-track df , and (c) along-track dB dy gradients

In regions without electric currents, such as along most part of the satellite orbits, the magnetic field, B, can be written as the (negative) gradient of a scalar potential V, B ¼
∇V

ð1Þ

and because of ∇B ¼ 0, the potential V has to obey Laplace’s equation: ∇2V ¼ 0. The scalar potential V can be split into an internal part V int and an external part V ext, which correspond to the internal and external magnetic field sources, that is, V ¼ V int þ V ext

ð2Þ

and from Eq. (1) we have: B ¼
∇V int
∇V ext ¼ Bint þ Bext :

ð3Þ

The internal and external potentials V int and V ext can be expanded in series of spherical harmonics (which are solutions of Laplace’s equation):

V int ¼ a

N int X n X n¼1 m¼0

 a nþ1 m m  gm cos mf þ h sin mf Pn ð cos yÞ ð4Þ n n r V

ext

¼a

N ext X n X n¼1 m¼0

  r n m m  qm cos mf þ s sin mf Pn ð cos yÞ, n n n

ð5Þ

where a ¼ 6371.2 km is the Earth’s mean radius, (r, θ, ’) are geocentric spherical coordinates, Pm n ð cos yÞ are the associated Schmidt semi-normalized Legendre functions of degree m m m n and order m, and gm n , hn and qn , sn are the spherical harmonic expansion coefficients describing the internal and external sources respectively. For more details see Chapman and Bartels (1940) and Langel (1987). Theoretically, the series extend up to infinite degree, but since in practice there is only a limited number of measurements at discrete points, the series are truncated at a maximum degree Nint (Next) for the expansion of the internal (external) potential. The estimation of the internal (or external) spherical harmonic coefficients m m m gm n , hn (or qn , sn ) from discrete satellite measurements of B is known as spherical harmonic analysis and the set of the estimated coefficients is referred to as a (global) magnetic field model. From the estimated spherical harmonic coefficients (the model), we can calculate the magnetic field values B at any given point. This process is called spherical harmonic synthesis. LEO satellites that measure Earth’s magnetic field orbit above the highly conducting layer of the ionosphere where the ionospheric currents flow (at about 110 km altitude) therefore they see the ionospheric field as an “internal” field. The separation of the lithospheric and ionospheric fields is challenging since both sources are internal to satellite observations. Therefore, lithospheric magnetic anomalies are often contaminated by ionospheric field signals in the satellite data. A standard technique to reduce contamination is to perform data selection and choose night time data when the conductivity of the ionosphere is low and the associated ionospheric magnetic field is significantly weak. An alternative, but also more complicated approach is to model the ionospheric magnetic field

Lithospheric Magnetic Anomalies from Satellite Data

simultaneously to the lithospheric field, an approach commonly known as comprehensive modeling (e.g., Sabaka et al. 2004). Another difficulty is the amalgamation of the lithospheric field with that originating in the Earth’s core. Specifically, at spatial scales of ~3000 km the core field overlaps the lithospheric field making the derivation of separate high-quality magnetic models for the core and lithosphere at those spatial scales very challenging. Various techniques for modeling the lithospheric field are reviewed by Thébault et al. (2010).

Global Models of Lithospheric Magnetic Anomalies Global models of the Earth’s magnetic field were derived from satellite observations more than 50 years ago using measurement of the field intensity. However, intensity measurements alone do not provide enough information to derive unique global models and vector observations are required. The CHAMP mission has provided high-quality vector data with increased sensitivity to Earth’s lithospheric field due to CHAMP’s low altitude ( 1) – such as for ferrimagnetic magnetite particles – ko approaches the “self-demagnetization” limit value of 1/N, and the magnetic anisotropy is controlled by particle shape. The degree of magnetic susceptibility shape anisotropy in a grain is defined as P ¼ ð1 þ ki N c Þ=ð1 þ ki N a Þ:

Magnetic Anisotropy, Fig. 1 Surface magnetic poles and internal demagnetizing field Hd for an ellipsoidal grain with uniform axial magnetization M

Therefore, the net effect of the shape anisotropy is to favor the orientation of the induced magnetization along the long

Magnetic Anisotropy

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axis of a magnetic grain, where the internal demagnetizing field is minimized. Magnetostriction: Stress Anisotropy The application of an external stress to a rock may cause reversible as well as irreversible changes to its magnetic anisotropy. In fact, straining a ferromagnetic crystal can rotate its spontaneous magnetization away from the easy axis given by magnetocrystalline anisotropy. Moreover, the application of a magnetic field which rotates the spontaneous magnetization of a ferromagnetic crystal away from its intrinsic preferred direction can strain the lattice and modify the crystal shape, producing changes in its demagnetization factors and shape anisotropy. Both processes are expression of the phenomenon known as magnetostriction and are due to changes in the exchange energy when modifications in the magnetization or in the crystal shape alter the spin-orbit coupling in neighboring atoms. Magnetostriction is defined as the spontaneous change in crystal dimensions that accompanies the process of magnetization or as the change in the magnetization of a crystal as a result of the application of stress. The linear saturation magnetostriction constant l is defined as the fractional change in the length ΔL/L of a demagnetized ferromagnetic crystal as its magnetization increases from zero to saturation. Magnetostriction is positive when the crystal expands in the direction of magnetization. In magnetite, at room temperature, the magnetostriction depends crucially on the direction, being positive for a magnetization along a easy axis but negative for magnetization along a hard axis. In a cubic crystal, the anisotropy of l is given by (e.g., Dunlop and Özdemir 1997)
 l ¼ 3=2 l100 a1 2 g1 2 þ a2 2 g2 2 þ a3 2 g3 2  2=3 þ 3 l111 ða1 a2 g1 g2 þ a2 a3 g2 g3 þ a3 a1 g3 g1 Þ, where αi (i ¼ 1, 2, 3) are the direction cosines of saturation magnetization (Ms) with respect to the , , and crystallographic axes, respectively, and γi are the equivalent direction cosines of the direction along which the magnetostriction is being measured. l100 and l111 are the crystal’s magnetostriction constants along the and crystallographic axes for M aligned in the same direction. In the case of isotropic magnetostriction (i.e., ls ¼ l100 ¼ l111), the magnetoelastic energy caused by the application of a stress s to a magnetic crystal is given by E ¼ 3=2ls s sin 2 y where θ is the angle between Ms and s.

In this simple case, the energy is minimum when Ms is parallel to s and lss > 0 or when Ms is perpendicular to s and lss < 0. Thus, the application of a uniaxial stress will cause Ms to rotate toward the stress axis if lss > 0 (i.e., under the combination of a compression and a negative ls or of a tension and positive ls). In general, the relationship between the change in the magnetic anisotropy and the increase of strain induced by a stress field is not simple, reflecting the complex behavior due to the superposition of the strain effects upon an already magnetically anisotropic specimen. Exchange Anisotropy Exchange anisotropy results from superexchange interaction across the interface of two different magnetically ordered phases. In the simplest model, the origin of exchange anisotropy is considered to be the coupling of a ferromagnetic spin system to an antiferromagnetic spin system, separated by a planar interface. To obtain a preferred direction of the coupling, it is necessary that Curie temperature TC of the ferromagnet is greater than the Néel temperature TN of the antiferromagnet. When a magnetic field is applied at a temperature TN < T < TC, the spins of the ferromagnet will orient parallel to the applied field. Then, as the material is cooled through TN, the spins of the antiferromagnetic lattice closest to the ferromagnet will align in the same direction as the ferromagnet, and subsequent spin planes in the antiferromagnetic lattice will orient antiparallel to each other. When the antiferromagnet is fully ordered, with high magnetocrystalline anisotropy, it holds the magnetization of the ferromagnetic material in the direction of the applied field, giving rise to a unidirectional magnetic anisotropy. Exchange anisotropy can be detected from a shifted hysteresis loop when the material is cooled in a magnetic field and from the presence of a sinθ term in torque curves measured in high fields due a rotational hysteresis loss. Exchange anisotropy in natural minerals has been found in titanomagnetite or in the intergrowth of maghemite with hematite and has been invoked to explain self-reversals in the direction of magnetization of natural rocks. In the last decades, there has been an increase of interest about this magnetic phenomenon because of the importance of its technological applications in spin-valve devices, such as hard disk read heads, and in magnetic thin layers.

Anisotropy of Magnetic Susceptibility (AMS) Magnetic anisotropy of rocks is often determined by means of the analysis of the anisotropy of the magnetic susceptibility (AMS), which is the property that has found the most applications in geophysical studies so far. The anisotropy of magnetic susceptibility of rocks depends on mineral content,

M

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their relative abundance, intrinsic susceptibility and anisotropy, preferential crystallographic orientation, and grain shape alignment. Therefore, knowing the composition of the rock-forming minerals and their magnetic anisotropy characteristics, it is possible to determine the spatial distribution of the grains and infer the geological processes that originated it. A preferential orientation-distribution of mineral grains is in fact typical of almost all rock types, and it develops during various geological processes, such as water flow in sediments, magma flow in igneous rocks, ductile deformation in metamorphic rocks, and even incipient strain in the paramagnetic clay matrix of apparently undeformed fine-grained sediments. The magnetic susceptibility is the capability of a material to be magnetized under the effect of an external magnetic field. Being all the materials “susceptible” to become magnetized in the presence of an applied magnetic field, the magnetic susceptibility describes this transient magnetism within a material sample. If the magnetic field is relatively weak, the magnetization of a rock is a linear function of the intensity of this field. The low-field magnetic susceptibility is defined as the ratio of the induced magnetization (M dipole moment per unit volume or J dipole moment per unit mass) to the applied low-intensity magnetic field (H). Only for isotropic substances, the induced magnetization is strictly parallel to the applied field, and the magnetic susceptibility is a scalar. In anisotropic media, like minerals and rocks, the induced magnetization is not parallel to the applied field, and the magnetization induced along the direction i is related to the magnetic field acting along the direction j by

Magnetic Anisotropy

can be represented by a set of coefficients (kij) that form a second-order symmetric tensor. Each coefficient quantifies how much a body can be magnetized along a certain Cartesian direction according to a Cartesian component of the applied field. The relationship between the components of the magnetization Mi (i ¼ 1, 2, 3) and the components of the magnetic field Hj (j ¼ 1, 2, 3) is expressed by the equations M1 ¼ k11 H1 þ k12 H 2 þ k13 H 3 M2 ¼ k21 H1 þ k22 H 2 þ k23 H 3 M3 ¼ k31 H1 þ k32 H 2 þ k33 H 3 which can be rewritten in subscript notation as M i ¼ kij H j where the coefficient kij defines a matrix that has six independent elements, since the second-order tensor must be symmetric to guarantee real eigenvalues, and kij is imposed equal to kji. Among all possible Cartesian reference systems, there exists one in which the non-diagonal terms of the tensor are zero so that the above equations simplify to M1 ¼ k11 H 1 M2 ¼ k22 H 2 M3 ¼ k33 H3

where in this case the three coefficients k11, k22, and k33 are the eigenvalues of the magnetic susceptibility tensor: they are called the principal susceptibilities and are generally indicated 3 J i ¼ wij H j ðmass specificÞ w is given in SI units of m =kg as kmax  kint  kmin (or k1  k2  k3), the maximum, intermediate, and minimum susceptibilities. or The corresponding eigenvectors are the maximum, intermediate, minimum principal susceptibility directions that are M i ¼ k ij H j ðvolume specificÞ k is dimensionless in SI units the directions along which the induced magnetization is strictly parallel to the direction of the applied field. The On the basis of the magnetic susceptibility, all substances magnetic susceptibility tensor may be represented geometrimay be classified as diamagnetic, paramagnetic, and cally by a triaxial ellipsoid, which is termed the magnitude ferromagnetic. ellipsoid, whose axes are parallel to the AMS tensor eigenIn diamagnetic minerals, the induced magnetization vectors and whose semi-axes lengths are proportional to the increases linearly with the increasing field, but in the opposite AMS tensor eigenvalues, (Fig. 2). We will refer to this direction, the magnetic susceptibility is therefore negative and ellipsoid in the following as the AMS ellipsoid. Since for typically very low (k ~ 105 SI). In paramagnetic minerals, diamagnetic minerals it is possible to have negative AMS the magnetization increases linearly with increasing field, eigenvalues making the magnitude ellipsoid difficult to visualong the same direction, and the susceptibility is positive alize, the magnetic susceptibility ellipsoid may also be (k ~ 104 – 105 SI). In ferromagnetic minerals, the induced expressed by the representation quadric, in which the length magnetization increases nonlinearly with increasing field, and of the radius vector in any direction from the origin is equal to the magnetic susceptibility is typically positive and much the inverse square root of the susceptibility in that direction. It higher (k ranges from 103 to 10 SI) than in diamagnetic has however a less direct relationship to the eigenvalues. and paramagnetic minerals. In practical measurements, the AMS eigenvectors are If the applied field is so low to approximate as linear the determined with two uncertainty angles, which define magnetic response of the body, the magnetic susceptibility the regions where each principal susceptibility direction lies

Magnetic Anisotropy

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Magnetic Anisotropy, Fig. 2 The magnetic susceptibility ellipsoid. The AMS tensor may be geometrically represented by a triaxial ellipsoid, in which the three orthogonal axes correspond to the AMS eigenvectors, kmax, kint, and kmin, respectively. The orientation of the AMS eigenvectors is then defined in a reference system of Cartesian coordinates (X, Y, and Z)

Z kmin

Y kmax kint X

with a probability of 95% (Fig. 3). The definition of a magnetic susceptibility ellipsoid identifies two important elements: a “magnetic foliation” (the plane orthogonal to the direction of minimum magnetic susceptibility) and a “magnetic lineation” (the direction of maximum magnetic susceptibility). AMS ellipsoid shapes are classified according to the relationships between the magnetic susceptibility eigenvalues: k1 ≈ k2 ≈ k3; isotropic susceptibility, the AMS ellipsoid is a sphere (Fig. 4a). k3 « k2 ≈ k1; the AMS ellipsoid has an oblate shape (i.e., the magnetic fabric is planar, Fig. 4b). k1 » k2 ≈ k3; the AMS ellipsoid has a prolate shape (i.e., the magnetic fabric is linear, Fig. 4c). k1 > k2 > k3; the AMS ellipsoid is triaxial (Fig. 4d). The orientations of principal susceptibilities for a set of samples are conventionally visualized in a Schmidt equalarea projection, lower hemisphere (Fig. 5). The bulk low-field magnetic susceptibility of a rock is a summation of the contribution from all mineral species composing the rock, weighted according to their relative abundance and susceptibilities. Obviously, the contribution of ferromagnetic minerals to the overall magnetic susceptibility of a rock specimen is overwhelming upon those of the paramagnetic and diamagnetic matrix. The contribution of the matrix will be negligible when the percentage of ferromagnetic minerals exceeds 0.1% (k > 3000–4000 mSI). Conversely, for rocks in which k < 200–300 mSI, the magnetic

M

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K3 Magnetic Anisotropy, Fig. 3 Relationship of the 95% confidence ellipses (e12, e23, e13) to the AMS eigenvectors. Around each principal susceptibility axis, the major and minor semi-axes of 95% confidence ellipse lay within the planes defined by the eigenvectors

susceptibility is almost completely controlled by the contribution of the paramagnetic matrix. For very small or even negative k values (i.e., k < 50 mSI), the contribution of the

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Magnetic Anisotropy, Fig. 5 Equal area projection of the principal AMS directions and the 95% confidence ellipses (dashed lines) for a single sampling site. Squares, kmax; triangles, kint; circles, kmin. Small symbols indicate individual specimens, and large symbols indicate the mean tensor

diamagnetic matrix to the bulk susceptibility of a rock specimen is not negligible. All the rock’s constituent mineral fractions contribute to the development of an AMS fabric depending on their intrinsic susceptibility, crystalline and shape anisotropy at the grain scale, and degree of preferential orientation-distribution. Among ferromagnetic minerals, hematite and pyrrhotite are the most anisotropic, both with anisotropy degree P > 100. Magnetite shows a typical range of P variation between 1.1 for nearly isometric grains and 3 for highly anisometric grains (Table 1). Paramagnetic minerals are characterized by a much smaller AMS degree. In biotite and most phyllosilicates, the kmax and kmin directions are, respectively, parallel and orthogonal to the basal plane, and their AMS ellipsoid is typically oblate. Finally, the main diamagnetic minerals typically show low magnetic susceptibilities and

low anisotropy (Table 1). A recent review of single crystal magnetic anisotropy for silicate and carbonate rock-forming minerals, as well as for accessory magnetic minerals (i.e., iron oxides and sulfides), is provided by Biedermann (2018). AMS is determined from a number of susceptibility measurements taken accordingly to specifically designed sets of directions in magnetic susceptibility bridges. Commonly used susceptibility bridges operate with alternating current (AC). AMS can be measured manually or utilizing an automated sample handler. The choice of measurement axes for the determination of AMS is critical. A minimum of six measurements is necessary to solve for the six unknown components of the AMS tensor. A 15-measurement position scheme is typically utilized for AMS measurements in manual mode. New 3D automatic specimen rotators instead require the user to insert the specimen in one position only, and the specimen is then automatically rotated around two axes, taking a total of 320 measurements. With the aim to enhance the paramagnetic contribution to the magnetic susceptibility and fabric, AMS measurements may be carried out also on samples kept at low temperatures (e.g., at liquid nitrogen temperature, ~77 K). This enhancement occurs because diamagnetic and, to the first order, ferromagnetic susceptibilities are temperature independent, whereas paramagnetic susceptibility increases by decreasing absolute temperature T according to the Curie–Weiss law, kpara ¼ C/(T – Θ), where C is a constant value and Θ is the paramagnetic Curie temperature. Recent instrumental implementation in susceptibility bridges allows the measurement of magnetic susceptibility and AMS in variable fields and frequency, as well as the simultaneous measurement of both the in-phase and out-of-phase susceptibilities and their anisotropies (Hrouda et al. 2017). Consequently, analytical techniques have been developed for direct determination of the magnetic subfabrics of magnetic minerals that exhibit low-field variation of susceptibility, frequency-dependent susceptibility, or measurable out-ofphase susceptibility, such as pyrrhotite, titanomagnetite, and magnetically viscous ultrafine superparamagnetic grains of magnetite or maghemite (Hrouda et al. 2017).

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Magnetic Anisotropy, Table 1 Grain AMS determinations reported for various common minerals Mineral Ferromagnetic (s.l.) Magnetite Hematite Pyrrhotite Paramagnetic Biotite Muscovite Phlogopite Chlorite Hornblende Tourmaline Siderite Diamagnetic Quartz Calcite Aragonite

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1.0  2.8 (103) 0.12  0.17 (103) 0.3  1.2 (103) 0.07  1.55 (103) 8.92 (103) 1.69 (103) 3.98 (103)

1.2  1.6 1.3  1.4 1.3  1.5 1.2  1.7 1.67 1.005 1.56

0.9  1 0.4  0.7 0.78  0.95 0.3  0.7 0.51 ~1 0.9

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13.4  15.4 (106) 13.8 (106) 19 (106)

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– 1 0.8

Magnetocrystalline Magnetocrystalline Magnetocrystalline

Adapted from Hrouda (1993, 2007); Tarling and Hrouda (1993), and Borradaile and Jackson 2004)

In comparison to other methods of petrofabric analysis, the study of AMS has the advantage to be quick, cost-effective, and highly sensitive, and above all, it can be performed systematically on all rock types. AMS data are usually employed to obtain information about the geological processes responsible of the fabric in rocks, both during lithogenesis and during any subsequent deformational event. AMS: Applications in Geology and Geophysics Magnetic anisotropy is an essential petrostructural phenomenon which can be used to study rock fabric, deformation, and various other processes that took place during the geologic history of a rock. The AMS of rocks can be determined with high accuracy, and the method is so sensitive that in rocks with a very weak preferred orientation-distribution of minerals, it is the only approach that gives reasonable results. The study of AMS has been therefore used as a tool of structural analysis for almost all rock types, and AMS analyses were applied to investigate a wide variety of geological processes capable to produce a preferential orientation-distribution of minerals in rocks. Moreover, AMS studies are an important complement to paleomagnetic researches, as a way to assess the reliability of paleomagnetic data and to determine possible deflections in the natural remanent magnetization components induced by deformation and compaction. For these reasons, the analysis of the AMS has experienced broad use in many branches of the Earth Sciences. A brief overview of the wide spectrum of potential applications of the AMS research is provided below. In undeformed sediments, the magnetic anisotropy may indicate the orientation of the sedimentation paleo-horizontal and eventual paleo-flow directions. The primary magnetic

fabric of sediments is strongly affected by the processes which govern deposition and compaction. In quiet water deposition, such as in a lagoon or lake environment, gravitydriven sedimentation brings platy grains to lay with their longer dimensions statistically parallel to the beddingcompaction plane. Then, with further sediment burial, the effect of diagenetic compaction on platy minerals by pressure and water expulsion reinforces the parallelism between the magnetic foliation and the bedding plane. Thus, if the magnetic fabric is purely depositional, or related only to compaction loading, the AMS ellipsoid shows an oblate shape, with the kmin axis perpendicular to the bedding plane (which can be determined through AMS analyses with high accuracy) and the kint and kmax axes mostly scattered in the bedding plane itself resulting in a wide uncertainty ellipse (Fig. 6a). The action of flow currents may cause the maximum susceptibility axis to be aligned either parallel or perpendicular to the paleocurrent direction (e.g., Hamilton and Rees 1970). In moderate currents, platy particles tend to imbricate, and elongated particles tend to align with their greatest axis parallel to the flow. On the AMS results obtained from a group of specimens, the imbrication induces a slightly vertical offset of the mean kmin and the clustering of the kmax axes in a direction antiparallel to the paleo-flow (Fig. 6b); the overall AMS fabric is still characterized by an oblate ellipsoid. In such a case, imbrication yields a better estimate of the flow direction than magnetic lineation, which may wander within the foliation plane and even be orthogonal to the flow direction. In fact, when deposition occurs under high current flow (traction transport mode of sedimentation), the distribution of kmin axes from various samples is streaked, the kmax axes cluster in a direction perpendicular to the flow (Fig. 6c),

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Magnetic Anisotropy, Fig. 6 Schematic representations of the AMS development for undeformed sediments deposited in quiet conditions (a) under the action of weak or moderate water currents (b) and under the

action of strong currents, sufficient to entrain particles and to induce the rolling of elongated grains with their long axis perpendicular to the water flow (c). (Modified from Tauxe 2005)

and the fabric is characterized by prolate or triaxial AMS ellipsoids. The study of AMS was also applied in aeolian deposits of Chinese loess to reconstruct paleo-wind directions

and in recent and contemporaneous marine sediments to reconstruct the dynamics of deposition following tsunami events.

Magnetic Anisotropy

In general, sedimentary depositional fabrics are comparatively low both in susceptibility (400 km) and strongly resembles a satellite-derived magnetic anomaly map. The short wavelength grid is computed by subtracting the long wavelengths from the original grid and shows finer details of the regional geology and structure of the crust (Singh 2011).

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To analyze the systematic errors in the long wavelengths between adjacent overlapping grids, the difference in the long wavelengths is computed. A visual inspection shows differences in the anomalies exceeding hundreds of nT and often shows a lack of continuity across two or more grids. Because it is difficult to judge the trustworthiness of an overlapping long-wavelength grid, it was best suited to replace the filtered long wavelength with a reliable satellite model. Techniques that merge two adjacent compilations often employ a weighted average of different-order surfaces such that the two adjoining maps are seamless within the merging area. These processes modify the grids within the overlaying regions and near the edges and alter the strength and pattern of anomaly features. In this process, anomalies are modified, or at times genuine crustal anomaly features are even entirely eliminated. The merging technique applied to merge shortwavelength magnetic anomaly data minimizes abrupt changes along boundaries of overlapping grids. Using the MOSAIC toolbox (http://www.esri.com/), the grid values within the overlapping regions were determined from Hermite Cubic functions (Franke 1982) and the short wavelength grids merged. The short wavelength anomaly differences between grids vary substantially; hence, it is difficult to prescribe the order of precedence for merging. However, the quality of an individual compilation is also reflected in the long-wavelength differences. Therefore, the order of precedence of the grid merging was based on the difference between long wavelengths and the quality of the parent grid. Thus, the shortwavelength grids were merged in the following order: Australia, North America, Austria, Canary Island, East Asia, Eurasia, Europe, Arctic, Spain, Fennoscandia, France, Russia, Argentina margin and inland, South Africa, Antarctica, Italy, India, Middle East, Africa, and South America, NGDC track line and Project Magnet data. The grids were merged beginning with the highest precedence. The accuracy and reliability of the merged grid is assessed by investigating the effect of merging on the continuity of anomaly patterns across two or more grids and the effects on the anomaly shapes and strengths within and near the grid boundary. The differences for the merged European and Eurasian grid ranged from 100 to 200 nT and 330 km) replacing the MF5 model used for WDMAM model. EMAG2 is 2 arc min grid defined at a height of 4 km above the geoid. The ocean data includes NGDC’s GEODAS marine data archive, Antarctic Digital Magnetic Anomaly Project data (Golynsky et al. 2001), and the Project Magnetic Airborne data of the Naval Research Lab (NRL). The line-leveling algorithm remains the same as described earlier. However, the search radius was reduced from Rs ¼ 100 to 8 km in order to minimize the crossover effects in processing the track line data. In addition, the number of correction coefficients per track is now Ni ¼ trunc (Xi/300 km) + 1. These efforts reduced the crossover errors from 92 to 70 nT and reduced the RMS misfit to the merged grid from 121 to 97 nT. Least squares collocation method requires estimation of the correlation function. No new correlation functions was computed for land areas as the gridded compilation remained the same. Notable changes for oceans, however, were made. For instance, for Australia, 100 m terrain clearance was used. Due to the inclusion of more accurate grids and original marine and airborne surveys, the assumed variance V0 was reduced from 40,000 to 33,000 nT and the correlation length rc from 15 to 14 km. The methodology of LSC was slightly modified for EMAG2. The correlation analysis now considered the anisotropy of oceanic magnetic anomalies. Track line data, just the way it was processed for NGDC candidate model, were first divided into small windows but along the constant heading direction of the track. Next, for each track, the azimuth for every pair of measurement points was calculated and compared with the direction of the oceanic isochrons at the locations of both points. The pair was removed from further processing if they met one of the following criteria: (1) the azimuth of the isochrons was not well defined, (2) the azimuth

Magnetic Anomaly Map, Global

differed by more than 5° between the points, or (3) the topographic gradient between the two points exceeded 3%. The NGDC ETOPO-2 bathymetry values were used to compute the topographic gradient. This gradient could detect and remove the magnetic anomalies due to seamounts, which otherwise would appear as noise. The anisotropic correlation function was then computed in 10° directional bins; from 0° (parallel) to 90° (perpendicular, i.e., in the spreading direction) to the isochrons. This anisotropic correlation model allowed the LSC method to be used as a directional gridding algorithm. From the analysis of covariance function in the isochron direction, it was noted that the function value reduced too rapidly. Furthermore, extrapolating the field far from a track line made the function reduce to zero. This produced erroneous stripes parallel to the track lines. To stabilize the covariance function, the correlation length was doubled when gridding the ocean areas. This made the anisotropy of the oceanic field in EMAG2 to be smoother than it perhaps is. EMAG2 also suffers inaccuracies from the oceanic crustal age model due to errors in the estimated ocean isochrons. This error could easily have contaminated those estimated oceanic magnetic anomalies of EMAG2, where the uncertainties in isochron age are large. To reduce these uncertainties, empirically, the anisotropy factor was increased fourfold for all oceanic isochrons of age 150 Ma was used. This procedure helped reduce the errors in EMAG2 within the oceans, in particular over the older crust. For compiling the final EMAG2 grid, the portion of the original total field anomaly up to degree 120 is replaced by that of the MF6 model. A spectral comparison between the original and modified grids shows a slight discrepancy above degree 120. This is perhaps due to the leakage of longwavelength power into shorter wavelengths. But the overall comparison generally validates the power content in EMAG2. The EMAG2 grid can be downloaded from http://earthref. org/cgi-bin/er.cgi?s¼erda.cgi?n¼970 or from http://geomag. org/models/EMAG2. New shipborne and trackline measurements were made available for the study of oceanic crust which led to the derivation of a significantly improved version of EMAG2, called the EMAG2V3. It represents magnetic anomalies at sea level in the ocean regions, and the wavelengths greater than 300 km were replaced with the CHAMP lithospheric field model, MF7 (Maus 2010a). The EMAG2V3 grid can be downloaded from https://www.ngdc.noaa.gov/geomag/emag2.html.

NGDC-720 Lithospheric Magnetic Model In the last decade, successful attempts have been made to improve the resolution of the global magnetic anomaly

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maps. The expansion of scalar magnetic potential into ellipsoidal harmonics in the least squares sense (Maus 2010b) led to the development of NGDC-720 model that represented anomaly of the total intensity of the field from degrees 16 to 719 corresponding to the variation of the wavelength from 2500 to 56 km. Newer versions of the models are available at https://geomag.us/models/ ngdc720.html. The wavelengths larger than 330 km were

from the MF6 et al. 2008).

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Enhanced Magnetic Model In the last half-a-decade, the Enhanced Magnetic Model (EMM) started replacing the NGDC-720 models. The

Magnetic Anomaly Map, Global, Table 1 Data sets, their statistics and sources, used for deriving the WDMAM2.0 model. (Courtesy: Lesur et al. 2016) Name Marine data France

N 7,193,664 1,488,585

Mean 4 0.54

Std dev. 110 32.00

Min/max 2057/2627 156/396

Italy Spain Germany Austria Fennoscandia United Kingdom Portugal Romania

439,320 494,171 20,815 5294 3,505,857 119,509 151,142 100,092

11.8 0.02 2.62 12.6 30 2.71 10 19

59.02 16.66 36.70 26.12 154 121 33 70

435/588 62/184 96/171 35/137 580/1500 962/1342 111/219 144/418

13 14 15 16 17 18 19 20

Japan (2007) East Asia Russia (VSEGEI) Dijibouti Middle-East (2007) India Circum-Arctic

91,094 958,040 168,8415 266,171 1,812,464 92,784 3,258,710

42 28 0.9 2.8 0.56 18.5 4

73 68 158 85 73 82 99

401/416 509/763 2274/5658 293/241 904/690 341/427 699/2754

21 22 23 24 25 26 27

353,672 99,066 329,975

24 3.9 0.9

46 72 75

76/190 283/143 410/681

28 29 30

Guadeloupe French Guiana Bahama-Cuba

Index 11 12

Nure-Namag Algeria Morocco Nigeria Argentina-Shelf Argentina-Inland Australia

2,012,238 2,126,820 394,179 132,407 1,497,947 34,134 7,089,164

11 0.2 1.4 26.3 9 1.5 41.2

123 48 43 40 45 104 137

3667/3998 189/274 194/207 163/177 150/318 3152/1173 1674/3360

31 32 33 34 35 36 37

South African DC Uganda Mozambique SaNaBoZi Acad Vernadsky Admap

427,270 123,020 419,248 83,380 69,123 56,125,472

3.2 83 159 77 170 15

120 116 171 109 166 104

1788/2030 673/301 409/1007 203/756 180/426 776/1371

38 39 40 41 42 43

5 2 1 8

152 102 59 55

1662/1639 891/819 1024/536 235/580

44 45 46 47

Marine Model Eurasia (2007) Getech (2007) GRIMM (2007)

901,539 540,898 1,038,080 3,645,256

Sources Quesnel et al. (2009) Le Mouël, Ann. Geophys., 26(2), 1970; Courtillot et al., Earth Planet Sci. Lett., 47, 1980 Chiappini et al., Ann Geophys, 45(5), 2000 Socias, Earth Planet Sci. Lett., 105, 1991 [email protected] GSA (http://www.geologie.ac.at/) J. Korhonen, GTK (http://en.gtk.fi/) British Geological Survey, NERC, IPR/132-01CT Miranda et al., Earth Planet Sci. Lett., 9, 1989 Dr. L. Besutiu, Institute of Geodynamics, Romania (http:// www.geodin.ro/) http://www.ccop.or.th/ http://www.ccop.or.th/ T. Litvinova, VSEGEI (http://www.vsegei.ru/en) Courtillot et al., Earth Planet Sci. Lett., 47, 1980 http://members.casema.nl/errenwijlens/itc/aaime/ GSI (http://www.gsi.gov.in/) Gaina, C., Werner, S. C. and the CAMP-GM group, NGU Report 2009.010 IPG, Paris, France (http://www.ipgp.fr/) BRGM, France (http://infoterre.brgm.fr/) Batista-Rodriguez et al., Geophysics, 72(3), https://doi.org/ 10.1190/1.2712425 Ravat et al., USGS open-file report 2009-1258 Ministry of Energy and Mines, Algiers, Algeria Ministry of Energy and Mines, Rabat, Morocco Nigerian Geological Survey, Abuja, Nigeria Max et al., Marine Petroleum Geology, 16, 1999 SEGEMAR (http://www.segemar.gov.ar/) Geosciences Australia (http://www.agso.gov.au/map/ magnetics info.html/) SADC (http://www.sadc.int/) Ministry of Energy and Mineral Development, Uganda Geologian Tutkimuskeskus, Espoo, Finland SADC (http://www.sadc.int/) http://wdmam.org/pdf/WDMAM_data_contrib.pdf http://www.antarctica.ac.uk/about antarctica/geography/ maps/admap.php/ Dyment et al., EPSL, 430, 2015 GSC (http://gsc.nrcan.gc.ca/) GETECH (http://www.getech.com/world data.php/) Lesur et al., Solid Earth, 4, 2013

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Magnetic Anomaly Map, Global, Fig. 2 The World Digital Magnetic Anomaly Map version 2.0 (WDMAM2.0) publicly released by 26th IUGG General Assembly in June 2015. (Courtesy: Lesur et al. 2016)

EMM2015 which extended to SH degrees 790 was able to resolve magnetic anomalies up to 51 km wavelengths, at least theoretically. The EMM2015 model was derived by merging ground magnetic measurements with the near-surface compilations and satellite data. An updated version, the EMM2017, was released on July 05, 2017, which included the European Space Agency’s (ESA) three-satellite Swarm mission data. The EMM2017 is also an improved version of EMAG2V3 model and represents anomalies to Spherical Harmonic degree 790. The two models EMM2015 and EMM2017 differ primarily in oceanic areas owing to the different methodologies used in the derivation of their respective parent, EMAG2 and EMAG2V3, models. The EMM2017 is valid from 2000.0

to 2022.0 and has been developed primarily for the purpose of navigation. The model can be downloaded from https://www. ngdc.noaa.gov/geomag/EMM/.

World Digital Magnetic Anomaly MAP Version 2.0 (WDMAM2.0) When the WDMAM project was conceptualized, it was envisaged to have WDMAM to be an ongoing project, with plans to produce updates of the map and grid every 4 years. As part of these updates, it was envisioned that more organizations will make additional data available for new versions of

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WDMAM, in particular over regions where large data gaps exist. Toward this, Maus et al. continued to collect and process the data and produced their own maps, e.g., NGDC-720, EMAG2, and higher versions of the same. Thus, a need to derive the candidate models for WDMAM2.0 with the inclusion of data with prior agreement of the data owners was agreed upon. After the release of the first version of the WDMAM (Korhonen et al. 2007), new aeromagnetic compilations over the continents and trackline data in the oceanic regions were released for the preparation of the second version of the WDMAM, or referred to as WDMAM2.0 (Table 1). Only one candidate model developed jointly by two independent groups, one at the GeoForschungsZentrum in Potsdam Germany, and the other at the Institut de Physique du Globe in Paris, France, was submitted for evaluation. The processing technique for the derivation of WDMAM2.0 (Lesur et al. 2016) was same as that followed for the derivation of the candidate GAMMA model (Hamoudi et al. 2007). All continental compilations were upward continued to 5 km above the WGS84 datum before merging them with the marine magnetic anomalies derived at the sea surface (0 km). The data gap over the marine areas has been filled up with the magnetization model of the ocean floor (Dyment et al. 2015). The

final merged grid is of resolution 0.05  0.05 degrees which are roughly equivalent to 5  5 km cell size on the Earth’s surface. The resulting WDMAM2.0 grid (Fig. 2) represents 25,927,200 total intensity values computed relative to the CM4 field calculated for epoch 1990 and is derived up to SH degrees 16–800, of which the long wavelengths corresponding to SH degrees 16–100 are corrected using lithospheric field model GRIMM_L120 model (Lesur et al. 2013). The WDMAM2.0 was approved by IAGA during the 26th IUGG assembly in Prague, June 2015, and was officially released for its use by the scientific community (http:// wdmam.org/). The power spectra of EMM2017 and WDMAM2.0 models for spherical harmonic degrees 1–800 (degrees 1–790 in the case of EMM2017) are plotted in Fig. 3. A prominent low power is noticed in WDMAM2.0 model from degrees 1 to 15 compared to EMM2017. This is because the main field (Geomagnetic field, IGRF) has been removed in WDMAM2.0 model while EMM2017 retains it. Instead, the long wavelength anomalies from the Earth’s lithosphere can be seen for degrees 1 in a uniaxial material, prism-shaped closure domains bounded by 90° walls may completely close off magnetic flux at the crystal surface (Fig. 4a). Because closure domains greatly reduce magnetostatic energy, the “body” domains can be several times broader than predicted for the “open” structure. When 2πMs2/K > 1.0, or cubic, such as iron, whose easy axes are along . Here, 90° walls separate closure domains from the principal “body” domains that fill most of the crystal. Arrows indicate the sense of spontaneous magnetization within the domains. (b) Wavy walls at the surface of a uniaxial material with 2πMs2/K < 1.0. Waviness dies out with increasing distance from the surface. (c) Prism-shaped closure domains at the surface of a cubic material, such as magnetite, whose easy directions of magnetization are along . Closure domains and body domains are separated by 71° and 109° walls

to the “easy” axis, making the style of surface closure shown in Fig. 4a energetically unfavorable. Instead, walls that are planar within the body of the crystal can become wavy at the surface (Fig. 4b). In extremely thick crystals, wavy walls can alternate with rows of reverse spikes. These elaborate surface domain structures lower magnetostatic energy by achieving a closer mixture of “positive” and “negative” free magnetic poles (e.g., see Szymczak 1968). Large crystals governed by cubic magnetocrystalline anisotropy also reduce surface flux through prism-shaped closure domains at the surface. When are easy directions, as in iron, closure domains are bounded by 90° walls (Fig. 4a). When are easy directions, as in

magnetite, closure domains are bounded by 71° and 109° walls (Fig. 4c). Temperature Dependence of Domain Structure Understanding how domain structure evolves during both heating to, and cooling from, the Curie point is crucial to understanding the acquisition and thermal stability of thermal remanent magnetization (TRM). If the number of domains changes significantly during cooling from the Curie point in a weak field, it is reasonable to hypothesize that TRM will not become “frozen in,” or blocked, until the overall domain structure reaches a stable configuration. According to Kittel’s original model (Fig. 2), grains will nucleate (add) domain walls and domains with heating in zero field, if the wall energy drops more rapidly with increasing temperature than does the magnetostatic term. Energy-wise, in this first case, a particle can “afford” to add domains with heating. Conversely, during cooling from the Curie point, a grain will denucleate (lose) domains and domain walls if wall energy rises more quickly than does the magnetostatic energy with decreasing temperature. If wall energy drops less rapidly with increasing temperature than does the magnetostatic term, then the opposite scenarios apply. This predicted behavior relies on the assumptions that the particle is able to maintain a global energy minimum (GEM) domain state at all temperatures and that the total magnetostatic energy of the walls themselves can be ignored. Using Amar’s (1958) model, Moskowitz and Halgedahl (1987) calculated the number of domains between room temperature and the Curie point in parallel-epipeds of TM60. As discussed earlier, they investigated two cases: dominant crystalline anisotropy (zero stress) and high stress (s ¼100 MPa). In TM60 grains larger than a few micrometers, most of their results gave an increase in the number of domains with heating. Exceptions to this overall pattern were cases in which walls broadened so dramatically with increasing temperature that they nearly filled the particle and rendered nucleation unfavorable. During cooling from the Curie point in zero field, the domain “blocking temperature” – the temperature below which the number of domains remained constant – increased both with decreasing grain size and with internal stress.

Micromagnetic Models In contrast to classical models of magnetic domain structure, micromagnetic models do not assume the presence of discrete domains. Instead, they allow the orientations of Ms-vectors to vary among extremely small subvolumes, into which a grain is divided. A stable configuration is obtained numerically when the grand sum of exchange, anisotropy, and magnetostatic energies over all subvolumes is minimum.

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Magnetic Domains, Fig. 5 Diagram illustrating the relative energies and energy barriers associated with local energy minimum (LEM) domain states. Each LEM state is characterized by a unique number of domains and is separated from adjacent states by nucleation and denucleation energy barriers. In this diagram, the global energy minimum, or GEM, domain state has five domains

In rock magnetism, micromagnetic models have focused on magnetite, and they assume unstressed, defect-free crystals. Moon and Merrill (1984, 1985) were the first in rock magnetism to construct one-dimensional (1D) micromagnetic models for defect-free magnetite cubes. Not only did they calculate the energies resulting from different numbers of domains within a grain of given size, but they also calculated the energy barriers associated with nucleation and denucleation of domains and domain walls. Their major breakthrough was the discovery that a particle can occupy a range of local energy minimum (LEM) domain states. Each LEM state is characterized by a unique number of domains and is separated from adjacent states by energy barriers. As illustrated in Fig. 5, the GEM state is the configuration of lowest energy but, owing to the energy barriers between states, a LEM state can be quite stable as well. Several authors have extended micromagnetic calculations for magnetite to two dimensions (2D) and three dimensions (3D) (e.g., Williams and Dunlop 1989, 1990, 1995; Newell et al. 1993; Xu et al. 1994; Fabian et al. 1996; Fukuma and Dunlop 1998; Williams and Wright 1998). These models yield a variety of exotic, nonuniform configurations of magnetization, such as “flower” and “vortex” states. Analogous to Moon and Merrill’s results, these models produce both LEM and GEM states, although with very different spin structures from those of 1D models. For example, Fig. 6 illustrates a “flower” state in a cube magnetized parallel to the z-axis. The flower state is reminiscent of a classical SD state of uniform magnetization, except that the Ms-vectors are canted at and near the crystal surface. In relatively large cubes of magnetite – for example, 4 or 5 mm – both 2D and 3D models yield configurations closely

Magnetic Domains, Fig. 6 Illustration of a “flower” state obtained through three-dimensional micromagnetic modeling of a cube largely magnetized along the cube’s z-axis

approaching those of classical domain structures expected for magnetite, such as “body” domains and closure domains at the surface (e.g., Xu et al. 1994; Williams and Wright 1998). In submicron magnetite undergoing hysteresis, magnetization reversal can occur through LEM–LEM transitions (e.g., flower to vortex state) (e.g., Williams and Dunlop 1995). Dunlop et al. (1994) used 1D micromagnetic models to investigate LEM states in stress-free magnetite particles during TRM acquisition. Their study was motivated by experimental results reported earlier by Halgedahl (1991) (see discussions below). Dunlop et al. (1994) calculated the energy barriers for all combinations among SD/two-domain/ three-domain transitions during cooling from the Curie point of magnetite in a weak external field. According to their results, after acquiring TRM under conditions of thermal equilibrium, populations would be overwhelmingly biased toward GEM domain states, and an individual particle should not exhibit a range of LEM states after several identical TRM runs. Using renormalization group theory, Ye and Merrill (1995) arrived at a different conclusion. According to their calculations, short-range ordering of spins just below the Curie point could give rise to a variety of LEM states in the same particle after replicate coolings.

Experimental Studies of Magnetic Domains Methods of Imaging Domains and Domain Walls Rock magnetists have mainly used three methods to image domains and domain walls: the Bitter pattern method,

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which uses liquid magnetic colloid to image walls, with a resolution limit of about 1 mm (e.g., see details in Halgedahl and Fuller 1983); the magneto-optical Kerr effect (MOKE), which images entire domains from the rotation of the polarization plane of light reflected by a magnetized specimen (e.g., Hoffmann et al. 1987; Worm et al. 1991; Heider and Hoffmann 1992; Ambatiello et al. 1999); and magnetic force microscopy (MFM), which can image magnetic features as small as 0.01 mm through the voltage induced in a vibrating magnetic needle situated very close to a particle’s surface (Williams et al. 1992; Proksch et al. 1994; Moloni et al. 1996; Pokhil and Moskowitz 1996, 1997; Frandson et al. 2004). Two other methods – transmission electron microscopy (TEM) and off-axis electron holography – have been used in some studies of magnetite, most notably magnetotactic bacteria and magnetite intergrowths in ulvospinel (Dunin-Borowsky et al. 2001; McCartney et al. 2001; Harrison et al. 2002). Styles of Domains Observed in Magnetic Minerals of Paleomagnetic Significance In rock magnetism, the majority of domain observation studies have focused on four magnetic minerals, all important to paleomagnetism: pyrrhotite (Fe7S8), titanomagnetite of roughly intermediate composition (near Fe2.4Ti6O4, or TM60), magnetite (Fe3O4), and hematite (Fe2O3). Owing to its high magnetocrystalline anisotropy constant and relative insensitivity to stress, pyrrhotite behaves magnetically as a uniaxial material. When studied with the Bitter colloid method, pyrrhotite often exhibits fairly simple domain patterns which suggest lamellar domains separated by 180° walls (Fig. 7a) (Soffel 1977b; Halgedahl and Fuller 1983).

Magnetic Domains

Despite being cubic, intermediate titanomagnetites rarely, if ever, exhibit the arrays of closure domains, 71°, and 109° walls predicted from theories. Instead, these minerals often display very complex patterns of densely spaced, curved walls (Appel and Soffel 1984, 1985). However, some grains do exhibit a simple array of parallel walls, such as that shown in Fig. 8a (e.g., Halgedahl and Fuller 1980, 1981), but it is not unusual to observe wavy walls alternating with rows of reverse spikes (Halgedahl 1987; Moskowitz et al. 1988). Both simple and wavy patterns suggest a dominant, internal stress that yields a uniaxial anisotropy, although the origin of this stress is still unclear. Small magnetite grains grown in a synthetic rocklike matrix with the glass-ceramic method (Worm and Markert 1987) generally display simple arrays of straight walls (e.g., Worm et al. 1991; Geiβ et al. 1996) (Fig. 8b). In such samples, there appears to be a paucity of closure domains, perhaps the result of high internal stress generated by quenching during synthesis. In contrast, domain images obtained from large crystals of natural magnetite and small magnetite grains in a variety of rocks reveal 180°, 71°, and 109° walls and closure domains, in accordance with theory (Bogdanov and Vlasov 1965, 1966; Smith 1980; Boyd et al. 1984; Ozdemir and Dunlop 1993, 1997, 2006; Ozdemir et al. 1995; Ambatiello et al. 1999). Evidently, internal stress is low in these natural magnetites, so that magnetocrystalline anisotropy is dominant. To date, domain studies on hematite have been limited to large (e.g., 100 mm to 1 mm) platelets. Even large crystals such as these contain very few walls and a fairly simple domain structure, owing to hematite’s weak spontaneous

Magnetic Domains, Fig. 7 Bitter patterns on a particle of natural pyrrhotite (a) after demagnetization in an alternating field of 1000 Oe and (b) in an apparently SD-like state after acquiring saturation remanence in 15 kOe

Magnetic Domains

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b

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20 μm

c

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Magnetic Domains, Fig. 8 (a) Bitter pattern on a grain of intermediate titanomagnetite (x ≈ 0.6) in oceanic basalt drilled near the Mid-Atlantic Ridge. (b) Bitter pattern on a grain of magnetite synthesized with the glass-ceramic method. In this particular state of magnetization, the grain contains four walls, whose lengths are commensurate with the particle’s length. Note that one wall is pinned very near the particle’s extreme lefthand edge. The small triangular patterns at the lower edge of the particle

represent walls which enclose small reverse spike domains. (c) Bitter pattern of a wall on a very large (approximately 150 mm width, 1 mm length) platelet of natural hematite from Elba, Italy. At most this platelet exhibits only 1–2 principal walls, although small edge domains are often observed. Apparently, the wall is bowing around a defect near the upper right-hand edge of the photograph. Only a small part of the crystal surface is shown

magnetization (about 2 emu/cm3) and low magnetostatic energy (Fig. 8c) (Halgedahl 1995, 1998).

transition size of 0.6 mm and a wall energy density of about 1 erg/cm2 for this composition. Similarly, from Bitter patterns on natural pyrrhotite in a Bavarian diabase, both Soffel (1977b) and Halgedahl and Fuller (1983) found that D / L0.40 to D / L0.45, depending on the magnetization state. For this pyrrhotite sample, the SD boundary fell between 1.5 and 2 mm. Geiβ et al. (1996) obtained D / L0 45 from Bitter patterns on synthetic magnetite particles grown in a glass-ceramic matrix (Worm and Markert 1987). They obtained a SD/two-domain transition size for magnetite of approximately 0.25 mm.

Number of Domains Versus Grain Size Both Kittel’s original model of domains in a semi-infinite platelet and calculations for finite grains by (e.g., Moskowitz and Halgedahl 1987) lead to the prediction that domain width D / L1/2, where L is plate or particle thickness. These predictions are supported by domain studies of natural magnetic minerals which generally yield a power-law dependence of D on L, although the power may differ somewhat from 0.5. Soffel (1971) was the first to study the grain-size dependence of the number of domains in a paleomagnetically important magnetic mineral. From Bitter patterns on grains of natural, intermediate (x ≈ 0.55) titanomagnetite in a basalt, he determined that, on average, both D and N (number of domains ¼ L/D) /L1/2, where L was average particle size. His results yielded a single domain/two domain

Domain Wall Widths When studied in the scanning electron microscope (SEM), patterns of dried magnetic colloid afford high-resolution views of domain walls. Moskowitz et al. (1988) applied this method to polycrystalline pellets of synthetic titanomagnetite substituted with aluminum and magnesium

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(Fe2.2Al0.1Mg0.1Ti0.6O4: “AMTM60”). Dried Bitter patterns on unpolished surfaces were virtually identical in style to those common to materials controlled by strong uniaxial anisotropy. Patterns indicated closely spaced stripe domains, sinusoidally wavy walls, and nested arrays of reverse spikes. Measurements from SEM images yielded wall widths between 0.170 and 0.400 mm. Very high resolution images of domains and domain walls in magnetite have been obtained with MFM. The first images were reported by Williams et al. (1992), who recorded a magnetic profile across a 180° wall on a {110} surface of a large natural magnetite crystal. Spins within the wall reversed their polarity of rotation along the length of the wall, demonstrating that walls in real materials can be much more complex than those portrayed by simple models. Using MFM, Pokhil and Moskowitz (1996, 1997) made a similar finding in glass-ceramic magnetite. Segments of opposite polarity were separated by Bloch lines, the transition regions where polarity changes sense. The number of Bloch lines within a wall varied among repeated alternating field (AF) demagnetization treatments. Thus walls, like particles, can occupy LEM states. Owing to its high-resolution capabilities, the MFM can provide estimates of wall width. Proksch et al. (1994) obtained MFM profiles across a 180° wall on a {110} surface of natural magnetite, thus obtaining a wall width of about 0.21 mm. Experimental Evidence for Local Energy Minimum (LEM) Domain States In their study of Bitter patterns on natural pyrrhotite, Halgedahl and Fuller (1983) noted that grains of virtually identical size could contain very different numbers of domains, despite these same particles having undergone the same magnetic treatments. Moreover, they found that an individual particle could arrive in very different domain states – that is, with different numbers of walls – alter different cycles of minor hysteresis. These observations led Halgedahl and Fuller (1983) to the conclusion that a particle could occupy domain states other than the ground state. Subsequent domain observation studies by Halgedahl on AMTM60 and by Geiβ et al. (1996) on magnetite provided strong evidence for LEM states in other magnetic minerals. LEM States and Thermomagnetic Treatments A particularly unexpected type of LEM state in pyrrhotite and intermediate titanomagnetite is a SD-like state that certain particles can occupy after being saturated in a strong external field, even though these same particles readily accommodate walls in other states of magnetization. Bitter patterns on intermediate (x ≈ 0.6) titanomagnetite in oceanic basalt and on natural pyrrhotite in diabase were studied during

Magnetic Domains

hysteresis by Halgedahl and Fuller (1980, 1983). In states of saturation remanence, most particles in a large population contained one or more walls, as expected on the basis of previous theories (Fig. 7a). However, several tens of percent of the finer (5–15 mm) particles appeared saturated after the maximum field was shut off (Fig. 7b). Halgedahl and Fuller (1980, 1983) proposed that grains that failed to nucleate walls could make a substantial contribution to saturation remanence. Particles in SD-like states remained saturated, until nucleation was accomplished by applying a back field of sufficient magnitude, and this nucleation field dropped off with increasing grain size L according to a power law in L–1/2. Similarly, Boyd et al. (1984) reported Bitter patterns on natural magnetite grains carrying saturation remanence, and these patterns suggested SD-like states. These results were surprising, in view of magnetite’s strong tendency to selfdemagnetize. Subsequent domain observations and hysteresis measurements of hematite platelets from Elba, Italy demonstrated that, after exposure to strong fields, even large crystals could arrive in states of near-saturation. Back fields were necessary to nucleate principal domains, and, similar to pyrrhotite, these nucleation fields also followed a power law in L–1/2 (Halgedahl 1995, 1998). Results from titanomagnetite and pyrrhotite suggesting SD-like states were interpreted by Halgedahl and Fuller (1980, 1983) in light of previous theoretical and experimental work on high-anisotropy, industrial magnets. In such materials, nucleation of walls is an energetically difficult process, because the internal demagnetizing field, which triggers nucleation, may be too weak to overcome the anisotropy field, which hinders nucleation (Brown 1963; Becker 1969, 1971a, b, 1976). Such particles may require an external back field, which aids the demagnetizing field in order to nucleate walls and accompanying domains. The style of domain structure and the range of LEM states that a particle can occupy can depend on thermomagnetic history. From Bitter patterns on natural, polycrystalline pyrrhotite, Halgedahl and Fuller (1981) discovered that crystallites displayed arrays of undulating walls on their surfaces after acquiring TRM in a weak external field. By contrast, after AF demagnetization in a strong peak field the same crystallites exhibited planar walls. Evidently, cooling locked in a high-temperature configuration of walls whose curved shapes promoted lower magnetostatic energy than would a planar geometry. As indicated by Moon and Merrill’s theoretical results for magnetite, a particular LEM state can be stable across a broad range of grain size. This prediction is born out by experiments by Halgedahl and Ye (2000), who investigated the effects of mechanical thinning on domain states in natural pyrrhotite. Individual pyrrhotite grains in a diabase were mechanically thinned and their Bitter patterns observed after

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Magnetic Domains, Fig. 9 Bitter patterns on a grain of natural pyrrhotite in a diabase before and after mechanical thinning. (a) Initial state, before thinning. (b) After thinning the particle to about one-half of its original length along a direction parallel to the trends of the Bitter lines. (c) Final state, after the particle has been thinned to about one-fourth of its original length

each thinning step. Despite some grains being thinned to about one-fourth of their initial diameter, the widths of surviving domains and positions of surviving walls remained unaffected (Fig. 9). Neither nucleations nor denucleations were observed, although calculations indicated that thinning would cause significant changes in GEM domain states. Bitter patterns on crystallites of polycrystalline AMTM60 were studied after many replicate TRM acquisition and AF demagnetization experiments (Halgedahl 1991). In each particle, the number of domains varied from one experiment to the next, defining a distribution of LEM states. For states of weak-field TRM, this distribution could be broad and could include single-domain-like states (Fig. 10). After replicate AF demagnetizations, however, typical distributions were narrow and clustered about a most probable state, possibly the GEM state. Evolution of Magnetic Domain Structures at Elevated Temperatures The manner in which domain structure evolves with temperature carries clear implications for TRM. Cooling-induced nucleations and denucleations of walls/domains could trigger sudden changes of the internal demagnetizing field. As a result, preexisting walls which survive a domain transition could be dislodged from imperfections where they were pinned initially at higher temperatures. The first Bitter patterns observed on magnetite above room temperature in the Earth’s field were made by Heider et al. (1988), using hydrothermally recrystallized crystals. Depending on the particle, Bitter patterns could be followed to approximately 200 °C, above which temperature the

patterns grew too faint to discern. Surprisingly, heating to very moderate temperatures drove certain walls across much of a particle; in some cases, denucleation occurred. Upon cooling, walls reassembled in a similar, though not identical, arrangement to that observed initially at room temperature. In some cases, repeated thermal cycling between room temperature and about 200 °C produced different numbers of domains in the same particle. Ambatiello et al. (1999) used the MOKE to study domain widths versus temperature in several large (several mm) crystals of natural magnetite. At room temperature, {110} planes were dominated by broad lamellar domains that terminated in closure domains at crystal edges. On {111} planes, they found complex, nested arrays of very small closure domains, which finely subdivided the main closure structure. Domain widths generally increased with heating, and such changes were thermally reversible. To explain their results, these authors hypothesized that heating promoted an increasingly finer subdivision into small domains near the crystal surface; this would reduce magnetostatic energy with heating and thus cause broadening of body domains (also see Muxworthy and Williams 2006). Bitter patterns observed at elevated temperatures on natural, intermediate titanomagnetites were reported by Soffel (1977a), Metcalf and Fuller (1987a, b, 1988), and Halgedahl (1987). During heating, patterns gradually faded to obscurity, with few significant changes in domain structure. Remarkably, some titanomagnetite particles studied by Metcalf and Fuller (1987a, b, 1988) displayed no Bitter lines after cooling from the Curie point in the Earth’s field. This observation provided evidence for Halgedahl

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Magnetic Domains, Fig. 10 Bitter patterns on a crystallite of titanomagnetite (“AMTM60”: see text) after each of eight replicate TRM acquisition runs in the Earth’s field

and Fuller’s (1983) earlier proposal that weak-field TRM acquired by populations of PSD grains could, in part, be attributable to particles which failed to nucleate walls during cooling. The possible importance of LEM states to TRM acquisition was raised by work on synthetic, polycrystalline AMTM60 with a Curie point near 75 °C (Halgedahl 1991). As shown in Fig. 10, replicate TRM experiments often produced different numbers of domains in the same particle. Some of these states suggested that the particle was entirely saturated, with no visible Bitter lines, or nearly saturated, with only small spike domains at grain boundaries. Patterns that evolved during cooling revealed that denucleation was one mechanism by which a particle arrived in a final LEM state. Denucleation occurred either

through contraction of large, preexisting spike domains, or by straight walls moving together and coalescing into spikes. Often, spikes would collapse altogether. In some cases, denucleation left behind large volumes which, apparently, contained no walls, although walls were present elsewhere in the grain. In other cases, denucleation left behind a grain that appeared nearly saturated, but for small edge domains. These results led to the idea of trans-domain TRM (Halgedahl 1991). Magnetic Domain Structure in Magnetite at Low Temperatures Magnetite undergoes two types of transitions at low temperatures, which can profoundly affect domain structure and remanence (e.g., see detailed discussions in Stacey and

Magnetic Domains

Banerjee (1974), Dunlop and Ozdemir (1997)). At the isotropic point (approximately 130 K), the first magnetocrystalline anisotropy constant, K1, passes through zero as it changes sign from negative at temperatures above the transition to positive below it. At the Verwey transition, Tv (approximately 120 K), magnetite undergoes a crystallographic transition from cubic to monoclinic. Thus at 130 K domain walls should broaden dramatically, because wall width is proportional to K1–1/2. It follows that, by passage through 130 K, walls may break free of narrow defects which pinned them at other temperatures. At the Verwey transition, the easy axis of magnetization changes direction. In multidomain and pseudosingle-domain magnetite, these thermal passages may cause demagnetization and complete reorganization of domain structure. Low-temperature demagnetization has proved useful in removing certain spurious components of NRM which are surprisingly resistant to thermal demagnetization above room temperature. Thus it is important to determine how these transitions affect domains. Using an MFM specially adapted to operate at low temperature, Moloni et al. (1996) were the first to image domains on {110} magnetite planes at temperatures near the two transitions. As the crystal was warmed from 77 K to a few degrees just below Tv, the domain structure disappeared entirely below the instrument’s noise level, evidently undergoing a complete reorganization near the crystallographic transition.

Cross-References ▶ Paleomagnetic Field Intensity ▶ Paleomagnetism, Principles ▶ Remanent Magnetism

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981 Becker JJ (1976) Reversal mechanism in copper-modified cobalt-rareearths. IEEE Trans Magn MAG-12:965–967 Bogdanov AK, Vlasov AY (1965) Domain structure in a single crystal of magnetite, (English trans.), Izv. Earth Phys 1:28–32 Bogdanov AA, Vlasov AY (1966) The domain structure of magnetite particles, (English trans.), Izv. Phys Solid Earth 9:577–581 Boyd JR, Fuller M, Halgedahl S (1984) Domain wall nucleation as a controlling factor in the behaviour of fine magnetic particles in rocks. Geophys Res Lett 11:193–196 Brown WF (1963) Micromagnetics. Wiley, New York, p 143 Chikazumi S (1964) Physics of Magnetism. Wiley, New York, p 664 Cullity BD (1972) Introduction to Magnetic Materials. Addison-Wesley, Reading, MA, p 666 Dunin-Borowsky RE, McCartney MR, Posfai M, Frankel RB, Bazylinski DA, Buseck PR (2001) Off-axis electron holography of magnetotactic bacteria: magnetic microstructure of strains MV-1 and MS-1. Eur J Mineral 13(4):671–684 Dunlop DJ, Ozdemir O (1997) Rock Magnetism: Fundamentals and Frontiers. Cambridge University Press, Cambridge, UK, 573 pp Dunlop DJ, Newell AJ, Enkin RJ (1994) Transdomain thermoremanent magnetization. J Geophys Res 99:19741–19755 Fabian K, Kirchner A, Williams W, Heider F, Leibl T (1996) Three-dimensional micromagnetic calculations for magnetite using FFT. Geophys J Int 124:89–104 Foss S, Moskowitz B, Walsh B (1996) Localized micromagnetic perturbation of domain walls in magnetite using a magnetic force microscope. Appl Phys Lett 69:3426–3428 Foss S, Moskowitz BM, Proksch R, Dahlberg ED (1998) Domain wall structures in single-crystal magnetite investigated by magnetic force microscopy. J Geophys Res 103:30551–30560 Frandson C, Stipp SLS, McEnroe SA, Madsen MB, Knudsen JM (2004) Magnetic domain structures and stray fields of individual elongated magnetite grains revealed by magnetic force microscopy (MFM). Phys Earth Planet Inter 141:121–129 Fukuma K, Dunlop DJ (1998) Grain-size dependence of twodimensional micromagnetic structures for pseudo-singledomain magnetite (0.2–2.5 mm). Geophys J Int 134:843–848 Geiβ CE, Heider F, Soffel HC (1996) Magnetic domain observations on magnetite and titanomaghemite grains (0.5–10 mm). Geophys J Int 124:75–88 Halgedahl SL (1987) Domain pattern observations in rock magnetism: progress and problems. Phys Earth Planet Inter 46:127–163 Halgedahl SL (1991) Magnetic domain patterns observed on synthetic Ti-rich titanomagnetite as a function of temperature and in states of thermoremanent magnetization. J Geophys Res 96:3943–3972 Halgedahl SL (1995) Bitter patterns versus hysteresis behavior in small single particles of hematite. J Geophys Res 100:353–364 Halgedahl SL (1998) Barkhausen jumps in larger versus small platelets of natural hematite. J Geophys Res 103:30575–30589 Halgedahl S, Fuller M (1980) Magnetic domain observations of nucleation processes in fine particles of intermediate titanomagnetite. Nature 288:70–72 Halgedahl SL, Fuller M (1981) The dependence of magnetic domain structure upon magnetization state in polycrystalline pyrrhotite. Phys Earth Planet Inter 26:93–97 Halgedahl S, Fuller M (1983) The dependence of magnetic domain structure upon magnetization state with emphasis upon nucleation as a mechanism for pseudo-single domain behavior. J Geophys Res 88:6505–6522 Halgedahl SL, Ye J (2000) Observed effects of mechanical grain-size reduction on the domain structure of pyrrhotite. Earth Planet Sci Lett 178:457–467

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982 Harrison TJ, Dunin-Borkowski RE, Putnis A (2002) Direct imaging of nanoscale magnetic interactions in minerals. Proc Natl Acad Sci 99:16556–16561 Heider F (1990) Temperature dependence of domain structure in natural magnetite and its significance for multi-domain TRM models. Phys Earth Planet Inter 65:54–61 Heider F, Hoffmann V (1992) Magneto-optical Kerr effect on magnetite crystals with externally applied fields. Earth Planet Sci Lett 108:131–138 Heider F, Halgedahl SL, Dunlop DJ (1988) Temperature dependence of magnetic domains in magnetite crystals. Geophys Res Lett 15:499–502 Heisenberg W (1928) Zur theorie de Ferromagnetismus. Z Phys 49:619–636 Hoffmann V, Schafer R, Appel E, Hubert A, Soffel H (1987) First domain observations with the magneto-optical Kerr effect on Ti-ferrites in rocks and their synthetic equivalents. J Magn Magn Mater 71:90–94 Kittel C (1949) Physical theory of ferromagnetic domains. Rev Mod Phys 21:541–583 McCartney MR, Lins U, Farina M, Buseck PR, Frankel RB (2001) Magnetic microstructure of bacterial magnetite by electron holography. Eur J Mineral 13(4):685–689 Metcalf M, Fuller M (1987a) Magnetic remanence measurements of single particles and the nature of domain patterns in titanomagnetites. Geophys Res Lett 14:1207–1210 Metcalf M, Fuller M (1987b) Domain observations of titanomagnetites during hysteresis at elevated temperatures and thermal cycling. Phys Earth Planet Inter 46:120–126 Metcalf M, Fuller M (1988) A synthetic TRM induction curve for fine particles generated from domain observations. Geophys Res Lett 15:503–506 Moloni K, Moskowitz BM, Dahlberg ED (1996) Domain structures in single crystal magnetite below the Verwey transition as observed with a low-temperature magnetic force microscope. Geophys Res Lett 23:2851–2854 Moon T, Merrill RT (1984) The magnetic moments of non-uniformly magnetized grains. Phys Earth Planet Inter 34:186–194 Moon TS, Merrill RT (1985) Nucleation theory and domain states in multidomain magnetic material. Phys Earth Planet Inter 37:214–222 Moskowitz BM, Banerjee SK (1979) Grain size limits for pseudosingle domain behavior in magnetite: implications for paleomagnetism. IEEE Trans Magn MAG-15:1241–1246 Moskowitz BM, Halgedahl SL (1987) Theoretical temperature and grain-size dependence of domain state in x ¼ 0.6 titanomagnetite. J Geophys Res 92:10667–10682 Moskowitz BM, Halgedahl SL, Lawson CA (1988) Magnetic domains on unpolished and polished surfaces of titanium-rich titanomagnetite. J Geophys Res 93:3372–3386 Muxworthy AR, Williams W (1999) Micromagnetic calculations of hysteresis as a function of temperature in pseudo-single domain magnetite. Geophys Res Lett 26:1065–1068 Muxworthy AR, Williams W (2006) Observations of viscous magnetization in multidomain magnetite. J Geophys Res 111:B01103. https://doi.org/10.1029/2005JB003902 Newell AJ, Dunlop DJ, Williams W (1993) A twodimensional micromagnetic model of magnetization and fields in magnetite. J Geophys Res 98:9533–9549 Ozdemir O, Dunlop DJ (1993) Magnetic domain structures on a natural single crystal of magnetite. Geophys Res Lett 20:1835–1838 Ozdemir O, Dunlop DJ (1997) Effect of crystal defects and internal stress on the domain structure and magnetic properties of magnetite. J Geophys Res 102:20211–20224

Magnetic Domains Ozdemir O, Dunlop DJ (2006) Magnetic domain observations on magnetite crystals in biotite and hornblend grains. J Geophys Res 111: B06103. https://doi.org/10.1029/2005JB004090. Ozdemir O, Xu S, Dunlop DJ (1995) Closure domains in magnetite. J Geophys Res 100:2193–2209 Pokhil TG, Moskowitz BM (1996) Magnetic force microscope study of domain wall structures in magnetite. J Appl Phys 79:6064–6066 Pokhil TG, Moskowitz BM (1997) Magnetic domains and domain walls in pseudo-single-domain magnetite studied with magnetic force microscopy. J Geophys Res 102:22681–22694 Proksch RB, Foss S, Dahlberg ED (1994) High resolution magnetic force microscopy of domain wall fine structures. IEEE Trans Magn 30:4467–4472 Rhodes P, Rowlands G (1954) Demagnetizing energies of uniformly magnetised rectangular blocks. Proc Leeds Philos Liter Soc Sci Sect 6:191–210 Sahu A, Moskowitz BM (1995) Thermal dependence of magnetocrystalline anisotropy and magnetostriction constants of single crystal Fe2.4 Ti0.6 O4. Geophys Res Lett 22:449–452 Smith PPK (1980) The application of Lorentz electron microscopy to the study of rock magnetism. Inst Phys Conf Ser 52:125–128 Soffel H (1971) The single-domain-multidomain transition in natural intermediate titanomagnetites. Z Geophys 37:451–470 Soffel HC (1977a) Domain structure of titanomagnetites and its variation with temperature. J Geomagn Geoelectr 29:277–284 Soffel H (1977b) Pseudo-single-domain effects and single-domain multidomain transition in natural pyrrhotite deduced from domain structure observations. J Geophys 42:351–359 Soffel HC, Aumuller C, Hoffmann V, Appel E (1990) Three-dimensional domain observations of magnetite and titanomagnetites using the dried colloid SEM method. Phys Earth Planet Inter 65:43–53 Stacey FD, Banerjee SK (1974) The physical principles of rock magnetism. Elsevier, Amsterdam, p 195 Szymczak R (1968) The magnetic structure of ferromagnetic materials of uniaxial structure. Electron Technol 1:5–43 Williams W, Dunlop DJ (1989) Three-dimensional micromagnetic modelling of ferromagnetic domain structure. Nature 337:634–637 Williams W, Dunlop DJ (1990) Some effects of grain shape and varying external magnetic fields on the magnetic structure of small grains of magnetite. Phys Earth Planet Inter 65:1–14 Williams W, Dunlop DJ (1995) Simulation of magnetic hysteresis in pseudo-single-domain grains of magnetite. J Geophys Res 100:3859–3871 Williams W, Wright TM (1998) High-resolution micromagnetic models of fine grains of magnetite. J Geophys Res 103:30537–30550 Williams W, Hoffmann V, Heider F, Goddenhenreich T, Heiden C (1992) Magnetic force microscopy imaging of domain walls in magnetite. Geophys J Int 111:417–423 Worm H-U, Markert H (1987) The preparation of dispersed titanomagnetite particles by the glass-ceramic method. Phys Earth Planet Inter 46:263–270 Worm H-U, Ryan PJ, Banerjee SK (1991) Domain size, closure domains, and the importance of magnetostriction in magnetite. Earth Planet Sci Lett 102:71–78 Xu S, Dunlop DJ, Newell AJ (1994) Micromagnetic modelling of two-dimensional domain structures in magnetite. J Geophys Res 99:9035–9044 Ye J, Halgedahl SL (2000) Theoretical effects of mechanical grain-size reduction on GEM domain states in pyrrhotite. Earth Planet Sci Lett 178:73–85 Ye J, Merrill RT (1995) The use of renormalization group theory to explain the large variation of domain states observed in titanomagnetites and implications for paleo-magnetism. J Geophys Res 100:17899–17907

Magnetic Gradiometry

Magnetic Gradiometry Harald von der Osten-Woldenburg National Heritage Department, Regional Government of Baden-Wuerttemberg, Esslingen am Neckar, Germany

Definition Magnetometer Magnetic gradiometer

Instrument with a single sensor that measures magnetic flux density. Pairs of magnetometers with sensors separated by a fixed distance.

The Earth’s magnetic field at any point on or near the Earth’s surface is the vector sum of the contributions from the primary field due to the dynamo in the Earth’s liquid core and the crustal field from the magnetic mineral content of local rocks. This vector has both an orientation and an amplitude. Slight changes in any direction influence the orientation and amplitude. Introduce a highly magnetic rock formation into an otherwise homogenous host, and the local magnetic vector will change. In a three-dimensional world, there are nine (3  3) spatial gradients forming a tensor which defines the anomalous field.

On the Conceptual Design of Gradiometers A gradiometer is a special measuring instrument that consists of more than one (usually two, rarely three or four) magnetometers. The differences between the readings of the sensors are seen as an approximation to the gradient of the magnetic field along their alignment. The (usually) two magnetometers of a gradiometer are arranged either vertically or horizontally. The distance of the two sensors can vary between several decimeters (for archaeological or environmental mapping) and several kilometers (for research of the deeper-lying geological structures of the Earth’s crust, see ▶ “Magnetic Methods, Airborne”) depending on the field of application.

Magnetometers Quantum magnetometers (see also ▶ “Magnetometers”) and saturable-core magnetometers are generally suitable for use in gradiometers. Alongside the fluxgate saturable-core magnetometers, cesium vapor magnetometers have become a standard in many areas (aeromagnetic survey, marine geophysics, environmental geophysics, and archaeological prospection) due to the improved noise signal levels against the

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Overhauser magnetometers or the helium-cooled sensors (Nabighan and Asten 2002). Thanks to new technological advances, for example, in the further development of magnetic gradiometer systems based on the superconducting gradiometric sensors SQUID (superconducting quantum interference device), systems are now available that enable very precise measurements of extremely small magnetic field changes. Due to this development, the importance of magnetic gradient tensor and field vector measurements has increased significantly in many applications (Clark 2012). Thus, a significantly higher magnetic and spatial resolution of magnetic maps is achieved than with conventional magnetometers. Fluxgate gradiometers measure a single component of the Earth’s magnetic field (usually the vertical component), while gradiometers consisting of quantum magnetometers (i.e., proton precession and cesium vapor sensors) measure the component of an anomaly in the direction of the Earth’s ambient magnetic field, as this is much larger than the anomaly itself (Blakely 1996). Additionally, fluxgate magnetometers are directionally sensitive and subject to a temperature-related drift, and therefore often have to be recalibrated during measurements. Instead of using a pair of different magnetic field sensors and differentiating their outputs in order to derive a magnetic gradient value, a single sensitive element can be used for measuring directly a magnetic gradient. This is done by the use of a stiff metallic string clamped at both ends and pumped with an AC current, the frequency of which is tuned to the second eigenmode of the string. The string is excited at that eigenmode in the presence of a quasi-static magnetic gradient. Then, the corresponding mechanical displacements of the string can be measured by the use of an inductive technique with an instrumental noise envelope of less than 1012 m, per 1 s measurement interval (Veryaskin 2001). Sheng et al. (2017) describe a new magnetic gradiometer based on optical spectroscopy of alkali atoms in the vapor phase. This type of magnetic gradiometer can be extremely interesting for applications in archeology and environmental sciences. In the field of environmental geophysics, there has been a trend towards the use of multi-channel gradiometer systems in recent years (Fig. 1). On a trailer, usually 10 or up to 20 probes are pulled over the measuring surface at a distance of 0.50 m (or 0.25 m if 20 probes are used) by an off-road vehicle. The georeferencing is done in real time by means of a differential GPS system.

Advantages and Limitations In measurements with a gradiometer, the measured signal amplitude of the magnetic anomaly of an interfering body decreases by the power of three or four depending on the distance between the two sensors. A gradiometer is therefore

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Magnetic Gradiometry, Fig. 1 The 10-channel fluxgate gradiometer system, manufactured by Eastern Atlas GmbH, Berlin, using the Fluxgate sensors of the Institute Dr. Foerster, Reutlingen, Germany. Magnetic field data are georeferenced in real time using a differential GPS system

somewhat less sensitive to deeper-lying features than a totalfield magnetometer, where the signal amplitude of an anomaly decreases by the power of three depending on the distance of the sensor to the interfering body. However, gradiometers react very sensitively to interfering features close to the surface and possess a higher spatial resolution in small-scale structures than total-field magnetometers. The variability in time of the Earth’s magnetic field and other sources of interference that are far enough away from the gradiometer, and whose influence on the sensors is equal or at least of a similar level during gradiometer measurements, can be eliminated – or at least significantly reduced – by subtracting the simultaneous readings of the two sensors from one another. An advantage of gradient measurements, arising from the ambiguity present in all potential field measurements, is the suppression of broad regional changes in the magnitude of the magnetic field – the long wavelength component. With gradiometer measurements, the local variations are enhanced, making small and weakly magnetic targets recognizable. Gradiometer surveys are particularly useful in areas that are geologically complex. Due to the resulting difference, the magnitude of the anomalies is, however, smaller than the amplitude that one would achieve with a total-field

magnetometer. This means that a total-field magnetometer can detect anomalies of smaller and deeper-lying features which cannot be detected by a gradiometer. Fluxgate gradiometers are very sensitive to their position in relation to the magnetic vector field: an ever so slight change in the inclination of the instrument during measurements can lead to incorrect results (Schmidt and Clark 2000). The use of vertical gradiometers for magnetic mapping effectively removes the time variations in the magnetic field map without having to establish base stations, networks, or tie lines and is the only insurance against elimination of the high-frequency component that contains the small or weakly magnetic targets. Gradiometers have shown that they can offer a high degree of immunity from diurnal and minor magnetic storm activity in the ambient magnetic field; they can enhance near-surface, small, or weakly magnetic anomalies.

Areas of Application Vertical axial gradiometers are used for measuring the vertical differences (the vertical gradient) in the vertical component of

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Magnetic Gradiometry, Fig. 2 The four-channel fluxgate gradiometer of the Institute Dr. Foerster, Reutlingen, Germany. Measurements in a grid of 0.05 by 0.06 m are possible with this device. The two fluxgate sensors are 0.65 m apart

the Earth’s magnetic field. This configuration is commonly used for ground-based geophysical surveys in the field of environmental geophysics, in areas with complex geology and archaeological prospecting, as the influences of deeper-lying interferences can be suppressed by this setup of the sensors. For practical reasons, the distance between the sensors of a gradiometer used for groundbased surveys lies somewhere between 0.5 and 1 m. As a rule, the ground penetration and detection ranges to about one to two meters below the surface. Deeper detection can be achieved if the interfering body is accordingly large and strongly magnetized. Fluxgate gradiometers are commonly found in environmental geophysics and in archaeological prospecting (Fig. 2). Surveys can be conducted at faster rates, and the highest requirements for spatial resolution of the smallest archaeologically relevant features can be met with measurement grids of 0.05 by 0.06 m without the fluxgate sensors influencing each other. Sometimes, these very small measuring intervals are necessary in archaeological prospecting in order to achieve the best possible spatial resolution of archaeological features (von der Osten-Woldenburg et al. 2006). The magnetic gradiometer map of a Roman fort and its adjacent settlement (Fig. 3) that was obtained with a measurement grid of 0.25 by 0.25 m shows a large number of diverse, often very characteristic anomalies which allow conclusions to be drawn on the position, shape, and size of defensive trenches, centrally located staff buildings with stone foundations, wooden barracks with their hearths, as well as buildings (stone-foundation buildings, street layouts, cellars, and pits) of the settlement adjacent to the fort.

Magnetic gradiometry is also employed in forensic geophysics, even if the radar method has been established as the standard procedure (Cheetham 2005; Hansen and Pringle 2013). However, certain general conditions such as a sufficient distance to the iron cores of reinforced concrete, fences, and other metal constructions must be provided, and it must be ensured that there is no metal refuse in the ground itself. The procedure is identical to that of archaeological prospecting: research is undertaken in small-scale, near-surface survey as the burials themselves are usually close to the surface. The human body only has a small measure of magnetic susceptibility, so a direct detection of the remains by magnetic gradiometry is not possible. But the excavation of a grave and the disruption of strata leads to disturbances of the layers of soil close to the surface that are usually magnetically homogenous, and the refilling of the grave with the excavated material – in which the magnetic particles are now statistically aligned – usually generates weak geomagnetic anomalies, often smaller than 2–3 nT (Fig. 4). The displacement of soil with a significantly higher susceptibility by the human body can intensify this effect. Axial gradiometers are also used in aeromagnetic surveys (see ▶ “Magnetic Methods, Airborne”), that is, by helicopter for the detection of ordnance remains close to the surface (Doll et al. 2006), in the prospection of mineral deposits (see ▶ “Magnetic Methods, Surface”) and for research of anomalies in the Earth’s magnetic field and deeper-lying geological structures of the Earth’s crust (see ▶ “Magnetovariation Studies”) from stratospheric balloons with a base length of sensors as far as 6 km apart (Webers et al. 2009).

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Magnetic Gradiometry, Fig. 3 Archaeological survey: Magnetic gradiometer map of a neolithic (5000 BC) to the La-Tène-period (sixth/ seventh century BC) sequence of settlements on the mountain Goldberg, part of the Nördlinger Ries, Germany. On the plateau of the mountain

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floor plans of the houses are clearly visible. The defensive structures in the west of this settlement date back to 4500 BC. West-East extension of the magnetogram: about 650 m

Transverse and longitudinal gradient magnetic measurements are applied in aeromagnetic surveys (Hogg 2004), in marine geophysics, and space exploration (see ▶ “Magnetic Methods, Principles”). The distance of the magnetometers from one to the other can range from 1.5 m (e.g., in cable and pipeline survey, or for detecting smaller metallic objects) to more than 500 m (for the detection of magnetic sediments close to the surface, shipwrecks, and the prospecting of mineral deposits), depending on the required area of application. Magnetic gradient tensor data are used, for example, for a better determination of the small-scale structure of the Earth’s lithospheric field (Kotsiaros and Olsen 2012). They can also be used to clean up areas which are contaminated with various types of unexploded ordonances (UXO) and landmines (Zuo et al. 2017).

Summary

Magnetic Gradiometry, Fig. 4 Forensic geophysics: Detection of a burial at the edge of a field (marked by a circle). The linear anomalies are produced by drainage pipes and geological structures in the substratum

Magnetics belong to the most commonly used methods for prospecting in applied geophysics when attempting to infer the position, geometry, and physical parameters of a body of interference or geological structure through the measurement of magnetic anomalies (Schmidt and Clark 2006).

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Gradiometers with two vertically arranged magnetometers that are positioned between 0.5 and 1 m apart, and which measure the vertical gradient and vertical component of the Earth’s magnetic field, have distinct advantages over totalfield magnetometers, especially in ground-based magnetic measurements in environmental geophysics (surveying of contaminated sites, examination of building construction sites, archaeological research of former settlements): they offer a high resolution in smaller features which is, however, gained at the cost of a lesser penetration of the ground in comparison to measurements made with total-field magnetometers. Additionally, they are very sensitive to close-by sources of interference. Background interference (such as the variability in time of the Earth’s magnetic field and other nearby sources of interference) can be eliminated or significantly reduced by subtracting the simultaneous readings of the two sensors in the gradiometer arrangement.

Cross-References ▶ Magnetic Gradiometry in Archaeo-geophysics ▶ Magnetic Methods, Airborne ▶ Magnetic Methods, Principles ▶ Magnetic Methods, Surface ▶ Magnetometers ▶ Magnetovariation Studies

Bibliography Blakely RJ (1996) Potential theory in gravity and magnetic applications. Cambridge University Press, Cambridge Cheetham P (2005) Forensic geophysical survey. In: Hunter J, Cox M (eds) Forensic archaeology: advances in theory and practice. Routledge, London/New York, pp 62–92 Clark DA (2012) New methods for interpretation of magnetic vector and gradient tensor data I: eigenvector analysis and the normalized source strength. Explor Geophys 43:267–282. https://doi.org/10.1071/ EG12020 Doll WE, Gamey TJ, Beard LP, Bell DT (2006) Airborne vertical magnetic gradient for near-surface applications. Lead Edge 25:50–53 Hansen JD, Pringle JK (2013) Comparison of magnetic, electrical and ground penetrating radar surveys to detect buried forensic objects in semi-urban and domestic patio environments. In: Pirrie D, Ruffeli A, Dawson LA (eds) Environmental and criminal geoforensics. Special publications, vol 384. Geological Society, London. https://doi.org/ 10.1144/SP384.13 Hogg S (2004) Practicalities, pitfalls and new developments in airborne magnetic gradiometry. First Break 22:59–66 Kotsiaros S, Olsen N (2012) The geomagnetic field gradient tensor. Int J Geomath 3:297–314. https://doi.org/10.1007/s13137-012-0041-6 Nabighan MN, Asten MW (2002) Metalliferous mining geophysics – state of the art in the last decade of the 20th century and the beginning of the new millennium. Geophysics 67:964–978 Schmidt PW, Clark DA (2000) Advantages of measuring the magnetic gradient tensor. Preview 85:26–30

987 Schmidt PW, Clark DA (2006) The magnetic gradient tensor: its properties and uses in source characterization. Lead Edge 25:75–78 Sheng J, Wan S, Sun Y, Dou R, Guo Y, Wie K, He K, Qin J, Gao J-H (2017) Magnetoencephalography with a Cs-based high-sensitivity compact atomic magnetometer. Rev Sci Instrum 88:094304 Veryaskin AV (2001) Magnetic gradiometry: a new method for magnetic gradient measurements. Sensors Actuators A Phys 91(1–2):233–235 von der Osten-Woldenburg H, Chaume B, Reinhard W (2006) New archaeological discoveries through magnetic gradiometry: the early Celtic settlement on Mont Lassois, France. Lead Edge 25:46–48 Webers W, Tsvetkov Yu, Brekhov O, Kraoivny A, Nikolaev A, Filippov S, Pchelkin A (2009) Complex satellite (“Swarm”) and stratospheric balloons geomagnetic researches. In: ESA’s second swarm international science meeting, Potsdam, 2009 Zuo B, Wang L, Chen W (2017) Full tensor eigenvector analysis on airborne magnetic gradiometer data for the detection of diploe-like magnetic sources. Sensors 17:1976. https://doi.org/10.3390/ s17091976

Magnetic Gradiometry in Archaeogeophysics Harald von der Osten-Woldenburg National Heritage Department, Regional Government of Baden-Wuerttemberg, Esslingen am Neckar, Germany

Definition Geophysical measurement procedures have become an important and integral part of surveying methods in archaeological disciplines. Unlike hard prospecting, where archaeological remains are analyzed during an excavation which ultimately leads to their destruction, geophysical measurement procedures are commonly referred to as soft prospecting. They are nondestructive and leave the ground undisturbed. Geophysical methods are therefore often used in the targeted preparation of rescue or research excavations (von der Osten-Woldenburg 2005). They can save time and money by helping to define the location and size, apparent state of preservation, and probable age of the surveyed archaeological remains. These measurement procedures are also widely used when documenting the sites of archaeological settlements: sites that can be preserved for coming generations as they remain undisturbed in the ground. Magnetic gradiometry is one of the most important methods when it comes to choosing from the range of geophysical measurement procedures that are suitable for archaeological surveys, and it is normally used as the initial prospecting technique. The reasons for this can be found in the usually very conclusive results, quick mapping of the site with a high level of data density, simple usage of the respective measuring instruments, and the relatively low

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requirements for data processing. The reasons for the existence of magnetic and therefore archaeologically relevant anomalies may include: buried artefacts containing ferromagnetic metals; long-term changes in magnetic properties as a result of the effects of fire, for example in the case of fires; changes in the distribution of magnetic properties in the ground due to the excavation of pits or defence trenches (Ernenwein and Hargrave 2009). Indications on the presence of yet undiscovered buried settlement sites can be provided by scanning the earth’s magnetic field in close proximity to the ground and recording the weakly defined, archaeologically caused geomagnetic anomalies. This geomagnetic mapping is an important tool for the discovery of archaeological settlement traces in unmapped terrain where there are no stray material finds, no indications in historical sources and no aerial photography findings. Additional methods that complement geomagnetic gradiometry – such as electrical resistivity tomography or ground penetrating radar – must be employed where necessary to verify the subareas in which the contrast of the magnetic susceptibilities, or the residual magnetization between the archaeological objects and the soil in which they are embedded, is faint and therefore appears in the magnetogram only as areas of weakly defined anomalies.

Advantages of Magnetic Gradiometers Gradiometers used in archaeological prospecting normally consist of an array of two sensors that are aligned at a vertical separation of 0.5 m, more commonly 0.65 m, and in some instances just over 1 m. They can detect magnetic anomalies that are weak, just beneath the surface, or of a limited size – all typical circumstances for archaeological objects – with more clarity than conventional measurements that are taken of the total intensity of the earth’s magnetic field in ground proximity while using only a single sensor. They also have a much greater spatial resolution than the single-sensor, total-field measuring magnetometer (GEMSystems 2019). For that reason alone, the gradiometer is the ideal instrument for detecting smaller anomalies that lie closer to the surface and extremely practical when it comes to archaeological mapping. It has also been shown that gradiometers have a high degree of insusceptibility toward the daily smaller magnetic storm activities that occur in the earth’s magnetic field (GEMSystems 2019). Measurements with an additional base station magnetometer that eliminates the variations in the earth’s magnetic field from the recorded data are therefore unnecessary. How deep down a gradiometer can detect archaeological structures depends on the one hand on the vertical separation between the two sensors, and on the other on the intensity of the contrast of the magnetic susceptibility, or the residual

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magnetization between the archaeological object and the soil in which it lies. Usually depths of 1–2 m can be reached under ideal conditions.

Instruments Magnetic gradiometry is carried out by using magnetometers that possess extremely high susceptibility and stability. A number of magnetometer sensors and configurations are available. Cesium vapor sensors or fluxgate sensors, used in pairs in a gradiometer configuration, are a common choice. In practice there are almost no differences in the capacity between the two types of sensor when they are used in a gradiometer configuration, apart from a higher sensitivity of cesium vapor sensors over the fluxgate variant. Cost and weight factors are the main reasons that fluxgate gradiometers are the preferred choice in archaeological field surveys. Standard gradiometers measure the vertical gradient of the earth’s magnetic field locally and in close proximity to the ground. This can be overlaid by faintly magnetic subsurface archaeological structures from former settlement sites. Composite maps, so-called magnetograms, which contain the magnetic image of subsurface archaeological structures, are then created from the measured variations of this magnetic gradient. As a rule, fluxgate gradiometry is carried out using a portable single or double gradiometer, or with up to four gradiometers that are mounted to a nonmetallic cart or carrying structure (Fig. 1). More recently, a growing number of vehicle-based measuring configurations have been used (Linford et al. 2015; Schneidhofer et al. 2016) that can carry up to 10 (Fig. 2) or even 20 fluxgate gradiometers (Eastern Atlas 2019; SENSYS Sensorik, & Systemtechnology 2019). These sensor carts are towed behind an off-road vehicle or an ATV and record the earth’s magnetic field in close proximity to the ground along a profile that can be up to five meters wide while the separation between the gradiometers can range from 0.25 m to 0.5 m.

Procedure in the Field Maps resulting from surveys that are made with an adequately high spatial resolution can provide quite recognizable and detailed contours of the buried structures. This allows a clear archaeological interpretation to be made. The necessary spatial resolution is, of course, dependent on the size of the structures that are under investigation. But as the actual size is usually not something that is known before the survey, the highest possible resolution must be chosen from the outset. Experience has shown that measurements that are taken with a maximum separation of 0.25 m along the profile line and at

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Magnetic Gradiometry in Archaeo-geophysics, Fig. 1 Using a four-channel fluxgate gradiometer (FEREX 4.032 by Institut Dr Foerster, Germany) on the site of the Lauchheim Roman fort (Fig. 5). (Photo Credit: Harald von der Osten-Woldenburg)

Magnetic Gradiometry in Archaeo-geophysics, Fig. 2 A geomagnetic survey of up 25 ha per day can be achieved – subject to soil conditions – with a ten-channel fluxgate gradiometer array (LEA D2

by Eastern Atlas, Germany) and a differential GPS system. (Photo Credit: Harald von der Osten-Woldenburg)

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an interval of 0.25 m are absolutely necessary to create and improve the composite image and therefore to address and interpret smaller magnetic anomalies. Surveys with gradiometer arrays, such as the single channel (viz. an instrument consisting of one gradiometer) FM36 or FM256 (both with a vertical sensor separation of 0.5 m) by Geoscan Research, Bradford/UK (Geoscan Research 2019), the two-channel Grad601 (vertical sensor separation: 1 m) by Bartington, Witney/UK (Bartington 2019), the FM256 Dual Gradiometer System by Geoscan Research (horizontal separation between the two gradiometers: 1 m), or the fourchannel FEREX 4.034 by Institut Dr Foerster, Reutlingen/ Germany (Foerster 2019), are usually performed within a regular grid and without the use of a GPS receiver attached to the gradiometer. In order to accomplish this, several grids must first be pegged with sizes that can range – depending on the instrument that will be used – from 20 m  20 m to 40 m  40 m. Guidelines are strung within these grids that Magnetic Gradiometry in Archaeo-geophysics, Fig. 3 Discovery of a prehistoric settlement site in Emerfeld (district of Biberach, Germany) with the ten-channel fluxgate gradiometer array LEA D2. (Photo Credit: Harald von der OstenWoldenburg)

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are one or two meters apart and allow the precise following of the parallel-aligned profiles. Gradiometers equipped with an audio signal (metronome) in combination with respective markings along the lines make it easier to follow a predefined speed when taking the measurements, which makes the magnetic recording of the buried structures as distortion-free as possible. Grids are not necessary in cases where a multichannel measurement system with at least ten fluxgate gradiometers is towed behind an off-road vehicle or ATV. Moreover, these surveys can also be seamlessly performed in irregularly shaped areas. This is made possible by using an RTK-DGPS system: The base station of a differential GPS system is set up at the edge of the survey area. The GPS-rover station is firmly attached to the measurement cart (Fig. 2). Such an array enables – depending on the soil conditions (e.g., meadow and ploughed field) – the mapping of relatively large areas of up to 25 ha per day with a high measurement density. In

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Magnetic Gradiometry in Archaeo-geophysics, Fig. 4 The example of a Roman building: Archaeological structures discovered with the help of magnetic gradiometry can be verified in more detail through

additional geophysical surveying methods (e.g., ground penetrating radar). (Photo Credit: Harald von der Osten-Woldenburg)

addition, there is also the advantage that the georeferencing of the measurements, which takes place at the same time as the gradiometer survey, allows for on-site processing of the data within minutes and the creation of the magnetogram. This provides important information already during the measurement process that can help to decide if the survey area needs to be extended or not.

cases where the initial discovery and localization is made in this manner. One such example is the discovery of a building from the Roman period (Fig. 4, left) that was recorded in just 2 min during the course of a geomagnetic gradiometry survey with a vehicle-based ten-channel fluxgate gradiometer. In a subsequent ground penetrating radar survey (SIR-4000 by GSSI, USA, 400 MHz antenna, 0.5 m profile interval), the floor plan became much clearer (Fig. 4, right). The expenditure of time for this latter survey, excluding the pegging of the grid, was higher by a factor of 10. Conversely, the initial use of the ground penetrating radar in this area would have been highly inefficient and the survey could have taken several days before the building would have been documented.

Case Studies Magnetic gradiometry is virtually predestined for the discovery of former settlement sites in until then archaeologically unidentified locations. The availability of multichannel gradiometers has enabled the geomagnetic surveys of larger suspected sites to be completed within a couple of hours. In the case of Emerfeld (in the district of Biberach, Germany), there were no indications whatsoever from either stray material finds, historical sources, or aerial archaeology. Measurements that were taken in a single day resulted in at least three locations being recorded that indicate toward the presence of a prehistoric settlement. They are marked in Fig. 3 by a circle and two ellipses. These are the characteristic geomagnetic anomalies associated with pit houses. The magnetogram also contains information on traces of ploughing (fine, parallel aligned lines), geological structures such as erosion gullies, and modern disturbances such as the location of utility poles for the power lines (marked as bright anomalies in squares) as well as a disused power line (the dark anomalies in squares). There are situations in which gradiometers may only provide an indication that archeologically relevant structures might be present. A more detailed documentation through the targeted use of other methods becomes necessary in

Do Different Measurement Instruments Deliver Different Results? During road works, the existence of a small, until then unknown Roman fort, was recorded in a magnetic gradiometer survey (Fig. 5a). The survey was done in October 1992, using a British Geoscan Research FM36. Profile separation and measurement-point interval were both 0.25 m. The outer ditch, foundations of the watchtowers, the fort’s main road, and post holes belonging to the fort’s wooden buildings are clearly recognizable in the magnetogram, as well as a Roman estate to the east of the fort and a medieval gallows in the northern part of the fort. In July 2009, the fort, which is located in an area of meadows and therefore lies protected in the ground, was reexamined with magnetic gradiometry (Fig. 5b). This time a four-channel German fluxgate gradiometer system FEREX 4.032 by Institut Dr Foerster was used (Fig. 1). The system was equipped with CON650 sensors (vertical separation between sensors: 0.65 m). The interval

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Magnetic Gradiometry in Archaeo-geophysics, Fig. 5 Comparison of surveys with different gradiometers in the example of a Roman earth and timber fort in Lauchheim, Ostalb district, Germany. Left: FM36, Geoscan Research, UK. Centre: FEREX 4.032, Institut Dr Foerster,

Germany. Right: LEA D2, Eastern Atlas, Germany. Multichannel gradiometers allow for larger areas to be surveyed in less time. The singlechannel gradiometer FM36 provided more detailed information on this small fort. (Photo Credit: Harald von der Osten-Woldenburg)

between the measuring points was 0.05 m, the profile separation 0.25 m. The data was resampled based on a grid of 0.25 m  0.25 m. The advantage of this instrument lies in the much higher prospecting speed over that of the FM36. This allowed a geomagnetic mapping of the entire meadows, and during the survey, a further Roman ditch system was discovered, the use of which is so far unknown (horse paddock?). A third survey of the site was undertaken in March 2015, using a ten-channel German Eastern Atlas LEA D2 system (Fig. 2). The sensors used in this survey were the same as those in the four-channel FEREX-system (CON650). The separation between the sensors was 0.5 m, and measurements were taken on average at an interval of 3 cm and a speed of 10 km/h. This data was also resampled based on a grid of 0.25 m  0.25 m (Fig. 5c).

archaeological objects would it be advisable to resort to targeted resurveying by foot.

Conclusion Even if a comparison between the FM36 sensors and the Foerster-CON650 sensors is not strictly allowed – since the separation of the fluxgate sensors of the FM35 is 0.5 m, and that of the CON650 is 0.65 m – it still shows that the measurements taken by foot were slightly more exact and that they contain more detail than the multichannel instrument surveys. The vehicle-based ten-channel Foerster gradiometer measurements, for example, do not show the fort’s eastern road, whereas they are clearly recognizable in the FM36 survey. However, the benefits of vehicle-based surveys – covering larger areas in less time – probably outweigh the disadvantages. Only for the actual documentation of suspected

Cross-References ▶ Electromagnetic Pulsations and Magnetic Storms ▶ Magnetic Gradiometry ▶ Magnetic Methods, Surface ▶ Magnetometers

Bibliography Bartington (2019). https://www.bartington.com/wp-content/uploads/ pdfs/datasheets/Grad-601_DS1800.pdf Eastern Atlas (2019). http://www.eastern-atlas.com/start/index_eng.php Ernenwein EG, Hargrave ML (2009) Archaeological geophysics for DoD field use: a guide for new and novice users report submitted to the environmental security technology certification program for project RC-200611: streamlined archaeo-geophysical data processing and integration for DoD field use. Center for Advanced Spatial Technologies, University of Arkansas, Fayetteville. https:// wayback.archive-it.org/6471/20150825215604/http://cast.uark.edu/ assets/files/PDF/ArchaeologicalGeophysicsforDoDFieldUse.pdf. Accessed 5 Mar 2014 Foerster (2019). https://www.fluxgate-magnetometer.com/de/produkte/ ferex-4034/ GEMSystems (2019). http://www.gemsys.ca/advantages-of-magneticgradiometers/ Geoscan Research (2019). http://www.geoscan-research.co.uk Linford N, Linford P, Payne A (2015) Chasing aeroplanes: developing a vehicle-towed caesium magnetometer array to complement aerial photography over three recently surveyed sites in the UK. Near Surf Geophys 13(6):623–631. https://doi.org/10.3997/1873-0604. 2015044

Magnetic Methods, Airborne Schneidhofer P, Nau E, Hinterleitner A, Lugmayr A, Bill J, Gansum T, Paasche K, Seren S, Neubauer W, Draganits E, Trinks I (2016) Palaeoenvironmental analysis of large-scale, high-resolution GPR and magnetometry data sets: the Viking Age site of Gokstad in Norway. Archaeol Anthropol Sci 9(6):1187–1213. https://doi.org/ 10.1007/s12520-015-0312-x SENSYS, Sensorik & Systemtechnology GmbH (2019). https:// sensysmagnetometer.com Von der Osten-Woldenburg H (2005) Applications of ground penetrating radar, magnetic and electrical mapping, and electromagnetic induction methods in archaeological investigations. In: Butler DK (ed), Near-surface geophysics: SEG, Society of Exploration Geophysicists, Tulsa, pp 621–626. https://doi.org/10.1190/1.978156080 1719.ch23

Magnetic Methods, Airborne Mike Dentith School of Earth Sciences (M004), The University of Western Australia, Crawley, WA, Australia

Synonyms Aeromagnetics

Definition Airborne magnetic surveys comprise measurements of the Earth’s magnetic field made with a magnetometer mounted on an aircraft.

Introduction Making measurements from the air of the Earth’s magnetic field is a well-established geophysical survey method. The normal practice is to measure the scalar amplitude of the magnetic field in the survey area, this being referred to as the total magnetic intensity (TMI), the name reflecting the fact that the measurements represent the resultant of all the magnetic fields, of whatever source, in the vicinity of the measurement. The major contributions come from the geomagnetic field and the fields due to magnetic rocks in the upper crust, the latter being the fields of interest. Initially used to infer the depth to magnetic rocks beneath nonmagnetic cover in sedimentary basin studies and for identification of major lineaments, the products from aeromagnetic surveys have evolved to now be a reliable means of creating detailed maps of the near-surface geology. Current developments in aeromagnetics are mostly related to the increasingly sophisticated inference of the 3D distribution of

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magnetism in the Earth’s crust using inverse modeling methods. Comprehensive accounts of aeromagnetic data, acquisition, processing, and interpretation are proved by Gunn (1997), Reeves (2005), Isles and Rankin (2013), and Dentith and Mudge (2014).

Making Magnetic Measurements from the Air Airborne magnetic data are acquired as a series of parallel “survey” or “flight” lines and more widely spaced orthogonal “tie” lines (Fig. 1). The purpose of the tie lines is to create repeat measurements at the locations where they cross the flight lines, which are required during processing of the data (see below). An advantage of making magnetic measurements from the air is that unless the survey area is small, typically less than a few square kilometers, it is much faster and cheaper to acquire the data than for a ground-based survey. The main disadvantage is that turbulence and variations in topography, especially if the terrain is rugged, may seriously affect the ability to position the aircraft as would be preferred and also affect the individual measurements in a hard to predict way. Equipment Airborne measurements are routinely made using fixed-wing aircraft and helicopters. Surveys using fixed-wing aircraft are significantly cheaper than helicopter surveys, but helicopters have the advantage of being able to fly lower and better maintain a consistent ground clearance in rugged terrain, but have the disadvantage of being less stable survey platforms. To reduce effects of magnetic fields originating in the aircraft the magnetic sensor, usually a cesium vapor magnetometer (see ▶ “Magnetometers”), is located as far as possible from the main body of the aircraft and hence the magnetic fields associated with ferrous components and electric currents. For fixed-wing aircraft the magnetic sensor is mounted in a “stinger” located aft of the aircraft. In helicopters the sensor may be in a “boom” that positions the magnetometer in front of the aircraft or it may be within a towed “bird.” The bird is a bomb-like device suspended on a cable beneath the aircraft. A “compensation” system within the aircraft is used to further reduce the effects of magnetic fields originating from the aircraft. In addition to carrying one or more magnetometers, the survey aircraft also measures terrain clearance using an altimeter and GPS position and time. One or more “base station” magnetometers are also deployed on the ground to measure temporal changes in the magnetic field in the survey area to enable such variations to be removed from the data (see below).

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Magnetic Methods, Airborne, Fig. 1 Schematic illustration of the flying pattern used in aeromagnetic surveys

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Magnetic Methods, Airborne, Fig. 2 Profile of total magnetic intensity across a dyke-like magnetic body. The four profiles correspond with the bodies at different depths in (c). The profiles in (b) represent

variations in TMI. In part (a) the amplitudes have been normalized to the profile with maximum amplitude (shallowest source)

The Importance of Source-Sensor Separation An important source of noise in aeromagnetic data is associated with changes in distance between the magnetic sensor

and the magnetic materials. Any changes in this distance will create variations in TMI, which could easily be confused with those caused by changes in the local geology. As the distance

Magnetic Methods, Airborne

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Magnetic Methods, Airborne, Fig. 3 Example of the flight path of a fixed-wing survey aircraft across rugged topography. (a) Topographic and altimetry profile, (b) vertical terrain clearance. (Based on data in Flis and Cowan 2000)

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b increases the measured magnetic field variation decreases in amplitude and increases in wavelength (Fig. 2b). The wavelength increase is most easily seen when the amplitude decrease is removed by rescaling the individual profiles so they have the same peak-to-peak amplitude (Fig. 2a). Ideally the survey aircraft maintains a constant ground clearance during the acquisition of the data, referred to as “drape” flying. In practice, aircraft are subject to the effects of wind and turbulence and the inaccuracies of human control. In addition, all aircraft have a limited ability to maneuver, especially climb. Figure 3 shows the flight path of a fixedwing aircraft across a ridge and adjacent areas. When short wavelength variations in topography (B on the diagram) are encountered, the aircraft cannot respond quickly enough to maintain the separation and positive relief will bring the magnetic sources closer to the sensor, with a corresponding increase in the amplitude of the magnetic field (c.f. Fig. 2), and vice versa. When large topographic features are encountered (A on the diagram) the aircraft will inevitably vary its ground clearance because of its limited ability to climb and to a lesser extent descend. If, as is the normal case, alternate flight lines are flown in opposite directions, adjacent parts of survey lines will have markedly different ground clearance adjacent to topographic features. This will introduce spurious variations into the data. Current practice is to “program” a preplanned flight path for the aircraft, which the pilot follows using a realtime display. This program accounts for the terrain and the aircraft’s performance. The aim is to achieve the best “loose drape,” whereby terrain clearance effects are

Position on flight-line

minimized as far as is practical. Note that even if a perfect drape was achieved, the effects of terrain would not be totally removed because even in the ideal scenario there are still the effects of variations in the distances to magnetic materials because it is not just the materials directly below the aircraft that are influencing the measurements. The measurement at X in Fig. 3 will be affected more by the magnetic materials to the side than beneath because of the lesser distance to these materials, although this scenario only occurs in severe terrain.

Survey Design Sampling theory requires individual measurements of TMI to be spaced at a maximum of half the wavelength of the shortest wavelength of variation. Economic and safety factors mean this is rarely practical and some aliasing of responses is the norm. This is not necessarily a major problem since during qualitative interpretation of the data it is relative changes in amplitude and texture that are used, and in fact accurate definition of the variations is only required when specific anomalies need to be analyzed quantitatively (see ▶ “Magnetic Anomalies: Interpretation”) and often there is a followup, more detailed, survey of the area of interest to improve anomaly characterization. Survey Specifications The survey geometry, illustrated in Fig. 1, requires definition of three key parameters: the survey line spacing and

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Magnetic Methods, Airborne, Table 1 Typical survey specifications for aeromagnetic surveys Survey line spacing (m) 50 100 200 400

Tie-line spacing (m) 500 1000 2000 4000

Flight height (m) 40 50 60 80

Sampling ratio: Survey/Tie linea 7.5 15 30 60

Shortest along-strike wavelength sufficiently sampled 100 200 400 800

Assumed flight speed is 240 km/h (67 m/s) and 10 Hz sampling resulting in a 6.7 m along survey line sampling interval. These parameters are typical for a survey with a fixed-wing aircraft

a

orientation and the flight height. Typical values of two of these variables for different survey types are given in Table 1. Note that the tie-line spacing is a dependent variable, typically being set to ten times the survey line spacing, although this may be reduced to five times or less for high resolution surveys. The survey line spacing controls the cost of the survey, which for fixed-wing aircraft is based on the total length of lines flown. The total line length of a survey in terms of the survey area and line spacing can be estimated from the equation below (Brodie 2002):   D lines 1000 1 þ DSurvey Survey area Tie lines Total line length ¼ DSurvey lines where ΔSurvey lines and ΔTie lines are the survey- and tie-line spacings in meters, respectively, the total line length is in kilometers, and the survey area is in square kilometers. To this cost must be added nonproduction costs such as mobilization and “stand by” costs associated with factors outside the acquisition company’s control, for example, bad weather, magnetic storms. With helicopter surveys the time spent in the air is also taken into account, and may be significant in mountainous areas where weather conditions severely restrict data acquisition both in terms of flying time and location (Mudge 1996). It is clear from Fig. 2 that the lower the flight height, the greater will be the amplitude of the targeted responses and also the shorter the wavelength of variation, and hence the greater the ability to resolve adjacent features. Figure 2 implies the lowest possible flight height should be used, but safety considerations mean flight heights are some tens of meters. “Crop-duster” fixed-wing aircraft or heli-borne surveys may fly lower but ultra-low flying is not necessarily an advantage from a technical point of view since it means maintaining consistent terrain clearance is more difficult. Besides, there may be very shallow magnetic sources whose anomalies mask those of deeper, more significant variations (Doyle and Lindeman 1985). Reid (1980) showed that accurate definition of a TMI anomaly requires the sample spacing to be a maximum of half the source-sensor separation. The combination of the

rate at which the aircraft’s magnetometers make a measurement (typically 0.1 s) and the velocity of the aircraft (typically 220–280 km/h for a fixed-wing aircraft) result in an along flight line sample spacing of about 7 m for fixedwing aircraft. Helicopter surveys with a 0.1 s sampling typically have a 4–5 m spacing, but the trend is to employ faster sampling magnetometers, reducing this spacing to even less. The important outcome from the above is that the along-line sampling exceeds the requirements of sampling theory for practical flight heights, and the variations in TMI in the direction of the flight path can be considered as more than adequately characterized. The same is not true for variations in the directions perpendicular to the survey line direction because, as shown in Table 1, the spacing of these lines is very much greater than the alongline sampling interval. Note that since anomaly wavelength decreases with source-sensor separation, a low flight height combined with wide survey lines will lead to greater aliasing in the direction perpendicular to the survey lines. This is reflected in the parallel increase in flight height with line spacing in Table 1. Conventionally airborne geophysical surveys are flown with the flight lines oriented perpendicular to the dominant geological strike direction. The rationale for this is that there will be shorter wavelength variations across strike than parallel to strike, and hence the greatest sampling density should be in the strike-perpendicular direction. For elongate anomalies the short wavelength variation parallel to strike is small, so significant aliasing does not occur. In practice economic considerations dictate the survey line spacing, meaning there is inevitably some aliasing of TMI variations in the direction perpendicular to the survey lines. Even if there is some variation in the strike across the survey area, and there is also a conflict caused by variations in stratigraphic and structural strike, it is best to use a single-line orientation. If this is not the case the data are very difficult to process into an easily interpreted form. For surveys in equatorial regions the east–west elongation of magnetic responses associated with low declinations (3

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Paleoseismology, Fig. 5 Two possible rupture sequences on the southern San Andreas fault (Weldon et al. 2004). Vertical grey bars are ranges in age for earthquakes at the sites listed at the lower margin and

400

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horizontal bars are rupture lengths. Open boxes represent multiple events age ranges;the individual event ages are unknown. Grey shading indicates regions and times with no data

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Paleoseismology

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Paleoseismology, Fig. 6 History of mega-thrust earthquakes along the Nankai trough, Japan (Sangawa 2010). The subduction zone is divided into five rupture segments A–E (F is the source of 1923 M ¼ 7.9 Kanto earthquake along the Sagami trough). The horizontal lines with number in lower panel indicate the rupture extents and occurrence years of large earthquakes. Dots with number denote archaeological sites with exposed significant liquefactions at specific ruins and their calendar years. Recent archaeo-seismological data filled gaps in long recurrence intervals originally estimated by only historic accounts (e.g., possible rupture on A and B segments at the 1,498 earthquake), and making the Nankai subduction earthquake cycles more regular and frequent. Tsunami deposits recovered from coastal lakes, lagoons, swamps, and lowlands are also filling the gaps (e.g., Takada et al. 2002) and also provide much longer rupture history back to the entire Holocene period (Komatsubara and Fujiwara 2007)

and their recurrence intervals along the 2004 SumatraAndaman (Jankaew et al. 2008), and along the Kuril trench (Nanayama et al. 2003), and relation with the M8 class predecessors of the 1960 M9.5 Chile earthquake (Cisternas et al. 2005), which breaks through the conventional idea on regular

occurrences of subduction earthquakes (see ▶ “Great Earthquakes”). Besides paleo-tsunami deposits, submarine turbidites are another geologic clue to reveal large submarine and coastal earthquakes. Goldfinger et al. (2007) demonstrated their validity and significance of sediments associated with earthquake-triggered turbidity currents by analyzing numerous piston cores collected from submarine channels and canyon systems draining the northern California continental margin. By studying synchronous triggering of the turbidity currents, they found 15 turbidites during the last ~2,800 years, equivalent to an average repeat time of ~200 years which is similar to the onshore value of ~230 years along the northern San Andreas fault. Along-strike correlation also suggests the ~320-km rupture extent of the most of the younger events. Broad deformation of off-fault areas is another indicator for paleoearthquakes (see ▶ “Earthquakes and Crustal Deformation”). Vertical deformation can induce changes in local rates of deposition and erosion that provide evidence of a paleoearthquake, particularly in fluvial and coastal environment (McCalpin and Carver 2009). Subduction associated mega-thrust earthquakes and dip-slip intraplate earthquakes largely change horizontal markers such as shorelines prior to an earthquake and record offfault coseismic uplift (e.g., Meghraoui et al. 2004) which are preserved as uplifted terraces and shorelines. Coastal areas facing subduction zones, such as southwest Japan, Taiwan, New Zealand, and Crete in Greece, have several steps of elevated terraces and shorelines associated with the mega-thrust earthquakes (see ▶ “Seismicity, Subduction Zone”). But significant coseismic uplifts are also inferred to have been formed by shallow near-coast offshore active faults that might be branched from subduction interface (e.g., Maemoku 2001). Although dating paleo-marine terraces and shorelines is not easy, coral reefs, coral microatolls (e.g., Zachariasen et al. 1999), fossils of sessile organisms inhabit intertidal level and become both paleo-shoreline markers and indicators of timing of earthquake occurrences. For older terraces over 50 ka, other dating techniques and regional tephra fallouts are used to estimate the emergent dates. In Japan, altitudes of uplifted marine terraces formed at marine oxygen isotope stage 5e (~120 ka) often provide us with the best estimates of long-term uplift rates associated with interplate events (Koide and Machida 2001). In contrast to coseismic uplift, studies for paleoseismic subsidence are much fewer. Subsidence associated with a subduction earthquake normally occurs inland from the rebounded uplift zone. Here, buried soil of subsided marsh and forest records the sudden vertical deformation (e.g., Atwater et al. 1995). Permanent deformation of landscapes associated with seismogenic faulting is sometimes modified or amplified by postseismic crustal movement. Recent mega-thrust subduction events caused not only coseismic uplift, but also postseismic transient deformation that lasts more than a few decades (see

Paleoseismology

▶ “Earthquakes and Crustal Deformation”). Although contribution of postseismic deformation associated with paleoearthquakes is hardly measured in the uplifted terraces, Sawai et al. (2004) used fossil diatom assemblages to infer the gradual changes in sedimentary environments corresponding to the transient postseismic deep creep that occurred in the seventeenth century along the Kuril subduction zone. Seismic Hazard Assessment Based on Paleoseismology Paleoseismic data are directly used for making probabilistic seismic hazard maps (see ▶ “Seismic Hazard”) which incorporate information on fault location, dimension, geometry, largest earthquake from such dimension, slip rates, and recurrence intervals (e.g., Working Group on California Earthquake Probabilities 1995). Two types of statistical models of earthquake occurrence are commonly used to estimate earthquake probability: a stationary Poisson model and a conditional quasi-periodic model (see ▶ “Statistical Seismology”). The former is used for active faults or active seismic regions where only frequency of large earthquakes is available. The latter considers fluctuations of repeat times (coefficient of variation in recurrence time) and elapsed time since the most recent event (see Fig. 3), which is time-dependent and compatible with the strain accumulation process. Several probability density functions such as normal, lognormal, Weibull, or gamma distributions are used for the recurrence process. Among them, the Brownian Passage Time function (Matthews et al. 2002) is evaluated as so far the best function because of its capability to account for the stress perturbations and has been adopted for the probabilistic seismic hazard studies in California and Japan (Field et al. 2009; ERC 2005). Time-dependent seismic hazard considering conditional probabilities on most major active faults provides us with more realistic estimates (Fujiwara et al. 2009). If further detailed paleoseismic information is available for a specific fault, a “scenario earthquake model (seismic hazard maps for specified seismic sources)” is often proposed to show the strong shaking for the areas being assessed when the specific earthquake occurs (ERC 2005). When providing paleoseismic data to seismic engineers, possible rupture patterns for the next earthquake, slip distribution, and plausible locations of asperities becomes additional information (Somerville et al. 1999). In addition to the shaking damage, surface rupture and displacement associated with a large earthquake often destroy man-made structures. Structures for human occupancy (e.g., hospital and school) and critical buildings (e.g., nuclear power plant) can be so located to avoid the traces of active faults (e.g., the Alquist-Priolo Earthquake Fault Zoning Act in California, see entry on ▶ “Seismic Zonation”). To mitigate direct damage due to surface-faulting, detailed positions of active faults and their paleoseismological properties are progressively been opened to the public through the internet.

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Issues in Paleoseismology and Perspectives There are fundamental limitations of the geologic approach to infer paleoearthquakes in time and space. The following are major caveats in using paleoseismological information: 1. Surface ruptures are commonly associated with M > 6 earthquakes (Wells and Coppersmith 1994) that occur at shallow depths (mostly 6.5) and localized seismicity (Stein and Okal 1978) along the northern portion of the Ninety East Ridge (see inset in Fig. 6), suggestive of left-lateral strike-slip motion. Figure 6 shows the observed convergence history of India and Australia relative to Eurasia since early Miocene based on geodetic (Sella et al. 2002) as well as paleomagnetic (Cande and Stock 2004; DeMets et al. 1994; Gordon and Jurdy 1986; Merkouriev and DeMets 2006) data collected along the Carlsberg and South East Indian ridges (labeled respectively as CBr and SEIr in the inset). Within error-bars, convergence rates are almost indistinguishable between 20 and 11 Myrs ago, suggesting that India and Australia behaved as one single plate with presumably little deformation occurring in between. Over the past 11 Myrs, however, their convergence to Eurasia differs distinctly. While India slowed down by almost 2 cm/year, convergence of Australia to Eurasia remained almost steady, with only some 0.5 cm/year of reduction. Timing of the India/Eurasia plate-motion change coincides reasonably well with the occurrence of diffuse deformation in the Indian Ocean. More relevant is the fact that Tibet had attained most of its current elevation (Tapponnier et al. 2001) prior to the slowdown of the Indian plate and prior also to the presumed formation of the India/

Plate Motion Changes in the Indian Ocean While instantaneous calculations of the plate tectonic momentum balance cannot be taken to model the temporal evolution of plate boundaries, they do allow us to test the effects of variations in plate geometry on global plate motions, and in particular the creation of new plate boundaries. From the principle of inertia, it follows that any such event would invariably trigger plate motion changes due to repartitions in the budget of basal drag and plate boundary forces. A recent such episode is thought to have occurred in the Indian Ocean, where a variety of evidence has been interpreted as the generation of a diffuse boundary between the India and Australia plates, dated between 8 and 20 Myrs ago (Wiens et al. 1985; Gordon et al. 1998). Ocean-floor

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Plate motion history in Southern Pacific and Southern Atlantic

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Plate Motions in Time: Inferences on Driving and Resisting Forces, Fig. 5 Predicted and observed relative plate motions in the South Atlantic and South Pacific over the past 10 Myrs for a set of adjacent plate pairs: PA/NZ, NZ/SA, NZ/AN, and SA/AF (abbreviations as in Fig. 6). Black bold segments in the small inset indicate positions along plate boundaries (thin black) at which relative motions have been computed. Observed plate motions (with error bars) inferred from paleomagnetic and geodetic data are represented by black dots, while empty squares indicate relative motions predicted from our simulations of the global coupled mantle/lithosphere system. The models explicitly account for the growth of the Andes over the past 10 Myrs, and demonstrate that the relative plate motion record can be entirely explained with the history of Andean orogeny. Our simulations thus point to the importance of far-field effects in plate tectonics and imply that resisting plate margin forces due to Andean growth account for about 18% of global plate motion changes over the past 10 Myrs (see Fig. 8). (Courtesy of Elsevier © 2009)

Australia plate boundary, implying that resistive plate boundary forces arising from the gravitational load of Tibet were already in place to act against convergence. Iaffaldano and Bunge (2009) tested explicitly whether plate-boundary forces from high Tibet are sufficient to explain the observed reduction of India/Eurasia plate convergence, once the former is separated from Australia by an additional plate boundary (Fig. 7). Specifically, they performed two distinct simulations of global plate motions, one with India and Australia cast as one single plate and the other with two plates built into the computational finite-element grid. A single India/Australia plate results in a predicted convergence of 5.2 cm/year at long 86°E, lat 27°N (Fig. 7a), whereas India being separated from Australia implies a convergence of 3.5 cm/year at the same position (Fig. 7b). The latter prediction compares remarkably well with the geodetic estimate (see Fig. 6). Finally, it is worth mentioning that simulations also predict an increased convergence between India and Australia,

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Plate Motions in Time: Inferences on Driving and Resisting Forces, Fig. 6 Observed convergence of India (red) and Australia (blue) relative to fixed Eurasia over the past 20 Myrs. Convergence rates are computed through rigid-rotation Euler poles at long 86°E, lat 27°N of the India/Eurasia margin. Present-day values (squares) are derived from geodetic techniques while the paleomagnetic record (solid lines) is computed by averaging finite rotations of magnetic anomalies identified along the Carlsberg and South-East Indian ridges (labeled respectively, as CBr and SEIr in inset. CIr is Central Indian ridge). Note that convergence rates are very similar between 20 and 11 Myrs ago, when India and Australia appear to behave as one single plate with presumably little internal deformation. The convergence relative to Eurasia differs more distinctly over the last 11 Myrs, when India and Australia slowed down by almost 2 cm/year and 0.5 cm/year, respectively. Inset shows locations of identified unconformities of sediments (magenta dots) as well as great (Ms > 6.5) earthquakes (green dots) indicating left-lateral strike-slip motion in the northern portion of the Ninety East Ridge (orange contours). Those evidences suggest diffuse deformation in the Indian Ocean particularly pronounced during late Miocene and have been interpreted as separation between the India and Australia. Plate boundaries are in white and continental topography in gray color scale. (Courtesy of Elsevier © 2009)

concentrated in the Indian Ocean, compatible with the aforementioned geologic and geodetic record.

Conclusions Recent results indicate that joint modeling of the mantle/ lithosphere system begins to achieve a level of maturity that allows explicit testing of a range of hypotheses on the force balance in plate tectonics and identifying key controlling parameters. While buoyancy forces from MCMs contribute significantly to the dynamics of plate motion, it is clear that plate boundary forces are of sufficient magnitude relative to these driving forces to affect plate motions and plate deformation, and to initiate rapid plate motion changes. One key controlling parameter in regulating plate velocity is the elevation of large mountain belts, because their topographic load

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Predicted plate motion of India relative to Eurasia IN and AU as one single plate

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Plate Motions in Time: Inferences on Driving and Resisting Forces, Fig. 7 Predicted India (IN) plate motion relative to Eurasia (EU) from two distinct simulations with India and Australia (AU) acting, respectively, (a) as one single and (b) as two separate plates, where plate boundaries in our computational mesh are shown in bold white and finite elements in thin white. Plate motions are computed at long 86°E, lat 27° N. Abbreviations of plate names as in Fig. 6. Note that a single India/

Australia plate results in a predicted convergence of 5.2 cm/year relative to EU, incompatible with the geodetic estimate. Two separate plates result in a convergence of 3.5 cm/year of IN relative to EU, similar to the present-day observation. In the latter scenario, resisting forces from the gravitational load of Tibet act only against the smaller India plate and are thus more effective in slowing the convergent motion. (Courtesy of Elsevier © 2009)

consumes a considerable amount of the driving forces available in plate tectonics, as much as 1013 N/m. Along the Nazca/South America plate boundary, these forces are sufficient to reduce the convergence rate over the past 10 Myrs by some 30%. This reduction is, however, not an isolated episode of a rapid plate motion slow down. Instead, many such variations are documented from the global compilation of Müller et al. (2008), which points to the importance of topography and erosion in the global tectonic system (Cloetingh et al. 2007). The fact that models accurately predict the spreading history of the Pacific/Nazca, Nazca/South America, Nazca/ Antarctica, and South America/Africa plate boundaries is of equal interest. The result is not entirely surprising and arises from the kinematic constraints of plate tectonics on the sphere. This suggests that far-field effects cannot be neglected in the geologic record at least in some cases. The strong influence of mountain belts on the plate tectonic force balance could have important implications. In an influential paper, Raymo and Ruddiman (1992) advanced the notion that Cenozoic climate change may have been caused by the uplift of Tibet. In other words, the rise of large mountain plateaus may act as a tectonic force on climate (Strecker et al. 2007). Low erosion rates have been implicated as a prerequisite for the creation of large mountain plateaus (Sobel et al. 2003; Clift and Vannucchi 2004). This implication suggests conversely that climate can act – through large topography – as a force in plate tectonics. Overall, a significant portion of recent

changes in global plate motions can be attributed to topography-related forcing along plate boundaries rather than to mantle buoyancy. These findings are summarized in Fig. 8, where the relative plate motion changes observed globally over the past 10 Myrs are plotted. Green and red bars show variations in plate motion that are related, respectively, to the growth of the high Andes or to the presumed recent separation between India and Australia, and amount to about 35% of the total change over the Earth surface. This remarkable first-order result clearly demonstrates the ability of plate boundary forces to affect the global plate velocity field. The level of maturity achieved by neo-tectonic simulations coupled with 3-D MCMs thus allows geodynamicists to make explicit predictions of the plate tectonic force balance that can be tested against the geologic record of present and past plate motions.

Cross-References ▶ Continental Drift ▶ Geodesy, Ground Positioning, and Levelling ▶ Geodesy, Networks, and Reference Systems ▶ GPS, Tectonic Geodesy ▶ Lithosphere, Oceanic ▶ Plate-Driving Forces ▶ Plates and Paleoreconstructions

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Observed variations of adjacent-plates motions for the past 10 Myrs

Adjacent-plates pairs

EU-NA AN-AF AR-EU NA-SA NA-AF EU-AF AN-SA Explicable through Andean growth solely AR-AF Explicable through IN/AU separation solely AU-AF AN-AU AF-SA CA-SA EU-PA NA-PA AU-PA CA-NA AN-PA AR-IN NZ-PA AN-NZ NZ-SA CA-NZ PA-PH IN-AF IN-AU IN-EU EU-PH 0

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Plate Motions in Time: Inferences on Driving and Resisting Forces, Fig. 8 Observed variations of adjacent-plates motions over the past 10 Myrs. Abbreviations of plate names as in Fig. 6. For each couple of adjacent plates, variations are computed as magnitude of difference between relative rotation poles at 10 Myrs, derived from paleomagnetic reconstruction, and at present day, obtained through geodetic techniques (GPS). The Cocos oceanic plate is not considered, since a geodetic estimate for its rotation pole is not available. Variations of AF/SA, AN/NZ, NZ/SA, and NZ/PA adjacent-plates systems (green bars) can be entirely explained through the effect of Andean growth

(see Figs. 6 and 7). They amount to about 18% of the global relative motions changes over the past 10 Myrs. Variations of IN/EU as well as IN/AU relative motions (red bars) can be entirely explained through the effect of separation between India and Australia (see Fig. 7). They amount to about 17% of the global relative motions changes over the past 10 Myrs. Thus, our models of mantle/lithosphere dynamics explicitly predict about 35% of the global plate motion changes observed over the past 10 Myrs from two well-identified tectonic variations. (Courtesy of Elsevier © 2009)

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Plate Tectonics, Precambrian Raymo ME, Ruddiman WF (1992) Tectonic forcing of late Cenozoic climate. Nature 359:117–122 Sella GF, Dixon TH, Mao A (2002) REVEL: a model for recent plate velocities from space geodesy. J Geophys Res 107:2081–2111 Sobel ER, Hilley GE, Strecker MR (2003) Formation of internally drained contractional basins by aridity-limited bedrock incision. J Geophys Res 108:2344 Stein RS (1987) Contemporary plate motion and crustal deformation. Rev Geophys 25(5):855–863 Stein S, Okal E (1978) Seismicity and tectonics of the Ninetyeast Ridge area: evidence for internal deformation of the Indian plate. J Geophys Res 83:2233–2246 Strecker MR, Slonso RN, Bookhagen B, Carrapa B, Hilley GE, Sobel ER, Trauth MH (2007) Tectonics and climate of the Southern Central Andes. Annu Rev Earth Planet Sci 35:747–787 Suppe J (2007) Absolute fault and crustal strength from wedge tapers. Geology 35(12):1127–1130 Tackley PJ, Stevenson DJ, Glatzmaier GA, Schubert G (1994) Effects of multiple phase transitions in a three-dimensional spherical model of convection in Earth’s mantle. J Geophys Res 99: 15877–15902 Tapponnier P, Zhiquin X, Roger F, Meyer B, Arnaud N, Wittlinger G, Jingsui Y (2001) Oblique stepwise rise and growth of the Tibet plateau. Science 294:1671–1677 van der Hilst RD, de Hoop MV, Wang P, Shim S-H, Ma P, Tenorio L (2007) Seismostratigraphy and thermal structure of Earth’s coremantle boundary region. Science 315:1813–1817 Weissel JK, Anderson RN, Geller CA (1980) Deformation of the IndoAustralian plate. Nature 287:284–291 Wiens DA, DeMets C, Gordon RG, Stein S, Argus D, Engeln JF, Lundgren P, Quible D, Stein C, Weinstein S, Woods DF (1985) A diffuse plate boundary model for Indian Ocean tectonics. Geophys Res Lett 12:429–432 Wilson JT (1965) A new class of faults and their bearing on continental drift. Nature 207:343–347 Zhong S, Zuber MT, Moresi L, Gurnis M (2000) Role of temperaturedependent viscosity and surface plates in spherical shell models of mantle convection. J Geophys Res 105:11063–11082

Plate Tectonics, Precambrian Y. J. Bhaskar Rao, T. Vijaya Kumar and E. V. S. S. K. Babu Geochronology Division, CSIR-National Geophysical Research Institute, Hyderabad, India

Definition Precambrian (>541 million years, Myr) is divisible into the Hadean Eon (>4.0 billion years, Gyr), Archean Eon (4.0–2.5 Gyr), and the Proterozoic Eon (2.5–0.54 Gyr). The Archean and Proterozoic Eons are further divisible into Eras: Eo-archean (4.0–3.6 Gyr), Paleo-archean (3.6–3.2 Gyr), Mesoarchean (3.2–2.8 Gyr), Neo-archean (2.8–2.5 Gyr), Paleoproterozoic (2.5–1.6 Gyr), Meso-proterozoic (1.6–1.0 Gyr), and Neo-proterozoic (1.0–0.541 Gyr). The Phanerozoic Eon corresponds to 1.2 GPa (>45 km) and T (>600–650 °C) with a characteristic pyropic-garnet and omphacitic-clinopyroxene-

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bearing assemblage. Different types of eclogites have been described, but Coleman’s group C (associated with glaucophane schists) and the “low-T eclogites” (Carswell 1990) could be reliable indicators of subduction process. Archean eclogites are rare and become abundant post Archean, mostly as mantle or crustal xenoliths in kimberlites and rarely in outcrop (Rollinson 1997). Study of eclogite inclusions in diamonds that originate in thick Archean SCLM suggests that the modern style subduction may not have occurred prior to ca 3.0 Gyr (Shirey and Richardson 2011). UHP (Ultra High Pressure) terrains form when continental crust initially subducted to greater than 100 km returns to the surface through subsequent tectonic processes. Typical assemblages include coesite and diamond, indicative of 700–900 °C and 3–4 GPa, or more (Liou et al. 2004). Some of the oldest known UHP terraines are in Mali, ca. 620 Myr (Jahn et al. 2001) and Kazakhisthan, 530 Myr (Maruyama and Liou 1998). Batholiths: Mere presence of magmatic “island arc” features suggests modern-style plate tectonics with deep subduction (100–250 km). In such a setting from older terrains, volcanic rocks are usually removed by erosion, leaving behind well-preserved mid-crustal batholiths (Hamilton and Myers 1967; Stern 2002; Hamilton 2011). Felsic plutonic lithologies, well represented by Tonalite-TrondhjemiteGranodiorite (TTG) suites, dominate Precambrian plutons coalesced into batholiths. Composition of igneous rocks: The relationship between tectonic setting and composition of igneous rocks has been a subject of considerable interest (see Pearce and Peate 1995; Pearce 2008). However, the diversity of sources, magmatic processes, and the problem of element mobility in secondary environments pose a serious challenge to “geochemical fingerprinting” of tectonic settings (Condie 2015). In general, the distribution of immobile trace elements and ratios (e.g., incompatible elements including Rare Earths, Y, Th, and High Field Strength Elements, such as Zr, Ta, Hf, and Nb) with similarity to modern arc magmas and the identification of certain magmatic associations like boninites and adakites in Archean granite-greenstone terranes have been used widely as evidence for operation of plate tectonics in the Archean (Foley et al. 2003; Kerrich and Polat 2006; Shirey et al. 2008). The petrogenesis and tectonic setting of the Archean sodic TTG complexes, the most voluminous lithology of the cratons, especially the early Archean (>3.2 Gyr); has been a subject of considerable discussion (Moyen and Martin 2012; Laurent et al. 2014; Hoffmann et al. 2014). Recent models based on geological and geochemical observations and thermomechanical-numerical modeling at conditions similar to Eo-Mesoarchean times invoke melting of hydrated basalt at garnet amphibolite, granulite, or eclogite facies conditions in different tectonic scenarios including subduction, as well as the range of nonplate tectonic (no subduction) scenarios

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and/or a combination of both (Rapp et al. 1991; Hill et al. 1992; Martin 1993; Abbott et al. 1994; Van Thienen et al. 2004; Smithies et al. 2005; Bédard 2006; Van Kranendonk et al. 2007; Van Kranendonk 2010; Moyen and Martin 2012; Sizova et al. 2015). Seismic reflections of deep crust: In general, deep-crustal seismic reflection studies in many cratons indicated: (1) gentle to moderately dipping crustal detachments identified based on the geometry of reflections and truncations; and (2) reflections projecting from the lower crust into the upper mantle suggesting that the crust may have been thrust into the mantle (e.g., Yilgarn Craton, Australia, Drummond et al. 2000; Goleby et al. 2004; Kaapvaal craton, South Africa, de Wit and Tinker 2004; Dharwar craton, India, Vijaya Rao et al. 2015; Mandal et al. 2018). Neoarchean (2.7 Gyr) plate tectonics was inferred in the Superior craton, Canada, by the discovery of “mantle reflections” and interpretation that these reflections reflect “subduction scars” or “fossil subduction” (Calvert et al. 1995; Calvert and Ludden 1999; van der Velden and Cook 2005). Palaeomagnetic data: The palaeomagnetic method offers a quantitative means to test relative motions between plates and the operation of Plate Tectonics during the Precambrian. In general, there is evidence in the Palaeomagnetic record that a modern-like geocentric axial dipole geomagnetic field with a weaker or comparable intensity to the present existed during the Phanerozoic and Proterozoic Eons. High quality palaeomagnetic data especially on magmatic associations like mafic dyke swarms could help in analyzing differential movements of continents at least since the Neoarchean (~3.0 Gyr) (McElhinny and Senanayake 1980; Pesonen et al. 2003; Evans and Pisarevsky 2008 and references cited therein). However, in the absence of precise geochronological data and poor constraints of field stability tests of remanence, the application of palaeomagnetic method to Archean terranes has been problematic.

Evolution of Earth’s Geodynamic Regime and the Onset of Plate Tectonics The Hadean Eon Thermal models of planetary evolution suggest three principal modes of nonradiative planetary heat transfer (Sleep 2000): (1) magma mush oceans, (2) plume and plate tectonics, and (3) stagnant-lid convection (illustrated in Fig. 1 in corresponding stages). The magma mush ocean mode is, by far, the most energetic mode and probably characterizes the early Hadean tectonic style. The stagnant-lid convection, the least energetic mode, characterizes the present Venus and Mars and was possibly present on Earth episodically. In general, several simulations of Hadean geodynamic environments suggest the difficulty of sustained plate motion and

Plate Tectonics, Precambrian Plate Tectonics, Precambrian, Fig. 1 Cartoon illustrating the evolution of global tectonic styles from the Hadean to Present involving three main stages. See text for more details. (Modified after Ernst 2007; Stern 2008 and references cited therein)

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Stage I: Hadean (4.55-4.0 Ga)

Early basaltic ‘bergs’

Convecting magma/mush ocean

Stage II: Archaean-Proterozoic (4.0-1.0 Ga) II.A. Proto-plate tectonics Plume and deep mantle-driven circulation and formation of platelet (4.4-2.7 Ga) and super cratonal stage (2.7-1.0 Ga)

II. B. Unstable Stagnant Lithospheric Lid Formation of oceanic and continental crust through mantle upwelling processes

Stage III: Modern-style plate tectonics 3.8 Gyr zircons indicate that the early enriched mafic proto-crust may have resided deep beneath the surface, much similar to the highly enriched silicate reservoir, the UrKREEP on the Moon, which eventually contributed to the generation of KREEP-rich lunar igneous rocks with ages between 4.4 and 3.9 Gyr (reviewed by Iizuka et al. 2017). Such an interpretation supports the stagnant-lid tectonics (Debaille et al. 2013), a single lid with no relative motion for the early Earth. Debaille et al. (2013) infer a dominantly stagnant-lid regime for about ~700 million years (between 4.5 and 3.8 Gyr). Thus, combined interpretation of zircon isotopic data and numerical models suggests the relevance of dominantly stagnant-lid with episodic transient subduction-like events for the Hadean crust. The Archean-Proterozoic Eons Significant cooling of the Earth’s crust (below the boiling point of water) by ~4.0 Gyr is inferred from preservation of the ~3.9 Gyr Isua volcano-sedimentary assemblages, western Greenland, which suggest the possibility that water-oceans marked the planet’s outer rind by the Late Hadean times (Fig. 1, stage II a, b). No record of the Hadean crust older than the 4.03 Gyr component of the Acasta gneiss (Slave craton, Canada) (e.g., Bowring and Williams 1999; Mojzsis et al. 2014) is preserved and Eoarchean crustal remnants are scarce. Archean cratons consist predominantly of crust generated in the Neoarchean, while Paleoarchean to Mesoarchean crust is documented from few cratons (e.g., Nutman et al. 2002; Bleeker 2003; Van Kranendonk et al. 2007; Wu et al. 2008; Zeh et al. 2011; Arndt 2013; Turner et al. 2014; Wan et al. 2015; Komiya et al. 2015; Santosh et al. 2016; Mueller and Nutman 2017; Jayananda et al. 2018; Sreenivas et al. 2019; Bhaskar Rao et al. 2020). Since the 1960s, Archean geologists recognize that the large-scale granitegreenstone structures in the Archean cratons are divisible into two broad types: the dome and keel type and the orogenic linear belt type and the geodynamic models comprise two broad scenarios: “vertical” and “horizontal” tectonic regimes (Windley 1984; de Wit and Ashwal 1995 for a review). The former has been generally ascribed to mantle upwelling

Plate Tectonics, Precambrian

(mantle plumes) and in some regions models invoking gravity driven Rayleigh-Taylor type instabilities and density inversions leading to sinking of dense volcanic-rich greenstone sequences and rising of felsic diapirs from the underlying crust producing dome and keel structures were proposed, typified by examples in Rhodesian and Pilbara cratons (Bédard 2006; Van Kranendonk et al. 2007; Wiemer et al. 2018). Horizontal tectonics in the Archean context is broadly analogous to Phanerozoic Wilson-type plate tectonics, but invokes subtle differences mainly due to higher mantle potential temperature and crustal heat flow during the Archean (Windley 1984; de Wit and Ashwal 1995; Kerrich and Polat 2006; Dewey 2007; Condie and Pease 2008; Ernst 2009). However, these “end-member” models are neither adequate nor mutually exclusive and a variety of complex mixed-mode scenarios involving, for example, multiple subduction zones and intermittent mantle plumes have been proposed (e.g., Wyman et al. 2002; Bédard 2006 and references therein). It is now widely appreciated that as much as 73% of the present volume of Earth’s continental crust may have been extracted from the mantle by the end of the Archean (Belousova et al. 2010; Dhuime et al. 2012; Cawood et al. 2013) and major peaks in the accretion of Archean juvenile crust centered around 3.0, 2.7 and 1.9 Gyr (Condie et al. 2011; Voice et al. 2011). Many researchers favor a view that the late Paleoarchean/early Mesoarchean (ca. 3.2–3.0 Gyr) marked a transition in the planet’s geodynamic regime from a dominantly vertical (driven by mantle upwelling) towards dominantly horizontal (some form of plate tectonics) geodynamic regime (summarized by Hawkesworth et al. 2017; Condie 2018). This view is increasingly in favor in the light of the convergence of many different datasets including field geological observations, paleomagnetic, geochemical, ore deposit studies, and global-scale zircon U-Pb age-HfO datasets (Dhuime et al. 2012, 2018; Cawood et al. 2013; Tang et al. 2016; Hawkesworth et al. 2017 and references therein). The numerous geological and lithotectonic changes documented during this interval are linked directly or indirectly to the cooling of Earth’s mantle and corresponding changes in convective style and the strength of the lithosphere. These data suggest a distinct change in the character, composition, and thickness of the continental lithosphere around ~3.2–3.0 Gyr. Some of the important changes include: (1) increasing proportion of basalts with “arc-like” mantle sources and the preponderance of K-rich granites; (2) the development of a stable continental lithosphere (Carlson et al. 2005; Condie and Kroner 2008; Moyen and Martin 2012); (3) the onset of a Great Thermal Divergence in the mantle (Condie et al. 2016); (4) appearance of eclogite inclusions in diamonds (Shirey and Richardson 2011); (5) an increase in the volume of crust generation marked by a change in the slope of the continental growth curve at ~3.0 Gyr estimated from the zircon age-Hf data (Belousova et al.

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Plate Tectonics, Precambrian, Table 1 Earth’s geodynamic transition towards plate tectonics. (Summarized from Hawkesworth et al. 2017) Stage-1 (the early ~15 Myr) Stage-2 (4.5–3.0 Gyr)

Stage-3 (3.0–1.7 Gyr) Stage-4 (1.7–0.75 Gyr)

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Plate-Driving Forces Alessandro M. Forte Department of Geological Sciences, University of Florida, Gainesville, FL, USA

Definition Plate-driving forces. The forces exerted on Earth’s tectonic plates that produce their observable horizontal motions.

Introduction Since the advent of the theory of plate tectonics in the 1960s, it has been postulated that the forces that drive the observed horizontal motions of Earth’s tectonic plates are produced by the process of thermal convection in the mantle. The proposition that such motions could be produced and sustained over very long geological time scales in a rocky shell, comprising the lithosphere and underlying deep mantle, that behaves as an imperfect elastic solid through which seismic waves travel, was initially viewed as a serious obstacle to the acceptance of the mantle convection hypothesis of plate motions (e.g., Jeffreys 1972). This obstacle was eliminated with the recognition, amply confirmed by high-temperature deformation experiments on minerals (e.g., Evans and Kohlstedt 1995), that the rheology of the mantle allows steady-state creeping flow when subjected to long-term shearing stresses, provided the ambient temperature is sufficiently high (Poirier 1985). The relationship between the mantle shear stress and deformation (measured in terms of strain rate) is characterized by an effective viscosity. Experimental and theoretical studies of the rheology of mantle minerals indicate that their effective viscosity is strongly dependent on pressure, temperature, and chemical composition (e.g., hydration). For sufficiently low ambient stress, mantle viscosity also depends on grain size, yielding linear (Newtonian) diffusion creep, whereas for high ambient stress, the viscosity depends on stress magnitude and is independent of grain size, yielding a nonlinear (power-law) dislocation creep (Karato and Wu 1993). The temperature dependence of mantle viscosity gives rise to a lithospheric layer that is stiffer and less deformable than the much hotter, less viscous upper mantle. Indeed, the very existence of tectonic plates, behaving as effectively rigid bodies, with little surface deformation is generally explained in terms of the strong temperature dependence of mantle viscosity. Nevertheless, plate-tectonic motions with observed speeds of several cm/yr could not exist unless localized rheological weakening mechanisms (e.g., plasticity, grain-size reduction, partial melting) also operate, yielding interconnected linear

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zones of weakness (i.e., plate boundaries) that continuously evolve in response to changing flow and stress distributions in the lithosphere and underlying mantle (e.g., Moresi and Solomatov 1998; Tackley 2000; Ghosh and Holt 2012; Rolf et al. 2018; Coltice et al. 2019). A rheological weakening mechanism that has been extensively employed to simulate plate boundary creation and evolution is viscoplasticity (e.g., Moresi and Solomatov 1998; Karato and Barbot 2018). The effective viscosity, eff, representing the combined operation of diffusion/dislocation creep and viscoplastic weakening may be mathematically expressed as follows:  1 ty 1 eff ¼ 1 , where pl ¼ cr þ  pl 2e_II

ð1Þ

in which cr is the Arrhenius temperature-dependent creep viscosity (e.g., Karato and Wu 1993) and pl is the viscoplastic viscosity that is defined in terms of a yield stress, ty, and the square root of the second invariant of the strain-rate tensor, e_II (e.g., Moresi and Solomatov 1998). Expression (1) implies that when pι is much smaller than cr, then the effective viscosity of the lithosphere will be locally dominated by viscoplastic weakening: eff ≈ pl (e.g., Spiegelman et al. 2016). The strong rheological contrast between the lithosphere and the underlying mantle is often used as the basis for models of plate-driving forces that treat the lithospheric plates and their descending limbs (subducted slabs) under the oceanic trenches (Fig. 1a) separately from the mantle (Elsasser 1969; McKenzie 1969; Solomon and Sleep 1974; Harper 1975; Forsyth and Uyeda 1975; Richardson et al. 1976; Chapple and Tullis 1977; Richter 1977; Davies 1978; Bird et al. 2008). Alternatively, the oceanic lithosphere can be regarded as an integral part of the convecting mantle. In the latter case, the lithosphere is modelled as the upper thermal boundary layer of mantle convection, such that the heat transported across the deep mantle by rising and sinking convective flow is finally transmitted to the Earth’s surface by vertical conduction in balance with horizontal advection of heat in the lithosphere (Turcotte and Oxburgh 1967; Jarvis and Peltier 1982). The vertical viscosity gradient across the lithospheric plates, from an effectively undeformable (highviscosity) surface to a more deformable (low-viscosity) base, is therefore a consequence of the thermal boundary-layer structure of the lithosphere. This boundary-layer model provides an alternative context for interpreting plate-driving forces, in which the horizontal motions of Earth’s oceanic lithospheric plates (Fig. 1a) are directly coupled to, and driven by, the mantle convective circulation (Turcotte and Oxburgh 1967; Richter 1973; Jarvis and Peltier 1982; Davies 1988). The horizontal plate motions are thus understood to be coupled to the underlying mantle flow generated by descending and ascending thermal plumes

(Hager and O’Connell 1981; Forte and Peltier 1987, 1994; Ricard and Vigny 1989; Gable et al. 1991; Vigny et al. 1991; Lithgow-Bertelloni and Richards 1998; Becker and O’Connell 2001; Lowman et al. 2008; Forte et al. 2009).

Balance Between Buoyancy Forces and Viscous Dissipation Two distinct approaches have therefore been adopted to develop models of plate-driving forces where, on the one hand, the plate-driving forces are analyzed separately from the dynamics of the underlying mantle and, on the other, the plate motions are modelled as an integral part of the convecting mantle. Both approaches must satisfy a fundamental constraint that expresses how work done by the body forces throughout the mantle is dissipated by the viscous deformation associated with the tectonic plate motions and flow in the underlying mantle. A mathematical expression of the balance between buoyancy and dissipation (e.g., Malvern 1969) over the entire volume, Vm, of the mantle (including the lithosphere) is: ð ð ð  ur dr go dV ¼ tij ϵij dV  ti Dui dS Vm

Vm

ð2Þ

Sd

in which tij and ϵij are the deviatoric stress and strain-rate tensors, respectively, ti is a traction vector acting on surfaces Sd across which there is a discontinuity in motion Δui, ur is the vertical (radial) component of mantle flow, δr are lateral perturbations in density, and go is the radial gravitational field. All indices i and j represent components in each of the three Cartesian coordinate directions, and repeated indices (in the terms on the right-hand side of Eq. 2) denote summation over all values (1–3) of the indices. The traction vector ti acting on the discontinuity surface Sd depends on the deviatoric stress as follows: ti ¼ tijnj, in which the nj are the Cartesian components of the local normal vector on Sd. Although the motion along Sd is discontinuous, the traction ti must be continuous by Newton’s third law (equal and opposite actions and reactions). The left-hand side of Eq. (2) is the (positive) rate of work done by the buoyancy forces δrgo where vertical flow occurs, as in hot plumes where δr < 0 & ur > 0, and in cold descending slabs where δr > 0 & ur < 0 (Fig. 1a). The first term on the right-hand side of Eq. (2) is the rate at which the energy released by the buoyancy forces is dissipated by the internal stresses as they continuously deform the mantle. Since the mantle rheology on plate-tectonic time scales is viscous, this energy dissipation is simply the viscous friction that converts mechanical work into heat. The second term on the right-hand side represents the rate of frictional dissipation of energy on internal discontinuity surfaces or faults, and

Plate-Driving Forces

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Continental Plate

Oceanic Plates

vp

Fcr

Transform

Ftr

vm

Slab

Fsp Fsr

Frp

60˚

90˚

vm

Fp

90˚

120˚ 150˚ 180˚ −150˚ −120˚ −90˚ −60˚ −30˚

(b) 0˚

EUR PHI

ARB

AFR 0˚

Plume

NAM

PAC

IND

CAR

COC

SAM

−30˚

NAZ

AUS

SCO

−60˚

Mantle

2nd Invariant of Strain Rate 30˚

60˚

JDF

Fsn

Fs

60˚ 30˚

Fbs

vs

vp

Oceanic Lithosphere

Fbs

30˚

90˚

vt

Continental Lithosphere

Observed & Predicted Plate Motions in NNR Reference Frame 0˚

Mid−Ocean Ridge

Trench

vp

(a)

ANT

−90˚ 8 cm/yr

(c)

Mentle Flow in the Asthenoshpere

120˚ 150˚ 180˚ −150˚ −120˚ −90˚ −60˚ −30˚

90˚ 60˚

60˚

30˚

30˚

0˚

0˚

−30˚

−30˚

−60˚

−60˚

0˚

30˚

60˚

90˚

120˚ 150˚ 180˚ −150˚ −120˚ −90˚ −60˚ −30˚

(d) 0˚

−90˚ −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

log10(e_II x 10^(16) s) Mantle Driven Basal Shear Tractions 90˚

0˚

30˚

60˚

90˚

−4

−3

−2

−1

0

1

2

3

4

Vertical Flow (cm/yr)

(e)

120˚ 150˚ 180˚ −150˚ −120˚ −90˚ −60˚ −30˚

Mantle drag of lithospheric movements 0˚

30˚

60˚

60˚

30˚

30˚

0˚

0˚

−30˚

−30˚

−60˚

−60˚

60˚

90˚

(f)

120˚ 150˚ 180˚ −150˚ −120˚ −90˚ −60˚ −30˚

P

−90˚ 8 cm/yr

5 MPa

−1.0

−0.5

0.0

0.5

1.0

Drag Coefficient

Plate-Driving Forces, Fig. 1 (a) Schematic illustration of forces acting on lithospheric plates. All plate-driving forces ultimately arise from the buoyancy forces in the convecting mantle, namely, the force Fp due to hotter and less dense upwelling plumes and the force Fs due to the colder and denser sinking slabs. These sublithospheric buoyancy forces generate the plate-driving forces Fsp (slab pull) and Frp (ridge push). Whether the force Fbs (basal shear) is a plate-driving or plate-resisting force depends on the relationship of the surface plate velocity vp to the underlying mantle flow velocity vm (see main text for discussion). The purely resistive plate forces are Fsr (slab resistance), Fcr (collision resistance), and Ftr (transform resistance). The balance between the slab-normal force Fsn and the opposing normal force generated by mantle flow will control the trench rollback or retreat velocity vt [Figure adapted and modified from Forsyth and Uyeda (1975)]. (b) The observed (green arrows) and predicted (blue arrows) present-day plate velocities, where the former are from the GEODVEL model (Argus et al. 2010) and the latter are obtained from a tomographybased model of mantle convection (Forte et al. 2015). The plate motions are with respect to a global No-Net-Rotation (NNR) frame of reference. The plate names are shown by the abbreviations in red font: AFR Africa, ANT

Antarctica, ARB Arabia, AUS Australia, CAR Caribbean, COC Cocos, EUR Eurasia, IND India, JDF Juan de Fuca, NAM North America, NAZ Nazca, PAC Pacific, PHI Philippine, SAM South America, SCO Scotia. (c) qffiffiffiffiffiffiffiffiffiffiffiffiffi A map of the second invariant of the strain-rate tensor ( ϵij ϵij =2 ) at the surface of the lithosphere predicted by the tomography-based mantle flow model employed in (b). (d) The mantle flow predicted at 250 km depth using the tomography-based convection model (Forte et al. 2015). The magenta arrows show the horizontal component of the predicted flow vector, and the color contours show the vertical component (scale at bottom). The blue arrows show the predicted surface plate motions from (b). (e) The horizontal traction vectors (magenta arrows) at 90 km depth (just below the mean depth of the lithosphere) exerted by the mantle flow predicted by the tomography-based convection model (Forte et al. 2015). The blue arrows show the predicted horizontal flow velocities at 90 km depth. Scale bars for vectors are at bottom. (f) A map of the drag coefficient Db determined using the basal traction Fbs at 90 km depth, shown in (e), and the predicted surface plate motion vp, shown in (b)

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Plate-Driving Forces

hence this term is only relevant in the upper mantle and especially the crust, where significant earthquake faulting occurs. Since the frictional tractions ti will always oppose the discontinuous fault motion Δui, the integral over the fault surfaces Sd must be negative (e.g., Malvern 1969). As noted above, a large number of models have evaluated the plate-driving forces by specifically focusing on the lithosphere itself, treated as distinct from the mantle. In this context, a buoyancy-dissipation balance analogous to Eq. (2) can be developed: ð ð  ur dr go dV  ½dP  ro df ui ni dS Vl

Sl

ð

þ

u j tji ni dS Sl

ð ¼

ð tij ϵij dV 

Vl

ti Dui dS

ð3Þ

plate velocities as a boundary condition, thereby implying corresponding externally applied tangential stresses on the lithosphere. This external forcing is sometimes figuratively referred to as the “hand of God” (e.g., Karpychev and Fleitout 1996). To avoid the nonphysical possibility that such external stresses do work on the lithosphere and underlying mantle, the models should be formulated so that compensating surface stresses, generated by the flowing mantle, are able to precisely cancel the work of externally applied stresses (e.g., Hager and O’Connell 1981). The principal idea that emerges from these general considerations of the plate-driving forces is that plate motions are produced by the buoyancy forces δrgo in the mantle (lefthand side of Eq. 2). Even when the focus is placed on the lithosphere itself, the contribution of the driving forces in the mantle is still present but “hidden” in the basal stresses that are exerted on the lithospheric plate (left-hand side of Eq. 3).

Sd

in which Vl represents the volume occupied by the lithospheric layer, Sl defines the lower surface of the lithosphere (where the local normal vector ni is approximately pointing in the negative radial direction br ), δΡ and roδf are, respectively, the dynamic pressure field and self-gravitation force acting on the base of the lithosphere, and tjini ¼ tj are the deviatoric shear tractions exerted on the base of the lithosphere by the underlying mantle flow. If the lithosphere has a sufficiently high viscosity, owing to its relatively colder temperatures compared to the mantle, the deformation rate ϵij in plate interiors may be assumed to be negligible. If this approximation is accepted, the first term on the right-hand side of Eq. (3) may be ignored, and the primary frictional dissipation of energy occurs on the fault surfaces Sd that define the boundaries of the plates. The vertical flow ur in the lithospheric layer is weak, owing to the proximity of the upper bounding surface, and thus the work done by the tangential shear stresses exerted by mantle flow (third term on the left-hand side of Eq. 3) is approximately balanced by the frictional dissipation of energy on plate boundary weak zones (e.g., faults). Since the dissipation on the right-hand side of Eq. (3) is always positive, this implies the tangential stresses on the base of the lithosphere are, on average, acting in the direction of the flow and hence do positive work on lithosphere. These approximations illustrate how plate motions may be governed by a balance between basal shear stresses, acting as plate-driving forces, and resistance to motion along plate boundaries (e.g., Hanks 1977; Davies 1978). Expressions (2) and (3) are obtained on the assumption that no external tangential stresses are applied to the upper and lower bounding surfaces of the mantle and that the local vertical flow vanishes on these bounding surfaces. Some models of plate-driving forces have employed prescribed

Plate-Driving Forces: Lithospheric Models Formulating the detailed physical mechanisms by which the vertically directed buoyancy forces in the mantle (Fs and Fp in Fig. 1a) are ultimately expressed as horizontal forces acting on and driving the plates is the central challenge in developing models that predict realistic plate motions. The earliest models explored the dynamics of the lithosphere and its descent into the mantle under deep ocean trenches (e.g., Elsasser 1969, 1971; McKenzie 1969; Richter 1973), and they suggested that the negative buoyancy forces due to the cold lithospheric slabs entering the mantle (Fs in Fig. 1a) are the most efficient drivers of the observed tectonic plate motions. These initial efforts set the stage for the subsequent development of parametrized plate-force models in which the forces driving the plate motions are represented by a limited number of discrete force vectors acting on the lithosphere (Fig. 1a). One of the most influential studies of the parametrized plate-force models was carried out by Forsyth and Uyeda (1975). The forces they considered are identified in Fig. 1a, and they include (1) the slab-pull force Fsp ¼ Fs sin f acting in the down-dip direction where f is the dip angle of the slab; (2) the slab-resistance force Fsr representing the opposing viscous shear stress acting on the sides of the descending slab; (3) the collision-resistance force Fcr representing the frictional stress acting on the upper surface of the slab as it slides past the fault surface of the shallow Benioff zone; (4) the basal shear force Fbs representing the tangential shear stress generated by the flowing mantle acting on the base of the lithospheric plates; (5) the transformresistance force Ftr representing the frictional stresses acting on segments of the plate boundaries where there is a component of relative motion that is tangential to the boundary, such

Plate-Driving Forces

as transform faults; and (6) the ridge-push force Frp corresponding to the lateral force exerted on the divergent plate boundaries due to the dynamic pressure field that produces the topographic elevation of the mid-ocean ridges. The orthogonal complement to the slab-pull force is the slab-normal force Fsn ¼ Fs cos f (Fig. 1a). This force component was not considered in the evaluation of the platedriving forces by Forsyth and Uyeda (1975), but the balance between this force and the opposing normal stress in the mantle will contribute to the retrograde motion (or “rollback”) of the trench (e.g., Elsasser 1971; Schellart 2008; Funiciello et al. 2008) with the relative velocity vt illustrated in Fig. 1a. The results obtained by Forsyth and Uyeda (1975), and earlier by Richter (1973), suggested that the balance of forces on subducting slabs is the primary control on plate motions, and this understanding has been subsequently reiterated through a long suite of published models (e.g., LithgowBertelloni and Richards 1998; Conrad and LithgowBertelloni 2004; Billen 2008; Funiciello et al. 2008; Schellart 2008; Stadler et al. 2010) up to the present day (e.g., Rolf et al. 2018; Coltice et al. 2019). A major source of uncertainty in the classic force-balance models of plate motions is the unknown relationship between the motion of the lithospheric plates vp and the underlying mantle flow vm (Fig. 1a) and hence the unknown local magnitude and direction of the basal shear force Fbs. The majority of models assumed the mantle is passive and that the flow below the lithosphere is entirely generated by the subducting slabs. This is equivalent to assuming that the ambient mantle temperature outside the sinking lithosphere is isothermal or adiabatic (e.g., McKenzie 1969; Hager and O’Connell 1981). For these passive-mantle models, the basal shear force acts in the opposite direction to the plate velocity and therefore opposes the plate-driving force provided by subduction. Recent global convection modelling of plate-like surface motions driven by subducting slabs (Coltice et al. 2019) indeed shows that most (in excess of e 2=3 of the global surface area) of the sublithospheric mantle is passive and driven by the overlying plate motions. The conclusions obtained from traditional force-balance modelling are nonunique and depend on which forces are assumed to be known. For example, whether basal shear forces contribute to driving or resisting the plate motions will depend on what forces (in Fig. 1a) are prescribed as known at the outset. In the study by Forsyth and Uyeda (1975), it was assumed that basal shear forces oppose the plate motions and have a direction that is antiparallel to the plate velocity. In contrast, Davies (1978) showed that if the frictional force Ftr acting on transform boundaries is assumed to be known in advance, on the basis of earthquake stressdrop data (Hanks 1977), then the basal shear force may act as a plate-driving force. The force balance considered by Davies

1271

(1978) also showed that the collision-resistance force Fcr may strongly oppose the slab-pull force Fsp, greatly reducing its contribution as a plate-driving force. The significance of basal shear stresses as a plate-driving force has also been highlighted by Bird et al. (2008) who employed a finiteelement model of the lithosphere that incorporates stressand temperature-dependent viscosity as well as an explicit treatment of plate boundary faults. Ghosh and Holt (2012) similarly found that basal shear tractions generated by buoyancy-driven mantle flow are the primary drivers of surface lithospheric velocities and yield close matches to present-day plate motions. The latter model results (see also Rowley et al. 2016) showing the importance of sublithospheric basal tractions as active drivers of plate motions differ from models that show a mainly passive mantle entrained by overlying plates driven by subducting slabs (e.g., Coltice et al. 2019).

Plate-Driving Forces: Whole-Mantle Models The popularity of parametrized plate-force models that focused on the lithosphere was based on the use of readily observed input variables derived from surface tectonics and observed plate velocities. Since the observed plate velocities were employed as an input, these models could not explain how the plate motions were generated by the internal dynamics of the mantle nor how they would evolve over time. Hager and O’Connell (1981) introduced 3-D spherical, whole-mantle flow models that explicitly included the interaction of the moving lithospheric plates with the underlying mantle. Although these models employed the present-day plate velocities as input, they predicted the 3-D flow throughout the mantle driven by a global distribution of subducted slabs derived from seismic catalogues of earthquake hypocenters. With the advent, in the 1980s, of seismic tomographic imaging of lateral heterogeneity in the mantle, it became feasible to apply these whole-mantle flow calculations to explore the detailed connection between surface plate motions and the 3-D flow in the mantle generated by thermal convection (e.g., Forte and Peltier 1987; Ricard and Vigny 1989). Seismic tomography provides the only direct in situ constraints on the 3-D lateral structure of the convecting mantle, and it thus provides a unique basis for developing realistic models of mantle flow that are no longer limited to an understanding of Earth dynamics derived only from surface observations. The models of mantle flow by Hager and O’Connell (1981) were therefore further extended in studies demonstrating that present-day plate motions can be successfully predicted by 3-D distributions of buoyancy forces derived from seismic tomography (e.g., Vigny et al. 1991; Forte and Peltier 1994; Becker and O’Connell 2001; Ghosh and Holt 2012; Forte et al. 2015). This confirmation that plate motions can be

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Plate-Driving Forces

predicted on the basis of stresses generated in the underlying mantle by a 3-D pattern of convection is a major step toward developing a complete understanding of the dynamics and temporal evolution of the mantle and lithosphere as a single, internally coupled system (e.g., Lithgow-Bertelloni and Richards 1998; Conrad and Lithgow-Bertelloni 2004; Lowman et al. 2008; Forte et al. 2009; Rowley et al. 2016; Rolf et al. 2018). A tomography-based model of present-day mantle flow (Forte et al. 2015) illustrates the detailed relationship between the convective circulation in the mantle and surface plate velocities. The model incorporates coupling of viscous mantle flow to overlying plates whose motions are limited to rigidbody rotations. The buoyancy-driven plate velocities predicted by this model (Fig. 1b) match the observations well (explaining 97% of data variance). The predicted lithospheric strain rates (Fig. 1c) are characterized by values ranging from less than 3  1016 s–1 in plate interiors to two orders of magnitude greater along plate boundaries. According to Eq. (1), plastic weakening would then predict viscosity reductions in plate boundary zones between one and two orders of magnitude. Below the lithosphere, in the underlying low-viscosity portion of the upper mantle referred to as the “asthenosphere,” the predicted flow velocities show significant deviations from the overlying plate motions, for example, under the western half of the North American plate and under the African plate (Fig. 1d). The lack of parallelism between the surface plate velocities and the upper-mantle flow circulation in the asthenosphere is a consequence of the complex 3-D distribution of buoyancy forces in the deep mantle that are not correlated with the surface plate geometry (e.g., subduction zones and mid-ocean ridges) in any straightforward manner. An important exception is the prediction of a mantle-wide upwelling centered directly below the East Pacific Rise (Rowley et al. 2016). The lack of parallelism between surface and sublithospheric motions is also a consequence of the strong viscosity reduction in the asthenosphere, which allows a partial decoupling between these two layers. The complex local variations in basal shear force relative to the surface velocities show that a simple correlation between these two, as assumed in the parametrized plateforce models (e.g., Forsyth and Uyeda 1975), is not appropriate. The local correlation between basal stresses and lithospheric motions may be quantified in terms of a drag coefficient, which provides a measure of the extent to which sublithospheric mantle flow actively “drags” the overlying lithosphere or vice versa. Two possible representations for the local mantle drag coefficient are:   v Db ¼ Fbs  vp = jFbs j vp jj or Dv ¼ m cos y  1 vp

ð4Þ

in which θ is the angular deviation between the sublithospheric velocity vm and the overlying plate velocity vp

(Fig. 1a). The drag coefficients Db and Dv are correlated, because the orientation and magnitude of the horizontal motions in the plates and sublithospheric mantle will determine the orientation and magnitude of the basal shear force Fbs (Fig. 1a). A simple 1-D analysis of purely horizontal flow shows that when the flow velocity increases with depth (i.e., the mantle “drags” the overlying plate), then both Db and Dv will be positive. The coefficient Dv is closely related, but opposite in sign, to the drag coefficient defined in Coltice et al. (2019). The tomography-based flow prediction of plate motions (Fig. 1d, e) yields the local drag coefficient Db shown in Fig. 1f. Positive mantle drag of the overlying plate motions occurs in 63% of the total surface area of the lithosphere. A similar calculation of Dv (not shown here) yields a positive mantle drag in 65% of the surface area. The theoretical convection-driven plate motions simulated in Coltice et al. (2019) yield positive mantle drag in 20 to 40% of the surface. The tomography-based whole-mantle flow models thus suggest that basal shear forces generated by a fully 3-D pattern of mantle convection drive the motions of the overlying plates (e.g., Vigny et al. 1991; Forte and Peltier 1994; Becker and O’Connell 2001; Ghosh and Holt 2012; Forte et al. 2015). The role of the plates is not entirely passive, however, because their intrinsic rigidity allows them to partially resist (and hence mask) a significant fraction of the deeper mantle flow patterns (e.g., Ricard and Vigny 1989; Forte et al. 2015). Furthermore, the buoyancy forces associated with subducted lithosphere do contribute to the basal shear forces that drive the plates (e.g., Coltice et al. 2019). Indeed, the continuous descent of lithosphere into the mantle ultimately generates a global distribution of subducted slabs that provide the negative buoyancy force Fs (Fig. 1a).

Summary Much progress has been achieved in our understanding of how mantle convection drives the surface plate motions since the earliest parametric force models were published over four decades ago. In particular, the increase and availability of high-performance computational resources over the past decade have permitted the development of global mantle convection simulations that incorporate increasingly Earthlike characteristics, including the strong lateral variations of viscosity needed to simulate plate-tectonic movements (e.g., Ghosh and Holt 2012; Rolf et al. 2018; Coltice et al. 2019). Furthermore, major improvements in seismic tomographic inversions for 3-D mantle structure have been incorporated into global models of mantle flow that yield very good matches to a wide suite of convection-related surface data, including the present-day plate motions (e.g., Forte et al. 2015). The interpretation of the seismic tomography models is however imperfect because individual published tomography

Plate-Driving Forces

models are derived from damped, smoothed inversions and they are nonunique. Moreover, the independent mineral physical modelling required to convert the seismic anomalies in the tomography models into corresponding density perturbations adds further uncertainties that may impact the reliability of the mantle flow predictions (e.g., Rowley et al. 2016). These outstanding challenges have led to the emergence of two contrasting views on how thermal convection in the mantle drives the tectonic plate motions. The long-held, predominant view is that the buoyancy forces due to subducted slabs are the primary drivers of surface plate motions (e.g., Richter 1973; Conrad and Lithgow-Bertelloni 2004). According to this view, the thermal evolution of the mantle is mainly controlled by cooling from above (i.e., subducting lithosphere), which occurs when the mantle heating is mainly internal, with little heat flux across the core-mantle boundary. In this case, the buoyancy flux associated with subducted slabs strongly dominates that of hot upwellings, and the slabs therefore determine the dynamics of surface lithospheric movements (e.g., Coltice et al. 2019). The second view that originated with the earliest mantle convection models (e.g., Turcotte and Oxburgh 1967) and some subsequent models (e.g., Jarvis and Peltier 1982; Forte et al. 2015; Rowley et al. 2016) is that a combination of active hot upwellings (“plumes”) and subducted slabs is required to explain the surface lithospheric motions. For such upwellings to be dynamically significant, there must be a large heat flux across the core-mantle boundary (CMB) and a welldeveloped lower thermal boundary layer that can sustain the activity of these hot plumes (e.g., Glišović et al. 2012). Timedependent convection modelling that integrates a mantle buoyancy distribution derived from global seismic tomography yields a constellation of strongly developed and longlived hot upwellings that remain geographically stable in time and are sustained by a large CMB heat flux ranging between 13 and 20 Terawatts (Glišović and Forte 2015). These tomography-based convection models are characterized by a present-day buoyancy flux from hot upwellings (i.e., plumes) and cold downwellings (i.e., slabs) that are of equal magnitude, such that the hot upwellings provide a substantial contribution to plate-driving forces (e.g., Rowley et al. 2016). The issue of heat flux across the CMB thus occupies a central role in determining the dynamical contribution of mantlewide hot upwellings on plate-driving forces. Independent constraints on this heat flux from other disciplines, such a mineral physics (e.g., Pozzo et al. 2012; Ohta et al. 2016), are important in addressing this debate.

Cross-References ▶ Continental Drift ▶ Earth’s Structure, Upper Mantle ▶ Energy Budget of the Earth

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▶ Lithosphere, Mechanical Properties ▶ Lithosphere, Oceanic ▶ Lithosphere, Oceanic: Thermal Structure ▶ Mantle Convection ▶ Mantle Plumes ▶ Mantle Viscosity ▶ Seismic Structure at Mid-Ocean Ridges ▶ Seismology, Global Earthquake Model ▶ Subduction Zones

Bibliography Argus DF, Gordon RG, Heflin MB, Ma C, Eanes RJ, Willis P, Peltier WR, Owen SE (2010) The angular velocities of the plates and the velocity of Earth’s centre from space geodesy. Geophys J Int 180(3):913–960 Becker TW, O’Connell RJ (2001) Predicting plate velocities with mantle circulation models. Geochem Geophys Geosyst 2. https://doi.org/10. 1029/2001GC000171 Billen MI (2008) Modeling the dynamics of subducting slabs. Annu Rev Earth Planet Sci 36:325–356 Bird P, Liu Z, Rucker WK (2008) Stresses that drive the plates from below: definitions, computational path, model optimization, and error analysis. J Geophys Res 113:B11406. https://doi.org/10.1029/ 2007JB005460 Chapple WM, Tullis TE (1977) Evaluation of the forces that drive the plates. J Geophys Res 82:1967–1984 Coltice N, Husson L, Faccenna C, Arnould M (2019) What drives tectonic plates? Sci Adv 5(10):eaax4295 Conrad CP, Lithgow-Bertelloni C (2004) The temporal evolution of plate driving forces: importance of “slab suction” versus “slab pull” during the Cenozoic. J Geophys Res 109:B10407. https://doi.org/10.1029/ 2004JB002991 Davies GF (1978) The roles of boundary friction, basal shear stress and deep mantle convection in plate tectonics. Geophys Res Lett 5:161–164 Davies GF (1988) Role of the lithosphere in mantle convection. J Geophys Res 93(10):451–10,466 Elsasser WM (1969) Convection and stress propagation in the upper mantle. In: Runcorn SK (ed) The applications of modern physics to the earth and planetary interiors. Interscience, New York, pp 223–246 Elsasser WM (1971) Sea-floor spreading as thermal convection. J Geophys Res 76:1101–1112 Evans B, Kohlstedt DL (1995) Rheology of rocks, in rock physics and phase relations: a handbook of physical constants, AGU Reference Shelf 3. American Geophysical Union, Washington, DC, pp 148–165 Forsyth D, Uyeda S (1975) On the relative importance of the driving forces of plate motion. Geophys J R Astron Soc 43:163–200 Forte AM, Peltier WR (1987) Plate tectonics and aspherical earth structure: the importance of poloidal–toroidal coupling. J Geophys Res 92:3645–3679 Forte AM, Peltier WR (1994) The kinematics and dynamics of poloidal-toroidal coupling in mantle flow: the importance of surface plates and lateral viscosity variations. Adv Geophys 36:1–119 Forte AM, Moucha R, Rowley DB, Quéré S, Mitrovica JX, Simmons NA, Grand SP (2009) Recent tectonic plate decelerations driven by mantle convection. Geophys Res Lett 36:L23301. https://doi.org/10. 1029/2009GL040224 Forte AM, Simmons NA, Grand SP (2015) In: Romanowicz B, Dziewonski AM (eds) Constraints on seismic models from other

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1274 disciplines – implications for the global mantle convective flow, in volume 1 of treatise of geophysics, 2nd edn. Elsevier (Amsterdam), pp 853–907 Funiciello F, Faccenna C, Heuret A, Lallemand S, Di Giuseppe E, Becker TW (2008) Trench migration, net rotation and slab-mantle coupling. Earth Planet Sci Lett 271:233–240 Gable CW, O’Connell RJ, Travis BJ (1991) Convection in three dimensions with surface plates. J Geophys Res 96:8391–8405 Ghosh A, Holt WE (2012) Plate motions and stresses from global dynamic models. Science 335(6070):838–843 Glišović P, Forte AM (2015) Importance of initial buoyancy field on evolution of mantle thermal structure: implications of surface boundary conditions. Geosci Front 6(1):3–22 Glišović P, Forte AM, Moucha R (2012) Time-dependent convection models of mantle thermal structure constrained by seismic tomography and geodynamics: implications for mantle plume dynamics and CMB heat flux. Geophys J Int 190(2):785–815 Hager BH, O’Connell RJ (1981) A simple global model of plate dynamics and mantle convection. J Geophys Res 86:4843–4867 Hanks TC (1977) Earthquake stress drops, ambient tectonic stresses and stresses that drive plate motions. Pure Appl Geophys 115: 441–458 Harper JR (1975) On the driving forces of plate tectonics. Geophys J R Astron Soc 40:465–474 Jarvis GT, Peltier WR (1982) Mantle convection as a boundary layer phenomenon. Geophys J R Astron Soc 68:385–424 Jeffreys H (1972) Creep in the earth and planets. Tectonophysics 13(1–4):569–581 Karato SI, Barbot S (2018) Dynamics of fault motion and the origin of contrasting tectonic style between earth and Venus. Sci Rep 8(1):11884 Karato S–i, Wu P (1993) Rheology of the upper mantle: a synthesis. Science 260:771–778 Karpychev M, Fleitout L (1996) Simple considerations on forces driving plate motion and on the plate-tectonic contribution to the longwavelength geoid. J Geophys Res 127:268–282 Lithgow-Bertelloni C, Richards MA (1998) The dynamics of Cenozoic and Mesozoic plate motions. Rev Geophys 36:27–78 Lowman JP, Gait AD, Gable CW, Kukreja H (2008) Plumes anchored by a high viscosity lower mantle in a 3D mantle convection model featuring dynamically evolving plates. Geophys Res Lett 35: L19309. https://doi.org/10.1029/2008GL035342 Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Cliffs McKenzie DP (1969) Speculations on the consequences and causes of plate motions. Geophys J R Astron Soc 18:1–32 Moresi L, Solomatov V (1998) Mantle convection with a brittle lithosphere: thoughts on the global tectonic styles of the earth and Venus. Geophys J Int 133(3):669–682 Ohta K, Kuwayama Y, Hirose K, Shimizu K, Ohishi Y (2016) Experimental determination of the electrical resistivity of iron at Earth’s core conditions. Nature 534(7605):95 Poirier J-P (1985) Creep of crystals. Cambridge University Press, Cambridge Pozzo M, Davies C, Gubbins D, Alfe D (2012) Thermal and electrical conductivity of iron at Earth’s core conditions. Nature 485(7398):355–358 Ricard Y, Vigny C (1989) Mantle dynamics with induced plate tectonics. J Geophys Res 94:17543–17559 Richardson RM, Solomon SC, Sleep NH (1976) Intraplate stress as an indicator of plate tectonic driving forces. J Geophys Res 81:1847–1856 Richter F (1973) Dynamical models for sea floor spreading. Rev Geophys Space Phys 11:223–287 Richter F (1977) On the driving mechanism of plate tectonics. Tectonophysics 38:61–88

Plates and Paleoreconstructions Rolf T, Capitanio FA, Tackley PJ (2018) Constraints on mantle viscosity structure from continental drift histories in spherical mantle convection models. Tectonophysics 746:339–351 Rowley DB, Forte AM, Rowan CJ, Glišović P, Moucha R, Grand SP, Simmons NA (2016) Kinematics and dynamics of the East Pacific rise linked to a stable, deep-mantle upwelling. Sci Adv 2(12): e1601107 Schellart WP (2008) Kinematics and flow patterns in deep mantle and upper mantle subduction models: influence of the mantle depth and slab to mantle viscosity ratio. Geochem Geophys Geosyst 9:Q03014. https://doi.org/10.1029/2007GC001656 Solomon SC, Sleep NH (1974) Some simple physical models for absolute plate motions. J Geophys Res 79:2557–2567 Spiegelman M, May DA, Wilson CR (2016) On the solvability of incompressible stokes with viscoplastic rheologies in geodynamics. Geochem Geophys Geosyst 17:2213–2238 Stadler G, Gurnis M, Burstedde C, Wilcox LC, Alisic L, Ghattas O (2010) The dynamics of plate tectonics and mantle flow: from local to global scales. Science 329(5995):1033–1038 Tackley PJ (2000) Mantle convection and plate tectonics: toward an integrated physical and chemical theory. Science 288:2002–2007 Turcotte DL, Oxburgh ER (1967) Finite amplitude convective cells and continental drift. J Fluid Mech 28:29–42 Vigny C, Ricard Y, Froidevaux C (1991) The driving mechanism of plate tectonics. Tectonophysics 187:345–360

Plates and Paleoreconstructions Alan G. Smith (Deceased)

Definition The lithosphere is the outer rigid part of the Earth, forming a shell whose thickness may range up to about 200 km. A tectonic plate is a part of the Earth’s lithosphere that is bounded by active plate margins. A present-day plate margin is a seismically active zone that cuts the lithosphere. Conservative plate boundaries are plate boundaries that separate two plates that are sliding past one another along a transform fault (which must cut the lithosphere). Divergent (extensional) plate boundaries mark where the two plates are separating from one another, generally forming passive (or Atlantic) continental margins (though extension can also occur behind island arcs). Convergent plate boundaries are located between a converging oceanic and continental plate forming an active (Pacific) continental margin, or between a converging oceanic plate and an island arc. A global paleoreconstruction shows a reassembly of the major continents (and oceans) relative to one another at some time in the past. If the reassembly is made using ocean-floor spreading magnetic anomalies and fracture zones, and derives

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seismic zones, particularly in the oceans (Fig. 1). Elsewhere there is little or no seismicity, showing that most of the lithosphere is not undergoing differential motion and is therefore rigid. The present-day surface extent of a plate is marked by zones of active seismicity that show the plate boundaries.

the geographic poles from paleomagnetism, it is a global paleomap. If only the continents are depicted, it is a global paleocontinental reconstruction (or map). If each continent is projected separately onto a global map frame (as in preMesozoic maps) is it is a global composite reconstruction. A paleo-plate reconstruction is simply a paleoreconstruction that shows the plate boundaries of the time concerned, that is, ridges, trenches, and major transform faults.

Plate Compositions, Sizes, and Shapes Plates are made up of Lithosphere, Continental, with a thickness that probably averages about 200 km (Priestley and McKenzie 2006), and Lithosphere, Oceanic, whose thickness varies from a small value at a mid-ocean ridge, where it is created, to about 100 km for old oceanic lithosphere (Stein and Stein 1992; Crosby et al. 2006). Most of the plates in the Pacific Ocean are almost entirely, or entirely, made of oceanic lithosphere, but other plates are made up of varying

Earthquakes and Plates Earthquakes are caused by the catastrophic release of accumulated elastic strain. The strain accumulates as a result of differential movement between parts of the lithosphere. A map of earthquakes over the past 40 years or so shows that differential movement is generally confined to narrow 150°W

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Plates and Paleoreconstructions, Fig. 1 Present-day earthquakes and simplified map of the present-day plates. The Philippine plate is also known as the Philippine Sea plate. The plate boundary between Eurasia and North America in northeast Asia is poorly defined. Cylindrical equidistant world map showing land areas as light gray. Areas between 2000 m water depth and the coastline are blue (colored version) or light gray (monochrome version). Deeper oceanic areas are uncolored. Over 400,000 earthquakes are shown. In the colored map, the color code is: 0–10 km (79,478) blue; 10–35 km (168,263) light blue; 35–100 km (79,890) green; 100–200 km (47,056) yellow-green; 200–300 km (11,144) yellow; 300–400 km (3874) orange-brown; 400–500 km (4674) red; 500–600 km (8427) magenta; 600–700 km (2477) deep magenta; >700 km (4) black. The monochrome version shows earthquakes ranging from light gray (shallow) to black (deepest). The rainbow-colored areas in the Australasia, together with smaller areas

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along the western coast of the Americas, indicate zones of seismic activity of increasing depth. The continuous black line is the plate network showing 14 plates, all labeled. It is unclear whether the Indo– Australian plate should be considered to be two plates with a boundary in the NW Indian Ocean. The plate margins in the oceans are very narrow and conform to ideal plate margins; those in the continents are much broader. In particular, the wide belt of active deformation in the Alpine– Himalayan belt shows that the deformation is not restricted to the narrow plate margin on the map between the African, Arabian, and Indo– Australian plates on the one hand, and the southern Eurasian plate on the other. Rigid plate tectonic descriptions of such areas may be inappropriate. (Copyright A. G. Smith). The earthquakes are those of magnitude 4 or greater listed in http://earthquake.usgs.gov/earthquakes/ eqarchives/epic/

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Plates and Paleoreconstructions

Plate tectonics describe the motions of the lithospheric plates and the effects they cause. It is now the major branch of terrestrial tectonics, which is the study of the origins of the large-scale structures of the Earth, particularly the lithosphere. The name tectonics is derived from the Greek word tektonikos (tεktonιkóB) “pertaining to building.”

given time interval is given by their Euler rotation (l, ’, θ). l varies from 90° N, the N pole, conventionally given the value of +90, to 90° S, the S pole, or  90. Longitudes are positive measured E from the Greenwich meridian, conventionally taken as 0°, or as 360°. If measured W from the Greenwich meridian, the longitudes are negative. Maps showing the whole world run from 0° E to 360° E, or 0 to 360. They could also run from 180° W, that is, 180, to 180° E, or +180. The sign of the rotation θ is found by fixing one plate as a reference and determining whether the rotation moves the second plate in an anticlockwise direction when looking down on it – a positive rotation – or moves clockwise – a negative rotation. Changing the reference plate changes the sign of the rotation.

Plate Motions and Euler’s Theorem

Sliding Motions

Plates are rigid bodies that can move relative to one another in three ways: one, they can slide past one another; two, they can move away from one another; three, they can move toward one another. According to a theorem attributed to Euler, the relative motion of two rigid bodies on a sphere can be described as a rotation about a line passing through the Earth’s center that cuts the Earth’s surface at a point known as the rotation pole. The pole will have a latitude, l, and a longitude, ’, with the rotation that takes place in a given time given by θ (Fig. 2). To a good approximation the Earth is a sphere and tectonic plates are rigid. Thus the motion of one plate relative to another in a

When plates slide past one another they neither create nor destroy lithosphere. They conserve the lithosphere and are described as conservative plate boundaries. On a sphere such boundaries are circular arcs forming what are known as transform faults as in the San Andreas fault. The Euler pole is the point at the center of the arc.

proportions of continental and oceanic lithosphere (Fig. 1). There are no wholly continental plates. The shapes of most plates are quite irregular, reflecting their past histories of growth and destruction.

Plate Tectonics

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When a continent breaks up and the pieces move away from one another the motion is divergent. At first the margins of the continents-to-be are stretched, forming what will become passive continental margins, as in the East African Rift. Eventually the stretching reaches the stage where new ocean-floor has to form, as in the Red Sea or the Gulf of Aden. The Euler rotations for each stage are given by the rotations needed to bring the different markers (passive margins; anomalies of the same age) together.

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Plates and Paleoreconstructions, Fig. 2 The line ABC on a rigid body on the surface of a sphere (e.g., Earth) is moved to A’B’C’ by a rotation through the angle θ, about the rotation axis. The rotation axis passes through the center of the Earth and cuts the surface at point P with geographic coordinates of latitude l and longitude ’ (not shown), leading to an Euler rotation of (l, ’, θ). (Copyright A. G. Smith)

Convergent Motions The local motions between present-day converging plates can be found from the slip vectors of earthquakes. If the convergent margin is long enough it is also possible to use these motions to estimate the position of the Euler pole. However, the local motions and Euler poles for ancient plate margins are difficult to estimate. In favorable circumstances it is possible to calculate them from a plate circuit. For example, the Cenozoic motion along a convergent plate boundary between India and the rest of Eurasia can be calculated by successively summing the motions from Eurasia to North America, North America to Africa, and Africa to the Indo–Australian plate (Fig. 1). All of these motions are known from the ocean-floor

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spreading patterns in the Atlantic and Indian Oceans. Their sum gives the required result. The absence of any Paleozoic and Precambrian ocean-floor means that the motions and Euler poles at all pre-Mesozoic convergent plate boundaries are highly uncertain.

results. The field is geocentric because it behaves as if it were at the Earth’s center; it is axisymmetric because the field is symmetric about the Earth’s spin (or rotation) axis; and it behaves like a bar magnet and is therefore a dipole. Such a field has a very simple relationship between the angle, I, known as the inclination that the lines of magnetic force make with the horizontal, and the latitude, l, of the point of observation:

Reference Frames tan I ¼ 2 tan l, Three classes of reference frames are relevant to global reconstructions: any plate, the paleomagnetic reference frame, and that given by “hot-spots.”

Plates as Reference Frames Any plate, large or small, that can be linked to all the other plates can be used as reference frame to make a global reassembly: there is nothing fundamental in the choice of a plate. These global reassemblies are not geographic maps because they do not show the geographic latitude and longitude of the time concerned. To turn a reassembly into a map one needs to know the position of the geographic pole at the time concerned, which is given by paleomagnetism.

Paleomagnetism as a Reference Frame Over a period of several tens of thousands of years or more, the Earth’s magnetic field averages to what is known as a geocentric axisymmetric dipole field, or GADF. Virtually all paleomagnetic measurements use this field model to interpret the

ð1Þ

where tan ¼ tangent of the angle (I or l). As a plate moves, the latitude of a point on it changes and the orientation of the plate varies. Both these effects are recorded in the remanent magnetization – or Paleomagnetism, Principles – preserved in rocks. The actual position of the north magnetic pole at the time must lie along the line through the observation point that coincides with the direction of the declination. It is at a distance of (90 – l) degrees from the observation point along that line. Thus, in principle, paleomagnetic data can be used to reposition that plate at the time that the old magnetism was superimposed on the rocks. The determination of the age and direction of magnetization in rocks becomes progressively more difficult as one goes back in time, but these problems will not be discussed here. However, any Mesozoic or Cenozoic global reassembly can be turned into a map by finding where the mean north magnetic pole lies on the reassembly and making that point the north geographic pole of the map (Smith et al. 1994). A map of Pangea just before break-up is shown in Fig. 3. Although paleomagnetic measurements give the past latitude and orientation of a plate, they do not give its past longitude. Thus it is generally not possible to make a

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Plates and Paleoreconstructions, Fig. 3 Mollweide projection of Pangea reassembled at the end of the Triassic period, ~200 Ma. The green areas are the present-day coastlines; the blue areas all lie above 2000 m water depth, showing the approximate present- day extent of the continental crust. The latitude-longitude grid is drawn at 30° and shows in particular the estimated position of the late Triassic equator. Pangea was assembled by the collision of the southern continents (Gondwana) with the northern continents (Laurasia). A large oceanic area, known as

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the Tethys, separated southern Eurasia from northeast Gondwana. The distribution of continental fragments lying within the Tethys is uncertain, particularly that of China and adjacent areas (the large fragment lying east of Eurasia). Some continental slivers and/or island arcs off western North America are shown schematically. Several other slivers probably lay adjacent to other continental margins of Pangea, but no attempt has been made to depict them. (Copyright A. G. Smith)

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paleoreconstruction from paleomagnetic measurements alone. However, paleo-longitude differences between any two points on different continents will be correctly given if the relative position of the two points has been determined via a plate tectonic circuit whose individual motions are known.

Hot-Spots as a Reference Frame The origin of linear volcanic chains like the Hawaiian Islands has been attributed to the slow movement of a tectonic plate over a fixed deep plume of hot mantle, giving rise to “hotspots” at the surface (Wilson 1963; Morgan 1981). A purely descriptive term for them is that they are examples of “large igneous provinces” created in the case of hot-spots by intense, generally basaltic, magnetism. From the point of view of making reconstructions they show an interesting property in that they move relative to one another at rates that are an order of magnitude slower than those of plates. Hot-spots have been used as a reference frame that is believed to provide longitudes as well as latitudes for a global reassembly, forming a so-called absolute reference frame (Duncan 1981; Müller et al. 1993). The paleomagnetic reference and hot-spots reference frames agree within the limits of error for much of Cenozoic time, then gradually diverge for reasons that are not clear (Livermore et al. 1983).

Plates and Paleoreconstructions

the passive margins have been stretched. When two continents converge, the continental lithosphere of one continent overrides that of the other, which is then partially subducted. But because continental lithosphere is less dense than the underlying mantle, it cannot be subducted to any great depth. Thus the continental crust in the collisional zone becomes thickened and the area of continental crust decreases. In the case of the collision between India and Asia, the Tibetan plateau, and the Himalayas may represent a loss of continental area in a strip that may be more than 500 km wide (Coward et al. 1988; Shackleton 1988). As one goes back in time, the subducted Indian crust should be pulled out from under Eurasia and added to the Indian continent. Similar adjustments, but on a generally smaller scale, are necessary to the shapes of all continents caught up in collision zones, but are rarely made. The continental shapes used for reconstructions generally show the edges of continents at about 2000 m water depth. This shape is unfamiliar where there is a wide shallow sea between the 2 km submarine bathymetry contour and the coastline. To aid in recognition, the present-day coastline is commonly shown on continents, but the coastline of the time concerned may be quite different, as well as being very difficult to pin down.

Record of Continental Separation in the OceanFloor Paleoreconstructions Paleoreconstructions attempt to show where the oceans and continents were in the geological past. That this is possible is due to several effects: (1) conservation of continental crust during plate motions; (2) the record of Mesozoic and Cenozoic continental separation in the intervening ocean-floor; and (3) the record of changes in continental latitude and orientation in Paleomagnetism, Principles.

Conservation of Continental Shapes The extension on passive continental margins is probably about 100 km for a margin that is 200 km wide at the present-day, such as parts of the eastern North American margin (e.g., Sawyer 1985). In other areas, the extension may be much higher: Powell et al. (1988) estimated a total of 360 km extension across the Australia–Antarctica margin before the formation of ocean-floor (180 km) equivalent to 1.6° of latitude for each margin, assuming symmetric extension. To a good approximation, continental shapes on a global scale are conserved during plate motions. In detail this is clearly not the case. For example, when continents separate

The ocean-floor generated during the break-up of Pangea and the creation of the Atlantic, India, Antarctic, and Arctic Oceans is still mostly preserved. Successively older oceanic transform faults, fracture zones, and ocean- floor magnetic anomalies and the continental edges themselves can be fitted together to give the Euler rotations needed to reposition all the major continents to a high degree of precision for most of Mesozoic and all of Cenozoic time, or roughly back to 200 Ma. The net results are paleocontinental reconstructions, that is, a display of the relative positions of the continents in the past (Smith et al. 1973).

Paleozoic and Precambrian Composites For pre-Mesozoic time the only ocean-floor that remains is tectonically highly deformed. Thus Paleozoic and Precambrian global “maps” are made by geographically repositioning individual continents using paleomagnetism. The uncertainties in paleomagnetic poles are considerable, particularly for Precambrian time. Although the order in which the continents are arranged on the globe is known, at least for Paleozoic time, the longitude separations of adjacent continents are not. These separations are subjectively

Plates and Paleoreconstructions

estimated from evidence such as the distribution of fossils, tectonic effects, and the like. The results have been called composites, rather than maps, to emphasize the fact that they include qualitative data (Smith et al. 1973). The Precambrian has few fossils that provide an accurate indication of age and only scattered age determinations. Their absence, together with the absence of ocean-floor data and the uncertainties in pole positions, means that there are no generally agreed composites for most of Precambrian time. The Precambrian is still a veritable terra incognita.

Paleo-Plate Reconstructions As noted above, a global paleoreconstruction is a reassembly of the major continents (and oceans) relative to one another at some time in the past. To turn this into a paleo-plate reconstruction, one needs to add the plate margins of the time concerned. For Jurassic and younger periods the former positions of ocean ridges are known from the ocean-floor record in the Atlantic, Indian, and Southern Oceans, but much of the Pacific ocean-floor has been subducted and for these areas former ridge positions have to be estimated. For Triassic and older periods the positions of former ocean ridges is largely a matter of informed guesswork (e.g., Stampfli and Borel 2002, 2004). Orogenic belts, that is, those areas made up of deformed, metamorphosed, and igneous rocks, mark the former positions of convergent margins. However, the margins are commonly difficult to locate precisely and many orogenic belts include more than one former convergent margin, together with the locations of former oceans that have since been subducted. Thus although one can join former ridges to orogenic belts to make a sketch plate boundary map for the time concerned, this map is necessarily imprecise. Precambrian global paleoreconstructions are generally schematic (e.g., Collins and Pisarevsky 2005). Because paleo-plate reconstructions build on an initial global reassembly, most Paleozoic and all Precambrian paleo-plate reconstructions are essentially cartoons that show the kinds of plate boundaries that may have existed at a given time but whose locations may have considerable errors.

Conclusions The ocean-floor magnetic anomalies, transform faults, and fracture zones, together with the paleomagnetic frame allow very precise global Cenozoic and Mesozoic maps to be made. The inferred relative longitudes of Paleozoic and Precambrian continents depend on qualitative data, giving rise to poorly defined “composites.” Fossils enable precise dating of Paleozoic features that help in the construction of composites, but

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such precision is not obtainable from Precambrian fossils, giving rise to considerable uncertainties in all Precambrian reconstructions. How to convert these global paleoreconstructions into verifiable paleo-plate reconstructions is a largely unsolved problem for pre-Mesozoic time.

Cross-References ▶ Lithosphere, Continental ▶ Lithosphere, Oceanic ▶ Magnetic Anomalies: Interpretation ▶ Magnetic Methods, Surface ▶ Paleomagnetism, Principles ▶ Subduction Zones

Bibliography Collins AS, Pisarevsky SA (2005) Amalgamating eastern Gondwana: the evolution of the Circum-Indian Orogens. Earth-Sci Rev 71:229–270 Cox A, Hart RB (1986) Plate tectonics: how it works. Blackwell, Oxford Crosby AG, McKenzie D, Sclater JG (2006) The relationship between depth, age and gravity in the oceans. Geophys J Int 166:553–573 Duncan RA (1981) Hotspots in the southern ocean – an absolute frame of reference for motions of the Gondwana continents. Tectonophysics 74:29–42 Isacks B, Oliver J, Sykes LR (1968) Seismology and the new global tectonics. J Geophys Res 73:5855–5899 Livermore RA, Vine FJ, Smith AG (1983) Plate motions and the geomagnetic field. 1: quaternary and late tertiary. Geophys J R Astron Soc 73:153–171 McKenzie DP, Parker RL (1967) The north pacific: an example of tectonics on a sphere. Nature 216:1276–1280 Morgan WJ (1968) Rises, trenches, great faults and crustal blocks. J Geophys Res 73:1959–1982 Morgan WJ (1981) Hotspot tracks and the opening of the Atlantic and Indian Oceans. In: Emiliani C (ed) The oceanic lithosphere. Wiley, New York, pp 443–487 Müller RD, Royer JY, Lawver LA (1993) Revised plate motions relative to the hotspots from combined Atlantic and Indian Ocean hotspot tracks. Geology 21:275–278 Powell CM, Roots SR, Veevers SJ (1988) Pre-breakup continental extension in East Gondwanaland and the early opening of the eastern Indian Ocean. Tectonophysics 155:261–283 Priestley K, McKenzie D (2006) The thermal structure of the lithosphere from shear wave velocities. Earth Planet Sci Lett 244:285–301 Sawyer DS (1985) Total tectonic subsidence: a parameter for distinguishing crust type at the U.S. continental margin. J Geophys Res 90:7751–7769 Shackleton RM (1988) Tectonic evolution of the Himalayas and Tibet. In: Shackleton RM, Dewey JF, Windley BF (eds) Philosophical transactions of the Royal Society of London, series A: mathematical and physical sciences. The Royal Society of London, London Smith AG, Briden JC, Drewry GE (1973) Phanerozoic world maps. In: Hughes NF (ed) Organisms and continents through time, volume 12: special paper. Palaeontological Association, London, pp 1–42 Smith AG, Smith DG, Funnell BM (1994) Atlas of Mesozoic and Cenozoic coastlines. Cambridge University Press, Cambridge

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Poroelasticity Ran Bachrach Geophysics and Planetary Sciences Department, Tel Aviv University, Tel Aviv, Israel

Definition Poroelasticity is a joint formulation for the behavior of a solid–fluid coupled porous system. Poroelasticity describes the behavior of porous continuum and pore fluid with respect to applied displacements and stresses. Poroelasticity jointly treats the solid frame and pore fluid to calculate changes in pore pressure, solid displacements, and fluid displacements due to stresses and displacements associated with external or internal processes. The theory of poroelasticity is based on the work of (Terzaghi 1941; Biot 1941, 1956a, b, 1962; Gassmann 1951). Poroelasticity is often called Biot theory as the work of M. A. Biot is considered to establish the complete foundation for the theory.

Basic Concepts Consider a connected network of pores fully saturated with fluid. The following are basic concepts that form the basis for the theory of poroelasticity.

Poroelasticity

in and out of the sample during the deformation process. Undrained (or jacketed) conditions refer to the condition in which the boundary of the porous solid (REV) is impermeable and pore fluid cannot exit the rock during the deformation process. Physical measurements on both states (laboratory measurements) will determine the poroelastic coefficients of the rock. Often “dry conditions” are associated with the drained experiment. In poroelasticity the term “dry” elastic modulus refers to the drained experiment and not the elastic modulus when the material is dry. Linear Stress–Strain Formulation for Poroelastic Media Consider the displacement vector in the solid u ¼ [ux, uy, uz,] and the average fluid displacement vector in the pore space U ¼ [Ux, Uy, Uz,]. The volume of fluid displaced through a unit area normal to the x, y, z, direction is fU and the fluid displacement field is w. The isotropic poroelastic stress–strain relations that relate stress, s, and pore pressure, P, to displacement and strain, e, can be written in an abbreviated notation as: sij ¼ 2meij þ dij ðlc y  aMzÞ P ¼ aM y þ M z y ¼ ∇  u, z ¼ ’∇  ðu  UÞ where lc is the lame’s coefficient for the closed (undrained) system and m is the shear moduli. Note that the shear strain and stress are not affected by pore pressure and fluid presence. M, α are the two poroelastic constants associated with the poroelastic stress–strain formulation. M is known as Biot modulus, or P wave modulus, and relates dry response to saturated response as follows: lC ¼ ldry þ a2 M, 1 a’ ’ ¼ þ : M K0 Kf where Kf is the pore fluid bulk modulus and ’ is the porosity. α is the Biot parameter and is defined as: a ¼ 1  K dry =K 0

Drained and Undrained Deformation For a fully saturated porous material, we consider two fundamental states (or thought experiments): Drained and undrained (sometimes these are called jacketed and unjacketed conditions). Drained conditions are achieved when pore pressure in a representative elementary volume (REV) is in equilibrium with the surrounding throughout the deformation of the saturated porous medium. A special case (described as a lab experiment) is when pore pressure and fluid can freely defuse

where Kdry is the bulk modulus of the rock measured under drained conditions and K0 is the bulk modulus of the frame mineral. Note that when the system is jacketed, that is, ζ ¼ 0 so there is no relative displacement between the fluid and matrix, the pore pressure and the total stress are related and the bulk modulus is defined in terms of “closed” elastic properties. When the matrix is open, the assumption is that there is a constant pore pressure in the system (pore pressure is in

Poroelasticity

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equilibrium with the surroundings) and thus fluid can flow freely in and out of the system. One special case of such conditions is when the pore pressure is equal to zero. The term “dry” frame conditions often refers to the states of zero pore pressure. The poroelastic linear stress–strain relation can be written in terms of pore pressure as the coupling variable rather than the relative fluid displacement (or divergence). A popular form of the linear isotropic poroelastic stress–strain relations is given by:

contribute to the shear stiffness of the system. Gassman’s equation can be derived directly from Biot’s equations or from basic elasticity considerations by accounting for mass conservation of pore fluid in a closed system. Gassmann’s equation assumes that pore fluid is in equilibrium throughout the sample and that the pore fluid does not interact with the mineral frame. For more information see Mavko et al. (1998). Anisotropic extension to Gassmann’s equation is derived by Brown and Korringa (1975) in terms of the elastic compliance tensor: 

sij þ aPdij ¼ 2meij þ ldry ydij

at Sdry ijkl  Sijkl

z ¼ ð1=MÞP þ ay Note that the pressure direction here is assumed to be negative, for example, the pore pressure resists a positive confining stress that pushes the grains together. The above equation can be written as an equivalent linear elasticity relation by substituting the total stress with Biot effective stress sij ¼ sij þ aPdij . as: sij  ¼ 2meij þ ldry ydij z ¼ ð1=MÞP þ ay In terms of the differential pressure (also known as Terzaghi’s effective stress or just the “effective” stress) defined as s0ij ¼ sij þ Pdij the linear stress–strain relations are given by: 0

sij  ð1  aÞPdij ¼ 2meij þ ldry ydij z ¼ ð1=MÞP þ ay When α ¼ 1 (soft materials where Kdry 0 and R ¼ x2 þ y2 þ z2 . Plane waves are then building blocks to construct solutions for the half-space problems and layered media. They play a central role in seismology. Certainly, it is the success of a simple, yet profound idea. Here g ¼

Green’s Function for Homogeneous Space The Green’s function is the displacement field produced within an elastic medium by a concentrated, impulsive unit load at a given point. For an infinite, homogeneous, isotropic, elastic medium Stokes found the analytical solution at the middle of the nineteenth century. If the body force is a unit concentrated impulsive, it can be represented with Dirac’s delta functions for space and time. Moreover, to specify the force direction, we add Kronecker delta δij (¼1 if i ¼ j, ¼0 for i 6¼ j). Therefore, we can define fi δijδ(|x  x|)δ(t) and ui(x, t) Gij(x, x; t) ¼ Green’s function for the infinite elastic domain. It is the displacement in direction i at x when a unit impulse is applied in direction j at point x. The Green’s function should satisfy Navier’s equation, which can be written as m

¼ dij dðjx  xjÞdðtÞ:

ð18Þ

The solution of this problem is due to Stokes (1849) and can be expressed by means of

ð16Þ

that represents a harmonic plane wave propagating and identify that the wave vector k ¼ (o/c)n that gives the direction of propagation of the plane wave, where n is a unit vector (n12 + n22 + n32 ¼ 1). It is possible to extend the idea that the unit vector n may have complex values then we have inhomogeneous plane waves. As an example of the significant representation power of generalized plane waves, we can express a harmonic spherical wave as a superposition of both homogeneous and inhomogeneous plane waves using the Weyl integral (see Aki and Richards 1980):

@ 2 Gij @ 2 Gkj @ 2 Gij þ ð l þ mÞ r 2 @xk @xk @xi @xk @t

Gij ðx, x; tÞ ¼

 

1 f 1 gi g j þ f 2 dij  gi g j , 4pmr

ð19Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where r ¼ jx  xj ¼ ðx1  x1 Þ2 þ ðx2  x2 Þ2 þ ðx3  x3 Þ2 and γj ¼ (xj  xj)/r is the unit vector form x to x. The functions f1 and f2 are given by b2 f 1 ðr, tÞ ¼ 2 dðt  r=aÞ þ 2b2 a

1=b ð

dðt  rkÞk dk, 1=a

and

ð20Þ

Propagation of Elastic Waves: Fundamentals

f 2 ðr, tÞ ¼ dðt  r=bÞ  b

2

1287

1=b ð

dðt  k r Þk dk:

ð21Þ

1=a

We notice in Eq. 19 that Stokes’ (1849) solution decays as 1/r and is modulated angularly by sines and cosines. This solution clearly shows up the emergence of both dilatational and distortional (P and S) spherical waves with their characteristic radiation patterns. Equations 20 and 21 show that both the dominant P and S pulses share the time dependence with the source. The S pulse has a larger amplitude. Between these P and S arrivals we find a disturbance decaying more rapidly away from the source. In the frequency domain (after applying Fourier transform) these functions can be written as follows:   f 1 ðr, oÞ ¼ b2 =a2 1  i2=qr  2=q2 r 2 exp ðiqrÞ  ð22Þ þ i2=kr þ 2=k2 r 2 exp ðikrÞ, and   f 2 ðr, oÞ ¼ b2 =a2 i=qr þ 1=q2 r 2 exp ðiqrÞ  þ 1  i=kr  1=k2 r 2 exp ðikrÞ:

ð23Þ

Here q ¼ o/α and k ¼ o/β are the wave numbers of P and S waves, respectively. If o equals zero, the static case, we obtain the constants f1 ¼ 1 and f2 ¼ (1 + β2/α2)/2 that correspond to the Kelvin solution for a unit static load in the full elastic space. The Stokes solution for the infinite space is frequently used in the integral BEM or IBEM formulations (see Bouchon and Sánchez-Sesma 2007). Other Green’s functions in two and three dimensions can be used to solve a variety of problems. A remarkable compendium of fundamental solutions is the one due to Kausel (2006). Fifty-five years after the Stokes’ (1849) solution, at the eve of the twentieth century, Lamb (1904) found the solution for an impulsive vertical point load at the surface of a half-space. This solution is based on integral representations, and the passage to time domain became premonitory of methods developed some decades later. A complete account of Lamb’s problem and summary can be found in Kausel (2013) who pointed out that the first truly complete solutions to Lamb’s problem were obtained by Pekeris (1955) and Chao (1960). They provided closed-form expressions for displacements engendered by a vertical and a horizontal load, respectively, but only when Poisson’s ratio is 1/4. Significant advances are due to Mooney (1974) and Richards (1979). Sundry numerical computations of the Green’s function associated to Lamb’s problem are due to Johnson (1974). Explicit formulae have been recently given for the displacement at the

surface of an elastic half-space for a buried point source in what can be called the 3D Lamb’s problem (Feng and Zhang 2018). Their results are expressed in terms of elementary algebraic expressions and elliptic integrals as well and are validated against Johnson’s (1974) solution. For a homogeneous or layered half space, the classical Fourier methods allow obtaining the elastodynamic Green’s function in integral form. Typically, the fields constructed in each layer by a set of plane waves with unknown coefficients that are found after boundary conditions are enforced. The strategies to solve this problem are diverse and we count among them the global matrix approach (Knopoff 1964), the Thomson-Haskell method (Haskell 1953), the stiffness matrix (Kausel 2006), and the Kennett reflection-transmission method (Kennett 2001, 2002). The computation of the displacement and stress fields requires integration along the horizontal wavenumber domain. Usually, in the discrete wavenumber method (DWN) the horizontal wavenumber is discretized and Fourier analysis permits to go to time domain (see Bouchon and Aki 1977; Bouchon 2003). Theory allows solving the problem of the homogeneous boundary conditions in terms of an eigenvalue problem. This led to the astounding subject of surface waves (See Aki and Richards 2002).

Green’s Function Retrieval from Correlations The pioneering studies of Aki (1957) on seismic ambient noise and coda are the basis of significant developments. Various scattering formulations have been developed in order to explain coda features (see Aki and Chouet 1975; Sato and Fehler 1998). When multiple scattering takes place, the intensities, which are related to energy densities, follow diffusion-like equations. These waves arrive at the receiver from different directions sampling the medium along their paths. It has been found that the imaginary part of the elastodynamic Green’s function is proportional to the average cross correlations within a diffuse field (for recent reviews, see Weaver 2010 and Perton and Sánchez-Sesma 2016). A diffuse field develops within a system if there is isotropic, equipartitioned illumination. One simple way to achieve this is using a set of elastic plane waves that fulfill the Principle of Equipartition (EQP) of Energy (Weaver 1982; SánchezSesma and Campillo 2006; Sánchez-Sesma et al. 2006; Weaver 2010). On the other hand, simple illumination from isolated sources may give rise to diffuse fields after multiple scattering takes place (see Hennino et al. 2001; Paul et al. 2005). The connection between this deterministic problem, namely, the computation of elastic Green’s function and the

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Propagation of Elastic Waves: Fundamentals

diffuse field theory, has been pointed out by Sánchez-Sesma et al. (2011a) in the framework of the partition of the energy injected into a half-space by surface loads (see Miller and Pursey 1955; Weaver 1985). In recent works, the computation of Green’s functions for 2D layered systems was achieved using a cocktail of equipartitioned homogeneous plane waves (P, SV, SH) and surface waves (Rayleigh and Love). Perton and Sánchez-Sesma (2016) discussed the case of a layered medium, while Baena-Rivera et al. (2016) considered an alluvial basin. In reality, for elastic inhomogeneous, anisotropic, irregular medium the role of multiple scattering should be invoked to assert that under very general circumstances the resulting seismic ambient field is very likely diffuse. Under these circumstances, the Green’s function can be approximately retrieved from averaging cross correlations of the recorded motions of such diffuse field (e.g., Weaver and Lobkis 2004; Wapenaar 2004; Gouédard et al. 2008). Seismic noise is produced by nearby, superficial low strength sources that we will generically call microtremors. Noise sources are 3D in nature and have significant geometrical spreading. This attenuation make that the relative sampling of the medium in depth is also much reduced.

In what follows, we review the Green’s function retrieval from correlations of field fluctuations. For the layered medium with free surface overlaying a half-space, we compute the imaginary part of the Green function when source and receiver coincide at the free surface. It has been demonstrated (e.g., Sánchez-Sesma et al. 2008) that if a 3D diffuse harmonic displacement vector field ui(x, o) is established within an elastic medium, the average cross-correlations of such motions at points xA and xB can be written as: D E  ui ðxA , oÞuj ðxB , oÞ ¼ 2pES k3 Im Gij ðxA , xB , oÞ : ð24Þ In this equation, the Green’s function Gij(xA, xB, o) ¼ displacement at xA in direction i produced by a unit harmonic point force acting at xB in direction j, o ¼ circular frequency, k ¼ o/β ¼ shear wavenumber, β ¼ shear wave propagation velocity, and ES ¼ ro2S2 ¼ average energy density of shear waves (r ¼ mass density, S2 ¼ average spectral density). Note that the asterix implies the complex conjugate and the angular brackets mean azimuthal average. Equation 1 is the

0.02 5,000 sources

0.01 0

–0.01

Im(G22(XA,XB)) –0.02 –6

–4

–2

0

2

4

6

8

2

4

6

8

2

4

6

8

0.02 100 sources

0.01 0 –0.01 –0.02 –6

–4

–2

0

0.02 20 sources 0.01 0 –0.01 –0.02 –6

–4

–2

0 Time (s)

Propagation of Elastic Waves: Fundamentals, Fig. 4 Correlations are computed with 20, 100, and 5000 sources, for the scalar case. We can see that as the number of sources considered increases the retrieval, the G22 function is better achieved

Propagation of Elastic Waves: Fundamentals

1289

analytical consequence of a correlation-type elastic representation theorem and has been verified recently in canonical examples of a full space (Sánchez-Sesma and Campillo 2006) and for an elastic inclusion embedded in a full space and for more general circumstances (Sánchez-Sesma et al. 2006, 2008). For the scalar case it is possible to compare the Green’s function retrieval from correlations. In Fig. 4, we show how the G22 function is retrieved taking into account 20, 100, and 5000 sources. The correlations are computed with these numbers of sources, and we can see that as the number of sources considered increases the retrieval is better achieved.

Energy Densities at Given Points and Directions When source and receiver are both at the same point, we have interesting consequences. For each component the autocorrelation is proportional to the directional energy density (DED) at a given point. On the other hand, the imaginary part of the Green function at the source is finite because the singularity of the Green’s function is restricted to the real part. Therefore, the DED is proportional to the imaginary part of Green’s function. In what follows we compute the theoretical energy density at a given point xA. In order to do so we rewrite Eq. 24 considering that xA ¼ xB:

 EðxA Þ ¼ ro2 um ðxA Þum ðxA Þ ¼ 2pmES k1  Im½Gmm ðxA , xA Þ

ð25Þ

where m ¼ shear modulus. The energy density of shear waves ES is a measure of the strength of the diffuse illumination. We see that the total energy density at a point is proportional to the imaginary part of the trace of the Green tensor for coincident receiver and source. The imaginary part is finite and regular and represents the rate of energy injected by the unit harmonic load at that point. This quantity “detects” the energy that goes back to the source-receiver and may be used to imaging. Equation 25 is valid even if the summation convention is ignored. In that case E(xA) Em(xA) and the energy density is associated to a particular direction (for discussions see Perton et al. 2009). The connection of the imaginary part of the Green’s function at the source with the optical theorem has been explored by Snieder et al. (2009). To illustrate these ideas from theoretical point of view, let us consider again the Stokes’ (1849) solution. After applying Fourier transform to Eqs. 20 and 21 and taking the limit r➔0, i.e., making that source and receiver coincide, we can write that

  o 1 2 þ Im Gij ðx, xÞ ¼  d : 12pr a3 b3 ij 



ð26Þ

This shows that the power injected into the infinite elastic medium by the unit harmonic load grows linearly with frequency, as expected, it is an isotropic tensor and the energy is explicitly labeled as being related to P and S waves follows the principle of equipartition. The relationships among energy densities and its partitions have been studied by Perton et al. (2009) for a half-space and by Margerin et al. (2009) for a stratified medium. The Green’s function may be useful to imaging the subsurface structure at the site. For horizontally layered systems and 1D excitation, the relationship between reflection response and autocorrelation of surface motion was discovered by Claerbout (1968). An algorithm to identify reflection coefficients based upon Claerbout’s results was proposed by Scherbaum (1987). In fact, this is a single-station inversion method, and, in principle, it allows imaging of the subsurface impedance structure. The idea was tested using small local recorded earthquakes.

The Horizontal-to-Vertical Spectral Ratio This concept, the duality of diffuse field energy density – medium impulse response or Green’s function, is being the engine of practical applications in exploration geophysics and engineering seismology. Indeed, the microtremor H/V spectral ratio (MHVSR) is frequently used to assess the dominant frequency of soil sites. It requires to measure ground motion produced by seismic ambient noise at a site and the processing is relatively simple. The diffuse field theory asserts that the average autocorrelations of motion components of a diffuse field at a given receiver give the directional energy densities (DEDs) which are proportional to the imaginary parts of the Green’s function components when source, receiver, and direction coincide. Therefore, the MHVSR can be modeled in terms of the Green’s function, and this opens the door to inversion of medium properties in layered systems (see Sánchez-Sesma et al. 2011b; García-Jerez et al. 2016; Sanchez-Sesma 2017; Kawase et al. 2011, 2015; Lontsi et al. 2015, 2019) and systems with lateral heterogeneity as well (Matsushima et al. 2014, 2017).

Concluding Remarks The fundamentals of dynamic elasticity have been reviewed. The Newton’s second law and the Hooke’s elastic idea led to the Navier-Cauchy governing equations. D’Alembert seminal solution in conjunction with Fourier analysis led to powerful descriptions in terms of plane waves. Dynamic elasticity is a

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science with almost two centuries of history. It is alive with significant impacts in seismology and engineering. It is central in the theory and applications of diffuse field concept.

Cross-References ▶ Body Waves ▶ Poroelasticity ▶ Seismic Signals in Well Water Observations ▶ Seismic, Ambient Noise Correlation ▶ Seismogram Interpretation ▶ Shear-Wave Splitting: New Geophysics and Earthquake Stress-Forecasting ▶ Statistical Seismology ▶ Surface Waves Acknowledgments Thanks are given to E. Kausel for his keen and constructive remarks. This work was partially supported by DGAPAUNAM Projects IN100917, IN121709, and CONACYT Mexico under PROINNOVA projects 251910 and 241763.

Bibliography Achenbach JD (1973) Wave propagation in elastic solids. North-Holland Publishing Co, Amsterdam/New York/Oxford Aki K (1957) Space and time spectra of stationary stochastic waves with special reference to microtremors. Bull Earthq Res Inst 35:415–456 Aki K, Chouet B (1975) Origin of coda waves: source, attenuation and scattering effects. J Geophys Res 80:3322–3342 Aki K, Richards PG (1980) Quantitative seismology. Theory and methods. W. H. Freeman, San Francisco Aki K, Richards PG (2002) Quantitative seismology. Sausalito, Calif: University Science Books Baena-Rivera M, Perton M, y Sánchez-Sesma FJ (2016) Surface waves retrieval from generalized diffuse fields in 2d synthetic models of alluvial valleys. Bulletin of the Seismological Society of America 106(6) Bouchon M (2003) A review of the discrete wavenumber method. Pure Appl Geophys 160:445–465 Bouchon M, Aki K (1977) Discrete wave number representation of seismic source wave fields. Bull Seismol Soc Am 67:259–277 Bouchon M, Sánchez-Sesma FJ (2007) Boundary integral equations and boundary elements methods in Elastodynamics. In: Wu R-S, Maupin V, Dmowska R (eds) Advances in wave propagation in heterogeneous earth. Advances in geophysicis, vol 48. ElsevierAcademic Press, New York/ Boston, pp 157–189 Chao CC (1960) Dynamical response of an elastic half-space to tangential surface loadings. J Appl Mech 27:559–567 Claerbout JF (1968) Synthesis of a layered medium from its acoustic transmission response. Geophysics 33:264–269 Feng X, Zhang H (2018) Exact closed-form solutions for Lamb’s problem. Geophys J Int 214:444–459. https://doi.org/10.1093/gji/ García-Jerez A, Piña-Flores J, Sánchez-Sesma FJ, Luzón F, Perton M (2016) A computer code for forward computation and inversion of the H/V spectral ratio under the diffuse field assumption. Comput Geosci 97:67–78 Gouédard P, Stehly L, Brenguier F, Campillo M, Colin de Verdiére Y, Larose E, Margerin L, Roux P, Sánchez-Sesma FJ, Shapiro NM,

Propagation of Elastic Waves: Fundamentals Weaver RL (2008) Cross-correlation of random fields: mathematical approach and applications. Geophys Prospect 56:375–393 Haskell NA (1953) The dispersion of surface waves in multilayered media. Bull Seismol Soc Am 43:17–34 Hennino R, Trégourès N, Shapiro NM, Margerin L, Campillo M, van Tiggelen B, Weaver RL (2001) Observation of equipartition of seismic waves in Mexico. Phys Rev Lett 86:3447–3450 Johnson LR (1974) Green’s function for Lamb’s problem. Geophys J Int 37:99–131. https://doi.org/10.1111/j.1365-246X.1974.tb02446.x Kausel E (2006) Fundamental solutions in elastodynamics. A compendium. Cambridge University Press, New York Kausel E (2013) Lamb’s problem at its simplest. Proc R Soc A 469:20120462. https://doi.org/10.1098/rspa.2012.0462 Kawase H, Sánchez-Sesma FJ, Matsushima S (2011) The optimal use of horizontal-to-vertical (H/V) spectral ratios of earthquake motions for velocity structure inversions based on diffuse field theory for plane waves. Bull Seismol Soc Am 101:2001–2014 Kawase H, Matsushima S, Satoh T, Sánchez-Sesma FJ (2015) Applicability of theoretical horizontal-to-vertical ratio of microtremors based on the diffuse field concept to previously observed data. Bull Seismol Soc Am 105:3092–3103. https://doi.org/10.1785/0120150134 Kennett BLN (2001) The seismic wavefield, volume I: introduction and theoretical development. Cambridge University Press, Cambridge, UK Kennett BLN (2002) The seismic wavefield, volume II: interpretation of seismograms on regional and global scales. Cambridge University Press, Cambridge, UK Knopoff L (1964) A matrix method for elastic wave problems. Bull Seismol Soc Am 54:431–438 Lamb H (1904) On the propagation of tremors over the surface of an elastic solid. Philos Trans R Soc London A 203:1–42 Lontsi AM, Sánchez-Sesma FJ, Molina-Villegas JC, Ohrnberger M, Krüger F (2015) Full microtremor H/V(z,f) inversion for shallow subsurface characterization. Geophys J Int 202:298–312. https://doi. org/10.1093/gji/ggv132. GJI Seismology Lontsi AM, García-Jerez A, Molina-Villegas JC, Sánchez-Sesma FJ, Molkenthin C, Ohrnberger M, Krüger F, Wang R, Fäh D (2019) A generalized theory for full microtremor horizontal-to-vertical [H/V(z,f)] spectral ratio interpretation in offshore and onshore environments. Geophys J Int 218(2):1276–1297. https://doi.org/10.1093/ gji/ggz223 Margerin L, Campillo M, van Tiggelen BA, Hennino R (2009) Energy partition of seismic coda waves in layered media: theory and application to Pinyon Flats Observatory. Geophys J Int 177:571–585 Matsushima S, Hirokawa T, de Martin F, Kawase H, Sánchez-Sesma FJ (2014) The effect of lateral heterogeneity on horizontal-to-vertical spectral ratio of microtremors inferred from observation and synthetics. Bull Seismol Soc Am 104:381–393 Matsushima S, Kosaka H, Kawase H (2017) Directionally dependent horizontal-to-vertical spectral ratios of microtremors at Onahama, Fukushima, Japan. Earth Planets Space 69:96. https://doi.org/10. 1186/s40623-017-0680-9 Miller GF, Pursey H (1955) On the partition of energy between elastic waves in a semi-infinite solid. Proc R Soc Lond A Math 233:55–69 Mooney HM (1974) Some numerical solutions for Lamb’s problem. Bull Seismol Soc Am 64:473–491 Paul A, Campillo M, Margerin L, Larose E, Derode A (2005) Empirical synthesis of time-asymmetrical Green function from the correlation of coda waves. J Geophys Res 110. https://doi.org/10.1039/ 2004JB003521 Pekeris CL (1955) The seismic surface pulse. Proc Natl Acad Scie USA 41:469–480. https://doi.org/10.1073/pnas.41.7.469 Perton M, Sánchez-Sesma FJ (2016) Green’s function calculation from equipartition theorem. J Acoust Soc Am 140:1309–1318

Propagation of Elastic Waves: Fundamentals Perton M, Sánchez-Sesma FJ, Rodríguez-Castellanos A, Campillo M, Weaver RL (2009) Two perspectives on equipartition in diffuse elastic fields in three dimensions. J Acoust Soc Am 126:1125–1130. https://doi.org/10.1121/1.3177262 Richards PG (1979) Elementary solutions to Lamb’s problem for a point source and their relevance to three-dimensional studies of spontaneous crack propagation. Bull Seismol Soc Am 69:947–956 Sanchez-Sesma FJ (2017) Modeling and inversion of the microtremor H/V spectral ratio: physical basis behind the diffuse field approach. Earth Planets Space 69:92. https://doi.org/10.1186/s40623-0170667-6 Sánchez-Sesma FJ, Campillo M (2006) Retrieval of the Green function from cross-correlation: the canonical elastic problem. Bull Seismol Soc Am 96:1182–1191 Sánchez-Sesma FJ, Pérez-Ruiz JA, Campillo M, Luzón F (2006) The elastodynamic 2D Green function retrieval from cross-correlation: the canonical inclusion problem. Geophys Res Lett 33:L13305. https://doi.org/10.1029/2006GL026454 Sánchez-Sesma FJ, Pérez-Ruiz JA, Luzón F, Campillo M, RodríguezCastellanos A (2008) Diffuse fields in dynamic elasticity. Wave Motion 45:641–654 Sánchez-Sesma FJ, Weaver RL, Kawase H, Matsushima S, Luzón F, Campillo M (2011a) Energy partitions among elastic waves for dynamic surface loads in a semi-infinite solid. Bull Seismol Soc Am 101:1704–1709

1291 Sánchez-Sesma FJ, Rodríguez M, Iturrarán-Viveros U, Luzón F, Campillo M, Margerin L, García-Jerez A, Suárez M, Santoyo MA, Rodríguez-Castellanos A (2011b) A theory for microtremor H/V spectral ratio: application for a layered medium. Geophys J Int 186:221–225 Sato H, Fehler M (1998) Wave propagation and scattering in the heterogeneous earth. Springer, New York Scherbaum F (1987) Seismic imaging of the site response using microearthquake recordings. Part 1. Method. Bull Seismol Soc Am 77:1905–1923 Snieder R, Sánchez-Sesma FJ, Wapenaar K (2009) Field fluctuations, imaging with backscattered waves, a generalized energy theorem, and the optical theorem. SIAM J Imag Sci 2:763–776 Stokes GG (1849) On the dynamical theory of diffraction. Cambridge Phil Soc Trans 9:1–62 Wapenaar K (2004) Retrieving the elastodynamic Green’s function of an arbitrary inhomogeneous medium by cross correlation. Phys Rev Lett 93:254301-1-4 Weaver RL (1982) On diffuse waves in solid media. J Acoust Soc Am 71:1608–1609 Weaver RL (1985) Diffuse elastic waves at a free surface. J Acoust Soc Am 78:131–136 Weaver RL (2010) Equipartition and retrieval of Green’s function. Earthq Sci 23:1–6 Weaver RL, Lobkis OI (2004) Diffuse fields in open systems and the emergence of the Green’s function. J Acoust Soc Am 116:2731–2734

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Radioactivity in Earth’s Core

Introduction

V. Rama Murthy (Deceased)

The Earth is a thermal engine driven by its internal heat. The bulk of this heat is produced by the elements potassium (K), uranium (U), and thorium (Th) through the radioactive decay of their isotopes 40K, 235U, 238U, and 232Th. Table 1 lists the atomic percentages, radioactive decay constants, half-life, and heat production characteristics of these radioactive isotopes. Because K, U, and Th are strongly lithophile, differentiation of the Earth into a metallic core and a silicate primitive mantle (BSE) will have partitioned them overwhelmingly into the BSE. A long-standing convention is that the radioactivity of the BSE represents the total radioactivity of the Earth and that the metallic core has no radioactivity (e.g., McDonough 1999, 2003). However, radioactivity in the core has come into focus due to recent developments in three areas: (1) modern high P, T experiments on the metal-silicate partitioning behavior of K and U; (2) theoretical and experimental advances in our understanding of the electronic structure of potassium under conditions of high pressure and temperature; and (3) thermal models of the core that satisfy the size and age of the inner core and the energy to sustain geomagnetic field. Also, it is known that the heat due to radioactivity in BSE, about 19 TW (e.g., compiled in Lodders and Fegley 1998), falls short of the surface heat flux estimated at 30–44 TW (Lee 1970; Hofmeister and Criss 2005; Pollack et al. 1993). This shortfall is suggestive of either a higher than BSE radioactive content of the mantle and/or radioactivity in the core (see Murthy 2006). In the discussion below, new experimental developments are emphasized, citing where necessary the models and calculations that bear upon the radioactivity in the core.

Definition Lithophile Chalcophile Siderophile BSE

P, T TW DK GPa Geoneutrinos

Oxygen fugacity, fO2 IW buffer

Affinity for silicates Affinity for sulfur Affinity for metals Bulk Silicate Earth refers to primitive silicate material in the Earth from which the core is separated. Corresponds to the total mantle + crust system of silicates now. Pressure, temperature Terawatt. 1 TW ¼ 1012 W Concentration of K in Fe-alloy/ Concentration of K in silicate melt. GigaPascal ¼109 Pascals (1 GigaPascal is equivalent to 10 kilobars of pressure) Electron antineutrinos produced inside the Earth due to β decay of naturally occurring radioactive elements in the Earth. A measure of the oxidation state of a system irrespective of the presence or absence of a gas phase containing free oxygen. Iron–Wustite buffer. A synthetic redox mineral reference buffer representing the oxygen fugacity of a system where Femetal is in equilibrium with FeO (wustite), according to the reaction Femetal + ½O2 ¼ FeO (wustite). The IW buffer corresponds to log fO2 ¼ 12.5 at 1,200 °C.

© Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

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Radioactivity in Earth’s Core

Radioactivity in Earth’s Core, Table 1 Important heat-producing radioactive isotopes in the Earth: atomic percentages, decay constants, halflives, and heat production values. (After Van Schmus 1995) Isotope 40

K

235

U U 232 Th 238

Atomic percentage 0.01167 0.7200 99.2743 100.00

Decay constant l (year1) 5.54  1010 9.85  1010 1.551  1010 4.95  1010

Potassium Radioactivity in the Earth’s Core Potassium radioactivity in the Earth’s core has been a topic of discussion for nearly four decades. Purely on geochemical grounds, Hall and Murthy (1971) and Lewis (1971) proposed that K enters the metallic core of the Earth as a consequence of the presence of sulfur in the core (Murthy and Hall 1970). Since then, the implications of K in the Earth’s core for the energetics of the core, convection and core cooling rate, growth rate of the inner core, the geomagnetic field, and thermal evolution of the Earth have been noted in several studies (e.g., Verhoogen 1973; Goettel 1976; Gubbins et al. 1979; Stevenson et al. 1983). Early experiments, however, concluded against the presence of K in the core (Oversby and Ringwood 1972; Ganguly and Kennedy 1977; Murrell and Burnett 1986). These low pressure (1 bar to 1.5 GPa) and low temperature (1,030–1,350 °C) studies yielded very low distribution coefficients, DK for potassium. The laboratory DK values applied to core formation led to the conclusion that K cannot be present in an S-bearing core. Since these first-generation experiments, we have made a substantial progress in understanding the partitioning behavior of elements. For example, we know now that the partitioning of an element between metal and silicate depends on a number of variables such as pressure, temperature, oxygen fugacity (fO2), and the composition of both the metal and silicate melts (Murthy 1991; Hillgren et al. 1996; Jana and Walker 1997; Righter et al. 1997; Gessmann and Rubie 1998, 2000; Gessmann et al. 1999; Li and Agee 2001; Bouhifd and Jephcoat 2003; Chabot et al. 2005). In addition, core formation models suggest metal-silicate partitioning at a pressure range of 30–60 GPa and a temperature of 3,000–4,000 °C (e.g., see Rubie et al. 2003 and Chabot et al. 2005 and the discussion and references cited therein). In the context of these developments, several new experiments have investigated the possibility of K-radioactivity in the core. Chabot and Drake (1999) made the first systematic study of the effect of some of the variables on K partition into metal alloys at 1.5 GPa and 1,900 °C. They showed that DK was dependent on the S content of the metal alloy and on the composition of the silicate melt. Their experiments yielded low values of DK (~6  103) leading to their conclusion that the K content in an S-bearing metallic core will be trivially small (< 1 ppm). The DK values obtained by Chabot and Drake (1999) are lower by an order of magnitude and at

Half-life (years) 1.251  109 7.038  108 4.468  109 1.401  1010

Specific isotopic heat production (mWkg1) 29.17 568.7 94.65 26.38

variance with a number of later studies. This discrepancy may be due to their capsule choice (Gessman and Wood 2002) or due to other experimental artifacts caused by the extreme lability of K under experimental conditions (Murrell and Burnett 1986; Murthy et al. 2003). Studies of K partitioning at high pressure and temperature (2.5–24 GPa and 1,500–1,900 °C) by Gessman and Wood (2002) found a significant entry of K into Fe-S-O alloys. K partition into the metallic liquids depended strongly on S and O contents of metal alloy and increased with temperature. K had little tendency to partition into S-free metal alloys, as was noted in many previous studies. They show clearly that the partitioning behavior of K is strongly influenced by anion/ cation ratio of the Fe-sulfide liquid as well as the depolymerization level of the silicate liquid noted previously by Chabot and Drake (1999). Evaluating the effect of these and other variables on DK, Gessman and Wood (2002) suggest the possibility of 100 ppm of K content in a core containing about 10% S and possibly up to a maximum of ~250 ppm for conditions relevant to core separation from a deep magma ocean. Murthy et al. (2003) found a significant partition of K into Fe–FeS liquids in experiments at 1–3 GPa at temperatures above the liquidus of both metal and silicate phases (1,200–1,800 °C) and at fO2 ~ 1.5 log units below IW buffer, corresponding to core separation conditions in a magma ocean (Walter and Thibault 1995; Chabot and Agee 2003; Chabot et al. 2005; Wade and Wood 2005). DK seemed positively correlated with temperature as noted in previous studies but not correlated with pressure in the pressure range of the experiments. Extrapolating to deep magma ocean conditions and a core with ~10% S, the inferred K content is 60–130 ppm, a little lower but comparable to that inferred by Gessman and Wood (2002). Bouhifd et al. (2007) studied the partition of K between silicates and FeS-rich alloy and pure Fe metal at 5–15 GPa and a temperature of 1,900 °C at an fO2 between 1.5 and 3 log units below IW buffer. DK was clearly positively correlated to S and O contents of the metal alloy and temperature as observed in previous experiments. In the pressure range of their experiments, no pressure dependence of DK was observed both in sulfur-free and sulfur-bearing metal alloys. A parameterization of all existing DK data showed a linear relationship between DK and temperature, a relationship useful to predict DK values at plausible

Radioactivity in Earth’s Core

temperature relevant to core formation if the S and O contents of the core are known. For reasonable assumptions of S and O contents, Bouhifd et al. (2007) found K content of ~250 ppm in the core. If the core is free of S and O, the K content falls as low as ~25 ppm. Partitioning experiments between peridotite silicate liquids and Fe-Ni-S-C-O melts at pressures of 1.0–7.7 GPa and temperatures of 1,650–2,200 °C by Corgne et al. (2007) did not find any relationship between DK and temperature, as observed by Gessman and Wood (2002), Murthy et al. (2003), and Bouhifd et al. (2007). Nor did they find any dependence of DK on the S content of metal alloy, as found in Chabot and Drake (1999), Gessman and Wood (2002), Murthy et al. (2003) and Bouhifd et al. (2007). A compositional control on DK by the O content of the metal alloy was observed leading to their conclusion that a core containing S, C, and about 2 wt.% O could only have up to 25 ppm of K irrespective of the S and C levels. For a possible O content of ~5 wt.% (Badro et al. 2007), the K content of the core is ~80 ppm. However, the discrepancy between this work and the other recent studies is unresolved at present. Quantum mechanical calculations predict a change in electronic structure of K at high pressures (from 4d-like orbitals to 3s-like orbitals), making K behave like a transition element (Bukowinski 1976; Lee et al. 2004). This prediction is experimentally confirmed by the work of Lee and Jeanloz (2003) that showed the alloying of K with a high-pressure polymorph of Fe at pressures >26 GPa and temperature 2,500 K. The change in chemical bonding behavior of K under pressure is noted in several other experiments (Ito et al. 1993; Parker et al. 1996; Hirao et al. 2006; Bouhifd et al. 2007). These experiments and calculations show that K entry into a Fe-metal core is facilitated by high temperatures and high pressures, making it siderophile at extreme conditions. Radioactivity in the core is suggested in a number of studies concerned with the age and size of the inner core, and the energy required to sustain the geomagnetic field. The estimated core–mantle boundary (CMB) heat flux is 6–12 TW (Buffett 2003) and controls the rate of core cooling. Recent studies (Buffett 2003; Nimmo et al. 2004; Costin and Butler 2006) have examined the question of how best to reconcile the CMB heat flux with the size of inner core, the ~3.5 Ga history of the magnetic field, and the CMB temperature. These authors note that the presence of K in the core at ~200–400 ppm is in best accord with the present size of the inner core and the power needs of a geomagnetic dynamo. Using somewhat different parameters, similar conclusions have been reached by others (for example, Labrosse 2003; Buffett 2003; Butler et al. 2005; Costin and Butler 2006). Thus, it appears that a variety of thermal models of the core call for an additional heat source in the core. In view of the

1295

recent experimental data, it is reasonable to attribute this at least in part to the radioactivity of K. However, no precise estimates are possible at present. A more definitive evaluation of the influence of temperature, pressure, and other variables on K partitioning into metal alloys is needed. There is also an uncertainty of the mantle K content at the time of core separation (Murthy 2006) although the convention is to use the model BSE content of ~240 ppm. Nor are the S and O contents of the core known. Given these caveats, a core K content of up to 250 ppm of K seems tenable. The corresponding radioactive heat production in the core, especially with any accompanying U and Th (see discussion below) would be ~2 TW, a significant fraction of the CMB heat flux. However, the uncertainty of these figures cannot be overemphasized. Table 2 summarizes the current estimates of K-radioactivity in the core from standard geochemical models, thermal modeling of the core, and recent high P, T partition experiments.

Radioactivity in Earth’s Core, Table 2 A comparison of inferred values of potassium abundance in the core from geochemical models, theoretical calculations, and recent experiments, and the corresponding heat production in terawatts (TW ¼ 1012 W) today K abundance in core Method (ppm) Geochemical models BSE models 0 E-chondrite Geophysical and Geodynamo models

Experiments Sulfurbearing alloys

Heat production in core (TW) 0

References

550  260 200–400

~4–5 –

McDonough (1999, 2003) Lodders (1995) Buffett (2003)

250–750

–

Labrosse (2003)

Up to 1,420 400 300

9

Roberts et al. (2003)

~3 –

Nimmo et al. (2004) Costin and Butler (2006)

3,000 K suggesting that significant U could be present in the core. In a series of detailed experiments, Murrell and Burnett (1986) measured the partitioning of K, U, and Th between silicate and sulfide liquids at low pressures (1 bar to 1.5 GPa) and found that U and Th tend to be less lithophile under reducing conditions and are markedly chalophile and partition at greater levels than K into Fe–FeS liquids. The high P, T partitioning experiments of U described below yield somewhat ambiguous results. Bao et al. (2005, 2006) investigated the partitioning of U between silicate melt and pure Fe liquids at pressures of 3.0–14.5 GPa and temperatures of 1,660–2,500 °C. They found that partition of U into the metal is dependent on whether the silicates were molten or solid and increases with both P and T. They found a similar P and T dependence of U in Fe-10 wt% S and Fe-35 wt% S liquids, with a positive dependence on S content of the metal phase. Inferred values for U content in core are in the range of 1–4 ppb for pure Fe and at least 10 ppb for 10 wt% S in core. At these latter levels, U alone can produce 1–2 TW of heat energy. Together with K, radioactivity in the core can then account for a significant fraction of the CMB heat flux. In contrast to these observations, Wheeler et al. (2006) found that U entry into core is not supported in their experiments for conditions up to 10 GPa, 2,300 °C and 28 wt% S at fO2 about 2 log units below IW. The discrepancy between these two studies is not understood. Malavergne et al. (2007) found that U partitioning into core metal is dependent on S content and highly reducing conditions (fO2 about 4–5 log units below IW buffer) and at temperatures up to 2,400 ° C and 20 GPa. These experiments permit a range of 0.0003–0.63 ppb U in core depending on whether the metal is S free or S rich. In experiments at temperatures of 2,200–2,400 °C and pressures up to 8 GPa, under very low oxygen fugacity (~4 log units below IW), Murthy et al. (2007) found U partitioning into metal to be positively correlated with S content. The data permit U concentration in the core of 1–2 ppb with a heat production of 0.1–0.3 TW. In a series of papers, Herndon (see for example, Herndon 1998, 2006, and references to his work cited therein) has argued that gravitationally segregated U in the core functions

Radioactivity in Earth’s Core

as a nuclear reactor to provide substantial energy. In order to segregate uranium into the core, the core–mantle differentiation needs to have occurred at fO2 of about 4–5 log units below IW buffer (Malavergne et al. 2007). Such a scenario would have reduced most iron in the planet to metallic form, leaving little oxidized iron in the mantle. This is not supported by the fact that the mantle contains ~8% of oxidized iron. The mantle would also have been strongly depleted in Ta under such a low oxidation, which is not the case (Mann et al. 2009). Current ideas of core formation based on siderophile element partitioning in the mantle suggest a much higher oxygen fugacity of about 2 log units below IW. Schuiling (2006) discusses other difficulties of the Herndon proposal.

Conclusions There have been many theoretical discussions of radioactivity in the Earth’s core in the past. But it is the new high P, T experiments of metal-silicate partitioning of K and U that urge a serious consideration of radioactivity in the Earth’s core. The experimental data could permit up to 250 ppm of K and up to 10 ppb of U in the core at the upper end. The corresponding heat production due to K and U (and associated Th) could be ~2–4 TW, a significant fraction of the CMB heat flux. As mentioned in the section on K-radioactivity, the uncertainty in these numbers is very large. It may not be possible to precisely determine the radioactivity of the core by currently available geochemical and geophysical analyses except to note the general possibility of radioactivity in the core. Away out of this impasse may now be available. Radioactive β decays of K, U, and Th in the interior of the Earth produce electron antineutrinos ðve Þ. These are termed geoneutrinos, and can serve as probes to directly measure the radioactivity of the Earth’s mantle and core (e.g., Raghavan et al. 1998; Rothschild et al. 1998; Mantovani et al. 2004; Enomoto et al. 2005; Fiorentini et al. 2005; Giammarchi and Miramonti 2006). Geoneutrino flux measurements use the antineutrino inverse β-decay reaction with protons, ve þ p ! eþ þ n in an appropriate liquid in a scintillator chamber. The resulting positron and neutron emissions are then measured by usual scintillation techniques. Geoneutrinos from each radioactive isotope can be distinguished by their characteristic energy spectrum. Neutrino detectors exist now in several countries, and new ones less susceptible to interference by nuclear reactors and other near surface radioactive sources are planned. These developments in neutrino geophysics should be able to measure the U and Th abundances in the deep Earth and the core. Present detection techniques do not allow measurement of K due to the lower energy of its antineutrinos compared to those from U and Th decay. Further developments are needed for K determination. Geoneutrino measurements are likely to

Radioactivity in Earth’s Core

provide a definitive answer about the level of radioactivity in the core and in addition clarify several questions regarding the Earth’s formation, differentiation, chemical composition, heat budget, and geodynamics (Sleep 2006).

Cross-References ▶ Core Dynamo ▶ Core-Mantle Coupling ▶ Energy Budget of the Earth ▶ Geodynamics ▶ Mantle Convection ▶ Radiogenic Heat Production of Rocks

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1297 Furst MJ, Stapanian MI, Burnett DS (1982) Observation of nonlithophile behavior for U. Geophys Res Lett 9:41–44 Ganguly J, Kennedy GC (1977) Solubility of K in Fe-S liquid, silicateK-(FeS)liq equilibria, and their planetary implications. Earth Planet Sci Lett 35:411–420 Gessman CK, Wood BJ (2002) Potassium in the Earth’s core? Earth Planet Sci Lett 200:63–78 Gessmann CK, Rubie DC (1998) The effect of temperature on the partitioning of nickel, cobalt, manganese, chromium, and vanadium at 9GPa and constraints on formation of the Earth’s core. Geochim Cosmochim Acta 62:867–882 Gessmann CK, Rubie DC (2000) The origin of the depletions of V, Cr, Mn in the mantles of the Earth and Moon. Earth Planet Sci Lett 184:95–107 Gessmann CK, Rubie DC, McCammon CA (1999) Oxygen fugacity dependence of Ni, Co, Mn, Cr, V, and Si partitioning between liquid metal and magnesiow¨ustite at 9–18 GPa and 2200°C. Geochim Cosmochim Acta 63:1853–1863 Giammarchi MG, Miramonti L (2006) Borexino: Geoneutrinos in Borexino. Earth Moon Planet 99:207–220 Goettel KA (1976) Models for the origin and composition of the Earth, and the hypothesis of potassium in the Earth’s core. Geophys Surv 2:369–397 Gubbins D, Masters TG, Jacobs JA (1979) Thermal evolution of the Earth’s core. Geophys J R Astron Soc 59:57–99 Hall HT, Murthy VR (1971) The early chemical history of the Earth: some critical elemental fractionations. Earth Planet Sci Lett 11:239–244 Herndon JM (1998) Composition of the deep interior of the Earth: divergent geophysical development with fundamentally different geophysical implications. Phys Earth Planet Inter 105:1–4 Herndon JM (2006) Solar system processes underlying planetary formation, geodynamics, and the georeactor. Earth Moon Planet 99:53–89 Hillgren VJ, Drake MJ, Rubie DC (1996) High pressure and high temperature metal-silicate partitioning of siderophile elements: the importance of silicate liquid composition. Geochim Cosmochim Acta 60:2257–2263 Hirao N, Ohtani E, Kondo T, Endo N, Kuba T, Suzuki T, Kikegawa T (2006) Partitioning of potassium between iron and silicate at the core-mantle boundary. Geophys Res Lett 33:L08303. https://doi.org/ 10.1029/2005GL025324 Hofmeister AM, Criss RE (2005) Earth’s heat flux revised and linked to chemistry. Tectonophysics 395:159–177 Ito E, Morooka K, Ujike O (1993) Dissolution of K in molten iron at high pressure and temperature. Geophys Res Lett 20:1651–1654 Jana D, Walker D (1997) The influence of silicate melt composition on distribution of siderophile elements among metal and silicate liquids. Earth Planet Sci Lett 150:463–472 Javoy M (1995) The integral enstatite chondrite model of the Earth. Geophys Res Lett 22:2219–2222 Javoy M (1999) Chemical Earth models. In: Earth and planetary sciences, vol 329. C. R. Academy Science, pp 537–555 Labrosse S (2003) Thermal and magnetic evolution of the Earth’s core. Phys Earth Planet Inter 140:127–143 Labrosse S, Macouin M (2003) The inner core and the geodynamo. Compt Rendus Geosci 335:37–50 Labrosse S, Poirier J-P, Le Mouel J-L (2001) The age of the inner core. Earth Planet Sci Lett 190:111–123 Lee WHK (1970) On the global variations of terrestrial heatflow. Phys Earth Planet Inter 2:332–341 Lee KKM, Jeanloz R (2003) High-pressure alloying of potassium and iron: radioactivity in the Earth’s core? Geophys Res Lett 30:2212. https://doi.org/10.1029/2003GL018515 Lee KKM, Steinle-Neumann G, Jeanloz R (2004) Abinitio high-pressure alloying of iron and potassium: implications for the Earth’s core. Geophys Res Lett 31:L11603. https://doi.org/10.1029/ 2004GL019839, 2004.

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1298 Lewis JS (1971) Consequences on the presence of sulfur in the core of the Earth. Earth Planet Sci Lett 11:130–134 Li J, Agee CB (2001) The effect of pressure, temperature, oxygen fugacity and composition on partitioning of nickel and cobalt between liquid Fe-Ni-S alloy and liquid silicate: implications for the Earth’s core formation. Geochim Cosmochim Acta 65:1821–1832 Lodders K (1995) Alkali elements in the Earth’s core: evidence from enstatite chondrites. Meteoritics 30:93–101 Lodders K, Fegley BJ Jr (1998) The planetary scientist’s companion. Oxford University Press, Oxford Malavergne V, Tarrida M, Combes R, Bureau H, Jones J (2007) New high-pressure and high-temperature metal/silicate partitioning of U and Pb: implications for the cores of the Earth and Mars. Geochim Cosmochim Acta 71:2637–2655 Mann U, Frost DJ, Rubie DC (2009) Evidence for high-pressure coremantle differentiation from the metal-silicate partitioning of lithophile and weakly-siderophile elements. Geochim Cosmochim Acta 73:7360–7386 Mantovani F, Carmignani GL, Fiorentini G, Lissia M (2004) Antineutrinos from the Earth: a reference model and its uncertainties. Phys Rev D 69:297–314 McDonough WF (1999) Earth’s core. In: Marshall CP, Fairbridge RW (eds) Encyclopedia of geochemistry. Kluwer Academic, Dordrecht McDonough WF (2003) Compositional model for the Earth’s core. In: Carlson RW (ed) Treatise on geochemistry. The mantle and the core, vol 2. pp 547–568 Murrell MT, Burnett DS (1986) Partitioning of K, U, and Th between sulfide and silicate liquids: implications for radioactive heating of planetary cores. J Geophys Res 91:8126–8136 Murthy VR (1991) Early differentiation of the Earth and the problem of mantle siderophile elements: a new approach. Science 253:303–306 Murthy VR (2006) Radioactivity of the Earth and the case for potassium in the Earth’s core. Earth Moon Planet 99:23–32 Murthy VR, Hall HT (1970) The chemical composition of the Earth’s core: possibility of sulphur in the core. Phys Earth Planet Inter 2:276–282 Murthy VR, van Westrenen W, Fei Y (2003) Experimental evidence that potassium is a substantial radioactive heat source in planetary cores. Nature 423:163–165 Murthy VR, Draper D, Agee C (2007) Uranium in the Earth’s core? Metal-silicate partitioning of Uranium at High Pressure and Temperature and Highly Reducing Conditions. In: Workshop on early planetary differentiation. Lunar Planetary Institute Contribution 1355, pp 78–79 Nimmo F, Price GD, Brodholt J, Gubbins D (2004) The influence of potassium on core and geodynamo. Geophys J Int 156:363–376 Oversby VM, Ringwood AE (1972) Potassium distribution between metal and silicate and its bearing on the occurrence of potassium in the Earth’s core. Earth Planet Sci Lett 14:345–347 Parker LJ, Atou T, Badding JV (1996) Transition element-like chemistry for potassium under pressure. Science 273:95–97 Pollack HN, Hurter SJ, Johnson JR (1993) Heat flow from the Earth’s interior: analysis of the global data set. Rev Geophys 31:267–280 Raghavan RS, Schoenert S, Enomoto S, Shirai S, Suekane F, Suzuki A (1998) Measuring the global radioactivity in the Earth by multidetector antineutrino spectroscopy. Phys Rev Lett 80:636–638 Righter K, Drake MJ, Yaxley G (1997) Prediction of siderophile element metal/silicate partition coefficients to 20 GPa and 2800°C: the effects of pressure, temperature, oxygen fugacity, and silicate and metallic melt composition. Phys Earth Planet Inter 100:115–134 Roberts PH, Jones CA, Calderwood AR (2003) Energy fluxes and Ohmic dissipation in the Earth’s core. In: Jones CA, Soward AM, Zhang K (eds) Earth’s core and lower mantle. Taylor, London Rothschild CG, Chen MC, Calaprice FP (1998) Antineutrino geophysics with liquid scintillation detectors. Geophys Res Lett 25:1083–1086

Radiogenic Heat Production in the Continental Crust Rubie DC, Melosh HJ, Reid JE, Liebske C, Righter K (2003) Mechanisms of metal–silicate equilibration in the terrestrial magma ocean. Earth Planet Sci Lett 205:239–255 Schuiling RD (2006) Is there a nuclear reactor at the center of the Earth? Earth Moon Planet 99:33–49 Sleep NH (2006) Strategy for applying neutrino geophysics to the Earth sciences including planetary habitability. Earth Moon Planet 99:343–358 Stevenson DJ, Spohn T, Schubert G (1983) Magnetism and thermal evolution of the terrestrial planets. Icarus 54:466–489 Van Schmus WR (1995) Natural radioactivity of the crust and mantle. In: Ahrens TJ (ed) Global Earth physics: a handbook of physical constants, AGU reference shelf 1. American Geophysical Union, Washington, DC, pp 283–291 Verhoogen J (1973) Thermal regime of the Earth’s core. Phys Earth Planet Inter 7:47–58 Wade J, Wood BJ (2005) Core formation and the oxidation state of the Earth. Earth Planet Sci Lett 236:78–95 Walter MJ, Thibault Y (1995) Partitioning of tungsten and molybdenum between metallic liquid and silicate melt. Science 270:1186–1189 Wheeler KT, Walker D, Fei Y, Minarik W, McDonough W (2006) Experimental partitioning of uranium between liquid iron sulfide and liquid silicate: implications for radioactivity in the Earth’s core. Geochim Cosmochim Acta 70:1537–1547

Radiogenic Heat Production in the Continental Crust Claude Jaupart1 and Jean-Claude Mareschal2 1 Université de Paris Institut de Physique du Globe, Paris, France 2 Centre GEOTOP-UQAM, University of Québec, Montréal, QC, Canada

Definition The crust is enriched in heat producing elements (HPE). Steady-state crustal temperature depends on Moho heat flux and the vertical distribution of HPE. Crustal differentiation with upper crustal enrichment in HPE cools and stabilizes the crust and makes it mechanically stronger.

Crustal Thickness and Heat Production On a global average, radiogenic heat production in the continental crust accounts for about two-thirds of the surface heat flow in geological provinces that are in thermal equilibrium, that is, 30–40 mW m2 over a total of 45–60 mW m2 (Jaupart and Mareschal 2015). Save for a minor contribution from heat sources in the mantle lithosphere, the other component of the surface heat flow is heat supply from the convecting mantle. The key difference between these two components is in their time-dependence. The former (crustal heat production) decays with time due to the rundown of the

Radiogenic Heat Production in the Continental Crust

1299

radioactive elements. The latter (mantle heat flux) does not vary much because continental lithosphere has remained thick over its whole history. As a consequence, the secular cooling of the mantle, which amounts to about 200 K over a total of 1600 K (see ▶ “Energy Budget of the Earth”), cannot induce large changes of heat flux at the base of the lithosphere. Thus, the secular evolution of the continental thermal structure is by and large a function of that of its crustal heat sources. This is consistent with the observed decrease of heat production with age (Fig. 1). In steady state, the surface heat flux q0 is the sum of the heat flux at the base of the crust qm and the total crustal heat production. ð q0 ¼ qm þ AðzÞdz

ð1Þ

with A(z) the heat production. In stable continents, several independent methods indicate that the Moho heat flux lies within a narrow range of 12–18 mW m2 (Jaupart and Mareschal 2015), implying that variations of surface heat flux are mostly due to differences in crustal heat production. If the mantle heat flux qm and the crustal thickness hm can be determined independently, one can estimate the bulk average heat production of the crust, noted Ac, as follows:

1.5 1.4

Ac ¼

qo  qm hm

ð2Þ

The thickness of the continental crust varies within a 20–60 km range (Fig. 2). Thus, assuming that the average crustal heat production is constant, one should observe large variations of the surface heat flux. This is clearly not the case (Fig. 3). For the global data sets, the total lack of correlation between heat flow and crustal thickness is partly due to transient thermal effects in tectonically perturbed provinces. A recent and accurate data set in the Precambrian Canadian Shield, which is in thermal steady-state, however, does not reveal any significant correlation between the two variables, implying that the thicker the crust, the lower the mean crustal heat production. One consequence is that one must determine the amount of crustal heat sources on a case-by-case basis. Calculating the total crustal heat production is relatively straightforward with sufficient surface heat flow coverage and even approximate knowledge of lithosphere thickness.

Crustal Differentiation Constraining the vertical distribution of heat sources is fraught with a larger uncertainty than that for the average. Various estimates concur that heat producing elements are vertically differentiated with enrichment of the upper crust (Table 1). From knowledge of the average value of heat production in surface rocks, AS, one can calculate a “differentiation index” (Perry et al. 2006):

1.3

μW m−3

1.2

7000

1.1 6000

1.0 0.9

R

5000 μ = 35.6 km

0.8 4000

3000

0.6 0.5 0.0

σ = 7.0 km

N

0.7

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Crustal age (Ga) Radiogenic Heat Production in the Continental Crust, Fig. 1 Average crustal heat production as a function of age, from Jaupart and Mareschal (2014). The large width of the age groups is due to the large spread of heat production values at any given age, which does not allow a fine scale separation. The curve illustrates the rundown of heat producing elements due to radioactive decay. After correction for this rundown, the crustal heat production at the time of crustal stabilization is essentially constant

2000

1000

0 10

20

30

50 40 thickness (km)

60

70

80

Radiogenic Heat Production in the Continental Crust, Fig. 2 Histogram of continental crust thicknesses sampled at 1°  1° from Laske et al. (2013). Most of values >50 km or < 30 km are in active regions. m and s are the mean and standard deviation

1300

Radiogenic Heat Production in the Continental Crust

If the lower layer has zero heat production, an extreme case of differentiation, the upper layer thickness is:

80

Q (mWm−2)

(a)

h¼

60

ð4Þ

and the crustal component of mantle temperature is: DT c ¼

40

1 Ac h2m DI 2l

ð5Þ

with l is thermal conductivity assumed constant. A more general two-layer model has nonzero heat production in the lower layer and an upper layer of thickness h  hm/ DI. In this case:    Ac hm ðhm þ hÞ D I Ac h 2 DI h DT c ¼ 1 þ 2l hm 2l

20

r=−0.24 20

30

40

50

60

70

80

60

Q (mW m−2)

hm DI

(b)

50 40 30

ð6Þ

This equation is useful to illustrate the effect of differentiation but precise calculations must include the temperature dependence of l which may vary by a factor of 1.5 over the crustal temperature range. A value of DI > 1 always results in lower value for the crustal component of Moho temperature than for an undifferentiated crust. In general, values of DI for different provinces are largest when heat flow is highest (Table 2). Values less than 1 are found in low heat flow subprovinces where the uppermost crustal layer has been emplaced tectonically.

r=0.03 Crustal Temperature and Strength

20 20

30

40

50

60

70

80

thickness (km) Radiogenic Heat Production in the Continental Crust, Fig. 3 Scatter plot of heat flux and crustal thickness. (a) Global data set comparing heat flux averaged over 1°  1° cells vs. crustal thickness from CRUST1.0. The solid line represents the best linear “fit” to the cloud of points (with a correlation coefficient r ¼ 0.24). (b) Data from eastern Canada. Heat flux data have been averaged over 1°  1° cells and their values are plotted against the corresponding crustal thickness values from Lithoprobe (Perry et al. 2002) and recent receiver function studies (Fiona Darbyshire, pers. comm.). The plot shows no significant trend and the correlation coefficient between the two data sets r ¼ 0.03

DI ¼

AS AS  h m ¼ Ac qo  q m

ð3Þ

where qo is a regional average. DI is equal to 1 if the crust is undifferentiated and is larger than 1 in most geological provinces because the upper crust is enriched compared to the bulk.

For a 40 km thick undifferentiated crust, lower crustal temperatures are near the melting temperature of the rocks at the time of crustal formation. Thermal stability of the crust can be achieved by crustal differentiation and remobilization of heat producing elements in the upper crust. For a differentiated crust, the temperature at the base of the crust increases with the thickness of the enriched layer h and decreases with increasing DI, but the two parameters are not independent as shown by Eq. 3. In the lower crust, melting starts when temperature reaches ≈800 °C. During the Archean with crustal heat production ≈1.5 mW m3, lower crustal temperatures always exceeded melting conditions for a >40 km thick crust. But a crust up to 50 km thick could stay below melting if it is sufficiently differentiated (DI  2) (Fig. 4). The process is well illustrated for the Appalachians and the ca. 400 My Acadian orogeny, which saw the emplacement of enriched plutons in the upper crust both during and after the orogeny (Table 3). In this case, the present-day distribution of heat sources is not relevant to the Acadian thermal conditions.

Radiogenic Heat Production in the Continental Crust

1301

Radiogenic Heat Production in the Continental Crust, Table 1 Different estimates of heat production in the continental crust (in mW m3). For each crustal model with the exception of the North Model Global geochemical models Rudnick and Gao (2014) Huang et al. (2013) Hacker et al. (2015)a Model A Model B Model C Model D Exposed cross-sections North American cordillera Lee et al. (2007)b Kohistan, Pakistan Jagoutz and Schmidt (2012)c Model 1 Model 2 Model 3 Heat flow data Jaupart and Mareschal (2014)

American Cordillera, the first line lists the thicknesses of the crustal layers and the total crustal thickness

Upper crust

Middle crust

Lower crust

Bulk

12 km 1.6 13 km 1.6 13.7 km 1.58 1.58 1.58 1.58

11 km 0.96 11 km 0.73 13 km 0.35 0.34 0.46 0.72

17 km 0.18 10 km 0.17 12.1 km 0.21 0.17 0.26 0.33

40 km 0.89 34 km 0.94 38.8 km 0.74 0.72 0.80 0.90

1.67

0.99

0.2

0.88

/ / / /

/ / / /

≈25 km 0.18 0.11 0.06

≈55 km 0.69 0.50 0.58 0.79–0.95

a

Models A–D correspond to different end-member compositions that are compatible with geophysical characteristics b No thicknesses for the three crustal layers are reported c Models 1–3 correspond to slightly different thicknesses and the inclusion or exclusion of the Chilas ultramafic-mafic complex & / No data for two different crustal layers at the top of the lower crust

Radiogenic Heat Production in the Continental Crust, Table 2 Average surface heat flux, qo , average crustal heat production, A, crustal thickness, hm, and differentiation index, DI, (Eq. 3) for different provinces Province Slave province, Can. Superior craton core, Can. Superior accreted terranes, Can. Wawa subprovince, Superior, Can. Abitibi subprovince (all), Superior, Can. Trans-Hudson Orogen, Can. Flin-Flon Snow Lake Belt (THO), Can. Wopmay Orogen, Can. Central Shield, Aust. Eastern Gawler craton, Aust. Grenville, Can. Namaqua, S. Africa Appalachians, N. Am.

Age (Gy) 3.1 >2.7 2.7 2.7 2.7 2.1–1.8 1.9–1.8 1.8 1.8 1.6 1.3–1.1 1.05 0.4

qo  s (mW m2)

A  s (mW m3)

51 31.8  5.2 41.0  8.7 45.1  8.0 39.9  7.0 42  11 40  5 90  15 72  24 78  19 41  11 61  11 57  13

2.0 0.78  0.37 0.8  0.8 0.85  0.79 0.5  0.4 0.7  0.5 0.32  0.2 4.8 3.6  1.9 5.0 0.8 2.3 2.6  1.9

hm (km) 36 40 40 40 38 40 40 32 35 40 40 43 40

DI 2.0 2.0 1.0 1.0 0.7 1.1 0.5 2.0 3.2 2.4 1.0 2.0 2.5

Reference (1) (2) (2) (3) (3) (2) (3) (3, 4) (5, 6) (5, 6) (3) (6, 7) (3)

References: (1) Perry et al. (2006), (2) Jaupart et al. (2014), (3) Perry et al. (2010), (4) Lewis et al. (2003), (5) Neumann et al. (2000), (6) Mareschal and Jaupart (2013), (7) Jones (1987, 1992)

Continental lithosphere thickened on a spatial scale of >100 km has higher potential energy relative to the surrounding and experiences a tensile stress regime. Strength of

the thickened lithosphere depends on its rheology which, below the brittle region, is controlled by temperature and composition. For standard values of the rheological

R

1302

Radiogenic Heat Production in the Continental Crust

Radiogenic Heat Production in the Continental Crust, Fig. 4 Moho temperature variations in function of crustal thickness and differentiation index DI for a mean crustal heat production 1.5 mm3 representing Archean conditions. The differentiated upper crust is 10 km thick. The Moho heat flux is assumed to be 15 mW m2

o

Moho temperature ( C) 11 00

10 00

0

13

0

00

14

00

D

I

12

00

10

11

00

0 90

80

60

70

0

0

0

50

2

00

12 00

90

80 0

60 0

70 0

2.5

00

14

00

12

00

11

00 10

0 90

700

800

13

00

1.5

1 35

40

45

50

55

60

Crustal thickness (km)

Radiogenic Heat Production in the Continental Crust, Table 3 Heat production data for the Appalachian province, USA Large-scale surface average Shalesa Syntectonic plutons (410–390 Ma)b Post-tectonic plutons (360 Ma)c Anorogenic granites (ca. 180 Ma)d

A (mW m3) 2.6  0.3 2.15–2.37 1.8–2.2 4.0 8.6

Reference (1) (2) (3, 4, 5) (6) (6)

References: (1) Jaupart and Mareschal (2014), (2) Della Vedova and Von Herzen (1987), (3) Chamberlain and Sonder (1990), (4) Lyons (1964), (5) Jaupart et al. (1982), (6) Roy et al. (1968) a Sediments from the US Atlantic continental margin (COST B-2 and B-3 boreholes) b New Hampshire plutonic suite (three major plutons: Kinsman, Spaulding, and Bethlehem gneiss) c Concord two mica granites (several plutons scattered throughout New England) d White Mountain anorogenic plutonic series

parameters, lithospheric strength can be calculated and compared with potential energy difference. Such calculations show that: (1) for an undifferentiated crust with average crustal heat production, the lithosphere cannot withstand the stress of more than a few km crustal thickening, and (2) with increased DI, the lithosphere can maintain greater differences in crustal thickness without collapsing (Fig. 5) (Jaupart et al. 2016).

Summary: Crustal Stabilization In steady state, on average, temperatures in the continental crust increase with depth due to two heat flow components (crustal heat production and heat input from the mantle). It turns out that these two components account for about the same temperature rise at Moho depth. The influence of crustal heat sources is large in young provinces with high heat flow, and it happens to also be large in older ones when one accounts for radioactive decay. It also becomes large in continental collision zones with thickened crust. Over the time span of an orogenic event, 100 My, say, crustal heat production overwhelms the effect of magmatic intrusions, which are short-lived. Thus, thermal models of continents are very sensitive to crustal heat sources. In addition, the vertical distribution of crustal heat sources changes during orogenic events and is yet another variable which must be treated with care. A particularly important fact is that the distribution that is observed today is a consequence of orogeny and hence is not representative of conditions during the orogenic event itself. Usually, an orogenic event leads to the generation of crustal melts that rise toward the surface and leave a depleted residue. These processes act to generate a thermally stable and mechanically strong crust.

Radiogenic Heat Production in the Continental Crust

strength(TNm−1)

15

1303

undiff

(a)

diff ΔU

10

5

0 40

50

60

70

20

undiff

strength(TNm−1)

(b)

diff

15

ΔU

10

5

0 40

50

60

70

crustal thickness (km) Radiogenic Heat Production in the Continental Crust, Fig. 5 Total strength of lithosphere vs. crustal thickness. The strength is calculated for DI ¼ 1 (undifferentiated) and DI ¼ 2.5 (differentiated crust). The potential energy difference relative to 40 km thick crust (ΔU) is given for comparison. (a) The average crustal heat production is 1.5 mW m3 (i.e., the average heat production in cratons at the end of Archean). (b) The average crustal heat production is 1.1 mW m3 (i.e., the average heat production of the Appalachians at the time of the Acadian orogeny)

Cross-References ▶ Energy Budget of the Earth ▶ Heat Flow, Continental ▶ Lithosphere, Continental: Thermal Structure ▶ Lithosphere, Mechanical Properties ▶ Radiogenic Heat Production of Rocks

Bibliography Chamberlain CP, Sonder LJ (1990) Heat-producing elements and the thermal and baric patterns of metamorphic belts. Science 250:763–769

Della Vedova B, Von Herzen RP (1987) Geothermal heat flux at the COST B-2 and B-3 wells, U.S. Atlantic continental margin. Technical report WHOI-87-27, Woods Hole Oceanographic Institution Technical Report Hacker BR, Kelemen PB, Behn MD (2015) Continental lower crust. Annu Rev Earth Planet Sci 43(6):1–6.39 Huang Y, Chubakov V, Mantovani F, Rudnick RL, McDonough WF (2013) A reference earth model for the heat-producing elements and associated geoneutrino flux. Geochem Geophys Geosyst 14:2003–2029 Jagoutz OE, Schmidt M (2012) The formation and bulk composition of modern juvenile continental crust: the Kohistan arc. Chem Geol 298–299:79–96 Jaupart C, Mareschal JC (2014) Constraints on crustal heat production from heat flow data. In: Rudnick RL (ed) Treatise on geochemistry, the crust, vol 4, 2nd edn. Elsevier, New York, pp 53–73 Jaupart C, Mareschal JC (2015) Heat flow and thermal structure of the lithosphere. In: Watts AB (ed) Treatise on geophysics, vol 6, The lithosphere, 2nd edn. Elsevier, Oxford, pp 217–253 Jaupart C, Mann JR, Simmons G (1982) A detailed study of the distribution of heat flow and radioactivity in New Hampshire (U.S.A.). Earth Planet Sci Lett 59:267–287 Jaupart C, Mareschal J-C, Bouquerel H, Phaneuf C (2014) The building and stabilization of an Archean Craton in the Superior Province, Canada, from a heat flow perspective. J Geophys Res Solid Earth 119:9130–9155 Jaupart C, Mareschal J-C, Iarotsky L (2016) Radiogenic heat production in the continental crust. Lithos 262:398–427 Jones MQW (1987) Heat flow and heat production in the Namaqua mobile belt, South Africa. J Geophys Res Solid Earth 92:6273–6289 Jones MQW (1992) Heat flow anomaly in Lesotho: implications for the southern boundary of the Kaapvaal craton. Geophys Res Lett 19:2031–2034 Laske G, Masters G, Ma Z, Pasyanos M (2013) Update on CRUST1.0 – a 1-degree global model of earth’s crust. In: EGU general assembly conference abstracts. Abstract EGU2013-2658 presented at 2013 Geophys. Res. Abstracts 15, vol 15. p 2658 Lee C-TA, Morton D, Kistler RW, Baird AK (2007) Petrology and tectonics of phanerozoic continent formation: from island arcs to accretion and continental arc magmatism. Earth Planet Sci Lett 263:370–387 Lewis TJ, Hyndman RD, Fluck P (2003) Heat flow, heat generation and crustal temperatures in the northern Canadian Cordillera: thermal control on tectonics. J Geophys Res Solid Earth 108:2316 Lyons JB (1964) Distribution of thorium, uranium in three early Paleozoic plutonic suites of New Hamphsire. Technical report 1144-F, U.S. Geological Survey Bulletin Mareschal J-C, Jaupart C (2013) Radiogenic heat production, thermal regime and evolution of continental crust. Tectonophysics 609:524–534 Neumann N, Sandiford M, Foden J (2000) Regional geochemistry and continental heat flow: implications for the origin of the South Australian heat flow anomaly. Earth Planet Sci Lett 183:107–120 Perry HKC, Eaton DWS, Forte AM (2002) A revised crustal model for Canada based on Lithoprobe results. Geophys J Int 150:285–294 Perry HKC, Jaupart C, Mareschal JC, Bienfait G (2006) Crustal heat production in the Superior Province, Canadian Shield, and in North America inferred from heat flow data. J Geophys Res Solid Earth 111:B04401 Perry H, Rosieanu C, Mareschal JC, Jaupart C (2010) Thermal regime of the lithosphere in Canada. Can J Earth Sci 47:389–408 Roy RF, Blackwell DD, Birch F (1968) Heat generation of plutonic rocks and continental heat flow provinces. Earth Planet Sci Lett 5:1–12 Rudnick RL, Gao S (2014) Composition of the continental crust. In: Rudnick RL (ed) Treatise on geochemistry, vol 4, 2nd edn. Elsevier, New York, pp 1–51

R

1304

Radiogenic Heat Production of Rocks rðE

Radiogenic Heat Production of Rocks Christoph Clauser Institute for Applied Geophysics and Geothermal Energy, RWTH Aachen University, Aachen, Germany

Ep ¼

rðE

dEp ¼ 0

0

Gm 16p2 r2 G dm ¼ r 3

rðE

r4 dr

0

 3 2 4prE r 3G 3 GM2E 16p r G r ¼ ¼ , ¼ 3 5rE 5 rE 3 5 |fflfflfflfflfflffl{zfflfflfflfflfflffl} 2 2

5

ð1Þ

ME

Synonyms Radiogenic heat generation rate

Definition Radiogenic heat production Geoneutrino

eV (electron Volt)

ppm (parts per million)

Physical property defining the amount of heat liberated in unit time in a unit volume of rock by the decay of unstable radiogenic isotopes; dimension: W m3. An electron antineutrino emitted in β-decay of nuclei during radiogenic heat production caused by the decay of the unstable isotopes 238 U, 232Th, and 40K. A non-SI unit of energy in nuclear physics, defined as the kinetic energy gained by an electron of elementary charge when accelerating through an electric potential difference of 1 V. Thus, one electron Volt equals one Volt, which is one Joule per Coulomb, multiplied by the electron charge of e ¼ 1.602176487(40)  1019 C. Therefore, 1 eV ¼ 1.602176487  1019 J. A non-SI unit of relative frequency (or abundance) in 106, similar to % (percent) or ‰ (per mil) in 102 and 103, respectively.

Radiogenic Heat Generation The main interior sources of heat in the Earth are the heat content of the infant Earth immediately after formation due to gravitational contraction and the decay of unstable, radioactive isotopes. The potential energy released as heat during the gravitational accretion of the infant Earth dwarfs all other heat sources. This energy is calculated from gravitational contraction. The mass accumulated in a spheroidal planet of constant density r and radius r equals m ¼ (4π/3) r3r. Addition of a further shell of thickness dr and mass dm ¼ 4π r2r dr results in the release of additional potential gravitational energy dEp ¼ G m dm/r ¼ (16/3) π2 G r2 r4 dr, where G ¼ 6673  1011 m3 kg1 s2 is the universal gravitational constant. Integration yields:

where ME ¼ 5.976  1024 kg and rE ¼ 6.371  106 m are mass and radius of the Earth, respectively. Accordingly, the accretion of the Earth results in the release of heat on the order of Ep ¼ 2.24  1032 J. A refined analysis considering the radial variation of density in the Earth modifies this value by only about 10 % to Ep ¼ 2.49  1032 J (Stacey and Davis 2008). An additional, similarly large energy input is assumed to have occurred in the wake of an impact into the young infant Earth by a Mars-sized protoplanet orbiting the sun close to the Earth’s orbit. The terrestrial Moon is generally believed to be the result of this impact, which released energy on the order of 1031 J (Melosh 1990; Canup and Righter 2000; Canup and Asphaug 2001). However, most of the vast energy liberated during accretion of the Earth or by this impact was radiated back into space already during accretion or immediately after the impact (Stacey and Davis 2008). For an average specific heat capacity of 1 088 J kg1 K1 and a cooling by 650 K over a lifetime of 4.6 billion years, the mean rate by which the Earth is losing its original heat is 29 TW (Vacquier 1991, 1992). This corresponds to a thermal energy of 4.2  1030 J. As the heat loss of the young and hot infant Earth was larger than today, the contribution of original heat to today’s heat loss is certainly smaller. A larger cooling by 1175 K would result in a mean cooling rate of 52 TW corresponding to a thermal energy of 7.6  1030 J (Stacey and Davis 2008). The annual production of radiogenic heat in the Earth, 6.3  1020 J (Jaupart et al. 2007), corresponds to more than twice the global production of primary energy in the year 2000. This huge energy source by itself clearly exceeds the world’s energy annual demands predicted through the year 2030 (IEA 2008). If it were used at great scale, it may satisfy a large proportion of the primary energy demand of the entire twenty-first century. Apart from the heat content of the infant Earth immediately after formation, the radiogenic decay of the unstable isotopes of uranium (238U, 235U), thorium (232Th), and potassium (40K) provides the largest internal source of heat. Most of these isotopes are enriched in the Earth’s crust and mantle (Table 1). During radioactive decay, mass is converted into energy. Except for the tiny amount associated with the antineutrinos and neutrinos generated in β- and β+-decay or electron capture, respectively, all of this energy is converted into heat. Certain peaks in the corresponding γ-spectra are characteristic for the different decay series while the continuous background spectrum is due to Compton scattering and photoelectric absorption.

Radiogenic Heat Production of Rocks

1305

Radiogenic Heat Production of Rocks, Table 1 Average radiogenic heat generation rate per mass, A0 , in geologic materials. (Data: McDonough and Sun 1995; Stacey and Davis 2008; Jaupart et al. 2007)

Material Igneous rocks

Meteorites Moon Earth

Granites Alkali basalt Tholeiitic basalt Eclogite Peridotite, dunite Carbonaceous chondrite Apollo samples Average crust (2.8  1022 kg) Average mantle (4.0  1024 kg) Average core Bulk silicate Earth (BSE)

Concentration (ppm by mass) CU CTh 4.6 18 0.75 2.5 0.11 0.4 0.035 0.15 0.006 0.02 0.007 4–0.008 0 0.029–0.030 0.23 0.85 1.2–1.3 4.5–5.6 0.013–0.025 0.040–0.087 0 0 0.020  20% 0.081  1 5%

The two uranium isotopes, 238U and 235U, decay into lead, 206 Pb and 207Pb, with a half-life of 4.5 and 0.71 billion years, respectively. Characteristic lines for the γ-spectrum of the 238U decay series, for example, are produced by the daughter element bismuth, 214Bi, at 609 keV, 1 120 keV, and 1 764 keV. Uranium is an abundant mobile trace element in many rocks. With a half-life of 14.1 billion years, thorium 232Th decays also into lead 208Pb. Characteristic lines for the γ-spectrum of the thorium decay series are produced by the daughter elements thallium, 208Tl, or actinium, 228Ac, at 584 keV and 2 615 keV or 912 keVand 966 keV, respectively. Thorium also occurs as a trace element, is relatively inactive chemically and frequently bound to clay minerals. The only unstable isotope of potassium is 40K. It disintegrates by electron capture or emission into argon, 40Ar, or calcium, 40Ca, respectively, with a corresponding characteristic line in the γ-spectrum of the potassium decay at 1 460 keV. Potassium occurs in many clay minerals at concentrations of several percent.

Tabulated Data The compilations of measured radiogenic heat generation rate which are reported in several research papers and reference books will not be duplicated here. Čermák and Rybach (1982) present a large collection of data arranged according to rock type. Van Schmus (1984, 1995) presents data on the abundance of radiogenic isotopes in various minerals and rocks of the Earth’s crust and mantle and discusses the variation of radiogenic heat generation over the lifetime of the Earth. Stacey and Davis (2008) present data on average mass specific heat generation rates for various geologic materials, and Jaupart et al. (2007) discuss radiogenic heat sources in the Earth’s crust and mantle. They provide an in-depth discussion on current models for the composition of a Bulk Silicate Earth (BSE), that is, the Earth’s crust and mantle without the core: These are based on data from samples of either (1) meteorites considered

CK 33 000 12 000 1 500 500 100 544–550 590 15 500 70–160 29 118  20%

CK/C U 7 000 16 000 13 600 14 000 17 000 20 000 2 500 13 000 2 800 – 5 400

Heat generation rate (1012 W kg1) A0 1 050 180 27 9.2 1.5 5.2 47 293–330 2.8–5.1 0.1 4.7  0.08

representative for the starting material and high-temperature processes in the early solar nebula when the Earth accreted or (2) upper mantle rocks formed by low-temperature processes. As can be expected, both show large variations in composition due to their different provenance and history. Chondrites are considered representative of undifferentiated silicate material from the solar system prior to melting and the formation of a (mainly) iron core. Different classes of chondrites correspond to perturbations in elemental abundance in the gas state caused mainly by volatility or condensation temperature. The important elements with respect to radiogenic heat generation are uranium, thorium, and potassium. As elaborated by Jaupart et al. (2007), the first two condensate at very high temperature, why they are called 'refractory lithophile' elements. They show the same ratio in all types of chondritic meteorites, which demonstrates that they behave similarly in the early solar system. In contrast, potassium is a 'moderately volatile' element with a lower condensation temperature. The best agreement with solar concentration ratios is found for carbonaceous chondrites of the CI type of which, however, McDonough and Sun (1995) report only five finds. Compositional data exist only for three of them, in particular for the 700 g meteorite found near Ivuna in Tanzania in 1938 from which the name 'CI' is derived for this type of carbonaceous chondrite. However, chondrites are known to be more enriched in volatiles such as H2O and CO2 than the Earth. A sample from the Earth’s mantle is named “pyrolite,” a contraction of the names of two principal mantle minerals, pyroxene and olivine, making up the complementary rocks peridotite and basalt. Because the former is the solid residue of the partial melting event which produced basalts, a mixture between the two was considered a suitable starting material, the pyrolytic mantle. Generally, pyrolite data suffer from leaching of uranium during low-temperature alteration. As most samples derive from the upper mantle, it has been questioned whether the pyrolytic composition adequately represents a bulk silicate Earth.

R

1306

Radiogenic Heat Production of Rocks

A third approach discussed by Jaupart et al. (2007) avoids a specific choice of a starting composition but determines it from the intersection of the two compositional trends of chondritic meteorites and peridotites. A source of error in this approach derives from the scatter in the two trends. A fourth method proceeds from the elemental ratios of uranium and thorium and determines the primitive bulk Earth abundances from measurements on peridotites. Again, Jaupart et al. (2007) provide a concise introduction into this method. Table 1 summarizes mass-specific radiogenic heat generation rates for terrestrial rocks, meteorites, Moon surface samples, and average values for the Earth’s crust, mantle, and a bulk silicate Earth (BSE).

Calculated Heat Generation Rate The energy emitted by all of these decay processes comprises the kinetic energy of the emitted particles and the γ-radiation associated with the different decay processes. It is absorbed in the rocks and finally transformed into heat. In general, the total heat generation rate A of a rock is the sum of the individual contributions AU, ATh, and AK by uranium, thorium, and potassium, respectively:   A ¼ r CU AU 0 þ CTh ATh 0 þ CK A0K ,

ð2Þ

where r is rock density, and A0 and C are heat generation rate per mass and concentration, respectively, of the corresponding element in the rock. Table 2 shows values for A0 reported by different authors. The variation is less than 3%, at most. Thus, if a rock’s density r and its concentrations in uranium (CU), thorium (CTh), and potassium (CK) are known, its radiogenic heat generation rate A can be determined. Inserting the values, for example, of Rybach (1988) from Table 2 into Eq. 2 yields:   A mW m3 ¼ 105  rðkg m3 Þ  f9:52  CU ðppmÞ þ2:56  CTh ðppmÞ þ 3:48  CK ð%Þg, (3)

where concentrations are given in weight-ppm (i.e., 106 kg kg1) for uranium and thorium and in weight-% for potassium. The natural γ-radiation of rocks can be measured, for instance, by spectrometry on rock samples in the laboratory (see ▶ “Geomagnetic Field, Measurement Techniques”). An alternative source of γ-spectra is the natural gamma spectrometer (NGS) borehole tool which yields as output the three logs URAN (in weight-ppm uranium), THOR (in weight-ppm thorium), and POTA (in weight-% potassium). In combination with density RHOB from the compensated density log [g cm3], this information can be used directly to obtain the bulk heat generation rate A from Eq. 2 using any of the values for A0 U, A0Th, and A0 K from Table 2. Alternatively, for lack of spectra, the total γ-ray emission GR may be used, measured either in the laboratory or by suitable logging tools in boreholes. A γ-ray (GR) log is often included in most logging runs as one of the basic geophysical measurements in boreholes. Its reading reflects the combined radioactive decay of uranium, thorium, and potassium. However, it does not yield information on the individual contributions as it records the total number of γ-rays detected by the tool during a time interval. The relative gamma activities of uranium, thorium, and potassium, that is, the relative number of γ-rays emitted by the same mass during the same time interval are shown in Table 3. Therefore, if POTA, THOR, and URAN are recorded as percent, ppm, and ppm, respectively, the gamma ray log’s reading GR is proportional to the corresponding contents (Beardsmore and Cull 2001): GR ¼ X  ðPOTA þ 0:13  THOR þ 0:36  URANÞ, ð4Þ

where the coefficient X varies with the radius of sensitivity of the log for detecting γ-rays. This varies with a number of factors, but lies within a few decimeters, at most, and can be approximated as constant for a given borehole (Beardsmore and Cull 2001). Then, bulk heat generation rate A can be related to GR by combining Eqs. 2 and 4 using the data of, for example, Rybach (1988) from Table 2:

  34:8  POTAð%Þ þ 25:6  THORðppmÞ þ 95:2  URANðppmÞ AðmW m3 Þ ¼ 103  RHOB g cm3 , GRðAPIÞ XðAPIÞ  POTAð%Þ þ 0:13  THORðppmÞ þ 0:36  URANðppmÞ that is :     GRðAPIÞ POTAð%Þ þ 0:736  THORðppmÞ þ 2:736  URANðppmÞ ð5Þ A mW m3 ¼ 0:034 8 RHOB g cm3 XðAPIÞ POTAð%Þ þ 0:130  THORðppmÞ þ 0:360  URANðppmÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 

¼ 0:034 8 RHOB  GR 

Y X



Y

Radiogenic Heat Production of Rocks

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Radiogenic Heat Production of Rocks, Table 2 Heat generation rate per mass, A0 , for uranium, thorium, and potassium (Clauser 2009) A0 U (mW kg1) 97.0

A0Th (mW kg1) 27.0

A0 K (mW kg1) 0.00 36

97.7

26.3

0.003 4

96.7

26.3

0.003 5

95.2

25.6

0.003 48

Data source Birch (1954) cited in Jessop (1990) Hamza and Beck (1972) cited in Jessop (1990) Emsley (1989) and Jessop (1990) cited in Beardsmore and Cull (2001) Rybach (1988)

Radiogenic Heat Production of Rocks, Table 3 Relative gamma activity of uranium, thorium, and potassium. (Data source: Adams and Weaver (1958) and Emsley (1989) referred to in Beardsmore and Cull (2001)) Element Half-life (Ga) Relative γ-activity

238

U 4.47 3 600

232

Th 14.1 1 300

40

K 1.28 1

Y in Eq. 5 still contains the unknown relative elemental abundances. However, it turns out that the range for Y is limited between 1  Y < 7.6 for the two extreme cases when either 0 ¼ THOR ¼ URAN and POTA >0, or 0 ¼ POTA ¼ THOR and URAN >0. Beardsmore and Cull (2001) argue that in regions in which sediments have been derived from a common source through time the relative proportions of the elements in Eq. 5, and hence, Y should remain relatively constant with depth. Thus, Beardsmore and Cull (2001) maintain that a plot of A (derived, e.g., from a NGS log) versus RHOB  GR should yield a gradient, which depends only on the relative proportions of the radioactive elements in the sediments and should be similar over any region containing sediment derived from the same source. An alternative empirical relationship was published by Bücker and Rybach (1996), which does not require information on the relative proportions of the radioactive elements in the rock. It relates bulk heat generation rate A to total γ-ray emission GR recorded by borehole tools as   A mW m3 ¼ 0:015 8 fGRðAPIÞ  0:8g: ð6Þ This relation has been successfully tested on a number of data sets where both NGS and GR logs were available. As the relations in Eqs. 3 and 6 are based on measurements mostly on igneous rocks, caution is required when applying them to sedimentary rocks.

Measuring Techniques A number of different analytical methods are available for determining the concentrations of uranium, thorium, and

potassium. An inter-laboratory comparison of different methods on selected materials yielded consistent results within a few percent (Rybach 1988). However, among all other methods, γ-ray spectroscopy is the only one which enables determining all three concentrations simultaneously. It implies secular equilibrium within the uranium and thorium decay series and a constant 40K/K ratio, both of which are satisfied for most rock types (Rybach 1988). The method is based on counting the number of decays per energy channel. Commercial spectrometers will offer, for instance, 2048 channels for measuring decays in the energy range of 0 keV–3 000 keV, yielding a resolution, in this particular case, of 1.46 keV per channel. In performing measurements, it is important to realize that radioactive decay is a stochastic process. Therefore, decays need to be recorded over a sufficiently long time. In view of the minute uranium, thorium, and potassium concentrations, and depending on crystal size and sensitivity, this may imply recording times on the order of several hours. Additionally, a possible sampling bias needs to be considered when selecting samples for measurements. The interpretation of γ-ray spectroscopy relates the elemental concentrations C linearly to the corresponding counting rates N in the energy channel x by the corresponding coefficient h(x): N(x) ¼ h(x) C. Thus, the total count rate due to the decay of uranium, thorium, and potassium in the energy channel x is given by: NðxÞ ¼ hU ðxÞ CU ðxÞ þ hTh ðxÞ CTh ðxÞ þ hK ðxÞ CK : ð7Þ The coefficients h depend mainly on the characteristic spectra of the radioactive elements, but also on the geometry of the sample, the geometric configuration of the measurement, and on the efficiency and sensitivity of the detectors. Thus, γ-spectroscopic measurements require calibration based on samples of known elemental concentrations. This enables accounting for the influence of the geometry of the detectors. Additionally, possibly variable sample geometry needs to be addressed separately. Three different effects require attention (Adams and Gasparini 1970): (1) Variation of the continuous background spectrum may occur if the background radiation was determined with an empty sample chamber in which no absorption of the natural background radiation occurs. This can be avoided by measuring the background radiation using a nonradioactive sample of equal size and similar density, such as water-filled plastic tubes; (2) Deformation of the spectrum may occur if a larger sample volume increases the likelihood for internal Compton scattering. Thus, larger samples are characterized by a decreased peak size and an increased level of the continuous spectrum; (3) Radiation self-absorption of a fraction of the γ-radiation emitted within the sample reduces the γ-radiation recorded at the detector. Self-absorption increases with sample thickness d and reduces the count rate N by a factor

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Radiogenic Heat Production of Rocks

F¼

1  emd , m d

where m is the absorption coefficient (Watt and Ramsden 1964). The joint effect of spectrum deformation and selfabsorption is quantified by measurements on potash salt standards. While the absolute 40K-concentrations are unknown, the relative variations reveal the influence of variable sample diameters. Theoretically, the total activity is proportional to the sample cross-section, that is, to the square of the sample diameter. In plotting the count rate versus sample diameter, one finds, however, a smaller exponent of only 1.5. Thus, the remainder of the radiation is either absorbed within the sample or transferred to other energy bands. The energy peaks in the spectrum are analyzed for finding the elemental concentrations. Theory predicts a Gaussian normal distribution for the energy peaks. The count rate N at energy x in the local neighborhood of such a Gauß peak comprises a contribution of the Gauß peak itself, G(x), and a linear trend ax + b, describing the background radiation and the Compton scattering (Adams and Gasparini 1970): NðxÞ ¼ GðxÞ þ a x þ b:

ð9Þ

Such linear trends need to be removed prior to further interpretation, for example, by linear regression. Following this detrending the raw data need to be smoothed for further processing so that the energy peak’s position can be clearly identified (Fig. 1). The area under the peak above the base line then corresponds to the sought total energy. In practice, the count rates within this area are summed up, or a Gaussian normal distribution function

2

ð10Þ

is fitted to the corrected peak, where x0 and s are mean energy and standard deviation of the peak, and A is a calibration constant. In a normal distribution, the peak’s full width at half maximum (FWHM) is related to s by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FWHM ¼ 2 s log 10 ð2Þ:

ð11Þ

The summed-up count rates analyzed this way then need to be calibrated by corresponding measurements on samples of known concentrations. Finally, this yields the concentrations of the radiogenic isotopes of uranium, thorium, and potassium. Heat Generation and Geoneutrinos Currently, general hypotheses on possible distributions of radiogenic heat generating elements in the mantle and core of the Earth are mainly based on analogies with chondritic material and direct evidence is lacking. However, as radiogenic sources produce heat, electron antineutrinos n are emitted in β-decay of nuclei in the decay chains of 238U, 232 Th, and 40K. Thus, the heat produced in nuclear decay is directly related to the antineutrino flux, as shown in Table 4 (Fiorentini et al. 2003). Neutrinos are elementary particles that travel close to the speed of light, are electrically neutral, and can pass through ordinary matter almost undisturbed. They have a very small, but nonzero mass. Therefore, neutrinos are extremely difficult to detect. As the emission of geoneutrinos is coupled to radiogenic heat generation, this enables, at least theoretically,

8

12 data smoothed base line

10 −3 −1 count rate (10 s )

2

GðxÞ ¼ A eðxx0 Þ =2s

ð8Þ

smoothed Emax 6

FWHM

8 4

6 4

2 2 0 400

500

600 energy (keV)

700

0 800 500

550

600 energy (keV)

Radiogenic Heat Production of Rocks, Fig. 1 Measurement and analysis of natural radioactivity shown here for the spectrum before (left) and after (right) removal of the linear base line trend (Eq. 10). FHWM computed after Eq. (11)

650

214

700

Bi-peak at 609 keV;

Radiogenic Heat Production of Rocks

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Radiogenic Heat Production of Rocks, Table 4 Maximum antineutrino energy and heat production rates in natural decay processes (de Meijer et al. 2006; Fiorentini et al. 2005) Decay reaction 238 U ! 206Pb + 8 4He + 6 e + 6 n 232 Th ! 208Pb + 6 4He + 4  e +4n 40 K ! 40Ca + e + n

Emax (MeV) 3.25

Heat generation rate (mW kg1) 95

2.25

27

1.31

0.003 6

neutrino detection sensors in a network of boreholes around the globe (de Meijer et al. 2006). Time will show whether the significant technical challenges associated with the construction of smaller sensors than those in the experiments at the Gran Sasso and Kamioka observatories can be overcome and a road opened for an antineutrino tomography of the Earth.

Summary a geoneutrino tomography of radiogenic heat distribution within the Earth. But this concept is as difficult to implement as it is intriguing. First, there is a huge flux of cosmic neutrinos which traverses the Earth from which geoneutrinos need to be distinguished: The neutrino flux at the Earth’s surface from the sun alone is estimated at about 60  109 s1 cm2 (e.g., Bahcall 1969). The flux of geoneutrinos is three orders of magnitude smaller, but still on the order of 7.4(+2.1/ 1.9)  106 s1 cm2 (Kamland Collaboration 2011). In spite of this immense flux density, the vanishing mass of neutrinos requires exceptional sensors for their detection: (1) At the Gran Sasso observatory in Italy, the detector is a tank of 100 t of liquid gallium trichloride (GaCl3) containing the equivalent of 30 t of gallium at a subsurface laboratory. Here, antineutrinos are observed indirectly through the conversion of a gallium nucleus into a germanium isotope by neutrino capture. For all the high neutrino flux through the tank, an interaction of a neutrino and a gallium nucleus occurs only every 35 h. (2) In the Kamioka Liquid Scintillator Antineutrino Detector (KamLAND) experiment (Araki et al. 2005), the detector comprises of a 13 m diameter nylon balloon filled with 1 kt of liquid scintillator consisting of mineral oil, benzene, and fluorescent chemicals. KamLAND is an experiment at the Kamioka Observatory, an underground neutrino observatory near Toyama, Japan, built to detect electron antineutrinos. The experiment is located in the old Kamiokande cavity in a horizontal mine drift in the Japanese Alps. The detector is housed in an 18 m diameter stainless steel spherical vessel with 1879 photomultiplier tubes mounted on the inner surface. KamLAND is the first detector to conduct an investigation on geoneutrinos and may yield important geophysical information. It has the sensitivity to detect electron antineutrinos produced by the decay of 238U and 232Th within the Earth. Earth composition models suggest that the radiogenic power from these isotope decays is 16 TW, approximately half of the total measured heat dissipation rate from the Earth. A successful antineutrino tomography requires global sampling and distinguishing between cosmic neutrinos traversing the Earth and geoneutrinos generated within the Earth. A project had been proposed some time ago (but unfortunately not funded) to deploy considerably smaller

The heat produced in the radioactive decay of the unstable isotopes of uranium (238U, 235U), thorium (232Th), and potassium (40K) is the largest internal heat source of the Earth. During radioactive decay, mass is converted into energy. Except for the tiny amount associated with the antineutrino and neutrinos generated in β- and β+-decay or electron capture, respectively, all of this energy ends up as heat. The annual production of radiogenic heat in the Earth equals 8.6  1020 J, which is about 1½ times the global production of primary energy in the year 2017 (IEA 2020). The distribution of radiogenic isotopes in the Earth controls to a large extent the thermal regime of the Earth. Unfortunately, this distribution is known only with great uncertainty. Recently, it has become possible to detect geoneutrinos, that is, antineutrinos emitted during radioactive decay of unstable isotopes, in large detectors. With decreased size and improved accuracy of detectors together with directional resolution power, an antineutrino tomography of the Earth appears to become possible. Ideally, this would enable locating and quantifying the distribution of unstable isotopes in the Earth, thus helping to resolve a number of open questions with regard to the state and evolution of the Earth’s thermal regime.

Cross-References ▶ Energy Budget of the Earth ▶ Heat Flow Determinations, Continental ▶ Heat Flow, Continental ▶ Heat Flow, Seafloor: Methods and Observations ▶ Radioactivity in Earth’s Core ▶ Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity

Bibliography Adams JS, Gasparini P (1970) Gamma-ray spectrometry of rocks. Elsevier, Amsterdam Adams JAS, Weaver CE (1958) Thorium to uranium ratios as indicators of sedimentary processes: examples of the concept of geochemical facies. Bull Am Assoc Pet Geol 42:387–430

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1310 Araki T et al (2005) Experimental investigation of geologically produced antineutrinos with KamLAND. Nature 436:499–503 Bahcall JN (1969) Neutrinos from the Sun. Sci Am 221(1):28–37 Beardsmore GR, Cull JP (2001) Crustal heat flow. Cambridge University Press, Cambridge Birch F (1954) Heat from radioactivity. In: Faul H (ed) Nuclear geology. Wiley, New York, pp 148–174 Bücker C, Rybach L (1996) A simple method to determine heat production from gamma-ray logs. Mar Pet Geol 13:373–375 Canup RM, Asphaug E (2001) Origin of the Moon in a giant impact near the end of the Earth’s formation. Nature 412:708–712 Canup RM, Righter K (eds) (2000) Origin of the Earth and Moon. University of Arizona Press, Tucson Čermák V, Rybach L (1982) Radioactive heat generation in rocks. In: Angenheister G (ed) Landolt-Börnstein, Group V: geophysics and space research. Physical properties of rocks, Subvol. A, vol l. Springer, Heidelberg/Berlin, pp 353–371 Clauser C (2009) Heat transport processes in the Earth’s crust. Surv Geophys 30:163–191. https://doi.org/10.1007/s10712-009-9058-2 De Meijer RJ, Smit FD, Brooks FD, Fearick RW, Wörtche HJ, Mantovani F (2006) Towards Earth AntineutRino TomograpHy (EARTH). Earth Moon Planet 99(1–4):193–206 Emsley J (1989) The elements. Clarendon, Oxford Fiorentini G, Mantovani F, Ricci B (2003) Neutrinos and energetics of the Earth. Phys Lett B 557:139–146 Fiorentini G, Lissia M, Mantovani F, Vanucci R (2005) Geo-neutrinos: a new probe of Earth’s interior. Earth Planet Sci Lett B 557:139–146 Hamza VM, Beck AE (1972) Terrestrial heat flow, the neutrino problem, and a possible energy source in the core. Nature 240(5380):343–344 IEA (2008) World energy outlook 2008. International Energy Agency (IEA), Paris. http://www.iea.org/textbase/nppdf/free/2008/weo2008. pdf. Retrieved 10 July 2010 IEA (2020) Data and statistics. International Energy Agency (IEA), Paris. https://www.iea.org/data-and-statistics/data-tables?country¼WORLD& energy¼Balances&year¼2017. Retrieved 17 Mar 2020 Jaupart C, Labrosse S, Mareschal J-C (2007) Temperatures, heat and energy in the mantle of the Earth. In: Bercovici D (ed) Mantle dynamics – treatise on geophysics, vol 7. Elsevier, Amsterdam, pp 253–303 Jessop AM (1990) Thermal geophysics. Elsevier, Amsterdam Kamland Collaboration (2011) Partial radiogenic heat model for Earth revealed by geoneutrino measurements. Nat Geosci 4(9):647–651 McDonough WF, Sun S-S (1995) The composition of the Earth. Chem Geol 120:223–253 Melosh HJ (1990) Giant impacts and the thermal state of the early Earth. In: Newsom HE, Jones JH (eds) Origin of the Earth. Oxford University Press, New York, pp 69–83 Rybach L (1988) Determination of heat production rate. In: Hänel R, Rybach L, Stegena L (eds) Handbook of terrestrial heat flow density determination. Kluwer, Dordrecht, pp 125–142 Stacey FD, Davis PM (2008) Physics of the Earth, 4th edn. Cambridge University Press, Cambridge Vacquier V (1991) The origin of terrestrial heat flow. Geophys J Int 106(1):199–202 Vacquier V (1992) Corrigendum to ‘the origin of terrestrial heat flow/ prime. Geophys J Int 111(3):637–638 Van Schmus WR (1984) Radioactivity properties of minerals and rocks. In: Carmichael RS (ed) Handbook of physical properties of rocks, vol III. CRC Press, Boca Raton, pp 281–293 Van Schmus WR (1995) Natural radioactivity in crust and mantle. In: Ahrens TJ (ed) Global Earth physics – a handbook of physical constants. AGU reference shelf, vol 1. American Geophysical Union, Washington, DC, pp 283–291 Watt DE, Ramsden D (1964) High sensitivity counting techniques. Pergamon, London

Recovery of Source Magnetization Direction from Magnetic Field Data

Recovery of Source Magnetization Direction from Magnetic Field Data Clive Foss Mineral Resources, CSIRO, Lindfield, NSW, Australia

Definition Magnetic field data are measurements of the strength, direction, or gradients of the magnetic field. Local variation between measurements is due to the external fields of nearby magnetizations. In favorable circumstances, the direction of those magnetizations can be estimated by analysis of magnetic field data.

Introduction Most rocks at shallow depth in the earth contain ferromagnetic minerals with both induced and remanent magnetizations (se Remanent Magnetism). The inverse problem of determining the spatial distribution and magnetization of rocks from magnetic field analysis is a form of remote sensing with substantial challenges of non-uniqueness (see ▶ “Inverse Theory, Linear). The direction of magnetization must either be known or be recovered as a part of the study. Induced magnetization is in most cases parallel to the local geomagnetic field. However, the strength of the induced magnetization is rarely known. For remanent magnetization, the strength and direction are also generally unknown. Remanent magnetization direction is of interest for age dating or correlation, global tectonic studies, and structural deformation investigations. It is also required to estimate the spatial distribution of magnetization. Analysis or inversion with an incorrect magnetization direction has long been recognized as a pitfall of magnetic field interpretation (see ▶ “Magnetic Anomalies: Interpretation). A review of the many methods to estimate magnetization direction is provided by Clark (2014) and a review of the issues in deriving magnetization direction from magnetic field data is provided by Foss (2017). The relative importance of induced and remanent magnetizations is traditionally quantified by the Koenigsberger ratio or “Q factor” (defined in Fig. 1) but this scalar statistic is insufficient to describe the vector relationship. Remanent magnetization parallel to the local geomagnetic field cannot be discriminated from induced magnetization by analysis of the static magnetic field, but also causes no error in estimation of the spatial distribution of magnetization. Many rocks contain some “soft” or “viscous” remanent magnetization in the present field direction carried by easily remagnetized

Recovery of Source Magnetization Direction from Magnetic Field Data

Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 1 The vector relationship between remanent, induced, and resultant magnetizations

multidomain grains (see ▶ “Magnetic Domains). The diagnostic expression of remanent magnetization in magnetic field data is best represented by the angle between the resultant magnetization and the geomagnetic field (the apparent resultant rotation angle or ARRA) as shown in Fig. 1. Angular difference between resultant magnetization and the geomagnetic field is caused by “hard” remanent components generally carried in single domain or pseudo-singe domain grains. Many rocks have a large assemblage of magnetic mineral grains carrying both hard and soft remanent magnetizations. If the resultant rotation angle is large, the pattern of the external magnetic field of the magnetization will deviate substantially from that due to an equivalent distribution of induced magnetization suggesting the presence of remanent magnetization. However, additional information such as the Koenigsberger ratio or magnetic susceptibility value is still required to resolve remanent magnetization direction from the resultant magnetization. An alternative approach to dealing with complications due to unknown magnetization direction is to invert transforms which have low sensitivity to magnetization direction (Paine et al. 2001). This provides an approximate mapping of the distribution of magnetization but does not recover information about the magnetization direction. Lelièvre and Oldenburg (2009) developed a method to recover magnetization direction by inverting for three orthogonal components of magnetization.

The Expression of Magnetization Direction in Magnetic Field Data For a thin planar (“2D”) distribution of magnetization, any components of magnetization in the strike direction of the sheet do not contribute to the magnetic field, and the angle of inclination of magnetization perpendicular to the sheet combines with the dip angle of the sheet into a single term from which neither can be independently recovered (Hood 1964).

1311

This limitation led to an incorrect but widely and long-held assumption that magnetization direction cannot be reliably recovered from magnetic field data. However, Helbig (1963) proved that the magnetization direction of a dipole (a homogeneously magnetized sphere or a source so small that its shape is irrelevant) can be recovered from magnetic field analysis provided the dipole field is isolated from other fields and the horizontal center of magnetization is known. Schmidt and Clark (1998) recognized the potential for modern computing capabilities to apply Hebig’s analysis to recovery of magnetization direction from magnetic field data and their publication led to a switch in acceptance of the ability to recover magnetization estimates from magnetic field data. In an early three-dimensional magnetic field modelling study (see ▶ “Magnetic Modeling, Theory, and Computation) Zietz and Andreasen (1967) noted that in steep inclination geomagnetic fields the declination of magnetization of their source models is indicated by the peak to trough azimuth of the magnetic anomalies, and the inclination is indicated by the peak to trough amplitude ratios. These relationships are illustrated in Fig. 2 for steep northern and southern hemisphere fields. Shallow inclination magnetizations give rise to anomalies with similar peak and trough amplitudes; steep magnetizations parallel to the field (“normal”) give rise to predominantly positive anomalies, and magnetizations anti-parallel to the field (“reverse” – see ▶ “Geomagnetic Field, Polarity Reversals) give rise to predominantly negative anomalies. In the southern hemisphere, the declination of magnetization is indicated by the trough to peak azimuth, and in the northern hemisphere by the peak to trough azimuth. A specified magnetization gives rise to a consistent external field, but as shown in Fig. 2 the total magnetic intensity (TMI) anomalies vary considerably in different geomagnetic settings because of their vector addition with those differently directed fields (see ▶ “Geomagnetic Field, Global Pattern). Because we can recognize the expression of magnetization direction in magnetic anomaly patterns we should expect to recover magnetization directions by magnetic field analysis or inversion. To investigate recovery of magnetization direction, we can consider the three conditions for Helbig analysis; namely, correct isolation of the magnetic field due to the magnetization, knowledge or estimation of the center of magnetization, and the influence of the shape of distribution of the magnetization.

Magnetization Contrast and Anomaly Separation The pattern of an anomaly depends on its isolation from background and other overlapping fields in what is known as “regional-residual separation” (see ▶ “Magnetic Data Enhancements and Depth Estimation). Regional-residual

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Recovery of Source Magnetization Direction from Magnetic Field Data

Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 2 Dipole TMI anomalies of identical magnetizations in steep southern and northern geomagnetic fields

Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 3 (a) TMI anomalies over holes in a volcanic sheet, (b) TMI computed from an inversion model of one of the holes, (c) the central section through the hole inversion model (shown in yellow)

separation is intrinsically interpretive and the output of any analytic method must be carefully checked. The choice of the regional field is also intrinsically related to the assumed background magnetization as illustrated in a study of magnetic field variations over part of the Beetaloo Basin of the Northern Territory of Australia (Foss and Dhu 2016) as shown in Fig. 3. This region is underlain by a wide volcanic

sheet delineated by a magnetic edge anomaly (beyond the area shown). The magnetic anomalies in Fig. 3 are interpreted as due to holes in the sheet; the background to each anomaly is the field above the sheet, and the direction of magnetization of the sheet is antiparallel to the magnetization derived from inversion of the hole anomalies. Because magnetic field anomalies are due to contrasts in magnetization and no

Recovery of Source Magnetization Direction from Magnetic Field Data

1313

Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 4 (a) TMI anomaly at Ethabuka, Queensland, Australia, (b) Standard RTP anomaly with borehole location, (c) RTP using the

inversion magnetization direction estimate. The top of the inversion model is outlined in magenta

absolute measure of the background field is available, it is not possible to reliably map the continuous distribution of magnetization beneath an area as is sometimes claimed of voxel inversion results. The moment integrals of Helbig analysis are particularly sensitive to error in anomaly separation but an approximate extension of Helbig’s theory to analysis of tensor gradients provides significant advantage in isolation of the fields for analysis (Phillips et al. 2007).

proposed magnetization is not tested by the borehole. Figure 4c shows the compact, essentially positive anomaly of an amended RTP transform using the resultant magnetization direction of the inversion results. Correlation of gravity anomalies and RTP transforms can provide a test of magnetization direction for bodies with co-located density and magnetization contrasts, and correlation of RTP with sourcecentric transforms such as the total gradient can similarly investigate whether the magnetization direction is as assumed in the RTP transform (Roest and Pilkington 1993). Dannemiller and Li (2006) used correlation between total gradient and RTP with test magnetization directions to automatically select the preferred direction. The normalized source strength (Beiki et al. 2012) provides a similar and in some cases superior central function to the total gradient transform (see ▶ “Magnetic Data Enhancements and Depth Estimation).

Magnetization Direction and Source Location Helbig analysis also requires supply of the horizontal center of magnetization. Inversion can seek to simultaneously recover estimates of magnetization direction and position but error in either of these parameters gives rise to error in the other. Alternatively, various magnetic field transforms can be used to estimate the center of magnetization. Figure 4a, b show a TMI anomaly measured at Ethabuka, Queensland, Australia, and the standard RTP transform as is commonly used to locate magnetic field anomalies above their source magnetizations. The TMI anomaly pattern is markedly different to that expected of an induced anomaly, disqualifying the assumption that magnetization is in the direction of the local geomagnetic field. A borehole into the RTP peak failed to intersect the magnetization. Parametric inversion of the anomaly recovered a model of low inclination southeastdirected magnetization (consistent with the magnetization and anomaly pattern relationships described above). The top of the inversion model plotted in Fig. 4b shows that the

Magnetization Direction and Source Shape Helbig analysis strictly applies only to a dipole magnetization which is at best an approximate representation of a geological body. As already noted, recovery of magnetization direction for thin sheet magnetizations is challenging and we can expect challenges arising from the shape of distribution of magnetization to progressively increase for magnetization distributions intermediate between a dipole and a sheet. Figure 5a shows a TMI anomaly at Coompana in South Australia and Fig. 5b shows four models of different

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Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 5 (a) the TMI northern anomaly at Coompana, South Australia with the center points of four alternative homogeneous magnetization models, (b) a perspective view of the models

geometry derived by inversions of the anomaly using independent forward modelling algorithms (Foss 2017). The computed fields of all the models closely match the measured field and all bodies are closely co-centered with near-identical resultant magnetization directions (Fig. 6a). Figure 6b shows a cross-plot of the volumes and intensity of magnetization of the models which both vary by a factor of 5, but their product (magnetic moment) values are consistent. These results reveal that for this body of magnetization and set of magnetic field data, estimates of the horizontal center of magnetization, mean magnetization direction, and magnetic moment are consistently recovered by inversion, but that the details of shape, intensity of magnetization, and volume are inconsistently estimated. The consistently recovered parameter values can be retained from any of the models with the models themselves discarded as improbable representations of the subsurface magnetization. Identical capabilities and limitations apply to voxel inversions. The consequence of

Recovery of Source Magnetization Direction from Magnetic Field Data

Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 6 (a) A stereo net plot of the four inversion model magnetization directions, (b) a cross-plot of volume and intensity of magnetization for the four models with a single magnetic moment contour

departure of shape from that of a dipole can be approximately quantified by a term nominated as “compactness” which is the ratio of the closest approach of a field measurement location to the body of the maximum extent of the body. This simple statistic does not address the full complexity of threedimensional shape, survey extent, and magnetization direction but importantly includes the influence of distance from source. At greater distances, the magnetic field of a body appears progressively more like that of a dipole. Alternatively, for a particular body of magnetization, measured magnetic fields can be classified as near-field or far-field (analogous to definitions of electro-magnetic fields due to an antenna) with magnetization direction most easily recovered from far-fields for which source shape is less significant. Fullagar and Pears (2015) developed methods to separately address the influence of shape and magnetization direction, but these methods require either independent constraints or assumptions.

Recovery of Source Magnetization Direction from Magnetic Field Data

Complex Distributions of Magnetization Study of complex distributions of magnetization requires application of site-specific strategies. A complex magnetic anomaly of approximately 12 km diameter measured at Coompana, South Australia (Foss 2017), and shown in Fig. 7a appears to be due to magnetizations of more than one direction. Figure 7b shows a close match to the measured field forward computed from a multi-body, multi-magnetization inversion model. The NSS transform shown in Fig. 7c highlights the circular pattern of the anomaly and a small anomaly at its Centre. In a multistage inversion, it was surmised that the anomaly can be explained with four homogeneous magnetization bodies (bodies “a” to “d” in Fig. 8a). This starting model was then inverted with gradually increased degrees of freedom. Figure 8b shows a perspective view of the resulting model which has a consistent limited depth extent and slight inward plunge, suggesting that it might represent a high-level, subvolcanic intrusive complex. To provide a drillsite depth prediction, a separate, focused inversion was performed of only the center anomaly (Fig. 8c, d) which is too small to be properly constrained in the main inversion. The main bodies of magnetization provide the background field, and the inversion model is of an excess magnetization superimposed on those magnetizations (which are themselves superimposed). This inversion successfully predicted the intersection depth.

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Homogeneity and Inhomogeneity of Magnetization If it is not feasible to reliably resolve the distribution bounds of magnetization from inversion of a far-field anomaly, it is unlikely to be feasible to determine the internal homogeneity or inhomogeneity of that magnetization. This is illustrated by two studies of the northwest Black Hill Norite anomaly as shown in Fig. 9 from the Australian Remanent Anomalies Database (http://bit.ly/2HvhZdP). Figure 10a shows two magnetization models which both closely match the measured anomaly. One is a homogeneous magnetization (Foss and McKenzie 2011) and the other has the same three magnetizations in similar distribution as a model by Macleod and Ellis (2016). The magnetization directions of both models are plotted in Fig. 10b. The magnetization direction of the single magnetization model is only 11° from a remanence direction measured at a nearby outcrop (the body causing the anomaly is not itself exposed). It is further from the resultant magnetization at that site but the two bodies may have different Koenigsberger ratios. The mean magnetization direction of the triple magnetization model summed from each of its component parts is close to that of the single magnetization model and there is only a small difference in their total magnetization strength. From the existing magnetic field measurements, it is difficult to discriminate between the two models and establish whether the magnetization is

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Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 7 (a) the TMI southeast satellite anomaly at Coompana, South Australia, (b) TMI forward computed from the inversion model with outline of the model bodies, (c) NSS anomaly

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Recovery of Source Magnetization Direction from Magnetic Field Data

Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 8 (a) Plan view of the top of the inversion model, (b) perspective view (body “c” excluded for clarity), (c) center TMI anomaly body “e” model and drill-hole location, (d) central cross-section

homogeneous or consists of three distinct components. However, in closer proximity (with near-field measurements) it might be possible to resolve individual magnetization directions for any inhomogeneities that may be present. Sun and Li (2019) propose an analysis for multiple magnetizations after first constructing a statistical model for the proposed number of magnetizations present.

Magnetic Field Studies and Paleomagnetism The major restrictions in application of magnetic field analysis to palaeomagnetic studies are ambiguity in recovering remanent magnetization direction from resultant magnetization and inability to resolve different remanent components from the total remanent magnetization (NRM) as is achieved in palaeomagnetic studies through alternating field and

thermal demagnetization. There are, however, opportunities for integrating magnetic field and palaeomagnetic studies arising from their very different sample volumes. Palaeomagnetic studies directly measure remanent magnetization of small volumes, in many cases collected from limited locations. Magnetic field analysis can be used to test validity in ascribing palaeomagnetic results to much larger volumes than those directly sampled. Combination of palaeomagnetic studies from outcrop and magnetic field studies of deeper magnetizations unavailable for palaeomagnetic sampling can provide more complete mapping of the distribution of magnetizations. The usefulness of magnetic field studies for mapping the distribution of magnetization depends on the precision with which magnetization directions can be recovered (in favorable circumstances to within 10° or possibly even 5°) and will be significantly enhanced if high precision can be achieved by an automated search algorithm to enable

Recovery of Source Magnetization Direction from Magnetic Field Data

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Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 9 Australian Remanent Anomaly Database image of the northwest Black Hill Norite anomaly in the AUSGIN web portal

processing of large data sets. Possible remanent magnetization directions known from a defined apparent polar wander (APW) curve (see ▶ “Paleomagnetism, Polar Wander) for an area can be used to guide inversion of magnetization direction (Cordani and Shukowsky 2009; Pratt et al. 2014). As a resource for both subsurface mineral exploration and palaeomagnetic studies, the Australian Remanent Anomalies Database (http://bit.ly/2HvhZdP) (see ▶ “Magnetic Anomaly Map, Global) has been established to compile and distribute magnetic field magnetization results, and this national coverage can be extended using global scale datasets such as EMAG2 (doi: https://doi.org/10.7289/V5H70CVX). At the other extreme of spatial scales, micro mapping of magnetic fields over individual mineral grains provides information

about the genesis of specific magnetizations within palaeomagnetic samples (e.g. Pastore et al. 2019).

Conclusions In favorable circumstances, estimates of magnetization direction can be recovered from analysis or inversion of magnetic field data. Analysis is best performed on fields of compact magnetization. Analysis of magnetic fields of planar sheet magnetizations is more problematic. Inversion of magnetic field measurements close to a magnetization depends on details of the distribution of that magnetization, but at increasing distance, the magnetic field progressively approximates to

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Recovery of Source Magnetization Direction from Magnetic Field Data

▶ Magnetic Anomaly Map, Global ▶ Magnetic Data Enhancements and Depth Estimation ▶ Magnetic Domains ▶ Magnetic Modeling, Theory, and Computation ▶ Paleomagnetism, Polar Wander ▶ Remanent Magnetism

Bibliography

Recovery of Source Magnetization Direction from Magnetic Field Data, Fig. 10 (a) alternative single and triple magnetization inversion models for the northwest Black Hill Norite anomaly, (b) stereonet display of the inversion model magnetizations

that of a dipole and inversion is simpler (provided that the lower amplitude distal field can still be reliably isolated). More complex distributions of magnetization can also be inverted, but only where fields of individual magnetization are at least partially separated, and generally with increased scope of non-uniqueness in the results. Magnetic field analysis provides estimates of resultant magnetization. Further information is required to resolve the resultant into induced and remanent components.

Cross-References ▶ Geomagnetic Field, Global Pattern ▶ Geomagnetic Field, Polarity Reversals ▶ Inverse Theory, Linear ▶ Magnetic Anomalies: Interpretation

Beiki, Clark, Austin, Foss (2012) Estimating source location using normalized magnetic source strength calculated from magnetic gradient tensor data. Geophysics 77:J23–J37. https://doi.org/10.1190/ geo2011-0437.1 Clark (2014) Methods for determining remanent and total magnetisations of magnetic sources – a review. Explor Geophys 45:271–304. https://doi.org/10.1071/EG14013 Cordani, Shukowsky (2009) Virtual Pole from magnetic Anomaly (VPMA): a procedure to estimate the age of a rock from its magnetic anomaly only. J Appl Geophys 69:96–102. https://doi.org/10.1016/j. jappgeo.2009.07.001 Dannemiller, Li (2006) A new method for determination of magnetization direction. Geophysics 71:L69–L73. https://doi.org/10.1190/1. 2356116 Foss (2017) Resultant-magnetization based magnetic field interpretation. In: Proceedings of exploration 17: sixth decennial international conference on nineral exploration edited by Tschirhart and Thomas, paper 44, pp 637–648. http://www.dmec.ca/Resources/Exploration17.aspx Foss, Dhu (2016) The Bark Without a dog — magnetic anomalies over holes in a volcanic sheet in the Greater McArthur Basin, NT. ASEG Ext Abstr 2016:1–5. https://doi.org/10.1071/ASEG2016ab276 Foss, McKenzie (2011) Inversion of anomalies due to remanent magnetisation: an example from the Black Hill Norite of South Australia. Aust J Earth Sci 58:391–405. https://doi.org/10.1080/ 08120099.2011.581310 Fullagar, Pears (2015) Remanent magnetisation inversion. In: 24th ASEG international geophysical conference, ASEG, extended abstracts. https://doi.org/10.1071/ASEG2015ab188 Helbig (1963) Some integrals of magnetic anomalies and their relation to the parameters of the disturbing body. Z Geophys 29:83–96 Hood (1964) The Königsberger ratio and the dipping dike equation. Geophys Prosp 12:440–456 Lelièvre, Oldenburg (2009) A 3D total magnetization inversion applicable when significant, complicated remanence is present. Geophysics 74(3):L21–L30. https://doi.org/10.1190/1.3103249 MacLeod, Ellis (2016) Quantitative magnetization vector inversion. ASEG Ext Abstr 2016:1–6 Paine, Haederle, Flis (2001) Using transformed TMI data to invert for remanently magnetised bodies. Explor Geophys 32:238–242. https:// doi.org/10.1071/EG01238 Pastore, Church, McEnroe (2019) Multistep parametric inversion of scanning magnetic microscopy data for modelling magnetization of multidomain magnetite. Geochem Geophys Geosyst 20:5334–5351. https://doi.org/10.1029/2019GC008542 Phillips, Nabigian, Smith, Li (2007) Estimating locations and total magnetization vectors of compact magnetic sources from scalar, vector, or tensor magnetic measurements through combined Helbig and Euler analysis. SEG Tech Program Ext Abstr 26:770–774. https://doi.org/10.1190/1.2792526 Pratt, McKenzie, White (2014) Remote remanence determination (RRE). Explor Geophys 45:314–323. https://doi.org/10.1071/ EG14031

Remanent Magnetism Roest, Pilkington (1993) Identifying remanent magnetization effects in magnetic data. Geophysics 58:653–659. https://doi.org/10.1190/1.1443449 Schmidt, Clark (1998) The calculation of magnetic components and moments from TMI: a case study from the tuckers igneous complex, Queensland. Explor Geophys 29:609–614. https://doi.org/10.1071/ EG998609 Sun, Li (2019) Magnetization clustering inversion – part 2: assessing the uncertainty of recovered magnetization directions. Geophysics 84: J17–J29. https://doi.org/10.1190/GEO2018-0480.1 Zietz, Andreasen (1967) Remanent magnetization and aeromagnetic interpretation. In: Mining geophysics, vol 2. SEG, Tulsa, pp 569–590

Remanent Magnetism Laurie Brown1 and Suzanne McEnroe2 1 Department of Geosciences, University of Massachusetts, Amherst, MA, USA 2 Norwegian Geological Survey, Trondheim, Norway

Definition Remanent magnetization

Permanent magnetization held in rocks and other earth materials, commonly, though not always, dating from cooling to below the Curie temperature (Tc) or Néel temperature (Tn) of the magnetic minerals in the rock, or the time of formation of the rock unit.

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Source of RM Remanence is a property unique to a small group of minerals possessing ferromagnetic or antiferromagnetic properties, mostly iron oxides and iron sulfides. Remanence is observable in hysteresis behavior, when, after saturation, the applied field is reversed and decreased to zero. If the magnetization does not also return to zero, but retains a measurable quantity, a remanent magnetization is present (Fig. 1). This characteristic is temperature dependent, with only some of the mineral phases possessing remanence at earth surface temperatures. The most common and important magnetic minerals on Earth belong to the cubic oxide series magnetite (Fe2+Fe23+O4) – ulvöspinel (Fe22+Ti4+O4), and the rhombohedral oxide series hematite (Fe23+O3) – ilmenite (Fe2+Ti4+O3). Both series have important solid solution versus temperature relationships, as well as magnetic complexities of key interest in the rock magnetism. Additionally, both series show remanence properties that vary with composition. Furthermore, the oxide bulk compositions in the vicinity of equilibrium tie lines between ilmenite and magnetite are found in a majority of igneous and metamorphic rocks, especially basalts. In these

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The fact that earth materials, particularly rocks, possess a permanent magnetization has been known for some time. Early observations of rocks producing large anomalies were commonly ascribed to lightning strikes. By the midnineteenth century remanence was accepted as a property of some rocks, and by the early twentieth century crude measurements of the direction of magnetization were possible. During the second half of the twentieth century, the measurement of remanent magnetization (RM) became a wellestablished research field (see entries under ▶ “Paleomagnetism, Principles”). RM is a vector quantity, having both direction, usually related to the direction of the Earth’s field at the time of origin, and intensity, related to both mineral phase and the external field. Descriptions of remanent magnetization, associated paleomagnetic techniques, and results are detailed in standard paleomagnetic textbooks such as Butler (1992), McElhinny and McFadden (2000), and Tauxe (2010). More detailed discussion of the theory and development of remanent magnetization can be found in rock magnetism books, including those by Nagata (1961), Stacey and Banerjee (1974), O’Reilly (1984), and Dunlop and Özdemir (1997).

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Remanent Magnetism, Fig. 1 Hysteresis loop of hematite-dominated sample of the EI Laco deposit, northern Chile. Horizontal axis is the external field, in Tesla; vertical axis is relative magnetization. Mr indicates the remanent magnetization left after the sample is saturated and returned to a zero field

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assemblages, the slopes of the tie lines are a function of temperature, whereas tie line positions are a function of the oxygen fugacity of equilibration according to the oxygen thermobarometer (Buddington and Lindsley 1964). End-member hematite is antiferromagnetically ordered with equal and opposite magnetic moments in alternate cation layers. However, at room temperature, a weak ferromagnetic moment is produced because the alternate moments are ~0.13° away from being perfectly antiparallel (Dzialoshinskii 1957). This effect is commonly referred to as “spin-canting” and such hematite is termed “canted-antiferromagnetic.” Many oxidized sediments contain hematite, and carry this weak, but very stable magnetization, which is mainly due to its high coercivity. The temperature and composition phase relations of the hematite-ilmenite series are a result of a complex interaction of Fe-Ti ordering and magnetic ordering at high temperature. Slow cooling to intermediate to low temperatures produces exsolution of discrete ilmenite and hematite (Burton 1991; Harrison et al. 2000; Ghiorso and Evans 2008). Exsolved grains contain two phases, a host (hematite or ilmenite) and lamellae of the second phase. Ilmeno-hematite (hematite with ilmenite exsolution), or hemo-ilmenite (ilmenite with hematite exsolution) commonly have a strong, and very stable magnetization, referred to as “lamellar magnetism.” Lamellar magnetism is due to a defect in the magnetic moment at the interface (or contact layer) between the two phases (Robinson et al. 2002, 2004) where a ferrimagnetic substructure is present because of the different quantity of Fe2+ and Fe3+ ions in contact Fe layers, as compared to standard Fe3+ layers in adjacent hematite. Intermediate members of this series rapidly cooled from high temperature produce intermediate metastable phases that are ferrimagnetic and show complicated magnetic properties near and below room temperature (Ishikawa and Akimoto 1957; Burton et al. 2008; Robinson et al. 2010) with some compositions with varied Fe-Ti ordering showing magnetic self-reversal (Harrison et al. 2005; Ishikawa and Syono 1963).

Natural Remanent Magnetization (NRM) The magnetization held by a rock sample prior to subjection to any laboratory procedures is referred to as Natural Remanent Magnetization (NRM). It is the sum total of all remanences held in the rock and may include primary as well as secondary remanences arising from different processes. The NRM is differentiated from induced magnetization, which a rock may acquire as it sits in an external field, and loses when the field is removed. Induced magnetization is dependent on the magnetic susceptibility of the material as well as the strength of the external field. NRM is traditionally described by the manner of acquisition of the remanences involved,

Remanent Magnetism

leading to a number of both natural and laboratory magnetizations, as described below.

Thermoremanent Magnetization (TRM) Igneous rocks, cooling down from a molten state through the Curie or Néel temperature (Tc or Tn), obtain a thermoremanent magnetization (TRM) parallel to the ambient field. For magnetite this temperature is 580 °C while for hematite it is 675 °C. At these temperatures the spontaneous nature of the ferromagnetic material occurs and remanence is blocked just below this temperature. The material must cool slightly further to the blocking temperature (Tb) before this magnetization becomes locked in with a relaxation time equivalent to geologic time spans. The direction of remanence, so important to the whole field of paleomagnetism, becomes permanent at this point, with a net direction parallel to the ambient field. The intensity of the magnetization is influenced by the size of the external field, and by the phase, composition, and grain size of the ferromagnetic minerals present. The TRM preserved in igneous rocks is a summation of the remanences of all the ferromagnetic grains present. This means there may be a range of Curie temperatures and related blocking temperatures, giving rise to partial thermal remanence (pTRM). As a simple case, if a rock had both pure hematite and pure magnetite grains present, the hematite grains would lock in the field slightly below 675 °C while the magnetite grains would provide additional remanence once the rock cooled to below 580 °C. Commonly, a combination of oxide minerals of varying composition provide for remanence acquired at a range of temperatures. In Fig. 2, a plot of pTRM is shown for a basalt with low Ti-magnetite. Metamorphic rocks heated to above their Curie temperatures will have the remanence reset as the material cools down. High grade metamorphic rocks, such as granulite- and amphibolite facies rocks, heated to above 600–700 °C will have all the existing magnetic minerals remagnetized in the external field at the time of cooling, whereas rocks of lower metamorphic grade will have only minerals with Tc below the peak metamorphic temperature completely reset. Néel (1955) introduced the theory of thermoremanence; Dunlop (1977) and Dunlop and Özdemir (1997) provide detailed treatment of the theory.

Detrital Remanent Magnetization (DRM Sedimentary rocks obtain a remanent magnetization in a very different way. Here the detrital grains settling in a water column are influenced by the external magnetic field producing a depositional remanent magnetization (DRM).) These grains are also influenced by procedures in the sedimentary

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The notion of individual grains settling in a water body has recently been investigated (Tauxe et al. 2006) and found to be more complicated. In marine environments in particular, finegrained sediments do not sink as discrete particles, but rather flocculate with coexisting nonmagnetic particles. The entire floc needs to be aligned with the external field, leading to variations in remanence, especially with respect to intensity, from that predicted by theory and redeposition experiments.

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Remanent Magnetism, Fig. 2 Thermoremanent magnetization plot from a Tertiary basalt, Meseta Lago Buenos Aires, Argentina. Upper curve labeled TRM represents the amount of magnetization left after increasing heatings up to 600 °C, with nearly 75% of the remanence being removed between 550 °C and 600 °C. Lower curve (pTRM) indicates the percentage of TRM residing within certain blocking temperatures

environment imparting a post-depositional remanence (pDRM). The general term of detrital remanent magnetization includes both processes of depositional as well as postdepositional magnetization. When eroded ferromagnetic grains enter the sedimentary regime, they retain their magnetic moments even though the original orientation of that moment is lost. These grains, as they settle through a water column, respond to the external field, and in a mechanical response, become aligned with the influencing field over a very short time period. As redeposition experiments in the laboratory have shown, this theory is not so simple as projected (Verosub 1977). There are a number of changes that can occur once the grain is deposited, but before magnetization is locked in. These effects lead to a pDRM, or post-depositional remanent magnetization, a prevalent but complicated process (Shcherbakov and Shcherbakova 1983). Common effects are the dewatering of the sediment, compaction of the grains, and bioturbation, all of which allow for ferromagnetic particles, particularly small ones, to realign. Magnetization is not considered final until the grains are no longer able to move about or realign, occurring at a depth referred to as the lock-in depth. This gives grain size an importance in DRM, with smaller grains more susceptible to realignment, but also yielding stronger and more stable remanence. To this end fine-grained sediments, as mudstones and siltstones, retain a remanence better than coarser-grained rocks such as sandstones or conglomerates.

Commonly rocks undergo chemical changes after formation, usually at temperatures below the Curie temperature of the minerals involved. These changes can be alterations to existing magnetic minerals and/or the creation of new iron oxides within the rock unit. These changes can result in new or altered remanences; this process is commonly referred to chemical remanent magnetization (CRM). A classic example of CRM is “red beds” or sedimentary rocks with strong reddish hues due to the ubiquitous presence of hematite. Such rocks may have a DRM from the time of formation, due to the presence of detrital grains of magnetite and/or hematite. After deposition, changes brought on by differing environments, a range of aqueous solutions, or variations in oxidation state may result in weathering of existing ferromagnetic minerals, or in the deposition by precipitation of new ferromagnetic minerals. Existing magnetic minerals may break down to weaker or nonmagnetic minerals, such as hematite altering to goethite or pyrrhotite to pyrite. Ferromagnetic minerals may also alter to other magnetic minerals with different remanence characteristics. One example is the oxidation of magnetite to maghemite. Another alteration common is that of nonmagnetic grains to magnetic ones, such as the case of Fe-bearing silicate grains altering to discrete hematite grains. New ferromagnetic minerals develop, such as the formation of hematite cements and coatings, which is pervasive in red beds develops. CRM can also be produced during progressive metamorphism, for example, when Fe-silicates breakdown to produce magnetite, in hematite-ilmenite solid solutions, or when Fe-bearing silicates react with pyrite (nonmagnetic FeS2) to produce pyrrhotite (magnetic FeS). Lamellar magnetism mentioned earlier is a CRM because it is produced during the chemical reaction of exsolution and not at the Curie temperature of the minerals; with slow cooling this can occur hundreds of degrees below the Tc. CRM is also recognized in igneous rocks, such as in the formation of secondary hematite in cracks and fractures in basalts or around the rims of discrete magnetite grains. In this case, the CRM is often hard to distinguish from the original TRM, as the CRM is quite stable and many of the magnetic properties of the two remanences are similar.

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Remanent Magnetism

Viscous Remanent Magnetization (VRM)

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At room temperature, in low ambient fields, rocks may acquire a component of magnetization in the direction of the “present” field. This remanence acquired over long time periods in low fields, such as the Earth’s field, is referred to as viscous remanent magnetization (VRM). It is generally assumed that the strength of the VRM is related to the log of the time involved, although this is a gross simplification of the situation. The importance of other factors, including grain size, initial magnetic state, and the angle of the ambient field relative to the original remanence vector have recently been investigated and shown to also affect the strength of VRM) (Yu and Tauxe 2006). Although VRM is ubiquitous in geologic examples and is found associated with many different rocks types, in many cases it is small and can easily be removed by demagnetization techniques. Due to its soft nature, VRM is often referred to as an “overprint” commonly representing the present magnetic field and must be removed prior to investigating the primary remanence. In natural situations where temperatures may be elevated for a considerable time, as in low-grade metamorphism, or in young oceanic lithosphere, there is commonly a thermal viscous remanent magnetization (TVRM) acquired. Here, increased temperature over extended time periods enhance the viscous remanence and produce a magnetization that may eventually replace the original NRM. Pullaiah et al. (1975) have investigated such magnetization changes over a range of temperatures and timescales, with additional discussion provided by Dunlop and Özdemir (1997).

other samples, the magnetization has been totally reset, giving rise to paleo-magnetic sites with random directions (Tauxe et al. 2003). However, this type of IRM is commonly accompanied by high heat; therefore, the term is slightly misused.

Isothermal Remanent Magnetization (IRM)

Applications of Remanent Magnetization

The presence of a large magnetic field over a short time period may also produce a remanent magnetization, referred to as isothermal remanent magnetization (IRM). An IRM can be imparted in the laboratory and used to help identify properties of magnetic materials. Samples are subjected to increasingly large external fields, usually to 1 T or larger, and the resulting magnetization recorded. Differences in composition, concentration, grain size, and saturation state effect the acquisition of IRM. For example, magnetite-bearing samples will saturate in fields of 0.1–0.3 mT, while hematite-dominated rocks will require fields larger than 1 T to reach saturation (Fig. 3). In the natural environment, a typically cited example of an IRM is the magnetization gained in a rock after being struck by lightning. The large magnetization produced by the electric currents in lightning imparts an intense, but chaotic remanence on surface rocks. First studied by Cox (1961), IRM is common in regions subject to intense thunderstorms, such as the desert southwest of the North America. In some cases, the effect of the IRM can be removed by demagnetization; in

RM is used extensively in the disciplines of magnetism, including paleomagnetism, rock magnetism, and the study of magnetic anomalies. The study of paleomagnetism centers on the identification and description of the remanent field held in rocks. Results from the measurement of RM are used widely in the study of plate tectonics, structure, regional tectonics, stratigraphy, volcanology, and archeology. Positions of the earth’s magnetic field at specific times in the geologic past provide us invaluable information on the position of continental bodies in the past, the interrelationship between different continents, and the paths they have followed in subsequent motions across the earth’s surface. RM can provide age information by comparison to established reversal timescales, as well as relative ages within a region. Regional tectonics, local rotations, and timing of folding can all be investigated using the remanence recorded in suitable rock units. Stratigraphic studies, especially in the wealth of deep-sea sediment cores from international drilling projects, lake sediments, or in thick sedimentary sequences on

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Remanent Magnetism Remanent Magnetism, Fig. 4 Aeromagnetic anomaly map from a fixed-wing survey over the Rogaland region in southwestern Norway. The magnetic total field was reduced to anomaly values by subtracting the International Geomagnetic Reference Field from the total field of 1965. Color shades: pink, large positive magnetic anomalies; blue, strong negative anomalies; yellow and green, intermediate values. Letters refer to specific bodies within the Rogaland Igneous Complex. (Redrawn from McEnroe et al. 2001)

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land, can be established and compared using the recorded remanence. Experimental studies in rock magnetism make excellent use of the remanent magnetization properties of rocks and minerals. Studies of hysteresis behavior, magnetization at high and low temperatures, and responses to acquisition and removal of laboratory remanences all provide important information on oxide composition, grain size, and magnetic properties of a wide range of natural and synthetic materials. Remanent magnetism plays an important part in the interpretation of magnetic anomalies. Well established in the 1960s is the study of marine magnetic anomalies from the seafloor, remanent polarity of extruded basalts played a pivotal role in the interpretation of observed anomalies (Vine and Matthews 1963; Heirtzler et al. 1968). The use of the polarity sequence preserved in the ocean crust allows for dating of the seafloor as well as calculations of spreading rates and investigations of plate histories. Anomalies on land show considerable complexities and many are assumed to be entirely related to induced fields due to the present magnetic field.

But, in many cases, there is a significant component of remanent magnetization interacting with the induced field, and depending on the remanent direction and intensity, enhancing or at times annihilating the induced field (McEnroe et al. 2009). An excellent case of remanent magnetization producing strong aeromagnetic anomalies comes from the Rogaland Igneous Complex in southern Norway (McEnroe et al. 2008), where anorthosites and a layered intrusion have intense remanence roughly antiparallel to the present field, producing large negative anomalies (Fig. 4). Exploration for natural resources by aeromagnetic surveys has been widely used since the 1950s. The classic work by Balsley and Buddington (1958) and later by McEnroe and Brown (2000) in the oxide rich area of the Adirondack Mountains of New York required a detailed knowledge of the RM for accurate interpretation of the data. Understanding the contribution of RM to magnetic anomalies will be crucial for future exploration as the accuracy and resolution of these surveys increase. Magnetic surveys will also be at the forefront for geological mapping of Earth and other planets.

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Summary Remanent magnetization is a permanent magnetization residing in earth materials that is gained when the material is formed or altered in the presence of the Earth’s magnetic field. Measurable RM is founding in most rocks, ranging from metamorphic gneisses to lava flows to marine and lake sediments. The remanence is carried by iron oxides and sulfides, but magnetite and hematite are by far the most common and the most important remanence carriers. RM is categorized and described by various methods of acquisition; TRM for remanence produced as minerals cool through their Curie temperatures, DRM for the remanence obtained as detrital grains of magnetic minerals align during the deposition process, and CRM for the process where remanence develops from chemical changes in the rock. RM forms the basis for research in paleomagnetism, where the direction, intensity, and age of the remanence are determined and interpreted. It is also useful in experimental studies of rock magnetism, and in the investigation of anomalies in global, regional, and local magnetic field measurements.

Cross-References ▶ Magnetic Anomalies: Interpretation ▶ Paleomagnetic Field Intensity ▶ Paleomagnetism, Magnetostratigraphy ▶ Paleomagnetism, Measurement Techniques and Instrumentation ▶ Paleomagnetism, Polar Wander ▶ Paleomagnetism, Principles

Bibliography Balsley JR, Buddington AF (1958) Iron-titanium oxides minerals, rocks and aeromagnetic anomalies of the Adirondack area, New York. Econ Geol 53:777–805 Buddington AF, Lindsley DH (1964) Iron-titanium oxide minerals and synthetic equivalents. J Petrol 5:310–357 Burton BP (1991) Interplay of chemical and magnetic ordering. In: Lindsley DH (ed) Oxide minerals. Mineralogical Society of America, Washington, DC, pp 303–321 Burton BP, Robinson P, McEnroe SA, Fabian K, Boffa Ballaran T (2008) A low-temperature phase diagram for ilmenite-rich compositions in the system Fe2O3-FeTiO3. Am Mineral 93:1260–1272 Butler RF (1992) Paleomagnetism: magnetic domains to geologic terranes. Blackwell Scientific, Boston. On-line version: http://www. geo.arizona.edu/Paleomag/book/ Cox A (1961) Anomalous remanent magnetization of basalt. US Geol Surv Bull 1083E:131–160 Dunlop DJ (ed) (1977) Origin of thermoremanent magnetization. Center for Academic Publications Japan, Tokyo Dunlop DJ, Özdemir Ö (1997) Rock magnetism – fundamentals and frontiers. Cambridge University Press, Cambridge

Remanent Magnetism Dzialoshinskii IE (1957) Thermodynamic theory of “weak” ferromagnetism in antiferromagnetic substances. Sov J Exp Theor Phys 5:1259–1272 Ghiorso MS, Evans B (2008) Thermodynamics of rhombohedral oxide solid solutions and a revision of the Fe-Ti two-oxide geothermometer and oxygen-barometer. Am J Sci 308:957–1039 Harrison RJ, Becker U, Redfern SAT (2000) Thermodynamics of the R to Rc phase transition in the ilmenite-hematite solid solution. Am Mineral 85:1694–1705 Harrison RJ, Kasama T, White T, Simpson ET, Dunn-Borkowski RE (2005) Origin of self-reversed thermoremanent magnetization. Phys Rev Lett 95. https://doi.org/10.1103/PhysRevLett95.268501 Heirtzler JR, Dickson GO, Herron EM, Pitman WC III, Le Pichon X (1968) Marine magnetic anomalies, geomagnetic field reversals, and motions of the ocean floor and continents. J Geophys Res 73:2119–2136 Ishikawa Y, Akimoto S (1957) Magnetic properties of the FeTiO3-Fe2O3 solid solution series. J Phys Soc Jpn 12:1083–1098 Ishikawa Y, Syono Y (1963) Order-disorder transformation and reverse thermoremanent magnetization in the FeTiO3-Fe2O3 system. J Phys Chem Solids 24:517–528 McElhinny MW, McFadden PL (2000) Paleomagnetism: continents and oceans. Academic, San Diego McEnroe SA, Brown LL (2000) A closer look at remanence-dominated anomalies: Rock-magnetic properties and magnetic mineralogy of the Russell Belt microcline-sillimanite gneisses, Northwest Adirondacks Mountains, New York. J Geophys Res 105:16437–16456 McEnroe SA, Robinson P, Panish P (2001) Aeromagnetic anomalies, magnetic petrology and rock magnetism of hemoilmenite- and magnetite-rich cumulates from the Sokndal Region, South Rogaland, Norway. Am Mineral 86:1447–1468 McEnroe SA, Brown LL, Robinson P (2008) Remanent and induced magnetic anomalies over a layered intrusion: effects from crystal fractionation and magma recharge. Tectonophysics. https://doi.org/ 10.1016/j.tecto.2008.11.021 McEnroe SA, Fabian K, Robinson P, Gaina C, Brown LL (2009) Crustal magnetism, lamellar magnetism and rocks that remember. Elements. https://doi.org/10.2113/gselements.5.4.241 Nagata T (1961) Rock magnetism. Maruzen, Tokyo Néel L (1955) Some theoretical aspects of rock magnetism. Adv Phys 4:191–243 O’Reilly W (1984) Rock and mineral magnetism. Blackie, Glasgow Pullaiah GE, Irving E, Buchan KL, Dunlop DJ (1975) Magnetization changes caused by burial and uplift. Earth Planet Sci Lett 28:133–143 Robinson P, Harrison RJ, McEnroe SA, Hargraves R (2002) Lamellar magnetism in the hematite-ilmenite series as an explanation for strong remanent magnetization. Nature 418:517–520 Robinson P, Harrison RJ, McEnroe SA, Hargraves R (2004) Nature and origin of lamellar magnetism in the hematite-ilmenite series. Am Mineral 89:725–747 Robinson P, Fabian K, McEnroe SA (2010) The geometry of ionic arrangements and magnetic interactions in ordered ferri-ilmenite solid solutions and its effect on low-temperature magnetic behavior. G e o c h e m G e o p h y s G e o s y s t . h t t p s : / / d o i . o rg / 1 0 . 1 0 2 9/ 2009GC002858 Shcherbakov V, Shcherbakova V (1983) On the theory of depositional remanent magnetization in sedimentary rocks. Geophys Surv 5:369–380 Stacey FD, Banerjee SK (1974) The physical principles of rock magnetism. Elsevier, Amsterdam Tauxe L (2010) Essentials of paleomagnetism. University of California Press, Berkeley. On-line version: http://magician.ucsd.edu/Essen tials/index.html

Remote Sensing and GIS Techniques for Tectonic Studies Tauxe L, Constable C, Johnson C, Miller W, Staudigel H (2003) Paleomagnetism of the Southwestern U.S.A. recorded by 0–5 Ma igneous rocks. Geochem Geophys Geosyst. https://doi.org/10.1029/ 2002GC000343 Tauxe L, Steindorf JL, Harris A (2006) Depositional remanent magnetization: toward an improved theoretical and experimental foundation. Earth Planet Sci Lett 244:515–529 Verosub KL (1977) Depositional and post-depositional processes in the magnetization of sediments. Rev Geophys Space Phys 15:129–143 Vine FJ, Matthews DH (1963) Magnetic anomalies over ocean ridges. Nature 199:947–949 Yu Y, Tauxe L (2006) Acquisition of viscous remanent magnetization. Phys Earth Planet Inter 159:32–42

Remote Sensing and GIS Techniques for Tectonic Studies Semere Solomon1 and Woldai Ghebreab2 1 Det Norske Veritas, DNV Research and Innovation, Høvik, Norway 2 Department of Geology and Environmental Science, University of Akron, Akron, OH, USA

Synonyms Remote sensing data; Space imagery

Definition Remote sensing

Geographic information system

Lineaments

Tectonics

Refers to small- or large-scale acquisition of information of an object or phenomenon (e.g., earth surface features), by the use of either recording or real-time sensing device (s) such as by way of aircraft, spacecraft, satellite, buoy, or ship that are not in physical contact with the object. Refers to spatial data management and analysis tools that can assist users in organizing, storing, editing, analyzing, and displaying positional and attribute information about geographical data. Any linear features that can be picked out as lines (appearing as such or evident because of contrasts in terrain or ground cover, tone, pattern, size, etc.) in aerial or space imagery. A field of study within geology concerned generally with the structures within the lithosphere of the Earth (or other planets) and particularly with the forces and movements that have operated in a region to create these structures.

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Neotectonics Active fault

Lithology

Geomorphic features

Strain meter

A subdiscipline of tectonics which deals with the current or recent geologic deformations. A planar fracture in rocks with visible displacement or seismic activity that has occurred during the geologically recent period, as opposed to passive faults that show no movements since their formation. The scientific study and description of rocks/ rock types in terms of compositional and physical character of the rock. Landform features such as rivers and peneplains formed by geomorphic processes (e.g., erosion and mass-wasting) that shape them. An instrument used by geophysicists to measure the deformation of the Earth in the field.

Introduction Mapping of lineaments from various space imagery are a commonly used initial step in tectonic studies. The surface expression of geological structures such as faults, joints, dykes, veins, and straight routes of rivers are often displayed or represented in the form of lineaments in aerial photographs or on remotely sensed images. In addition, remote sensing data are useful for detecting lithology and geomorphic features, which are, in turn, important in assessing the mechanisms of faulting. The most common and widely used remote sensing techniques include: Satellite Pour l’Observation de la Terre (SPOT), which is a high-resolution optical imaging Earth Observation Satellite (EOS), Landsat satellite both multispectral scanner (MSS), and thematic mapper (TM) including enhanced thematic mapper (ETM), Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER), Shuttle Radar Topographic Mission (SRTM) and/or SAR Interferometry (InSAR), and Global Positioning System (GPS). Data acquired from optical, thermal, radar images or digital terrain elevation at the same or different sensor platforms or devices are all useful for tectonic studies. Lineament mapping and analysis is a combined effort of technology and field evidence (Fig. 1). The availability of multispectral and multi-sensor data with synoptic coverage and use of different image enhancement techniques provides an opportunity to prepare more reliable and comprehensive lineament maps useful for tectonic studies. Field studies or ground truthing of orientation measurements of structural features and identifying rock types provide support for verifying remote sensing derived tectonic interpretation. Geographic information system (GIS) facilitates integration

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Remote Sensing and GIS Techniques for Tectonic Studies

Tectonic study Remote sensing methods

Field methods

Lineaments Geomorphicfeatures Lithology

Structural field data

Data integration and analysis in a GIS

Tectonic interpretation maps

Remote Sensing and GIS Techniques for Tectonic Studies, Fig. 1 A simplified flow chart showing the methodology of tectonic study using remote sensing and field data in a GIS framework

of all data types and provides a strong basis for tectonic interpretation of structural features.

emits microwaves or radio waves of the electromagnetic energy to detect objects. The Global Positioning System (GPS) is a US space-based global navigation satellite system. GPS is made up of three parts: satellites orbiting the Earth; controlling and monitoring stations on Earth; and the GPS receiver that reads broadcasted signals from space sent by GPS satellites. Each GPS receiver then provides threedimensional location based on latitude, longitude, and altitude, and the time that are useful for tectonic studies. Digital elevation models (DEM) may be prepared in a number of ways, but they are frequently obtained by remote sensing rather than by direct survey. One powerful technique for generating digital elevation models is InSAR: two passes of a radar satellite (such as RADARSAT-1 or TerraSAR-X), or a single pass if the satellite is equipped with two antennas (like the SRTM instrumentation), suffice to generate a digital elevation map with extensive areal coverage and high resolution (10–30 m). Alternatively, other kinds of stereoscopic pairs can be employed using routine photogrammetric techniques, where two optical images acquired with different angles taken from the same pass of an airplane or an EOS such as the high-resolution stereo of SPOT5 or the visible and near-infrared band of ASTER (e.g., Hiranoa et al. 2003).

Remote Sensing Data and Method Mapping of geological structures from various space imageries is a commonly used initial step in tectonic studies. The routine procedure for extracting tectonic features from digital remote sensing data usually involves initial digital image enhancement followed by manual interpretation (Suzen and Toprak 1998). There have been significant approaches for the evaluation and automatic detection of lineaments and curvilinear features from satellite images (Karnieli et al. 1996). However, the human expert judgment still remains to be an asset for lineament detection and interpretation. Satellite-based remote sensing data may be optical image (e.g., SPOT and Landsat), which usually senses the behavior of visible, ultraviolet, and infrared light components of the electromagnetic energy used in imaging. The data could also be thermal image in which the devices sense the thermal energy of the electromagnetic spectrum. The Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) is an example of modern remote sensing device that senses the behavior of visible, ultraviolet, infrared light, and thermal infrared components of the electromagnetic energy spectrum. The remote sensing satellite may also acquire radar images such as the Shuttle Radar Topographic Mission (SRTM) and/or Interferometric Synthetic Aperture Radar (InSAR) in which the system has a transmitter that

The surface expression of lineaments displayed in aerial photographs or remote sensing data (Fig. 2) may be passive faults, showing no movements since their formation or may be active faults with movements created by either earthquakes or ongoing uplifting. Earthquakes and uplifts are related to tectonic movements in the subsurface. The surface expression of these movements as structures visible on satellite imageries can be mapped and used to investigate the tectonic movements. Time series thermal satellite remote sensing datasets were used to detect pressure buildup due to tectonic activities and associated subsurface degassing that created changes in the thermal regime prior to an earthquake event (Saraf and Choudhury 2005). Such studies are useful in locating earthquakes’ epicenters and understanding the fault mechanics involved in the deformation during earthquake occurrences and uplifting processes. DEM acquired from satellites such as SRTM data or optical images are useful for derivation of topographic parameters such as slope and surface curvatures (geomorphic indices) for mapping drainage networks and lineament extraction (Zizioli 2008; Demirkesen 2008). These morphotectonic parameters can be used as indicators of active tectonics in a region. For instance, rivers are sensitive to changes in tectonic deformation, adjusting their routes over different periods of time depending on the physical properties of the host rocks, climatic effects, and tectonic

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Remote Sensing and GIS Techniques for Tectonic Studies, Fig. 2 Different types of remote sensing data showing conspicuous NE-SW trending lineaments from 12 km  12 km area (a) SPOT band

3, (b) digitally enhanced Landsat TM band 5, and (c) minimum curvature image derived from SRTM digital elevation model. The lineaments represent both extensional and strike-slip tectonics

activity. Thus, the drainage system of a region records the evolution of tectonic deformation. Quantitative measurements of a number of geomorphic indices are commonly used as a reconnaissance tool in tectonic geomorphology studies to identify areas experiencing tectonic deformation. Combinations of different geomorphic indices derived from DEM data allow the quantification of surface deformation both on maps and on stream profiles. It is possible to assess and quantify the rate of uplift by studying the influence of faults on stream geometry and changes in direction as a result of neo-tectonic activity (Gloaguen et al. 2008). Such thematic geomorphic indices can easily be incorporated into a GIS tool for further analysis and interpretation. Interferometric Synthetic Aperture Radar (InSAR) is a radar technique used in geodesy and remote sensing. This geodetic method measures the line-of-sight displacements and uses two or more synthetic aperture radar (SAR) images to generate maps of surface deformation or digital elevation, using differences in the phase of the waves returning to the satellite, or aircraft. The technique can potentially measure millimeter-scale or even lower changes in deformation over time spans of days to years. InSAR can be used to measure ground movements due to earthquakes and to monitor creep and strain accumulation on faults (e.g., Stramondo et al. 2005). Due to its easy integration within a GIS environment, DEM data is important in such analyses. Time series data, a sequence of space imagery shots taken at different times, is a requirement in such studies. Then the changes are detected at observation points that have a fixed geographical location for later comparison and monitoring purposes. GPS measurements, among other applications, are now in use to determine the motion of the Earth’s tectonic plates and deformation around active faults and volcanoes. It provides a powerful means to directly measure the kinematic pattern of present-day crustal deformation by setting up several regional

GPS network stations and monitoring the changes in geographical locations and heights at the control points (Jouanne et al. 1998). GPS can measure movement of faults or tectonic plates to within the precision of a few millimeters to approximately 1 cm over baseline separations of hundreds of meters to thousands of kilometers (Segall and Davis 1997). The threedimensional nature of GPS measurements allows one to determine vertical as well as horizontal displacement at the same time and place. Field methods such as the use of strain meters can provide far greater strain sensitivity than does GPS. However, strain meters cannot offer the spatial coverage and longterm stability as GPS do. InSAR measurements are tremendously exciting because of their unparalleled spatial coverage. InSAR and GPS are complementary in that GPS provides long-term stability, vector displacements, and better temporal coverage as compared to the extensive spatial coverage provided by SAR/InSAR and thus results from one system are directly relevant to the others (Segall and Davis 1997).

Tectonic Studies Tectonics deals with large-scale deformations of the Earth’s lithosphere. The lithosphere is broken up into several tectonic plates that move relative to one another. Plate tectonics theory explains the formation of the plates and the origin of the forces responsible for the motion of these plates. There are three types of plate boundaries characterized by their relative motion: each having different types of surface phenomena including volcanic activities, earthquakes, and faults. The tectonic styles at the different plate boundaries are: • Extensional tectonics that results in the formation of normal faults (Fig. 3) and associated tectonic processes due to the thinning/stretching of the lithosphere.

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1328 Remote Sensing and GIS Techniques for Tectonic Studies, Fig. 3 Fault block models and their equivalent outcrop-scale structural features from field study formed by different tectonic processes: (a) Normal faults, (b) strike-slip faults, and (c) thrust faults. In the sketches, the maximum principal stress (s1), the vertical stress (sv), and minimum principal stress, (s3) are shown. Field structures in (a) and (c) are cross-sectional views and (b) is plain view

Remote Sensing and GIS Techniques for Tectonic Studies σv

σ3 σ1

a

Normal fault: σv > σ1 > σ3 σv σ3

b

σ1 Strike-slip fault: σ1 > σv > σ3 σv σ1

c

σ3 Reverse fault: σ1 > σ3 > σv

• Strike-slip tectonics that results in strike-slip faults (Fig. 3b) in zones of lateral displacement within the lithosphere. • Compressional tectonics that results in the formation of thrust faults (Fig. 3c) and associated tectonic activities due to the shortening and thickening of the lithosphere.

The difference between structural geology and tectonics is the scale of observation. Structural geology deals with smallscale rock deformation, while tectonics is more concerned with larger features. Thus, remote sensing methods help in detecting large-scale tectonic features while field studies are necessary in investigating how the mapped lineaments control the tectonics of a region by looking into the details at mesoscopic-scale. All remote sensing methods used for lineament extraction cannot describe the nature of the tectonics in a region without ground truthing. It is generally recognized that lineament or fracture trace analysis used in conjunction with tectonic investigations is not credible without field verification. In regional scale studies, although field checking of each lineament is impossible due to significant cost and time involved (Mabee et al. 1994), systematic field checking is still necessary to decipher and derive confidence estimates of the nature of lineaments. The field data include observation and measurement of orientations of joints, dykes, and faults. Field evidences presented in the form of figures and sketches (e.g., Fig. 3a–c) can provide good support for tectonic

interpretation. The structural data collected from the field need to be grouped according to rock type and proximity of sampling points to take into account the geographic locations of observation points. In order to understand the structural– geological significance of fracture arrays, it is a common practice to subdivide them into separate sets on the basis of orientation. Rose diagrams are commonly used to reveal the orientations of vertical and steeply dipping joints, faults, and dykes observed at different scales. Many workers (e.g., Arlegui and Soriano 1998; Solomon and Ghebreab 2006) have effectively used remote sensing data supported by field studies to map regional spatial distributions of lineaments toward understanding their tectonic origin.

Geographic Information System (GIS) GIS is a tool in spatial data management and analysis that can assist users in organizing, storing, editing, analyzing, and displaying positional and attribute information about geographical data. GIS facilitates integration of all data types and allows a better understanding of the tectonic interpretation of a region under study. The full potential of remote sensing and GIS can be utilized when an integrated approach is adopted. Integration of the two technologies has proven to be an efficient tool in many studies, specifically in tectonic studies (e.g., Ratzinger et al. 2006; Gloaguen et al. 2008).

Remote Sensing and GIS Techniques for Tectonic Studies

For instance, different geomorphic indices derived from DEM and a number of factors influencing each index can be integrated in a GIS that will allow a better interpretation of the results. Backed with remote sensing data and groundchecking, it is possible to outline the deformed area and the local strain intensity (relative uplift rates) and thus infer the neotectonic history of a region. Jain and Verma (2006) applied a combination of remote sensing and GIS for mapping active tectonic intensity zone in India. Superimposing a lineament map over a lithological map in a GIS environment provides an insight into the nature of tectonics of a region. Visible horizontal displacement along lineaments (e.g., Fig. 3b) as evidenced from movement in rock units can be used to depict strike-slip fault tectonism. Draping the lithological map or digitally enhanced color image over a three-dimensional DEM data in a GIS environment can also reveal the faulting mechanisms such as normal faulting due to visible subvertical displacement of lithological units (Fig. 4). In order to classify the fault mechanics involved in deformation, an overlay of historical earthquake data on mapped tectonic features or lineaments can be integrated in a model derived from remote sensing data with results such as from differential radar interferometry in a GIS environment. A GPS-derived parameters can be easily integrated in a GIS framework to provide a powerful means of directly measuring the dynamics of present-day crustal deformation by monitoring the changes in geographical (horizontal) locations and heights (vertical displacements) at well-positioned control points. All these examples demonstrate the application of remote sensing and GIS techniques for mapping both active and passive faults. However, to better understand the types of faults involved, field checking still remains a requirement.

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Summary Remote sensing and GIS are complementary to each other and efficient techniques for lineament mapping, and hence for tectonic studies. Advanced remote sensing techniques such as InSAR and GPS have revolutionized tectonic studies by directly measuring the motion of faults associated with earthquakes and rates of uplifts. Field studies are, however, needed to identify the types of lineaments and correlate them to the remotely acquired data. Comparison of orientations of major structures or lineaments such as joints, dykes, and faults obtained from field measurements, and plotted on rose diagrams facilitates their tectonic interpretation. GIS is a very time-saving and cost-effective tool once the database is established. Integrating data of different layers such as lineament and lithological maps, and geomorphic indices acquired, for example, from DEM, GPS measurements and field studies in a GIS environment followed by spatial analysis of the data allows correlation between different parameters for unraveling the nature of the tectonics of a region. Moreover, integrating historical dataset from past earthquake events, if any, including an understanding of the regional tectonic setting is essential in facilitating interpretation of results. Overall, combining remote sensing, GIS, and field techniques provides a powerful tool for understanding the tectonic framework of a region, and assessing the resulting fault mechanisms.

Cross-References ▶ Remote Sensing, Applications to Geophysics ▶ SAR Interferometry

Bibliography

Remote Sensing and GIS Techniques for Tectonic Studies, Fig. 4 Three-dimensional synoptic view of a digitally processed remote sensing data created by draping from ASTER bands (13, 5, and 3) in red, green, and blue order over the SRTM DEM data in a GIS environment. Note the east-west normal fault that dips to the north

Arlegui LE, Soriano MA (1998) Characterizing lineaments from satellite images and field studies in the central Ebro basin (NE Spain). Int J Remote Sens 19:3169–3185 Demirkesen AC (2008) Digital terrain analysis using Landsat-7 ETM+ imagery and SRTM DEM: a case study of Nevsehir province (Cappadocia), Turkey. Int J Remote Sens 29:4173–4188 Gloaguen R, Käßner A, Wobbe F, Shazah F, Mahmood A (2008) Remote sensing analysis of crustal deformation using river networks. In: IEEE international geoscience & remote sensing symposium, paper: TH1.101.1, Boston, 6–11 July Hiranoa A, Welcha R, Langb H (2003) Mapping from ASTER stereo image data: DEM validation and accuracy assessment. ISPRS J Photogramm Remote Sens 57:356–370 Jain S, Verma PK (2006) Mapping of active tectonics intensity zones using remote sensing and GIS. J Indian Soc Remote Sens 34:131–142 Jouanne F, Genaudeau N, Ménard G, Darmendrail X (1998) Estimating present-day displacement fields and tectonic deformation in active mountain belts: an example from the Chartreuse Massif and

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1330 the southern Jura Mountains, Western Alps. Tectonophysics 296:403–419 Karnieli A, Meiseis A, Fisher L, Arkin Y (1996) Automatic extraction and evaluation of geological linear features from digital remote sensing data using the Hough transform. Photogramm Eng Remote Sens 62:525–531 Mabee SB, Hardcastle KC, Wise DW (1994) A method of collecting and analyzing lineaments for regional-scale fractured bedrock aquifer studies. Ground Water 32:884–894 Ratzinger K, Neuhäuser B, Papthoma M (2006) Hazard mapping of earthquake triggered landslides. In: First European conference on earthquake engineering and seismology, Geneva, p 1405, 3–8 Sept Saraf AK, Choudhury S (2005) Thermal remote sensing technique in the study of pre-earthquake thermal anomalies. J Indian Geophys Union 9:197–207 Segall P, Davis JL (1997) GPS applications for geodynamics and earthquake studies. Annu Rev Earth Planet Sci 25:301–336 Solomon S, Ghebreab W (2006) Lineament characterization and their tectonic significance using Landsat TM data and field studies in the central highlands of Eritrea. J Afr Earth Sci 46:371–378 Stramondo S, Moro M, Tolomei C, Cinti FR, Doumaz F (2005) InSAR surface displacement field and fault modelling for the 2003 Bam earthquake (southeastern Iran). J Geodyn 40:347–353 Suzen ML, Toprak V (1998) Filtering of satellite images in geological lineament analysis: application to a fault zone in Central Turkey. Int J Remote Sens 19:1101–1114 Zizioli D (2008) DEM-based morphotectonics analysis of Western Ligurian Alps. Scientifica Acta 2:44–47

Remote Sensing, Applications to Geophysics Hojjatollah Ranjbar Department of Mining Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

Definition Remote sensing is defined as the acquisition of data using a remotely located sensing device, that relies on the measurements of electromagnetic energy (EM) in the range of 0.4 mm to 1 m, and extraction of information from the data (Gupta 2003; McCloy 2006). Remote sensing can be multilevel (i.e., sensing the surface of material from a distance of few centimeters to millions of kilometers). Taking a spectrum of a leaf by using a spectroradiometer to taking an image by Hubble telescope from a star at a distance of few light years fall in the realm of remote sensing. However, here, we mean the imaging of the Earth’s surface.

Introduction The remote sensing systems can acquire images from the surface of the Earth by either passive or active methods.

Remote Sensing, Applications to Geophysics

Passive systems use the electromagnetic energy that is produced by the sun or any other process like anomalous heat buildup within the Earth. Active remote sensing systems create their own electromagnetic energy that (1) is transmitted from the sensor toward the terrain, (2) interacts with the terrain producing a backscatter of energy, and (3) is recorded by the remote sensor’s receiver (Jensen 2000). The sending and receiving of signals are largely unaffected by atmospheric conditions. Therefore, radar satellites can record the details of the Earth’s surface in cloudy weather condition and any time of the day and night. LIDAR (LIght Detection and Ranging) and SONAR (SOund NAvigation Ranging) are also the active remote sensing systems. It should be noted that remote sensing is a geophysical method as it uses the electromagnetic spectrum as a medium for surveying the Earth’s surface. It is now recognized as a separate branch of science and engineering. The electromagnetic spectrum that is used by remote sensing systems is named according to the wavelength ranges. Gupta (2003) has classified the electromagnetic spectrum into visible (0.4–0.7 mm), near infrared (0.7–1.0 mm), Shortwave infrared (1.0–3.0 mm), mid-infrared (3.0–35 mm), far-infrared (35–1.0 mm), and RADAR (1–100 cm). A remote sensing system that allows sensing the Earth in any wavelength and in any time makes an ideal remote sensing system. An ideal remote sensing system cannot be achieved due to the absorptions of the electromagnetic energy from the Earth’s atmosphere. Each object has unique sets of absorption and reflectance features concerning the wavelengths. The plot is known as spectral signature. Figure 1 shows the spectra of the common surface materials. Although each material at the surface of the Earth has many absorption and reflection features, the present remote sensing systems can only image these materials in limited wavelengths because the atmosphere is not clear for all the wavelengths. There are different sensors that are used onboard the satellites or in an aeroplane. The present remote sensing systems that orbit the Earth are imaging the Earth’s surface in visible, near infrared, shortwave infrared, thermal infrared, and RADAR regions of EM spectrum. The reader can refer to Jensen (2000) and Lillesand et al. (2004) for different sensors and platforms specifications. A digital image consists of pixels. Each pixel is representative of a surface area that defines the spatial resolution of the system. Enhanced Thematic Mapper Plus (ETM+) multispectral data has a resolution of 30 m (each pixel is equivalent to 900 m2 ground area), while Quickbird has a resolution of 60 cm for its panchromatic image (each pixel is equivalent to 0.36 m2 on the ground). As these images are in digital formats, several simple or complex mathematical operations can be applied on them in order to enhance specific features from these images. This is

Remote Sensing, Applications to Geophysics

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Remote Sensing, Applications to Geophysics, Fig. 1 Typical spectral reflectance curves for selected common natural objects (Gupta 2003)

100

80 Limonite

Reflectance (%)

60 Dry soil 40 Vegetation 20 Water (clear) 0

0.4

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0.8

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1.2

1.4

1.6

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Remote Sensing, Applications to Geophysics, Fig. 2 Enhancement of faults and other linear features by using directional filter. (a) Original satellite color image, (b) filtered image

called image processing operation. Image filtering (Mather 2001) is a common image processing technique that enhances the linear features. Figure 2 shows the enhanced faults in the southern part of Iran in a filtered image.

Application of Remote Sensing to Geophysics Remote sensing data can be used for designing the survey grid and line in geophysical surveys, verifying, and helping a better interpretation of the geophysical methods. Remote sensing images can be combined to form color images in various scales that depend on the spatial resolution of the

remote sensing system. These color images can be draped over a digital elevation model (DEM) to give a threedimensional (3-D) model of the area (Fig. 3). These 3-D models can be used in both land and airborne geophysical surveys for designing the survey lines. The 3-D model can provide information such as the terrain ruggedness, manmade installations, surface conditions, transportation routes, etc. The surface of the Earth is continuously deformed due to the tectonic activities over geological time that causes the formation of structures such as fold and fault. Geophysical methods such as magnetic, magneto-telluric, gravity, and seismic methods are used for studying these structures.

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Remote Sensing, Applications to Geophysics 58 10 E

58 30 E

29 10 N

29 00 N

Remote Sensing, Applications to Geophysics, Fig. 3 3-D view of folding due to the movement of Arabian plate against Iranian microplate. The force direction is shown by an arrow. Landsat color image (Band7 in red, band 4 in green, and band 2 in blue) is draped over DEM of the area

These methods are subsurface methods. Remote sensing techniques are useful in order to study these structures and to interpret the tectonic activities in the regional and local scale at the surface of the Earth. In order to obtain a clear picture of the structures, the combined interpretation of remote sensing and geophysical data is helpful, if the geological structures have any trace in the satellite images. Remote sensing applications to tectonic and structural studies are reported by many researchers (e.g., Harris 1991; Philip 1996; Saintot et al. 1999; Chernicoff et al. 2002; Chatterjee 2003; Raharimahefa and Kusky 2006; Walker 2006). Figure 3 shows a part of Zagros simply folded belt that the folds axis directions indicate a force direction because of the subduction of Arabian plate beneath the Iranian plate. These structures are also studied by using gravitational, magnetic, and seismic methods. Remote sensing is also useful for interpretation of geophysical data. Alteration process causes the changes in mineralogical content of rocks due to the reaction of hot solution with the rocks. Porphyry copper mineralization is associated with these alteration zones (Evans 1993). Both radiometric and remote sensing methods can be used for alteration detection (Ranjbar et al. 2001, 2010; Ranjbar and Honarmand 2004). The minerals present in sericite and k-feldspar have a higher gamma radiation. We cannot differentiate these minerals by geophysical methods. Remote sensing method is capable of recognizing the minerals responsible for the higher radiation in geophysical data. There are also minerals in the alteration zones that have very low radiation to be detected by geophysical methods. Kaolinite, Calcite, epidote, and chlorite are such minerals. These minerals are recognized by remote sensing methods. Remote sensing techniques can also be used in combination with other geophysical methods such as magnetic, electromagnetic, and electrical methods in helping with the interpretation.

Remote Sensing, Applications to Geophysics, Fig. 4 An interferogram that shows the deformation along a fault due to the Bam earthquake in Iran. The rings that are closer, indicating more deformation. (After Ye 2005)

Another area that received much attention is the field of active faults study by the combined use of interferometry and geophysical data. RADAR (Radio Detection And Ranging) images are more useful for detection of the geological structures than the images acquired in the optical ranges of EM spectrum, because the RADAR pulses are illuminating the ground from a specific direction. The RADAR wavelengths are in order of few tens of centimeters. Synthetic Aperture Radar interferometry (INSAR) is a branch of remote sensing that has emerged in the recent years that deals with the deformation in the Earth’s crust. There are several papers published about INSAR method in structural and tectonic fields (e.g., Klees and Massonnet 1999; Wright et al. 2001, 2004; Ye 2005; Bürgmann et al. 2006; Taylor and Peltzer 2006). Figure 4 shows an interferogram of a fault in Bam area that triggered an earthquake in 2003. This fault is detected by remote sensing and geophysical methods.

Summary Remote sensing images due to their multi-temporal, multispectral, and synoptic views are extensively used for solving problems related to mineral exploration, tectonics, vegetation cover studies, surface changes, earthquake studies, weather and volcanic eruption forecasting, etc. The images can be acquired in different wavelength ranges such as visible, near infrared, thermal, and Radar ranges of electromagnetic spectrum. We are now able to recognize many minerals from the space by using their spectral properties. Remote sensing data can be used for designing the survey grid and line in

Remote Sensing, Applications to Geophysics

geophysical surveys, verifying, and helping a better interpretation of the geophysical methods. Small and large structures are easily recognizable in the images, although we often need to enhance the images for recognizing these structures.

Cross-References ▶ Earthquake Precursors and Prediction ▶ Earthquake Rupture: The Inverse Problem ▶ Earthquakes and Crustal Deformation ▶ Gravity Method, Airborne ▶ Magnetic Anomalies: Interpretation ▶ Magnetic Methods, Airborne ▶ Plate Tectonics, Precambrian ▶ Remote Sensing and GIS Techniques for Tectonic Studies ▶ SAR Interferometry ▶ Seismic Data Acquisition and Processing ▶ Subduction Zones ▶ Very Long Baseline Interferometry

Bibliography Bürgmann R, Hilley G, Ferretti A, Novali F (2006) Resolving vertical tectonics in the San Francisco Bay Area from permanent scatterer InSAR and GPS analysis. Geology 34:221–224 Chatterjee RS (2003) Structural pattern of Holenarsipur Supracrustal Belt, Karnataka, India as observed from digitally enhanced highresolution multi-sensor optical remote sensing data aided by field survey. Int J Appl Earth Obs Geoinf 4:195–215 Chernicoff CJ, Richards JP, Zappettini EO (2002) Crustal lineament control on magmatism and mineralization in northwestern Argentina: geological, geophysical, and remote sensing evidence. Ore Geol Rev 21:127–155 Evans AM (1993) Ore geology and industrial minerals: an introduction. Blackwell, Oxford Gupta RP (2003) Remote sensing geology. Springer, Berlin Harris JR (1991) Mapping of regional structure of eastern Nova Scotia using remotely sensed imagery: implications for regional tectonics and gold exploration. Can J Remote Sens 17:122–136 Jensen JR (2000) Remote sensing of the environment: an earth resource perspective. Prantice Hall, New Jersey

1333 Klees R, Massonnet D (1999) Deformation measurements using SAR interferometry: potential and limitations. Geol Mijnb 77:161–176 Lillesand T, Keifer RW, Chipman JW (2004) Remote sensing and image interpretation. Wiley, New York Mather PM (2001) Computer processing of remotely-sensed images. An introduction. Wiley, Chichester McCloy KR (2006) Resource management information systems: remote sensing, GIS and modelling, 2nd edn. Taylor and Francis, Boca Raton Philip G (1996) Landsat thematic mapper data analysis for quaternary tectonics in parts of the Doon valley, NW Himalaya, India. Int J Remote Sens 17:143–153 Raharimahefa T, Kusky TM (2006) Structural and remote sensing studies of the southern Betsimisaraka Suture, Madagascar. Gondwana Res 10:186–197 Ranjbar H, Honarmand M (2004) Integration and analysis of airborne geophysical and ETM + data for exploration of porphyry type deposits in the central Iranian Volcanic Belt, using fuzzy classification. Int J Remote Sens 25:4729–4741 Ranjbar H, Hassanzadeh H, Torabi M, Ilaghi O (2001) Integration and analysis of airborne geophysical data of the Darrehzar area, Kerman province, Iran, using principal component analysis. J Appl Geophys 48:33–41 Ranjbar H, Masoumi F, Carranza EJM (2010) Evaluation of geophysics and spaceborne multispectral data for alteration mapping in Sar Cheshmeh mining area, Iran. Int J Remote Sens. https://doi.org/10. 1080/01431161003745665 Saintot A, Angelier J, Jean Chorowicz J (1999) Mechanical significance of structural patterns identified by remote sensing studies: a multiscale analysis of tectonic structures in Crimea. Tectonophysics 313:187–218 Taylor M, Peltzer G (2006) Current slip rates on conjugate strike-slip faults in central Tibet using synthetic aperture radar interferometry. J Geophys Res 111:1–16 Walker RT (2006) A remote sensing study of active folding and faulting in southern Kerman province, S.E. Iran. J Struct Geol 28:654–668 Wright T, Fielding E, Parsons B (2001a) Triggered slip: observations of the 17 August 1999 Izmit (Turkey) earthquake using radar interferometry. Geophys Res Lett 28:1079–1082 Wright T, Parsons B, Fielding E (2001b) Measurement of interseismic strain accumulation across the North Anatolian Fault by satellite radar interferometry. Geophys Res Lett 28:2117–2120 Wright TJ, Lu Z, Wicks C (2004) Constraining the slip distribution and fault geometry of the Mw 7.9, 3 November 2002, Denali fault earthquake with interferometric synthetic aperture radar and global positioning system data. Bull Seismol Soc Am 94:S175–S189 Ye X (2005) Bam earthquake: surface deformation measurement using radar interferometry. Acta Seismol Sin 18:451–459

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SAR Interferometry Masato Furuya Department of Natural History Sciences, Hokkaido University, Sapporo, Japan

Synonyms Differential InSAR (abbreviated as D-InSAR); Interferometric SAR (abbreviated as InSAR); Radar interferometry; SAR interferometry

Definition Radar

InSAR

Acronym standing for Radio Detection and Ranging. A technique to detect any targets and measure the distance to them, based on the roundtrip time of microwave (radio wave) pulses between the antenna and the targets. SAR. Acronym standing for Synthetic Aperture Radar. A technique to image any ground surfaces, using airborne or spaceborne radar sensor. Its high spatial resolution is achieved by collecting numerous return pulses from each target in sight and by effectively synthesizing large antenna size. Acronym standing for Interferometric SAR. A technique to image surface topography and ground displacements, using phase values of two or more SAR images.

Introduction Crustal deformation data have been traditionally acquired by ground-based geodetic techniques such as leveling, triangulation, and electro-optic distance measurement. More © Springer Nature Switzerland AG 2021 H. K. Gupta (ed.), Encyclopedia of Solid Earth Geophysics, https://doi.org/10.1007/978-3-030-58631-7

recently, global positioning system (GPS) has become a standard tool for high-precision crustal deformation measurement, and provided us with a wealth of data to study plate tectonics, earthquakes, volcanic activities, and atmospheric and hydrological loading deformation. All these techniques, however, require in situ benchmarks, and thus prevent us from observing inaccessible areas. Interferometric SAR (InSAR) was, therefore, regarded as a surprising and revolutionary technique when Massonnet et al. (1993) first showed an image of the co-seismic deformation associated with the 1992 M7.3 Landers earthquake, because the raw data was completely acquired on a spaceborne sensor. Another big surprise for the community was its incredibly high spatial resolution, which no other geodetic techniques were possible to achieve in practice. Nowadays, InSAR users have proliferated in a worldwide community and applied to a variety of geophysical problems. A number of excellent and extensive reviews for advanced students and researchers are already published (Bamler and Hartl 1998; Massonnet and Feigl 1998; Burgmann et al. 2000; Hanssen 2001; Pritchard 2006; Simons and Rosen 2007; Zhou et al. 2009). I therefore tried to make this article much shorter and more introductory, but it still includes necessary and useful concepts, ranging from the fundamentals of SAR/InSAR imagery to more up-to-date topics.

Fundamentals of SAR Imaging and SAR Data SAR satellite flies over at an altitude of hundreds of km, repeating transmission and reception of microwave pulses. The along-track and across-track axes are almost identical to the azimuth and range axis in the acquired radar image. The area illuminated on the ground is called swath, whose width spans roughly 50–100 km in the standard stripmap (or strip) mode with an incidence angle of 20–50
(Fig. 1). While previous SAR applications are mostly derived from the stripmap mode, another imaging mode, ScanSAR, is also promising because it covers much wider swath width,

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ScanSAR mode

k trac ion) und direct o r h G t imu (Az

θoff.nadir

Stripmap mode

Sla

H

nt r an

Sla

H

ge

θoff.nadir

nt r an ge

idth

hw

t Swa nge

r ra

Nea

Far

Swath w

idth

Far

e

rang

e

rang

Ground range

SAR Interferometry, Fig. 1 Geometry of SAR imaging. SAR sensor transmits microwave pulses in slant range direction, and receives their reflected pulses. While stripmap mode achieves high spatial resolution

with a fixed off-nadir angle, ScanSAR mode achieves wider imaged area (swath) with multiple off-nadir angles at the expense of the resolution

300–500 km, by illuminating multiple swaths at the expense of reducing the resolution. ScanSAR is useful for imaging long-wavelength signals associated with, for instance, a magnitude-8-class earthquake (Motagh et al. 2008). Although it was not strictly necessary, satellite-based SAR system has been often placed on a sun-synchronous and nearpolar orbit with an inclination angle of slightly greater than 90
. When the satellite moves to the north (south), we call it is in ascending (descending) orbit. The raw data acquired on SAR sensor are impossible to visually interpret, and require a bit involved processing algorithms; those algorithms are detailed in a couple of text books (e.g., Curlander and McDonough 1991; Cumming and Wong 2005). The first interpretable SAR image is a single-lookcomplex (SLC) image, whose important difference from other optical images is that each pixel consists of a complex (real and imaginary) value, i.e., amplitude and phase. This is because the waveform of each repeated pulse is precisely controlled to be identical, and hence the received pulse provides us with not only a scattering (reflection) intensity but also a phase. The phase data do contain the geometric information from the antenna to the ground targets, and are fully exploited in generating InSAR image as discussed later. However, the phase image itself is usually not as useful as the intensity image because it is again impossible to visually interpret the physical meaning. Meanwhile, the intensity image is often useful and derived from a square-root magnitude of SLC data with spatial averaging called multi-looking. By single-look, it means the finest spatial resolution for both

range and azimuth axis. In the standard stripmap mode, the range and azimuth resolutions are derived as, Dr ¼

c L , and Da ¼ , 2B 2

ð1Þ

respectively; the c, B, and L are the speed of light, the frequency bandwidth of the microwave pulse, and the antenna length along azimuth axis, respectively (Curlander and McDonough 1991; Cumming and Wong 2005). The waveform of each microwave pulse is called chirp signal, whose instantaneous frequency linearly changes by as much as the frequency bandwidth B over the duration of each pulse. It should be noted that the spatial resolution depends neither on the sensor altitude nor the carrier frequency of microwave. Intensity images are often shown in gray scale images, in which strongly (weakly) reflected objects/areas are usually colored as bright (dark). Although they simply look like black-and-white photographs, we should keep in mind that they could be acquired regardless of weather and time because SAR is actively transmitting and receiving microwaves. Also, intensity images are indispensable for high-precision image matching prior to a generation of InSAR image.

Fundamental Principles of InSAR Interferometric SAR (InSAR) is a technique to generate a digital elevation model (DEM) or a ground displacement

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SAR Interferometry, Fig. 2 (a) Geometry of the Young’s experiment. Depending on the path difference, the two coherent waves from the slit, S1 and S2, are in-phase or out-of-phase on the screen, and interference fringes are observed on the right screen. (b) Orbital fringe (flat earth fringe) can be regarded as a 3-D analogue of the Young’s experiment

Young’s fringe

Orbital fringe (Flat earth fringe) P

r1 r2

S1

O

d S2 Path difference

a

image from a pair of SLC images. The term interferogram is often used to represent InSAR image. We can understand the principle of InSAR, recalling the classical Young’s experiment that is known to be a proof of the wave characteristics of the light (Ghilia and Pritt 1998). Two coherent waves out of the slits will generate “stripes” on the wall, called interference fringe (Fig. 2a). We can simulate the fringe if we know the separation of the slits, the distance from each slit to the wall, and the wavelength of the coherent wave. Depending on the path difference, the two coherent waves are in-phase or outof-phase when they reach the screen. Namely, the difference of the phases generates the interference fringe. We may regard the imaging geometry of InSAR as the 3-D Young’s experiment (Fig. 2b). The repeat orbit tracks, the ground surface, and the microwave correspond to the double slits, the screen, and the coherent wave, respectively. Once we get two SLC images, we can generate an initial interferogram, multiplying one SLC image with the complex conjugate of the other SLC image. We then observe similar fringes in the initial interferogram as illustrated in Fig. 2b, which is literally a map of the difference of two SLC phases. For descriptive purposes, the former SLC image is often denoted as master, and the latter SLC image is called slave. At this moment, the slave image must be precisely co-registered (or matched) to the master image (Fig. 3); we will come back to this image co-registration (or image matching) procedure later on. While Fig. 2b shows an initial interferogram over flat areas with parallel orbits, the fringe will appear undulated if the areas are not flat. The fringe over flat areas is called flat Earth fringe (or, orbital fringe), and can be precisely simulated from the pair of orbit data. If we subtract the flat Earth fringes from the initial interferogram, we can extract topographic fringe that can be used to generate DEM. The Shuttle Radar Topography Mission (SRTM) was carried out along this idea in 2001, and generated 3-s resolution DEM over +/ 60
latitudes (Farr et al. 2007). In the case of SRTM, they carried two SAR antennas on the same platform, and thus were able to generate DEM without repeating the previous orbit track. In contrast, all the present SAR satellite systems carry only one

b

Master image

Image

Slave image

Matching

Residual offset is displacement.

SAR Interferometry, Fig. 3 Image registration (matching) of the master and slave images prior to interferogram generation, and the principle of pixel-offset technique to derive large displacements. While longwavelength distortion can be corrected, localized huge displacement remains as residual offset. Courtesy of Tobita et al. (2001a)

antenna with a repeat-pass period of several weeks, which are 11 days for TerraSAR-X, 16 days for COSMO-SkyMed, 24 days for Radarsat-1/2, 35 days for Envisat, and 46 days for ALOS. Therefore, if ground surface undergoes significant deformation during the repeat orbit cycles due, for instance, to earthquake and volcanic eruption, the interferogram will include deformation fringe as well. To extract deformation fringe, we must take out both orbital fringe and topographic fringe, which can be simulated from satellite orbit data and DEM. The deformation fringes represent slant range changes along the radar line-of-sight (LOS), and thus projections of the 3-D displacement vector on the ground along the unitary vectors toward the radar LOS (Fig. 4). The range changes should be interpreted as relative displacements to the

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reference point(s) inside each interferogram. Depending on literatures, they denote differential interferometric SAR (D-InSAR) when the technique is used to detect deformation signals. Recently, however, the term InSAR is often and simply used to represent D-InSAR. Even if no significant ground displacements take place during the repeat-pass period, however, we usually encounter other non-negligible fringes due to the spatial heterogeneities in the propagation delay of microwaves through the atmosphere, the errors in satellite orbit data, and those in DEM. Because these fringes limit the precision and accuracy of SARbased crustal deformation measurement, a couple of correction approaches have been proposed. More advanced time-series analysis techniques have also been developed to overcome the issues, which will be introduced in the last section.

InSAR Processing Image registration (Matching): Before we get an initial interferogram, we must register (or, match) each imaged target in one SLC image to the same target in the other SLC image with a sub-pixel level accuracy, because any ground objects do not usually locate at the same pixel coordinates in each SLC image. This pre-processing is called image registration (or image matching) and prerequisite to be performed prior to generating an initial interferogram. Although a simple polynomial transformation between the range and azimuth coordinates of two SLC images is sufficient in most cases, we need to take into account the effects of 3-D topography when the terrain surface is rugged to eliminate a stereoscopic effect (Michel et al. 1999). When large ground displacements on the order of meters or more take place locally, and if we correct for the longwavelength image distortion using the polynomial transformation, we can detect and quantify those localized displacements as a by-product of image registration without viewing InSAR image (Fig. 3; Tobita et al. 2001a). This approach to detect large displacements is called pixel offset or feature tracking technique, and has been applied to earthquakes, volcanic eruptions, and glacier movements. The advantages of pixel-offset data are twofolds. First, pixel-offset data can quantify large displacements even in such areas that completely loses interferometric coherence, where InSAR data cannot be unwrapped; we describe coherence and unwrapping later below. Secondly, in contrast to InSAR data, pixel-offset data provide us with not only range offset but also azimuth offset component. While the range offset has the same sensitivity to the 3-D displacement vector as InSAR data (Fig. 4), the azimuth offset is a projection of the displacement vector onto the unitary vector perpendicular to the LOS. Hence, the azimuth offset data are complementary to the range offset or InSAR data. Taking advantage of this property,

SAR Interferometry

Radar LOS

Displacement vector

InSAR observable

SAR Interferometry, Fig. 4 InSAR observable is a projection of the displacement vector along the radar line-of-sight (LOS) direction

Fialko et al. (2001) derived a full 3-D displacement map for the 1999 M7.1 Hector Mine earthquake, combining the InSAR data from both ascending and descending track with the azimuth offset data. Using pixel-offset data from both descending and ascending track, Tobita et al. (2001a, b) inferred a 3-D displacement map associated with the 2,000 eruption episode at Usu volcano. Interferometric phase and its relation to geometry: Suppose we have two co-registered SLC images, E1 and E2, acquired from different ranges r1 and r2: E1 ¼ e

jfScatter 

4pr 1 l

E2 ¼ e

jfScatter 

4pr 2 l

e e

ð2aÞ ð2bÞ

Here we assume that the reflection magnitude and scattering phase are constant during the data acquisition time. Then, the interferometric phase f is derived as E1 E2 ¼ e

4pðr1 r 2 Þ l

ð3Þ

or f¼

4p ðr  r 2 Þ l 1

ð4Þ

The last one is the fundamental equation for InSAR, which describes “unwrapped” phase in the initial interferogram. The actual phase in the initial interferogram is “wrapped” into an interval [π, π], and thus has ambiguities of 2πN; N is integer. In order to quantify the ground displacement along radar LOS, we have to perform 2-D phase unwrapping on the interferogram, which is not necessarily straightforward (Bamler and Hartl 1998; Ghilia and Pritt 1998). While the interferometric phase is strictly a phase “difference” of two SLC phases, it is conventional to simply call phase. The factor 4 is to take into account the round-trip distances. Figure 5 is a cross section that is perpendicular to the satellite repeat tracks and passes through the Earth’s center,

SAR Interferometry

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A2

B α

A1 B// H

called off-nadir angle, and is identical to incidence angle if the Earth’s curvature is negligible. The other baseline component B⊥ (or Bperp) is perpendicular to radar LOS and gives us an important criterion for successful InSAR processing as we discuss below. Decorrelation (Coherence): In the actual InSAR data processing, we do not necessarily get clear fringes over the entire area. Depending on the data pairs and places, it is not uncommon that no fringes are observed. To detect clear fringes, the reflected waves received at master and slave acquisitions must be more or less correlated to each other. The degree of correlation is quantified as coherence, and there are two independent decorrelation sources. The first source of decorrelation originates in the imaging geometry. As Fig. 6 indicates, we observed higher (fewer) fringe density as becomes longer (shorter); imagine the case of zero baseline length. The fringe density can be derived from the gradient of phase (Eq. 5) along the range axis:

θ

B⊥

r2

r1

Spheroid

re

@f 4pB⊥ 4pB⊥ 4pB⊥ ¼ þ :  @R lR tan y lðr e þ H Þ sin y lR tan y

r0

ð6Þ

Namely, the fringe density is proportional to the perpendicular baseline B⊥, and inversely proportional to the wavelength l; see Simons and Rosen (2007) for the case with topography. If the fringe density becomes too high to be counted within a range resolution of SAR image, we will not be able to identify any orbital fringes. This type of decorrelation is termed baseline decorrelation (or spatial decorrelation). The critical baseline is given as such a perpendicular baseline that gives a phase gradient 2π over the range resolution Δr; SAR Interferometry, Fig. 5 Geometry of InSAR data acquisition and its relation to the baseline. The A1 and A2 are the satellite’s repeat orbits, and the spatial distance between the A1 and A2 is the baseline B. The initial InSAR phase is proportional to the difference between the ranges, r1, and r2, and hence the Bpara (Eq. 5). The fringe rate (density) along the range axis is proportional to the Bperp (Eq. 6)

and shows a geometry of InSAR data acquisition. The spatial separation of the repeating orbits is termed baseline (or spatial baseline), B; the temporal separation of data acquisition is sometimes called temporal baseline. Because the baseline B is usually much shorter than the ground range distance R, a parallel ray approximation holds (Zebker et al. 1994) and the fundamental Equation 4 can be approximated as follows: f¼

4p 4p ðr  r 2 Þ   B== ¼ B sin ðy  aÞ, l 1 l

ð5Þ

where f and α are defined in Fig. 5, and B// (or Bpara) is a baseline component parallel to the radar LOS. The angle θ is

Bc⊥ ¼

lR tan y : 2Dr

For a typical value of ALOS/PALSAR with l ¼ 23(cm), R ¼ 870(km), θ ¼ 34
, Δr ¼ 5(m), the critical baseline becomes Bc⊥ ¼ 135, 000 (m), which gives an upper limit of B⊥. However, we practically prefer much shorter B⊥, generally less than ~2,000 m for ALOS/PALSAR, because in more realistic situations the effect of topography also comes in. The longer the B, the more sensitive to rugged terrain as Fig. 6 indicates. To eliminate topographic fringes, we need more accurate and higher resolution DEM if the B⊥ becomes longer. Massonnet et al. (1996) proposed an alternative approach that could effectively reduce the B⊥ by a combination of integer multiplied (wrapped) interferograms. For instance, if one interferogram with perpendicular baseline of 300 m is combined with the other interferogram with perpendicular baseline of 290 m with factors 1 and – 1, the effective perpendicular baseline becomes 10 m. The scaling operation,

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shorter

Bperp

SAR Interferometry, Fig. 6 The fringe rate (density) depends on the Bperp; see Eq. (6). The shorter the Bperp, the fewer the observed fringes, and thus better to detect deformation signals. In order words, there is a

SAR Interferometry, Fig. 7 Comparison of two interferograms at Izu-Osima volcano, derived from (left) L-band HH JERS data and (right) C-band VV ERS data. While clear fringes are observed to the left even with 2.5 years temporal baseline, we can recognize the fringes only around the caldera that are covered with few vegetations

longer

limit in the Bperp over which we cannot count the number of fringes. The InSAR image is based on JERS data over Izu-Oshima volcano island, Japan. Original SAR data is copyrighted by JAXA and MITI, Japan

JERS(L-band, 23.5 cm, HH)

ERS(C-band, 5.6 cm, VV)

Izu-Oshima

1992.10.15-1995.5.14

however, also scales the amount of noise, and thus the approach is limited to small integer numbers. The second type of decorrelation is termed temporal decorrelation, which is related to the scattering phase in the Eq. (2a), and originates in how the microwave pulses interact with the physical objects near the ground. We often encounter the temporal decorrelation problem over vegetated areas with C-band (shorter-wavelength) SAR data and/or snow-covered areas; see Fig. 7. It should be recalled that each pixel value in SLC image is a superposition of all the reflected microwaves from all scatterers inside each resolution cell (~5  10 m). Short-wavelength microwave pulses tend to be reflected on the vegetation canopies before reaching the ground surface, and their random motion will result in different scattering phases at different acquisition time, causing temporal decorrelation. On the contrary, longwavelength microwave pulses can more easily reach the ground, which does not move as rapidly as vegetations, and thus the resulting scattering phases will be also stable over time. Besides the selection of wavelength, the polarization of microwave is also essential for better coherence over time. While, most presently, operated satellite-SAR sensors are capable of multi-polarization modes, it was shown that HH-polarization gives better coherence than VV-polarization (Cloude and Papathanassiou 1998). This is because the HH-polarized pulses can more easily penetrate through vegetations.

Izu-Oshima

1995.10.1-1997.9.1

Outlook for InSAR Geodesy Limitations of present InSAR: Although it has a potential to detect tens of km-scale or even larger-scale secular deformation signals on the order of mm/year, InSAR technique has been most successfully applied to detection of spatially localized signals on the order of centimeters or more, such as those associated with earthquakes, volcanic eruptions, and ground subsidence. This is because the artifacts due to inaccurate satellite orbit data and/or microwave propagation delays (advances) in the troposphere (ionosphere) can mask smallamplitude, long-wavelength deformation signals that are similar in both their amplitude and the spatial scale. Although high-precision orbit data are indispensable to correct for the orbital fringes in the initial interferograms, their errors even on the order of 10 cm or less will generate non-negligible long-wavelength artifacts, which usually look like curved surfaces in the entire interferogram (e.g., Hanssen 2001). Conventionally, they are fitted with low-order polynomials and simply taken out unless any sort of stacking or time-series analysis discussed below is applied. While this procedure works to eliminate the artifacts due to orbit errors, it will also take out any long-wavelength geophysically interesting signals such as the inter-seismic, post-seismic, ocean tidal loading, solid- Earth tide, and post-glacial rebound signals. Alternatively, if the ground control points (GCP) are available, where the precision ground deformation data are

SAR Interferometry

available, we can reestimate the baseline, based on those GCP data (e.g., Rosen et al. 1996), but such data are often unavailable in remote areas. One approach to correct for the tropospheric delay signals is to employ the other independent estimates derived from either the GPS-based tropospheric delay estimates (e.g., Onn and Zebker 2006) or the output results from high-resolution numerical weather model (e.g., Foster et al. 2006). These so-called calibration approaches are, however, not easily applicable. The dense ground-based GPS network is limited to a few areas in the world. Also, high-resolution numerical weather model still needs significant computational resources. Besides the tropospheric delay problem, the effects of ionosphere on both interferograms and pixel-offset images were clearly recognized in the results of the 2008 Wenchuan earthquake (Mw7.9), China, derived from ALOS/PALSAR (Kobayashi et al. 2009; Raucoules and de Michele 2010), although they were pointed out in polar region many years ago (e.g., Matter and Gray 2002). It is well known that the lower the carrier frequency is, the more significant the ionospheric dispersion impacts on the propagation delay. Thus, in many of the previous applications of C-band SAR data, the effects of ionosphere could have been neglected. While GPS also employs L-band, the high-precision GPS geodetic survey corrects for the ionospheric effect with the use of dual frequency, L1 and L2, observation data. In contrast, PALSAR is a single frequency SAR sensor and incapable of the standard ionosphere-correction approach. Empirically, however, we will encounter the ionospheric signals more frequently in the ascending data acquired in the local nighttime than in the descending data acquired in the local daytime. We also recall that the JERS, the other L-band SAR operated during 1992–1998, did not reveal any significant ionospheric signals at least in mid-latitude regions, and that most of the JERS data were acquired in the descending track. Besides the latitude, the effects of ionosphere on SAR image might, therefore, significantly depend on the data acquisition time. Like the tropospheric effects, detailed studies of ionospheric impacts on the SAR data are also currently underway. A simple approach to eliminate those noises is stacking, in which several interferograms are stacked to isolate smallamplitude signals, because those noises can be regarded as temporally random, whereas the deformation signals are spatially persistent. Two important prerequisites for successful stacking are: (1) the data acquisition dates of those interferograms should not be overlapped, in order not to enhance the noises of any particular acquisition date(s), and (2) each temporal baseline should be as long as possible so that each interferogram can include as much deformation signals as possible. In reality, it is not easy to gather many independent interferograms that have desirably long temporal baselines because the available data often encounter the spatial and temporal decorrelation. Also, the simple stacking approach

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inherently assumes temporally linear evolution in the ground deformation, preventing us from deriving time-series data. Time-Series Analysis: Ferretti et al. (2000, 2001) proposed a new analysis technique called Permanent Scatterer InSAR (PS-InSAR), in which they take advantage of even such data pairs whose spatial baselines are longer than the critical values. Thereby, they could expand the temporal coverage, and thus could estimate the long-term deformation signals on the order of mm/year. Key idea of PS-InSAR is to pick up only such pixels that will exhibit long-term coherence due to the existence of corner-reflector-like targets, which Ferretti et al. called “permanent scatterers.” Based on those pixels alone, they generate a stack of differential interferograms, using available DEM and orbit data. The phase values include not only deformation signals, but also such topographic signals that were not initially taken into account, because the longer spatial baseline pairs are so sensitive to the topography that the available DEM could not account for. In PS-InSAR and its variants (Werner et al. 2003; Hooper et al. 2004), they fit the differential interferogram stack to a phase model that describes not only temporal evolution of deformation but also corrections to the available DEM. Deviations from the phase model can be filtered into either non-linear deformation or atmospheric signals because the former signals are correlated and thus low-pass filtered along temporal axis, while the latter signals are temporally random; the orbit data must be assumed to be correct. A known limitation of PS-InSAR is its rather lower sampling density over non-urban areas. However, despite a lack of man-made objects, Furuya et al. (2007) succeeded in detecting active salt tectonic motion, applying a similar technique to Canyonlands National Park, Utah, presumably because the area was non-vegetated and the exposed surface rocks behaved like corner-reflector-like targets. Another time-series analysis approach was devised and known as small baseline subset (SBAS) algorithm (Berardino et al. 2002). Key idea of the SBAS algorithm is least-squares inversion of unknown deformation at each SAR data acquisition epoch, based on the available unwrapped differential interferograms (e.g., Lundgren et al. 2001; Schmidt and Burgmann 2003). Using small baseline interferometric pairs, the SBAS approach is free from spatial decorrelation and allows us to take advantage of the fine spatial resolution of InSAR data. If the number of interferograms is greater than or equal to the number of SAR acquisitions, the inversion problem becomes an over-determined or well-determined problem, and can be easily solved in a leastsquares approach. It is uncommon, however, that all the available interferometric pairs have short baselines, and accordingly the temporal sampling rate will decrease. Berardino et al. (2002) proposed to employ several groups of “small baseline subset” to overcome the lower temporal resolution issue, and solved the rank-deficient problem with the use of singular value decomposition (SVD) technique.

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The SVD gives the minimum-norm least-squares solution, which is equivalent to minimizing the estimated velocities at any time intervals. Time-series analysis of SAR data is a promising technique, but almost all previous analyses are based on the C-band ERS1/2 and Envisat data, because not only more-thandecade-long data but also high-precision, well-controlled satellite orbits are available for these satellites. As noted before, not all geophysically interesting phenomena could be detected by C-band and shorter-wavelength SAR data. If the L-band ALOS/PALSAR data are archived for a much longer time, and if the follow-on ALOS-2 and the DESDynI are launched as scheduled, the time-series analysis of SAR data will become feasible even in areas that have never been monitored before. The time-series analysis with ScanSAR data should also be possible. Long-term continuous monitoring with L-band SAR will provide us with more opportunities for new discoveries.

Cross-References ▶ Earthquake Rupture: The Inverse Problem ▶ Earthquake Source Theory ▶ Earthquakes and Crustal Deformation ▶ Geodesy, Ground Positioning, and Levelling ▶ GPS, Tectonic Geodesy ▶ Inverse Theory, Global Optimization ▶ Inverse Theory, Linear ▶ Inverse Theory, Monte Carlo Method ▶ Inverse Theory, Singular Value Decomposition ▶ Remote Sensing and GIS Techniques for Tectonic Studies ▶ Remote Sensing, Applications to Geophysics ▶ Slow Earthquake

Bibliography Bamler R, Hartl P (1998) Synthetic aperture radar interferometry. Inverse Problems 14:R1 Berardino P, Fornaro G, Lanari R, Sansosti E (2002) A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms. IEEE Trans Geosci Remote Sens 40:2375 Burgmann R, Rosen PA, Fielding EJ (2000) Synthetic aperture radar interferometry to measure Earth’s surface topography and its deformation. Annu Rev Earth Planet Sci 28:169 Cloude SR, Papathanassiou KP (1998) Polarimetric SAR interferometry. IEEE Trans Geosci Remote Sens 36:1551 Cumming IG, Wong FH (2005) Digital processing of synthetic aperture radar data: algorithm and implementation. Artech House, Boston Curlander JC, McDonough RN (1991) Synthetic aperture radar: systems and signal processing. Wiley interscience, New York Farr TG et al (2007) The shuttle radar topography mission. Rev Geophys 45:RG2004 Ferretti A, Prati C, Rocca F (2000) Nonlinear subsidence rate estimation using permanent scatterers in differential SAR interferometry. IEEE Trans Geosci Remote Sens 38:2202

SAR Interferometry Ferretti A, Prati C, Rocca F (2001) Permanent scatterers in SAR interferometry. IEEE Trans Geosci Remote Sens 39:8 Fialko Y, Simons M, Agnew D (2001) The complete (3-D) surface displacement field in the epicentral area of the 1999 Mw7.1 hector mine earthquake, California, from space geodetic observations. Geophys Res Lett 28:3063 Foster J, Brooks B, Cherubini T, Shacat C, Businger S, Werner CL (2006) Mitigating atmospheric noise for InSAR using a high resolution weather model. Geophys Res Lett 33:L16304 Furuya M, Mueller K, Wahr J (2007) Active salt tectonics in the Needles District, Canyonlands (Utah) as detected by interferometric synthetic aperture radar and point target analysis: 1992–2002. J Geophys Res 112:B06418 Ghilia DC, Pritt MD (1998) Two dimensional phase unwrapping: theory, algorithms, and software. Wiley, New York Hanssen RF (2001) Radar interferometry: data interpretation and error analysis. Kluwer, Dordrecht Hooper A, Zebker H, Segall P, Kempes B (2004) A new method for measuring deformation on volcanos and other natural terrains using InSAR persistent scatterers. Geophys Res Lett 31:L23611 Kobayashi T, Takada Y, Furuya M, Murakami M (2009) Location and types of ruptures involved in the 2008 Sichuan earthquake inferred from SAR image matching. Geophys Res Lett 36:L07302 Lundgren P, Usai S, Sansosti E, Lanari R, Tesauro M, Fornaro G, Berardino P (2001) Modeling surface deformation observed with synthetic aperture radar interferometry at Campi Flegrei caldera. J Geophys Res 106(B9):19355 Massonnet D, Feigl KL (1998) Radar interferometry and its application to changes in the earth’s surface. Rev Geophys 36:331 Massonnet D, Rossi M, Carmona C, Adragna F, Peltzer G, Feigl K, Raboute T (1993) The displacement field of the landers earthquake mapped by radar interferometry. Nature 364:138 Massonnet D, Vadon H, Rossi M (1996) Reduction of the need for phase unwrapping in radar interferometry. IEEE Trans Geosci Remote Sens 34:489 Matter KE, Gray AL (2002) Reducing ionospheric electron density errors in satellite radar interferometry applications. Can J Remote Sens 28:583 Michel R, Avouac J-P, Taboury J (1999) Measuring ground displacements from SAR amplitude images: application to the landers earthquake. Geophys Res Lett 26:875 Motagh M, Wang R, Walter TR, Bürgmann R, Fielding E, Anderssohn J, Zschau J (2008) Coseismic slip model of the 2007 august pisco earthquake (Peru) as constrained by wide swath radar observations. Geophys J Int 174:842 Onn F, Zebker HA (2006) Correction for interferometric synthetic aperture radar atmospheric phase artifacts using time series of zenith wet delay observations from a GPS network. J Geophys Res 111:B09102 Pritchard ME (2006) InSAR, a tool for measuring Earth’s surface deformation. Phys Today 59(7):68 Raucoules D, de Michele M (2010) Assessing ionospheric influence on L-band SAR data: implications on Coseismic displacement measurements of the 2008 Sichuan earthquake. IEEE Geosci Remote Sens Letters 7:286 Rosen PA, Hensley S, Zebker HA, Webb FH, Fielding EJ (1996) Surface deformation and coherence measurements of Kilauea volcano, Hawaii, from SIR-C radar interferometry. J Geophys Res 101(E10):23109 Schmidt DA, Burgmann R (2003) Time-dependent land uplift and subsidence in the Santa Clara valley, California, from a large interferometric synthetic aperture radar data set. J Geophys Res 108(B9):2416 Simons M, Rosen PA (2007) Interferometric synthetic aperture radar geodesy. In: Herring TA (ed) Treatise on geophysics, vol 3. Elsevier, New York, pp 391–446 Tobita M, Murakami M, Nakagawa H, Yarai H, Fujiwara S (2001a) Twodimensional field of three-dimensional components of deformations

Satellite Altimetry and velocities, and volume change around Usu volcano associated with the 2000 eruption by matching of SAR images (in Japanese). J Geogr Survey Inst 95:37 Tobita M, Murakami M, Nakagawa H, Yarai H, Fujiwara S, Rosen PA (2001b) 3D surface deformation of the 2000 Usu Eruption measured by matching of SAR images. Geophys Res Lett 28:4291 Werner CL, Wegmuller U, Strozzi T, Wiesmann A (2003) Interferometric point target analysis for deformation mapping, paper presented at IGARSS’03. Geoscience Remote Sensing Society, Toulouse Zebker HA, Rosen PA, Goldstein RM, Gabriel A, Werner CL (1994) On the derivation of coseismic displacement fields using differential radar interferometry: the landers earthquake. J Geophys Res 99(B10):19617–19634 Zhou XB, Chang NB, Li SS (2009) Applications of SAR interferometry in Earth and environmental science research. Sensors 9:1876

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Measurements of altimetric heights are thus creating profiles of the Earth’s surface along satellite orbits with respect to Earth’s center of mass. Satellite observations coming from altimeters, like those of Sentinel-3, Jason-3, CryoSat-2, IceSat-2, HY-2, and others, address several applications in solid Earth geophysics. Gravity determination, natural resources exploration, ocean circulation and variability, ice topography, hydrology, geodesy, climate change, bathymetry, sea-floor topography, etc., are a few applications of satellite altimetry.

Measurement Principles in Altimetry

Satellite Altimetry Stelios P. Mertikas and Constantine Kokolakis Geodesy and Geomatics Engineering Laboratory, School of Mineral Resources Engineering, Technical University of Crete, Chania, Greece

Synonyms Earth observation; Marine geodesy; Remote sensing; Satellite monitoring; Sea surface height

Definition Altimetry is a technique for measuring heights. Satellite altimetry measures the time taken by an electromagnetic pulse to propagate from the satellite to the surface and back to the satellite

Satellite Altimetry Satellite altimetry is a technique of measuring heights of the Earth’s surface from a satellite flying approximately at an attitude of about 1000 km. The word “altimetry” originates from the Latin “altus,” meaning “high” and the Greek suffix “metry” (“mεtrία,” as in Geo-metry, Trigono-metry, etc.) denoting the science or scientific discipline related to measuring heights. At the beginning, the objective of altimetry has been to measure height principally of sea surface to an accuracy of about 1 cm, continuously, objectively, and on a global scale. Altimeters measure distances, or ranges, from a satellite to the Earth’s surface and principally onto the sea, inland water levels (lakes, rivers) and ice surfaces as well (Fig. 1).

Altimeters on board a satellite transmit radar or laser pulses towards the surface of the Earth. Propagated signals are then reflected by this surface and finally are received by the altimeter on the satellite. Ranges are realized by measuring time differences between satellite transmission and reception of these electromagnetic pulses. Each range value is an average of all measurements made within the footprint of the altimeter. Time series of surface height and its change can be constructed for a location along the satellite ground track during its lifetime. Altimeter ranges require proper measurement reductions to eliminate various error sources arising from orbit determination, atmosphere delays along the propagation path of the signals (wet and dry troposphere, ionosphere), antenna gain pattern, satellite clocks, geoid height, tides (ocean, solid earth, pole), and so on. Analysis and evaluation of return signals and specifically of their waveforms provides information upon measured range, wind speed, wave heights, and backscatter coefficient (sigma-naught) of observed targets (Fig. 2). Precise heights for oceans, lakes, rivers, sea ice, glaciers, and land surfaces are thus measured from an altitude of 800–1300 km with an accuracy of about 1 cm (or better). Altimetric observations are made independent of night or day, of strong winds or hurricanes, cloudy or clear weather conditions. In this manner, altimetry is producing a wealth of Earth observations continuously, accurately, objectively, repeatedly, and on a global scale. As of 1992 with its first TOPEX/Poseidon, it has been generating precise time series of altimetric observations for building up a climate data record of ocean topography, sea level, and inland water levels. Time-averaged heights of the sea surface essentially represent the geoid, showing the variability of the gravity field at the earth’s surface, caused by the uneven distribution of the density of earth’s crust (Fig. 3). The slope of ocean surface, related to changes in gravity and submarine features, is estimated by differencing height measurements along altimeter tracks. Altimetry has in that

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Satellite Altimetry

Satellite Altimetry, Fig. 1 Principle of satellite altimetry. The satellite altimeter transmits signals towards the Earth, which are later received after their reflection onto the Earth’s surface. After orbit determination, and several measurement reductions, height profiles of the Earth’s surface are established with respect to its center of mass

way revealed subsea mountains and features and made it possible to construct a uniform resolution map of the seafloor topography. A detailed knowledge of seafloor topography is fundamental to the understanding of several Earth processes.

Also, gravity and bathymetry data derived from altimetry have been used to construct global models of marine gravity to a latitude of 85
with accuracies of 1 mGal at 7 km resolution. Such models have helped to determine gravity models for remote and inaccessible regions and also

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Satellite Altimetry, Fig. 2 Examples of satellite altimeters for Earth observation ((a) Upper image). The lower (b) diagram presents an example of a waveform for a return signal on an altimeter. Ranges are determined by measuring time difference at the half-power point (point B0). The slope of the leading edge (AC leg) of waveform defines wind speed. The level of return power represents wave heights (Point C). The rise time of the leading edge (A0C0) corresponds to significant wave height

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Satellite Altimetry

Satellite Altimetry, Fig. 3 The sea surface exhibits slopes, tiny riddles and variations (enlarged in the cartoons above), commonly invisible to human eye, but related to bathymetry and gravity changes underneath it (cartoon by Nikitas Papadoulakis)

Satellite Altimetry, Fig. 4 A global map of marine gravity as produced by combining satellite altimetry and gravity data (Sandwell and Smith 2009; Sandwell et al. 2014)

Satellite Altimetry

to identify submarine canyons, faults, and plate boundaries and oceanic plateaus (Fig. 4). These geomorphic features provide evidence to where to look for deposits of natural resources.

Oceanography by Altimetry Oceans cover around 71% of the Earth’s surface and contain about 97% of its water. Their role is crucial in the balance of Earth’s climate. They control the amount of CO2 in the atmosphere through plankton, absorb 90% of the thermal energy of the Earth, form microclimates, regulate fish population by ocean circulation and currents, shape and curb the greenhouse effects, etc. Before Seasat in 1978, which was the first satellite to observe Earth’s ocean, measurements for ocean monitoring were constrained to coastlines with tide gauges and/or along ship’s routes. Such ocean observations on the surface are commonly biased as they are subject to and contaminated by local and dynamic effects, bathymetry, and so on. Up to that point, observations made at the Earth’s surface have prevented geophysicists from understanding global oceans’ mechanisms. So, satellite altimetry came to fill in this gap by providing the global picture with accuracy and objectivity. In altimetry, information about the radiated ocean surface originates from the received echo of the transmitted satellite pulse. Over ocean, this response has a particular shape, a waveform of the Brown model as shown in Fig. 2 (bottom). The epoch at half-power point (corresponding to the midheight of sea level) defines the altimeter range; the maximum power return provides the reflexivity coefficient of the altimeter signal on the surface (called sigma-naught and related to properties of the scattering surface); the leading edge slope of the signal return determines the amplitude of sea waves (significant wave height); and the slope of its trailing edge is connected with the off-nadir pointing of the altimeter antenna (Passaro et al. 2014). Satellite altimetry is thus able to observe ocean surface topography, which is nothing else but the difference between the ocean surface and the geoid (the equipotential surface of the sea level). Mapping ocean surface topography with altimetry enables us to observe ocean currents and circulation. Any variation on ocean surface topography, such as a high (hill) or a low (valley) at a location, implies that currents swirl around it, primarily driven by thermodynamic processes. The Gulf Stream which flows from the Caribbean to the Grand Bank off Newfoundland is such an example. This Stream has been regularly monitored from the onset of altimetry with the TOPEX/Poseidon satellite. During its course, the Gulf Stream warms the ocean and releases heat and water vapors into the atmosphere. So it modifies and

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regulates the exchanges of heat between the ocean and the atmosphere and has a straight through impact on the climate and its changes. Altimetry has also contributed to the improvement of tidal models. Nowadays, these models have an uncertainty of 2 cm in open seas. This level of uncertainty allows us to examine and better understand gravity interactions between the Moon (main cause of tidal effects) and the Earth. Daily sea level measurements from altimetry have given the opportunity to adjust model parameters according to real satellite measurements in open seas where no previous observations could have been validated them (Fu and Cazenave 2000). These accurate tidal models are essential in forecasting extreme ocean events, such as tsunamis. Satellite altimetry supports absolute and undisputable sea level monitoring over regional to global scales, accurate to some [mm/yr] and with respect to the center of mass of the Earth (Mertikas et al. 2018). Sea-level is a key indicator of the impact of climate change over oceans. It is one of the essential climate variables (Downy 2016) that needs to be systematically monitored for maintaining a global climate observing system.

Hydrology by Altimetry Hydrology is composite word which originates from the Greek “hydro” (ύδor), meaning water and the ending “logy” (lóγoB), corresponding to “thorough investigation.” Thus, hydrology is a scientific field of Geophysics that studies the movement, distribution, and quality of water on Earth. Although the earliest altimetry missions have been dedicated to observing the open ocean and ice, recent altimetry has moved into monitoring every stretch of water on Earth (enclosed seas, lakes, rivers, flooding areas, and so on) provided satellites fly over them. An altimeter is capable of monitoring these water levels of lakes and their variations. To our advantage altimetry monitors regions difficult or even impossible to access for in situ measurements. Levels of lakes rise and drop throughout seasons, depending upon how much water flows in (e.g., rain, snow melting) or flows out their water bodies (e.g., evaporation, pumping for irrigation). Thus, the level of water is an indicator of regional climate variations, and it is important to monitor it to be able to access quantity of water and stretch of a lake. This is achieved by altimetry (water height and volume) in combination with imagery (lake stretch). The same holds true for enclosed seas, such as the Caspian Sea. Some altimeters with specific ground tracks are capable of monitoring adjacent lakes, as the African lakes of Victoria, Tanganyika and Malawi, and also correlate their water levels. The European Remote Sensing satellite (ERS) and its

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successor Envisat had observed 215 large lakes out of 842 that exist globally. River levels and flooded areas can also be observed by altimetry. Satellite altimeters can observe heights of large rivers such as the Amazon or the Brahmaputra and monitor their changes over seasons. The recent observation mode of synthetic aperture radar (SAR) altimetry takes advantage of Doppler effect of earth targets with respect to the moving satellite and achieves an improvement in spatial resolution, along track, of about 300 m contrary to 1–5 km in conventional altimetry (also called low resolution mode). The CryoSat-2 satellite and Sentinel-3A and Sentinel-3B implement this SAR technique which allows them to monitor narrow straits, coastal lines, and inland water resources reliably. In SAR mode, the satellite emits a series of coherent pulses (called bursts) and thus multiple signal returns (echoes) are received by the altimeter for the same target on the surface of the Earth. After processing these multiple returns, better resolution and more precise altimeter ranges are produced. SAR altimetry has applications for inland water observations, such as the CryoSat-2 SAR altimetry over the Mekong River in Southeast Asia (Bercher et al. 2013). Finally new algorithms, called fully-focused algorithms, are applied for processing altimeter signals to further improve the along track resolution down to half of the antenna length, at about 0.5 m spatial resolution.

Ice and Cryosphere Monitoring Altimetry can also measure thickness of polar sea ice, floating ice, and glacier topography as well as monitor changes in ice sheets. Altimeters of ERS-1, ERS-2, Envisat, CryoSat-2, IceSat, and IceSat-2 have observed the cryosphere and have allowed us to map Greenland and Antarctica, where about 77% of the Earth’s fresh water is located and accessibility has been difficult for surface measurements. Before satellite altimetry, in situ measurements in polar caps were sparse in space and time. This limitation had not permitted us to appreciate the true extend of ice, melting progression, and variations. As of 2010, the synthetic interferometric radar altimeter of CryoSat-2 (as well as IceSat and IceSat-2) has been acquiring elevation data for glaciers and ice caps to assess the consequences of receding ice cover as a result of global warming (McMillan et al. 2014). Also, the interferometric synthetic aperture radar of CryoSat-2 has allowed a smaller footprint (~250 m) in the along-track (fore- and aft) direction and detection of the position of the source of the echo in its across-track (left and right) direction; thus, CryoSat-2 has been able to distinguish slopes. This characteristic has produced accurate height

Satellite Altimetry

measurements in regions of complex topography. Signal parameters in the waveform of the echo are similar with these in Fig. 1, with the exception that the leading-edge width gives information about signal penetration into the ice and about ice roughness. The amount of ice sheet and their variations are not only indicators of the current climate situation of the Earth but are reliable predictors for the future. If the ice of Greenland and Antarctica melts, for example, the sea level will rise about 80 m (Gardner et al. 2013).

Summary Satellite altimetry is a powerful observation tool for Geophysics. It has proven extremely useful for uninterrupted, global, and accurate observation of marine geodesy, oceans, ice caps, lakes, rivers, coastlines, natural resources, and many more. It has revealed mechanisms of the Earth, such as global bathymetry and ice melting that no other technique could do in the past. Recognizing that progress, diverse satellite altimeters along with advanced measuring techniques (Nadir, Delay-Doppler, interferometry altimeters, wide swath, Ku-band, Ka-band frequencies, etc.) are placed into orbits and some of them such as Sentinel-6/Jason-CS are to become operational missions. New missions SWOT (Morrow et al. 2019), Quanlan (Chen et al. 2019), CRISTAL (Kern et al. 2020), etc.) are planned by various nations, organizations, and space agencies and are on their way to support Earth observation and drive the advancement of Geophysics.

Acknowledgments We thank Professor David Sandwell of Scripts Institute of Oceanography, USA, for giving us permission to publish the map of Fig. 4.

Bibliography Bercher N, Dinardo S, Lucas BM, Fleury S, Calmant S, Crétaux JF, Benveniste J (2013, March) Applications of CryoSat-2 SAR & SARIn modes for the monitoring of river water levels. In: Proceedings of the CryoSat third user workshop, Dresden, Germany, pp 12–14 Chen G, Tang J, Zhao C, Wu S, Yu F, Ma C, Xu Y, Chen W, Zhang Y, Liu J, Wu L (2019) Front Mar Sci. https://doi.org/10.3389/fmars. 2019.00194 Downy C (2016) What is an ECV? ESA-Climate Change Initiative. Retrieved 16 Feb 2019, from http://cci.esa.int/content/what-ecv Fu LL, Cazenave A (eds) (2000) Satellite altimetry and earth sciences: a handbook of techniques and applications, vol 69. Elsevier, New York

Seafloor Spreading Gardner AS, Moholdt G, Cogley JG, Wouters B, Arendt AA, Wahr J, Ligtenberg SR (2013) A reconciled estimate of glacier contributions to sea level rise: 2003 to 2009. Science 340(6134): 852–857 Kern M, Cullen R, Berruti B, Bouffard J, Casal T, Drinkwater MR, Gabriele A, Lecuyot A, Ludwig M, Midthassel R, Traver IN, Parrinello T, Ressler G, Andersson E, Martin-Puig C, Andersen O, Bartsch A, Farrell S, Fleury S, Gascoin S, Guillot A, Humbert A, Rinne E, Shepherd A, van den Broeke M R, Yackel J (2020) The Copernicus polar ice and snow topography altimeter (CRISTAL): expected mission contributions. Cryosphere. https://doi.org/10.5194/ tc-2020-3 McMillan M, Shepherd A, Sundal A, Briggs K, Muir A, Ridout A, Wingham D (2014) Increased ice losses from Antarctica detected by CryoSat-2. Geophys Res Lett 41(11):3899–3905 Mertikas S, Donlon C, Féménias P, Mavrocordatos C, Galanakis D, Tripolitsiotis A, Guinle T (2018) Fifteen years of Cal/Val service to reference altimetry missions: calibration of satellite altimetry at the permanent facilities in Gavdos and Crete, Greece. Remote Sens 10(10):1557 Morrow R, Fu LL, Ardhuin F, Benkiran M, Chapron B, Cosme E, d’Ovidio F, Farrar JT, Gille ST, Lapeyre G, Le Traon PY, Pascual A, Ponte A, Qiu B, Rascle N, Ubelmann C, Wang J, Zaron ED (2019) Global observations of fine-scale ocean surface topography with the surface water and ocean topography (SWOT) mission. Front Mar Sci. https://doi.org/10.3389/fmars. 2019.00232 Passaro M, Cipollini P, Vignudelli S, Quartly GD, Snaith HM (2014) ALES: a multi-mission adaptive subwaveform retracker for coastal and open ocean altimetry. Remote Sens Environ 145: 173–189 Sandwell TD, Smith WHF (2009) Global marine gravity from retracked Geosat and ERS-1 altimetry: ridge segmentation versus spreading rate. J Geophys Res 114:B0141 Sandwell TD, Muller RD, Smith WHF, Garcia E, Francis R (2014) New global marine gravity model from CryoSat-2 and Jason-1 reveals buried tectonic structure. Science 346(6205): 65–67

Seafloor Spreading Richard N. Hey Hawaii Institute of Geophysics and Planetology, School of Ocean and Earth Science and Technology, University of Hawaii, Honolulu, HI, USA

Definition Seafloor spreading is the mechanism by which new oceanic lithosphere is created at and moves away from the divergent plate boundaries known as mid-ocean ridges. The seafloor spreading hypothesis led to one of the most important paradigm shifts in the history of the Earth sciences, the plate tectonics scientific revolution.

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Introduction The revolutionary seafloor spreading hypothesis improved and subsumed the continental drift hypothesis, and rapidly culminated in what is now known as plate tectonic theory. First hypothesized by Harry Hess in a 1960 preprint and paper (Hess 1962), he considered so speculative he called it “an essay in geopoetry,” and named in another influential early paper (Dietz 1961), it offered a simple explanation for many problems with the prevailing paradigm that the Earth was a mostly static, slowly contracting planet, with fixed continents and old ocean basins, and no large-scale horizontal displacements. This paradigm had previously been challenged, most notably by Alfred Wegener with his continental drift hypothesis (Wegener 1912), and by paleomagnetic measurements in the 1950s that were consistent with continental drift, but before the 1960s these ideas were not generally accepted.

The Revolution Hess realized that if mantle convection carried seafloor and continents away from seafloor spreading centers (mid-ocean ridges) toward trenches (subduction zones), with new seafloor formed at ridge axes and destroyed at trenches, this would explain the shallow bathymetry, earthquakes, high heat flow, lack of sediments, and extensional structures characterizing ridge axes, as well as the deep trenches, earthquakes, compressional structures, mountain ranges, and volcanoes characterizing subduction zones. A key step in the confirmation of seafloor spreading was the recognition by Fred Vine and Drum Matthews (Vine and Matthews 1963) (and independently by Lawrence Morley, in a paper unfortunately turned down as too speculative by both Nature and the Journal of Geophysical Research, later published in Morley and Larochelle 1964), that it offered a simple explanation for the existence of puzzling magnetic “stripes” on the seafloor. The Vine– Matthews hypothesis proposed that a combination of seafloor spreading and episodic reversals of the Earth’s magnetic field (at the time another very speculative idea) would create alternating zones of normally and reversely magnetized crust, and thus, linear positive and negative magnetic anomalies in a pattern symmetric about the spreading axis. The next key step in the revolution occurred in 1965, when Tuzo Wilson noted that deformation of the Earth’s crust is concentrated in narrow mobile belts, and postulated that these features are all interconnected in a global network, the first qualitative model of plate tectonics (Wilson 1965). The zones

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Seafloor Spreading

Seafloor Spreading, Fig. 1 Raff and Mason (1961) magnetic stripes recognized as symmetric about seafloor spreading axes (arrows) by Vine and Wilson (1965). (From Vine 1966. Reprinted with permission from AAAS)

of extension and compression are connected by a new class of faults called transform faults by Wilson, which required relative plate motion and turned out to be the most important type of fault on Earth. He showed that seafloor spreading occurring on offset mid-ocean ridge axes would produce relative motion exactly opposite to the motion the sense of offset would predict without seafloor spreading, and that earthquakes should only occur between the offset seafloor spreading axes. These radical predictions, completely different from prevailing wisdom, were soon confirmed seismically. Furthermore, by correctly interpreting the San Andreas fault as a transform fault between the Pacific and North American plates, he predicted the existence of previously unrecognized seafloor spreading offshore western North America. Vine and Wilson (1965) showed that the predicted magnetic symmetry existed in this area, where magnetic stripes had been discovered in the mid-1950s

(Mason 1958; Mason and Raff 1961; Raff and Mason 1961), over what they recognized as the Juan de Fuca Ridge (Fig. 1), and furthermore, that the pattern of stripes corresponded perfectly with the pattern of magnetic field reversals (Cox et al. 1963; McDougall and Tarling 1963), once the Jaramillo anomaly was discovered. Vine (1966) also demonstrated similar symmetry and correlation with the reversal timescale in another important data set, the Project Magnet aeromagnetic data collected over the Reykjanes Ridge south of Iceland. His color figures of these classic data sets and correlations (Vine 1968) became iconic. The smoking gun for many scientists was the discovery by Pitman and Heirtzler (1966) of near perfect symmetry in the Eltanin-19 profile collected over the Pacific–Antarctic Ridge. Essentially every tiny wiggle seen in the magnetic anomaly profile on the Pacific plate was mirrored on the Antarctic

Seafloor Spreading

plate, and correlated perfectly with the magnetic reversal time scale (Fig. 2). The symmetry in these data required a symmetric axial process, with new seafloor carried away on both plates, and thus provided compelling evidence for both the Vine–Matthews and seafloor spreading hypotheses. Vine convincingly summarized this evidence in influential symposia and publications (Vine 1966, 1968), and, by the end of 1966, seafloor spreading was generally accepted by marine geophysicists, who quickly extrapolated the magnetic reversal time scale from ~10 Ma to ~80 Ma and worked out at least the basic recent evolutionary history of every ocean basin. This scientific revolution culminated in plate tectonics the following year when Jason Morgan (Morgan 1968) and Dan McKenzie (McKenzie and Parker 1967) made the key assumption that plates behave rigidly, and extended Wilson’s transform fault geometry to the sphere. They showed that transforms are small circles about rotation poles describing relative plate motion, that seafloor spreading rates increase as the sine of the angular distance away from these poles, and that it was possible to use known patterns of seafloor spreading to quantitatively predict other plate motions and plate boundary evolution.

Ridge Axis Geometry, Morphology, and Crustal Formation Volcanism along the mid-ocean ridge system has formed the longest mountain chain on Earth (Fig. 3). As the plates slowly (at present ~0–150 mm/yr, or km/Myr, DeMets et al. 2010, with rates remarkably similar whether using magnetic anomalies or geodetic measurements such as GPS) move apart by seafloor spreading, magma from the hot asthenosphere rises to fill the crack at mid-ocean ridge axes, elevated because of the higher temperatures. Both axial depth and morphology correlate with spreading rate (Macdonald 1982). Fast-spreading ridge axes such as the East Pacific Rise generally have shallow depths and relatively smooth morphologies, with very small extensional axial summit troughs (except where these are completely buried by the latest eruption in the neovolcanic rift zone). Slow spreading ridges such as the Mid-Atlantic Ridge generally have much deeper, rougher, and higher-amplitude axial valleys (Fig. 4), except near hotspots such as Iceland where magma supply is unusually large and even slowspreading ridges have shallow axes with fast-spreading morphology. The asthenosphere typically melts to become mid-ocean ridge basalt in a magma chamber under the ridge axis.

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East Pacific Rise Profile Reversed

51°S

500 γ

4.4 cm/yr

Model

100

0

100 KM

S.L. 3.3 5 KM

Seafloor Spreading, Fig. 2 Eltanin-19 magnetic anomaly profile (top) from Pitman and Heirtzler (1966), shown compared with its mirror image (middle) and magnetic anomaly model calculated from the magnetic reversal timescale (bottom), assuming seafloor spreading at a constant 44 km/Myr. (From Vine 1966. Reprinted with permission from AAAS)

Some basalt is erupted onto the seafloor to form pillow basalts and lava flows, which are progressively covered by sediments as the lithosphere ages and moves away from the axis. Below the basalts are the sheeted dikes, the paths through which the lava moved from magma chambers to the surface, and below the dikes are the layered gabbros which cooled and crystallized in place (intrusives) rather than erupting at axial volcanoes (extrusives). This typical sequence of mantle ultramafics – gabbro – sheeted dikes – extrusive basalts – sediments is observed on the seafloor in tectonic windows where existing lithosphere is rifted apart, and also where slices of old ocean lithosphere are thrust into ophiolite mountain belts as in Oman and Cyprus (Karson 2002).

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3

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−120

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0 30 Longitude (°)

Seafloor Spreading, Fig. 3 Global distribution of seafloor spreading axes (heavy black lines) with known (black dots) or inferred (gray dots) hydrothermal fields. Stars are near-ridge hotspots. (From NOAA/PMEL

Neovolcanic zone

Fast

60

90

120

150

180

Vents Program website: https://www.pmel.noaa.gov/pubs/outstand/ bake2544/images/fig01a.gif (Baker and German 2004))

(Fig. 5). White and clear smokers occur at lower temperatures. The chemical reactions provide energy for an unusual kind of life that does not depend on photosynthesis but flourishes as chemosynthetic communities at many of these vents, rare deep-ocean oases in an otherwise extremely barren environment (Kelley et al. 2002).

Intermediate Slow

Summary VE~4x AXIS 20

10

0 km

10

20

Seafloor Spreading, Fig. 4 Ridge axis fault patterns and morphology as a function of spreading rate. (Modified from Macdonald 1982)

Seafloor spreading was a critical step in the contentious scientific revolution from the previous static Earth paradigm to the now universally accepted plate tectonic paradigm. Today it refers to the processes creating new oceanic lithosphere where plates move apart. Seafloor spreading replaces the lithosphere destroyed by subduction and exerts important influences on Earth’s chemical and biological evolution.

Hydrothermal Vents Cross-References As the new seafloor cools it contracts and cracks. Water goes down these cracks, is heated, reacts with the surrounding rocks, and comes back up as buoyant hydrothermal vents (Baker and German 2004). These occur at a range of chemistries and temperatures, up to ~400
C in black smokers, so-called because of the sulfide-rich plumes that precipitate suddenly when the superheated water is injected into the surrounding ~2
C seawater, forming sulfide chimneys

▶ Continental Drift ▶ Continental Rifts ▶ Lithosphere, Mechanical Properties ▶ Lithosphere, Oceanic ▶ Plate-Driving Forces ▶ Plates and Paleoreconstructions ▶ Subduction Zones

Sedimentary Basins

Seafloor Spreading, Fig. 5 A black smoker community comprised of giant red tubeworms and hundreds of squat lobsters. This vent is located in Strawberry Fields of the Main Endeavour hydrothermal field on the Juan de Fuca Ridge. Vibrant colonies of tube worms with red gills thrive on this vent which is predominantly composed of iron- and sulfurbearing minerals. (Credit: University of Washington; NOAA/OAR/ OER)

1353 Macdonald KC (1982) Mid-ocean ridges: fine scale tectonic, volcanic and hydrothermal processes within the plate boundary zone. Annu Rev Earth Planet Sci 10:155–190 Mason RG (1958) A magnetic survey over the west coast of the United States between latitudes 32
and 36
N, longitudes 121
and 128
W. Geophys J R Astron Soc 1:320–329 Mason RG, Raff AD (1961) A magnetic survey off the west coast of North America 32
N to 42
N. Bull Geol Soc Am 72:1259–1265 McDougall I, Tarling DH (1963) Dating of polarity zones in the Hawaiian islands. Nature 200:54–56 McKenzie DP, Parker RL (1967) The North Pacific: an example of tectonics on a sphere. Nature 216:1276–1280 Morgan WJ (1968) Rises, trenches, great faults and crustal blocks. J Geophys Res 73:1959–1982 Morley LW, Larochelle A (1964) Paleomagnetism as a means of dating geological events. In: Geochronology in Canada. Royal Society of Canada special publication, vol 8. Royal Society of Canada, Toronto, pp 39–50 Pitman WC III, Heirtzler JR (1966) Magnetic anomalies over the PacificAntarctic ridge. Science 154:1164–1171 Raff AD, Mason RG (1961) Magnetic survey off the west coast of the United States between 40
N latitude and 52
N latitude. Bull Geol Soc Am 72:1267–1270 Vine FJ (1966) Spreading of the ocean floor: new evidence. Science 154:1405–1415 Vine FJ (1968) Magnetic anomalies associated with mid-ocean ridges. In: Phinney RA (ed) The history of the Earth’s crust. Princeton University Press, Princeton, pp 73–89 Vine FJ, Matthews DH (1963) Magnetic anomalies over oceanic ridges. Nature 199:947–949 Vine FJ, Wilson JT (1965) Magnetic anomalies over a young oceanic ridge off Vancouver Island. Science 150:485–489 Wegener A (1912) Die entstehung der kontinente. Geol Rundsch 3:276–292 Wilson JT (1965) A new class of faults and their bearing on continental drift. Nature 207:343–347

Sedimentary Basins Bibliography Baker ET, German CR (2004) On the global distribution of hydrothermal vent fields. In: German CR et al (eds) Mid-ocean ridges: hydrothermal interactions between the lithosphere and oceans. Geophysical monograph, vol 148. American Geophysical Union, Washington, DC, pp 245–266 Cox A, Doell RR, Dalrymple GB (1963) Geomagnetic polarity epochs and Pleistocene geochronometry. Nature 198:1049–1051 DeMets C, Gordon RG, Argus DF (2010) Geologically current plate motions. Geophys J Int 181(1):1–80. https://doi.org/10.1111/j.1365246X.2009.04491.x Dietz RS (1961) Continent and ocean basin evolution by spreading of the sea floor. Nature 190:854–857 Hess HH (1962) History of ocean basins. In: Engel AEJ, James HL, Leonard BF (eds) Petrologic studies: a volume to Honor A.F. Buddington. Geological Society of America, New York, pp 599–620 Karson JA (2002) Geologic structure of the uppermost oceanic crust created at fast- to intermediate-rate spreading centers. Annu Rev Earth Planet Sci 30:347–384 Kelley DS, Baross JA, Delaney JR (2002) Volcanoes, fluids and life at mid-ocean ridge spreading centers. Annu Rev Earth Planet Sci 30:385–491

Magdalena Scheck-Wenderoth Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany RWTH Aachen University, Aachen, Germany

Definition Sedimentary basins are regions of prolonged subsidence of the Earth’s surface that provide the accommodation space for mineral and organic material (Allen and Allen 2013). These deposits – the sedimentary rocks – are the record of the past geological history including tectonic events, climatic conditions, changes in sea level, and other environmental modifications. In addition, sedimentary basins are long-lived, low-temperature geo-reactors in which the accumulated material experiences a variety of transformations (Bjorlykke 2010; Littke et al. 2008; Roure et al. 2009; Welte et al. 1997). As a result of these processes, basins contain our resources of fossil

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Sedimentary Basins

fuels, groundwater, and inorganic commodities. Moreover, they are important reservoirs of heat and provide repositories for different socioeconomically relevant fluids such as CO2, H2, and CH4.

Basin Types Basins can be classified in terms of their plate-tectonic setting. The plate-tectonic Wilson cycle (Fig. 1) describes that the relative movements of plates on the sphere of the Earth result in a closed loop of continental rifting and breakup, ocean basin development, closure of oceans at subduction zones, and, in consequence of plate convergence, continental growth. Basins thus can form as continental rifts that may either evolve into intracontinental basins or lead to continental breakup and ocean basin formation. New oceanic lithosphere (see entry ▶ “Lithosphere, Oceanic”) is produced along the mid-ocean ridges due to oceanic spreading. If oceanic lithosphere cools, it becomes denser and subsides in the oceanic basins that are bordered by passive continental margins. Of the latter “hot” or volcanic passive margins are distinguished from “cold” or nonvolcanic passive margins (White et al. 2003). If oceanic lithosphere cools beyond a specific threshold, it becomes too heavy to be sustained by the less dense asthenosphere and will finally descend back to the mantle at subduction zones. In these convergent settings, the descending plate is flexed downward and deep oceanic trenches develop above the down-going plate. Subduction may culminate in continent–continent collision if the oceanic lithosphere is entirely subducted and collisional fold and

Sedimentary Basins, Fig. 1 Basins in their platetectonic setting

thrust belts form. Loading by collision-related fold and thrust belts also causes a downward flexure of the lithosphere and foreland basins to form. The Wilson cycle may stop at any evolutionary step, because the causative forces cease to be effective. Accordingly, a continental rift not necessarily develops into an ocean basin, but may survive for hundreds of millions of years as an intracontinental basin (Heine et al. 2008; Littke et al. 2008; Cacace and Scheck-Wenderoth 2016). Finally, horizontal movements along strike-slip faults may also cause local extension and related pull-apart basins (Allen and Allen 2013; Gürbüz 2014; Petrunin and Sobolev 2008; Smit et al. 2010; Weber and DESERT Working Group 2009). These different types of basins not only have a specific subsidence history (Xie and Heller 2009) but also a characteristic structure of the sediment fill as well as of the underlying crust and mantle lithosphere. This concerns the geometric configuration, the distribution of physical properties, and the resulting isostatic (see entry ▶ “Isostasy”) and thermal configuration. To assess the configuration of a basin, a wide spectrum of methods has to be integrated. Observations obtained from field measurements, from deep seismic imaging (see entry ▶ “Deep Seismic Reflection and Refraction Profiling”), and from wells drilled into the basin fill, potential field data, and heat flow (see entry ▶ “Heat Flow Determinations, Continental”) data, and new data sets from remote sensing need to be integrated with numerical models that simulate processes in basins at different scales. Figure 2 shows exemplary crustal sections across the Norwegian passive margin and across the intracontinental Central European Basin System.

intracontinental rift

post-rift deposits continental breakup

passive margin

ocean basin MOR

syn-rift deposits

passive margin

strike-slip basin passive margin

trench

foreland basin

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restraining bend releasing bend

subduction

Sedimentary Basins

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Sedimentary Basins, Fig. 2 (a) Lower panel: Line drawing of a crustal-scale integrated cross section with three generations of basins from the Norwegian passive margin modified after Faleide et al. 2008. Upper panel: Seismic close-up showing the breakup unconformity and pre-breakup extensional faulting during phase syn-rift 2. (b) Lower panel: Line drawing with superimposed seismic p-wave velocities

PQ2-9.1

PQ2-5

Vp (km/s) 6 km/s in the upper crystalline crust and can range up to 7.4 km/s (Christensen and Mooney 1995) depending on the composition of the crust. There is a negative correlation between the quartz content of crystalline rocks and seismic p-wave velocity as well as density. Silicic (quartz-rich) rocks are characterized by smaller velocities and lower densities than mafic rocks (Table 1). Seismic shear wave velocities are also higher for crystalline than for sedimentary rocks but are also dependent of additional parameters (Christensen and Mooney 1995). Accordingly, the Vp/Vs ratio is an additional parameter helping to interpret crustal structure (Afonso et al. 2010; Mjelde et al. 2003). As also the magnetic properties are different for mafic (Fe-rich) and silicic (Fe-poor) rocks, the interpretation of magnetic anomalies is an additional technique for the evaluation of the crustal structure beneath basins. At the crust–mantle boundary (Moho), often a characteristic reflection and a sudden increase of seismic p-wave

velocity to values >8 km/s are observed in deep seismic data marking the transition to the lithospheric mantle (Bauer et al. 2000; DEKORP-BASIN Research Group 1999; Fowler 1996; Hirsch et al. 2009; Levander et al. 2006; Meissner and DEKORP Research Group 1991; Thybo and Nielsen 2009; Turcotte and Schubert 2002; Weber and DESERT Working Group 2009). This is related to the change in composition from various crustal rocks to the mantle consisting mainly of peridotite. As less dense sediments replace a denser crystalline crust, sedimentary basins should be characterized by a negative Bouguer anomaly with highest amplitude in the basin center after re-equilibration of the isotherm. Many basins, however, are characterized by a long-wavelength negative Bouger anomaly with positive anomalies superimposed. This observation, together with strong variations in the crustal velocity structure beneath sedimentary basins, challenges the classical concepts of crustal thinning and related Moho uplift in that they commonly occur in concert with a flat Moho (Thybo and Nielsen 2009). Crustal bodies with seismic velocities >7 km, generally referred to as “high-velocity bodies,” are observed in the lower crust beneath many intracontinental rift structures, for example, the Baikal and the Kenya Rifts (Thybo and Nielsen 2009); beneath intracontinental basins, for example, in the

Sedimentary Basins, Table 1 Overview on physical properties of rocks relevant to sedimentary basins: seismic properties (After Christensen and Mooney 1995), thermal properties. (After Cermak and

Rybach 1982; Fernàndez et al. 2005; Förster and Förster 2000; Ritter et al. 2004; Scheck-Wenderoth and Maystrenko 2008)

S E D I M E N T S Crust Continent

Crust Ocean Mantle

Dominant lithology Sea water Uncompacted fine-grained siliciclastics Slightly compacted fine-grained siliciclastics Medium compacted fine-grained siliciclastics Compacted fine-grained siliciclastics Strongly compacted fine-grained siliciclastics Compacted siliciclastics with carbonates Carbonates: chalk Carbonates: limestones, dolomites Rock salt Granites and gneisses Mafic granulites/gabbros/eclogites Gabbros Basalt Sheeted dikes/gabbroic intrusions Gabbro Peridotite continent Peridotite ocean

Thermal conductivity l [W/mK] 0.563 1.2 (2.0)

Heat production S [mW/m3] 0 1

Density [kg/m3] 1,030 1,950

p-wave velocity [km/s] 1.48 2.05

1.8 (2.2)

1

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1

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1 1

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0.9

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0.6 0.6 0.1 0.8 0.3 0.5 0.4 0.2 0.2 0.03 0.03

2,000 2,600 2,150 2,790 3,120 3,150 2,580 2,890 3,150 3,330 3,180

3.4 6.0 4.0–5.5 6.0–6.7 6.8–7.0 7.1–7.6 4.0–5.0 5.0–6.7 7.1–7.6 8.0–8.3 7.8

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Danish Basin (Nielsen and Thybo 2006); and beneath extended passive margins, for example, at the conjugate margins of the North and South Atlantic (Bauer et al. 2000; Contrucci et al. 2004; Faleide et al. 2008; Franke et al. 2007; Hirsch et al. 2009; Mjelde et al. 2002, 2005; Sibuet et al. 2007). Gravity analysis indicates that these bodies are also characterized by higher densities than the average crust (Fernàndez et al. 2005, 2010; Franke et al. 2007; Hirsch et al. 2009; Maystrenko and Scheck-Wenderoth 2009; Osmundsen and Ebbing 2008). Differential thinning with depth (Huismans et al. 2005; Huismans and Beaumont 2008; Kusznir and Ziegler 1992; Lavier et al. 1999; Lavier and Steckler 1997; Pascal and Cloetingh 2002; Sclater and Christie 1980; Steckler and Watts 1978), emplacement of magmatic material in the crust during the thinning process, and magmatic underplating (Thybo and Nielsen 2009; Burg and Gerya 2008) are some possible processes during basin formation and evolution that can be responsible for the actual configuration of the crust beneath a specific basin. The relative contributions of the processes vary strongly for different basins, and, accordingly, their relative importance in general terms is still under debate. Moreover, structural inheritance may have a fundamental role in defining the crustal and lithosphere structure. Successive suturing of different plates in Earth history may result in a mosaic of crustal domains with contrasting physical properties, possibly also including lower crustal bodies predating the rifting process (Ebbing et al. 2009; Faleide et al. 2008; Plomerová and Babuska 2010; ScheckWenderoth and Lamarche 2005; van Wijk 2005; Vauchez et al. 1998; Sippel et al. 2017). In basins developing on such a substrate, such older rheological discontinuities may be reactivated to localize deformation in areas of reduced strength, thus also facilitating discontinuous stretching with depth.

The Configuration of the Mantle Lithosphere Much less is known on the configuration of the mantle lithosphere below sedimentary basins due to the limited amount of direct observations. Though reflections and lateral variations in seismic wave velocities in the lithospheric mantle have been observed in deep seismic data, the energy used in active seismic experiments is generally not sufficient to reach depth intervals below the Moho. Also, the gravity signal from the lithospheric mantle is difficult to isolate from the cumulative signal of the entire upper mantle. Recent development in the acquisition and evaluation of passive seismological data (Dalton and Faul 2010; Fichtner et al. 2018; Fishwick 2010; Geissler et al. 2010; Heintz and Kennett 2005; Hieronymus and Goes 2010; Plomerová and Babuska 2010; Priestley and

Sedimentary Basins

McKenzie 2013; Priestley et al. 2006; Ritzmann and Faleide 2009; Rychert and Shearer 2009; Schaeffer and Lebedev 2013; Zhang and Lay 1996); electromagnetic and magnetotelluric methods (see entries ▶ “Magnetotelluric Interpretation” and ▶ “Magnetotelluric Data Processing”) (Jones et al. 2010), together with remote sensing gravity observation (Kaban et al. 2003; Schotman et al. 2009); as well as geochemical data (O’Reilly and Griffin 2010; Trumbull et al. 2002; Wang 2010; Kovács et al. 2018) and thermal studies (Artemieva 2009; Hasterok and Chapman 2007; Hieronymus and Goes 2010; Schaeffer et al. 2016; Lucazeau 2019) indicates that the lithospheric mantle is less homogenous than previously thought. Apart from strong differences in thickness, also considerable lateral variations in surface heat flow (see entry ▶ “Heat Flow Determinations, Continental”) are observed, with a positive correlation between the two. The LAB may be located deeper than 250 km in cratonic areas, lies at around 100 km beneath Phanerozoic continental domains and old oceans, and is close to the seafloor at mid-ocean ridges. Agreement is established that the LAB is a fundamental boundary in plate-tectonic theory that separates the rigid plates from ductile convecting material below the plates (Artemieva 2009; Eaton et al. 2009; Jones et al. 2010). Three broad definitions in terms of a mechanical boundary layer, a thermal boundary layer, and a chemical boundary layer are based on different types of data and partly in geometrical conflict. For the evolution of sedimentary basins, the depth of the thermal LAB is especially relevant as it determines the thermal and mechanical state of the lithosphere subjected to any of the basin-forming mechanisms. Systematic mapping of lithosphere thickness beneath sedimentary basins is, however, still patchy and partly contradictory depending on the method used to derive it. The thermal LAB is interpreted as an isotherm of about 1,300
C, and its depth corresponds to the depth where an average continental conductive geotherm intersects the mantle adiabat. This is corroborated by heat flow inversion studies in continental lithosphere (see entry ▶ “Lithosphere, Continental”) (Artemieva 2009), by combined thermal and gravity modelling (Fernàndez et al. 2005, 2010; Freymark et al. 2017; Hasterok and Chapman 2007; Hyndman et al. 2009; Maystrenko et al. 2013; Przybycin et al. 2015; Sippel et al. 2017; Scheck-Wenderoth and Maystrenko 2008) and by cooling models in oceanic lithosphere (see entry ▶ “Lithosphere, Oceanic”) (Crosby et al. 2006; McKenzie 1978; Sclater 2003; Stein and Stein 1992). Isostatically, thinning of the lithospheric mantle results in net surface uplift as heavier lithospheric mantle is replaced by less dense asthenospheric material (active rift model). Only if the interaction with surface processes takes place (erosion), net subsidence takes place due to subsequent cooling of the asthenospheric material.

Sedimentary Basins

Heat Flow in Sedimentary Basins The thermal field in sedimentary basins has been of primary interest in the exploration of fossil fuels (Welte et al. 1997) and gains increasing importance for the use of geothermal energy (Huenges 2010). Moreover, together with composition, temperature is a main controlling factor for the rheological behavior of the lithosphere and accordingly its deformation (Burov 2011; Violay et al. 2015a, b). The hotter the lithosphere, the weaker is its rheology and the easier it is thinned to form a sedimentary basin. Accordingly, the main controlling factors for the thermal state of a basin are its platetectonic setting and its evolutionary stage. In terms of observables, surface heat flow measurements and temperatures measured in wells characterize the presentday thermal state of a basin, whereas the maturity of organic matter (Welte et al. 1997) and thermochronological data (Andriessen 1995; Braun 2005; Braun and van der Beek 2004; Kounov et al. 2007; Simoes et al. 2010; Van der Beek 2007; Willenbring and von Blanckenburg 2010) provide a record of the thermal history. Different families of integrated process-oriented models attempt to reproduce these observables and indicate that several processes contribute to the heat flow in sedimentary basins. There is a first-order contrast in thermal properties between sedimentary and crystalline rocks, in that sediments are thermally less conductive and produce less radiogenic heat than crystalline crustal rocks (Table 1). The amount of heat entering the basin at the base of the sediment fill depends on the thickness and composition of the crystalline crust as well as on the depth of the thermal LAB. The mafic components of the crust and the lithospheric mantle are characterized by a high thermal conductivity but low radiogenic heat production. Accordingly, the shallower the LAB and the thicker and radiogenic the crystalline crust, the more heat arrives at the base of the sediments. Due to their higher porosities, the thermally low-conductive sediments act as a thermal blanket, causing heat storage in the basin (Cacace et al. 2010; Theissen and Rüpke 2009; Van Wees et al. 2009). In addition, the sediments contribute a modest but, in the sum, considerable amount of radiogenic heat to the system. Also within the sediment fill, the thermal properties may vary (Table 1) with the thermal conductivity of salt being two times larger than that of clastic sediments. The upper part of sedimentary basins may store paleoclimatic signals of previous glaciations as present-day permafrost down to more than 700 m depth attest (Majorowicz and Šafanda 2018; Szewczyk and Nawrocki 2011). In response to the distribution of thermal properties, longwavelength variations in temperatures in sedimentary basins (scale of hundreds of kilometers) are determined by the crustal structure and composition as well as by the thermal thickness of the lithosphere (Cacace et al. 2010; Hasterok and

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Chapman 2007; Hyndman et al. 2009; Scheck-Wenderoth and Maystrenko 2008, 2013; Sclater 2003). In contrast, the short-wavelength pattern of temperature distribution (scale of kilometers and less) is controlled by the internal configuration of the sediment fill. Heat transfer by conduction is assumed to dominate the transport of heat in the lithosphere in contrast to the convecting mantle (see entry ▶ “Mantle Convection”). Geological and geochemical data together with models of coupled fluid and heat transport (Scheck-Wenderoth et al. 2014) show that the thermal field in sedimentary basins is additionally influenced by heat transport due to moving pore fluids. Advective (fluid pressure-driven) heat transport can influence the thermal field of the sedimentary fill on the regional scale. In contrast, buoyancy-driven free convective heat transport and heat transport along faults is only locally important but needs consideration for exploration on the reservoir scale. Summarizing, the heat flow in a specific basin may vary spatially, though some ranges can be given for different types of basins (Allen and Allen 2005, 2013). Typically, the surface heat flow in intracontinental basins varies between 40 and 70 mW/m2, can be up to 150 mW/m2 in active rifts or close to volcanic arcs, and can reach values higher than 180 mW/m2 at mid-ocean ridges and oceanic rifts. In oceanic basins, surface heat flow decreases with increasing distance from the midocean ridges according to the cooling of the lithosphere with age (Parsons and Sclater 1977; Stein and Stein 1992). While magma-poor passive margins are characterized by a rather low surface heat flow, the range of surface heat flow at magma-rich passive margins depends on the age of the adjacent oceanic lithosphere (Gholamrezaie et al. 2018). At young margins as in the 55-my-old North Atlantic, the surface heat flow is still controlled by the cooling of the oceanic lithosphere that is considerably thinner than the lithosphere of the continental margin. This step in the thermal LAB is consistent with observed heat flow values increasing from 45 mW/m2 at the continental side to 80 mW/m2 at the oceanic side of the margin (Ritter et al. 2004; Scheck-Wenderoth and Maystrenko 2008). At older passive margins as in the 130-my-old South Atlantic, an opposite trend is observed (Goutorbe and Bonneville 2008; Maystrenko et al. 2013) with about 45 mW/m2 at the oceanic side of the margin to up 65 mW/m2 at the continental side. As the oceanic lithosphere had sufficient time to cool and thicken, the depth of the thermal LAB is continuous and similar between the continental and the oceanic part of the system. Instead, the radiogenic heat contribution of the continental crust (see entry ▶ “Earth’s Structure, Core”) thickening toward the continent becomes the controlling factor for heat input from the lithosphere. Flexural basins are generally colder than extensional basins with average heat flow values around 40 mW/m2 in oceanic trenches and foreland basins. The thermal signature of strike-slip basins is highly variable as it depends on the

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attributes of the lithosphere on which they develop and if hydrothermal heat transport takes place along the fault zones.

Summary The geodynamics of sedimentary basins results from the interaction of a variety of processes acting on different spatial and temporal scales. Geophysical and geological observations indicate that intra-plate stress and heat input from the asthenospheric mantle are first-order controlling factors. On the continental lithosphere (see entry ▶ “Lithosphere, Continental”), structural inheritance and its impact on rheology is a third major player controlling the development of sedimentary basins. Intra-plate stress determines if an extensional or a compressional basin is formed. Temperature dominantly controls the rheological behavior of the lithosphere in addition to composition. These two parameters determine if the lithosphere is thinned uniformly, depth dependently, and in a symmetric or asymmetric mode. The more layered the lithosphere rheology is, the more discontinuous and asymmetric the stretching process will be. In contrast, a lithosphere deforms in a uniform stretching mode if the crust is strongly coupled to the mantle and no significant vertical or horizontal rheology contrasts are present. In both cases, the magnitude and rates of effective stress as well as the magnitude of the heat anomaly determine if stretching takes place slow enough to allow for cooling related strain hardening or fast enough to result in continental breakup. These deep factors interact continuously with surface processes such as deposition or erosion of sediments influenced by climatic conditions. Deposition leads to isostatic loading and enhanced subsidence, whereas erosion results in isostatic unloading and enhanced uplift.

Cross-References ▶ Continental Crustal Structure ▶ Continental Rifts ▶ Deep Seismic Reflection and Refraction Profiling ▶ Geodynamics ▶ Geophysical Well Logging ▶ Gravity Anomalies, Interpretation ▶ Heat Flow, Continental ▶ Heat Flow, Seafloor: Methods and Observations ▶ Isostasy ▶ Lithosphere, Continental ▶ Lithosphere, Mechanical Properties ▶ Lithosphere, Oceanic ▶ Lithosphere, Oceanic: Thermal Structure ▶ Magnetic Anomalies: Interpretation ▶ Magnetotelluric Interpretation

Sedimentary Basins

▶ Mantle Convection ▶ Mantle Plumes ▶ Radiogenic Heat Production of Rocks ▶ Seafloor Spreading ▶ Seismic Imaging, Overview ▶ Seismic Properties of Rocks ▶ Seismology, Global Earthquake Model ▶ Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat ▶ Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity

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Seismic Data Acquisition and Processing Kabir Roy Chowdhury Department of Earth Sciences, Utrecht University, Utrecht, The Netherlands

Definition Seismic data acquisition

Generation of (artificial) seismic signals on land (on surface, or, buried) or in water, reception of the signals after their travel through the interior of the earth, and their (digital) recording for later analysis. Analysis of recorded seismic signals to filter (reduce/eliminate) unwanted components

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Seismic data processing

Seismic Data Acquisition and Processing

(noise) and create an image of the subsurface to enable geological interpretation and eventually to obtain an estimate of the distribution of material properties in the subsurface (inversion).

Introduction Reflection seismics is akin to the “echo-in-the-well” experiment, it involves calculating the depth of the geological boundary from the two-way traveltime (TWT) of the seismic signal and its speed. Seismic data acquisition and processing aims mainly to obtain an image of the sedimentary basins in interior of the earth, using waves generated by “artificial” earthquakes. These images can then be used to identify locations favorable for accumulation of hydrocarbons (oil and gas), which may then be drilled to determine the ground truth – and eventually to exploit the resources. Since the first known reflection seismic experiment in 1921 near Oklahoma City, USA (Fig. 1.3, Sheriff and Geldart (1995)), reflection seismics has established itself as the most accurate technique to image the sedimentary basins for the exploration of hydrocarbons. The phrase “seismic” instead of “seismological” in the following stresses the “man-made” nature of the waves used. Both seismics and seismology use the basic theory of wave-propagation through the earth, for which Aki and Richards (2002) is a good resource. Table 1 summarizes the important differences between the two approaches though; let us briefly look at two. Frequency vs. period: Due to the spectral range of the signals involved, seismology traditionally uses period (s) to

describe the waves, whereas in seismics, frequency (Hz.) is used. Waves provide information about the medium through which they propagate at the scale of their wave-length, and use of higher frequencies in seismics (shorter wavelengths) leads therefore to a greater resolution (of the structure) compared to seismology. Wave-propagation: Seismology – again historically – mostly uses refracted energy, whereas exploration seismics is often synonymous with reflection seismics, although refraction seismic exploration predates the latter. It may be noted, however, that recently there has been a blurring of the boundaries between the two fields. This has been driven by progress both in the instrumentation (acquisition) and in theory (processing). Seismologists are increasingly using higher frequencies and reflected wavefields, and practitioners of reflection seismics are using lower frequencies and refracted phases – both with the aim to improve the imaging of their respective targets. Italicized items in Table 1 are thus changing with time. A similar comment applies also to “noise.” Considered earlier to be a bane of imaging, it is being increasingly used in innovative ways to optimize acquisition and improve imaging; this aspect will be briefly dealt with later. This essay will be mainly concerned with acquisition and processing of reflection seismic data. Note, however, that seismics is being increasingly applied to both shallower depths (high resolution seismics) and crustal-scale studies down to Moho and beyond (deep seismics), see ▶ “Deep Seismic Reflection and Refraction Profiling” for details of the latter. Seismic Data Acquisition and Processing is a broad subject, the treatment here will have to make choices based upon space constraints, etc. Some subtopics, for example, ▶ “Seismic, Migration” are, however, dealt with in separate essays in this volume. This essay has been updated for the 2nd

Seismic Data Acquisition and Processing, Table 1 Imaging the earth using natural/artificial earthquakes Keyword Wave source Energy penetration Max imaging depth Location of source Time of occurrence Energy involved Wave propagation Frequencies mostly excited/used Receivers Wave-field sampling Data volume Accuracy Main application Other applications Investment $$$$ Very ! expensive; $$ ! Expensive

Seismics Explosions, vibrations Shallow Base of crust Precisely known Precisely known Small-medium Mostly vertical 1–100 Hz Geophones Dense Terrabytes Large Oil and gas Civil engg., crustal $$$$

Seismology Natural earthquakes Deep Whole earth Estimated postfacto Estimated postfacto Can be huge Mostly horizontal 0.01–1 Hz. Seismometers Sparse (getting better) Gigabytes Small-medium Earth-structure Civil engg. $$

Seismic Data Acquisition and Processing

edition by briefly indicating relevant advances and including references for the same. The adjective “classical” has been used at places to indicate possible recent changes of the subject/term under discussion. The reader is assumed to be familiar with the basic theory of elasticity and wave-propagation and the related concepts of reflection, refraction, and scattering. The concept of rays will be frequently used – especially in illustrations – for convenience; real seismic signals are of course associated with wave-fronts. Similarly, the figures will depict a 2-D section of the 3-D earth. There are many good resources available even for the narrower field of Reflection Seismic Data Acquisition and (Signal) Processing, for example, Vermeer (2002), Yilmaz (2001), Menke (1989), Liner (2004), and Sheriff and Geldart (1995); the last one also contains some historical background and material over refraction seismics. Recently, some resources have also been made available for downloading on the Internet, for example, Claerbout (1985a, b). In this entry, all-capitals will be used to denote acronyms for jargons, of which there are quite a few (e.g., TWT above), and italicized phrases within double quotes will refer to articles elsewhere in this volume, for example, ▶ “Propagation of Elastic Waves: Fundamentals”.

Seismic Data Acquisition Before seismic signals can be processed, an artificial wavefield has to be generated using suitable sources at appropriate locations, measured by suitable receivers at other locations after getting reflected back from within the earth, and stored using recorders. Several technical and financial parameters have to be considered to obtain optimal data, for example, dense spatio-temporal sampling, power-supply, timing accuracy (GPS?), and – increasingly – HSE (Health–Safety– Environment) aspects. Design of a seismic survey (geometry) needs inputs from regional geology, exploration objectives, and logistical considerations. At first confined to land, seismic surveys are now-a-days carried out mostly in marine environments in round-the-clock operations using large vessels and a lot of instrumentation; single-channel seismics has faded away in favor of multichannel acquisition, allowing much more information to be obtained (see ▶ “Single and Multichannel Seismics”). Table 2 gives an overview of the equipment used under different field environments. Several interesting theoretical- and practical-advances have been made recently in the field of seismic data acquisition and processing. These include simultaneous recording of wavefields overlapping in time and use of noise as signal. These will be briefly described towards the end of this entry.

1367 Seismic Data Acquisition and Processing, Table 2 sources and receivers used in seismic surveys Environment Land Marine Water-bottom Onshoreoffshore

Sources Explosives/vibrators/ impact/noise Air/water-guns Explosives/guns Explosives/guns

Receivers Geophones Hydrophones/nodes Nodes/geo/hydrophones Geo/hydro-phones

Seismic Sources One needs a signal that is high in energy (amplitude) to ensure a good depth penetration and short in duration to ensure accurate determination and separation of the travel-times – a Dirac-Delta spike would be ideal, possessing perfect resolution, it is however a-causal. In practice, a sharp, compact, and repeatable signal is preferred. This quasi-idealized waveform, possessing finite temporal duration and frequency band-width (both with respect to the ambient noise), is called a wavelet. The source wavelet changes form as it travels through the earth due to several physical processes to be briefly discussed below – much of the later data processing aims to undo these changes. Repeatability of the source wavelet – that is, that of its amplitude and phase content is an important pre-requisite for the later processing steps. Explosives were the initial choice for source on land, providing large energy (good depth penetration) but having non repeatable signal shape and negative environmental impact. Development of large, truck-mounted electro-mechanical vibrators have led since 1960s to their increasing use in land-seismics, with both the above disadvantages of impulsive sources reduced significantly. In marine environment, compressed air is used – with explosion (air gun) or implosion (water gun) – to create the acoustic waves. The sources are towed by a ship together with the receivers (single vessel seismic) or by a separate ship (twoship seismics). There have also been experiments with shear-wave sources – both impact-type and vibratory. These – either alone or together with compressive sources – can provide extra information about the subsurface medium. For investigating shallower structures in engineering, environmental, and archeological applications, small impact-based sources, for example, weight-drops, rifles, and even portable vibrators are being frequently used and provide the required higher resolution. Seismic Receivers All land-seismic (and seismological) receivers (geophones, seismometers) are electro-mechanical devices that transform the relative motion of the medium in which they are embedded, into electrical voltages. Fidelity of this transformation –

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both in amplitude & phase is important to ensure maximum information retention for later retrieval – a flat amplitude response, with no phase distortion within the band of frequencies that are of interest, would be ideal. The output of such devices can be made to be proportional to the displacement/ velocity/acceleration associated with the causative motion. Dennison (1953) provides an overview of the physicomathematical underpinnings of geophone design; see also ▶ “Seismic Instrumentation”. Originally, geophones were designed to move – and hence record information – only in the vertical direction. Later, the importance of recording and analyzing the entire threedimensional elastic wavefield came to be realized. Multicomponent receivers, enabling recording/analysis of both horizontal components, or all three spatial components of the ground movement are being increasingly used even in large scale surveys. For use in water, small piezo-electric elements – hydrophones – are employed to record pressure variations – modern deployments typically consist of thousands of such elements being towed near the water surface by streamers, several kilometers long, which are liquid-filled plastic tubes, fitted with fins (for buoyancy), gps receivers (for location information), and fiber-optic cables (to transfer the data) to the ship. Three-component receivers may be deployed together with hydrophones at the water bottom (4C), to record the wavefield across it, see ▶ “Ocean Bottom Seismics”. Advances in this area have resulted in cable-free autonomous “nodes” – for both marine- and land-environments, for example, Dean et al. (2018). Besides saving cables and manpower (read: cost) for their deployment, they allow for complex acquisition geometries, large recording distances, and are also environmentally friendly. Note that both sources and receivers may be deployed in groups, using specific patterns, which affect the generation and sampling of the wavefield due to their directiondependent radiation/reception characteristics. Seismic Recorder The time-varying electrical signals output by the receivers represent arrivals back-scattered from different depths, all juxtaposed in time, and embedded within the ever present background noise and require storage for later processing. In the beginning, photographic films and later magnetic tapes were used for this purpose. The digital revolution starting in the 1960s, itself partly driven by the needs of seismic dataacquisition and processing, caused a complete shift to in situ digitization and digital storage. Similarly, the wires connecting the receivers to the recorder have been mostly replaced by fiber-optic cables or wireless. Modern 3-D seismic surveys record four-dimensional data – two each for sources and receivers; repeat surveys (time-lapse seismics) even add a fifth dimension.

Seismic Data Acquisition and Processing

Preserving the frequency, amplitude, and phase of the signal and the desired dynamic range are important considerations in designing the digitizing (sampling) unit. Digitization of this vast amount of data (wavefield) has traditionally been done honoring the Nyquist Sampling Criterion, which requires a uniform sampling rate of more than twice the highest frequency – or wavenumber – depending upon the domain (time/space) – present in the data. Sampling below the Nyquist rate produces artifacts (called aliases) in the digitized data at lower frequencies (or wavenumbers) and can eventually interfere with real signals that may be present there – see Claerbout (1985a) or Menke (1989) for details. Theoretically, Nyqist rate sampling allows the reconstruction of the uniformly sampled wavefield using an infinite integral – which, clearly, is not practical. Hence, the signal received at each receiver location is usually over-sampled (e.g., at twice- or higher-Nyquist rate) to maintain highfidelity while reconstructing the wavefield from the traces with a finite effort, without loss of information. Spatial sampling of the seismic wavefield is seldom regular and uniform and produces “gaps,” which need to be filled by interpolation; this will be briefly discussed in a later section. Recently, the necessity of acquiring seismic data at- or above-the Nyquist criterion has come under critical review – especially in view of the resultant massive increase in the data volume. An interesting development in this field has been included towards the end of this entry. As each receiver (group) corresponds to a different channel in the recorder, digitization in such systems (typically consisting of thousands of channels) must preserve the timebase, to enable comparison of the arrival-times between different traces. Also, the actual instant of the shot – t0 – must be transferred from the source to the recorder and recorded; it is often used to start the recording process itself, as seismics is only interested in travel-times, that is, arrivals-times with respect to t0. The digitized data – uniformly sampled timeseries – from each individual experiment (shot), consisting of multiple traces (output of receivers) is called a seismic record. Acquisition Geometry Assuming a layer-cake model (sedimentary beds parallel to the surface), early surveys deployed a number of sources and receivers along a straight line on the earth-surface to obtain a vertical cross-section of the geology below this line (2-D). Figure 1 shows schematically the approach in such a survey and is – in spite of its simplifications – useful in understanding several basic ideas. All the five panels show the earth-surface at the top, and a reflecting boundary (target), parallel to it, at some depth. Numerals on the surface represent surveyed equi-distant flag-positions to denote locations. The top panel shows the first measurement, with the source at “0” and eight receivers at locations “1” thru “8”. Assuming a

Seismic Data Acquisition and Processing Seismic Data Acquisition and Processing, Fig. 1 Schematics of seismic data acquisition by common-mid-point (CMP) profiling. Panels CSG0 through CSG3 represent common-sourcegathers (CSG); CMP4 is a common-mid-point gather for one common-depth- point (CDP). See text for details

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CSG ... CMP ... FOLD 0

2

4

6

8

10

12 surface locations

CSG0 Reflector

0

2

4

6

8

10

12 surface locations

CSG1 Reflector

0

2

4

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12 surface locations

CSG2 Reflector

0

2

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CSG3 Reflector

0

2

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6

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CMP

12 surface locations

S

CMP4 Reflector

Fold = 4

CDP

homogeneous and isotropic medium, the paths predicted by Snell’s law for a part of the source energy to first travel downwards to the target and then reflect upwards from it to reach the receivers are indicated by the oblique lines. The signals output from the receivers are digitized in the recorder to yield a seismic record, that is, a collection of seismic traces. Such an ordered collection of seismic traces

coverage = 400%

is called a gather; having a common source, the record resulting from our first measurement is a common source gather (CSG0), the suffix denoting source position. Under the twin-idealizations of no background noise, and a spike-like source signal, each channel in the recorder (seismic trace) will consist of one single blip corresponding to the arrival time of the signal; in reality, the arrivals will have

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random background oscillations due to noise, and one wavelet corresponding to the single reflection arrival. Assuming constant speed of propagation v and depth to the target H, it is trivial to show (e.g., Sheriff and Geldart 1995) that
the travel-times to the receivers can be written as  t2x ¼ x2 þ 4H 2 =v2 ¼ t20 þ x2 =v2 , tx being the arrival-time recorded by a receiver at a source-receiver offset of x. The travel-time curve for such a situation is thus a hyperbola – this simple relationship underlies much of seismic processing. t2 plotted against x2 thus yields a straight line, the slope being v2, that is, square of the slowness of the medium. Note that in seismics the velocity, which sensu-stricto is a vector, is almost always used to denote the local wave speed (a scalar), which is a property of the medium (rocks) . . . we shall follow this usage. Our aim is to find H, the depth to the target (¼ t0/2v). We have thus to estimate t0 from the rest of the reflection hyperbola. Note that the reflection points on the target for the different receivers are different. Hence, in what has become almost universal practice, the measurement is repeated after shifting the whole set-up laterally along the measurement line, keeping all the relative distances the same. In panel 2 of Fig. 1, the source and the receivers have been shifted right by one unit; only a few raypaths are shown for this gather (CSG1). Similarly, gathers CSG2 and CSG3 are also measured and recorded. During these measurements, the same receiver locations recorded signals from different sources, so that a postmeasurement re-arrangement of the traces could also yield common receiver gathers (CRG), in our case we would obtain CRG1–CRG11; these are useful for certain processing situations. The lowest panel of this figure shows a special kind of resorting, collecting the traces from the four shots with one common reflection point (CRP). Four traces corresponding to source/receiver combinations of 0/8, 1/7, 2/6, and 3/5 were selected respectively from the four gathers. For our simple geometry, the four ray-paths shown share two things – a common mid-point (CMP4) between their respective source and receiver locations and the common depth point (CDP) at the target depth, the latter being the same as CRP. Such a gather is called a CMP-gather and indexed by the position of the CMP. The travel-time plot of the reflection arrivals in a CMP-gather is also a hyperbola. The four ray-paths shown for the gather CM P4 all have the same reflection point and thus contain information about the same subsurface geology. The arrival times of the reflection signal in the four traces are of course different, as the travel paths are different. If this difference is corrected for, then adding the four traces should increase the coherent signal (information regarding the CRP) with respect to the random noise. The improvement of S/N by adding N traces is given by

Seismic Data Acquisition and Processing

P pffiffiffiffi ðN traces with identical signa1Þ N P  pffiffiffiffi ¼ N : ðN traces with random signa1Þ N The improvement of the signal-to-noise (S/N) ratio is thus roughly proportional to the square-root of the number of traces added. This number (4 in our case) depends upon the survey geometry and is called the fold of the survey. Starting from fold 1 for CMP0.5 (not shown), it gradually builds up to its nominal value (4 in this case) and again drops off at the other end of the survey. Acquisition configuration can be specified by expressions describing the position of the source relative to the receivers viz. end-on, split-spread, broad-side, etc. Depending upon the geology and the noise regime, these configurations, as also varying fold, leave subtle but important footprints on the data. In reality, the geology is of course not as in Fig. 1, and presence of structure (dips, faults, folds, etc.) is what makes hydrocarbon accumulation possible in the first place. Processing of 2-D data can remedy this situation – though only partially. Availability of more equipment and data-processing power led therefore to development of 3-D acquisition, with receivers laid out on the surface in a 2-D pattern, and sources also positioned in a different 2-D pattern, thus causing a better illumination of the subsurface by the seismic waves. Here too, the basic concept of adding fold number of traces in a CMPgather holds sway – point-shaped CMPs and CDPs being replaced by finite bins, their sizes depending upon the survey design and objectives (see Vermeer 1990, 2002 for further insight into acquisition design). In areas with structural complexity, the simplifying assumptions of CMP- processing break down, and the availability of computer power may make it possible – nay desirable – to process each trace of the recorded CSG separately, to try to obtain a better image (see also ▶ “Seismic Imaging, Overview” and ▶ “Seismic, Migration”). Restricting the deployment to the surface of the earth implies – as we shall see later – a bias for horizontal structures; this was eventually removed by carrying out measurements inside bore-holes called VSP; see ▶ “Vertical Seismic Profiling” for details. Finally, better recording instrumentation coupled with the need to detect changes in the hydrocarbon reservoirs resulting from exploitation have given rise to time lapse seismic (4D), whereby repeat imaging of the same area carried out after several years of production is used to validate/improve production models for reservoirs.

Seismic Data Processing Introduction Reflection seismic data, acquired in the field, has to be taken through several processing steps, before it can be interpreted

Seismic Data Acquisition and Processing

in terms of the subsurface structure. The source signal, on its way down, and back up to the receivers is modified by many factors; the aim of processing is to undo (i.e., correct for) as many/much of these effects as possible, leaving only the effects due to the causative structure of interest (geology) to be interpreted. Seismic data is a spatio-temporal sampling of the backscattered seismic wavefield, an ordered collection of traces, and can be considered to be a 2/3-D data matrix along with some auxiliary information regarding location, etc. As indicated before, analysis of the recorded wavefield requires it to have been uniformly and properly sampled in all relevant dimensions. Field logistics, however, frequently causes the spatial dimension(s) of the wavefield to violate this criterion. The resulting data will then be irregularly sampled in that dimension and will need resampling. Gaps may exist even within areas of otherwise equidistant traces due to problems associated with some receiver locations (houses, factories, etc.). Interpolating the recorded data to fill up these gaps to enable further processing requires theoretical underpinnings, for example, Gülünay (2003), Zwartjes and Sacchi (2007), and Liu and Fomel (2011). These are regularly employed to preprocess the seismic data before the real work (processing) can begin. It may be mentioned here that even after recovering uniformly sampled (spatial) wavefield obtained from an irregular acquisition geometry, the data may be aliased for coherent noise, for example, groundroll and multiples; see below for a brief explanation of these terms and ▶ “Seismic Noise” for details. Recently there have been some interesting advances in this field, which economize by intentionally undercutting the Nyquist criterion during acquisition before recovering the wavefield by suitable preprocessing – these will be briefly mentioned at the end of this entry. Presently, the spatial dimension too will be assumed to have been uniformly- and properly sampled. The traces themselves are an ordered collection of uniformly sampled amplitude values (time-series), with relevant information contained in their respective headers in (internationally) agreed formats. All processing steps aim to improve the spatio-temporal S/N ratio of the data by reducing the noise and/or by sharpening the wave-form (to improve the resolution). Signal vs. Noise Before proceeding further, it is useful to reflect on the terms signal & noise. That it is a matter of perspective is clear from this relative definition: signal is useful noise and noise is useless signal. In other words, someone’s noise is someone else’s signal, and vice versa. For example, the ground-roll, hated in reflection seismics, is useful in surface-wave

1371

seismology and shallow-seismics. Amazingly, using noise for seismic imaging has now become a field of active research (see the section “Noise as a Seismic Source” for references). In classical reflection seismics, signal is synonymous with primary reflection. Primaries, as these are often referred to, represent seismic energy reflected only once during its travel from source to receiver. Everything else, present in the traces, is taken to be noise. This includes multiply-reflected energy (multiples), diffractions (caused by sharp structures in the subsurface, for example, faults, pinch-outs), refracted arrivals, surface-waves (ground-roll). Nongeological noise sources include nature (wind, waves, animals) and man (traffic, drilling, industry, etc.). From processing point of view, noise could be coherent (ground-roll, water-pump, multiples), or, incoherent, each needing a different strategy. See ▶ “Seismic Noise” for details. Kinematics of the Seismic Signal (Primaries) Starting with some simple (but inaccurate) assumptions, for example, horizontal layering, constant speed, useful structural information can be extracted – a large data volume contributing to the robustness of the processing algorithms (also see ▶ “Seismic, Migration”). In this section, we focus on the travel-times of the waves (visualized as rays), see also additional information in ▶ “Seismic Ray Theory”. NMO

The travel-time for a primary-reflection from a horizontal reflector, shown earlier to be hyperbolic, can be rewritten as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ 4H2  2H : tx  t0 ¼ v

ð1Þ

The quantity on the left is the difference (see Fig. 2) between the oblique reflection-time at source-receiver offset (distance) x and the vertical TWT and leads to the relation: 2H Dtx ¼ v

! ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 x2 1 þ 2  1 : ð2Þ 1 þ 2  1 ¼ t0 4H 4H

Expanding the expression under square-root, and recognizing that in most seismic measurements offset targetdepth, we obtain the approximate relation 3, which could be improved by retaining additional higher order terms. Dtx 

x2 x2 ¼ 4vH 2v2 t0

ð3Þ

Δtx(¼tx  t0) is called the normal moveout (NMO) associated with the reflection travel-time. NMO can be used to

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Seismic Data Acquisition and Processing

align the primary reflection in all traces at t0 (TWT) by removing the effect of source-receiver distance (offset), that is, by flattening the reflector. NMO, an important concept in seismics, is used both to first identify primaries and later to align them for imaging the reflector. Note that to use 3 we need to know x (source-receiver offset), v (speed), and H (target depth); in practice, x is known and iteration is used to obtain optimal values for v and H. Dipping Bed

For a dipping reflector (Fig. 2, below), travel-time for the primary reflection is still hyperbolic, given by v2 t2y ¼ x2 þ 4H 2 þ 4Hx sin y:

ð4Þ

The minimum of the hyperbola is now shifted updip; the quantity t+x  tx is a measure of the asymmetry and can be used to estimate the dip.

Many Reflectors: Layer-Cake

Dix (1955) considered the case of many reflectors parallel to the surface – a good starting model for sedimentary sequences – and showed, that here too, the travel-time curve can be approximated at short offsets by a hyperbola: t2x



t20

x2 þ 2 , with vrms ¼ vrms

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 v Dt P i i: Dti

ð5Þ

The homogeneous velocity v (¼ vnmo) is now replaced by vrms (root-mean- square velocity), which depends upon the velocities of the layers vi and the vertical transit times ti through them. Vrms plays a role similar to vnmo in flattening the primaries in the multilayer case. Individual layervelocities may then be computed from the Dix’ equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2rmsn tn  v2rmsn1 tn1 vn ¼ : tn  tn1

ð6Þ

Velocities in Seismics

In seismics different terms are used to denote “velocity” depending upon the context. Table 3 lists a few, along with brief explanations, some of these will be elaborated later.

t Normal Move Out tx t0

tx− t0

NMO Stretch

x

O V

V’

X

H

O’ Z t

Semblance: A Measure of Signal Alignment

To apply optimal NMO correction, a quantitative measure of alignment of amplitudes, across several traces, is useful. Such a measure of similarity between n (amplitude) values, called semblance, is defined by

t t−x +x −x

x

O

X

V

H θ

Dip Move Out t+x − t−x

After NMO correction, a time-interval Δtx, say corresponding to a period of the wavelet recorded on a trace at an offset x, becomes Δt0; the wavelet is thus distorted (stretched). The expression 100 (Δt0  Δtx)/Δtx is a percentage measure of this stretch – 0% implying no distortion. In practice, a threshold percentage is specified to exclude parts of data – relatively large offsets and small arrival times – from being NMOcorrected (and taking part in further processing).

V’ O’ Z

Seismic Data Acquisition and Processing, Fig. 2 NMO of a primary reflector; horizontal (above), dipping (below). See text for details

Seismic Data Acquisition and Processing, Table 3 Jargons associated with the term velocity in seismics Jargon vint vav vapp vnmo vrms vstk vmig

Brief description Speed in a geological interval (assumed constant) Average speed between two points along a ray path Apparent speed measured by receivers in field (¼ dx/dt) Speed used for NMO-correction (strictly, only for one layer) Dix’ root-mean-square NMO velocity for layer-cake situation best velocity to stack CMP gathers best velocity to migrate the seismic data

Seismic Data Acquisition and Processing


P

n val

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P

P

2

S¼ P , n n val 2

and Sgate ¼ P

gate gate


P

n val n val

2 : 2

ð7Þ

Note that semblance is a dimensionless number between 1 (perfect match) and is 0 (perfect mismatch). The second form uses a time gate along the traces, generally having the width of the dominant period of the signal, for increased robustness. Semblance is used extensively in modern reflection velocity analysis, to evaluate the goodness of alignments of primary reflections along move-out curves computed for a range of trial velocities (Fig. 3). Velocity: Processing Point of View Wave-speed (called velocity in seismics) in the medium is the missing link needed to convert the travel-time information to depths required for structural interpretation – and eventually drilling. Note that velocity is needed to find the structure (geology), but structure is needed to find the velocity. This catch-22 situation is solved iteratively – shown schematically in Fig. 4. Velocity is a macroscopic (wave-length-scale average) property of the rock depending upon the density and elastic properties of the minerals making up the lithology (see ▶ “Seismic Properties of Rocks”). In rocks of interest in seismics (sandstone, shale, limestone), velocity is not a good indicator of lithology, with considerable overlap in 0

C M P gather

x

values, with some exceptions, for example, salt, anhydrite (relatively higher velocity). Presence of porosity and porefluids (water, oil, gas) are the most important factors for this overlap and are in turn caused by the burial history of the rocks. Wave-propagation in fluid-filled porous media is described by Biot-Gassman theory, see Lee (2008) for references and recent developments. Propagation velocity (the missing link) can be estimated by direct measurements (see, e.g., Sheriff and Geldart 1995 for details), which have shortcomings though (see Table 4). The velocity used for processing seismic reflection data is usually determined iteratively from the data itself and will be described later. Amplitude Changes along the Propagation Path Several factors cause the amplitude of the seismic waves to change as they travel from source to receiver. These can be corrected for, so as not to mask the weaker changes (signals) of interest. Geometrical Spreading

Conservation of energy requires a continuous reduction of amplitude, as a seismic wave-front spreads through a medium – hence the term geometrical spreading. The loss depends upon the mode of spreading and the distance travelled (r). For pffiffiffiffiffiffiffiffiffiffiffiffiffi primaries (body-waves), amplitude ð/ energyÞ decreases / r1, whereas for ground-roll (surface-wave), the decrease is / r1/2, the latter shows why ground-rolls, with their (relatively) large amplitudes, are a big problem in seismics. Absorption

Tgate Vtrial

t Seismic Data Acquisition and Processing, Fig. 3 Schematical drawing showing calculation of multichannel semblance. Curved bold line represents the move-out curve for a trial velocity, and the two surrounding lines represent the boundaries of the time-gate; see text for details Seismic Data Acquisition and Processing, Fig. 4 Iteratively solving for both structure and velocity in seismics

The propagating wave continuously loses energy due to absorption too, which is a physical property of the medium, and can be described by several equivalent parameters, for example, absorption coefficient, damping factor, the most common being the quality factor of the material, Q ¼ 2π/[fractional energy lost per cycle]. It is a dimensionless quantity, with a value of 0 implying perfect absorption and 1 implying perfect elasticity. Absorption, with Q considered to be frequencyindependent within the band-width of interest in seismics, causes relatively greater attenuation of higher frequencies – leading to a change in the waveform during propagation. See ▶ “Seismic Viscoelastic Attenuation” for more details. Energy Partitioning at Interfaces

Boundaries of geological heterogeneities (layering, faults, etc.) also cause changes in the amplitude of the wavelet; such

Man−made

Structure

Velocity

Wave−field

Depth Image

Field

Seismic Record

Elastic Disturbance

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Seismic Data Acquisition and Processing

Seismic Data Acquisition and Processing, Table 4 Direct determination of seismic velocities and their shortcomings Method Uphole-time Check-shots, well-shoot VSP Sonic log Lab measurements

Shortcoming Useful only for the weathering layer Limited depth-range, destructive Available late in exploration, expensive Available late, noisy (high-frequency) Limited availability

changes are, indeed, of prime interest in seismics. As in optics, the interaction between the wave-fronts and the geological structure depends upon their relative dimensions, that is, their radii of curvature – with specular reflections and pointscattering building the two end-members, both of which are encountered in seismics. Another concept from optics, diffraction, is useful to understand the complexity of the interaction between the wave-front and the medium. See ▶ “Seismic Diffraction”, ▶ “Seismic Waves, Scattering”, and ▶ “Energy Partitioning of Seismic Waves” for additional details. Waveforms: Convolution, Deconvolution Factors modifying the source signal along the path of the seismic wave may be divided as: near-source (ns), that is, weathering layer, earth (e), that is, the target geology, nearreceiver (nr), receiver (r) and recorder (rec), with the output trace (o) as the final result. Each of these, denoted in Eq. 8 below by the expression in parentheses, affects (filters) the source wavelet (s). In a series of papers/reports (Robinson 2005; Treitel 2005), the MIT geophysical analysis group (GAG) laid the foundation of the digital revolution in seismic data processing, by examining the nature of these filters and developing methods to undo their effects. These resulted in major advances in time-series analysis and digital filtering (Robinson and Treitel 1964) and a critical evaluation of the (statistical) nature of earth’s reflectivity (target geology). Convolutional Model of the Seismic Trace

As the source – and recorded – signals are both time-series (uniformly sampled, ordered collection of amplitudes), it is useful to represent all the other elements mentioned in the above-paragraph also as such. For a column of vertically layered reflectivity, such a time series would correspond to values equal to RCs placed at times converted from depths using velocities. Now, making the crucial assumption that all these filter elements are linear systems, the recorded trace can be expressed as: oðtÞ ¼ sðtÞ  nsðtÞ  eðtÞ  nrðtÞ  r ðtÞ  recðtÞ þ nðtÞ: ð8Þ In Eq. 8,  (star) is the convolution operator, well-known in the theory of linear systems; n(t) represents some additive noise which does not follow this model, hopefully, it is mostly removed early in the processing. The time-series that transform s(t) into o(t) can also be interpreted as the impulse

response of the corresponding elements, for example, r(t) is the response of the receiver to a sudden spike signal. Using Fourier-Transforms to change the time-series into their spectra, and remembering that convolution in time- domain corresponds to multiplication in frequency domain, one obtains: OðoÞ ¼ SðoÞ NS ðoÞ EðoÞ NRðoÞ RðoÞ REC ðoÞ, ð9Þ where the noise term has been neglected (see Sheriff and Geldart 1995 for introduction to linear operators and Fourier theory). Equation 9 clearly shows the filtering effect of the different elements, each one modifying the spectrum of the incoming signal by modifying/removing a part of its frequencies. Our aim, in seismic data processing, is to extract e(t), the geological structure from the recorded signal o(t). Deconvolution as Inverse Filtering

Undoing the act of the filterings implied in Eqs. 8 and 9 is called deconvolution (decon), or, inverse filtering. Equation 9 can be rewritten as O(o) ¼ E(o) · REST (o), where REST(o) groups together all the elements on the right besides the geology. Then, E(o), or e(t), can be estimated from EðoÞ ffi OðoÞ=REST ðoÞ,

or,

1

eðtÞ ffi oðtÞ  rest ðtÞ :

ð10Þ

The approximation sign, for both forms of Eq. 10 – in frequency-domain (first), or, in time-domain (second) – is necessary, even in the noise-free case. Spectral division needs precautions to avoid zero-division in parts of the spectrum, where frequencies have been weakened/removed. Fortunately, addition of noise helps, since signals of interest in seismics exist – by definition – only above the ambient noise level. See Liner (2004), Sheriff and Geldart (1995) and Yilmaz (2001) for the stabilizing role of spectral whitening in decon. Wavelet Processing

Wavelets: Let’s take a closer look at seismic wavelet, introduced in the section about “Seismic Sources,” as a signal of finite frequency band-width and temporal duration. Using standard concepts from time-series analysis (Sheriff and Geldart 1995; Yilmaz 2001), simple examples of wavelets are: a : ð3, 2, 1Þ, b : ð2, 3, 1Þ and c : ð1, 2, 3Þ, the numbers representing uniformly sampled amplitudes starting from t ¼ 0. Remembering that squares of the amplitudes in a wave(let) are measures of energy, we see that these three wavelets, while looking very different, have the same total energy. Depending upon the energy build-up, wavelet a is called minimum-delay (energy is front loaded), b is mixeddelay, and c is maximum delay; physical (causal) wavelets are

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minimum delay, although in the example, a is not strictly causal, due to the instantaneous build-up of energy at t ¼ 0. In frequency domain, the expressions minimum/mixed/ maximum-phase are used instead. Wavelet estimation: Auto-correlation of the wavelets a-c are all symmetrical about t ¼ 0, that is, have no phase information, for example, ’bb ¼ (2, 3, 14, 3, 2); these are Fourier Transforms of the respective power-spectra. In seismics, an estimate of the power spectrum is often available from the data. The question then arises whether an estimate of the wavelet may be obtained from it – an outline follows. Using Z-transform notation, one can write the wavelet, say c, and its auto-correlation as polynomials:

time-limited (typically, 10–20s long) signal with the frequency continuously varying between given start- and endvalues and comes in many flavors, for example, up-, down-, linear-, nonlinear-sweeps. Neglecting other terms, one could write from Eqs. 8 and 9: o(t) ¼ s(t)  e(t). The recorded signal is thus the convolution of earth reflectivity with the sweep signal. We could remove its effect (deconvolve) by crosscorrelating the observed signal with the sweep (which we know precisely), a process, which is equivalent to convolving with its time-reversed version, and get

CðZÞ ¼ 1 þ 2Z þ 3Z 2 , and, Fcc ðZÞ ¼ 3Z 2 þ 4Z 1 þ 14 þ 4Z  3Z2 ,

Due to the sweep-signal being time-limited, its autocorrelation is not a Delta-spike (ideal), but is a symmetrical (zero-phase) signal called Klauder wavelet. The result is thus not quite the desired earth reflectivity (although it has the correct phase) and needs further processing for improvement (see Yilmaz 2001; Liner 2004; Sheriff and Geldart 1995). De-ghosting: The effect of large RCs in the shallow subsurface has been mentioned earlier. Figure 5 (above) shows one such situation; here the source is placed below the weathering layer, for better energy transmission towards the deeper target (ray going directly downwards). A part of the

Z being the unit-delay operator, its powers denoting timeshifts with respect to t ¼ 0. According to the fundamental theorem of algebra, a polynomial of degree n in Z must have n roots, that is, it can be expressed as a product of n factors of the form: (Z  Z1)(Z  Z2) . . . (Z  Zn), each factor representing a basic wavelet (doublet). Half the doublets of an auto-correlation polynomial are minimum-delay; their product represents the Z-transform of the unique minimumdelay causative wavelet. See Yilmaz (2001) and Sheriff and Geldart (1995) for details, assumptions, critical remarks, and alternate approaches (e.g., homomorphic deconvolution) to deconvolution of time-series. Wavelet manipulation: Much of seismic processing is involved with manipulating the wavelet (deconvolution in a general sense). While very powerful, it contains potential for pitfalls, if applied without a proper understanding of the suitability of the particular technique, as each decon step also causes artifacts. Spiking decon aims to sharpen the shape of the signal, to improve temporal resolution – and interpretation. Ideally, it involves convolving the wavelet with its inverse operator, to yield a spike, that is, perfect resolution. Zero-phasing converts the signal to one with zero-phase; the result is a symmetrical signal (a-causal) and is primarily useful for interpretation if the peak can be made to coincide with the reflecting boundary. Any-phasing is used in merging seismic datasets of different vintages and with differing source wavelets. General shaping groups methods to convert the signal to any desired shape optimally – using some statistical criteria. Depending upon whether a model is available for decon, the methods could also be divided in deterministic, that is, model-based and statistical. Deterministic Deconvolution

Vibroseis processing: Vibrators (see the section on “Seismic Sources”) use a repeatable source signal, called sweep. It is a

sðtÞ  oðtÞ ¼ sðtÞ  sðtÞ  eðtÞ  dðtÞ  eðtÞ  eðtÞ: ð11Þ

surface −R TWT nΔt

source

Ghost

target

surface (−1) water hard bottom (R)

Reverberation primary single multiple double multiple target

Seismic Data Acquisition and Processing, Fig. 5 deterministic deconvolution applied to ghost (above) and reverberation (below). The near vertical ray-paths are shown obliquely for better visualization, see text for details

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wave-energy also travels upwards and gets reflected down from the base of the weathering layer (ray going first up, and then down). In certain cases, the RCweathering could be quite large and negative. The energy reflected downwards follows with a short delay behind the direct wave and is called a ghost; the observed record is thus corrupted by that caused by a delayed ghost. Removing the latter from the recorded trace is called deghosting and is an example of model-based decon. Assuming the TWT between the source and the base of the weathering to be n samples (¼ nΔt), one can write: o(t) ¼ s(t)  Rs(t  nΔt), or, using Z-transforms, OðZ Þ ¼ SðZÞ  RS ðZ Þ Zn ¼ SðZÞð1  RZ n Þ: (1  RZ ) is, clearly, the Z-transform of the ghost-operator. Hence, n

SðZÞ ¼ RðZ Þð1  RZ n Þ1 , or, sðtÞ ¼ oðtÞ þ sðt  nÞ: The last form above implies recursive filtering in the timedomain to achieve deghosting. Alternately, expanding (1  RZn)1, the inverse-filter operator in time-domain can be written as
 gðtÞ1 ¼ 1, 0, 0, . . . þ R, 0, 0, . . . þ R2 , 0, 0, . . . De-reverberation The lower part of Fig. 5 shows another situation, where strong reflectivity associated with the water-bottom causes long trains of high-amplitude reverberation of signals in the water-layer. The ray-paths shown schematically are: one primary reflection from the target, two multiples reflected once in the water layer, and three multiples reflected twice; there could be many more, posing a serious problem in marine seismics. Depending upon the depth of water, certain frequencies will, as a result, experience severe distortion (enhancement or suppression). In the simplified case of a water-column with a TWT equal to the sampling interval, and remembering that the negative reflectivity causes phase-change, the total operator (signal + reverberation) can be written as: wðtÞ ¼ ð1, 2R, þ3R . . .Þ ! W ðZÞ ¼ 1  2RZ þ 3R2 Z2  . . . ¼ ð1 þ RZ Þ2 : It follows that the deconvolution in this case can be achieved by the operator
 W ðZ Þ1 ¼ ð1 þ RZ Þ2 , or, wðtÞ1 ¼ 1, 2R, R2 : This elegant operator is called the Backus filter (see Backus 1959).

Statistical Deconvolution

In the absence of a deterministic model, one could attempt to change the signal wavelet to any desired shape, by designing filters that are optimal in a statistical sense. Based upon work in information theory by Norbert Wiener and others, the applications in seismics were pioneered by the MIT-GAG group, for example, Robinson (1967). Schematically, the basic approach is: assume filter ! compute output ! compute error ! minimize to get normal equations ! solve for filter coefficients If errors are assumed to be Gaussian, and l2 norms are used, the operators obtained are called Wiener filters. Such optimum filters are used widely, for example, in • Zero-lag spiking – to increase resolution • Zero-phasing – to ease interpretation • Prediction Filtering – to remove multiples which are predictable, the remnant being the prediction error, corresponding to the deeper signal Wiener optimum filter The normal equations for the filter-coefficients f are given by the matrix equation shown in Eq. 12 in its compact form finput,input  f ¼ finput,output ,

ð12Þ

which relates the auto-correlation of the recorded (input) wavelet to its cross- correlation with the desired (output) wavelet. For the derivation of Eq. 12, and a detailed treatment of statistical deconvolution, see, for example, Yilmaz (2001) or Sheriff and Geldart (1995) – an example is shown below to illustrate the approach. Spiking filter If the wavelets are all n-sample long, the Matrix Eq. 12 can be expanded as 0 B B B B B @

fi,i ð0Þ

fi,i ð1Þ

...

fi,i ð1Þ

fi,i ð0Þ

...

fi,i ðn  1Þ fi,i ðn  2Þ . . . 1 0 fi,d ð0Þ C B B fi,d ð1Þ C C B ¼B C C B A @ fi,d ðn  1Þ

fi,i ðn  1Þ

10

f0

1

C CB C B fi,i ðn  2Þ C CB f 1 C C CB C CB A A@ fi,i ð0Þ f n1

ð13Þ The auto-correlation matrix fi,i in Eq. 12, with the same element in each diagonal descending from left to right, is a

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Toeplitz matrix; f is a column vector with the filtercoefficients to be determined and fi,o is a column-vector with elements from the cross-correlation matrix. Equations with Toeplitz matrices can be efficiently solved by a procedure called Levinson recursion. Wiener filter: a simple example Given the input wavelet it ¼ (1, 1/ 2), let us find the optimum two-element Wieneroperator to transform it to the desired wavelet dt ¼ (1, 0), that is, a zero-delay unit-spike. We get, ’i,i ¼ (5/4, 1/2), and, ’i,d ¼ (1, 0). Equation 13 then becomes 5=4

1=2

1=2

5=4

!

f0

!

f1

yielding f Wiener

¼

1

!

, 0   20 20 ¼ , : 21 21

Applying this filter to the input, we obtain the output (20/21, 2/21, 4/21), which compared to the desired output, gives a squared-error of 1/21. The ideal filter for this decon is the inverse filter for the input wavelet. Writing I(Z) ¼ 1  Z/2 for the Z-transform of the input, the Z-transform of the inverse-filter (which will convert the input to an ideal unit-spike) is ¼ (1  Z/2)1 ¼ 1 + Z/2 + Z2/4 + . . ., which is an infinitely long operator! For an honest comparison of its performance with that of the Wiener-filter, we apply its first two terms to the input, getting the filtered version as (1, 0, 1/4); although looking better at the first glance, its squared error is 1/16, that is, larger than that of the Wiener-filter! It can be shown that the Wiener filter is the best two-element filter for this problem. Suppose the input wavelet is (1/2, 1), that is, not minimum delay, which we want to transform to a zero-delay spike. Normal equations now give the Wiener-filter as (10/21, 4/21), with the output (5/21, 8/21, 4/21) and the squared error as 6/21. Inverse filter is now (2, 4, 8, . . .), which is extremely unstable! Its first two filter-elements give the output (1, 0, 4) with 16 as error! Wiener filter performs here worse than in the first case, because it was trying to convert a maximum-delay wavelet to a minimum- delay spike, but it still does better than the (finite) inverse filter. In this case, if a maximum delay spike (0, 1) was desired, Wiener-filter coefficients would be (16/21, 2/21), giving a filtered output of (8/21, 17/21, 2/21) with a squared error 4/21, which is better than that for a zero-lag spike output. Table 5 summarizes the results. Seismic Data Acquisition and Processing, Table 5 Performance of Wiener- and Inverse-filters Input wavelet (1, 0.5) (1/2, 1) (1/2, 1)

Desired wavelet (1, 0) (1, 0) (0, 1)

2-point Wiener Filter (20/21, 8/21) (10/21, 4/21) (16/21, 2/21)

Error 1/21 6/21 4/21

2-point inverse Filter Error (1, 0.5) 1/16 (2, 4) 16

The Processing Flow: Putting it all Together Most of the processing modules (filters) operate on the data (time-series) sequentially, the entire process resembling a flow, though there are a few stand-alone modules too. The operations could be on individual traces (single-channel), or on a gather of traces (multichannel). Schematically, a seismic processing flow looks like: Input module ! aseries of processing  modules ! Output module

Modules have been developed for carrying out specific tasks within the flow, for example, static correction, bandpass filtering, stacking, migration. Usually, there is a choice of modules (algorithms) available for a specific step – each with slightly different characteristics (and artifacts) – and the proper selection of the modules for a flow needs both expertise and experience. This point is illustrated in the Fig. 6, which shows six different results of processing the same data. An overview of commonly applied corrections (processing module) is shown in Fig. 7. Space constraints will permit us to briefly describe only selected items from this list, which itself is not exhaustive; see Yilmaz (2001) for a more detailed treatment, and ▶ “Seismic Imaging, Overview” for additional information. Note that some modules may be applied more than once in the flow, and also, that parts of the flow may be iteratively repeated, till a reasonable result is obtained. The latter shows the importance of quality control (Q/C), by means of visual-display and other (quantitative) tools. The decision as to whether the processing of a dataset is finished depends often on the geological objectives, technical possibilities, and managerial constraints of time and money. Preprocessing

Editing of seismic traces is an important first step, in view of the largely automated processing sequences later. Geometry assignment is also an essential step at this stage and attaches acquisition information to the traces, for example, source- and receiver-coordinates. Each seismic trace is assigned a header, to store such and other information to enable efficient interprocess communication. Prestack Processing

Static corrections These are time-invariant corrections applied to the traces, due to, for example, elevation differences, and involve up-, or, down-shifting the entire trace in time; an example of their dramatic effect can be seen in Fig. 8. The static effects due to slow lateral changes (long wavelength statics) are particularly difficult to model and can cause imaging problems. Residual statics involves small time-shifts applied late in the flow to improve the result; it uses the powerful concept of surface-consistency to try to correct for near-surface errors that were not modeled properly in the earlier stages. Its implementation by Rothman (1985)

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Seismic Data Acquisition and Processing, Fig. 6 Seismic Data Processing has no perfect answer. Seismic cross-sections produced from the same data processed by six different contractors. (Figure from Yilmaz 2001, courtesy SEG and the author)

heralded the use of nonlinear optimization (simulated annealing, genetic algorithm) in seismics. Amplitude corrections Loss of amplitude due to geometrical spreading and absorption can be corrected for using the theory described earlier; the latter needs a Q-model, in the absence of which empirical relationships based on the total travel path/time are used. A part of the record may be removed

from processing due to the presence of noise, or suspected nonprimaries; depending upon the part of the data-volume removed, one then talks about top-mute, bottom-mute, or a generalized mute. Similarly, a balancing (amplitude equalization) may be applied to several adjacent traces to compensate, in an ad hoc manner, for local variations, for example, bad receiver coupling.

Seismic Data Acquisition and Processing Seismic Data Acquisition and Processing, Fig. 7 Components of seismic processing flow

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Pre-processing QC: edit bad traces – assign field-geometry – resample – notch-filter Pre-stack processing statics:

elevation – datum – residual

amplitude: filtering:

spreading – Q-compensation – mute – balance band-pass – f-k – f-x – τ -p – median – notch

deconvolution:

deterministic – predictive – statistical

velocity analysis: stack:

CMP sort – const. velocity stack – semblance analysis

NMO correction – DMO correction – CMP stack – AGC – display

Post-stack processing migration velocity analysis – migration – time-to-depth – display No-stack processing migration velocity analysis – prestack depth migration – display Special processing True Amplitude (AVO, DHI) – VSP – Anisotropy – image rays

Filtering, sharpness, taper Any process that removes/ reduces a part of the suspected noise from the data is a filter. Frequency-filters (high-cut, low-cut, band-pass) are the simplest examples. Data f(t) is transformed using Fourier theory to its spectrum F(o) ¼ A(o) expiot in the frequency domain, the amplitudes mainly corresponding to noise are zeroed-out, and the data is transformed back to the time-domain. Development of algorithms for fast and efficient Fourier transform of time-series (FFT) have caused large scale application of digital filters. Multichannel data enables double-transformation of f(x, t) to F(o, k), making filtering possible based upon slopes (apparent velocities) in the o–k plane; this is particularly effective in eliminating, for example, slow travelling ground-roll (large amplitude surface waves), which often mask the primaries. An example of such filtering is shown in Fig. 9. Notch filters are used to remove a narrow band of frequencies, for example, a 50 Hz noise from overhead transmission line. t -p transforms are useful in filtering multiples, and in untangling far-offset data for velocity analysis, these use the Radon domain for the decomposition (Phinney et al. 1981). A few general comments apply to all filters:

• Filtering is effective only to the extent of signal-noise separation in the transformed domain. • For any filtering, there is a trade-off between sharp cut-offs in the transform- domain and oscillatory artifacts in timedomain – and vice versa. A compromise solution to this unavoidable problem is to apply tapers to smoothen the cut-off and thus minimize edge-effects. Deconvolution This important aspect has been dealt with in some detail in an earlier section. Stacking velocity analysis This is almost always carried out in the CMP-domain, after re-sorting the data. The aim is to determine the velocity model, that is, vrms(TWT), to be used for computing the best move-out correction for the CMPgathers. For a suite of velocity models, hyperbolic move-out curves are computed for the range of TWTs of interest; semblances are then computed to determine how well the arrivals in the gather line-up along these curves and displayed in a contour plot in the vtrial-TWT domain, allowing interactive picking of an improved velocity model (Fig. 8). The process is repeated at as many CMPs as possible – sometimes grouping neighboring CMPs together for averaging, velocity being a macroscopic property. The result is a laterally varying

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Seismic Data Acquisition and Processing, Fig. 8 Stacking velocity analysis using semblance (color contours). Semblance values are shown for a dataset for a range of trial velocities (horizontal- axis) and enable interactive velocity picking as a function of TWT (vertical- axis). The

right panel shows a dramatic improvement in resolution as a result of proper static correction. (Figure from Yilmaz 2001, courtesy SEG and the author)

model of vrms. Equation (6) can now be used to infer interval velocities. CMP-stack Once a reasonable velocity function has been determined, each trace in the CMP-gather (say, CMP4 in Fig. 1) is shifted in time by subtracting the corresponding move-out corrections. The move-out corrected traces in the CMP-gather are then added (stacked) together to produce one trace. This process, CMP-stack, reduces random noise – which does not line-up, while strengthening the reflection signal – which does, and thus improves S/N ratio of the data. Note that stacking reduces the data volume too – by a factor of fold! Much of the power of the seismic imaging derives from this simple step, which enhances the primary reflections (those only once reflected) at the expense of everything else.

Zero-offset traces/sections The stack traces are also called zero-offset traces, the move-out correction having made the source and receiver coincident. A collection of stack traces is a stack- or zero-offset section and represents the first (albeit approximate) 2-D cross-section of the subsurface. For display purposes, CMP-stack sections may be subjected to automatic gain control (AGC), an extremely nonlinear time-variant amplitude scaling, to balance weaker/ deeper signals and stronger/shallower ones. Poststack Processing: Positioning Properly

The CMP-stack has one big drawback, and dips were neglected throughout, which is what we are really after. This results in many artifacts in the section, for example, crossing

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Seismic Data Acquisition and Processing, Fig. 9 Use of two-dimensional Fourier Transform as an apparent-velocity filter for four marineseismic records brings out (weaker) reflections. (Figure from Yilmaz 2001, courtesy SEG and the author)

S

M

M’

N

P MIG

t’

NMO

DMO

S’

R

θ

t

Seismic Data Acquisition and Processing, Fig. 10 Effect of dip in positioning the reflector

layering, diffraction tails. Anticlinal structures are somewhat flattened, and synclinal structures could give rise to bow-ties. Migration Fig. 10 shows the problem schematically in the CMP-domain, for the case of a trace recorded from source S at receiver R from a reflector with a dip θ. After conventional prestack processing, the zero-offset trace would be plotted on the t axis below the mid-point M. This is clearly an error, as the zero-offset ray for M should be incident normally on the reflector – at N, this correction is called migration (Yilmaz 2001; Sheriff and Geldart 1995). Migration steepens and shortens energy alignments and moves these up- dip, clarifying the tangled image. Figure 11 shows an example of a successful migration. For further details, please see ▶ “Seismic, Migration”. DMO Fig. 10 shows yet another error to be considered – the actual reflection point for the source-receiver combination SR is P, and not N. Even worse, for the different SR pairs

making-up the CMP-gather with mid-point M, the reflection points are all different, that is, are smeared along the reflector, the amount of smear being dip-dependent. The process used to correct for this dip-dependent part of the move-out correction is called DMO. In practice, this step is applied before migration as indicated in Fig. 7; Figure 10 shows the sequence: • Reflection time is NMO corrected and plotted below M. • NMO corrected time is DMO corrected and plotted below M’, the true zero-offset point. • NMO + DMO corrected time is MIGRATED and plotted at P, the reflection point. Time-to-depth conversion For final structural interpretation, the TWTs in the seismic section (stacked, or, migrated) need to be converted to depths, with velocity again playing the key role. For a homogeneous medium, this is just a rescaling of the vertical axis; with the velocity varying smoothly only in vertical direction (e.g., for flat sedimentary sequences), a nonuniform stretch of the vertical axis may suffice. Laterally varying velocities present depth-conversion problems though, increasing with the strength of the heterogeneity; ray-bending now needs to be taken into consideration. No-Stack Processing: Imaging Complex Structures

In the presence of strong lateral velocity variations (e.g., below salt structures), the conceptual model used to process CMP-gathers breaks down. Removing the simplifying assumptions makes the imaging physically more reasonable, albeit at the cost of substantially increased computational effort.

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Seismic Data Acquisition and Processing, Fig. 11 Migration positions energy from dipping structures properly. Here bow-tie like artifacts in the top part of panel (a) are imaged back into causative synclinal

structures in panel (b). The artifacts persisting in the bottom of panel (b) probably point to lack of interest in imaging deeper structures in this case. (Figure from Yilmaz 2001, courtesy SEG and the author)

Prestack depth migration and Migration velocity analysis Simply put, prestack depth migration involves tracing the seismic energy from the source to the receiver for every recorded trace, with the philosophy that every seismic trace should be computable if the structure and the velocity model were both known. A detailed velocity model is essential for the success of PSDM; often a simplified model is assumed and iteratively improved using migration velocity analysis (MVA). For details/issues regarding 2-D vs. 3-D, time- vs. depth- and poststack vs. prestack migration, see ▶ “Seismic, Migration” and Yilmaz (2001).

Anisotropy Many seismic media are anisotropic, a common example being shales, which exhibit faster speeds parallel to the layering than across it, and require modification of procedures for proper imaging, for example, the move-out curve would no more be hyperbolic. This field is proving important for reservoir studies too, see Helbig and Thomsen (2005) for an overview and also ▶ “Magnetic Anisotropy”.

Special Processing

True amplitude – AVO, DHI Observed variations of the RC with respect to angle of incidence may be interpreted in terms of changes in lithology across the reflecting boundary (amplitude versus offset, or, AVO) and may even indicate the nature of the pore-fluids. Such direct hydrocarbon indicators (DHI) include bright-spots, flat-spots, polarity-reversals, etc. (see Yilmaz 2001; Sheriff and Geldart 1995). A prerequisite for such analyses is true amplitude processing, avoiding modules that remove differential amplitude information, for example, balancing, stacking, AGC. Converted waves Using multicomponent receivers, it is possible to identify waves that have been converted at the reflection boundary and hence possess asymmetrical up- and down-ray-paths. Proper processing of such data, with CCP (common conversion point) replacing CDP, provides a better constraint for imaging. VSP & Cross-well tomography Bore-holes can be used for placing receivers (and sources), resulting in significant noise-reduction. The first processing step now is to separate up- and down-going wavefields, for details, see ▶ “Vertical Seismic Profiling”.

Some Recent Developments Recording overlapping wavefields Several efforts have been initiated recently to gather more (denser) data without significantly adding to the recording time. Commonly referred to as simultaneous/continuous sources, these use rapidly firing sources, for example, airguns/vibrators in marine surveys, and record wavefields overlapping in time (blending), which are later separated using various flavors of deblending. This results in a considerable increase in spatial bandwidth due to much closer spacing of the sources. Not having to repeatedly reshoot the lines later from the intermediate source locations also leads to significant savings of time (read: costs) besides improving data quality by reducing/ eliminating unavoidable location errors. Another variation of this approach is “simultaneously” recording sources from different azimuths for both land- and marine-surveys. Blending/deblending in seismic acquisition mentioned above involves coding/decoding different parts of the recorded wavefield in a manner enabling their later separation. The subject is well known in communications theory; the “Cocktail Party Effect” uses differential characteristics of the desired signal (voice of a single person at a distance) to make it understandable from out of a cacophony of many others including some (loud) near ones (Cherry (1953)).

Seismic Data Acquisition and Processing

The related but different issue of intentionally recording overlapping signals in seismics is not new (Garotta 1983), although it is currently undergoing rapid development – both in theoretical and practical aspects (Beasley 2008). Carrying the idea farther, an interesting case has been reported, wherein wavefields generated by individual elements of the airgun firing at rapid but randomized intervals are being continually recorded (Hegna et al. 2019). Simultaneous sources could be a game changer in the field of seismics due to its positive impact on several fronts – faster, cheaper, safer, and better, for example, Nakayama et al. (2019). Randomly undersampling the wavefield The importance of properly sampling the seismic wavefield in both temporal and spatial dimensions has been stressed before, as the first step for deriving accurate structural information regarding the subsurface. Sampling in the time domain is usually not a problem albeit at the cost of the increased data volume (by the factor) associated with oversampling with respect to the Nyquist criterion. Proper sampling in the spatial domain à la Nyquist, however, is a different story and is seldom achieved in a modern acquisition setting, owing partly to the scale of operations – the necessary preprocessing step of interpolation of traces to fill-up the “gaps” has already been mentioned. The necessity – or otherwise – to adhere to the Nyquist criterion while sampling analog signals has been studied also by signal processing engineers. Recent work in information theory has shown that signals can indeed be recovered even from (severely) undersampled data under certain circumstances. If the signal of interest exhibits a certain structure in a transformed domain, it can be sampled below the Nyquist criterion and recovered in that domain – fortunately, this characteristic is shown by reflection seismic data too. One of the earliest contributions in this field, specifically for the case of spatial undersampling in reflection seismics, was by Bednar (1996). In it, the possibility of randomly sampling the wavefield spatially was discussed and was hinted to be more efficient and economical than uniform undersampling. This is perhaps not surprising, given that imaging reflectors is essentially summing up (integration) of energy along certain specified trajectories (see ▶ “Seismic, Migration”), and efficient evaluation of integrals by summing values at random points is known in applied mathematics. In practice, random undersampling of the signal is supposed to perform superior to regular undersampling as it will convert aliased energy (due to undersampling) into incoherent noise, which can then be easily filtered out. Hennenfent and Herrmann (2008) showed jittery undersampling (with respect to the Nyquist criterion) to perform even better, as it also enables control of the omnipresent gaps that need to be filled by interpolation for later reconstruction. Herrmann (2010) discusses this idea of randomized undersampling

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further. Developed from the technique of compressive sampling, commonly used in modern electronics, e.g., digital camera and magnetic resonance imaging, it is shown to lead to a significant reduction in data volume and hence to be able to master the “curse of dimensionality” – a phrase which points to the exponential increase in data volume with each added dimension of seismic acquisition. It is again shown to be particularly effective, if the signal is sparse in a suitable transform-domain. Further studies about the applicability of this technique to related fields, for example, ▶ “Seismic, Waveform Modeling and Tomography” are subjects of active current research. One may dare to state that the Nyquist sampling criterion has at last been conquered – at least as far as seismics is concerned! Noise as a seismic source As hinted in the first edition, it is now an exciting and established field of research under the general banner of seismic interferometry & daylight imaging. Starting from early conjectures (Claerbout 1968) and later breakthroughs (Fink 1993, 1997), there have been both theoretical and practical advances understanding its possibilities and limits. Although this topic belongs to some other essays in this volume (▶ “Seismic, Ambient Noise Correlation”, ▶ “Seismic Noise”), a very brief introduction follows. The basic idea is that “noise” – also in seismics – contains useful information, which can be extracted by – and this is somewhat surprising – some pretty straightforward processing. Noise recorded at two locations can be used to obtain relevant medium properties in the intervening medium. This approach can also be used to place virtual sources anywhere in the medium. For an introduction to this fascinating subject, see Curtis et al. (2006); Wapenaar and Snieder (2007); Wapenaar et al. (2010a, b); Snieder and Wapenaar (2010). Imaging vis-a-vis Inversion Imaging tries to obtain useful (drillable) structural information using large data-redundancy and simple conceptual models, whereas inversion aims at getting values for the physical parameters of the medium, using more involved theory, see ▶ “Seismic Imaging, Overview”. Buske et al. (2009) give a good introduction to the two approaches, whereas Weglein et al. (2009) discuss some critical underlying questions. Till recently, resolution in seismic imaging suffered from an information-gap as pointed out by Claerbout (1985b): velocity information was being obtained by kinematic analyses for apparent frequencies below ca. 2 Hz and reflectivity was being derived for frequencies above ca. 10 Hz. Recent improvements in tomographic velocity analysis and broadband data acquisition have resulted in increasingly narrowing this gap, which results in improved resolution (Nichols 2012). As shown in the Fig. 12, there is now even an information overlap between the results obtained from these two processing techniques. This, in turn, gives rise to interesting questions regarding the consistency of the high- and low-frequency results that have finally to be combined.

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Seismic Data Acquisition and Processing

Seismic Data Acquisition and Processing, Fig. 12 Sketch showing the improvement in resolution in reflection seismics. In black, the famous sketch by Claerbout (1985b), showing the gap between information (accuracy) from velocity analysis and reflectivity. Advances in

tomography (red) and broadband acquisition (blue) have changed the situation essentially to that of an informational overlap. (Figure from Lambare et al. (2014), courtesy EAGE and the author)

Full wave-form inversion Obtaining the visco-elastic properties of the medium so as to be able to reproduce each seismic/seismological trace completely remains the ultimate goal of inversion of seismic/seismological data (see ▶ “Seismic, Waveform Modeling and Tomography”). This subject is currently the subject of major theoretical and applied research, the latter also profiting from considerable increase in available computational power. In the beginning, efforts were limited to achieving a reasonable match between a selected part (phase) of the observed waveform and the computed synthetic. Given improvements in acquisition, theory, and the computer power, progress is being made to reproduce larger parts of the entire wavetrain – a complete reproduction may though be never achievable. For a quick introduction to this topic, including underlying problems, and related developments in exploration geophysics, see Virieux and Operto (2009); Virieux et al. (2017); Brittan and Jones (2019). Note that perfect inversion implies perfect imaging – and vice versa!

▶ Seismic Imaging, Overview ▶ Seismic Instrumentation ▶ Seismic Noise ▶ Seismic Properties of Rocks ▶ Seismic Ray Theory ▶ Seismic Viscoelastic Attenuation ▶ Seismic Waves, Scattering ▶ Seismic, Ambient Noise Correlation ▶ Seismic, Migration ▶ Seismic, Waveform Modeling and Tomography ▶ Single and Multichannel Seismics ▶ Vertical Seismic Profiling

Summary The simple echo-in-the-well experiment mentioned at the start needs many physico-mathematical supports to analyze data obtained from the earth’s subsurface. Starting at data acquisition, these acquisition/processing modules yielding the final image resemble a pipeline (flow). Several of these have been explained briefly; for others, cross-references elsewhere in this volume have been provided.

Cross-References ▶ Energy Partitioning of Seismic Waves ▶ Magnetic Anisotropy ▶ Ocean Bottom Seismics ▶ Propagation of Elastic Waves: Fundamentals ▶ Seismic Diffraction

Bibliography Aki K, Richards PG (2002) Quantitative seismology, 2nd edn. University Science Books, CA, USA Backus M (1959) Water reverberations – their nature and elimination. Geophysics 24(2):233–261 Beasley C (2008) Simultaneous sources: a technology whose time has come. Soc Expl Geophys. Expanded abstracts, 2796–2800 Bednar J (1996) Coarse is coarse of course unless . . .. The Leading Edge 15(6):763–764 Brittan J, Jones I (2019) FWI evolution – from a monolith to a toolkit. The Leading Edge 38(3):179–184 Buske S, Lecomte I, Nemeth T, Operto S, Sallares V (2009) Imaging and inversion – introduction. Geophysics 74(6):WCA1–WCA4 Cherry E (1953) Some experiments on the recognition of speech, with one and with two ears. J Acoust Soc Am 25(5):975–979 Claerbout JF (1968) Synthesis of a layered medium from its acoustic transmission response. Geophysics 33(2):264–269 Claerbout JF (1985a) Fundamentals of geophysical data processing. Blackwell. http://sepwww.stanford.edu/sep/prof/fgdp5.pdf Claerbout JF (1985b) Imaging the earth’s interior. Blackwell. http:// sepwww.stanford.edu/sep/prof/iei2/ Curtis A, Gerstoft P, Sato H, Snieder R, Wapenaar K (2006) Seismic interferometry – turning noise into signal. The Leading Edge 25(9):1082–1092 Dean T, Tulett J, Barnwell R (2018) Nodal land seimsic acquisition: the next generation. First Break 36:47–52 Dennison AT (1953) The design of electromagnetic geophones. Geophys Prosp 1(1):3–28 Dix C (1955) Seismic velocities from surface measurements. Geophysics 20:68–86

Seismic Diffraction Fink M (1993) Time reversal mirrors. J Phys D Appl Phys 26:1333–135. http://iopscience.iop.org/0022-3727/26/9/001 Fink M (1997) Time reversed acoustics. Phys Today 50:34–40. https:// doi.org/10.1063/1.881692 Garotta R (1983) Simultaneous recording of several vibroseis© seismic lines. Soc Expl Geophys. Expanded abstracts, 308–310 Gülünay N (2003) Seismic trace interpolation in the Fourier transform domain. Geophysics 68(1):355–369 Hegna S, Klüver T, Lima J, Wisloff J (2019) Making the transition from discrete shot records to continuous seismic records and source wavefields, and its potential impact on survey efficiency and environmental footprint. Geophys Prospect 67(6):1472–1485 Helbig K, Thomsen L (2005) 75-plus years of anisotropy in exploration and reservoir seismics: a historical review of concepts and methods. Geophysics 70:9ND–23ND Hennenfent G, Herrmann F (2008) Simply denoise: Wavefield reconstruction via jittered undersampling. Geophysics 73(3):V19–V28 Herrmann F (2010) Randomized sampling and sparsity: getting more information from fewer samples. Geophysics 75(6):WB173–WB187 Lambare G, Guillaume P, Montel J (2014) Recent advances in ray- based tomography. In: Extended abstracts, 76th annual meeting. EAGE Lee M (2008) Comparison of the modified Biot-Gassmann theory and the Kuster-Toksöz theory in predicting elastic velocities of sediments. U.S. Geological survey scientific investigations Report Liner C (2004) Elements of 3D seismology. PennWell corporation, OK Liu Y, Fomel S (2011) Seismic data interpolation beyond aliasing using regularized nonstationary autoregression. Geophysics 76(5): V69–V77 Menke W (1989) Geophysical data analysis, rev edn. Academic, New York Nakayama S, Blacquière G, Ishiyama T, Ishikawa S (2019) Blendedacquisition design of irregular geometries towards faster, cheaper, safer and better seismic surveying. Geophys Prospect 67(6):1498–1521 Nichols D (2012) Resolution in seismic inversion. In: Expanded abstracts, 74th annual meeting – workshop. EAGE Phinney R, Roy Chowdhury K, Frazer LN (1981) Transformation and analysis of record sections. J Geophys Res 86(B1):359–377 Robinson E (1967) Predictive decomposition of time series with application to seismic exploration. Geophysics 32:418–484 Robinson E (2005) The MIT Geophysical Analysis Group (GAG): 1954 and beyond. Geophysics. https://doi.org/10.1190/1.2000287 Robinson E, Treitel S (1964) Principles of digital filtering. Geophysics 29:395–404 Rothman D (1985) Nonlinear inversion, statistical mechanics, and residual statics estimation. Geophysics 50(12):2784–2796 Sheriff RE, Geldart LP (1995) Exploration seismology, 2nd edn. Cambridge University Press, Cambridge, UK Snieder R, Wapenaar K (2010) Imaging with ambient noise. Phys Today 63(9):44–49 Treitel S (2005) The MIT Geophysical Analysis Group (GAG): 1954 and beyond. Geophysics. https://doi.org/10.1190/1.1993707 Vermeer G (1990) Seismic wavefield sampling. Society of Exploration Geophysicists, Tulsa Vermeer GJ (2002) 3-D seismic survey design. Society of Exploration Geophysicists, Tulsa Virieux J, Operto S (2009) An overview of full-waveform inversion in exploration geophysics. Geophysics 74(6):wcc1–wcc26 Virieux J, Asnaashari A, Brossier R, Métivier L, Ribodetti A, Zhou W (2017) An overview of full-waveform inversion in exploration geophysics. In: Encyclopedia of exploration geophysics, SEG. pp R1–1–R1–40. https://doi.org/10.1190/1.9789560803027.entry6 Wapenaar K, Snieder R (2007) Chaos tamed. Nature 447:643 Wapenaar K, Draganov D, Snieder R, Campman X, Verdel A (2010a) Tutorial on seismic interferometry, part I. Geophysics 75(5):75A195–75A209 Wapenaar K, Slob E, Snieder R, Curtis A (2010b) Tutorial on seismic interferometry, part II. Geophysics 75(5):75A211–75A227

1385 Weglein A, Zhang H, Ramírez A, Liu F, Lira J (2009) Clarifying the underlying and fundamental meaning of the approximate linear inversion of seismic data. Geophysics 74(6):WCD1–WCD13 Yilmaz O (2001) Seismic data analysis, processing, inversion and interpretation of seismic data, 2nd edn. Investigations in geophysics, vol I. Society of Exploration Geophysicists, Tulsa Zwartjes P, Sacchi M (2007) Fourier reconstruction of nonuniformly sampled, aliased seismic data. Geophysics 72(1):V21–V32

Seismic Diffraction Enru Liu China University of Mining and Technology, Xuzhou, China

Definition Diffraction

Diffraction wave-field

Diffraction tomography

Redistribution in space of the intensity of waves resulting from the presence of an object. It is also referred as the penetration of wave energy into areas forbidden by geometrical optics, e.g., the bending of wave energy around obstacles without obeying Snell’s law as explained in Huygens’ principle (generation of secondary sources). An event observed on seismic data produced by diffracted energy, resulting at the termination of reflectors (as at faults and other abrupt changes in seismic impedance), and it is characterized on seismic records and sections by a distinctive alignment. An inverse technique that is used in seismic exploration to reconstruct the physical properties under investigation using waveequation propagation.

Introduction When a wave (elastic wave, electromagnetic wave, or sound wave) meets an object, particle, or obstacle, it is diffracted due to scattering of energy of the propagating wave (Fig. 1). However, in the literature, the terms of diffraction and scattering are often used interchangeably, and it can be sometimes confusing. Scattering and diffraction are two physical phenomena that any kind of waves can experience, but they are not the same thing. Scattering is understood in terms of particles, and behaves similarly for waves. Scattering is effectively bouncing off something. For waves, it is being absorbed and then almost immediately released in another direction. Scattering occurs because an object gets in the way of the wave. The part of the wave that strikes the object

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Receiver Source

uo + us uo

us S

L

n

u’s n

Seismic Diffraction, Fig. 1 Problem configuration: a scattering object S bounded by the curve L with outward normal n. Upon an incidence of u0 located at source, the total wave-field received at receiver is the superposition of the incident wave-field u0 and the scattered wave-field us

must either pass through (e.g., light through glass), be absorbed (sunburn), or be scattered (light bouncing off the wall, so we can see the wall). Diffraction is due to part of a wave being removed. It is an action taken by the part of the wave that does not strike an object. Here is an example of diffraction: Imagine a straight wave traveling forward along the surface of water. If you block the left half of the wave, the right half will not just keep moving forward, and it will expand toward the left, toward where the blocked wave would have been. A wave seems to continuously regenerate itself, always pushing forward on itself. When a section is removed, parts of the wave get pushed into the empty spot. This, in some ways, correlates to your pushing a wide row of blocks. If many people push on a few blocks each, a straight line can be maintained. If one person tries to do so, the blocks in front will tend to spread out. In a sense, diffraction and scattering refer to a wave being redirected as a result of interacting with objects. However, a more precise definition used in optics distinguishes these two. In optics, scattering generally implies interaction of waves (or photons) with spatially uncoordinated (unordered) atoms, i.e., wave being scattered off the particles and surfaces. This means that if one looks at a picture of the scattered radiation, one would see a spatially continuous footprint. Diffraction, on the other hand, occurs when the object or part of the object is made up of ordered atoms. These atoms, being neatly arranged, “scatter” the waves or photons in a coordinated way, i.e., in specific directions, giving rise to what we can see on a film as bright spots rather. In other words, diffraction is a special type of scattering that leads to large-scale interference effects. Usually this is because the surface causing the scattering has some overall organization, such as a ruled diffraction grating or the knife-edge of a slit.

Although scattering and diffraction are not logically separate, the treatments tend to be separated, with diffraction being associated with departures from geometrical optics caused by the finite wavelength of the waves. Thus, diffraction traditionally involves apertures or obstacles whose dimensions are large compared to a wavelength. To lowest approximation the interaction of waves is described by ray tracing (geometrical optics). The next approximation involves the diffraction of the waves around the obstacles or through the apertures with a consequent spreading of the waves. Simple arguments show that the angle of diffraction of the waves are confined to the region θ l/a where l is the wavelength and a is linear dimension of the aperture or obstacle (approximations considered work well if l/a 1). Note that diffraction may cause the localization of seismic waves – a phenomenon that is similar to the localization of lights in crystal (Larose et al. 2004). This phenomenon is caused by the focusing and defocusing of energy when seismic wave propagates through media with distributions of periodical or random distribution of scattering bodies. In order to gain some perspective on these two extremely complex phenomena (diffraction and scattering), various theories and models have been developed in physics, and the limitations and validity of these theories are controlled by two ratios: object dimension α to wavelength (α/l) and pathlength L to wavelength l(L/l). In contrast to other branches of physics, in geophysics path-length is also important as we are interested both in near field as well as far field (and often near and far fields are treated differently). We also often use the dimensionless parameters kα and kL (where k ¼ o/l is the wave number, o is frequency).

Diffraction Theories Application of any diffraction model can be divided into two separate tasks. First, one must obtain the fields exiting a diffracting object (i.e., the near fields, or the boundary field values), or a reasonable approximation thereof. The second step involves propagating those fields to the desired observation point. These are distinct and separate parts of the diffraction models. Most texts do not make this separation clear. Instead, the boundary value assumptions and subsequent propagation into the far field are lumped together into one theoretical treatment. If the resulting diffraction pattern is at all inaccurate, it is difficult to determine how much of that error is due to incorrect boundary fields and how much is the result of the propagation calculation. Because of this, it is often difficult to know which model is appropriate for a particular problem. There are a number of different models to compute diffraction wave-field due to wave scattering, including the Huygens’ Principle, the Rayleigh-Sommerfeld theory, the

Seismic Diffraction

Kirchhoff’s diffraction theory, Taylor series perturbation theory or the high-order Born approximation, Rytov phase approximation, and a model referred to as angular spectrum of plane waves. The well-known Fraunhofer and Fresnel approximations, as they appear in most introductory texts, are derived from the Kirchhoff model. Several other methods are also available for treating diffraction problems: discrete wave-number techniques; generalized ray techniques; and various numerical methods (finite difference methods; finite element method; and boundary integral or element methods). Each of these theoretical methods and models is based on some assumptions and has its strengths and weaknesses, and each can be satisfactorily employed for some ranges of problems. The choice of an appropriate model is based on what is known about a specific problem. If there are several objects or particles, the scattering from one object will induce further scattered fields from all the other objects, which will induce further scattered fields, from all the other objects, and so on. This process is called multiple scattering. Not all theories are applicable to multiple scattering problems (in physics, this is called many-body problem). The Kirchhoff approximation ignores multiple scatterings between any two surface points. In general, it has been considered valid for the large-scale objects. Perturbation theory based on the Taylor series expansion, also sometimes called the high-order Born approximation, is valid for the smallscale objects whose dimensions are less than a wavelength (or objects whose physical properties are not too different from background solids). The Rytov phase approximation to large-scale object is not subject to the stringent restrictions that apply to the Kirchhoff approximation. Studies have shown that the Rytov approximation improves the Kirchhoff approximation in both amplitude and phase. To some degrees, the high-order Born series approximation can account for multiple scattering between closely-spaced objects. For instance, the second-order Born approximation might be sufficient to guarantee the accuracy for general rough surfaces without infinite gradients and extremely large surface heights. In contrast to other branch of field, e.g., optics, in seismology two kinds of waves exist, compressional and shear waves. These two waves can convert to each other when one meets an object. When multiple objects exist, the conversion and interaction between different wave types due to multiple scattering can be very complex. Therefore, care must be taken when one uses any diffraction theory to solve specific geophysical problems.

Geometrical Theory of Diffraction The geometrical theory of diffraction (GTD) is an extension of geometrical optics that accounts for wave diffraction by edges. It was introduced in 1953 by Keller (the most

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commonly cited reference was published by Keller in 1962). The geometrical theory of diffraction was devised to eliminate many of the problems associated with geometrical optics. The strongest diffracted fields arise from edges, but ones of lesser strength originate from point discontinuities (tips and corners). The total field u ¼ (u1, u2, u3) at an observation point * x is decomposed into geometrical optic rays (the incident or reference field) u0i and diffracted components udi       ! ! ! ui x ¼ u0i x þ udi x :

ð1Þ

The behavior of the diffracted field is based on the following postulates of GTD: 1. 2. 3. 4. 5.

Wavefronts are locally plane waves. Diffracted rays emerge radially from an edge. Rays travel in straight lines in a homogeneous medium. Polarization is constant along a ray in an isotropic medium. The diffracted field strength is inversely proportional to the cross sectional area of the flux tube. 6. The diffracted field is linearly related to the incident field at the diffraction point by a diffraction coefficient (see Achenbach et al. 1982, for various analytic solutions). GTD is a high frequency method for solving wave scattering problems from large-scale discontinuities or discontinuities in more than one dimension at the same point, and it uses ray diffraction to determine diffraction coefficients for each diffracting object-source combination. These coefficients are then used to calculate the field strength and phase for each direction away from the diffracting point. These fields are then added to the incident fields and reflected fields to obtain a total solution. Multiple scattering wave-fields cannot be easily computed using GTD.

Kirchhoff Approximation In the Kirchhoff representation of diffracted wave-fields, the ith component of diffracted wave-field ui is computed using 2 p  ! !3 ð @Gi x , X
 
  ⇀ ⇀ 5nj dS ! , udi x ¼  P uk x c0kjpq 4 x @x

(2)

q

where nj is the jth component of the normal n to the surface of scattering object, X is a point on the face of the scattering object, [uk] is the kth displacement discontinuity across the object in the direction of n (object normal), and c0kjpq is the elastic tensor of the background, which are often assumed to be isotropic. Equation 2 provides a means of evaluating the diffracted field so long as the displacement discontinuity [u]

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on the object can be estimated accurately. Since the amplitudes and phases of [u] are unknown, in the Kirchhoff approximation, these are taken to be the same as if the object were infinitely long, that is, the effect of the boundary is ignored (see Douglas and Hudson 1990; Liu et al. 1997). Therefore, the Kirchhoff approximation is a high frequency approximation, which is only applicable to diffraction from objects whose dimension is larger than the wavelength.

Perturbation Theory: Born and Rytov Approximation The diffracted wave-field in Eq. 1 can be written as an infinite series of the Taylor series expansion and is derived by assuming that the physical property of scattering objects is written as a perturbation of background media. The Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. It is the perturbation method applied to scattering by an extended body. It is accurate if the scattered field is small, compared to the incident field, in the scatterer. It is only valid for weak scattering or when the obstacles are small compared to both the wavelength and the spacing between the objects. Clearly, it has serious limitations when dealing with large-scale objects. The simplest approximation, called single scattering or the first-order Born approximation, is to ignore the multiply scattered field between objects. This approximation has been widely used in geophysics (Hudson and Heritage 1981; Wu 1982, 1989). The Rytov approximation linearly perturbs the phase of a wave-field with respect to model parameters such as velocity whereas the Born approximation perturbs the amplitude. When the Green’s functions for point sources are replaced by Eikonal approximations, the Rytov perturbed wave-field becomes a scaled, differentiated, time-delayed version of the reference or incident wave-field.

Seismic Diffraction

(see Bouchon 1987; Pointer et al. 1998; Liu and Zhang 2001; Sanchez-Sesma and Campillo 1991, 1993).

An Example of Seismic Diffraction from a Single Fracture Some practical applications of diffraction theories in seismology include scattering from cavities; topographic variation, fractures, cracks (Fig. 2). Here we give three examples to demonstrate the application of various theories in tacking diffraction problems. An example is given here for diffraction from a single fracture as computed using the finite difference method (Vlastos et al. 2003, 2007). The model geometry is shown in Fig. 2. The source, receivers, and fracture are situated in an ideal elastic full space (Vp ¼ 3300 m/s, Vs ¼ 1800 m/s, density r ¼ 2.2 g/m3). The receiver array at which vertical and horizontal particle displacements are recorded is horizontal and 340 m above the fracture. The fracture is 300 m long. The source is located at the center of the receiver array. The source type is a vertical force. The source signal is a Ricker wavelet with a peak frequency of 25 Hz and a pulse initial time of 0.1 s. Figure 3 also shows the different kinds of waves generated by the interaction of the waves generated by the source and the fracture. The source generates both P and S waves. When they reach the fracture boundary those waves are reflected and we have PPr, PSr, SPr, and SSr waves. We calculate the theoretical ray travel-times and overlap them on the synthetic seismograms. Figure 3a, b

10 20 30

Diffracted waves SSd PSd PPd

Reflected waves PPr PSr SSr

Numerical Methods Used to Compute Diffraction Wavefield Various numerical methods can also be used to compute diffracted wave-fields, particularly now, when highperformance computers are widely available. In geophysics, finite difference methods have been used widely in the study of scattering of elastic waves by crustal heterogeneities with continuous variation of physical properties and they have also been used to model scattering by thin cracks and large fractures (see Coutant 1989; Fehler and Aki 1978; Coates and Schoenberg 1995; Vlastos et al. 2003, 2007). Elastodynamic boundary integral equation or boundary element method has also been widely used to compute wave-fields from discrete inclusions with various spatial distributions as well as rough-surface topographic variations

Depth in metres

40 50 60 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120 Horizontal distance in metres

Seismic Diffraction, Fig. 2 Schematic representation showing diffraction from a fracture, and representation of the ray paths of the different kind of waves generated by the source that interact with the fracture

Seismic Diffraction

a 0

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Receiver numbers 20 40

b 60

0

60

0.2

0.2

Time in seconds

Receiver numbers 20 40

0.4

PP

0.4

PP

PPr

PPd PS

PPr

PPd PS

0.6

SS PSd

PSr SSr 0.8

SSd

1.0

0.6 PSr

SS PSd

SSr 0.8

SSd

1.0 Horizontal component

Horizontal component

Seismic Diffraction, Fig. 3 Comparison between the theoretical ray travel-times and the synthetic seismograms generated by the modeling method: (a) the horizontal (x) and (b) the vertical (z) components

show the horizontal (x) and the vertical (z) components, respectively, of the synthetic seismograms together with the theoretical ray travel-times. As we can see from both figures, we have very good agreement between the theoretical ray travel-times and the synthetic seismograms. All types of waves are accurately represented in the synthetic seismograms. Owing to the type of source that we implement, we have strong arrivals at short offsets on the horizontal component and strong arrivals at long offsets on the vertical component. In addition to that, the diffracted waves from the tips of the fracture and the PPr and PPd waves are not visible in the horizontal component, but they are very clearly demonstrated in the vertical component and follow the theoretical travel-times. This is expected because the source causes vertical displacements on the medium, so very close to the source and very far away from it, the horizontal displacement is negligible. Another aspect of the comparison between the theoretical and the modeled data is that they give us further insight into the waveform patterns. For instance, we can see in both Fig. 3a, b that in the areas of superposition between the reflected waves from the fractures and the diffracted waves from the tips we have maximum amplitude in the wave-field, as a result of constructive interference. This gives us valuable information concerning the medium we are examining.

Diffraction Tomography Seismic tomography is emerging as an imaging method for determining subsurface structure. When the viewangle

coverage is limited and the scale of the medium inhomogeneities is comparable with the wavelength, as is often true in geophysical applications, the performance of ordinary ray tomography becomes poor. Other tomographic methods are needed to improve the imaging process, e.g., diffraction tomography. It has been widely used in surface reflection profiling (SRP), vertical seismic profiling (VSP), and crosshole measurements. Theoretical formulations are derived by Wu and Toksoz (1987) for two-dimensional geometry in terms of line sources along a source line and line receivers along a receiver line. The theory for diffraction tomography is based on the Born or Rytov approximation. Multisource holography, which is similar to Kirchhoff-type migration, often gives distorted images of the object. This distortion causes long tails of the image in the case of SRP and a strong noise belt in the case of VSP and is due to incomplete and nonuniform coverage of the object spectrum. The filtering operation of diffraction tomography helps in correcting the nonuniform coverage (including duplication) of the object spectrum in the reconstruction process and therefore reduces the distortions. On the other hand, multisource holography is better suited for imaging sharp boundaries with large acoustic impedance contrasts since diffraction tomography is restricted, to weak inhomogeneities. In addition, multisource holography has the flexibility to be used with an arbitrary number of sources (including a single source). Its sampling interval is not restricted by the Nyquist frequency. Numerical examples show that combined data sets (such as surface reflection data combined with VSP data or cross-hole data combined with surface data) improve the image quality.

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Summary Diffraction refers to the spatial distribution of the intensity of seismic waves resulting from the presence of an object (e.g., a hill- or valley-like topographic feature on the surface, or a fracture, cavity, or cave in the subsurface). It is also referred as the penetration of wave energy into areas forbidden by geometrical optics, e.g., the bending of wave energy around obstacles without obeying Snell’s law as explained in Huygens’ principle (as secondary sources). In the geophysical literature, the words of diffraction and scattering are often used interchangeably and it can be confusing. Diffraction and scattering are two different physical phenomena, but they are related to each other. Several analytic diffraction theories have been developed, e.g., geometrical theory of diffraction and Kirchhoff diffraction theory. More recently, numerical methods, such as finite difference and boundary element or boundary integral methods, are becoming increasingly used by geophysicists to simulate wave diffractions by complex variation of Earth’s topography or subsurface cavies (cavities), fractures, irregular layers, etc. Geophysicists now often use the diffracted wave-field to reconstruct the subsurface physical properties (diffraction tomography) to solve the so-called inverse diffraction problem.

Cross-References ▶ Deep Seismic Reflection and Refraction Profiling ▶ Earthquake Rupture: The Inverse Problem ▶ Propagation of Elastic Waves: Fundamentals ▶ Seismic Ray Theory ▶ Seismic Waves, Scattering ▶ Seismic, Migration

Bibliography Achenbach JD, Gautesen AK, McMaken H (1982) Ray methods for waves in elastic solids, with application to scattering by cracks. Pitman Learning, London Bouchon M (1987) Diffraction of elastic waves by cracks or cavities using the discrete wave-number method. J Acoust Soc Am 81:1671 Coates RT, Schoenberg M (1995) Finite-difference modelling of faults and fractures. Geophysics 60:1514 Coutant O (1989) Numerical study of the diffraction of elastic waves by fluid-filled cracks. J Geophys Res 94:17805 Douglas A, Hudson JA (1990) The effect on teleseismic P of the zone of damage created by an explosion. Geophys J Int 103:111 Fehler M, Aki K (1978) Numerical study of diffraction of plane elastic waves by a finite crack with application to location of a magma lens. Bull Seismol Soc Am 68:573 Hudson JA, Heritage JR (1981) The use of the Born approximation in seismic scattering problems. Geophys J Int 66:221 Keller JB (1962) Geometrical theory of diffraction. J Opt Soc Am 52:116 Larose E, Margerin L, van Tiggelen BA, Campillo M (2004) Weak localization of seismic waves. Phys Rev Lett 93:048501-1–048501-4

Seismic Discontinuities in the Transition Zone Liu E, Zhang ZJ (2001) Numerical study of elastic wave scattering by distributed cracks or cavities using the boundary integral method. J Comput Acoust 9:1039 Liu E, Crampin S, Hudson JA (1997) Diffraction of seismic waves by cracks with application to hydraulic fracturing. Geophysics 62:253 Pointer T, Liu E, Hudson JA (1998) Numerical modelling of seismic waves scattered by hydrofractures: application of the indirect boundary element method. Geophys J Int 135:289 Sanchez-Seama FJ, Campillo M (1993) Topographic effects for incident P, SV, and Rayleigh waves. Tectonophysics 218:113 Sanchez-Sesma FJ, Campillo M (1991) Diffraction of P, SV, and Rayleigh waves by topographic features: a boundary integral formulation. Bull Seismol Soc Am 81:2234 Vlastos S, Liu E, Main IG, Li XY (2003) Numerical simulation of wave propagation in media with discrete distributions of fractures: effects of fracture sizes and spatial distributions. Geophys J Int 152:649 Vlastos S, Liu E, Main IG, Narteau C (2007) Numerical simulation of wave propagation in fractured media: scattering attenuation at different stages of the growth of a fracture population. Geophys J Int 171:865 Wu RS (1982) Attenuation of short period seismic waves due to scattering. Geophys Res Lett 9:9 Wu RS (1989) The perturbation method in elastic wave scattering. Pure Appl Geophys 131:605 Wu RS, Toksoz MN (1987) Diffraction tomography and multisource holography applied to seismic imaging. Geophysics 52:11

Seismic Discontinuities in the Transition Zone Lev P. Vinnik Institute of Physics of the Earth, Moscow, Russia

Definition The transition zone (TZ) is the mantle layer bounded by the 410- and 660-km seismic boundaries. The high P-wave and S-wave velocity gradients within the TZ are caused by a series of polymorphic phase transitions, the depths (pressures) of which are controlled by temperature and composition. Structure of the TZ plays an important role in the heat/mass transfer between the upper and the lower mantle.

Mineral Physics Data on the Phase Transitions in the TZ The most frequently used model of mantle composition is pyrolite which contains 60% of olivine (Mg,Fe)2SiO4. At a depth of 410 km olivine (α) transforms to wadsleyite (β, modified spinel). The Clapeyron slope of this transition is positive (4.0 MPa/K, Katsura et al. 2004); the increase of the S-wave velocity is ~12%. At a depth of 550 km wadsleyite transforms to ringwoodite (γ, silicate spinel). At a depth of 660 km ringwoodite transforms to a mixture of

Seismic Discontinuities in the Transition Zone

perovskite (Mg,Fe)SiO3 and magnesiowüstite (Mg,Fe)O. The S velocity contrast of this transition is comparable to that of the α/β transition. The post-spinel transition is sharp and has a negative Clapeyron slope of 2.8 MPa/K (e.g., Hirose 2002). This value is disputed and more recent estimates are in a range from 0.4 to 1 MPa/K (Ohtani and Sakai 2008). The other components of pyrolite are pyroxenes and garnet. These components experience in the TZ more gradual transformations: the pyroxenes dissolve into garnet (majorite) and majorite transforms to perovskite near the bottom of the TZ. The Clapeyron slope of the post-majorite transition is 1.3 MPa/K (Hirose 2002). The post-spinel and post-majorite phase boundaries intersect at 1,700–1,800
C, and the corresponding seismic discontinuity may be an effect of both transitions. At the temperatures less than 1,700
C the discontinuity is formed mainly by the post-spinel transition, while at the higher temperatures the post-majorite transition becomes dominant (Hirose 2002).

Seismic Methods The models of composition and temperature of the TZ are constrained by seismic data. An increase of the seismic velocity with depth is mirrored by the increase of the apparent velocity of the P and S seismic arrivals with distance. Details can be obtained from the analysis of the related loop (triplication) in the travel times and modeling the related waveforms with synthetic seismograms. The triplications corresponding to discontinuities in the TZ are observed at epicentral distances between 1,700 and 3,200 km, whereas properties of the mantle may change significantly on a scale of a few hundred kilometers. Observations of reflected and mode-converted phases of distant earthquakes provide better lateral resolution. In the neighboring field of exploration seismology, accurate mapping of discontinuities is based on observations of reflected phases. These signals are small relative to noise, and the detection is performed by using receiver arrays and stacking the recordings of different sensors with move-out time corrections. In seismology adequate receiver arrays still are few, especially in the oceans. The idea of receiver function approach in seismology is to replace receiver arrays by seismic-source arrays that can be very large and dense. The differences in the waveforms of individual earthquakes that are recorded at the same station are eliminated by appropriate frequency filtering (deconvolution). P-wave receiver functions (PRFs) present the oldest and most usable variety of receiver functions. This technique is based mainly on observations of Ps (converted from P to S) phases. The delay of the Ps phase relative to P depends on the depth of the discontinuity and velocities along the wavepropagation paths. The amplitude of the Ps phase is

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proportional to the S velocity contrast at the discontinuity. The detection of small seismic phases in PRFs is accompanied by measurements of their slowness. In a laterally homogeneous Earth, Ps phases from the TZ discontinuities differ by slowness from lithospheric reverberations that arrive in the same time window, and this difference helps to separate the signals from noise. The best results can be obtained by combining source and receiver arrays. Unfortunately, practical detection of the TZ seismic phases in noise is often based on relaxed criteria. Some of the widely cited results are obtained from noisy receiver functions which show a continuous train of positive “swells” and negative “troughs.” This interference pattern may be mistaken for a sequence of separate seismic arrivals. S-wave receiver functions (SRFs) are complementary to PRFs and deal with the Sp phases converted from S to P. Multiple scattering at shallow discontinuities, which presents the major source of noise in PRFs, is practically absent in SRFs, because the Sp phases from large depths arrive much earlier than the scattered phases from shallow discontinuities. Another useful method is based on observations of ScS reverberations. Detection of the phases reflected from TZ discontinuities in this method is in principle similar to that employed in receiver functions and this technique can be viewed as a variety of receiver function techniques. A shortcoming of this method is its low (in comparison to PRFs and SRFs) lateral resolution. Precursors to the seismic phases SS (SH component), PP, and P0 P0 (SH, P, and P0 waves reflected from the Earth’s surface between the source and the receiver) include phases reflected from TZ discontinuities. These phases can be detected by using the receiver function approach. Lateral resolution of the SS-precursor technique (a few thousand kilometers) is by an order of magnitude lower than of P410s and P660s phases in PRFs, but the SS precursors are useful in the studies of the TZ on a global scale.

Topography and Sharpness of TZ Discontinuities The amplitudes of seismic phases converted or reflected from the TZ discontinuities at 410- and 660-km depths vary laterally but on the average are approximately two times lower than predicted for olivine. This relationship implies that the actual composition of the TZ is broadly similar to pyrolite which contains 60% of olivine. In the literature there are numerous reports on observations of reflected phases from a discontinuity at a depth around 520 km (e.g., Flanagan and Shearer 1998). The most popular interpretation relates this discontinuity to the phase transition from wadsleyite to ringwoodite. However, the theoretical amplitudes of reflected and converted phases from the 520-km discontinuity in pyrolite should be several times lower than the observed ones and, for this reason, practically

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unobservable. There are indications of a low S-wave velocity in a depth range from 450 to 500 km in the vicinities of several hotspots. Hence the seismically detectable 520-km discontinuity in hotspot regions may in reality present the base of this low-velocity layer (e.g., Vinnik et al. 2012). The width of the 410-km discontinuity has been estimated by seismic studies since about 1993. The estimates range from 4 km to 35 km. According to Katsura et al. (2004) the width of the olivine-wadsleyite transition is from 7 to 13 km. The largest width is possible in regions of rapid mantle flow. The smallest value is characteristic of stagnant mantle with a slow vertical movement. The width of 4 km and less has no accepted explanation. There are many observations of short-period P waves reflected from the 660-km discontinuity, which indicate that this discontinuity is sharp. However, the spectra of P660s converted phase are shifted to longer periods relative to P with implication that the 660-km boundary is a few tens of kilometers wide. This complexity may be explained with a model of two phase transitions (post-spinel and post-majorite) of different width in about the same depth range. Analysis of PRFs demonstrates that the large, in a range of a few seconds, lateral variations of travel times of the TZ seismic phases are caused mainly by lateral heterogeneity of the upper mantle outside the TZ. These variations are similar for P660s and P410s seismic phases and can be removed by taking the difference between the P660s and P410s arrival times at the same station. Lateral variations of this difference are sensitive to changes in the TZ thickness. The data on the P660s-P410s differential times suggest that the depths to the 410- and 660-km discontinuities usually vary in a range of not more than a few kilometers or a fraction of a second in time almost everywhere, except in the anomalously hot and cold regions: hotspots and subduction zones, respectively. These normal depths of the major boundaries practically coincide with those in the IASP91 model (410 and 660 km), and correspond to the normal TZ thickness of 250 km. The results for hotspots, most of which are located in oceans, are controversial. The related data are highly variable in quality. Therefore, instead of relying on statistics, I prefer to single out the results for Iceland, which is one of the most thoroughly investigated hotspots. The network of seismograph stations in Iceland is large, and there are several independent tomographic studies of this region. Seismic tomography and receiver functions reveal beneath Iceland a narrow columnar low-velocity body extending into the TZ, in which the 410-km discontinuity is depressed by 15 km, while the 660-km discontinuity is flat. The conclusion on the flat 660-km boundary is by Du et al. (2006). The depressed 410-km discontinuity suggests that the TZ temperature is elevated by 150
C. The flat 660 km discontinuity means that either the temperature at this depth is close to normal or,

Seismic Discontinuities in the Transition Zone

as suggested by high-pressure data (Hirose 2002), a sensitivity of depth of the transition to temperature is low. This model is disputed by Jenkins et al. (2016) who propose that the 660-km boundary is depressed like the 410-km boundary. This depression implies that the Clapeyron slope at a depth of 660 km is positive. The positive slope is possible if the 660-km discontinuity is related with the phase transition in garnet. It should be noted that the dual interpretation of the data on topography of the 660-km discontinuity beneath Iceland depends on some poorly resolved details. The seismic data for another hotspot in the Atlantic (Cape Verde, Vinnik et al. 2012) provide clear evidence of a depression on the 410-km discontinuity and uplift of the 660-km discontinuity of comparable amplitude. Note that for the ray paths outside the columnar body beneath Iceland, the time difference between the P660s and P410s seismic phases is close to the standard time with implication that the TZ of the normal oceanic mantle has the same 250-km thickness as beneath the continents. Broadly similar results are obtained for several other hotspots. For example, similar measurements at several stations in the region of the South Pacific superswell (Suetsugu et al. 2007) show that beneath one station the thickness of the TZ is 216  19 km. The anomalous region cannot be larger than a few hundred kilometers. The average for all other stations is 248  5 km, very close to the standard value of 250 km. By comparison, the SS-precursor data portray in the South Pacific a large (several thousand kilometers) region, where the TZ thickness is reduced to 230 km (e.g., Flanagan and Shearer 1998). A similar region is found in the Atlantic. These data, if accepted at face value, imply that the TZ beneath oceans differs from that beneath continents, and the TZ beneath the hotspots is the normal oceanic TZ, contrary to the receiver function data. Substantial discrepancies between the results of the two methods exist in continental regions where numerous seismograph stations provide good-quality data. Subduction zones demonstrate another kind of complexity. Temperature in subducted slabs is anomalously low and the equilibrium depth of the olivine-wadsleyite phase transition should be 100 km less than in ambient mantle. However, the 410-km discontinuity in most of the presently active subduction zones apparently cannot be detected with either PRFs or other methods, most likely because the olivine-wadsleyite phase transformation is kinetically hindered and a wedgeshaped zone of olivine may persist at depths greatly exceeding 410 km. The signals from the 410-km discontinuity that are sometimes obtained in seismic studies of these regions are most likely formed outside the cold subducted lithosphere. Observations of short-period phases converted from S to P near the hypocenters of deep earthquakes show that the 660-km discontinuity in subducted slabs varies in depth but sometimes is depressed by up to 50 km. This may be

Seismic Discontinuities in the Transition Zone

explained by the negative Clapeyron slope of the post-spinel transition and a low temperature of the subducted slab. The temperature anomalies may be up to 600
C. Seismic observations sometimes reveal a low S-wave velocity layer over the 660-km discontinuity. This feature can be explained by accumulation of oceanic crust in subduction zones above the 660-km discontinuity. Robust evidence for an elevated differential time P660sP410s and a depression of the 660-km discontinuity is provided by PRFs in the Alps where the active phase of Cenozoic subduction is over, but remnants of the oceanic lithosphere are present in the TZ. Similar anomalies are either found or can be expected in other regions of the Alpine belt.

The Issue of Water in the TZ The cited results of mineral physics were obtained for the dry TZ. Many researchers examined solubility of water in minerals of the TZ (see, e.g., Ohtani and Sakai 2008). Hydration means modification of structures of the TZ minerals by incorporation of hydroxyl (OH). It appears that wadsleyite and ringwoodite may incorporate up to 2.0 wt% of water, and the TZ is thought to be the main reservoir of water in Earth (e.g., Karato 2011). Pressure of the olivine-wadsleyite transition decreases by hydration by up to about 1 GPa (30 km in depth), whereas the pressure of the post-spinel transition increases. The width of the two-phase loop between olivine and wadsleyite increases with increasing water content, and may reach a few tens of kilometers in depth relative to several kilometers for dry conditions. Hydration of 1 wt% lowers S velocity by 1–2%, whereas P velocities remain practically the same. The predicted anomalies in the depth (30 km) and thickness (a few tens of kilometers) of the 410-km discontinuity are sufficiently large to be detected by PRFs. These predictions stimulated attempts to detect the effects of water in seismic data. So far the results are controversial. Karato (2011) stressed striking incompatibility between the results of different studies and concluded that frequently used seismological observations are more sensitive to the temperature and major element chemistry than to water content. The estimates of the Clapeyron slope for the post-spinel transition (between 0.4 and 1.3 MPa/K) in the dry TZ may explain only part of the observed topography on the 660-km discontinuity, but the discrepancy can be reduced in the wet TZ (Litasov et al. 2005). A low S velocity layer atop the 410-km discontinuity may present another possible effect of hydration of the TZ. Owing to the large water solubility, the TZ may have higher water concentration than water storage capacity of the upper mantle. Then upwelling mantle material entering the upper mantle from the TZ may undergo dehydration melting.

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Evidence of the low-velocity layer atop the 410-km discontinuity is obtained in PRFs and SRFs at a number of locations that include Antarctica, Africa, Siberia, North America, China, and Arabian Peninsula. Most locations of this layer seem to be associated with Cenozoic and Mesozoic mantle upwellings.

Summary Seismic observations of the major TZ discontinuities near 410- and 660-km depths on a global scale are broadly consistent with the pyrolite mantle model. The 410-km discontinuity is related to the olivine-wadsleyite phase transition with a positive Clapeyron slope. The 660-km discontinuity is related to the post-spinel transition in the olivine component and post-majorite transition in the other components, with a negative and positive Clapeyron slope, respectively. In the normal mantle, the depths of the discontinuities are stable and in good agreement with the IASP91 model, where the thickness of the TZ is 250 km. Anomalous topography of the TZ discontinuities is related to hot and cold regions (hotspots and subduction zones, respectively). The anomalies beneath oceans are related mainly to hotspots, where the 410-km discontinuity is depressed by 20 km. The corresponding temperature anomalies are up to 200
C. Beneath continents, except Africa and Antarctica, the anomalies are related mainly to Cenozoic subduction zones, where the 660-km discontinuity can be depressed by up to 50 km. The related temperature anomalies are in a range of several hundred degrees Celsius. In spite of high water solubility in wadsleyite and ringwoodite, credible observations of seismic effects of hydration in the TZ are few. The thin low-Svelocity layer atop the 410-km discontinuity, found at a number of locations, is the exception. The low velocity can be an effect of a hydrous melt. On a regional scale the TZ may contain less well-understood complexities such as the 520-km discontinuity or a low S velocity layer between 450- and 500-km depths.

Cross-References ▶ Body Waves ▶ Earth’s Structure, Global ▶ Earth’s Structure, Upper Mantle ▶ Geodynamics ▶ Mantle Convection ▶ Mantle Plumes ▶ Seismic Phase Nomenclature: The IASPEI Standard ▶ Seismic, Receiver Function Technique ▶ Seismology, Global Earthquake Model ▶ Subduction Zones

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Bibliography Du Z, Vinnik LP, Foulger GR (2006) Evidence from P-to-S mantle converted waves for a flat “660-km” discontinuity beneath Iceland. Earth Planet Sci Lett 241:271–280 Flanagan MP, Shearer PM (1998) Global mapping of topography of transition zone velocity discontinuities by stacking SS precursors. J Geophys Res 103:2673–2692 Hirose K (2002) Phase transitions in pyrolitic mantle around 670-km depth: implications for upwelling of plumes from the lower mantle. J Geophys Res 107(B4):2078. https://doi.org/10.1029/2001JB00 0597 Jenkins J, Cottaar S, White RS, Deuss A (2016) Depressed mantle discontinuities beneath Iceland: evidence of a garnet controlled 660 km discontinuity? Earth Planet Sci Lett 433:159–168 Karato SI (2011) Water distribution across the mantle transition zone and its implications for global material circulation. Earth Planet Sci Lett 301(3–4):413–423 Katsura T, Yamada H, Nishikawa O, Song M, Kubo A, Shinmei T, Yokoshi S, Aizawa Y, Yoshino T, Walter MJ, Ito E, Funakoshi K (2004) Olivine-wadsleyite transformation in the system (Mg,Fe)2SiO4. J Geophys Res 109:B02209. https://doi.org/10. 1029/2003JB002439 Litasov KD, Ohtani E, Sano A, Suzuki A, Funakoshi K (2005) Wet subduction versus cold subduction. Geophys Res Lett 32(13). https://doi.org/10.1029/2005GL022921 Ohtani E, Sakai T (2008) Recent advances in the study of mantle phase transitions. Phys Earth Planet Inter 170:240–247 Suetsugu D, Shiobara H, Sigioka H, Fukao Y, Kanazawa T (2007) Topography of the mantle discontinuities beneath the South Pacific superswell as inferred from broadband waveforms on seafloor. Phys Earth Planet Inter 160(3–4):310–318 Vinnik L, Silveira G, Kiselev S, Farra V, Weber M, Stutzmann E (2012) Cape Verde hotspot from the upper crust to the top of the lower mantle. Earth Planet Sci Lett 319:259–268

Seismic Hazard Andrzej Kijko Natural Hazard Centre, Africa, University of Pretoria, Pretoria, Republic of South Africa

Definition Seismic hazard

Seismic hazard analysis

Any physical phenomenon associated with an earthquake (e.g., ground motion, ground failure, liquefaction, and tsunami) and its effects on land, man-made structures, and socioeconomic systems that have the potential to produce a loss. The terms are used also without regard to a loss to indicate the probable level of ground shaking occurring at a given point within a certain period of time. Quantification of the ground motion expected at a particular site or selection of sites.

Deterministic seismic hazard analysis Probabilistic seismic hazard analysis Ground motion model (known also as ground motion prediction equation)

Quantification of a single or relatively small number of individual earthquake scenarios.

Quantification of the rate (or probability) that a specified level of ground motion will be exceeded at least once at a site (or a map of sites) given all possible earthquakes during the specified exposure time. A mathematical equation that provides a means of predicting the level of ground shaking and its associated uncertainty at any given site or location, based on earthquake magnitude, source-to-site distance, local soil conditions, and focal mechanism.

Introduction Estimating the ground motion that could occur at a particular site is crucial in the design of both vital structures, such as nuclear power plants, bridges, and dams, and ordinary structures, such as houses and commercial buildings. The process of assessing future earthquake ground motion parameters is called seismic hazard assessment or seismic hazard analysis. Seismologists and earthquake engineers distinguish between seismic hazard and seismic risk, although the two phrases have the same meaning in everyday usage. Seismic hazard is used to characterize the severity of the ground motion at a site regardless of the consequences, whereas risk refers exclusively to the consequences to humans and infrastructure from the occurrence of the hazard. Therefore, even a strong earthquake could have negligible risk potential if it occurred in sparsely populated and undeveloped areas. In contrast, a relatively weak seismic event in a highly developed and densely populated location poses a high risk owing to the possible loss of life and extensive infrastructure damage. Seismic hazard analysis is performed deterministically when a particular earthquake scenario is considered and probabilistically when the likelihood or frequency of specified earthquake size and location are evaluated. Deterministic seismic hazard analysis (DSHA) involves an initial assessment of the maximum possible earthquake magnitude mmax for each of the seismic sources, such as active faults or seismic source zones (SSHAC 1997). The maximum earthquake magnitude for a particular seismic source is often referred to as the maximum credible earthquake (MCE). An area with a radius of up to 450 km around the site of interest can be investigated. Assuming that each of these earthquakes will occur at the minimum possible distance from the site, the ground motion is calculated using approximate attenuation equations, known as the ground motion model (GMM).

Seismic Hazard

However, this straightforward and intuitive procedure is overshadowed by the complexity and uncertainty in selecting the appropriate earthquake scenario. This creates the need for an alternative, probabilistic methodology free from selecting an earthquake occurrence scenario. Probabilistic seismic hazard analysis (PSHA) quantifies the rate (or probability) of exceeding various ground motion levels at a site (or a map of sites) from all earthquakes of all possible magnitudes and at all significant distances from the site of interest. Therefore, deterministic earthquake scenarios are special cases of the probabilistic approach. Depending on the scope of the project, DSHA and PSHA could be complementary, i.e., providing additional insights into the seismic hazard. The current study discusses PSHA. In principle, any natural hazard caused by seismic activity can be described and quantified by the formalism of PSHA. As the damage caused by ground shaking quite often results in the largest economic losses, this presentation of the basic concepts of PSHA is illustrated by quantifying the likelihood of ground shaking generated by earthquakes. The classic (Cornell 1968; McGuire 2004) procedure, known as the Cornell–McGuire procedure for PSHA, includes four steps (Fig. 1). Step 1. Identification and parameterization of the seismic sources (also known as source zones, seismogenic zones, or seismic zones) that can affect the relevant site. These can be represented as area, fault, or point sources. Area sources are often used when a specific fault cannot be identified. In classic PSHA, uniform distribution of seismicity is assigned to each seismic source, implying that earthquakes are equally likely

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to occur at any point within the source zone. Combining earthquake occurrence distributions with the source geometry results in the space, time, and magnitude distributions of earthquake occurrences. Seismic source models can be interpreted as a list of potential scenarios, each with associated earthquake magnitude, location, and seismic activity rate. Step 2. Specification of the temporal and magnitude distributions of seismicity for each seismic source. The Cornell– McGuire approach assumes that earthquake occurrence in time is random and follows the Poissonian process. This implies that earthquake occurrences in time are statistically independent and occur at a constant rate, i.e., the occurrence of future earthquakes does not depend on the occurrence of past earthquakes. The model of earthquake magnitude recurrence used most often is the Gutenberg–Richter frequency– magnitude relation log ðnÞ ¼ a  bm,

ð1Þ

where n is the number of earthquakes with a magnitude of m and a and b are parameters. It is assumed that earthquake magnitude m belongs to the domain , where mmin is the level of completeness of the earthquake catalogue and magnitude mmax is the upper limit of the earthquake magnitude for a given seismic source. Parameter a is the measure of the level of seismicity, whereas parameter b describes the ratio between the number of small and large events. The Gutenberg–Richter relations can be interpreted either as a cumulative relationship where n is the number of

Seismic Hazard, Fig. 1 Four steps of PSHA

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events with magnitude equal or larger than m or as a density law, i.e., n is the number of earthquakes in a specific, small magnitude interval around m. Under these assumptions, the seismicity of each seismic source is described by four parameters, i.e., the (annual) rate of seismicity l, which is equal to the parameter of the Poissonian distribution, the lower and upper limits of earthquake magnitude mmin and mmax, and the b-value of the Gutenberg–Richter relation. Step 3. The relevant parameters in PSHA include peak ground acceleration, peak ground velocity, peak ground displacement, spectral acceleration, macroseismic intensity, strong ground motion duration, and the like. Most ground motion models available today are empirical and depend on the earthquake magnitude, source-to-site distance, type of faulting, and local site conditions. Therefore, the third step, i.e., choosing an appropriate ground motion model, is crucial, as, quite often, it is a major contributor to the formal uncertainty in the estimated PSHA. Step 4. Calculations of the ultimate result of PSHA in terms of a seismic hazard curve, i.e., the annual probability of exceeding a specified ground motion parameter at least once. An alternative definition of the hazard curve is the frequency of exceedance vs ground motion amplitude. This calculation integrates the uncertainties in the earthquake location, earthquake magnitude, and ground motion model into the probability that the relevant ground motion parameter will be exceeded at the specified site during the specified time interval.

Cornell–McGuire PSHA Methodology Conceptually, the computation of a seismic hazard curve is fairly simple (Kramer 1996). Assuming that seismic hazard is characterized by ground motion parameter Y, the probability of exceeding a specified value y, P[Y y], is calculated for an earthquake of particular magnitude located at a possible source and then multiplied by the probability that such a particular earthquake will occur. The computations are repeated and summed for the entire range of possible magnitudes and earthquake locations. The resulting probability P [Y y] is calculated by utilizing the total probability theorem P½Y y ¼

X

P½Y yjEi P½Ei 

ð2Þ

where ð

ð

P½Y yjEi  ¼ P½Y yjx1 , x2 x3  f i ðx1 Þ f i ðx2 jx1 Þ f i ðx3 jx1 , x2 Þ dx3 dx2 dx1 ð3Þ P[Y y| Ei] denotes the probability of ground motion parameter Y y at the relevant site when an earthquake

occurs within seismic source i. The variables xi (i ¼ 1, 2, . . .) are uncertainty parameters that influence Y. In the classic approach, developed by Cornell (1968) and later extended to accommodate ground motion uncertainty, the parameters of ground motion are earthquake magnitude M and earthquake distance R. Functions f(∙) are probability density functions (PDFs) of parameters xi. Assuming that x1  M and x2  R, the probability of exceedance (3) takes the form: P½Y yjEi  ¼

ð mmax ð m0

P½Y yjm, r  f M ðmÞ f RjM ðrjmÞdrdm, RjM

ð4Þ where P[Y y| m, r] denotes the conditional probability that the chosen ground motion level y is exceeded for a given magnitude and distance, fM(m) is the probability density function (PDF) of earthquake magnitude, and fR j M(r| m) is the conditional PDF of the distance from the earthquake for a given magnitude. The conditional PDF of the distance fR j M(r| m) arises in specific instances, such as a seismic source being represented by a fault rupture. As the earthquake magnitude depends on the length of the fault rupture, the distance to the rupture and the resulting magnitude are correlated. In Eq. (4) the magnitude m0 is chosen on the basis of the minimum magnitude that will cause damage or loss. In most PSHA, the m0 is set in the range of 4.0–5.0. The standard choice for the probability P[Y y| m, r] is a normal, complementary, cumulative distribution function (CDF), based on the assumption that the ground motion parameter y is a log-normal random variable, ln(y) ¼ g(m, r) + ε, where ε is a random error. The mean value of ln(y) and its standard deviation are known and are defined as ln ðyÞ and sln(y), respectively. In most engineering applications of PSHA, it is assumed that earthquake magnitudes follow the Gutenberg–Richter relation (1), which implies that fM(m) is a negative exponential distribution, shifted from zero to level of completeness mmin, and truncated from the top by mmax f M ðmÞ ¼

b exp ½bðm  mmin Þ 1  exp ½bðmmax  mmin Þ

ð5Þ

In Eq. (5), β ¼ b ln 10, where b is the parameter of the Gutenberg–Richter frequency–magnitude relation (1). It should be noted that the truncation of the classic Gutenberg– Richter frequency–magnitude relation is obligatory to ensure that the seismic energy released by the seismogenic process is finite. If nS seismic sources {Ei, i ¼ 1,. . ., ns}can be distinguished in the vicinity of the relevant site, each with an annual average rate of earthquake occurrence li, the total average annual rate of events with a site ground motion level y or more takes the form

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lðyÞ ¼

XnS

l P½ Y i¼1 i

yjEi ,

ð6Þ

After assuming that in every seismic source, earthquake occurrences in time follow a Poissonian distribution, the probability that y, a specified level of ground motion at a given site, will be exceeded at least once within any time interval t is P½Y > y; t ¼ 1  exp ½lðyÞt:

ð7Þ

For t ¼ 1 year, the plot P[Y > y; t] vs ground motion parameter y is the hazard curve – the ultimate product of PSHA (Fig. 2). For small probabilities, less than 0.05, P½Y > y; t ¼ 1 ¼ 1  exp ½l   1 ffi 1  1  l þ l2  ffi l, 2

ð8Þ

which means that the probability (7) is approximately equal to l(y).This shows that PSHA can be characterized interchangeably by the annual probability (7) or by the rate of seismicity (6). Often, practitioners prefer to describe seismic hazard in terms of frequency rather than probability. Frequencies have a clear physical meaning and, following Eq. (6), they are additive. Therefore, frequencies facilitate assessing the individual hazard contribution from each seismic source/area and support the optimal design decision and choice of alternative sites for structure location. In the Cornell–McGuire procedure for PSHA, it is assumed that the earthquakes in the catalogue are independent events. As clusters of seismicity, or foreshocks and

Annual exceedance frequency

100 10–1 10–2 10–3 10–4 10–5 10–6 10–3

10–2

10–1

100

PGA [g] Seismic Hazard, Fig. 2 Example of a peak ground acceleration (PGA) seismic hazard curve and its confidence intervals

aftershocks, violate this assumption, these dependent events are removed from the catalogue before computing PSHA. However, removing such dependent events is one of the most controversial practices in current PSHA. Estimation of Seismic Source Parameters Following the Cornell–McGuire PSHA procedure, each seismic source is characterized by four parameters: • Level of completeness of the seismic data, mmin. • The annual rate of seismic activity l, corresponding to the magnitude mmin • B-value of the Gutenberg–Richter frequency–magnitude relation • The upper limit of earthquake magnitude mmax Estimation of mmin. The level of completeness of the seismic event catalogue, mmin, can be estimated in at least two different ways. The first approach is based on information provided by the seismic event catalogue, where mmin is determined from the deviation of the empirical and assumed earthquake magnitude distribution model. In most instances, the model is based on the Gutenberg–Richter relation (1). Occasionally, mmin is estimated from a comparison of the day-to-night ratio of events. Despite being used widely, the approach based on the seismic event catalogue has several weak points. One of these is the prerequisite knowledge of the spatial and temporal earthquake occurrence model. The second approach utilizes information on the detection capabilities and signal-to-noise ratio of the seismic stations that record seismic events. The approach is free from assumptions that seismic events are statistically independent and that they form a stationary process. Choosing the most appropriate procedure for estimating mmin depends on several factors, such as knowledge of the time history and evolution of the seismic network and data collection and processing. Estimation of the rate of seismic activity l and the bvalue of Gutenberg–Richter. The maximum likelihood procedure is the conventional approach to estimate the seismic source recurrence parameters l and b. If successive earthquakes are independent in time, the number of earthquakes with magnitude equal to or exceeding a level of completeness mmin follows the Poissonian distribution, with the parameter equal to the annual rate of seismic activity l. The maximum likelihood estimator of l is then equal to n/ t, where n is the number of events occurring within time interval t. For specified mmax, the maximum likelihood estimator of the b-value of the Gutenberg–Richter relation can be obtained from the recursive solution of the equation:

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ðmmax  mmin Þ exp½bðmmax  mmin Þ ∙, 1  exp½bðmmax  mmin Þ (9)

In most instances, estimating l and the b-value by the above simple formulas cannot be performed because seismic event catalogues are incomplete. Generally, seismic event catalogues contain three types of data, namely, prehistoric (paleo-) events, the incomplete largest historical events that usually occurred over a period of a few hundred years, and the instrumental data, recorded over the last 100 years. The completeness of instrumental data often varies with time (Fig. 3). Obviously, combining and utilizing all three types of data would be the optimal procedure to apply in such an instance. Additionally, such a procedure should accommodate information on highly uncertain sizes (magnitudes/focal intensities) of paleo- and historical earthquakes and unavoidable uncertainties associated with the applied earthquake occurrence models. The procedure developed by Smit et al. (2019) extends the well-known approach developed by Weichert (1980), which does not allow for the paleo- and historic part of the catalogue. Estimation of mmax. The maximum magnitude mmaxis defined as the upper limit of magnitude for a given seismic source. Often, the magnitude of the largest possible earthquake or maximum credible earthquake is synonymous with the upper limit of earthquake magnitude. This terminology assumes a sharp cutoff earthquake magnitude distribution at a maximum magnitude mmax. As a rule, mmax is crucial in PSHA, particularly in assessing long return periods, although, currently, there is no generally

accepted method for estimating mmax. It is estimated by combining several factors, based on two types of information, namely, the seismicity of the area and the geological, geophysical, and structural information of the seismic source. Using The seismological information focuses on the maximum observed earthquake magnitude within a seismic source and the statistical analysis of the available seismic event catalogue. The geological and geophysical information is used to identify distinctive tectonic features, which control the value of mmax. Current assessments of mmax are divided between deterministic and probabilistic procedures, based on the nature of the tools applied. Deterministic procedures. The deterministic procedure most often applied is based on the tectonic features of the investigated area, revealed by empirical relationships between magnitude and various parameters, such as the fault rupture length, rupture dimension, or slip rate. Relationships differ for different seismic areas and different types of faults. Despite empirical relationships between magnitudes and fault rupture parameters being used extensively in PSHA (particularly for assessing the maximum possible magnitude generated by fault-type seismic sources), the weak point of the approach is the requirement to specify the highly uncertain length of a future rupture. An alternative approach to determining earthquake recurrence on singular faults with a segment-specific slip rate is the so-called cascade model, where segment rupture is defined by the individual cascadecharacteristic rupture dimension. Another deterministic procedure that has strong intuitive appeal is based on records of the largest historical or paleoearthquakes. This approach is particularly applicable to areas of low seismicity, where large events have long return periods. In the absence of any tectonic and geological indications, it is assumed that the maximum possible earthquake magnitude is equal to the largest magnitude observed mobs max, plus an increment. Typically, the increment varies from ¼ to 1 magnitude unit. Another deterministic procedure for mmax evaluation, particularly for area-type seismic sources, is based on extrapolation of the Gutenberg–Richter frequency–magnitude relation

Seismic Hazard, Fig. 3 Illustration of a typical seismic event catalogue. Prehistoric (paleo-) data suffer from uncertainty related to the time of occurrence and the event size. The historic part of the catalogue contains only the largest events, the size of which is also highly

uncertain. The third part of the catalogue contains complete data, with different levels of completeness. Tg denotes gaps in the catalogue where records are absent or the seismic networks were out of operation. (After Smit et al. 2019)

  mmin þ 1=b ¼ m

where β ¼ b ln 10 and m is the sample mean of earthquake magnitude. If the range of earthquake magnitudes exceeds two magnitude units, the solution of Eq. (9) can be approximated by the well-known Aki–Utsu estimator b ¼ 1=ðm  mmin Þ:

ð10Þ

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when the frequency–magnitude curve is truncated at the specified value of the annual probability of exceedance (e.g., 0.001). However, currently, the procedures based on such extrapolation are not used often, and they are replaced by procedures based on the physical parameters of fault rupture. The value of mmax can be estimated also from tectonic and geological features, such as the strain rate or the rate of seismic moment release. Similar approaches have also been applied in evaluating the maximum possible magnitude of anthropogenic seismicity, particularly that generated by fluid injection or deep mining. However, mostly, the uncertainty of mmax as determined by any deterministic procedure is large, often reaching a value of one unit on the Richter scale. Probabilistic procedures. The first probabilistic procedure for maximum regional magnitude was developed in the late 1960s and is based on the formalism of the extreme values of random variables. Statistical tools required for the estimation of the end point of distribution functions have only been used recently (Kijko 2004, and references therein). Available statistical tools for estimating mmax vary significantly. Selecting the most suitable procedure depends on the assumptions of the statistical distribution model and/or the information available on past seismicity. Some of the procedures can be applied in extreme cases when no information is available on the nature of the earthquake magnitude distribution. Some of the procedures can also be used when the earthquake catalogue is incomplete, i.e., when only a limited number of the largest magnitudes are known. Two estimators are presented here. Broadly speaking, the first estimator is straightforward and simple in application, whereas the second requires a more computational effort but provides more accurate results. It is assumed that both the analytical form and the parameters of the distribution functions of earthquake magnitude are known. Based on the distribution of the largest among n observations, and on the condition that the largest observed magnitude mobs max is equal to the largest magnitude to be expected, the “simple” estimate of mmax is of the form (Pisarenko et al. 1996) bmax ¼ mobs m max þ

1
, n f M mobs max

1  exp ½bðmobs max  mminÞ  : obs nb exp bðmmax  mmin 

ð mobs max

½FM ðmÞn dm,

ð13Þ

mmin

where FM(m) denotes the CDF of random variable m. If applied to the Gutenberg–Richter frequency–magnitude relation (1), the respective CDF is 8 0 m < mmin > > < 1  exp ðbðm  m Þ min mmin m mmax , FM ðmÞ ¼ 1  exp ðbðmmax  mmin Þ > > : 1 m mmax

ð14Þ and the mmax estimator (13) takes the form bmax ¼ mobs m max þ

E1 ð n 2 Þ  E1 ð n 1 Þ þ mmin exp ðnÞ, b exp ðn2 Þ

ð15Þ



  ,n2 ¼ n1 exp where n ¼ n= 1  exp b mobs max  mmin 
obs 1  b mmax  mmin , and E1(∙) denotes an exponential integral function. Despite these probabilistic procedures providing powerful tools for evaluating mmax, they have a significant weak point, namely, most of the catalogues of tectonic origin events are too short and provide insufficient information for reliable estimations of mmax. Therefore, the Bayesian extension of statistical procedures, allowing the inclusion of alternative and independent information, such as local geological conditions, tectonic environment, geophysical data, paleo-events, a similarity with another seismic area, and the like, is superior to the procedures based only on knowledge of the seismicity of the area. Finally, the recent trend in mmax assessment must be mentioned. As most of the current techniques provide quite uncertain point estimates of mmax, several researchers suggest using instead the assessment of the confidence level of the maximum magnitude to be observed in the future time interval T. However, many engineering applications require knowledge of the point estimates of mmax. Therefore, the formalism of the quantile assessment T-maximum cannot be used.

ð11Þ


 is the PDF of the earthquake magnitude where f M mobs max distribution. If applied to the Gutenberg–Richter relation with PDF (5), it takes the simple form bmax ¼ mobs m max þ

bmax ¼ mobs m max þ

ð12Þ

The second (“advanced”) procedure often used in assessing mmax is based on the formalism derived by Cooke (1979)

Numerical Computation of PSHA Excepting a number of special cases, the hazard curve cannot be computed analytically. For the most realistic distributions, the integrations can only be evaluated numerically. The common practice is to divide the possible ranges of magnitude and distance into nM and nR intervals, respectively. The average annual rate (6) is then estimated as lðY > yÞ ffi

XnS XnM XnR i¼1

j¼1

k¼1

 
 li P Y > yj mj , rk f M j mj f Rk ðrk ÞDmDr,

(16)

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where mj ¼ mmin + (j  0.5)(mmax  mmin)/nM, rk ¼ rmin + (k  0.5)(rmax  rmin)/nR, Δm ¼ (mmax  mmin)/nM, and Δr ¼ (rmax  rmin)/nR. Applying the procedure to a grid of points will result in a map of PSHA, on which the contours of the expected ground motion parameter during the specified time interval can be mapped. An example of applying the above formalism is shown in Fig. 4. This figure shows the worldwide distribution of seismic hazard expressed in terms of the PGA, with a 10% probability of being exceeded at least once in 50 years. The maps were created in the framework of the Global Earthquake Model (GEM) project.

Controlling earthquakes are calculated for different structural frequency vibrations, typically for the fundamental frequency of a structure. In the deaggregation process, the hazard for a reference probability of exceedance of specified ground motion is portioned into magnitude and distance bins. The relative contribution to the hazard for each bin is calculated. The bins with the largest relative contribution identify those earthquakes that contribute the most to the total seismic hazard.

Deaggregation of Seismic Hazard By definition, the PSHA aggregates ground motion contributions from earthquake magnitudes and distances of significance to a site of engineering interest. Therefore, the PSHA results are not representative of a single earthquake. However, an integral part of the design procedure of any critical structure is examining the most relevant earthquake acceleration time series, which are generated by earthquakes at specific magnitudes and distances. Such earthquakes are called “controlling earthquakes,” and they are used to determine the shapes of the response spectral acceleration at the site. Controlling earthquakes are characterized by mean magnitudes and distances derived from so-called deaggregation analysis (McGuire 2004). During the deaggregation procedure, the results of PSHA are separated to determine the dominant magnitudes and the distances that contribute to the hazard curve at a specified reference probability.

Source-Free PSHA Procedures The concept of seismic sources is the core element of the Cornell–McGuire PSHA. However, seismic sources or specific faults often cannot be identified and mapped, and the causes of seismicity are not understood. In such instances, the delineation of seismic sources is highly subjective and is a matter of an expert opinion. In addition, seismicity within the seismic sources is often not distributed uniformly, as is required by the classic Cornell–McGuire procedure. The difficulties in dealing with seismic sources have stimulated the development of alternative techniques, which are free from the delineation of seismic sources (Veneziano et al. 1984; Woo 1996; Molina et al. 2001). The core of source-free procedures is calculation of the empirical distribution of the specified seismic hazard parameter, normalized for the duration of the seismic event catalogue. Alternatively, the nonparametric PSHA is based on

Modifications of Cornell–McGuire PSHA Procedure and Alternative Models

Seismic Hazard, Fig. 4 Example of the product of PSHA. Map of seismic hazard of the world. Peak ground acceleration expected at 10% probability of exceedance at least once in 50 years. The Global Earthquake Model (GEM) project

Seismic Hazard

seismicity distributions, which are assessed by data-based kernel functions. By their nature, nonparametric procedures function well in areas of high seismic activity and when the record of past seismicity is considerably complete. However, the nonparametric approach has significant weak points, as it provides an unreliable estimate of small probabilities for areas of low seismicity. Additionally, the procedure is not recommended for an area where the seismic event catalogues are incomplete. Moreover, in its present form, the procedure is not capable of using any additional geophysical or geological information to supplement the seismological data. Therefore, a procedure that accommodates the incompleteness of the seismic event catalogues and, at the same time, does not require the specification of seismic sources would be an ideal tool for analyzing and assessing the seismic hazard. Such a procedure, which can be classified as a parametric–historic procedure for PSHA, has been used successfully in several parts of the world (see, e.g., Giardini 1999). In some instances, during the computation process of seismic hazard, the Gutenberg–Richter frequency– magnitude relation is extended to account for the contribution of tectonic faults and characteristic earthquakes. The procedure accepts the potential contribution of seismicity from active faults and compensates for the incompleteness of seismic event catalogues. The final maps of seismic hazard are smoothed by a Gaussian-type kernel function. This conceptually simple and intuitive parametric–historic approach combines the best of the deductive and nonparametric historical procedures and, in many instances, is free from the disadvantages characteristic of each of the procedures. Another approach to assessing seismic hazard is based on Monte Carlo simulations of long-term seismic catalogues. Despite this approach having several advantages and being straightforward and easy to apply, its role in PSHA is highly underestimated. The Monte Carlo-based procedure allows easy incorporation of alternative earthquake space–time– size occurrence models, such as Markovian, non-Poissonian, and non-Gutenberg–Richter frequency–magnitude relation. In addition, it allows for the assessment of uncertainties associated with the input data and model parameters. The potential disadvantage of the Monte Carlo-based procedure is the requirement for extensive computation power. Alternative Earthquake Recurrence Models Time-dependent models. In addition to the classical assumption that earthquake occurrence in time follows a Poissonian process, alternative approaches are used occasionally. These procedures attempt to assess the temporal or spatiotemporal dependence of seismicity. Time-dependent earthquake occurrence models specify the distribution of the time to the next earthquake, where this distribution depends on the magnitude of the most recent earthquake. In order to incorporate the memory of past events, non-Poissonian distributions or

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Markov chains are applied. In this approach, the seismogenic zones that recently produced strong earthquakes become less hazardous than those that did not rupture in recent history. Time-dependent models are also applied to quantify the occurrence of large earthquakes on segments of active faults. Kramer (1996) provides an overview of non-Poissonian models. Several time-dependent models play an important role in PSHA, of which the best-known are the two models by Shimazaki and Nakata (1980). These two models of earthquake occurrence, namely, a time-predictable and a slippredictable model, are based on the correlation of seismic activity with earthquake-related coastal uplift in Japan. The time-predictable model states that earthquakes occur when the accumulated stress on a fault reaches a critical level. However, the stress drops, and the magnitudes of subsequent earthquakes vary among seismic cycles. Therefore, assuming a constant fault slip rate, the time to the next earthquake can be estimated from the slip of the previous earthquake. The slip-predictable model is based on the assumption that irrespective of the initial stress on the fault, an earthquake occurrence always causes a reduction in stress to the same level. Therefore, the fault slip in the next earthquake can be estimated from the time since the previous earthquake (Shimazaki and Nakata 1980; Scholz 1990; Thenhaus and Campbell 2003). The second group of time-dependent models is based less firmly on the physical considerations of earthquake occurrence, and these models attempt to describe intervals between the consecutive events by specified statistical distributions. At least two such models play a significant role in the current practice of PSHA, namely, the log-normal and Brownian passage time (BPT) renewal model. Employing a log-normal model was justified by the discovery that normalized intervals between the consecutive large earthquakes in the Circum-Pacific region follow a lognormal distribution, with an almost constant standard deviation. Following this discovery, the log-normal model became a key component of time-dependent PSHA procedures. More often applied nowadays than the log-normal model is the Brownian passage time (BPT) distribution. This model is described by two parameters, m and s, which, respectively, represent the mean time interval between the consecutive earthquakes and the standard deviation. The aperiodicity of earthquake occurrence, as described by the BPT model, is controlled by the variation coefficient α ¼ s/m. For a small α, the aperiodicity of earthquake occurrence is small, and the shape of the distribution is almost symmetrical. For a large α, the shape of the distribution is similar to the log-normal model, i.e., skewed to the right and peaked at a smaller value than the mean. The straightforward control of aperiodicity of earthquake occurrence by parameter α makes the BPT model extremely attractive, and it has been used to model earthquake occurrence in many parts of the world.

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characteristic, the maximum possible earthquake magnitude mmax. The PDF of this distribution is given by Eq. (5). Generally, the Gutenberg–Richter relation describes magnitude distributions within seismic source zones adequately; however, in some instances, it appears not to apply. In such an instance, the relationship (5) must be modified. In some places, particularly in areas known for paleo-earthquakes, seismic belts, and large faults, the Gutenberg–Richter relation underestimates the occurrence of large magnitude events. Consequently, the continuity of the distribution (5) breaks down. The distribution is adequate only for events up to a magnitude of 6.0–7.0. Larger events tend to occur within a relatively narrow range of magnitudes (7.5–8.0) but with a frequency higher than that predicted by the Gutenberg–Richter relation (5). These events are known as characteristic earthquakes (Youngs and Coppersmith 1985, Fig. 5). However, it must be noted that the characteristic earthquake model is not free from controversy. Some researchers believe that the frequency–magnitude Gutenberg–Richter relation satisfies the earthquake occurrence characteristics equally well as does the earthquake characteristic model. The discrepancy between the two models would disappear if a sufficiently long record of seismicity were available. When a seismic hazard is caused by seismicity of anthropogenic origin, an alternative to the Gutenberg–Richter 100

Cumulative Number ≥ M per Year

Several comparisons between time-dependent and timeindependent earthquake occurrence models have shown that the time-independent (Poissonian) model is applicable to most engineering applications of the PSHA. The exception is when the time interval from the occurrence of the last earthquake exceeds the mean time interval between consecutive events related to a seismic hazard dominated by a single seismic source with a significant component of characteristic earthquakes. However, in many instances, the information on strong seismic events provided by current earthquake databases is insufficient to distinguish between different models. Therefore, using non-Poissonian models is justified only if sufficient data are available. Clearly, such time-dependent models can result in a more realistic PSHA, but many of them have not yet reached the level of development required by routine engineering applications. Spatiotemporal models. Often, a sequence of foreshocks and a sequence of aftershocks after the main event can be observed. The fore- and aftershock sequences cannot be described by the homogenous Poissonian process, as their rate depends on time, the magnitude of the main event, and its location. Some areas are notorious for the generation of earthquake swarms. Frequently, individual tectonic faults interact with each other, and such seismic activity leads to earthquakes being clustered in both time and space. The best-known spatiotemporal model of earthquake occurrence is ETAS (epidemic-type aftershock sequence) and its mutations. The core assumption of the ETAS model is that each earthquake, however small, triggers its own aftershocks (offspring) independently of the others. The probability distributions of location and magnitude of the triggered event are dependent on the magnitude and location of the triggering event. The rate of aftershock occurrence decays with time according to an inverse power law known as the modified Omori law. The magnitudes of all the events, including background events and their offspring, are independent random variables following the same probability distribution. An alternative spatiotemporal model is one in which each mainshock and its aftershocks are treated as spatially and temporally independent clusters. However, the mathematical formulation does not have a closed form, which restricts the usage of the model. An elegant, straightforward, and statistically consistent incorporation of foreshocks and aftershocks into standard PSHA is done by Iervolino et al. (2014). The proposed formalism provides a remedy to one of the most controversial current practices of PSHA, known as declustering (i.e., removal) of “dependent” events from the seismic event catalogue. Alternative frequency–magnitude models. In the classic Cornell–McGuire procedure for PSHA assessment, it is assumed that earthquake magnitudes follow the Gutenberg– Richter relation truncated from the top by a seismic source

Seismic Hazard

10

1.0 seismicity data

0.1

geologic data

0.01

2

3

4

5

6

7

8

9

Magnitude, M Seismic Hazard, Fig. 5 Gutenberg–Richter characteristic earthquake magnitude distribution. The model combines the Gutenberg–Richter frequency–magnitude relation with a uniform distribution of characteristic earthquakes. The model predicts higher rates of exceedance at magnitudes near the characteristic earthquake magnitude. (After Youngs and Coppersmith 1985)

Seismic Hazard

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frequency–magnitude relation (1) is often required. For example, the magnitude distributions of tremors generated by mining activity are multimodal and change their shape in time (Gibowicz and Kijko 1994). Often, the only possible method that could lead to a satisfactory PSHA for mining or fracking areas is replacing the analytical, parametric frequency–magnitude distribution by its model-free, nonparametric counterpart. Two additional modifications of the recurrence models are introduced regularly: one is when earthquake magnitudes are uncertain and the other when the seismic occurrence process comprises temporal trends, cycles, short-term oscillations, and purely random fluctuations. The effect of error in earthquake magnitude determination (particularly significant for historical events) can be minimized by the simple procedure of correcting the earthquake magnitudes in a catalogue (e.g., McGuire 2004). Modelling random fluctuations in earthquake occurrence is often done by introducing compound (mixed) distributions, in which the parameters of earthquake recurrence models are treated as random variables (Kijko et al. 2016).

Ground Motion Models The assessment of seismic hazard at a site requires knowledge of the GMM, i.e., the prediction equation of strong-motion parameter as a function of distance, earthquake magnitude, faulting mechanism, and, often, the local site condition. The most common form of the GMM is ln ðyÞ ¼ const þ g1 ðmÞ þ g2 ðr Þ þ g3 ðfaulting systemÞ þ g4 ðsoilÞ þ e:

In general, GMMs are built in three distinct ways: (a) empirically, by application of observed ground motion characteristics; (b) by stochastic methods, which are based on the physical properties of the earthquake source and seismic wave travel path; and (c) by the hybrid empirical method. Empirical methods are applied in areas of high seismicity in the presence of dense instrumental networks, where a significant number of ground motion records are available. Examples of such empirical GMMs are those developed as part of the Next Generation Attenuation Phase 2 (NGA-West2) project, managed by the Pacific Earthquake Engineering Research Center (PEER). Stochastic methods are used mainly in areas of low seismicity, where only a limited number of observations are available. The hybrid empirical method, like the stochastic method, is used to develop GMM for areas with limited ground motion records; however, the technique used is different. In the hybrid empirical method, the ground motion in the target area is predicted from the empirical GMM in the host region with the help of adjustment factors between the two regions. These factors consider the differences in earthquake source, wave propagation, and site response characteristics between the two regions. The ground motion models remain a particularly important component of PSHA, as ground motion uncertainty is a major contributor to the uncertainty of the PSHA. An excellent discussion on all the aspects of the estimation, application, and role of GMMs in PSHA is provided by McGuire (2004).

Uncertainties in PSHA ð17Þ

In Eq. (17), y is the amplitude of the designated ground motion parameter. This parameter can be the peak ground acceleration (PGA), spectral acceleration, displacement, the Modified Mercalli (MM) Intensity, seismic record duration, and the like. Functions g1(m) and g2(r) are the function of earthquake magnitude and the function of the shortest earthquake distance from the site to the earthquake source, respectively. Function g3(.) represents the faulting mechanism, g4(.) describes the site effect, and ε is the random error with zero mean and standard deviation sln(y). Assuming that ln(y) has a normal distribution, and following the properties of lognormal distribution, regression of (17) provides the mean value of ln(y), the exponent of which corresponds to the median value of y, y. As the log-normal distribution is positively skewed, the mean value  of y, y ,exceeds the median value y by a factor of exp 0:5s2ln ðyÞ . This indicates that the seismic hazard for a particular site is higher when expressed in terms of y when the hazard for the same site is expressed in terms of y.

Contemporary PSHA distinguishes between two types of uncertainty, namely, aleatory and epistemic. Aleatory uncertainty derives from the randomness in nature and is inherent for any random phenomenon. It represents unique details of an earthquake, such as its source, path, and site, and cannot be known before the earthquake’s occurrence. It is sometimes referred to as “randomness,” “stochastic uncertainty,” or “inherent variability” (SSHAC 1997). Typical examples of aleatory uncertainties are the number of future earthquakes in a specified area, parameters of future earthquakes as origin times, epicenter coordinates, depths and their magnitudes, size of the fault rupture, and associated stress drop. Additionally, the uncertainty of ground motion parameters at the given site is also aleatory by their nature. The aleatory uncertainties are characteristic of the current model and cannot be reduced by the incorporation of additional data. Aleatory uncertainties are included in the PSHA by means of integration over these uncertainties and are represented by the hazard curve. Epistemic uncertainty is the uncertainty deriving from insufficient knowledge about the model or its parameters. In

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the past, epistemic uncertainty was known as statistical or professional uncertainty (McGuire 2004). Examples of epistemic uncertainties are the boundaries of seismic sources, distributions of seismic source parameters (e.g., annual rate of seismic activity l, b-value, and mmax), or median value of the ground motion parameter, given the source properties. The model (in the broad sense of its meaning, as, e.g., a particular statistical distribution) can be inexact and, therefore, predicts values that differ from the observed values by a fixed, but unknown, amount. When uncertainties are associated with the numerical values of the parameters, they are also epistemic by nature. Epistemic uncertainty can be reduced by incorporating additional information or data. The epistemic distributions of model parameters can be updated using the Bayes’ formalism. Epistemic uncertainties are included in the PSHA through the use of alternative hypothesis, different sets of parameters with different numerical values, different models, or a logic tree. The major benefit of the separation of uncertainties into aleatory and epistemic is potential guidance in the preparation of input for PSHA and the interpretation of the results. However, the division of uncertainties into aleatory and epistemic is model dependent and, to a large extent, arbitrary, indefinite, and often confusing.

Seismic Hazard, Fig. 6 Example of a logic tree. The alternative hypothesis accounts for uncertainty in the ground motion attenuation relation, magnitude distribution model, and the assigned maximum magnitude mmax

Seismic Hazard

Logic Tree The mathematical formalism of PSHA computation integrates over all the random (aleatory) uncertainties of a particular seismic hazard model. In many instances, however, because of the lack of information on the factors controlling the mechanism of the earthquake generation and wave propagation processes, the best choices for the elements of the seismic hazard model are not clear. The uncertainty could originate from the choice of alternative seismic sources and earthquake generation mechanisms, rupture type and directivity, competitive earthquake recurrence models and their parameters, particular site and its topography, settings of the recording instruments, and the chosen ground motion model. The standard approach to the explicit treatment of alternative hypotheses, models, and parameters is applying the logic tree formalism (Coppersmith and Youngs 1989). Although not without controversy, the logic tree formalism provides a convenient tool for the quantitative treatment of any alternatives. Each node of the logic tree (Fig. 6) represents uncertain assumptions, models, or parameters, and the branches extending from each node are the discrete uncertainty alternatives (McGuire 2004). In logic tree analysis, each branch is weighted according to its probability of being correct. As a result, each end branch

Attenuation model

Magintude Distribution GutenbergRichter

Maximum Manitude 7.0 p=0.25 7.5 p=0.5

p=0.6 8.0 p=0.25 Attenuation model 1 p=0.5 7.0 p=0.25 Characteristic earthquake p=0.4

7.5 p=0.5 8.0 p=0.25

GutenbergRichter

7.0 p=0.25 7.5 p=0.5

p=0.6 8.0 p=0.25 Attenuation model 2 p=0.5 7.0 p=0.25 Characteristic earthquake

7.5 p=0.5

p=0.4 8.0 p=0.25

Seismic Hazard

represents a hazard curve with an assigned weight, where the sum of weights of all the hazard curves is equal to 1.0. The derived hazard curves are used to compute the final (e.g., mean) hazard curve and its confidence intervals. Figure 6 shows an example of a logic tree, accounting for uncertainty in the ground motion attenuation model, magnitude distribution model, and assigned maximum magnitude mmax. It must be noted that the choice of the structure of the logic tree and its respective weights is critical and often extremely subjective.

Controversy Although the PSHA procedure was formulated more than 50 years ago, controversy still surrounds it. Among others, the disagreements are related to (1) the requirement of declustering of the seismic event catalogue, (2) absence of the upper limit of ground motion parameters, (3) division of uncertainties between aleatory and epistemic, (4) correctness of the characteristic earthquakes model, and (5) accounting for epistemic uncertainty by applying the logic tree formalism. Declustering, i.e., removing earthquakes that occur in clusters, such as foreshocks, aftershocks, and swarms, from the seismic catalogue (in order to create a catalogue of “independent” events) is probably the most controversial factor in current PSHA practice. The procedure violates the fundamental rule of physics taught to first-year students, i.e., “never twist data to suit theory; instead, design such theory, which suits the data.” However, declustering does the exact opposite. Declustering affects the estimation of the earthquake occurrence parameters (the magnitude activity rate and the b-value), and, thereby, it affects (usually underestimates) the true hazard and risk. Iervolino et al. (2014) show that in terms of the annual rate of exceedance of PGA, declustering leads to 30% underestimation of the true hazard. Among others, Cornell and Winterstein (1988) contend that, in most instances, correct seismic hazard analysis can be performed without declustering. They show that the classic, time-independent (Poissonian) model is applicable even when the temporal clustering of earthquake occurrence is close to 30%. In addition, declustering can be avoided by replacing the Poissonian model (as well as the exponential Gutenberg–Richter magnitude distribution) with their compounded (mixed) counterparts. Further criticism of the current PSHA procedure concerns the absence of the upper limit of the ground motion parameter in the hazard curve. The core of the classic Cornell–McGuire PSHA is the assumption that the logarithm of the ground motion parameter is distributed normally. As the domain of the normal distribution is unlimited from the right side, it results in a nonzero probability of unrealistically high values

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for the ground motion parameter. Several solutions have been suggested; however, none of these is physically justifiable. The absence of the upper bound of the earthquake-generated ground motion parameter has been identified as the “missing piece” in current seismic hazard formalism. However, the problem of estimating the site-characteristic upper limit of PGA has been solved by Pavlenko (2017), who contends that, as a maxima of acceleration time series, PGA should follow an extreme value distribution, not the normal distribution. Based on extensive statistical tests of PGA records by K-NET and KiK-net networks, Pavlenko demonstrates that the distribution of ln(PGA) converges to the generalized extreme value (GEV) distribution. Replacing the normal distribution in Eqs. (4) and (5) by the GEV distribution allows for the assessment of the upper bound value of PGA. Therefore, the GEV distribution provides a model that is physically justifiable and describes the scatter of the logarithm of PGA significantly better than does the currently used normal distribution, and its application provides the required upper bound of the seismic hazard curve. Furthermore, criticism surrounds the division of uncertainties into aleatory and epistemic in the current PSHA procedure. As noted in section “Uncertainties in PSHA,” the division remains an unresolved issue. Different criticism is directed at the ergodic assumptions that underlie the formalism of the PSHA procedure. The ergodic process is a random process in which the distribution of a random variable in space is the same as the distribution of that variable at a single point when sampled as a function of time. It has been shown that the major contribution to PSHA uncertainty derives from the uncertainty of the ground motion model. The uncertainty of the ground motion parameter y is characterized by its standard deviation, sln(y), which is calculated as the misfit between the observed and the predicted ground motions at several seismic stations for a small number of recorded earthquakes. Therefore, sln(y) mainly characterizes the spatial and not the temporal uncertainty of ground motion at a single point. This violates the ergodic assumption of the PSHA procedure, leading to an overestimation of seismic hazard, particularly when the exposure times are longer than the earthquake return times. Castanos and Lomnitz (2002) and Mulargia et al. (2017) have voiced equally strong criticism about the application of the logic tree formalism. They describe the application of the logic tree formalism as a misunderstanding in probability and statistics, as it is fundamentally wrong to admit “expert opinion as evidence on the same level as hard earthquake data.” The science of seismic hazard assessment is, therefore, subject to much debate. Currently, PSHA represents a besteffort approach to quantify an issue of which not enough is known to provide definitive results and, as indicated by the numerous estimations, considerably more time and measurements will be needed before these issues can be resolved.

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Further reading: Several excellent studies are available that describe all aspects of modern PSHA. Thenhaus and Campbell (2003) and McGuire (2004) present outstanding overviews covering theoretical, methodological, and procedural issues of modern PSHA. The most comprehensive treatment to date of all aspects of PSHA, including the treatment of aleatory and epistemic uncertainties, is provided by the SSHAC (1997) report. Seismic hazard from a geologist’s perspective is described in a book by Yeats et al. (1997). Kramer (1996) provides an elegant, coherent, and understandable description of the mathematical aspects of both DSHA and PSHA. The information presented here benefited from all the sources quoted above, particularly the exceptional work by Kramer (1996).

Seismic Hazard

they are represented by the hazard curve. In contrast, epistemic uncertainties are included through the use of alternative models, different sets of parameters with different numerical values, or a logic tree. The PSHA procedure as known in its current form is not without controversy. The controversy arises from issues such as (1) the absence of the upper limit of the ground motion parameter, (2) division of uncertainties between aleatory and epistemic, and (3) the methodology itself, particularly the application of the logic tree formalism. An extended version of this article is available at the author’s ResearchGate profile https://www.researchgate.net/ profile/Andrzej_Kijko/research.

Cross-References Summary The term seismic hazard refers to any physical phenomenon associated with an earthquake (e.g., ground motion, ground failure, liquefaction, and tsunami) and its effect on the land, infrastructure, and socioeconomic systems that, potentially, can produce a loss. The term is also used, without regard to any loss, to indicate the probable level of ground shaking occurring at a given point within a certain period of time. Seismic hazard analysis involves the quantification of the expected ground motion at particular sites. Seismic hazard analysis can be performed deterministically when a particular earthquake scenario is considered or probabilistically when the likelihood or frequency of a specified level of ground motion at a site during a specified exposure time is evaluated. In principle, any natural hazard caused by seismic activity can be described and quantified in terms of the probabilistic methodology. Classical probabilistic seismic hazard analysis includes four steps, which are (1) identification and parameterization of the seismic sources, (2) specification of temporal and magnitude distributions of earthquake occurrence, (3) calculation of ground motion prediction equations and their uncertainty, and (4) integration of uncertainties in earthquake location, earthquake magnitude, and ground motion prediction equations into the hazard curve. An integral part of PSHA is the assessment of uncertainties. Contemporary PSHA distinguishes between two types of uncertainty, namely, aleatory and epistemic. Aleatory uncertainty stems from the randomness in nature and is the probabilistic uncertainty inherent in any random phenomenon. Aleatory uncertainties are characteristic of the currently used model and cannot be reduced by the incorporation of additional data. Epistemic uncertainty derives from insufficient knowledge about the model or its parameters. Epistemic uncertainty can be reduced by incorporating additional information or data. Aleatory uncertainties are included in the probabilistic seismic hazard analysis because of the integration over these uncertainties, and

▶ Characteristic Earthquakes and Seismic Gaps ▶ Earthquake, Magnitude ▶ Earthquakes, Early and Strong Motion Warning ▶ Earthquakes, Intensity ▶ Earthquakes, ShakeMap ▶ Earthquakes, Strong-Ground Motion ▶ Seismic Zonation ▶ Statistical Seismology

Bibliography Castanos H, Lomnitz C (2002) PSHA: Is it Science? Opinion Paper. Eng Geol 66:315–317 Cooke P (1979) Statistical inference for bounds of random variables. Biometrika 66:367–374 Coppersmith K, Youngs RR (1989) Issues regarding earthquake source characterization and seismic hazard analysis with passive margins and stable continental interiors. In: Gregersen S, Basham PW (eds) Earthquakes at North Atlantic passive margins: neotectonics and postglacial rebound. Kluwer, Dordrecht, pp 601–631 Cornell CA (1968) Engineering seismic risk analysis. Bull Seismol Soc Am 58:1583–1606 Cornell CA, Winterstein SR (1988) Temporal and magnitude dependence in earthquake recurrence models. Bull Seismol Soc Am 78:1522–1537 Giardini D (1999) The global seismic hazard assessment program (GSHAP) 1992–1999. Ann Geofis 42:957–1230 Gibowicz SJ, Kijko A (1994) An introduction to mining seismology. Academic, San Diego Iervolino I, Giorgio M, Polidoro B (2014) Sequence-based probabilistic seismic hazard analysis. Bull Seismol Soc Am 104:1006–1012 Kijko A (2004) Estimation of the Maximum Earthquake Magnitude mmax. Pure Appl. Geophys 161:1655–1681 Kijko A, Smit A, Sellevoll MA (2016) Estimation of earthquake hazard parameters from incomplete data files. Part III. Incorporation of uncertainty of earthquake-occurrence model. Bull Seismol Soc Am 106:1210–1222 Kramer SL (1996) Geotechnical earthquake engineering. Prentice-Hill, Englewood Cliffs McGuire RK (2004) Seismic hazard and risk analysis. Earthquake Engineering Research Institute, Oakland. MNO-10

Seismic Imaging, Overview Molina S, Lindholm CD, Bungum H (2001) Probabilistic seismic hazard analysis: zoning free versus zoning methodology. Boll Geofis Teor Appl 42:19–39 Mulargia F, Stark PB, Geller RJ (2017) Why is probabilistic seismic hazard analysis (PSHA) still used? Phys Earth Planet Inter 264:63–75 Pavlenko VA (2017) Estimation of the upper bound of seismic hazard curve by using the generalized extreme value distribution. Nat Hazards 89:19–33. https://doi.org/10.1007/s11069-017-2950-z Pisarenko VF, Lyubushin AA, Lysenko VB, Golubieva TV (1996) Statistical Estimation of Seismic Hazard Parameters: Maximum Possible Magnitude and Related Parameters. Bull. Seism. Soc. Am. 86:691–700 Scholz CH (1990) The Mechanics of Earthquakes and Faulting. Cambridge: Cambridge University Press Shimazaki K, Nakata T (1980) Time-predictable recurrence model for large earthquakes. Geophys Res Lett 7:279–282 Smit A, Stein A, Kijko A (2019) Bayesian Inference in Natural Hazard Analysis for Incomplete and Uncertain Data. Environmetrics, p. e2566. https://doi.org/10.1002/env.2566 SSHAC – Senior Seismic Hazard Committee (1997) Recommendations for probabilistic seismic hazard analysis: guidance on uncertainty and use of experts. NUREG/CR-6372, UCR-ID-122160, Main Report 1. Prepared for Lawrence Livermore National Laboratory Thenhaus PC, Campbell KW (2003) Seismic hazard analysis. In: Chen WF, Scawthorn C (eds) Earthquake engineering handbook. CRC Press, Boca Raton, pp 8-1–8-50 Veneziano D, Cornell CA, O’Hara T (1984) Historic method for seismic hazard analysis. Report, NP-3438. Electric Power Research Institute, Palo Alto Weichert DH (1980) Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes. Bull Seismol Soc Am 70:1337–1346 Woo G (1996) Kernel estimation methods for seismic hazard area source modeling. Bull Seismol Soc Am 86:353–362 Yeats RS, Sieh K, Allen CR (1997) The geology of earthquakes. Oxford University Press, New York Youngs RR, Coppersmith KJ (1985) Implications of fault slip rates and earthquake recurrence models to probabilistic seismic hazard estimates. Bull Seismol Soc Am 75:939–964

Seismic Imaging, Overview Gerard T. Schuster Division of Environmental and Earth Sciences, King Abdullah University of Science and Technology, Thule, Saudi Arabia

Definition Seismic imaging or tomography (tomo ¼ slice and graph ¼ picture) is a procedure for estimating the earth’s rock parameters from seismic data. These rock parameters can be represented by the spatial distribution of, e.g., P-wave velocity, S-wave velocity, porosity, density, or anisotropic parameters. The result of inversion is graphically presented as a 2-D or 3-D grid of pixels, where each pixel contains the value of the model parameter of interest, which is P velocity in

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Fig. 1g–h. Such tomograms are used to estimate the geometry and lithology of geologic layers, and can help exploration geophysicists and earthquake seismologists understand the evolution of the earth’s interior. There are five main types of seismic imaging: seismic migration, least squares migration, full waveform inversion (FWI), phase-like inversion, and migration velocity analysis. Four of these methods can be derived as special cases of finding the optimal model that minimizes a waveform or a phase-related misfit function. This entry will present these methods as they are applied to body wave events in seismic data, and overview the current best practices in waveform tomography.

Inversion Theory There are four steps to inverting for the model m from seismic data d. We will assume that the input data are either seismic traces, generated by man-made or earthquake sources, or some skeletonized part of the traces, such as first arrival traveltimes or the phase of an event at some frequency. Step 1: L(m) 5 d. Establish the mathematical relationship L(m) 5 d between the seismic data d and model m. d ¼ LðmÞ,

ð1Þ

where L represents the forward modeling operator for the actual model. Equations A1–A3 in Table 1 show the three steps in establishing a linearized version of Eq. 1 for the acoustic wave equation, where the extensions to the elastic (Mora 1987) and anisotropic (Barnes et al. 2008; Operto et al. 2009) wave equations are tedious but straightforward. Step 2: Discretize m, d, and L. Discretize the 3D earth model into a 3D grid of N physical parameters (e.g., unknown slowness in each cell) and assemble the unknowns into the N  1 vector m. Discretize the seismic traces in both space and time (or frequency) into an M x 1 vector d of data measurements. In this case, L reduces to a M  N matrix. The forward modeling for FWI requires knowledge of the source wavelet, which can be estimated in a variety of ways: stacking of the direct arrival at different near-offset hydrophones in marine data, iterative inversion of the source wavelet (Mora 1987; Zhou et al. 1997; Pratt 1999), or deconvolution of the shot gather with a time-windowed near-offset trace (Sheng, personal communication). The time window can be a few periods long centered about the direct arrival. Step 3: Linearize Lδm ≈ δd. Linearize the nonlinear relationship between the data and model. Expanding the ith data measurement di(m) to first order in δm by a Taylor series about a first-guess model mo (close to the true model) gives the linearized estimate:

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Seismic Imaging, Overview

a

b

Marmousi model 0

Blended CSGs 0

km/s

Z (km)

4.5

Time (s)

Seismic Imaging, Overview, Fig. 1 (a) Marmousi velocity model, (b) time-shifted and blended shot gathers, (c) zerooffset Kirchhoff migration image, (d) zero-offset least squares migration image after 30 iterations (courtesy of Naoshi Aoki), (e) reverse time migration (RTM) image, (f) least squares RTM image (Dai and Schuster 2009), (g) waveform tomogram after 50 iterations using the prestack gathers, and (h) waveform tomogram where the inputs are eight supergathers, each with 12 phase-encoded shot gathers (Zhan and Schuster in press); images are at different scales from one another

1.5

3 0

3

16

c

0

d

ZO Kirch. mig. image

X (km)

2

ZO LSM image

Z (km)

0

3 6

12

e

6

f

RTM image

12 LSM RTM image

Z (km)

0

3 0

g

16

h

Waveform inversion image

16

0 Multisource inversion image

Z (km)

0

0.6

d i ðm Þ  d i ðm o Þ þ

X ddi ðmo Þ dm j

j

0

X (km)

dm j ! dd i ðmÞ

wavepathfunction

X zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{   ¼ @di ðmo Þ@m j dm j ,

ð2Þ

j

or in matrix-vector notation dd ¼ Ldm: Here,

ð3Þ

h   @d i ðmo Þ=@m j ¼ lim Dm j !0 di mo þ bjDm j 

di ðmo Þ=Dm j is the Fréchet derivative with respect to the jth parameter; the data residual δdi ¼ [di(m) – di(mo)] is the difference between the ith components of the predicted data vector d(mo) and the observed data vector d(m); and bj is the jth unit

2

0

X (km)

2

vector in the finite-dimensional model space. The model perturbation δm ¼ m – mo is the difference between the actual model m and the guessed model mo, and L is now interpreted as the Jacobian matrix. Its elements [L]ij ¼ @di(mo)/@mj determine the sensitivity of the data to the perturbations in the model. For a windowed arrival, the Jacobian plots out in model space as a wavepath for a single source-receiver pair (Woodward 1992; Luo 1992; Dahlen et al. 2002; Marquering et al. 2002; Montelli et al. 2004; van der Hilst and de Hoop 2006; Xu and Xie 2009); and the velocity variations within its first Fresnel zone mostly influence the event of interest. Eq. A3 shows that the linearized equations take the form of the Born approximation (Stolt and Benson 1986) to the Lippmann–Schwinger equation, and its kernel is the Jacobian given by Eq. A4. For a single source and a single receiver in a smoothly increasing velocity medium, Eq. A4 plots out as a curved

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Seismic Imaging, Overview, Table 1 Symbols and specific formulas for inverting the Helmholtz equation for the perturbed slowness distribution δs(x), where δs(x) is assumed to be small compared to the background slowness model s(x). The harmonic pressure field is denoted by p(x|s) for a source localized about s, and the Green’s function for the Helmholtz equation is given by G(x|s) for a point source at s and observer

at x. The source wavelet spectrum is W(o) at the angular frequency o and the body force term localized about the point s is denoted by f(x|s). The predicted (observed) traveltime of an event received at g for a source at s is denoted as t(g|s) (t(g|s)obs.); and ε ¼ ||d – dobs.||2 is the misfit function for waveform inversion

Symbol (A1) Helmholtz equation: A(d, m) ¼ f

Mathematical formula   ∇ 2 þ o2 sðxÞ2 pðxjsÞ ¼ f ðxjsÞ m ! sðxÞ; d ! pðxjsÞ

(A2) linearized Helmholtz equation: δA(d, m) ¼ 0

ð∇ 2 þ o2 sðxÞ2 dpðxjsÞ ¼ 2o2 sðxÞdsðxÞpðxjsÞ dm ! dsðxÞ; dd ! dpðxjsÞ ð dpðgjsÞ ¼ 2o2 GðgjxÞsðxÞdsðxÞW ðoÞGðxjsÞdx3

(A3) Lippmann-Schwinger equation with born approx.: δd ¼ Lδm

where pðxjsÞ ¼ W ðoÞGðxjsÞ δp(g| s)/δs(x) ¼ – 2s(x)o2W(o)G(g| x)G(x| s) ð de=dsðxÞ ¼ g GðgjxÞ  DdðgjsÞGðxjsÞ  dgds

(A4) Jacobian or wavepath function: δd/δm (A5) misfit gradient or reverse time migration or L†[Lδm – δd]: δε/δm

where g ¼ 2o2 sðxÞW ðoÞ (A6) least squares migration or linearized inversion: m (A7) waveform inversion: m(k) – δε/δm (A8) wave equation traveltime inversion: m(k) – δε/δm

(k)

and DdðgjsÞ ¼ pðgjsÞ  pðgjsÞobs: Ð m(x)(k) – γ G(g| x)  Δd(g| s)(k)G(x| s)  dgds Ð m(x)(k) – γ G(k)(g| x)  Δd(g| s)(k)G(k)(x| s)  dgds ð mðxÞðkÞ  g GðkÞ ðgjxÞ  DdðgjsÞðkÞ GðkÞ ðxjsÞ  dgds   where DdðgjsÞ  pðgjsÞobs: tðgjsÞ  tðgjsÞobs:

– δε/δm

“fat” ray (trace a ray that connects the source and receiver and honors Snell’s law. Surround this ray with a “fat finitefrequency ray” such that the propagation of events from the source to receiver within the “fat” ray differs in traveltime by no more than 1/2 the period of the source wavelet. This fat ray region is that portion of the earth which mostly influences the traveltime of the event of interest) that connects the source and receiver points. This fat ray is denoted as a wavepath by Woodward (1992) and forms the basis of finite-frequency traveltime tomography. There seems to be a general (Montelli et al. 2004), but not a universal (van der Hilst and de Hoop 2006), agreement that finite-frequency tomography can be superior to that of ray-based tomography. As an example, earthquake seismologists use these wavepaths (renamed as banana-doughnuts) with finite-frequency tomography to invert earthquake traveltimes and surface wave data for deep mantle velocity variations attributed to, for example, ascending plumes of rocks (Dahlen et al. 2002; Marquering et al. 2002; Montelli et al. 2006). Step 4: Solve Lδm ¼ δd by an iterative gradient method. Eq. 3 is typically an overdetermined, inconsistent, and poorly conditioned system of equations. Therefore, the solution we seek is the one that minimizes the sum of the data misfit jjLm  djj2K and model penalty l2 jjCmjj2W functions in the K and W norms (Clinthorne et al. 1993): 1 l2 e ¼ kLdm  ddk2K þ kCdmk2W : 2 2

ð4Þ

where kLdm  ddk2K ¼ ðLdm  ddÞ† KðLdm  ddÞ and K dimes taken as the inverse of the data covariance matrix (Tarantola 1987) that is real and symmetric. The penalty function (also known as the regularization term) is defined as kdCmk2W ¼ ðCmÞ† WðCmÞ , where the real symmetric matrix W might strongly weight certain regions in the model because they mostly influence the data. The matrix C might be a spatial second-derivative operator in one or several directions so as to encourage solutions with smoothly varying model parameters along selected directions. A simplified steepest descent solution for the ith model parameter mi is given by

S

gradient Equation A5 ðkþ1Þ

mi

ðk Þ

¼ mi  a

z}|{ de dmi

! mðkþ1Þ

ð5Þ

migration of data residual

¼ mðkÞ 

zfflfflfflfflffl}|fflfflfflfflffl{ aL† ddðkÞ

,

where l ¼ 0; K ¼ I; α is the step length; and preconditioning is used (Beydoun and Mendes 1989; Clinthorne et al. 1993; Causse et al. 1995) to accelerate convergence. A sequence of models is generated until the data residual falls below some acceptable level. In practice, a preconditioned conjugate gradient method (Mora 1987; Luo and Schuster 1991; Epanomeritakis et al. 2008), a Gauss-Newton Krylov solver (Erlangga and Hermann 2009), or a limited memory quasi-

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Newton method (Pratt et al. 1998; Plessix 2006) for different implementations of the adjoint method (Plessix 2009) is implemented; and sometimes a direct matrix solver is used to find the Hessian inverse if the problem size is small enough (Pratt and Goulty 1991). Approximations to the Hessian by a systematic procedure can be found in Thierry et al. (1999). There is strong evidence (Crase et al. 1990; Brossier et al. 2010) that using the L1 norm, misfit function is noticeably more resistant to data noise than the L2 norm misfit function, and the combination of the two is sometimes the best choice.

Five Types of Seismic Imaging Methods Five types of seismic imaging methods and their resolution properties will be discussed (there are many seismic inversion methods, but the five types discussed here are often used in the geophysical community): migration (modest resolution of reflectivity), least squares migration (high resolution of reflectivity), full waveform inversion (high resolution of velocity), phase-like inversion (modest resolution of velocity), and migration velocity analysis (modest resolution of velocity). The high-resolution methods typically pay the price of increased computational cost and decreased robustness, compared to the moderate-resolution methods with relatively low cost and desirable robustness. For reflection migration and inversion, the spatial resolution limits are approximately defined by a generalized Radon transform analysis (Beylkin et al. 1985); and the resolution limits for ray-based transmission tomography (Williamson 1991) can be estimated by considering the width of the transmission Fresnel zone. Migration is the first iterate solution of Eq. 5, and least squares migration, sometimes known as linearized inversion, is the final iterative solution where the operator L is not updated after each iteration; also, m(x) represents the reflectivity model at the position x. In contrast to migration, the waveform inversion tomogram is the final iterative solution, where the velocity model m and L are updated after every iteration; waveform inversion falls under the class of nonlinear optimization methods. It can be shown under certain assumptions that waveform inversion reduces to either wave equation traveltime tomography (Luo and Schuster 1991) or ray-based tomography. These last two methods are classified as phase-like inversion methods. Unlike minimizing the data misfit function in Eq. 4, migration velocity analysis updates the velocity model to minimize a model misfit function, which is the normed difference between the predicted migration image and the actual migration image in the, e.g., common image gather. Convergence problems associated with local minima in the misfit function are reduced by emphasizing flatness in the CIG misfit function (Symes and Carazone 1991; Shen et al. 1993).

Seismic Imaging, Overview

Migration If the starting model is a smoothly varying velocity distribution that only generates the accurate first arrival, then the input data d – d(0) residual becomes the scattered data. In this case, the desired model is the reflectivity distribution, which is similar to the slowness perturbation function δs(x), and the background velocity model is the inverse of slowness s(x). If a finite-difference solution to the wave equation, for example, is used to calculate G(x|s) and G(g|x) in Eq. A5, then the first model update δm(1) in Eq. 5 is known as the reverse time migration image (Whitmore 1983; McMechan 1983). If a oneway wave equation method is used to generate the Green’s function, then the migration method is a phase-shift or FX-type algorithm (Stolt and Benson 1986; Claerbout 1992; Etgen et al. 2009). A diffraction-stack migration method results if the Green’s function is replaced by its asymptotic approximation GðxjsÞ ¼ Aðx, sÞeiotxs (here, A(x, s) accounts for geometric spreading losses, and txs is the first arrival time for a ray that connects the source at s with the observer at x. A ray-tracing method can be used to compute these traveltimes for a sufficiently high frequency and smoothly varying medium [Bleistein et al. 2001]). An example of standard poststack migration is depicted in Fig. 1c, where the velocity model is shown in Fig. 1a. The migration image depicts the reflectivity distribution computed by a Kirchhoff migration method with an eikonal traveltime solver. Seismic migration is also used to migrate teleseismic body waves processed to form so-called receiver functions. Examples include Ryberg and Weber (2000), Sheehan et al. (2000), and also Bostock et al. (2001) who used a ray-Born inversion approach. More recently, Nowack et al. (2007) applied the Gaussian beam migration approach of Hill (2001) to the migration of teleseismic body waves. Despite its widespread use and its robustness, migration is considered to be a moderateresolution method because it approximates the inverse Hessian matrix [L†L]1 by a diagonal matrix. Least Squares Migration If the background model 1/s(x) is not updated after each iteration (i.e., L in Eq. 5 is independent of the k index), then m(k+1) for large k is known as the least squares migration image (Nemeth et al. 1999; Duquet et al. 2000). As in standard migration, the model to be iteratively updated is the reflectivity distribution and not the velocity model. The iterative least squares migration Eq. A6 is interpreted as a sequence of standard migrations, where the data residual is back projected into the earth model by the migration operator L†. Least squares migration (LSM) is also known as linearized waveform inversion (Lailly 1984; Tarantola 1986, 1987; Jin et al. 1992; Lambaré et al. 1992) and is superior to standard migration by reducing migration artifacts caused by a poor acquisition geometry; it also can provide a spatial resolution that is more than twice (Nemeth et al. 1999; Yu

Seismic Imaging, Overview

et al. 2006) that of standard migration if the migration velocity is sufficiently accurate. Its main drawbacks are that its effectiveness is very sensitive to the accuracy of the migration velocity model, and it can be more than an order of magnitude more expensive than standard migration. (Recent developments [Dai and Schuster 2009] in phase-encoded migration suggest a great reduction in the cost of LSM.) As an example, Fig. 1d depicts the LSM image obtained from zero offset (ZO) data, which is more accurate than the ZO standard migration image in Fig. 1c. Figure 1e–f depict the RTM and LSM RTM images obtained from the prestack shot gathers.

Full Waveform Inversion If the background model is updated after each iteration, then Eq. A7 is known as nonlinear waveform inversion (Tarantola 1987; Mora 1987; Mora 1989; Song et al. 1995; and many others); a common designation for waveform inversion is FWI or full waveform inversion. Unlike LSM or standard migration, Eq. A7 for FWI iteratively updates the velocity and reflectivity models so that the final velocity image can be much improved in both resolution and accuracy. Its main drawbacks are a tendency to get stuck in local minima, it is computationally expensive compared to standard migration, and there might be more than one model that can explain the same data, i.e., a nonunique solution. For example, elastic isotropic modeling codes can sometimes generate predicted traces that adequately fit the observed data, but the estimated isotropic velocity model is inconsistent with the actual anisotropic rocks in the real earth. Successful examples of waveform tomography images are shown in Fig. 1g–h and were obtained from the same shot gathers used for migration. The final tomogram shows the velocity distribution that is almost identical to that of the actual model. One of the keys to success in waveform inversion is that a good starting model is often required for an accurate result. This starting model can be obtained by migration velocity analysis (Stork 1992; Jousselin et al. 2009), reflection traveltime tomography (Langan et al. 1985; Bishop et al. 1985; Nolet 1987; Zelt 2002), or refraction tomography (Pratt and Goulty 1991; Luo and Schuster 1991; Min and Shin 2006; Sheng et al. 2006). A major challenge to the success of waveform inversion is the limited offset range between sources and receivers and the lack of low-frequency information in the recorded data (Sirgue and Pratt 2004; Barnes et al. 2008; Boonyasiriwat et al. 2009; Kelly et al. 2009). Such deficiencies can prevent waveform inversion from reconstructing the low wavenumber parts of the model, and sometimes prevent convergence to any type of reasonable model. Remedies being explored include the possibility of recording data at much lower frequencies with more capable recording devices, and

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obtaining very wide-offset data. Other challenges address the validity of the acoustic approximation versus the reality that the recorded data are largely of elastic (Vigh et al. 2009) or viscoelastic nature (Causse et al. 1999; Askan 2006). Elastic inversion of elastic seismograms (Mora 1987; Mora 1989; Zhou et al. 1997; Brossier et al. 2009) have been presented, but the acoustic approximation can still provide useful results. One approach to viscoelastic inversion is to invert for the acoustic velocity first, then follow this with inversion for attenuation parameters (Kamei and Pratt 2008); another approach is to use a ray-based method to invert for the attenuation factor Q and then use this Q to correct for the attenuation in the data (Pratt et al. 2005; Sheng et al. 2007; Boonyasiriwat et al. 2009). Wave Equation Traveltime Inversion If the data residual δd(g|s) is replaced by the traveltime residual δt(g|s) weighted by the recorded trace d(g|s)obs., then this is known as wave equation traveltime (Luo and Schuster 1991) tomography (WT); it is a variant of Rytov inversion (Woodward 1992) and updates the velocity model by smearing weighted traveltime (not waveform) residuals over the associated wavepaths (Woodward 1992; Luo 1992). In the high-frequency limit, it reduces to ray-based traveltime tomography (RT). The advantage of WT over RT is that it does not require a high-frequency approximation and accounts for the band-limited nature of waves as they propagate through the earth. Shin et al. (2002, 2007) use a modified logarithm norm to invert for phase data, which bears a close relationship to the Rytov inversion method; and Effelsen (2009) compares phase inversion to traveltime tomography for inverting refraction events. Figure 2 illustrates how the above seismic imaging methods are related to one another. The main disadvantage of WT is that it is at least an order of magnitude more expensive than RT because it requires a finite-difference solution of the wave equation rather than a ray-traced approximation. Earthquake seismologists often see wave equation traveltime as a major improvement in estimating the earth’s velocity parameters from earthquake records (Dahlen et al. 2002; Marquering et al. 2002; Montelli et al. 2006; van der Hilst and de Hoop 2006). Migration Velocity Analysis The previous imaging methods can be described as estimating the earth model by predicting synthetic data that best matches the observed data in the data domain. In contrast, migration velocity analysis (MVA) finds the velocity model that best flattens or focuses the migration sections in the image domain; here, the image domain is the migration cube in the (x, y, z) indices and some other index s such as shot index, receiver index, source-receiver offset, (Yilmaz and Chambers 1984; Faye and Jeannot 1986; Al-Yahya 1989; Toldi 1989; Stork 1992; Lafond and Levander 1993), common angle

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Seismic Imaging, Overview

Seismic data dj = Aje iwφj

Phase misfit ||dφ ||2

Reflection misfit ||ddscatt.||2

Waveform misfit ||dd||2

Born approx.

Born approx.

Rytov approx.

m(k+1) = m(k) − aC(k)L†(k)δφ(k) Wave eq. traveltime inversion L = Woodward−Rocca integral

scatt.

m(k+1) = m(k) − aC(k )L† dd(k)

m(k +1) = m(k) − aC(k )L†(k)dd(k )

Least squares migration L = Lippmann−Schwinger integral

Waveform inversion L = Lippmann−Schwinger integral

a =1; k=0 C = Illum. comp.

Hi-freq. approx.

G−>ray m(k+1) = m(k) − aC(k)L†(k)dφ(k ) Ray-based traveltime inversion L = Traveltime integral

(1)

(0)

†

Kirchhoff migration

scatt.

m = m − C L dd G−>1−way Reverse-time migration L = Lippmann−Schwinger integral

Phase-shift class migration

G−>beam Beam migration

Seismic Imaging, Overview, Fig. 2 Three types of seismic imaging methods: phase inversion, least squares migration (LSM), and nonlinear waveform inversion. Note that LSM is a linearized inversion so that the modeling operator L does not get updated after each iteration. This compares to nonlinear waveform inversion which updates L after each iteration. The Lippmann-Schwinger-type and Woodward-Roca-type integrals are displayed in Table 1, and details about various migration methods are in the Seismic Migration section in the Encyclopedia of

Solid Earth Geophysics. The traces that only contain scattered arrivals are obtained by muting the direct arrivals; and the ensemble of these traces is symbolized by the vector dscatt. Instead of computing the Green’s functions in the integral equations by a finite-difference solution to the wave equation, various approximations such as ray-based Green’s functions (Kirchhoff migration), one-way wave equation approximations (phase-shift type migration), and Gaussian Beam (beam-like migration) are used for migration

parameter (Xu et al. 1998), or subsurface offset coordinate (Rickett and Sava 2002; Sava and Biondi 2004; Robein 2010). To understand why we seek a velocity model that flattens events in the image domain, consider the 2D migration image m(x, z, s0) in Fig. 3a obtained by migrating one shot gather (the source is at the surface with shot index s0). If the migration velocity is correct, then all of the single-shot migration images should be similar in appearance. This means that all of the migrated reflection events should be flat in the common image gather (CIG) given by m(xCIG, z, s) for all values of s, z, and a fixed value of xCIG (see Fig. 3b). Note that if the migration velocity is accurate, the reflector boundary denoted by the dotted horizontal line will be flat for a common image gather. If the migration velocity is too slow, then the imaged reflector boundary will curve upward as illustrated by the curved dashed line, and if too fast, the imaged reflector will curve downward.

With MVA, the goal is to find the velocity model so that, † ideally, the predicted migration image mpred: mig: ¼ L LM best † fits the actual migration image mmig. ¼ L d. The associated misfit function can be constructed so that it is similar to that of Eq. 4, except that the norm of the migration residual

2 1

l2

e ¼ mpred:  m kCmk2W mig: þ mig: 2 2 K

ð6Þ

is minimized rather than the data residual; in this case, m represents the velocity model. To find the velocity model, we can use the unregularized steepest descent equation: migrationmodelingofmigrationresidual

mðkþ1Þ ¼ mðkÞ  ðkÞ

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ ðkÞ aL† Ldmmig:

,

ð7Þ

where dmmig: ¼ mpred: mig:  mmig: is the migration residual at the kth iteration. The importance of this formula is that it shows

Seismic Imaging, Overview

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Migration image mmig. (x,z,s)

xCIG Shot axis

xCIG

xCIG X Z

Flat mmig. (xCIG,z,s)

a

Shot axis Common image gather mmig. (xCIG,z,s)

Δzi

Migration velocity too slow

xCIG Flat events indicate accurate migration velocity Z

b Seismic Imaging, Overview, Fig. 3 (a) Migration image mmig. (x,z,s) cube in the model coordinates of x and z and the surface shot offset index s; the common image gather mmig. (xCIG, z, s) is computed by migrating the shot gathers and displaying the migration image at the fixed value of

x ¼ xCIG for all shot s and depth z values. (b) Common image gather mmig. (xCIG, z, s) for the common image point at xCIG. The migration image should only be non-zero along the interface between the brick region and open region shown above

that the gradient term is computed by modeling the migration ðk Þ residual to get the filtered data residual Ldmmig: , and then the velocity model is updated by smearing this filtered (the filtered data residual is constructed by migrating and modeling the actual data residual; hence, it is a filtered version of the actual data residual) data h residual i into the model by the ðkÞ migration operation L† Ldmmig: . These filtered residuals are smeared along wavepaths for each source-receiver pair of traces, and the migration and modeling operators are updated after each iteration. This leads to a moderate resolution of the velocity model because the filtered, not unfiltered, data are kinematically fitted. Moreover, MVA is largely a curve fitting exercise that is mostly insensitive to subtle amplitude variations in the migration traces; ignoring such information will blind MVA to subtle variations in the impedance distribution. To reduce computational costs and increase robustness at the expense of reduced model resolution, MVA is sometimes implemented with the following steps:

: 1. Automatically pick the depth residual Dzi ¼ zi  zref of a i coherent CIG event in Fig. 3b at the ith shot position; here, z is the picked depth of the targeted reflection at the xCIG : offset in the ith migrated shot gather. The depth zref of the i reference event for that reflector, is estimated from the near-offset trace in that CIG. A computer algorithm can window about the near-offset reflection of interest and use cross-correlation with neighboring traces in the CIG to : estimate the depth lag zi  zref associated with the stroni gest correlation energy. 2. The misfit function is then defined as

e ¼ 1=2

X ðDzi Þ2 þ model smoothness constraints,

ð8Þ

i

for each CIG and the summation is over the shot index in Fig. 3b. Sometimes the misfit function is the sum over all CIGs.

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Seismic Imaging, Overview

3. The velocity model is iteratively updated by a gradient optimization method until e is minimized. An example of the above procedure is shown in Fig. 4, where the left column of images depict the (a) migration image obtained with an inaccurate velocity model, (b) the CIG with curved events, and (c) the migration velocity model. After seven iterations of MVA, the right column of figures is obtained. Note the flat events in the CIG, and the final velocity model is almost the same as the actual velocity model. The Fréchet derivative @c@zðxi Þ associated with the gradient of

ε can be numerically computed by using a ray-tracing scheme (Stork 1992; Chauris et al. 1998) to determine the change in depth zi, of a migrated reflection point with respect to a change in the velocity parameter at x. Sometimes, the depth residual Δzi for the ith ray is converted into a time residual Δti, and, similar to traveltime tomography, this converted time residual is smeared along the reflection ray to iteratively update the velocity model. For arbitrary reflector geometries in a homogeneous media, an analytic conversion formula was derived

a

" # 2

X XX



e ¼ 1=2

g @mðx, y, z, hÞmig: =@h2



x, y z h

that rewards flat events (i.e., accurate velocity models) in the CIG domain. Here, γ is a normalization term that depends on (x, y). Empirical tests by Chauris et al. (1998) suggest that the semblance based ε is much smoother than that in Eq. 6 and is nearly devoid of local minima in misfit function. Some links between MVA and FWI are established in Symes (2008), and MVA compared to several tomography methods is presented

Mig. image w/o MVA

d

Mig. Image with MVA

CIG w/o MVA

e

CIG with MVA

Starting V (x,z)

f

V (x,z) by MVA

Depth (m)

0

40

b

Depth (m)

0

40

c 0

40

0

ð9Þ

þconstraints:

Depth (m)

Seismic Imaging, Overview, Fig. 4 (a) Migration image (Sun 2001) obtained by prestack migration using the smooth homogeneous velocity model in (c) that is far from the true model approximated in (f); (b) common image gather in the shot offset index, where the curved events indicate an incorrect velocity model; (c) incorrect migration velocity model used to compute (a); (d) migration image obtained after seven iterations of MVA; (e) CIG after seven iterations of MVA; and (f) velocity model inverted by seven iterations of MVA. This result closely resembles the true velocity model

by Al-Yahya (1989), and a ray-tracing method was used by Stork (1992) and others (Robein 2010). Similar to waveform inversion, MVA seeks to predict the wiggly migration traces seen in the migration image L†d and, therefore, can easily get stuck in the many local minima of ε. To avoid this problem, Symes and Carazone (1991) proposed the smoother differential semblance misfit function (Chauris et al. 1998; Shen et al. 2003):

X (m)

40

0

X (m)

40

Seismic Imaging, Overview

by Le Bégat et al. (2004). The subsurface offset domain can be exploited for MVA (Sava and Biondi 2004) and extraction of scattering angle information (Rickett and Sava 2002), and an excellent summary of MVA research is given in Robein (2010). In summary, the exploration community heavily favors MVA over many other velocity estimation methods because it is robust, efficient, and the picking of depth residuals is easily automated in the migration image domain. (Traveltimes picked from traces are notoriously difficult to automate because waveforms often interfere with one another. In comparison, migration untangles these events and focuses them to their origin along the reflectors so that automatic picking is easier in the CIG. Sometimes semblance methods are used to find the best fit hyperbolic or parabolic curve to the data [Robein 2010].) Its chief disadvantage is that an MVA image lacks the detailed resolution of waveform inversion, which suggests that MVA should be used to estimate the starting velocity models for waveform tomography.

Recent Advances in Seismic Imaging In the last 15 years, several breakthroughs have enabled practical waveform inversion of seismic data. One of these advances is the relentless increase in computational capabilities of cluster computers and GPU-like processors, and two others are multiscale inversion and phase-encoded waveform inversion. Multiscale Waveform Inversion One of the main difficulties with waveform inversion is that the misfit function is plagued by many local minima. If the starting velocity model is moderately far from the actual model (an erroneous velocity might be one where the modeled events arrive by more than a period after the actual arrivals), then the iterative gradient solution gets stuck there and never reaches the global minimum or actual model. The partial cure to this local-minima problem is a multiscale approach (Bunks et al. 1995), where the initial iterations estimate a coarse-grid velocity model from low-frequency data. For a reasonable starting model, this often prevents getting stuck in local minima because the predicted lowfrequency arrivals are more likely to arrive within a period of the arrivals in the low-pass filtered data. After a number of iterations, the velocity model is refined to a finer scale (the grid interval is halved) and intermediate frequency data are iteratively inverted to update the velocity model. After suitable reduction of the data misfit, the model grid is refined again and higher-frequency data are inverted until a satisfactory model is reconstructed. One of the first relevant demonstrations of multiscale FWI applied to real data in a complex environment was performed by Ravaut et al. (2004). Other

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results with both synthetic and field data (Sirgue and Pratt 2004; Sirgue et al. 2007; Plessix 2009; Vigh and Starr 2007, 2008; Sirgue et al. 2010) convincingly demonstrate the effectiveness of this approach. As mentioned earlier, improved model reconstructions can be achieved if more accurate starting models are employed, lower frequency data are recorded, and wider offset data are acquired. The new challenges are to employ modeling and inversion codes that robustly take into account the effects of viscoelasticity and anisotropy in the data. Phase-Encoded Multisource Waveform Inversion A major difficulty, until recently, is the enormous computational expense of waveform inversion. Each shot gather of residuals must be migrated at each iteration, which for 3D problems can be too demanding even for the most powerful computers. To relieve this problem, Krebs et al. (2009), Operto et al. (2009), and Dai and Schuster (2009) proposed summing phase-encoded shot gathers into supergathers and migrating the supergathers at each iteration; a supergather is modeled with just one finite-difference simulation, where the computational cost is the same as that for one shot gather. This is similar to the phase-encoding RTM of Romero et al. (2000), except iterations are used to reduce the crosstalk noise in supergather migration. The result can be an enormous cost savings compared to conventional waveform inversion or migration. Figure 1h shows an example of phase-encoding shot gathers, where each trace in a shot gather has the same random time shift but different shots have different time shifts. In this case, 12 shot gathers were time-shifted and blended together into one supergather; there were 192 traces per shot gather. A total of 16 nonoverlapping supergathers were input into the iterative waveform inversion code, where each supergather migration costs the same as the migration of one shot gather because 12 shots were excited at nearly the same time. Hence, the Fig. 1h tomogram costs 1/12 that of the Fig. 1g tomogram. For 3D inversion, the computational cost savings can be more than two orders of magnitude compared to conventional waveform inversion. For a 3D example, Fig. 5a shows results after waveform inversion of 3D synthetic data, where the model size is 800  800  186 grid points and the spatial sampling interval is 20 m. There are 1089 sources evenly distributed along the surface with an interval of 500 m in the inline (X) and crossline (Y) directions. Multisource waveform inversion using the Krebs method (Krebs et al. 2009), static QuasiMonte Carlo (QMC) phase encoding, and dynamic QMC phase encoding (Boonyasiriwat and Schuster 2010) are applied to this data set. Figure 5b–d show that dynamic QMC phase encoding provides a higher-quality tomogram than the other methods, yet the computational cost is two orders of magnitude less

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Seismic Imaging, Overview

a

Velocity (m/s)

True depth slice 0

Velocity (m/s)

True cross section

5,500

6,000

0

5,500 5,000 Z (km)

Y (km)

5

1

10

5,000 4,500

2

4,000

4,500

3,500 3

15 0

b

5

10 X (km)

15

0

Velocity (m/s)

Krebs depth slice

0

4,000

3,000 5

10 X (km)

15

Velocity (m/s)

Krebs cross section

5,500

2,500

6,000

0

5,500 5,000 Z (km)

Y (km)

5

1

10

5,000 4,500

2

4,000

4,500

3,500 3

15 0

c

5

10 X (km)

15

4,000

3,000 0

Velocity (m/s)

Static QMC depth slice 0

5

10 X (km)

15

Velocity (m/s)

Static QMC cross section

5,500

2,500

6,000

0

5,500 1

5,000 Z (km)

Y (km)

5

10

5,000 4,500

2

4,000

4,500

3,500 3

15 0

d

5

10 X (km)

15

4,000

3,000 0

Velocity (m/s)

Dynamic QMC depth slice

10 X (km)

15

2,500

Velocity (m/s)

Dynamic QMC cross section

5,500

0

5

6,000

0

5,500 1

5,000 Z (km)

Y (km)

5

10

5,000 4,500

2

4,000

4,500

3,500 3

15 0

5

10 X (km)

15

4,000

Seismic Imaging, Overview, Fig. 5 Depth slices at z ¼ 2.1 km and cross-sections at y ¼ 8 km of (a) true model, and inverted models using (b) the Krebs method (1089 CSGs/supergather), (c) static QMC method (99 CSGs/supergather), and (d) dynamic QMC method (99 CSGs/

3,000 0

5

10 X (km)

15

2,500

supergather) after 40 iterations. A greater number of CSGs/supergather require a greater number of iterations to get the same accuracy in the final tomogram; the dynamic encoding strategy is more effective than the static strategy. (Figures courtesy of C. Boonyasiriwat)

Seismic Imaging, Overview

than that of conventional waveform inversion. The Krebs strategy is the most efficient because 1089 shot gathers were blended into one supergather, compared to the QMC strategies that used 99 CSGs/supergather. Dynamic phase encoding changed the time shifts of each shot gather after each iteration, while the static strategy kept the phase encoding the same. Current Status and Future of Seismic Imaging Reverse time migration is becoming the preferred means for seismic imaging beneath complex regions such as salt bodies. Earthquake seismologists are now recognizing the benefits of migration imaging earthquake records for tectonic structures such as subduction zones (Bostock et al. 2001) or using wideangle seismic experiments to image the crust (Brenders and Pratt 2007) and mantle (Montelli et al. 2006). Least squares migration and waveform inversion are now being considered as viable upgrades to standard migration because of the tremendous speed up from phase-encoded multisource methodology. There is still a debate about whether waveform inversion should be computed in the frequency (there are additional advantages by formulating the problem in the Laplace transform domain [Shin and Ha 2008]) or time domains (Vigh and Starr 2007; Warner 2008), but there is no debate that we eventually need to account for viscoelastic and anisotropic effects in the data. 3D waveform inversion of earthquake records for whole earth tomograms greater than 1 Hz is still too computationally demanding except at very low frequencies, and the same can be said for 3D exploration geophysics at frequencies above 50 Hz. Challenges still remain, especially in the critically important area of anisotropic RTM (Zhang and Zhang 2009; Fowler et al. 2010) and waveform inversion; estimation of accurate anisotropic migration velocity models is an ongoing line of research. Earthquake seismologists are now testing the possibility of using earthquake records for inverting basin structures and velocity models so as to improve their simulation-based predictions of earthquake hazard. Passive seismic recordings and the use of interferometry to extract surface wave records (Shapiro and Campillo 2004; Shapiro et al. 2005), followed by inversion for S-velocity tomograms are playing an increasingly important role in earthquake seismology.

Cross-References ▶ Body Waves ▶ Free Oscillations of the Earth ▶ Inverse Theory, Global Optimization ▶ Inverse Theory, Linear ▶ Inverse Theory, Monte Carlo Method ▶ Numerical Methods, Finite Difference ▶ Seismic Tomography

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▶ Seismic, Ambient Noise Correlation ▶ Seismic, Migration ▶ Seismic, Waveform Modeling and Tomography ▶ Single and Multichannel Seismics ▶ Traveltime Tomography Using Controlled-Source Seismic Data ▶ Vertical Seismic Profiling

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1418 Crase E, Pica A, Noble M, McDonald J, Tarantola A (1990) Robust elastic non-linear waveform inversion: application to real data. Geophysics 55:527–538 Dahlen F, Hung S, Nolet G (2002) Frechet kernels for finite-frequency traveltimes I Theory. Geophys J Int 141:157–174 Dai W, Schuster GT (2009) Least-squares migration of simultaneous sources data with a deblurring filter. In: Expanded abstracts of SEG international meeting, pp. 2990–2993 Duquet B, Marfurt K, Dellinger J (2000) Kirchhoff modeling, inversion for reflectivity, and subsurface illumination. Geophysics 65:1195–1209 Effelsen K (2009) A comparison of phase inversion and traveltime tomography for processing of near-surface refraction traveltimes. Geophysics 74:WCB11–WCB24 Epanomeritakis I, Akçelik V, Ghattas O, Bielak J (2008) A Newton-CG method for large-scale three-dimensional elastic full waveform seismic inversion. Inverse Problems 24:975–987 Erlangga Y, Hermann F (2009) Seismic waveform inversion with gaussNewton-Krylov method. In: Expanded abstracts of SEG international meeting, pp 2357–2361 Etgen J, Gray S, Zhang Y (2009) An overview of depth imaging in exploration geophysics. Geophysics 74:WCA5–WCA17 Faye J-P, Jeannot J-P (1986) Prestack migration velocities from focusing depth analysis. In: Expanded abstracts of SEG international meeting, pp 438–440 Fowler P, Du X, Fletcher R (2010) Coupled equations for reverse time migration in transversely isotropic media. Geophysics 75: S11–S22 Hill NR (2001) Prestack Gaussian beam depth migration. Geophysics 66:1240–1250 Jin S, Madariaga R, Virieux J, Lambaré G (1992) Twodimensional asymptotic iterative elastic inversion. Geophys J Int 108:575–588 Jousselin P, Duquet B, Audebert F, Sirgue J (2009) Bridging the gap between ray-based tomography and wave-equation migration image gathers. In: Expanded abstracts of SEG international meeting, pp 3979–3983 Kamei R, Pratt G (2008) Waveform tomography strategies for imaging attenuation structure for cross-hole data. In: 70th conference and technical exhibition, EAGE Expanded Abstracts, p F019 Kelly S, Ramos-Martinez J, Tsimelzon B, (2009) The effect of improved, low-frequency bandwidth in full-wave form inversion for velocity. In: Expanded abstracts of SEG international meeting, pp 3974–3977 Krebs JR, Anderson JE, Hinkley D, Neelamani R, Lee S, Baumstein A, Lacasse MD (2009) Fast full-wavefield seismic inversion using encoded sources. Geophysics 74:WCC177–WCC188 Lafond C, Levander A (1993) Migration moveout analysis and depth focusing. Geophysics 58:91–100 Lailly P (1984) Migration methods: partial but efficient solutions to the seismic inverse problem. In: Santosa F, Pao YH, Symes W, Holland CH (eds) Inverse problems of acoustic and elastic waves. SIAM, Philadelphia Lambaré G, Virieux J, Madariaga R, Jin S (1992) Iterative asymptotic inversion in the acoustic approximation. Geophysics 57:1138–1154 Langan R, Lerche I, Cutler RT (1985) Tracing of rays through heterogeneous media: an accurate and efficient procedure. Geophysics 50:1456–1465 Le Bégat S, Chauris H, Devaux V, Nguyen S, Noble M (2004) Velocity model estimation for depth imaging: comparison of three tomography methods on a 2D real data set. Geophys Prospect 52:427–438 Luo Y (1992) Calculation of wavepaths for band-limited seismic waves. In: Expanded abstracts of SEG international meeting, pp 1509–1512 Luo Y, Schuster GT (1991) Wave-equation traveltime inversion. Geophysics 56:645–653 Marquering H, Dahlen FA, Nolet G (2002) Threedimensional sensitivity kernels for finite-frequency traveltimes: the banana-doughnut paradox. Geophys J Int 137:805–815

Seismic Imaging, Overview McMechan G (1983) Migration by extrapolation of time-dependent boundary values. Geophys Prospect 31:413–420 Min D, Shin C (2006) Refraction tomography using a waveforminversion back-propagation technique. Geophysics 71(3):R21–R30 Montelli R, Nolet G, Masters G, Dahlen F, Hung SH (2004) Global P and PP traveltime tomography: rays versus waves. Geophys J Int 158:637–654 Montelli R, Nolet G, Dahlen F (2006) Comment on “Banana-doughnut kernels and mantle tomography” by van der Hilst and de Hoop. Geophys J Int 167:1204–1210 Mora P (1987) Elastic wavefield inversion. PhD dissertation, Stanford University, pp 143 Mora P (1989) Inversion¼migration+tomography. Geophysics 54:1575–1586 Nemeth T, Wu C, Schuster GT (1999) Least-squares migration of incomplete reflection data. Geophysics 64:208–221 Nolet G (1987) Seismic tomography: with applications in global seismology and exploration. Springer, Dordrecht Nowack RLWP, Chen UK, Dasgupta S (2007) Imaging offsets in the Moho: synthetic tests using Gaussian beams with teleseismic waves. Pure Appl Geophys 164:1921–1936 Operto S, Virieux J, Ribodetti A, Anderson J (2009) Finite-difference frequency-domain modeling of viscoelastic wave propagation in two-dimensional tilted transversely isotropic media. Geophysics 74:T75–T95 Plessix RE (2006) A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys J Int 167:495–503 Plessix RE (2009) 3D frequency-domain full-waveform inversion with an iterative solver: geophysics. Geophysics 74:WCC149–WCC157 Pratt G (1999) Seismic waveform inversion in the frequency domain, part I: theory and verification in a physical scale model. Geophysics 64:888–901 Pratt RG, Goulty NR (1991) Combining wave-equation imaging with traveltime tomography to form high-resolution images from crosshole data. Geophysics 56:208–224 Pratt RG, Shin C, Hicks GJ (1998) Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion. Geophys J Int 133:341–362 Pratt RG, Hou F, Bauer K, Weber M (2005) Waveform tomography images of velocity and inelastic attenuation from the Mallik 2002 crosshole seismic surveys. In: Dallimore SR, Collett TS (eds) Scientific results from the Mallik 2002 gas hydrate production research well program, Mackenzie Delta, Northwest Territories, Canada. Geological Survey of Canada, Canada Ravaut C, Operto S, Improta L, Virieux J, Herrero A, dell’Aversana P (2004) Multi-scale imaging of complex structures from multi-fold wide-aperture seismic data by frequency-domain full-wavefield inversions: application to a thrust belt. Geophys J Int 159:1032–1056 Rickett J, Sava P (2002) Offset and angle-domain common-image gathers for shot-profile migration. Geophysics 67:883–889 Robein E (2010) Seismic Imaging. EAGE Publications, The Netherlands Romero L, Ghiglia D, Ober C, Morton S (2000) Phase encoding of shot records in prestack migration. Geophysics 65:426–436 Ryberg T, Weber M (2000) Receiver function arrays: a reflection seismic approach. Geophys J Int 141:1–11 Sava P, Biondi B (2004) Wave-equation migration velocity analysis-1: theory. Geophys Prospect 52:593–606 Shapiro N, Campillo M (2004) Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise. Geophys Res Lett 31: L07614. https://doi.org/10.1029/ 2004GL019491. Shapiro N, Campillo M, Stehly L, Ritzwoller M (2005) High-resolution surface-wave tomography from ambient seismic noise. Science 307:1615–1618 Sheehan AF, Shearer PM, Gilbert HJ, Dueker KG (2000) Seismic migration processing of P-SV converted phases for mantle discontinuity

Seismic Instrumentation structure beneath the Snake River plain, Western United States. J Geophys Res 105:055–065 Shen P, Symes W, Stolk C (2003) Differential semblance velocity analysis by wave equation migration. In: Expanded abstracts of SEG international meeting, pp 2135–2139 Sheng J, Leeds A, Buddensiek M, Schuster GT (2006) Early arrival waveform tomography on near-surface refraction data. Geophysics 71(4):U47–U57 Shin C, Ha W (2008) A comparison between the behavior of objective functions for waveform inversion in the frequency and Laplace domains. Geophysics 73:VE119–VE133 Shin C, Min D-J, Marfurt KJ, Lim HY, Yang D, Cha Y, Ko S, Yoon K, Ha T, Hong S (2002) Traveltime and amplitude calculations using the damped wave solution. Geophysics 67:1637–1647 Shin C, Pyun S, Bednar B (2007) Comparison of waveform inversion, part 1: conventional wavefield vs logarithmic wavefield. Geophys Prospect 55:449–464 Sirgue L, Pratt RG (2004) Efficient waveform inversion and imaging: a strategy for selecting temporal frequencies. Geophysics 69:231–248 Sirgue L, Etgen J, Albertin U (2007) 3D full-waveform inversion: wideversus narrow-azimuth acquisitions. In: Expanded abstracts of SEG international meeting, pp 1760–1764 Sirgue L, Barkved OI, Dellinger J, Etgen J, Albertin U, Kommedal JH (2010) Full waveform inversion: the next leap forward in imaging at Valhall. First Break 28:65–70 Song Z, Williamson P, Pratt G (1995) Frequency-domain acoustic-wave modeling and inversion of crosshole data, part 2: inversion method, synthetic experiments and real-data results. Geophysics 60:786–809 Stolt R, Benson A (1986) Seismic migration: theory and practice. In: Handbook of geophysical exploration, vol 5. Geophysical Press, London, UK Stork C (1992) Reflection tomography in the postmigrated domain. Geophysics 57:680–682 Sun H (2001) Wavepath migration for depth imaging and velocity analysis. PhD dissertation, University of Utah Symes W (2008) Migration velocity analysis and waveform inversion. Geophys Prospect 56:765–790 Symes W, Carazone J (1991) Velocity inversion by differential semblance optimization. Geophysics 56:654–663 Tarantola A (1986) Linearized inversion of seismic reflection data. Geophys Prospect 32:998–1015 Tarantola A (1987) Inverse problem theory: methods for data fitting and model parameter estimation. Elsevier Science, Amsterdam Thierry P, Operto S, Lambaré G (1999) Fast 2D ray-born inversion/ migration in complex media. Geophysics 64:162–181 Toldi J (1989) Velocity analysis without picking. Geophysics 54:191–199 van der Hilst R, de Hoop M (2006) Reply to comment by R. Montelli, G. Nolet, and F.A. Dahlen on “Banana-doughnut kernels and mantle tomography”. Geophys J Int 167:1211–1214 Vigh D, Starr EW (2007) Comparisons for waveform inversion, time domain or frequency domain? In: Expanded abstracts of SEG international meeting, pp 1890–1894 Vigh D, Starr EW (2008) 3D prestack plane-wave full-waveform inversion. Geophysics 73:135–144 Vigh D, Starr EW, Elapavuluri P (2009) Acoustic waveform inversion vs. elastic data. In: Expanded abstracts of SEG international meeting, pp 2298–2301 Virieux J, Operto S (2009) An overview of full-waveform inversion in exploration geophysics. Geophysics 74:WCC1–WCC26 Warner M (2008) 3D wavefield tomography: synthetic and field data examples. In: Expanded abstracts of SEG international meeting, pp 3330–3334 Whitmore ND (1983) Iterative depth migration by backward time propagation. In: Expanded abstracts of SEG international meeting, pp 827–830

1419 Williamson P (1991) A guide to the limits of resolution imposed by scattering in ray tomography. Geophysics 56:202–207 Woodward MJ (1992) Wave-equation tomography. Geophysics 57:15–26 Xu S, Chauris H, Lambar G, Noble M (1998) Common angle image gather: a strategy for imaging complex media. In: Expanded abstracts of SEG international meeting, pp 1538–1541 Xu W, Xie X (2009) How serious is the nonlinear effect on traveltime delays predicted by sensitivity kernels. In Expanded abstracts of SEG international meeting, pp 4049–4053 Yilmaz O, Chambers R (1984) Migration velocity analysis by wave-field extrapolation. Geophysics 49:1664–1674 Yu J, Hu J, Schuster G, Estill R (2006) Prestack migration deconvolution. Geophysics 71:S53. https://doi.org/10.1190/ 1.2187783 Zelt C (2002) Modelling strategies and model assessment for wide-angle seismic traveltime data. Geophys J Int 139:183–204 Zhan G, Schuster GT (in press) Multisource phase-encoded waveform inversion. J Explor Geophys Zhang Y, Zhang H (2009) A stable TTI reverse time migration and its implementation. In: Expanded abstracts of SEG international meeting, pp 2794–2798 Zhou C, Schuster GT, Hassanzadeh S, Harris JM (1997) Elastic waveequation traveltime and waveform inversion of crosshole seismic data. Geophysics 62:853–868

Seismic Instrumentation Duncan Carr Agnew Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA, USA

Synonyms Seismometry

Definition Datalogger. Device for recording electrical signal from a seismometer, usually in digital form and with accurate time information. Geophone. Another name for seismometer, used in geophysical exploration. Seismometer. Device for providing a record of ground motion, usually by converting it into an electrical signal. Strainmeter. Device for continuously measuring the deformation of the Earth, either as extension along a line, or volume change.

Introduction Seismic instruments give a time record of ground motion caused by elastic waves. The first instruments were built at

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the end of the nineteenth century; subsequent developments have evolved towards wider coverage of both the frequency and amplitude of the waves that could be recorded. Over most of the twentieth century much of this evolution, and of the diversity of instrument types, was related not to how ground motion was sensed but to how it was recorded; the last three decades have seen such rapid development in digital technology, especially in data storage, that recording techniques unique to seismology are no longer needed. Modern seismic systems all consist of a seismometer for converting some aspect of ground motion into an electrical signal, and a datalogger for recording this signal. In exploration geophysics, the name geophone is a common synonym for seismometer; now that data recording is clearly separated from sensing ground motion, the old term seismograph could probably be abandoned. Since dataloggers are not unique to seismology, this article discusses only the general requirements for them, focusing instead on the designs of seismic sensors, or seismometry. Three recent reviews cover seismometry in more detail: Wielandt (2002), Havskov and Alguacil (2004), and Bormann (2009).

Seismic Instrumentation

Inertial Seismometers: Basic Principles Almost all seismometers measure motion of the ground using the inertia of an internal mass, and so are called inertial sensors. This mass, of amount m, is subject to five forces: (1) constraints that restrict the mass motion to be in a particular direction (denoted by a unit vector e) making the system one with a single degree of freedom; (2) the gravitational force vector, gm, along that direction, namely, e·gm; (3) an elastic restoring force (from a spring),  k(x – x0), where k is the spring constant and x the displacement (in the direction e) away from an equilibrium position x0; (4) a viscous damping force proportional to velocity dx_ ; (5) additional forces Fb that may be applied by a feedback system (discussed below). The displacement of the mass relative to an inertial frame is x + e · u, where x is the motion of the mass relative to the Earth, which is what we can measure, and u is the (vector) motion of the Earth relative to the inertial frame, which is that of the Earth in the absence of seismic waves (we can neglect non-inertial effects from the Earth’s rotation). Combining forces (2), (3), and (4), and applying the momentum equation, the acceleration of the mass is given by

Requirements for Instruments €Þ ¼ e gm  kðx  x0 Þ  dx_ mðx€ þ e u The frequencies to be measured by seismometers range from 0.31 mHz (the slowest free oscillation) up to 1 kHz (in nearsurface geophysics): roughly six orders of magnitude. Fig. 1 shows the amplitudes that can occur over most of this frequency range, from the background noise level at quiet sites (Berger et al. 2004) to the large signals that have been observed at moderate distances from very large earthquakes, to the even larger, and damaging, strong ground motions close to an earthquake. The large signals are infrequent, but also of considerable scientific importance – but so are very small signals, which can be seen only if the seismometer has lower noise than the ground does. Since digital dataloggers can record over a much wider range of amplitudes and frequencies than the older analog recorders could, modern seismic systems can be characterized as broadband (covering a wide band of frequencies) and high dynamic range (a wide range of amplitudes). But the range from noise to the largest signal covers over 10 orders of magnitude, too much for any single instrument and datalogger. Since most seismometer designs require practical tradeoffs between low-frequency capability and other performance goals, seismometers are still classified by what frequencies and amplitudes they cover best. Systems for recording the largest signals without distortion are called strong-motion seismometers, and have usually been designed to meet the needs of earthquake engineers rather than seismologists, though digital technology is blurring this distinction.

where we allow the possibility of variations in both e and g. With no motion, we assume that the mass position is x0, so that e0 g0m ¼ kx0. Then the above equation becomes, for small variations in e and g, and after some rearrangement of terms, k d €  e 0 ð g  g0 Þ  ð e  e 0 Þ g0 x þ x_ þ x€ ¼ e0 u m m more usually written as €  e0 ðg  g0 Þ x€ þ 2gx_ þ o20 x ¼ e0 u ðe  e0 Þ g0

ð1Þ

pffiffiffiffiffiffiffiffiffi where o0 ¼ k=M is the natural frequency of the seismometer (T0 ¼ 2π/o0 is called the free period); γ ¼ d/2 M is the damping constant. These names describe the behavior of the seismometer if the right-hand side of the equation is zero and x ¼ x0 at t ¼ 0; the subsequent motion of the seismometer mass is then  2pt gt xðtÞ ¼ x0 cos e , T0 which is a decaying sinusoid with period T; the damping g needs to be large enough to avoid ringing (a sign of resonance effects), and is usually set to about 0.8.

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The right-hand side of Eq. 1 shows that an inertial sensor responds to three inputs:

X o2 ¼ 2 U o0  gio þ o2

€ ; this is why inertial 1. The acceleration of the ground, u sensors are often called accelerometers. 2. Changes in the gravitational vector g along the direction of sensitivity; if this is the dominant signal, the sensor is usually referred to as a gravimeter (see Gravimeters). 3. Changes in the direction of sensitivity relative to the direction of gravity. If e0 and g0 are parallel (the mass moves vertically), then to first order e – e0 is perpendicular to g0, and this contribution is zero. However, if e0 and g0 are perpendicular (the mass moves horizontally), then to first order e – e0 is parallel to g0, and this contribution can be significant; if this is the dominant signal, the sensor would be referred to as a tiltmeter, since changes in e reflect tilting of the sensor (or the ground it is attached to).

is the frequency response of the seismometer. For ground motion at frequencies much higher than the natural frequency o0, X ≈ U: the mass motion looks like ground displacement. For ground motion at much lower frequencies, X  Uo2 =o20: the mass motion looks like ground acceleration, scaled by the inverse of the natural frequency of the seismometer. The minimum noise level of any inertial sensor is set by the thermal (Brownian) motion of the mass in equilibrium with its surroundings. This noise level can be expressed as an equivalent ground acceleration, with a flat power spectral density of 8o0lkB0/m, where kB is Boltzmann’s constant and θ is the temperature (Aki and Richards 2002). For a temperature of 310 K and γ ¼ 0.8, this expression is 1.72  1019/(mT0) m4 s3, with m in kilograms and T0 in seconds. The dotted lines in Fig. 1 show noise levels for two cases that cover the range of designs: mT0 ¼ 104 (1 g mass and free period of 0.1 s), which cannot resolve ground noise at the longer periods, and mT0 ¼ 3 (300 g and 10 s), which shows that a moderately large mass and long period are necessary and sufficient for this thermal noise limit to fall below ground noise at all frequencies.

Much confusion has been created by the use of these different terms for what is the same kind of sensor, when the difference actually refers to the type of signal being measured. Ignoring the tilt and gravity terms, suppose that the ground displacement is purely sinusoidal, u(t) ¼ Ueiot (U being complex, and the real part being taken). Then the mass motion will be Xeiot, where

100 10–1 10–2 RMS amplitude in 2/3 octave (m/s2)

Seismic Instrumentation, Fig. 1 Seismic and instrument noise levels. The solid and long dashed lines show the lowest levels of vertical and horizontal ground motion observed by the stations of the Global Seismographic Network. The dotted lines show the seismometer Brownian noise limit for two different values of the product of the sensor mass and free period. The dashed lines show the noise levels for a high-quality shortperiod system using a moving-coil transducer (Rodgers 1994), for a MEMS sensor used for measuring strong motion (Evans et al. 2005), for a sensitive broadband system (Ringler and Hutt 2009), and for GPS used at seismic frequencies

ð2Þ

10–3 10–4

MEMS strong-motion GP

Sp

os

10–5 10

itio

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g

mT0 = 1 –4 0

Short-period

10–8 10–9 10–10

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nin

–6

Ground noise

mT0 = 3

10–11 STS-1 10–12 10–1

0

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101 Period (s)

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Mechanical Design In an inertial sensor, the mass needs to be constrained to move in one direction only, with a restoring force that is both exactly linear in the mass displacement and of whatever amount needed for the sensor to have the desired free period. The free period enters into the design partly because of Eq. 2: the longer the period, the greater the displacement of the mass for a given acceleration at periods longer than the free period; that is to say, the higher the long-period sensitivity, and the lower the noise. Sensors designed to measure only high frequencies, such as the geophones used in geophysical exploration, can have a short free period, which means that the springs used can be relatively stiff (large k) and rugged. Seismometers for high frequencies use elastic elements both to constrain mass motion to a straight line and to provide a linear restoring force (Fig. 2a). The latest innovation for shortperiod instruments is the MEMS (Micro-Electronic Mechanical Systems) sensor, in which the mass-spring system is machined from monolithic silicon and packaged with integral electronics; these are mass-produced for non-seismological applications and so are relatively inexpensive. While not as Seismic Instrumentation, Fig. 2 Mechanical designs for seismometers, shown in cartoon form. (a) is a simple mass on a spring. (b) is the “garden-gate” design used for long-period horizontal instruments; note that the support post is not vertical. (c) shows the geometry of a Lacoste suspension; the restoring force of the spring must be equal to its physical length. (d) shows a vertical sensor that uses a leaf spring (elastica) to supply the restoring force. (e) and (f) show the design of a single sensor in a triaxial seismometer, and how three such sensors are arranged to record orthogonal motions

sensitive as traditional seismometers, they are already useful for measuring strong ground motions. The high sensitivity needed for scientific purposes makes the mechanical design more challenging. If the mass moves vertically, it must be stably suspended against gravity while the restoring force is also kept low. In most long-period seismometers, the mass is constrained by hinges and stops to move along a small part of the circumference of a circle, which for most applications is an adequate approximation to a straight line (departures from this cause cross-coupling (Rodgers 1968; LaCoste 1967)). For sensing horizontal motion, the circle is slightly tilted from a horizontal plane, in which case the hinged mass is a horizontal pendulum (also called a garden-gate suspension, shown in Fig. 2b). The restoring force then comes from gravity and is easily adjusted by tilting the instrument. Long-period sensors that measure in other directions use a spring arranged to give a weak, but still linear, restoring force. One common design is the LaCoste suspension (Fig. 2c), which uses a zero-length spring: a helical spring that exerts a force exactly proportional to its physical length. The

a

b

c

d

e

f

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geometry of the spring and hinged mass is chosen to make the restoring force nearly independent of mass position, giving a very long period. An alternative method uses a single sheet of elastic material, bent to form a leaf spring (Fig. 2d), to create a long-period system with adequate linearity over the actual range of motion. The traditional arrangement of seismometers is an orthogonal pair for horizontal motion, and a single sensor of vertical motion; if only one sensor is used, it is usually a vertical to give the best signal-to-noise ratio for P waves. Many systems now use a “triaxial” arrangement, with three identical systems measuring motion at an angle of 54.73
to the vertical, and at azimuths 120
apart; this also gives three orthogonal directions, namely those of the edges of a cube balanced on a corner (Fig. 2e, f). Any seismometer requires careful design to ensure that the only vibration that takes place is the mass motion, or at least that other modes of vibration (called parasitic resonances) will not produce any signals if excited by ground motion. Long-period seismometers need special materials that do not creep much under load and whose dimensions and elastic constants are insensitive to temperature changes; even then, isolation and active environmental control may be needed to reduce the effects of changes in air temperature, air pressure (which causes varying buoyancy of the mass), and magnetic fields (which affect spring materials). Electronic Design Sensing the mass motion in a seismometer is now almost always done electronically, in one of two ways (Agnew 1986). In the first, electrodynamic method, permanent magnets and coils of wire are arranged so that the motion of the _ where G is the sensor produces an induced voltage V ¼ Gx, generator constant; this voltage is then amplified and recorded. This method is simple and rugged, and provides damping if some of the induced current flows through a resistor. However, because the voltage produced depends on the velocity of the mass, and because the noise in the amplifiers rises with low frequency, electrodynamic sensing cannot resolve ground noise at periods longer than about 10 s. The second method is active sensing of the mass motion. An oscillating voltage is input to a variable capacitor or inductor attached to the mass, which produces an output voltage with the same frequency and amplitude proportional to x; this output voltage is then demodulated to produce V ¼ Sx, where S is the sensitivity. Up to the point of demodulation, the signal is at a frequency (several kHz) at which electronics noise is relatively low, so such displacement sensors can be made sensitive enough to measure motions less than an atomic diameter. This output voltage can be recorded directly, but more often is used in a feedback system: the integral of this voltage (over some frequency band) is used to apply an additional

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force to the mass, usually with a coil and magnet, acting against the mass displacement. Because this force acts against the apparent force of ground acceleration, this is called a force-balance accelerometer. Feedback does not decrease noise, but does have three other merits. First, it can be used to vary the seismometer response much more than mechanical modifications can; for example, a mechanical system with a free period longer than 10 s is difficult to make, but this is easy to create electronically; and the response of the electronic system will be much more stable over time. Second, because the feedback force reduces the mass motion, the mechanical system needs to be linear over a smaller range. Finally, the calibration depends on the relation between voltage and feedback force, and this is very stable over time. Calibration Equation 2 is an example of the transfer function of a seismometer: the relation, as a function of frequency, between the ground motion and the output. In that example, the output was assumed to be mass motion x; a more realistic example would include a displacement sensor that generates a voltage V ¼ Sx followed by a datalogger that digitizes the voltage and produces a number N ¼ AV. The total transfer function of the system, from input Earth displacement to digital output, would then be

o20

ASo2 :  gio þ o2

ð3Þ

Seismometer calibration consists of determining all the parameters in expressions like this one. Some components, such as the datalogger, are calibrated by putting a known signal in and recording the output. For the seismic sensor, this requires producing a known ground motion, which can be done with a shake table before the seismometer is installed. After installation, a seismometer calibration can be checked by putting a signal into the seismometer electronics that will apply force to the mass without otherwise disturbing the output. Calibration information is usually provided in a form like Eq. 3:a leading constant, and the (complex) polynomials that give the response as a function of a frequency. Often the polynomials are specified by giving their roots in the complex plane, referred to as poles and zeros. The frequency response can usually be found quite accurately, though the absolute response to ground motion is usually difficult to estimate to better than 1%. Sometimes mundane aspects of the calibration, such as the actual direction of measurement, can be in error by significant amounts. Installation An ideal seismometer installation should maximize signals of interest whether distant earthquakes or the shaking of a building, while minimizing noise that might hide these signals.

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Any actual installation will be a compromise between this ideal and such practical matters as access and cost. For earthquake signals, putting an instrument deeper below the Earth’s surface will give better results, but if this requires a deep drillhole, a shallower installation may be all that can be done. An extreme example of low-quality installations being accepted for reasons of cost is ocean-bottom seismometers, for which the standard installation is the seismometer sitting on the bottom, sometimes on very soft sediments, however it happens to land after being dropped off a ship. On land, the shallowest installations are for temporary instruments used in geophysical exploration, with seismometers planted on the surface with an attached spike to hold them in place. More permanent installations range from instruments set on surface rock exposures or in shallow holes in soil, to purpose-built deep vaults, caves (if available), and drill-holes. Deeper installations reduce the noise from wind and the range of temperature variation, and even a meter of soil will attenuate daily fluctuations. It is advisable, though not always possible, for instruments to be installed away from noise sources such as roads, machinery, rivers, and trees; in an urban setting borehole installations at depths of a few hundred meters may be the only way to reduce the noise to reasonable levels.

Displacement Seismometry Using Satellites Because the mass motion is proportional to acceleration for motions with periods longer than the seismometer free period, data from inertial seismometers has to be integrated twice to find long-period displacements. This integration can introduce significant error, especially if the response is slightly nonlinear or the sensor tilts. Direct sensing of displacement can now be done using repeated distance measurements between a ground sensor and Earth satellites, since the satellites define an inertial reference frame unaffected by ground motion. Distances between satellites of the Global Positioning System (GPS) and a receiver on the ground can be measured to within less than a millimeter precision, even over short times and in the presence of large accelerations; the actual ground displacement can be found after some processing (Bilich et al. 2008). The accuracy of GPS distance measurements is lower than their precision because of propagation effects, including interference from signals reflected from nearby objects (known as multipath), but it has proved possible to determine horizontal ground displacements to within a few millimeters over time spans up to several minutes and at rates up to 10 Hz. This sensitivity is too low to measure most earthquake signals, but can provide high-quality displacement data for ground motions near large earthquakes; a combination of inertial and GPS sensors is the optimal way to measure strong motion.

Seismic Instrumentation

Deformation Seismometers A final class of seismometers measures, not the displacement at a point, but quantities related to the displacement gradient. For a vector displacement u, the gradient ∇ u is a tensor, which can be decomposed into two parts: a symmetric part, the strain tensor, which describes local deformation; and an antisymmetric part, which is equivalent to a vector that describes local rotation, tilting about horizontal axes and rotating about the vertical axis. Two types of instruments, both known as strainmeters, can measure components of the strain tensor (Agnew 1986). Extensometers measure relative displacement along a line between two end points; this displacement divided by the baseline (the distance between the points) gives the extensional or compressional strain. Extensometers with baselines as short as a few centimeters are installed in boreholes, and ones with baselines up to tens of meters in tunnels. In both of these, the measurement is made relative to a solid length standard. Much longer instruments use optical methods to measure strain over hundreds of meters. The other class of strainmeter is the volumetric type, in which the change in a volume of fluid is found by sensing the displacement of fluid in and out of a container cemented in a borehole: such an instrument measures the volume strain, or dilatation. Measuring the rotation vector requires either a stable direction of reference or some way of measuring rotation directly with respect to inertial space. Until recently, no technology existed that do could this with the low noise levels required. A few instruments measure rates of rotation around a vertical axis using Sagnac interferometers, also called laser gyroscopes (Schreiber et al. 2009). These sense the difference in frequency between two light beams propagating in opposite directions around a closed loop.

Seismic Data Recording Three specialized requirements for seismological dataloggers are: (1) a high dynamic range, to capture all the signals possible; (2) large amounts of storage, since it may be necessary to record for a long time to capture unpredictable events; and (3) accurate absolute timing, to relate seismic-wave travel times to a common system. In all three areas, the progress of electronics has meant rapidly improving performance at ever-lower cost. Many seismic dataloggers use specialized systems to provide 24 bits of range (about 5  107); this is accomplished by oversampling followed by digital filtering. The amount of storage depends on the sample rate; except in limited situations, a rate of about 200 Hz will capture all seismic data without aliasing. A day of 3-component data with this resolution and sample rate comes to just over 50 Mb, an amount that in the last two decades has gone from requiring specialized and

Seismic Microzonation

bulky storage to something easily dealt with. Cross-correlation methods can find time delays to a precision of 0.1 of the sample interval, and so the time of each sample point should be known well: for 200 Hz sampling, this requirement would be 500 ms. This too is a level of accuracy that until recently was not easily attained. It can be reached without much difficulty if radio signals from the GPS satellites are available, though providing it over long times in the absence of such a signal still requires expensive equipment. Lower levels of accuracy can be obtained from other radio signals, and, over intervals of a few days, from inexpensive crystal oscillators. Often, a single system records data telemetered from multiple sensors, forming an array (if the region covered spans only a few wavelengths of the waves being recorded) or network (if larger). The most precise timing then requires corrections for the transmission time of the data (latency) unless a separate datalogger is used for each sensor.

Summary With over a century of development, inertial seismic sensors are a mature technology, usually capable of recording ground motion much better than it can be modeled. While no single sensor can cover the full range of amplitudes and periods of seismic waves, only a few instruments are needed to provide a faithful record of ground motion. Since other developments in electronics have largely routinized digital recording of seismic data, most users treat seismometers as a “black box” system that can be acquired and used with little specific expertise: a usually justifiable assumption, though as always it is important to know enough to recognize poor performance when it does occur.

1425 Bilich A, Cassidy JF, Larson KM (2008) GPS seismology: application to the 2002 MW 7.9 Denali fault earthquake. Bull Seismol Soc Am 98:593–606 Bormann P (ed) (2009) New manual of seismological observatory practice. http://www.iaspei.org/projects/NMSOP.html Evans JR, Hamstra RH, Kundig C, Camina P, Rogers JA (2005) TREMOR: a wireless MEMS accelerograph for dense arrays. Earthq Spectra 21:91–124 Havskov J, Alguacil G (2004) Instrumentation in earthquake seismology. Springer, Dordrecht LaCoste LJB (1967) Measurement of gravity at sea and in the air. Rev Geophys 5:477–526 Ringler AT, Hutt CR (2009) Self-noise models of seismic instruments. EOS Trans Am Geophys Union Fall Meet Suppl 90:S23A–1736 Rodgers PW (1968) The response of the horizontal pendulum seismometer to Rayleigh and love waves, tilt, and free oscillations of the earth. Bull Seismol Soc Am 58:1384–1406 Rodgers PW (1994) Self-noise spectra for 34 common electromagnetic seismometer/preamplifier pairs. Bull Seismol Soc Am 84:222–229 Schreiber KU, Hautmann JN, Velikoseltsev A, Wassermann J, Igel H, Otero J, Vernon F, Wells JPR (2009) Ring laser measurements of ground rotations for seismology. Bull Seismol Soc Am 99:1190–1198 Wielandt E (2002) Seismometry. In: Lee WHK (ed) International earthquake and engineering seismology Part A. Elsevier Academic Press, New York, pp 283–304

Seismic Microzonation Fumio Yamazaki1 and Yoshihisa Maruyama2 1 National Research Institute for Earth Science and Disaster Resilience, Tsukuba, Ibaraki, Japan 2 Graduate School of Engineering, Chiba University, Inage-ku, Chiba, Japan

Definition Cross-References ▶ Earthquakes, Location Techniques ▶ Earthquakes, Strong-Ground Motion ▶ Free Oscillations of the Earth ▶ Gravimeters ▶ Seismic Noise ▶ Seismogram Interpretation ▶ Seismology, Rotational

Seismic microzonation

The mapping of an area on the basis of various factors that can affect the intensity of ground shaking, such as seismic hazard, geological conditions, and topographical features, so as to account for the effects of local conditions on earthquake-induced damage.

Introduction Bibliography Agnew DC (1986) Strainmeters and tiltmeters. Rev Geophys 24:579–624 Aki K, Richards PG (2002) Quantitative seismology. University Science Books, Sausalito Berger J, Davis P, Ekström G (2004) Ambient earth noise: a survey of the global seismographic network. J Geophys Res 109:B11307. https:// doi.org/10.1029/2004JB003408

Local site conditions affect the intensity of ground shaking, and as a consequence, the extent of earthquake-induced damage. The amplitude, frequency content, and duration of strong ground motion are significantly influenced by local site conditions. A well-known example is the 1985 Mexico City earthquake. Although the fault rupture of the earthquake was about 350 km away from Mexico City, the city sustained

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catastrophic damage due to the strong amplification of the ground motion by soft soil deposits (Seed et al. 1988). The 1989 Loma Prieta earthquake caused extensive damage in the San Francisco Bay Area. The San Francisco Bay mud significantly influenced the amplitude, frequency content, and duration of ground shaking and resulted in the collapse of the northern portion of the I-880 Cypress Viaduct (Earthquake Engineering Research Institute 1990; Kramer 1996). Seismic microzonation provides the basis for site-specific risk analysis, which can assist in the mitigation of earthquake-induced damage.

Methodology Seismic microzonation typically involves the mapping of predominant periods, soil amplification factors, topographical conditions, liquefaction susceptibility, etc. To draft microzonation maps for a particular region, various data such as existing geological maps, borehole survey data, seismic observation data, and microtremor observation data are collected. Since seismic microzonation entails spatial classification of soil conditions in a small area (e.g., a city), geological data are required for not just a single location but for many locations. In this regard, geological classification maps are most often used as one of the data sources. However, to classify the target area in a more quantitative manner, actual soil profiles obtained from borehole survey data or seismic observation data are better sources. Unfortunately, in most cases, the borehole survey data and/or seismic observation data available for a small area are insufficient. Thus, microtremor observation data have emerged as a popular source for dense spatial information on site amplification characteristics. Three examples of seismic microzonation are described hereafter. Example 1: Seismic Microzonation Based on Geomorphological Classification Maps Several seismic microzonation studies in Japan have employed geomorphological and geological data from the Digital National Land Information (DNLI), which is a geospatial information system (GIS) database that covers the whole of Japan with a 1  1 km mesh, to estimate site amplification characteristics (Matsuoka and Midorikawa 1995; Fukuwa et al. 1998; Yamazaki et al. 2000). Wakamatsu et al. (2004) drafted the Japan Engineering Geomorphologic Classification Map (JEGM) on the basis of the analysis of local geomorphological features at scales of 1:50,000, and all the attributes were digitized and stored in a GIS database. Wakamatsu and Matsuoka (2013) extended the JEGM to 250  250 m grid cells that were categorized into 24 classes on the basis of geomorphological characteristics. The shear-wave velocity averaged over the upper 30 m (Vs30) is often used as a simplified index of site conditions

Seismic Microzonation

(Building Seismic Safety Council 2003). Region-wide site condition maps for California were constructed on the basis of Vs30 and the classification of geological units (Wills et al. 2000). The Next Generation of Ground-Motion Attenuation Models (NGA) project was launched in an attempt to collect all publicly available site condition information at strong motion stations. Vs30 is used in the absence of site condition information (Chiou et al. 2008). Matsuoka et al. (2006) constructed a nationwide Vs30 distribution map using the nationwide shear-wave velocity datasets for Japan, which were obtained from 1000 K-NET and 500 KiK-net seismic stations and the JEGM. The National Research Institute for Earth Science and Disaster Resilience (NIED), Japan, has developed an open web system that interactively provides seismic hazard maps for Japan; this system is called the Japan Seismic Hazard Information Station (J-SHIS) (Fujiwara et al. 2006). J-SHIS uses the JEGM and Vs30 distribution map to draw probabilistic seismic hazard maps for the whole of Japan made by the Headquarters of Earthquake Research Promotion, Japan (Fig. 1). Example 2: Seismic Microzonation Based on Dense Borehole Data and GIS Since 2001, the Tokyo Gas Co., Ltd. has been operating the Super-Dense Real-time Monitoring of Earthquakes (SUPREME) system, having about 4000 seismometers (SI-sensors), in order to control natural gas supply soon after the occurrence of earthquakes (Shimizu et al. 2006). This system employs a GIS to interpolate the monitored spectral intensity (SI) values by using subsoil data from 60,000 boreholes. The digitized borehole data specify the location, depths of soil layers, classification of subsurface soil, standard penetration test (SPT) blow counts, surface elevation, and elevation of the ground water table. Thus, microzonation of the area on the basis of individual borehole data is possible. Shear-wave velocities are estimated from an empirical relationship by using the SPT-N values; then, the average shear-wave velocities in the top 20 m of soil at a borehole site are used to estimate the amplification factors of the SI values (Fig. 2). The accuracy of seismic microzonation can be confirmed after several years of operating a dense seismic network by evaluating the seismic records obtained for moderate to small earthquake events occurring in that period. The SUPREME system was activated during the 2011 Great East Japan Earthquake, and Tokyo Gas confirmed its effectiveness in emergency response after an earthquake (Inomata and Norito 2012). Example 3: Seismic Microzonation Based on Microtremor Measurements Microtremor measurements have emerged as a popular tool for determining the dynamic properties of soil layers and,

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Seismic Microzonation, Fig. 1 Vs30 distribution map for the whole of Japan

Seismic Microzonation, Fig. 2 Site amplification map of Tokyo and surrounding areas, developed using dense borehole data

S hence, are being widely employed for seismic microzonation. In this method, ambient vibrations (of the order of microns) on the earth’s surface are measured. The main sources of these vibrations are traffic and industrial and human activities (Kanai 1983; Lermo and Chavez-Garcia 1994). Microtremor measurements can be used to determine the predominant period of vibrations at a site. Nakamura (1989) proposed the horizontal-to-vertical (H/V) spectral ratio method, in which the predominant periods of ground vibrations are determined from the ratio of horizontal and vertical Fourier spectra of the microtremors recorded at a site. Konno and Ohmachi (1998) drafted a map of fundamental periods and amplification

factors for the 23 wards of Tokyo on the basis of microtremor measurements carried out at 546 stations. Tuladhar et al. (2004) drew a seismic microzonation map for the greater Bangkok area, Thailand, on the basis of microtremor observations carried out at 150 sites. The predominant periods of these sites were obtained by using the H/V method. The estimated predominant periods were validated by comparing them with the transfer functions obtained from onedimensional wave-propagation analysis conducted at eight sites. According to the variation in the predominant period of the ground, the greater Bangkok area was classified into four zones as follows: Zone I (period less than 0.4 s), Zone II

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Seismic Microzonation, Fig. 3 Microzonation of greater Bangkok area on the basis of variation in predominant period

(0.4–0.6 s), Zone III (0.6–0.8 s), and Zone IV (longer than 0.8 s). Figure 3 illustrates the microzonation of the greater Bangkok area on the basis of variation in the predominant period. Based on microtremor array measurement, Jirasakjamroonsri et al. (2019) also evaluated site response characteristics of soft soil deposits in the Bangkok Metropolitan region using the horizontal-to-vertical (H/V) spectral ratio method and the spatial autocorrelation (SPAC) method. Spatial variations of the predominant period and shear-wave velocity profile are presented in a seismic microzonation map.

Summary The objectives and methodologies to perform seismic microzonation are described and some examples are presented. The three major methods introduced to achieve seismic microzonation are the uses of geomorphological classification maps, dense borehole datasets, and microtremor measurements. The results of seismic microzonation are compiled for a GIS to draft microzonation maps, and they can be used to predict ground motions during disastrous earthquakes and thus can assist in the mitigation of earthquake-induced damage.

Cross-References ▶ Earthquakes, Intensity ▶ Earthquakes, Strong-Ground Motion

▶ Seismic Hazard ▶ Seismic Zonation ▶ Seismicity, Intraplate ▶ Seismology, Global Earthquake Model

Bibliography Building Seismic Safety Council (BSSC) (2003) The 2003 HEHRP recommended provisions for the development of seismic regulations for new buildings and other structures. FEMA, Washington, DC Chiou B, Darragh R, Gregor N, Silve W (2008) NGA project strongmotion database. Earthquake Spectra 24(1):23–44 Earthquake Engineering Research Institute (1990) Loma Prieta earthquake reconnaissance report. Earthquake Spectra 6:1–448 Fujiwara H, Kawai S, Aoi S, Morikawa N, Senna S, Kobayashi K, Ishii T, Okumura T, Hayakawa Y (2006) National seismic hazard maps of Japan. Bull Earthq Res Inst Univ Tokyo 81:221–231 Fukuwa N, Arakawa M, Nishizaka R (1998) Estimation of site amplification factor using digital national land information. J Struct Eng 44(B):77–84. (in Japanese) Inomata W, Norito Y (2012) Result of SUPREME (Super-Dense Real time Monitoring Earthquake system for city gas supply) in the great east Japan earthquake. In: Proceedings of the international symposium on engineering lessons learned from the 2011 great east Japan earthquake. pp 1508–1513 Jirasakjamroonsri A, Poovarodom N, Warnitchai P (2019) Seismic site characteristics of shallow sediments in the Bangkok metropolitan region, and their inherent relations. Bull Eng Geol Environ 78(3):1327–1343 Kanai K (1983) Engineering seismology. University of Tokyo Press, Tokyo, pp 128–139 Konno K, Ohmachi T (1998) Ground-motion characteristics estimated from spectral ratio between horizontal and vertical components of microtremor. Bull Seismol Soc Am 88(1):228–241

Seismic Monitoring of Nuclear Explosions Kramer SL (1996) Geotechnical earthquake engineering. Prentice Hall, Upper Saddle River Lermo J, Chavez-Garcia FJ (1994) Are microtremors useful in site response evaluation? Bull Seismol Soc Am 84(5):1350–1364 Matsuoka M, Midorikawa S (1995) GIS-based integrated seismic hazard mapping for a large metropolitan area. In: Proceedings of the fifth international conference on seismic zonation, II. pp 1334–1341 Matsuoka M, Wakamatsu K, Fujimoto K, Midorikawa S (2006) Average shear-wave velocity mapping using japan engineering geomorphologic classification map. J Struct Eng Earthq Eng Japan Soc Civ Eng 23(1):57s–68s Nakamura Y (1989) A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface. Q Rep Railw Res Inst 30:25–33 Seed HB, Romo MP, Sun JI, Jaime A, Lysmer J (1988) The Mexico earthquake of September 19, 1985-relationships between soil conditions and earthquake ground motions. Earthquake Spectra 4:687–729 Shimizu Y, Yamazaki F, Yasuda S, Towhata I, Suzuki T, Isoyama R, Ishida E, Suetomi I, Koganemaru K, Nakayama W (2006) Development of real-time control system for urban gas supply network. J Geotech Geoenviron 132(2):237–249 Tuladhar R, Yamazaki F, Warnitchai P, Saita J (2004) Seismic microzonation of the greater Bangkok area using microtremor observations. Earthq Eng Struct Dyn 33(2):211–225 Wakamatsu K, Matsuoka M (2013) Nationwide 7.5-arc-second Japan engineering geomorphologic classification map and Vs30 zoning. J Disaster Res 8(5):904–911 Wakamatsu K, Matsuoka M, Hasegawa K, Kubo S, Sugiura M (2004) GIS-based engineering geomorphologic map for nationwide hazard assessment. In: Proceedings of the 11th international conference on soil dynamics & earthquake engineering and 3rd international conference on earthquake geotechnical engineering, vol 1. pp 879–886 Wills CJ, Petersen M, Bryant WA, Reichle M, Saucedo GJ, Tan S, Taylor G, Treiman J (2000) A site-conditions map for California based on geology and shear-wave velocity. Bull Seismol Soc Am 90(6B):S187–S208 Yamazaki F, Wakamatsu K, Onishi J, Shabestari KT (2000) Relationship between geomorphological land classification and site amplification ratio based on JMA strong motion records. Soil Dyn Earthq Eng 19(1):41–53

Seismic Monitoring of Nuclear Explosions Paul G. Richards1, Zhongliang Wu2, Won-Young Kim1 and David P. Schaff1 1 Lamont-Doherty Earth Observatory, Columbia University, Palisades, NY, USA 2 Institute of Earthquake Forecasting, China Earthquake Administration, Beijing, China

Acronyms CTBT

CTBTO DPRK

Comprehensive Test Ban Treaty or Comprehensive Nuclear-Test-Ban Treaty (its formal name) CTBT Organization Democratic People’s Republic of Korea

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IDC IMS LTBT OSI TTBT UNE

International Data Centre (of the CTBTO) International Monitoring System (of the CTBTO) Limited Test Ban Treaty On-site inspection Threshold Test Ban Treaty Underground Nuclear Explosion

Definition We take this title to mean the methods of acquiring records of ground motion (seismograms) and of analyzing them for purposes of detecting and identifying those seismic signals that originate from a nuclear explosion and the work of characterizing that explosion.

Introduction The original development of nuclear weapons, and their first use in 1945, was followed by several decades of further weapon development in which more than two thousand nuclear test explosions were conducted. About 500 of these were carried out in the atmosphere, mostly in the 1950s and 1960s. They generated radioactive fallout that was detected worldwide with some regional concentrations and aroused widespread public opposition to nuclear testing. A few nuclear tests were carried out underwater and in space. Most nuclear test explosions, about 1500, were conducted underground in ways that greatly reduced fallout – the first of them in 1957, in Nevada, USA – generating signals that have been intensively studied by seismologists. Hundreds of these individual nuclear tests consisted of multiple nuclear devices, exploded almost simultaneously. A ban on nuclear testing in the atmosphere, underwater, or in space was negotiated and went into effect in 1963 between the USA, the USSR, and the UK. Known as the Limited Test Ban Treaty (LTBT), it has since been ratified or acceded to by more than a hundred countries. Though France and China did not sign, and China carried on with nuclear testing in the atmosphere up to 1980, eventually both these countries came to abide by its terms. The concept of a Comprehensive Test Ban Treaty (CTBT) emerged in the 1950s, intended as a restraint upon nuclear weapon development. It was debated in many forums for more than 40 years and finalized in terms of specific treaty text in September 1996. But this treaty is not in effect (as of 2019), due to continuing debate in specific countries that have not ratified this treaty and whose ratification is needed as a condition for the CTBT to enter into force. They include the Democratic People’s Republic of Korea (DPRK), India, and Pakistan (not signed or ratified) and China, Israel, and the USA (signed but not ratified). Those countries that have signed the treaty are effectively adhering to a moratorium on

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nuclear testing. They include the five countries recognized as nuclear weapon states by the Non-Proliferation Treaty which went into effect in 1970. Listing them in the order in which they acquired nuclear weapons capability, these are the USA, the USSR (whose CTBT obligations have been assumed by Russia), the UK, France, and China. The two countries that by far have conducted the most nuclear test explosions – the USA with 51% of the world total and the USSR/Russia with 35% – ended nuclear testing in the early 1990s (see Yang et al. 2003, for lists of nuclear explosions conducted in the twentieth century). India and Pakistan carried out nuclear explosive tests in May 1998. To date (November 2019), only the DPRK has conducted such tests in the present century – six of them, in October 2006, May 2009, February 2013, January and September 2016, and September 2017. Seismic monitoring of nuclear explosions has been an important activity ever since the first nuclear test in July 1945 in New Mexico. Such monitoring is driven by two different objectives that have engaged a range of different institutions and organizations. The first objective, which dominated for the early decades of nuclear testing up to the early 1990s when nuclear explosions were being conducted on average about once a week, was to acquire basic information about military weapons being tested, especially if (from the point of view of the monitoring organization) the tests were being carried out by a potential adversary. Relevant questions were: What countries had nuclear weapon programs, developed to the level of carrying out nuclear explosive tests? And how big were these explosions? The second objective has been in the context of a major initiative in nuclear arms control, namely, to achieve confidence in the capability to monitor compliance with a CTBT, recognizing that many countries – considering whether or not to support such a treaty and to be bound by its terms – would need to have confidence in the monitoring system to some adequate degree. Given that monitoring cannot be done all the way down to zero yield, evaluation of progress toward this second objective entails questions such as: Down to what small size can nuclear explosions be detected, located, identified, and attributed with high confidence? And what are the specific capabilities of different types of monitoring program, applied to different parts of the world, to catch evidence of a nuclear test, should one occur? Seismology is the most effective technology for monitoring nuclear tests carried out underground, which is the one environment that was not covered by the LTBT and which is also the hardest environment to monitor. The importance of achieving the two objectives stated above has shaped modern seismology itself, in that much of the funding that has led to the facilities and bodies of knowledge now used widely in seismological research, including studies of seismic hazard, was stimulated by government programs intended to improve capabilities for seismic monitoring of nuclear

Seismic Monitoring of Nuclear Explosions

explosions. These facilities and methods include high-quality ▶ “Seismic Instrumentation”, global networks that monitor for earthquakes as well as explosions, quantitative methods of characterizing seismic sources (various magnitude scales, the moment tensor), theoretical understanding of seismic wave propagation in Earth models of increasing and more realistic complexity, our knowledge of the Earth’s internal structure, and methods of seismic signal detection and interpretation. The technical capability to monitor explosions, or a perceived lack of such capability, has played a role in the development of policy options on weapons testing and/or arms control and the content of international treaties. A key technical question arising in debates has been down to what value of yield can monitoring be accomplished – and with what level of confidence? Some seismologists claim now that there is no fundamental technical problem with monitoring explosions down to 1 kt, even if determined efforts at evasion must be considered. But there have been assertions that it is possible to muffle and thus hide (or confuse the procedures for identifying) the seismic signal even from a substantial underground explosion at the level of 10 kt or more. These latter assertions do not appear plausible after review of the technical difficulties (NAS 2012); but, as assertions, one finds that they continue to survive. Seismic monitoring for underground nuclear explosions must be done with recognition of the great variety and number of earthquakes, chemical explosions, and other nonnuclear phenomena that generate seismic signals every day. Efforts to sort out and identify signals from underground nuclear explosions in the midst of signals from these other phenomena have made great progress since they commenced in the 1950s, and improvements in monitoring capability will surely continue to be made. Sections below describe basic properties of earthquake and explosion signals and different steps in seismic monitoring for nuclear explosions. A review is given of methods used for decades in the era when thousands of kilometers separated nuclear weapon testing activity and monitoring stations (“teleseismic monitoring”), when nuclear weapon testing was commonplace and there was little incentive to hide testing activity. Descriptions are then given of modern methods (“regional monitoring”) that monitor for very small explosions and the possibility of tests conducted in ways intended to evade discovery. Examples are given of special events that were important in developing effective and in some cases new discriminants; and finally a brief summary is given of monitoring capabilities, as of 2019, emphasizing the utility of data and data products from the International Monitoring System and its associated International Data Centre that are operated today by the Preparatory Commission of the CTBT Organization, headquartered in Vienna, Austria (www.ctbto.org).

Seismic Monitoring of Nuclear Explosions

Basic Properties of Earthquake and Explosion Signals Seismic monitoring for underground nuclear explosions has to face the reality of hundreds of earthquakes, chemical explosions, and other nonnuclear phenomena, generating seismic signals daily that will be recorded at multiple stations by any effective monitoring network. But after decades of effort, an extensive infrastructure of national and international agencies now sorts out and identifies the signals from earthquakes, chemical explosions, and the occasional underground nuclear explosion. Modern methods of nuclear explosion monitoring are vastly more capable than they were when this work began in the late 1950s. The improvements have mostly been steady as data quality and quantity from monitoring networks increased but with occasional jumps in capability as new types of analysis have been developed, validated, and eventually applied by different agencies. Seismic signals are traditionally grouped into teleseismic waves and regional waves, depending on the distance at which they are observed. Teleseismic waves propagate either as ▶ “Body Waves” through the Earth’s deep interior, emerging with periods typically in the range 0.2–10 s at distances greater than about 1500 km, or as ▶ “Surface Waves”, analogous to the ripples on the surface of a pond, with periods of about 15–100 s. Teleseismic waves were the basis of most US monitoring of foreign nuclear tests prior to 1987. Teleseismic body waves are further subdivided into P-waves and S-waves. (P-waves, which are the fastest-traveling seismic waves and are therefore the first to arrive, are excited efficiently by explosions. Earthquakes tend to excite S-waves and surface waves more efficiently.) For small (subkiloton) explosions, teleseismic signals can be too weak for detection at distant stations, and monitoring then requires regional signals. Regional seismic waves are of several types, including P-waves and S-waves, all propagating only at shallow depths (less than 100 km below the Earth’s surface) with periods as short as 0.05 s (frequencies as high as 20 Hz, i.e., cycles per second). Regional waves reach distances up to 1500 km and sometimes beyond, depending on source size and whether the propagation path is an attenuating one or not. They are regional also in the sense that they have speeds and attenuation properties that vary according to details of local structures in the Earth’s crust and uppermost mantle – so they can vary from place to place within continents and oceans. Figure 1 shows a regional seismogram of a Soviet underground nuclear explosion in Kazakhstan recorded in July 1989 at a distance of slightly less than 1,000 km by a highquality seismographic station in northwestern China. The original recording is shown in red. Different signals derived from it are shown in blue, each of them filtered to pass information in a particular band of frequencies.

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Seismologists characterize the size of seismic signals by means of logarithmic magnitude scales (see Earthquake, Magnitude), with each scale based on a different type of seismic wave. A magnitude scale using teleseismic surface waves was first described in the 1930s based on the logarithm (to the base 10) of amplitude of maximum ground displacement due to surface waves with periods about 20 s. It is known as the Ms scale. Another widely used magnitude scale is that based on the amplitude of teleseismic P-waves. Known as mb, it entails measurement of ground motion at about 1 s period. As part of the assignation of Ms and mb values, for a particular seismic event as recorded at a particular station, a standard correction is applied to account for the distance between the source and the receiver at which the data was obtained. Magnitudes range from about –3 for the smallest observable microearthquakes up to above 9 for the largest earthquake. A 1 kt underground explosion has an mb roughly about 4, and each year there are about 7500 shallow earthquakes worldwide with mb > 4 (Ringdal 1985). Although use of seismic moment has superseded use of mb and Ms in much of modern seismology, and magnitude is only an empirical estimator of seismic event size, magnitude scales are still often used in discussion of seismic monitoring because this is a practical way to relate that discussion directly to properties of signal strength. For example, monitoring capability is often characterized in terms of contour maps or shaded maps indicating the magnitude levels down to which detection or identification is deemed possible with given resources, such as signals from a particular network. We conclude this article with such a map (see Fig. 8). Explosion energy is measured in kilotons. A kiloton is formally defined as a trillion calories and is roughly the energy released by exploding a thousand tons of TNT.

The Different Steps in Explosion Monitoring Nuclear explosion monitoring entails a series of steps, beginning with detection of signals (did a particular station detect anything?) and association (can we gather all the different signals, recorded by different stations, that originate from the same “event”?). The next steps involve making a location estimate and an identification (did it have the characteristics of an earthquake, a mining blast, a nuclear weapon test?). Then follow the steps of yield estimation (how big was it?) and attribution (if it was a nuclear test, what country carried it out?). Once the CTBT is formally in effect, the possibility arises of conducting an on-site inspection (OSI) on the territory of a CTBT signatory state, at and near the location where objective evidence indicates that a nuclear test has occurred. Extensive rules govern procedures for making an OSI request, for its consideration and possible approval by a 51-member Executive Council of the CTBO, for its conduct if approved

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Seismic Monitoring of Nuclear Explosions Pn

Surface waves

Filtered seismograms, all derived from the original broadband record

Raw broadband

20–100 s 0.01–0.05 Hz Long-period surface waves 10–20 s 0.05–0.1 Hz

2–10 s 0.1–0.5 Hz Lg-waves, trapped in the crust 0.5–1 Hz

1–5 Hz High-frequency P-waves 5–10 Hz

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Seismic Monitoring of Nuclear Explosions, Fig. 1 The verticalcomponent seismogram recorded at station WMQ in northwestern China, for an underground nuclear explosion on July 8, 1989, in Kazakhstan at a distance of almost 1,000 km, is shown in red (top). Filtered versions of the original trace in different frequency bands are shown in blue. Time in seconds at bottom is with respect to the time the explosion occurred. Different types of seismic wave propagate at different frequencies, and hence their ground motions show up in different bands. P-waves, in this case the regional wave called Pn which travels in the uppermost mantle, arrive about 120 s after the explosion at this distance,

involving short-period (high-frequency) motions. Long-period surface waves can be seen in the top two blue traces. Some surface waves arrive up to 600 s after the explosion at this distance and thus travel as much as five times slower than P-waves. S-waves (weak in this example) are shear waves, traveling slower than P waves. A high-frequency wave marked as Lg, which is often the largest wave at regional distances from an earthquake but is only weakly excited by explosions, is dominated by shearing motions and is largely trapped in the Earth’s crust. The amplitude of ground motion in the longest period band is less than 2% of the amplitude in the short period band from 1 to 5 Hz

(at least 30 votes in favor are required), and for consequential actions if an OSI supports the accusation of a treaty violation. OSIs can involve seismic monitoring for explosion-induced aftershocks, which (from experience in the USA and the former Soviet Union) are known to occur after some underground nuclear test explosions.

In making assessments of detection capability, one of the key concepts widely used in seismology is the magnitude threshold, above which 90% of the seismic events can be detected at more than three stations (the least number of stations for making routine estimates of the location of the event). Transferring from magnitude to yield, one infers the capability for detecting nuclear tests above a certain size (NAS 2002; NAS 2012).

Detection Concerning detection, nuclear explosion monitoring is often done with arrays of sensors, deployed as a group spread out over an area roughly 10 km across, that facilitate methods to enhance signal-to-noise ratios. This is done typically by stacking signals from independent sensors, often with appropriate delays to increase signal strength and reduce noise. Array data can also give estimates of the direction from which signals are arriving.

Association Association is the effort to identify those sets of detected signals, from different stations, which all originate from the same seismic event. It is one of the hardest steps in practice, particularly when multiple seismic sources around the world are active at roughly the same time (within minutes), resulting in signals from different events that are interlaced in the

Seismic Monitoring of Nuclear Explosions

waveforms recorded by each station. In such cases, array data can be helpful in resolving which signals correspond to which event. Location To obtain a location estimate, typically the arrival times of various seismic waves are measured from the recorded waveforms such as shown in Fig. 1. They are used to find four parameters: the latitude, longitude, depth, and origin time. In this work, it is necessary to know the travel time from any hypothesized source location to any particular seismographic station for any type of seismic wave that the station might observe. In practice, locating seismic events accurately on a global basis (say, to within 10 km of their true location) using networks with stations several hundred km apart requires extensive efforts in station calibration. Thus, it is important to include path-specific travel time corrections to standard travel time models to account for lateral variations of Earth structure (Murphy et al. 2005; Myers et al. 2010). Many authors have shown that greatly improved precision of location estimates can be achieved for a given region if seismic events are located relative to each other and in large numbers – preferably thousands of them or more, all at the same time – rather than one at a time (Waldhauser and Schaff 2008; Schaff et al. 2018b). Methods of Identification Identification of the nature of a seismic source on the basis of its seismic signals – that is, making a determination from seismograms as to whether it could be a nuclear explosion, or a natural earthquake, or a mine blast, or something more exotic such as a bolide impacting our planet and exploding in the atmosphere – is a large subject in view of the many possibilities. See, for example, Richards (1988), OTA (1988), Dahlman et al. (2009), and Bowers and Selby (2009). Seismic events generate many different types of seismic wave, in various different frequency bands as shown in Fig. 1, and different types of seismic source generate a different mix of seismic waves. We can make an analogy here with sound waves and the capability of the human ear and brain to analyze them. A deep bass voice, a gunshot, a whistle, and rolling thunder constitute a set of sound sources that are easily distinguished from each other on the basis of their different frequencies, their emergent or impulsive nature, and their duration. It is the mix of information in both the time domain and the frequency domain that is effective. Seismic methods for discriminating between earthquakes and explosions are based on interpretation of the event location (including its depth); on the relative excitation of a variety of body waves and surface waves; and on features of the frequency spectrum for signals associated with each of these two different types of source. Within these three broad categories, many different methods have been tried, with various degrees of success. As the capabilities of each method

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are probed, the question of interest is often: “Down to what size of seismic event does this method of discrimination work?” In some cases discrimination is unambiguous even at very small event size. (For example, however small an event, it may be presumed to be an earthquake if it is located at a depth greater than 15 km below the Earth’s surface. And, even a small event will attract attention, if it occurs in an area that is geologically stable and that for decades has had no seismic activity.) The most useful methods for discrimination can be listed as follows: • Interpretation of the location. Is the event in a seismic or an aseismic area? Below the floor of an ocean? At depth below a continent? There is an important role here for common sense: seismic events in Canada attract less attention from western monitoring agencies than such events in the DPRK (though a seismic event in the Canadian Shield would still attract attention and intensive study). • Relative amplitude of body waves and surface waves. This can be studied by plotting the event of interest on an Ms: mb diagram, as shown in Fig. 2. The surface-wave amplitude is read typically on the vertical component from signals with period about 20 s and the body-wave amplitude at about 1 s period. (Though effective for large enough events, an explosion with mb much below 4.5 may not have large enough surface-wave signals at teleseismic distances to apply this method dependably.) • Use of the observed “first motion” of the ground. Does the initial P-wave motion of the ground indicate compressions radiated to all directions from the source, leading to upward motions, as would be the case for a simple explosion? Or, are dilatations recorded at some azimuths, leading to downward motions – as would sometimes be expected from earthquakes but not from explosions? The methods described so far in this section have concerned the use of teleseismic signals, which can be used to monitor effectively for high magnitudes and on down to somewhere in the magnitude range from 4.0 to 4.5. Since the late 1980s/early 1990s, there has been growing recognition of the merits of regional waves, to enable monitoring down to far lower magnitudes, often well below magnitude 3. The method is based upon the general observation that explosion signals, when compared to earthquakes, have much stronger P-waves at high frequency, whereas signals from earthquakes have stronger S-waves (and surface waves). This modern method is being studied with frequencies in the range 0.5–20 Hz. and sometimes even higher. An example is shown in Fig. 3 comparing regional seismic signals of a very small earthquake and a small explosion. The method has been demonstrated by seismologists in China, even down to around mb 2 (Pan et al. 2007).

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Seismic Monitoring of Nuclear Explosions, Fig. 2 An Ms: mb diagram from Bowers and Walter (2002). It can be seen here that for seismic events of the same Ms value, earthquakes have a significantly smaller mb value than do the explosions. The offset is about 0.8 mb units, at Ms ¼ 5. Because magnitudes are based on logarithmic scales, and 100.8 ~ 6, it follows that at frequencies near those at which body wave magnitude is measured (about 1 Hz), the P-waves from an underground nuclear explosion are typically about 6 times larger than such waves from an earthquake having the same strength of surface waves. Also, indicated by the red star are the body-wave and surface-wave magnitudes of an interesting but fortunately rare event, a large mine collapse with P-wave magnitude greater than 5. This event, which plots with the explosion population, is discussed further below – see Fig. 6

Seismic Monitoring of Nuclear Explosions, Fig. 3 Typical vertical-component records from an earthquake and an explosion. Traces plotted are unfiltered (top); low-frequency band-pass filtered (middle); and high-frequency band-pass filtered (bottom). (From Kim et al. 1993)

As an important example of this development, Fig. 4 shows the results of an analysis of the P-wave and S-wave spectra, pertinent to identifying the very small underground nuclear explosion conducted by the DPRK on October 9, 2006, and a larger test nearly 3 years later on May 25, 2009. The smaller explosion took place at 0135 hours (GMT), and by 0706 hours the US Geological Survey (USGS) had issued a report based on seismic signals from 20 stations around the world including sites in China, the Republic of Korea (ROK), Russia, Japan, Kazakhstan, Kyrgyzstan, Alaska, and Nevada. Its magnitude, about 4, indicated a subkiloton yield (see Koper et al. 2008, who discuss the uncertainty of estimating yield in view of the variability of seismic signal excitation for shots of different depth). From teleseismic signals, the explosive nature of such a small seismic event can be difficult to recognize (they could be signals from an earthquake). Fortunately, discrimination for events such as this is very clear providing high-quality regional data is available. In this analysis, the original seismograms from station MDJ, located in China, are filtered to pass narrow frequency bands as illustrated in blue in Fig. 1 but this time with bands centered on each of the frequencies from 1, 3, 5, 7, 9, 11, 13, to 15 Hz as indicated for the horizontal axis in Fig. 4. Here the amplitudes of the Pg and Lg waves are measured in each narrow band. The amplitude ratio is formed (the “spectral ratio”), and the quantitative comparison can begin. Figure 4 shows how this ratio varies with frequency for the set of eight earthquakes and for the set of four small chemical explosions. The ratio differs for these two populations as frequency rises, and the separation between them is very clear at the high frequencies (from 9 to 15 Hz in this case). The spectral ratios

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Seismic Monitoring of Nuclear Explosions, Fig. 4 Spectral ratios are shown, for nuclear explosions carried out by the DPRK in 2006 and 2009, as measured from waveforms recorded at station MDJ in China (distance, about 370 km). They are compared with these ratios for a small group of earthquakes and another group of chemical explosions, all in the vicinity of the DPRK’s nuclear test site (about 370 km to the south of MDJ). Colored bars represent  one standard deviation in the ratios for chemical explosions (yellow) and small earthquakes (magenta). The spectral ratios for these two events in the DPRK are both explosion-like

of the signals recorded for the nuclear events of 2006 and 2009 are like those of the known chemical explosions. This successful seismic discriminant based upon regional waves is important in enabling capability to be extended down to lower magnitudes than can be monitored teleseismically. In practice, there is often very little difference between the magnitude thresholds for detection (at enough stations to enable a useful location estimate) and identification, since so many regions of the Earth are now monitored to low magnitude for earthquakes as part of investigations into seismic hazard. It may take only one regional seismogram to enable discrimination to be carried out with high confidence (provided the recording is of adequate quality and is for a station that has an archive of signals from previous known earthquakes and explosions). Along with the use of regional seismic waves and their spectral ratios at 5 Hz and higher, another successful method for distinguishing between earthquakes and explosions is the use of observed seismic waveforms to make estimates of the set of forces that appear to be acting at the seismic source. The set of forces here is quantified by what seismologists call the moment tensor. As shown by Ford et al. (2009) from study of numerous earthquakes and underground explosions, seismic events separate into specific populations as determined by the way their moment tensors behave – whether they are more

representative of the all-around (isotropic) features of an explosion, or of the type of shearing motions more typical of an earthquake. In general for underground tests, seismic data alone cannot distinguish between nuclear explosions and chemical explosions in which all the material making up the explosive was fired within less than about a tenth of a second. But chemical explosions, if large, are very rarely fired in this way. In the case of the DPRK tests of 2006 and 2009, both of which were announced as nuclear, objective evidence for the nuclear nature of the 2006 explosion came from several different detections of radionuclides that are diagnostic of a nuclear explosion. Although such radionuclides were not detected from the 2009 explosion, it was so large as to be implausible as a chemical explosion, since it would have to have consisted of literally thousands of tons of explosive material. We return later in this article to a discussion of additional nuclear tests in the DPRK up to 2017, some of which generated small earthquakes (explosion-induced aftershocks). Yield Estimation Yield estimation was of particular importance in the years following 1974 when a bilateral treaty between the USA and the USSR was negotiated, intended to go into effect in 1976. This was the Threshold Test Ban Treaty (TTBT), limiting the size of underground nuclear explosions conducted by these two countries to a yield of not more than 150 kt. The TTBT proved contentious, with each side sending the other several inquiries asserting that the agreed-upon limits had possibly been exceeded (Timerbaev 2007). But this treaty, and a related treaty placing yield limitations on so-called Peaceful Nuclear Explosions, finally entered into force in 1990. Both treaties have become less important since the CTBT was finalized and a nuclear testing moratorium by the signatory countries began in 1996. Yield estimation is however still important as an exercise in the interpretation of signals from the few underground explosions since that date, specifically those of India and Pakistan in 1998 and of the DPRK in the present century. For several tens of underground nuclear explosions, most of them at the Nevada Test Site in the USA or at Kazakhstan’s Semipalatinsk Test Site when it was part of the Soviet Union, the yield has subsequently been announced by agencies involved in the conduct of the test. It has therefore been possible to calibrate observed seismic magnitudes for these tests against the announced yields, and an example is given in Fig. 5 using mb values and yields reported for Nevada explosions in tuff and rhyolite. The line mb ¼ 4.05 + 0.75 log (yield) fits the data well (yield in kilotons). Such a calibration curve can be applied to obtain a seismic yield estimate for Nevada explosions with unannounced yield. But it requires correction, prior to its use in obtaining a seismic yield estimate for an explosion at a

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different site. This must be done, to allow for physical and geological differences between the sites. For example, in different rock types, there can be different efficiencies in the coupling of nuclear yield into seismic energy and differences in the propagation efficiencies as seismic waves travel out from the source of interest, as compared to seismic signals from a Nevada explosion. In this connection, it is of interest to note the mb value and yield for the US nuclear explosion LONGSHOT (conducted in 1965 in the volcanic breccias of an Aleutian island). The mb value is 5.9, corresponding to a yield of about 300 kt. if the Nevada curve of Fig. 5 is applied directly. But the announced yield for LONGSHOT is 80 kt. One way to obtain a calibration curve for the Aleutians is therefore to add a correction of about 0.4 mb units to the Nevada values of mb at a given yield, before the curve of Fig. 5 is used to supply a seismic yield estimate in this new location. This mb correction, for a site differing from that where a calibration curve is directly available, is called the test site bias. If the bias correction is not applied, then a Nevada magnitude–yield curve can give too high a seismic yield estimate for a non-Nevada explosion. Note that the Nevada Test Site is in a region of active tectonics, with significant episodes of volcanism in the last few million years, resulting in high temperatures within the upper mantle and thus anomalous attenuation of seismic waves propagating through the hot and partially molten upper layers of the Earth, 100 or 200 km in thickness beneath the Nevada Test Site. Such propagation through an attenuating medium is presumed to be a contributing cause of bias. The existence of mb bias has long been known in seismology in connection with what is called “station bias.” By this 6.40 6.20 6.00 5.80 mb

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Seismic Monitoring of Nuclear Explosions, Fig. 5 Seismic magnitude mb vs announced yield, for 17 Nevada Test Site nuclear explosions in tuff and rhyolite. The straight line here, which fits the data quite well, can be used to make a yield estimate of other events at this test site, in similar rock, if the seismic magnitude is known. (Data from Nuttli 1986)

term is meant the systematic difference between mean mb values (obtained for a particular seismic event by averaging reported mb from seismometers all over the globe) and mb reported by just one station. For example, the station BMO in Oregon (another region of active tectonism) has reported mb values that for a given earthquake are typically about 0.3 units below the global average; and station KJN in Finland (in a stable shield region) reports values about 0.15 mb units higher than the average. Their station bias values are thus 0.3 and +0.15, respectively. Station bias values commonly range over  0.3 mb units, so it may be expected that source region bias (which is what must be applied when a standard mb–yield curve is used for different source regions) can also range over about 0.6 mb units. The nuclear weapon test site of the USSR that conducted the most underground nuclear explosions was near the city of Semipalatinsk, in northeastern Kazakhstan. Several multimegaton underground explosions were conducted on Russia’s Novaya Zemlya island test site, far to the north of continental Eurasia (see Khalturin et al. 2005). But these were all prior to the intended date of entry-into-force of the TTBT (March 1976). After that date, the magnitude of the largest underground tests at Semipalatinsk rose higher and higher over several years, with some magnitudes exceeding 6.1. Such magnitudes, according to the Nevada Test Site formula discussed above, mb ¼ 4.05 + 0.75 log (yield), implied yields greater than 500 kt, far in excess of the TTBT limit (150 kt). Intensive discussion in political and technical areas ensued, with stronger and stronger evidence accumulating to indicate a substantial test site bias between the Nevada and Semipalatinsk test sites. For example, it was of great interest that teleseismic signals from the largest underground explosions from these two tests, if recorded at the same station in a shield region, looked significantly different. The teleseismic P-wave from a large underground explosion at the site in Kazakhstan would routinely have frequency content at the 5 Hz level and sometimes higher (Der et al. 1985). The signal from Nevada would not contain such high frequencies. It was as if the signal from Nevada had passed through some type of filter, which of course would reduce its amplitude. Correcting for that effect would mean that the appropriate relation between magnitude and yield for an underground nuclear explosion at Semipalatinsk had the form mb ¼ 4.05 + bias + 0.75 log (Yield), and Ringdal et al. (1992) and Murphy (1996) among many others concluded that the appropriate formula relating teleseismic P-wave magnitude and yield at Semipalatinsk should be this equation with a bias of 0.4. Support for their conclusion came from many arguments (see Richards 1988, for a review). But in the political realm, the most persuasive was the very practical one associated with a Joint Verification Experiment of September 14, 1988, in which a team from the USA at the Semipalatinsk Test Site was allowed to make close-in measurements (within

Seismic Monitoring of Nuclear Explosions

a few tens of meters) of a large Soviet underground nuclear explosion, in particular of the speed and extent of the shock wave it sent out into rock near the source at that test site. From such shock wave measurements, a reliable non-seismic method provided an accurate yield estimate (it was in the range 100–150 kt). Stations around the world provided measurements teleseismically, giving a seismic magnitude around 6.1 – comparable with the largest magnitudes of Semipalatinsk explosions since 1976, indicating that they too had been conducted in a way that respected the 150 kt limit of the TTBT. A reciprocal Joint Verification Experiment had been conducted at the Nevada Test Site, on August 17, 1988, with a Russian team making its own close-in measurements of the shock wave from a large US underground nuclear test intended to be in the range 100–150 kt. Apparently, the yield of this explosion somewhat exceeded 150 kt. Timerbaev (2007) and news reports give it as 180 kt.

Special Events The work of monitoring – for both earthquakes and explosions – is done in practice by hundreds of professionals who process the vast majority of seismic events routinely and who also look out for the occasional events that, in the context of monitoring for the possibility of underground nuclear explosions, exhibit interesting characteristics and which may then become the subject of special study. These special events have stimulated the development of effective new discrimination techniques and a better appreciation of overall monitoring capability. Examples include a mine collapse in 1989 in Germany and two such collapses in 1995, in the Urals (Russia) and in Wyoming (USA); a small earthquake of magnitude 3.5 and its smaller aftershock in 1997 beneath the Kara Sea near Russia’s former nuclear test site on Novaya Zemlya; the series of underground nuclear test explosions carried out by India and Pakistan in 1998; underwater explosions in 2000 associated with the loss of a Russian submarine in the Barents Sea; and a number of small seismic events near the DPRK’s nuclear test site in addition to the six announced nuclear test explosions at that location from 2006 to 2017. The mining collapses were seismically detected all over the world. For example, teleseismic P-waves from the Wyoming event of 1995 were detected in South and North America, Europe, and Asia. Mining collapses such as these have caused concern because their mix of surface waves and body waves as recorded teleseismically can appear explosionlike using the classical Ms: mb discriminant, as shown in Fig. 2 (see above). But a careful analysis of regional and teleseismic waves from these events has showed that although the surface waves were quite weak, and in this respect seemed explosionlike, they had the wrong sign. Therefore the motion at the

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source was implosive (the ground had moved inward toward the source and thus downward), rather than explosive. Indeed, mining collapses are an implosion phenomenon, and it was important to learn that their implosive nature could be reliably determined from seismic recordings. Teleseismic waveforms from the Wyoming mine collapse are shown in Fig. 6. This is an example of the use of what seismologists call the “first motion” of the P-wave, which is clearly downward in these data. The Kara Sea earthquake was too small to apply the Ms: mb discriminant (the surface waves were too small to measure reliably). This event showed the importance of accurate locations and of using spectral ratios of regionally recorded P- and S-waves to discriminate small events (Richards and Kim 1997). As we have discussed earlier, the DPRK nuclear test of 2006 was of interest as an example of a nuclear explosion that was promptly detected globally, though its yield has been estimated as less than 1 kt. This event required regional seismic data in order to determine that indeed an explosion had been carried out and that the signals were not from an Mine collapse signals, converted to broadband seismograms KIEV

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earthquake (Kim and Richards 2007). Subsequently, xenon radionuclides were detected that decisively identified this explosion as nuclear. The DPRK at this time of writing (November 2019) has conducted six nuclear explosive tests, the first (October 9, 2006) with mb about 4 and the last (September 3, 2017) with mb greater than 6. The magnitude of the largest test has been assigned values lying mostly in the range from 6 to 6.3, reflecting the fact that the number of seismographic instruments providing data today is roughly a hundred times the number available in the 1980s and 1990s when magnitude–yield relations were being worked out, and the effects of station bias (for many new stations) are not yet quantified. Of interest as a demonstration of capability to monitor for even small seismic events is that the huge 2017 explosion (with yield around 200 kt) was followed by tens of seismic events that were detected and intensely analyzed by groups in neighboring countries – including China and the Republic of Korea (ROK) – as well as in distant countries and by the Viennabased IDC using IMS data. These events have been identified as small earthquakes, down to magnitude of about 2.2, apparently induced by the very large explosion of 2017 (e.g., Kim et al. 2018; Schaff et al. 2018a). As shown in Fig. 7, thirteen of these small earthquakes occurred from Sept 2017 to April 41.40˚

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Seismic Monitoring of Nuclear Explosions, Fig. 7 A map showing locations for the summit of Mount Mantap (black triangle); the first of the DPRK’s UNEs (small star), to the east and south of this summit; five subsequent UNEs conducted within the mountain (larger stars) including the large test explosion of September 3, 2017; and a series of small aftershocks aligned over several hundred meters, about 8 km to the north of Mount Mantap (red circles). (Adapted from Kim et al. 2018)

2018 close to a northeast–southwest line about 700 m long, lying about 8 km to the north of the main mountain – Mount Mantap – in which most of the DPRK’s test explosions took place. Low-magnitude seismic activity has continued in the region, with events in January and June of 2019. The earthquakes have relative locations determined via analysis of whole waveforms recorded regionally in China and ROK (Schaff et al. 2018b). Their overall location to the north of the test site is determined from work of Tian et al. (2018), who had access to data from numerous seismic stations in China. The overall point here is that the small 2006 nuclear explosion in the DPRK was promptly well-characterized; and in later years it has become possible to detect, locate, and identify seismic events in the region that are a hundred times smaller.

Evasion Several methods have been proposed, by which underground explosions might be concealed. One method is simply to make them small enough; but then there would be relatively little to learn, from the point of view of a weapon designer. The more important methods are those which combine as many features as possible, designed to reduce seismic signalto-noise ratios at all relevant monitoring stations. Proposed methods include emplacement of the nuclear device in material such as dry alluvium, to reduce the coupling of explosion energy into seismic signal (but that method is likely to result in leakage of detectable radioactivity); waiting until a sufficiently large natural earthquake occurs fairly near a test site (which presents the formidable challenge of identifying the event within a few minutes of its occurrence as large enough and then within a couple of minutes executing the weapon test so that its seismic signals would hopefully be swamped by the large and prolonged signals from the earthquake); and setting off a sequence of several explosions that are designed to simulate a natural earthquake signal. Careful study of each of these methods indicates that they are relatively ineffective in comparison with the methods known as cavity decoupling and mine masking, which we next discuss and which are widely regarded as setting the practical levels down to which seismic monitoring of nuclear explosions is possible. When an underground explosive device is tightly packed into its hole (“tamped” or “fully coupled”) and is detonated at sufficient depth to contain all radioactive products, a shock wave travels some distance from the shot point out into the surrounding rock at speeds that exceed the normal P-wave speed. This nonlinear phenomenon reduces at sufficient distance from the shot point, and thereafter the wave propagation can be regarded as elastic. The so-called elastic radius for a tamped explosion, i.e., the radius beyond which wave

Seismic Monitoring of Nuclear Explosions

propagation is linear, is roughly 100 m times the cube root of the yield (in kilotons). If the explosion is set off inside a large underground cavity instead of being tamped, then the shock wave set up in the rock can be weakened or even eliminated, in which case only elastic waves are radiated. The explosion is said to be fully decoupled if only elastic waves result, and theoretical work begun in 1958 has addressed the question of how much weaker the seismic signal might be made. Theoretical work has indicated that signals could thereby be reduced by factors in the range 50–100, compared to a tamped explosion. The cavity radius itself is the “elastic radius” for a fully decoupled shot. For salt, the cavity radius required for full decoupling has been estimated as about 25 m times the cube root of the yield (in kilotons). For hard rock the cavity size for full decoupling is comparable; for weak salt it is somewhat greater. See Sykes (1996) for further discussion and Denny and Goodman (1990) for estimates of the decoupling factor derived from the practical experience in 1966 of carrying out a small nuclear explosion (about 0.38 kt) in the cavity produced by a tamped shot of 5.3 kt conducted 2 years earlier in a Mississippi salt dome. They conclude that the amplitude reduction is about 70, at low frequencies, for salt. At frequencies that have conventionally been used for seismic monitoring, the seismic signal strength is proportional (very roughly) to the volume within the elastic radius. This volume is substantially reduced by fully decoupling, which is the reason why cavity decoupling has been proposed as offering the technical possibility of a clandestine program of nuclear testing. However, the signal strength is not nearly so strongly reduced, by decoupling, at frequencies above that associated with resonances of the internal surface at the elastic radius. In practice, the frequency above which decoupling is likely to be substantially less effective is around 10–20 Hz, divided by the cube root of the yield (in kt). The overall effect on the seismic signals from a fully decoupled shot of yield Y, given the results of Denny and Goodman, is to make these signals look like those from a tamped shot of yield Y/70. A thorough discussion of decoupling as an evasion scenario would have to include several non-seismological considerations. These are reviewed in NAS (2012) and include the military significance of being able to carry out nuclear tests up to various different yield levels (e.g., 0.1, 1, or 10 kt.) and the political consequences if an attempted clandestine test program was uncovered. Technical considerations include methods of (clandestine) cavity construction and the capabilities of non-seismological surveillance techniques. Preventing the leakage of radioactivity from an underground cavity would be a challenge, given that much of the energy of a decoupled explosion goes into pumping up the pressure in the cavity. While some assert that clandestine use of cavity decoupling would be so difficult to execute that it belongs to the realm of fantasy, others have been persuaded that the risk

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might indeed be manageable and that estimates of concealable yields, under this evasion scenario, must be made. The NAS (2002) report describes ten “layers of difficulty” with successfully hiding an underground nuclear explosion via cavity decoupling, concluding that even a nation committing significant resources to this work could not have confidence in being able to get away with tests above 1 or 2 kt. The evasion scenario known as mine masking hypothesizes the execution of an underground nuclear weapon test explosion in a mining region, concurrently with a large mine blast. Such blasts in a big commercial operation consist of hundreds of separate charges, fired in sequence to break and/or move as much rock as possible, in a procedure known as ripple firing (Khalturin et al. 1998). Regardless of the logistical difficulties of such a scenario, estimates of the possibilities of concealment via this approach can come from taking examples of signals from large mine blasts, and signals from small underground nuclear explosions, and then adding them together before subjecting them to the methods used to discriminate between various types of seismic events. What is typically found is that the maximum size of the identifiable waves (e.g., the P-waves) from the mine blast is about that expected from individual sub-blasts (commonly called “delays”), and these amplitudes are spread out over a longer time in seismograms. A study of mine masking possibilities by Smith (1993) used several different examples of mine blast seismograms together with single-fired explosion records and found a number of features that could be used to identify a simultaneous shot within a ripple-fired blast. He concluded that to conceal a single-fired deep detonation (depth is required for containment of radionuclides), the single explosive shot should not exceed 10% of the total explosive. The conclusion here is that mine blasts are not effective for concealing large releases of energy at the level associated with kiloton-scale nuclear weapons tests, unless the nuclear explosion was subject to efforts at decoupling. Again nonseismic considerations arise, including an assessment of the plausibility of carrying out a complicated decoupled nuclear explosion at the same time and location as a large mine blast that would itself attract some level of monitoring attention – particularly if the seismic signals seemed unusual in comparison with those from prior blasting in the region.

Event Detection Capability of the International Monitoring System In 1976, a group of international scientists was established at the Conference on Disarmament in Geneva, for the study of monitoring technologies and data analysis methods in the context of supporting a future test ban treaty. This group of scientific experts (GSE) played an essential role in laying

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Seismic Monitoring of Nuclear Explosions

the scientific groundwork for the final stage of CTBT negotiations conducted from 1994 to 1996. Prior to the final stage, GSE organized a series of technical tests which contributed significantly to the development of the international system being built today to support treaty verification. The text of the CTBT includes a description of networks to monitor treaty compliance using hydroacoustic, infrasound, and radionuclide technologies as well as seismological methods. The CTBT Organization (CTBTO) operates an International Monitoring System, as well as an International Data Centre to analyze signals sent via satellite to headquarters in Vienna. Extensive descriptive material on these networks is available online (see http://www.ctbto.org). To implement the CTBT seismic monitoring system, a sequential four-step process is needed to build each station

(CTBTO PrepComm 2009): (1) site survey; (2) installation; (3) certification; and (4) operation. It must be demonstrated for IMS stations that data received at the International Data Centre (IDC) are authentic. This is achieved through a special digital “signature” embedded in the data flow from each station. The IMS station must be certified to ensure that all of its equipment, infrastructure, and settings meet the technical specifications set by the CTBTO and to ensure also that all data are transmitted to the IDC through the Global Communication Infrastructure (GCI) in a timely manner. Here, we note that the primary seismographic network is to consist of 50 stations, many of them arrays; and that location estimates are based upon detection of signal at three stations or more. An auxiliary network of 120 continuously operating stations is to provide seismic waveform

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Magnitude Seismic Monitoring of Nuclear Explosions, Fig. 8 A map illustrating detection capability of the IMS primary seismographic network. This network is intended to have 50 stations, but it is incomplete as of 2019. Here, is shown the capability of the network, based on practical experience with 43 stations. The capability is represented by the magnitude of the smallest seismic event that would be detected with a 90% probability

by three stations or more. The method used to develop such maps is described by Kværna and Ringdal (2013). A complete IMS network (50 stations) may be expected to lower the illustrated magnitude threshold by about 0.1 magnitude units, in some regions. (Figure, courtesy of T. Kværna.)

Seismic Monitoring of Nuclear Explosions

data, again via satellite, in order to help characterize the events detected by the primary network. Although these two networks are not completely built (as of August 2019, 44 primary and 108 auxiliary stations have been certified), enough stations operate to provide good indications of what the detection capability will be when all stations are installed and providing data. Their performance today, although the networks are incomplete, is considerably better than what was envisaged for the IMS and IDC during the CTBT negotiations of the 1990s. Figure 8 shows a map of the detection capability of the primary seismic network of the IMS, based on practical experience with 43 operating stations. This network is planned eventually to have 50 stations. Capability is expressed in terms of magnitude thresholds, above which 90% of the seismic events are expected to be detected at enough stations to provide a location estimate. The work of identifying events is left to member states. We note that event identification is not just a technical matter since it is a political act for one country to make an allegation that another country has committed a treaty violation. The evidence in support of such an allegation can come from the IMS and IDC, as well as from the National Technical Means of member states, and/or from a subset of the thousands of seismographic stations operated around the world for purposes not directly related to monitoring for nuclear explosions.

Summary We have described the basic steps in monitoring for nuclear explosions and have emphasized the seismic monitoring system specified by the Comprehensive Nuclear-Test-Ban Treaty of 1996. When the treaty was being negotiated, the goal for the International Monitoring System was that it be capable of detecting and identifying treaty violations – nuclear explosive tests – at the 1 kt level and higher, if they are not evasively tested. Recognizing that a 1 kt underground nuclear explosion has a magnitude in the range about 4–4.5, if it is conducted in the way that almost all the more than 1500 prior underground nuclear explosions were carried out (i.e., well-tamped and not with intent to reduce the signals picked up by monitoring networks), the evidence from Fig. 8 is that this design capability has been significantly exceeded. For almost all of the northern hemisphere, including Eurasia and North America, capability is good down to about magnitude 3.3. This corresponds to a yield of less than 100 t (0.1 kt) for a well-tamped explosion in hard rock. Only time will tell whether this capability, combined with other monitoring assets, is deemed adequate to support entry into force of the CTBT.

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Cross-References ▶ Body Waves ▶ Earthquake, Magnitude ▶ Seismic Instrumentation ▶ Seismological Networks ▶ Seismology, Monitoring of CTBT ▶ Surface Waves

Bibliography Bowers D, Selby ND (2009) Forensic seismology and the comprehensive Nuclear-Test-Ban Treaty. Annu Rev Earth Planet Sci 37:209–236 Bowers D, Walter WR (2002) Discriminating between large mine collapses and explosions using teleseismic P-waves. Pure Appl Geophys 159:803–830 CTBTO PrepComm (2009) Verification regime. http://www.ctbto.org/ verification-regime/ Dahlman O, Mykkeltveit S, Haak H (2009) Nuclear Test Ban: converting political visions to reality. Springer, Berlin Denny MD, Goodman MD (1990) A case study of the seismic source function: Salmon and Sterling reevaluated. J Geophys Res 95:19705–19723 Der Z, Mcelfresh T, Wagner R, Burnetti J (1985) Spectral characteristics of P waves from nuclear explosions and yield estimation. Bull Seismol Soc Am 75:379–390. also erratum, 75, 1222–1223 Ford SR, Dreger DS, Walter WR (2009) Identifying isotropic events using a regional moment tensor inversion. J Geophys Res 114: B01306. https://doi.org/10.1029/2008JB005743 Khalturin VI, Rautian TG, Richards PG (1998) The seismic signal strength of chemical explosions. Bull Seismol Soc Am 88:1511–1524 Khalturin VI, Rautian TG, Richards PG, Leith WS (2005) A review of nuclear testing by the Soviet Union at Novaya Zemlya, 1955–1990. Sci Glob Secur 13:1–42 Kim W-Y, Richards PG (2007) North Korean nuclear test: seismic discrimination at low yield. EOS Trans Am Geophys Union 88(14):157–161 Kim W-Y, Simpson DW, Richards PG (1993) Discrimination of Earthquakes and explosions in the Eastern United States using regional high-frequency data. Geophys Res Lett 20:1507–1510 Kim W-Y, Richards PG, Jo EY, Ryoo YG (2018) Identification of seismic events on and near the North Korean test site following the underground nuclear test explosion of 2017 September 3. Seismol Rev Lett 89:2220–2230 Koper KD, Herrmann RB, Benz HM (2008) Overview of open seismic data from the North Korean event of 9 October 2006. Seismol Rev Lett 79:178–185 Kværna T, Ringdal F (2013) Detection Capability of the Seismic Network of the International Monitoring System for the Comprehensive Nuclear-Test-Ban Treaty. Bull of the Seismol Soc Am 103:759– 772. https://doi.org/10.1785/0120120248 Murphy JR (1996) Types of seismic events and their source descriptions. In: Husebye ES, Dainty AM (eds) Monitoring a comprehensive Nuclear Test Ban Treaty, NATO ASI series E, vol 303. Kluwer Academic, Dordrecht, pp 225–245 Murphy JR, Rodi W, Johnson M, Sultanov DD, Bennett TJ, Toksöz MN, Ovtchinnikov V, Barker VW, Reiter DT, Rosca AC, Shchukin Y (2005) Calibration of International Monitoring System (IMS)

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1442 stations in Central and Eastern Asia for improved seismic event location. Bull Seismol Soc Am 95:1535–1560 Myers SC, Begnaud ML, Ballard S, Pasyanos ME, Phillips WS, Ramirez AL, Antolik MS, Hutchenson KD, Dwyer JJ, Rowe CA, Wagner GS (2010) A crust and upper mantle model of Eurasia and North Africa for Pn travel time calculation. Bull Seismol Soc Am 100:640–656 NAS (2002) National Academy of Sciences report, technical issues related to the comprehensive Nuclear-Test-Ban Treaty. National Academy Press, Washington, DC NAS (2012) National academy of sciences report, the comprehensive Nuclear Test Ban Treaty – technical issues for the United States. National Academy Press, Washington, DC Nuttli OW (1986) Yield estimates of Nevada Test Site explosions obtained from seismic Lg waves. J Geophys Res 91:2137–2151 OTA (1988) Office of technology assessment, congress of the United States, seismic verification of nuclear testing treaties, OTA-ISC-361. U. S. Government Printing Office, Washington, DC Pan C-Z, Jin P, Wang H-C (2007) The applicability of P/S amplitude ratios for the discrimination of low magnitude seismic events. Acta Seismol Sin 20:553–561. https://doi.org/10.1007/s11589-0070553-6 Richards PG (1988) Seismic methods for verifying test ban treaties. In: Schroeer D, Hafemeister D (eds) Chapter 4 of nuclear arms technologies in the 1990s, AIP conference proceedings, vol 178. American Institute of Physics, New York, pp 54–108 Richards PG, Kim W-Y (1997) Testing the Nuclear Test-Ban Treaty. Nature 389:781–782 Ringdal F (1985) Study of magnitudes, seismicity and earthquake detectability using a global network. In: Kerr, AU (ed) The VELA Program: A Twenty-Five Year Review of Basic Research, published for the Defense Advanced Research Projects Agency by Executive Graphics Services, pp 611–624 Ringdal F, Marshall PD, Alewine RW (1992) Seismic yield determination of Soviet underground nuclear explosions at the Shagan River Test Site. Geophys J Int 109:65–77 Schaff DP, Kim W-Y, Richards PG, Jo E, Ryoo Y (2018a) Using waveform cross-correlation for detection, location, and identification of aftershocks of the 2017 nuclear explosion at the North Korea test site. Seismol Res Lett 89:2113–2119 Schaff DP, Richards PG, Slinkard M, Heck S, Young C (2018b) Lg-wave cross correlation and epicentral double-difference location in and near China. Bull Seismol Soc Am 108:1326–1345 Smith AT (1993) Discrimination of explosions from simultaneous mining blasts. Bull Seismol Soc Am 83:160–179 Sykes LR (1996) Dealing with decoupled nuclear explosions under a comprehensive test ban treaty. In: Husebye ES, Dainty AM (eds) Monitoring a comprehensive Nuclear Test Ban Treaty, NATO ASI series E, vol 303. Kluwer, Dordrecht, pp 247–293 Tian D, Yao J, Wen L (2018) Collapse and earthquake swarm after North Korea’s 3 September 2017 nuclear test. Geophys Res Lett 45:3976–3983. https://doi.org/10.1029/2018GL077649 Timerbaev R (2007) On the “Threshold” Test Ban Treaties of 1974–76. Secur Index Russ J Int Secur 13:155–162. https://doi.org/10.1080/ 19934270.2007.9756509 Waldhauser F, Schaff DP (2008) Large-scale relocation of two decades of Northern California seismicity using cross-correlation and doubledifference methods. J Geophys Res 113. https://doi.org/10.1029/ 2007JB005479 Yang X, North R, Romney C, Richards PG (2003) Worldwide nuclear explosions. Chapter 84 of International Handbook of Earthquake and Engineering Seismology, part B. In: Lee WHK, Kanamori H, Jennings P, Kisslinger C (eds) On behalf of the International Association of Seismology and Physics of the Earth’s Interior. Academic Press, London and San Diego, pp 1595–1599

Seismic Noise

Seismic Noise Dhananjay Kumar1,2 and Imtiaz Ahmed2 1 Chevron, Houston, TX, USA 2 BP, Houston, TX, USA

Definition Seismic noise Multiple SNR

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Passive seismic

Noise is the undesirable part of seismic data that is not signal, and signal is what fits our conceptual model. A seismic event that experiences more than one reflection in the subsurface. Signal-to-noise ratio – is a measure of strength of signal compared to noise, and it is a measure of seismic data quality. Seismic data recordings of artificial manmade sources such as dynamite and vibroseis in land surveys and air gun in marine surveys. Example: exploration and engineering seismology. Seismic data recordings of natural sources such as earthquake, solar waves, and ocean waves. Example: earthquake seismology.

Introduction Seismic noise comprises all of the unwanted recorded energy that contaminates seismic data. A part of the seismic energy is considered noise if it does not fit the conceptual model of seismic signal. Seismic noise can be random or coherent. The identification of seismic noise depends on the type of data analysis and the type of data available – a part of data treated as noise in one application can be signal in another application. For example, the S-wave energy is generally considered noise in a P-wave processing project; vice versa, the P-wave energy is considered noise in an S-wave processing project. Historically, only the traveltime information of the seismic data was used to study the subsurface. For example, the knowledge about the earth’s deep interior was primarily derived from seismic traveltime information recorded during deep earthquakes. Also, only the traveltime information was used to derive the structural image of subsurface for exploration projects. As seismic technology has advanced and the appetite for understanding complicated geological features has increased, there has been a push toward technologies using the complete waveforms (amplitude and traveltime) in seismic analysis. Full waveform inversion technology (Tarantola 1986) is one such example that uses the complete waveform to estimate properties of the subsurface from the

Seismic Noise

seismic data. The success of such technologies is understandably very dependent on clean, noise-free seismic data. It is essential that seismic data are carefully processed to derive high quality seismic images (see Yilmaz 2001 for seismic data processing). One of the major challenges in seismic data processing is to separate noise from signal or to attenuate noise. In practice, noise cannot be completely attenuated and occasionally it is not even desired to attenuate noise but to use it as signal. The objective of the noise attenuation or noise separation process in seismic data processing is to enhance signal-to-noise ratio (SNR). There have been significant progress in data processing to improve SNR; advances have been made in random noise attenuation (Yilmaz 2001) and coherent noise attenuation (see Weglein and Dragoset 2005 for multiple attenuation methods). Recently, there have been various efforts to use seismic noise as signal, for example: (1) using multiple reflected energy (multiples) in seismic migration and inversion to image subsurface; and, (2) using very low and very high frequency passive seismic signal for reservoir monitoring. In the following sections, we write brief descriptions about the types of seismic noise, the noise attenuation techniques, and how seismic noise can be useful.

Types of Seismic Noise There are two types of seismic noise: random noise and coherent noise. In a multichannel seismic dataset, random noise does not correlate either with the neighboring channels (i.e., no spatial correlation) or along the same channel (i.e., no temporal correlation). Coherent noise, however, is part of the data that correlates spatially and/or temporally. Random noise is easier to attenuate during seismic data processing. Coherent noise is difficult to attenuate in processing; therefore, residual coherent noise can interfere with real signal and be misinterpreted as signal. The possible sources of these seismic noises can be placed under four categories: (1) ambient sources, (2) wave propagation related noise, (3) data acquisition related noise, and (4) data processing artifacts. The severity and types of noise can differ between marine and land acquisition environment. Ambient noise is the noise from unwanted sources like wind, swell, power line, activities on nearby road, surface facility, marine activities like ships and marine animals, and other cultural noise. Ambient noise can be present in various forms on seismic data, such as linear features, localized very high amplitude response, and mono-frequency events. Ambient noise can be random noise and coherent noise. Wave propagation related noise includes the surface waves, multiples, and geologic noise. Seismic response for an active source survey include primary reflection event (e.g.,

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incident P-wave reflected back as P-wave), refraction, ground roll, mode converted event (e.g., incident P-wave reflected as S-wave), and several events reflected multiple times in the subsurface (multiples). In reflection seismology, we are interested in P-wave (and/or S-wave) primary reflections. All the other coherent energies recorded are considered noise. Multiples are considered the major source of coherent noise in the seismic experiment and are really difficult to attenuate. Wave propagation related noise also includes seismic response from unwanted geology (complicated stratigraphy, shallow gas, and faults) not considered in seismic analysis; they are known as geologic noise. Data acquisition related noise is due to poor source and receiver coupling to the earth, source and recording instruments-generated noise, and acquisition footprint related to the acquisition design. Noise from poor coupling and noise related to instruments can be easily identified and attenuated. Acquisition footprint related to the acquisition design, observed as linear trend in seismic amplitude map view, is common in seismic data and can be suppressed in processing. Seismic data processing is another source of noise in processed seismic data. There are various factors affecting seismic reflection amplitude that do not contain subsurface information (Sheriff 1975), but it is impossible to correct for all the factors affecting amplitudes. Many approximations and assumptions are made in seismic data processing depending on computer resources availability, project timeline, understanding of the physics of wave propagation, and the type of seismic data available. Some of these factors may alter the data and may introduce noise in the recorded data. Some examples are: (1) noise introduced in seismic data due to poor multiple attenuation and poor normal move-out (NMO) correction (Fig. 1); (2) noisy subsurface image due to imperfect velocity model used and/or approximate physics used for migration; (3) noise introduced due to inaccurate amplitude calibration of the raw amplitudes from the processed seismic data with synthetics seismic amplitude for quantitative AVO/AVA (amplitude variation with offset/angle) analysis; (4) artifacts introduced from the process of frequency enhancement to broaden frequency bandwidth for better depth resolution; and (5) artifacts introduced from the process of interpolation and regularization to compensate for the irregular acquisition geometry.

Enhancing Signal Over Noise There is always some noise present in the seismic data. Figure 2 schematically shows amplitude spectra of various seismic signals and noise, and noise is present in the whole signal bandwidth. Thus, the objective is to enhance SNR by noise attenuation or separation so that data can be effectively

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Seismic Noise, Fig. 2 Schematic amplitude spectrum: for conventional broadband seismic data (active seismic for exploration range 10–60 Hz, and passive seismic for earthquake range from 10–100 Hz), low frequency passive seismic data (typically less than 10 Hz), high frequency passive seismic (microseismic) data (typically 30–300 Hz), and noise. Because noise is present at all frequencies, there is a need to do noise attenuation so that we get signal-to-noise ratio greater than 1. Spectrum is plotted for near zero frequency to Nyquist frequency. Nyquist frequency is the highest frequency without aliasing; beyond this frequency signal cannot be reliably estimated. Note that passive low and high frequency data can be treated as noise in broadband seismic data

c value as a function of angle;the RMS value is computed in entire time gate shown here. In (c), the dipping coherent noises are multiples and the strong amplitudes at far angles are residual data after NMO. Thus, part of the data became residual data (un-flattened data after NMO in (b) is noise in AVA analysis) due to imperfect seismic velocity and/or seismic modeling method used in NMO (see red lines marked at 3,100 ms across three gathers for residual data)

used for analysis. An important step in noise attenuation/ separation is to identify signal from noise. White (1984) discusses spectral methods for signal and noise estimation on seismic reflection data. Kanasewich (1990) and Chap. 6 of Yilmaz (2001) have good discussion on noise and multiple attenuation methods with data examples. Noise level in data compared to signal can be estimated as the ratio of autocorrelation of data to a cross correlation of data. This is because autocorrelation of data represents both signal and noise but cross correlation of data with a representative signal trace (can be local partial stack of data) represents signal in data. Signal and noise separation process can be broadly divided into two methods: prediction based methods and separation based methods. By performing seismic simulation over a known (conceptual) model and comparing synthetics to the field seismic data, signal and noise can be identified on data, and therefore noise can be attenuated. Also, signal and noise representation differs in different data domains. Therefore, a suitable data domain can be identified for the optimum separation of the two. Figure 3 schematically shows various seismic events for a three-layer model in native acquisition (X-T) domain and transformed domains: (1) X-T domain (represents spatial-time domain), (2) F-K domain (represents frequency – wave number domain), and (3) t-p domain

Seismic Noise Model

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Seismic Noise, Fig. 3 Seismic responses in various data domains for a three-layer model. The time domain earth model is shown in left (a) with seismic velocity for three layers as V1, V2, and V3. The second from left is the seismic response fora three-layer model in X-T domain (b), where D represents direct arrival, L represents linear noise, such as ground rolls, R1 and R2 are primary reflection events from 1st and 2nd layer interfaces, and H1 and H2 are refraction events. The third from left shows the

seismic response in F-K domain (c), where noise (D, L events) is on the upper right corner with apparent velocity less than the reference apparent velocity (V) and signal (R1, R2, H1, H2 events) are on the lower left corner corresponding to lower wave numbers. The right most plot (d) shows the seismic response in the t-p domain, where the linear event in X-T domain plots on a point in the t-p domain (Diebold and Stoffa 1981)

(representing zerooffset traveltime – ray parameter domain). The key to successful noise attenuation is large separation between signal and noise in a specific data domain. Different types of noise may require different domains for better separation between signal and noise. For example, a linear event on a shot gather (X-T domain) maps to a radial line in the F-K domain and it maps onto a point in the t-p domain, and thus can be rejected by F-K filtering and t-p filtering, respectively. Caution should be observed while performing the forward and inverse transforms, as some implementations may not be completely reversible and may also introduce artifacts. Random noise is not correlated and can be attenuated easily during data processing. One of the most robust methods to attenuate random noise on multichannel data by a factor of square root of N is by stacking (compositing) data from N channels. If prestack data is needed, however, F-X deconvolution (predictive deconvolution) is effective in random noise attenuation, where F-X corresponds to frequency – space domain. Deconvolution operation (Yilmaz 2001) is designed in space domain to predict coherent signals, and then coherent signals can be subtracted from the seismic record. Coherent noise is relatively more difficult to attenuate. An appropriate data domain (Fig. 3) can be selected to distinguish signal from noise using the noise characteristics, like velocity and frequency. Radon filtering in t-p domain and wavelet transform filtering in F-K domain is very effective method for coherent noise attenuation. In many field data cases, signal

and noise are not well separated in any data domain and therefore it is very difficult to attenuate the noise. Recently a version of F-K filtering called Curvelet transform filtering has shown better results for both random and coherent noise attenuation (Neelamani et al. 2008). Multiples are example of coherent noise present in the data. There has been extensive research in the field of multiples suppression. The surface related multiple elimination (SRME) has been heavily relied upon to suppress multiples (Berkhout and Verschuur 1997). This technique is based on the stationary phase theory of the seismic signal and the multiples are predicted by convolving traces with one another and then stacking these convolved traces. Another way to have better SNR in seismic data is via improved seismic acquisition techniques. There have been various developments in seismic acquisition on land and in marine environment. For example, over/under marine seismic acquisition promises better SNR in both low and high frequency range (Moldoveanu et al. 2007). In over/under towedstreamer acquisition pairs of streamer are deployed at two different depths in the same vertical plane. Other advances in seismic acquisition promise cleaner and better seismic images, for example, Multicomponent Ocean Bottom recording, Wide Azimuth recording, and Simultaneous Sources recording. The multiples suppression is strongly dependent on the acquisition effort. For example, the suppression of multiples in the case of Wide Azimuth acquisition is the result of a natural weighting of the traces going into the stack

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because of the areal nature of the acquisition (VerWest and Lin 2007). Some noise will always remain in the seismic data even after careful seismic processing. Also, there are some noises that are still not understandable and difficult to attenuate. Therefore, it is important to incorporate noise in the seismic data analysis – for example, in seismic inversion the data should not be over fitted as it might be fitting the noise.

Use of Noise as Signal There have been significant efforts recently to use all of the recorded energy in the seismic data. Traditionally, only the primary reflection wavefield is used in reflection seismic imaging, while the multiple/ghost is discarded. However, the ghost/multiples can be used as signal, for example, as an input to mirror migration (Verm 1987) to produce superior shallow images for the sparse receiver ocean bottom data where illumination holes deteriorate the primary images (Clarke et al. 2006; Dash et al. 2009). Note that multiples are also generated from the same source as primary, they travel longer paths and contain more information than primary, and therefore in some circumstances multiples can be more useful than primaries. Full waveform inversion is another technique that uses both the primary and the multiples to invert for the subsurface parameters more effectively. Multiples have also been used to interpolate for missing near offsets in seismic recording using the technique of interferometry (Schuster 2009). Seismic interferometry can also be used to extract primary signal from background noise and multiples by simply cross-correlating recorded data at two different stations (Sabra et al. 2005; Curtis et al. 2006; Draganov et al. 2007). Some new seismic acquisition methods have been useful in using conventional multiples to improve SNR and/or extract primary. For example, over/under towed- streamer acquisition technology uses multiples to improve seismic data quality (Moldoveanu et al. 2007), and there is a possibility to estimate primaries from surface multiples from data recorded with simultaneous sources also called blended data (van Groenestijn and Verschuur 2010). In the simultaneous source acquisition (Berkhout 2008; Hampson et al. 2008) multiple sources are fired in a short time interval to speed up the acquisition. Converted S-wave data, regarded as noise in reflected P-wave data imaging, has been quite successful in imaging gas reservoirs and areas where we have shallow gas anomalies (Tatham and McCormack 1991). The property that S-wave does not get as attenuated as P-wave when the wavefield travels through these porous medium helps

Seismic Noise

create better images through pure S-wave or converted S-wave imaging. Low frequency earth’s ambient passive response caused by natural phenomenon (such as wind, ocean waves, and humanmade noise) and high frequency passive seismic response due to small earthquakes (microseisms) caused by induced fractures in a petroleum reservoirs are considered noise in a broadband active seismic data, but they can be effectively used to study subsurface. Low frequency earth’s ambient noise can be used to extract signal (Draganov et al. 2007), and the low frequency passive seismic anomaly can be correlated with the presence of hydrocarbon (Saenger et al. 2009); however the research is still in its early stages. High frequency passive seismic data (also called microseismic data) are effectively used in hydraulic fracture monitoring (Warpinski 2009) by locating microseisms induced by hydrocarbon production related activities.

Summary Seismic noise is an integral part of the seismic record and is defined as all unwanted seismic energy on data. It can be divided into two categories: random and coherent noises. Random noise is not correlated among traces and is easier to attenuate compared to coherent noise that is spatially and/or temporally correlated. Multiples and geologic noise that are coherent noise are more difficult to attenuate and often interfere with seismic signal and makes seismic analysis challenging. Strategies for seismic noise attenuation are needed to preserve the seismic signal of interest and to improve signalto-noise ratio (SNR). The success in noise attenuation lies in identification and then separation or prediction of signal and noise. Transformation of data to different data domains (X-T, F-K, t-p, curvelet, wavelet domains) have helped in better separating noise from signal. Wave-equation extrapolations, inverse scattering methods, surface related multiple elimination, deconvolution, etc., model the noise and/or data in the process of noise attenuation. Advances in seismic data acquisition and processing have been made to improve SNR. Recently various efforts have been made to use noise as signal and are an active topic of research. This includes using multiples as well as primaries in seismic migration and inversion, and using low and high frequency passive seismic data in imaging subsurface.

Cross-References ▶ Earthquake Source Theory ▶ Seismic Data Acquisition and Processing ▶ Single and Multichannel Seismics

Seismic Phase Nomenclature: The IASPEI Standard

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Bibliography Berkhout AJ (2008) Changing the mindset in seismic acquisition. Lead Edge 27:924–938 Berkhout AJ, Verschuur DJ (1997) Estimation of multiple scattering by iterative inversion, Part I: theoretical considerations. Geophysics 62:1586–1595 Clarke R, Xia G, Kabir N, Sirgue L, Mitchell S (2006) Case study: a large 3D wide azimuth ocean bottom node survey in deepwater GOM. In: 76th annual international meeting, Society of Exploration Geophysicists, Expanded abstracts, pp 1128–1132 Curtis A, Gerstoft P, Sato H, Snieder R, Wapenaar K (2006) Seismic interferometry – turning noise into signal. Lead Edge 25:1082–1092 Dash R, Spence G, Hyndman R, Grion S, Wang Y, Ronen S (2009) Widearea imaging from OBS multiples. Geophysics 74(4):Q41–Q47 Diebold J, Stoffa PL (1981) The traveltime equation tau-p mapping and inversion of common midpoint data. Geophysics 46:238–254 Draganov D, Wapenaar K, Mulder W, Singer J, Verdel A (2007) Retrieval of reflections from seismic background-noise measurements. Geophys Res Lett 34:L04305. https://doi.org/10.1029/ 2006GL028735 Hampson G, Stefani J, Herkenhoff F (2008) Acquisition using simultaneous sources. Lead Edge 27:918–923 Kanasewich ER (1990) Seismic noise attenuation. handbook of geophysical exploration, seismic exploration, vol 7. Pergamon, New York Moldoveanu N, Combee L, Egan M, Hampson G, Sudora L, Abriel W (2007) Over/under towed-streamer acquisition: a method to extend bandwidth to both higher and lower frequencies. Lead Edge 26:41–58 Neelamani R, Baumstein AI, Gillard DG, Hadidi MT, Soroka W (2008) Coherent and random noise attenuation using the curvelet transformation. Lead Edge 27:240–248 Sabra KG, Gerstoft P, Roux P, Kuperman WA (2005) Extracting timedomain Green’s function estimates from ambient seismic noise. Geophys Res Lett 32:L03310. https://doi.org/10.1029/ 2004GL021862 Saenger EH, Schmalholz SM, Lambert M-A, Nguyen TT, Torres A, Metzger S, Habiger RM, Muller T, Rentsch S, Mendez-Hernandez E (2009) A passive seismic survey over a gas field: analysis of lowfrequency anomalies. Geophysics 74(2):029–040 Schuster GT (2009) Seismic interferometry. Cambridge University Press, Cambridge, UK Sheriff RE (1975) Factors affecting seismic amplitudes. Geophys Prospect 23:125–138 Tarantola A (1986) A strategy for nonlinear elastic inversion of seismic reflection data. Geophysics 51:1893–1903 Tatham RH, McCormack MD (1991) Multicomponent seismology in petroleum exploration. Society of Exploration Geophysicists, Tulsa van Groenestijn GJA, Verschuur DJ (2010) Using surface multiples to estimate primaries by sparse inversion from blended data. Geophys Prospect. https://doi.org/10.1111/j.1365-2478.2010.00894.x Verm R (1987) Imaging VSP’s 3 kilometers beyond the borehole receiver. In: Offshore technology conference proceedings, Paper 5570 VerWest BJ, Lin D (2007) Modeling the impact of wide- azimuth acquisition on subsalt imaging. Geophysics 72:241–250 Warpinski N (2009) Microseismic monitoring: inside and out. J Pet Technol 61:80–85 Weglein AB, Dragoset WH (eds) (2005) Multiple attenuation, Geophysics reprint series no. 23. Society of Exploration Geophysicists, Tulsa White RE (1984) Signal and noise estimation from seismic reflection data using spectral coherence methods. Proc IEEE 72(10):1340–1356 Yilmaz O (2001) Seismic data analysis: processing, inversion, and interpretation of seismic data. Society of Exploration Geophysicists, Tulsa

Seismic Phase Nomenclature: The IASPEI Standard Johannes Schweitzer 1, Dmitry A. Storchak 2 and Peter Borman (Deceased) 1 NORSAR, Kjeller, Norway 2 International Seismological Centre (ISC), Thatcham, UK

Definition Different types of seismic waves can travel through the Earth. Their propagation depends on the wave type and the seismic velocities in the Earth. Due to the internal structure of the Earth (propagation velocities), the different wave types may be split at internal discontinuities (reflection, refraction, and conversion of waves) and will be recorded at seismic stations separated in time. The onsets of the different, separated waves are called seismic phases. Depending on their type and path through the Earth, seismic phases are labeled with different names, which are standardized for use by the seismological community.

Introduction The working group (hereinafter WG) on the standard seismic phase names was set up by the IASPEI Commission on Seismological Observation and Interpretation (CoSOI) in 2001. The WG was chaired by D. A. Storchak and included R. D. Adams, P. Bormann, E. R. Engdahl, J. Havskov, B. L. N. Kennett, and J. Schweitzer. The WG put together a modified standard nomenclature of seismic phases that was meant to be concise, consistent, and self-explanatory on the basis of agreed rules. The list was not meant to satisfy specific requirements of seismologists to name various phases used in a particular type of research. Instead, it was hoped that the new list would ensure an expanded standardized data reporting and exchange by data analysts and other users. After numerous consultations with the seismological community, the standard seismic phase list was finalized and adopted by the CoSOI/IASPEI at its meeting in Sapporo on July 04, 2003. The original list of standard seismic phase names was first published as part of the New Manual of Seismological Observatory Practice (Storchak et al. 2002), and then the version formally approved by the IASPEI was published in the Seismological Research Letters (Storchak et al. 2003). Various updates (e.g., in the revised second edition of the New Manual of Seismological Observatory Practice (Bormann et al. 2013)) to the list were required due to progress in observational

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seismology and relevant changes in other observational standards. This entry accommodates the advances made in the nomenclature since its last publication. The new nomenclature partially modified and complemented an earlier one published in the 1979 edition of the (Willmore 1979). It is more in tune with modern Earth and travel-time models. As opposed to former practice, the WG tried to make sure that the phase name generally reflects the type of wave and the path it has traveled. Accordingly, symbols for characterizing onset quality, polarity, etc. are no longer part of the phase name. The WG was also aware that seismic phases exist that are common in some regions but are only rarely or not found in other regions, such as Pb (P), PnPn, PbPb, etc. The names and definitions of acoustic and hydroacoustic phases are likely to be reviewed and revised based on new analysis practices being established in the data centers. The extended list of phase names as presented below reflects significantly increased detection capabilities of modern seismic sensors and sensor arrays, even of rather weak phases, which were rarely observed on the classical analog records. It also accounts for improved possibilities of proper phase identification by means of digital multichannel data processing such as frequency-wavenumber (f-k) analysis and polarization filtering, by modeling the observations with synthetic seismograms or by showing on the records the theoretically predicted onset times of phases. Finally, the IASPEI Seismic Format (ISF) (http://www.isc. ac.uk/standards/isf/download/isf.pdf) is much more flexible than the older formats previously used by the ISC, the NEIC, and other data centers. It also allows reporting, computer parsing, and archiving of phases with long or previously uncommon names. ISF also accepts complementary parameters such as onset quality, measured backazimuth and slowness, amplitudes and periods of other phases in addition to P and surface waves, for components other than vertical ones, and for instruments with nonstandard response characteristics. This increased flexibility of the parameter-reporting format requires improved standardization, which limits the uncontrolled growth of incompatible and ambiguous parameter data. Therefore, the WG agreed on certain rules. They are outlined below prior to the listing of the standardized phase names. To facilitate the understanding of the phase names, ray diagrams are presented below. They have been calculated for local seismic sources on the basis of an average onedimensional two-layer crustal model and for regional and teleseismic sources using the global 1D Earth model ak135 (Kennett et al. 1995). Before assigning abbreviated shortcut seismic phase names, one should agree first on the language to be used and its rules. As in any other language, we need a suitable alphabet (here plain Latin letters); numbers (here Arabic

Seismic Phase Nomenclature: The IASPEI Standard

numbers and +/ signs); an orthography, which regulates, for example, the use of capital and lowercase letters; and a syntax, which describes the rules of correct order and mutual relationship of the language elements. One should be aware, however, that like any historically developed language, the seismological nomenclature will inevitably develop exceptions to the rules and depend on the context in which it is used. Although not fully documented below, some exceptions will be mentioned. Note that our efforts are mainly aimed at standardized names to be used in international data exchange so as to build up unique, unambiguous global databases for research. Many of the exceptions to the rules are related to specialized, mostly local research applications. The identification of related seismic phases often requires specialized procedures of data acquisition and processing that are not part of seismological routine data analysis. Also, many of these exceptional phases are rarely or never used in seismic event location, magnitude determination, source mechanism calculations, etc., which are the main tasks of international data centers. We focus, therefore, on phases that are particularly important for seismological data centers as well as for the refinement of regional and global Earth models on the basis of widely exchanged and accumulated parameter readings. In addition, we added references to the first definition of some wave types and phase names.

Standard Letters, Signs, and Syntax Used for Describing Seismic Phases Capital Letters Individual capital letters that stand for primary types of seismic body waves include: P

K

I S

T J

Longitudinal wave that has traveled through the Earth’s crust and mantle, from undae primae (Latin) ¼ first waves (Von Dem Borne 1904) Longitudinal wave that has traveled through the Earth’s outer core, K, from Kern (German) ¼ core (Sohon 1932; Bastings 1934) Longitudinal wave that has traveled through the Earth’s inner core (Jeffreys and Bullen 1940) Transverse wave that has traveled through the Earth’s crust and mantle, from undae secundae (Latin) ¼ second waves (Von Dem Borne 1904) Wave that has partly traveled as sound wave in the sea, from undae tertiae (Latin) ¼ third waves (Linehan 1940) Transverse wave that has traveled through the Earth’s inner core (Bullen 1946)

Exceptions • A capital letter N used in the nomenclature does not stand for a phase name but rather for the number of legs traveled

Seismic Phase Nomenclature: The IASPEI Standard

(or N1 reflections made) before reaching the station. N should usually follow the phase symbol to which it applies. For examples see syntax below. • The lowercase letters p and s may stand, in the case of seismic events below the Earth’s surface, for the relatively short upgoing leg of P or S waves, which continue, after reflection and possible conversion at the free surface, as downgoing P or S wave. Thus seismic depth phases (e.g., pP, sP, sS, pPP, sPP, pPKP, etc.) are uniquely defined. The identification and reporting of such phases are of utmost importance for source depth determination (Scrase 1931; Stechschulte 1932; Gutenberg et al. 1933; Macelwane et al. 1933). • Many researchers working on detailed investigations of crustal and upper mantle discontinuities denote both the up- and downgoing short legs of converted or multiplereflected P and S phases as lowercase letters p and s, respectively. Individual or double capital letters that stand for surface waves include: L R Q G

LR LQ PL

(Relatively) long-period surface wave, unspecified, from undae longae (Latin) ¼ long waves (Von Dem Borne 1904) Rayleigh waves (short- to very long-period waves in crust and upper mantle) (Angenheister 1921) Love waves, from Querwellen (German) ¼ transverse waves (Angenheister 1921) (Very long-period) global (mantle) Love waves, firstly observed and reported by Gutenberg and Richter (1934); in honor of Gutenberg, Byerly proposed the usage of G for these waves (Richter 1958) Long-period Rayleigh waves, usually relating to the Airy phase maximum in the surface wave train Long-period love waves Fundamental leaking mode following P onsets, firstly observed and reported by Somville (1930, 1931)

Lowercase Letters and Signs Single lowercase letters generally specify the part of Earth’s crust or upper mantle in which a phase has its turning point or at which discontinuity it has been reflected and/or eventually converted: g

b

n

Following the phase name characterizes waves “bottoming” (i.e., having their turning point in the case of P or S body waves) or just travel (surface waves) within the upper (“granitic”) Earth’s crust (e.g., Pg, Sg, Rg), (Jeffreys 1926) Following the phase name characterizes body waves bottoming in the lower (“basaltic”) Earth’s crust (Jeffreys 1926) (e.g., Pb, Sb; alternative names for these phases are P, S (Conrad 1925)); also used for phases radiated directly to the Earth’s surface from events in the lower crust Following the phase name characterizes a P or S wave that is bottoming in the Earth’s uppermost mantle or traveling as head wave below the Mohorovičić discontinuity (e.g., Pn, Sn), (continued)

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m c

i z

introduced after Andrija Mohorovičić discovered the Earth’s crust and separated the crustal travel-time curve from the normal (¼n) mantle phase (Mohorovičić 1910); also used for phases radiated directly to the Earth’s surface from events in the uppermost mantle (Upward) reflections from the outer side of the Mohorovičić (Moho) discontinuity (e.g., PmP, SmS) Reflections from the outer side of the core-mantle boundary (CMB), usage proposed by James B. Macelwane (see Gutenberg 1925) Reflections from the outer side of the inner core boundary (ICB) Reflections from a discontinuity (other than free surface, CMB or ICB) at depth z (measured in km). Upward reflections from the outer side of the discontinuity may additionally be complemented by a + sign (e.g., P410+P; this, however, is not compulsory), while downward reflections from the inner side of the discontinuity must be complemented by a  sign (e.g., P660P)

An exception from these rules is the use of lowercase p or s to indicate arrivals of longitudinal or transverse waves that were first radiated to go up toward the free surface to be reflected/converted back into the Earth as normal P or S waves (see near-source surface reflections and conversions section of the phase list below). Naming phases radiated directly to the Earth’s surface in the near-source region from events in the lower crust (Conrad discontinuity) as Pb and from below the Moho as Pn is a pragmatic decision, which helps analysts and source-location programs to separate different branches of the travel-time curves. Double lowercase letters following a capital letter phase name indicate the travel-time branch to which this phase belongs. Due to the geometry and velocity structure of the Earth, the same type of seismic wave may develop a triplication of its travel-time curve with different, in some cases, well-separated, branches. Thus, it is customary to differentiate between different branches of core phases and their multiple reflections at the free surface or the CMB. Examples are PKPab, PKPbc, PKPdf, SKSac, SKKSac, etc. The separation of the different PKP branches with letters ab, bc, and df was introduced by Jeffreys and Bullen (1940). Three lowercase letters may follow a capital letter phase name to specify its character, e.g., as a forerunner (pre) to the main phase, caused by scattering (e.g., PKPpre) or as a diffracted wave extending the travel-time branch of the main phase into the outer core shadow (e.g., Pdif in the outer core shadow for P). Syntax of Generating Complex Phase Names Due to refraction, reflection, and conversion in the Earth, most phases have a complex path history before they reach the station. Accordingly, most phases cannot be described by a single capital letter code in a self-explanatory way. By

S

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combining the capital and lowercase letters as mentioned above, one can describe the character of even rather complex refracted, reflected, or converted phases. The order of symbols (syntax) regulates the sequence of phase legs due to refraction, reflection, and conversion events in time (from left to right) and in space.

Examples for Creating Complex Standard Phase Names Traditional examples of complex phase names are as follows. Refracted and Converted Refracted Wave • PKP is a pure refracted longitudinal wave. It has traveled the first part of its path as P through the crust and mantle, the second through the outer core, and the third again as P through the mantle and crust. An alternative name for PKP is P0 (Angenheister 1921), which should be read as “P prime.” • PKIKP (alternative to PKPdf) is also a pure refracted longitudinal wave. It has traveled the first part of its path as P through the crust and mantle, the second through the outer core, the third through the inner core, and the fourth and fifth parts back again through the outer core and mantle/crust. • SKS is a converted refracted wave. It has traveled as a shear wave through the crust and mantle, being converted into a longitudinal P wave when refracted into the outer core and converted back again into an S wave when entering the mantle. • SKP or PKS are converted refracted waves with only one conversion from S to P when entering the core or from P to S when leaving the core, respectively. Pure Reflected Waves • In the case of (downward only) reflections at the free surface or from the inner side of the CMB, the phase symbol is just repeated, e.g., PP, SS (Geiger 1909), PPP, SSS, KK, KKK, etc. • In the case of (upward) reflections from the outer side of the Moho, the CMB, or the ICB, this is indicated by inserting symbols m, c, or i, respectively, between the phase symbols, e.g., PmP, PcP, ScS, and PKiKP. • Reflections from any other discontinuity in the mantle or crust at depth z may be from the inner side (; i.e., downward back into the mantle) or from the outer side (+; i.e., back toward the surface). To differentiate between these two possibilities, the sign has to follow z (or the

Seismic Phase Nomenclature: The IASPEI Standard

respective number in km), for example, P410+P or P660P. • To abbreviate names of multi-leg phases due to repeated reflections, one can also write Phasename N. This type of abbreviation is customary in the case of multiple phases with long phase names such as PmP2 for PmPPmP (free-surface reflection of PmP), SKS2 for SKSSKS (the alternative name for S0 2, the free-surface reflection of SKS), PKP3 for PKPPKPPKP (double free-surface reflection of PKP; alternative name to P0 3), or P4KP for PKKKKP (triple reflection of P at the inner side of the CMB). Two additional notes are to be made. First, PKP2 ¼ PKPPKP are now alternative names for P0 2 or P0 P0, respectively. This should not be mistaken for the old usage of PKP2 for PKPab. Secondly, in the case of multiple reflections from the inner side of the CMB, the WG followed the established tradition of placing the number N not after but in front of the related phase symbol K. Reflected Waves with Conversion at the Reflection Point In the case that a phase changes its character from P to S, or vice versa, one writes: • PS (first leg P, second leg S) or SP (first leg P, second leg S) in the case of reflection/conversion from the free surface downward into the mantle (Geiger and Gutenberg 1912a, b). • PmS or SmP, respectively, for reflection/conversion from the outer side of the Moho. • PcS or ScP for reflection/conversion from the outer side of the CMB. • Pz+S or SzP for reflection/conversion from the outer (+) side or inner () side, respectively, of a discontinuity at depth z. Note that the  is compulsory, the + is not. • pS or sP reflection/conversion at the free surface for body waves with a direct upgoing first leg. In this context, it is worth mentioning that mode conversion is impossible for reflections from the inner side of the CMB back into the outer core because the liquid outer core does not allow the propagation of S waves. The WG determined the new IASPEI standard phase names along these lines and rules. Where these deviate from other traditionally used names, the latter are given as well. Either the traditional names are still acceptable alternatives (alt) or they are old names (old), which should no longer be used.

Seismic Phase Nomenclature: The IASPEI Standard

Ray-Path Diagrams for Some of the IASPEI Standard Phases We show ray paths through the Earth for many of the mentioned phases. The three diagrams for crustal phases are sketches illustrating the principal ray paths in a twolayer crust (Fig. 1). The rays in all other figures (Figs. 2, 3, 4, 5, and 6) were calculated by using the ray picture part of the WKBJ3 code (Chapman 1978; Dey-Sarkar and Chapman 1978); as velocity model, we chose the standard Earth model ak135 (Kennett et al. 1995). For some types of P and S phases, the ray paths through the Earth are very similar

Seismic Phase Nomenclature: The IASPEI Standard, Fig. 1 Seismic “crustal phases” observed in the case of a two-layer crust in local and regional distance ranges (0
< D < approximately 20
) from the seismic source in the (a) upper crust, (b) lower crust, and (c) uppermost mantle

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because the velocity ratio VP/VS does not change enough to give very different ray pictures. In these cases, we calculated only the ray paths for the P-type ray (i.e., P, Pdif, pP, PP, P660P, P660P, PcP, PcP2, and PcP4) and assumed that the corresponding ray paths of the respective S-type phases were very similar. To show the different ray paths for phases with similar phase names, we show on many figures rays leaving the source once to the left and once to the right in different colors. The three most important discontinuities inside the Earth are indicated as black circles (i.e., the border between the upper and lower mantle, the CMB, and the ICB).

Pg/Sg

Upper crust

Pb/Sb PmP/SmS Pn/Sn

Lower crust

Uppermost mantle

P/S

Pb/Sb

Upper crust Pb/Sb PmP/SmS Pn/Sn

Lower crust

Uppermost mantle

P/S

S Upper crust

Pn/Sn

Lower crust

Pn/Sn

P/S

Uppermost mantle

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Seismic Phase Nomenclature: The IASPEI Standard P/S and Pdif/Sdif

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Seismic Phase Nomenclature: The IASPEI Standard, Fig. 2 Mantle phases observed at teleseismic distance ranges (D > approximately 20
)

Seismic Phase Nomenclature: The IASPEI Standard

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Seismic Phase Nomenclature: The IASPEI Standard, Fig. 3 Mantle phases reflected from the Earth’s core

S IASPEI Standard Seismic Phase List Crustal Phases

Pg

Pb Pn PnPn

At short distances, either an upgoing P wave from a source in the upper crust or a P wave bottoming in the upper crust; also, at larger distances, arrivals caused by multiple P-wave reverberations inside the whole crust with a group velocity around 5.8 km/s (alt:P) Either an upgoing P wave from a source in the lower crust or a P wave bottoming in the lower crust Any P wave bottoming in the uppermost mantle or an upgoing P wave from a source in the uppermost mantle Pn free-surface reflection (continued)

PgPg PmP PmPN PmS Sg

Sb Sn SnSn

Pg free-surface reflection P reflection from the outer side of the Moho PmP multiple free-surface reflection; N is a positive integer. For example, PmP2 is PmPPmP P to S reflection/conversion from the outer side of the Moho At short distances, either an upgoing S wave from a source in the upper crust or an S wave bottoming in the upper crust; also, at larger distances, arrivals caused by superposition of multiple S-wave reverberations and SV to P and/or P to SV conversions inside the whole crust (alt:S) Either an upgoing S wave from a source in the lower crust or an S wave bottoming in the lower crust Any S wave bottoming in the uppermost mantle or an upgoing S wave from a source in the uppermost mantle Sn free-surface reflection (continued)

1454 SgSg SmS SmSN SmP Lg

Rg

Seismic Phase Nomenclature: The IASPEI Standard Sg free-surface reflection S reflection from the outer side of the Moho SmS multiple free-surface reflection; N is a positive integer. For example, SmS2 is SmSSmS S to P reflection/conversion from the outer side of the Moho A wave group observed at larger regional distances and caused by superposition of multiple S-wave reverberations and SV to P and/or P to SV conversions inside the whole crust. The maximum energy travels with a group velocity of approximately 3.5 km/s Short-period crustal Rayleigh wave

ScP ScSN Sz+S

SzS

Sz+P SzP

Mantle Phases ScSP Sdif P

PP PS

PPP PPS PSS PcP PcS PcPN Pz+P

PzP

Pz+S PzS PScS Pdif S

SS SP

SSS SSP SPP ScS

A longitudinal wave, bottoming below the uppermost mantle; also an upgoing longitudinal wave from a source below the uppermost mantle Free-surface reflection of a P wave leaving a source downward P, leaving a source downward, reflected as an S at the free surface. At shorter distances, the first leg is represented by a crustal P wave Analogous to PP PP which is converted to S at the second reflection point on the free surface; travel time matches that of PSP PS reflected at the free surface P reflection from the core-mantle boundary (CMB) P converted to S when reflected from the CMB PcP reflected from the free surface N  1 times; N is a positive integer. For example, PcP2 is PcPPcP (alt:PzP) P reflection from the outer side of a discontinuity at depth z; z may be a positive numerical value in km. For example, P660+P is a P reflection from the top of the 660 km discontinuity P reflection from the inner side of a discontinuity at depth z. For example, P660P is a P reflection from below the 660 km discontinuity, which means it is precursory to PP (alt:PzS) P converted to S when reflected from the outer side of discontinuity at depth z P converted to S when reflected from the inner side of a discontinuity at depth z P (leaving a source downward) to ScS reflection at the free surface (old:Pdiff) P diffracted along the CMB in the mantle Shear wave, bottoming below the uppermost mantle; also an upgoing shear wave from a source below the uppermost mantle Free-surface reflection of an S wave leaving a source downward S, leaving a source downward, reflected as P at the free surface. At shorter distances, the second leg is represented by a crustal P wave Analogous to SS SS converted to P when reflected from the free surface; travel time matches that of SPS SP reflected at the free surface S reflection from the CMB (continued)

S converted to P when reflected from the CMB ScS multiple free-surface reflection; N is a positive integer. For example, ScS2 is ScSScS (alt:SzS) S reflection from the outer side of a discontinuity at depth z; z may be a positive numerical value in km. For example, S660+S is an S reflection from the top of the 660 km discontinuity S reflection from the inner side of discontinuity at depth z. For example, S660S is an S reflection from below the 660 km discontinuity, which means it is precursory to SS (alt:SzP) S converted to P when reflected from the outer side of a discontinuity at depth z S converted to P when reflected from the inner side of a discontinuity at depth z ScS to P reflection at the free surface (old:Sdiff) S diffracted along the CMB in the mantle

Core Phases

PKP PKPab PKPbc PKPdf PKPpre PKPdif PKS PKSab PKSbc PKSdf P0 P0 P0 N P0 zP0

P0 S0

PS0 PKKP PKKPab PKKPbc PKKPdf PNKP PKKPpre PKiKP PKNIKP PKJKP

(alt:P0 ) unspecified P wave bottoming in the core (old:PKP2) P wave bottoming in the upper outer core; ab indicates the retrograde branch of the PKP caustic (old:PKP1) P wave bottoming in the lower outer core; bc indicates the prograde branch of the PKP caustic (alt:PKIKP) P wave bottoming in the inner core (old:PKhKP) a precursor to PKPdf due to scattering near or at the CMB P wave diffracted at the inner core boundary (ICB) in the outer core Unspecified P wave bottoming in the core and converting to S at the CMB PKS bottoming in the upper outer core PKS bottoming in the lower outer core PKS bottoming in the inner core (alt:PKPPKP) Free-surface reflection of PKP (alt:PKPN) PKP reflected at the free surface N  1 times; N is a positive integer. For example, P0 3 is P0 P0 P0 PKP reflected from the inner side of a discontinuity at depth z outside the core, which means it is precursory to P0 P0 ; z may be a positive numerical value in km (alt:PKPSKS) PKP converted to SKS when reflected from the free surface; other examples are P0 PKS and P0 SKP (alt:PSKS) P (leaving a source downward) to SKS reflection at the free surface Unspecified P wave reflected once from the inner side of the CMB PKKP bottoming in the upper outer core PKKP bottoming in the lower outer core PKKP bottoming in the inner core P wave reflected N  1 times from the inner side of the CMB; N is a positive integer A precursor to PKKPdf due to scattering near the CMB P wave reflected from the inner core boundary (ICB) P wave reflected N  1 times from the inner side of the ICB P wave traversing the outer core as P and the inner core as S (continued)

Seismic Phase Nomenclature: The IASPEI Standard PKKS PKKSab PKKSbc PKKSdf PcPP0

SKS SKSac SKSdf SPdifKS SKP SKPab SKPbc SKPdf S0 S0 S0 N 0

S zS

0

S0 P0

S0 P SKKS SKKSac SKKSdf SNKS SKiKS SKJKS SKKP

SKKPab SKKPbc SKKPdf ScSS0

P wave reflected once from the inner side of the CMB and converted to S at the CMB PKKS bottoming in the upper outer core PKKS bottoming in the lower outer core PKKS bottoming in the inner core (alt:PcPPKP) PcP to PKP reflection at the free surface; other examples are PcPS0 , PcSP, PcSS0 , PcPSKP, and PcSSKP (alt:S0 ) unspecified S wave traversing the core as P SKS bottoming in the outer core (alt:SKIKS) SKS bottoming in the inner core (alt:SKPdifS) SKS wave with a segment of mantle-side Pdif at the source and/or the receiver side of the ray path Unspecified S wave traversing the core and then the mantle as P SKP bottoming in the upper outer core SKP bottoming in the lower outer core SKP bottoming in the inner core (alt:SKSSKS) Free-surface reflection of SKS SKS reflected at the free surface N  1 times; N is a positive integer SKS reflected from the inner side of discontinuity at depth z outside the core, which means it is precursory to S0 S0 ; z may be a positive numerical value in km (alt:SKSPKP) SKS converted to PKP when reflected from the free surface; other examples are S0 SKP and S0 PKS (alt:SKSP) SKS to P reflection at the free surface Unspecified S wave, traversing the core as P with one reflection from the inner side of the CMB SKKS bottoming in the outer core SKKS bottoming in the inner core S wave reflected N  1 times from the inner side of the CMB; N is a positive integer S wave traversing the outer core as P and reflected from the ICB S wave traversing the outer core as P and the inner core as S S wave traversing the core as P with one reflection from the inner side of the CMB and then continuing as P in the mantle SKKP bottoming in the upper outer core SKKP bottoming in the lower outer core SKKP bottoming in the inner core (alt:ScSSKS) ScS to SKS reflection at the free surface; other examples are ScPS0 , ScSP0, ScPP0, ScSSKP, and ScPSKP

Near-Source Surface Reflections and Conversions (Depth Phases)

pPy

All P-type onsets (Py) as defined above, which resulted from the reflection of an upgoing P wave at the free surface or an ocean bottom. WARNING: The character “y” is only a wild (continued)

1455

sPy

pSy

sSy

pwPy pmPy

card for any seismic phase, which could be generated at the free surface. Examples are pP, pPKP, pPP, pPcP, etc. All Py resulting from the reflection of an upgoing S wave at the free surface or an ocean bottom, for example, sP, sPKP, sPP, sPcP, etc. All S-type onsets (Sy) as defined above, which resulted from the reflection of an upgoing P wave at the free surface or an ocean bottom, for example, pS, pSKS, pSS, pScP, etc. All Sy resulting from the reflection of an upgoing S wave at the free surface or an ocean bottom, for example, sSn, sSS, sScS, sSdif, etc. All Py resulting from the reflection of an upgoing P wave at the ocean’s free surface All Py resulting from the reflection of an upgoing P wave from the inner side of the Moho

Surface Waves

L LQ LR G GN

R RN

PL

SPL

Unspecified long-period surface wave Love wave Rayleigh wave Mantle wave of Love type Mantle wave of Love type; N is an integer and indicates wave packets traveling along the minor arcs (odd numbers) or major arc (even numbers) of the great circle Mantle wave of Rayleigh type Mantle wave of Rayleigh type; N is an integer and indicates wave packets traveling along the minor arcs (odd numbers) or major arc (even numbers) of the great circle Fundamental leaking mode following P onsets generated by coupling of P energy into the waveguide formed by the crust and upper mantle S wave coupling into the PL waveguide; other examples are SSPL and SSSPL

Acoustic Phases

H HPg HSg HRg I IPg ISg IRg T

TPg TSg TRg

A hydroacoustic wave from a source in the water, which couples in the ground H phase converted to Pg at the receiver side H phase converted to Sg at the receiver side H phase converted to Rg at the receiver side An atmospheric sound arrival, which couples in the ground I phase converted to Pg at the receiver side I phase converted to Sg at the receiver side I phase converted to Rg at the receiver side A tertiary wave. This is an acoustic wave from a source in the solid Earth, usually trapped in a low velocity oceanic water layer called the SOFAR channel (SOund Fixing And Ranging) T phase converted to Pg at the receiver side T phase converted to Sg at the receiver side T phase converted to Rg at the receiver side

S

1456

Seismic Phase Nomenclature: The IASPEI Standard PKPab, PKPbc, and PKPdf

.

80.

280.

90.

270.

100.

260.

80. 90.

100.

.

250

0.

12

0. 0.

15

160

170.

180.

190.

.

0. 21

200 .

22

0.

0.

0. 14

0. 15

. 160

14

0.

13

170.

0.

23

13

0.

24

12

.

110

180.

0.

190.

.

70. 200 .

70.

.

110

. 0.

.

50

60

0.

.

22

50

60

21

.

2

40

. 30

.

0.

24

SKSac

290

PKPab PKPbc

30

260.

.

SKSdf

0.

270.

250

0.

30

PKPdf

20.

280.

18.

290 .

0.

0.

350.

0.

. 31

0.

32

40

0. 0.

30

. 340

33

.

0.

30

.

20.

18.

0.

350.

340

33 32 31

SKSac and SKSdf

Seismic Phase Nomenclature: The IASPEI Standard, Fig. 4 Seismic rays of direct core phases

Amplitude Measurements The following set of amplitude measurement names refers to the IASPEI Magnitude Standard of 2013 (see http:// w w w. i a s p e i . o r g / c o m m i s s i o n s / c o m m i s s i o n - o n seismological-observation-and-interpretation/Summary_ WG_recommendations_20130327.pdf), compliance to which is indicated by the presence of leading letter I. The absence of leading letter I indicates that a measurement is nonstandard. Letter A indicates a measurement in nm made on a displacement seismogram, whereas letter V indicates a measurement in nm/s made on a velocity seismogram. All amplitude measurements for standard magnitudes, with the exception for ML, are made on vertical-component seismic records.

IAML IAMs_20 IVMs_BB

IAmb

IVmB_BB

AX_IN

VX_IN

Displacement amplitude measured according to the IASPEI standard for local magnitude ML Displacement amplitude measured according to the IASPEI standard for surface-wave magnitude MS(20) Velocity amplitude measured according to the IASPEI standard for broadband surface-wave magnitude MS(BB) Displacement amplitude measured according to the IASPEI standard for short-period teleseismic bodywave magnitude mb Velocity amplitude measured according to the IASPEI standard for broadband teleseismic body-wave magnitude mB(BB) Displacement amplitude of phase of type X (e.g., PP, S, etc.), measured on an instrument of type IN (e.g., SP, short-period; LP, long-period; BB, broadband) Velocity amplitude of phase of type X and instrument of type IN (as above) (continued)

A V AML AMs Amb AmB END

Unspecified displacement amplitude measurement Unspecified velocity amplitude measurement Displacement amplitude measurement for nonstandard local magnitude Displacement amplitude measurement for nonstandard surface-wave magnitude Displacement amplitude measurement for nonstandard short-period body-wave magnitude Displacement amplitude measurement for nonstandard medium to long-period body-wave magnitude Time of visible end of record for duration magnitude

Unidentified Arrivals

x rx tx Px Sx

(old:i, e, NULL) unidentified arrival (old:i, e, NULL) unidentified regional arrival (old:i, e, NULL) unidentified teleseismic arrival (old:i, e, NULL, (P), P?) unidentified arrival of P-type (old:i, e, NULL, (S), S?) unidentified arrival of S-type

Summary Since the early days of instrumental seismology, seismograms showed distinguished onsets of different seismic phases with different characteristics. Seismologists developed an own nomenclature for these phases, which is basic for any seismogram interpretation. This entry includes the historical development and currently established definitions of the seismic phase nomenclature as recommended by IASPEI.

Seismic Phase Nomenclature: The IASPEI Standard PKiKp and SKiKS

PKiKS and SKiKP

80. 90.

100. 80. 90.

100.

0.

14

0. 15

. 160

170.

180.

1

. 40

0. 15

. 160

170.

0. 0.

1

0. 15

. 160

170.

180.

200 .

190.

0. 21

22

0.

. 40

0.

23

13

24

0.

.

110

.

12 0

0.

13

1

250

12

100. . 40

.

110 .

90.

0. 15

260.

100.

. 160

270.

90.

70.

80. 170.

280.

80.

.

. 180.

70.

50

60

0.

.

.

0.

50

0.

60

190.

180.

.

0. 21

1

22

. 40

0. 15

. 160

170.

0.

0.

13

180.

200

12 0.

200 .

190.

.

110

.

.

13 0

.

22

12 0.

190.

110 .

200

100.

70.

0.

80. 90.

.

. 0.

70.

50

60

22

. 40

21

60

.

30

.

20.

10.

0.

350.

0.

.

0.

30 0. 290 .

340

32

40

.

250

PKKPab + PKKPbc

0. 33

.

260.

23

.

21

50

0.

31

270.

0.

190.

22

0.

0. 14

21 0. 200 .

12 0. 13

0.

24

23

280.

24

1

4 0.

.

250

30

20.

10.

0.

.

340

350.

PKKPdf

30 0. 290 .

3

S’S’ac

SKKSac

0.

32

0. 33

0.

0.

S’S’df

PKKP

31

20.

0.

10.

2

0.

260.

350.

50.

270.

.

260.

0.

30 0. 290 . 280.

340

270.

. 30

0.

32

.

31

280.

2

0. 33

40

.

30

20.

10.

0.

. 340

350.

0. 33

0.

24

30 .

.

110

0.

PKiKS

0.

23

12 0.

0. 15

0.

24

S⬘S⬘ac and S⬘S⬘df

0.

32 31

110 .

. 160

90.

SKiKP

.

250

P⬘P⬘

30 0. 290 .

100.

170.

80.

70. 180.

70.

.

. 190.

.

0.

60

50

60

22

0.

.

21 0. 200 .

50

PKiKP

.

0 23

4

0.

0.

2

30 .

SKiKs

20.

260.

260.

10.

270.

0.

280.

270.

. 340

31

30 0. 290 .

280.

24

0.

.

32

40

0. 0.

50.

0. 33

30 .

.

20.

10.

0.

340

350.

32

0. 33

31

30 0. 290 .

350.

Seismic Phase Nomenclature: The IASPEI Standard, Fig. 5 Seismic rays of singlereflected core phases

1457

S

1458

Seismic Phase Nomenclature: The IASPEI Standard

.

90.

100.

90.

100. .

110 .

70.

80.

80.

250

110 .

0.

170.

0. 14

15 0. 14

0. 15

. 160

.

0.

22

21

0. 15

1

0.

0.

50

.

60

0.

.

280.

90.

270.

100.

260.

90. 100. .

10.

250

1

0.

.

12 0

24

0.

1

0 14

. 50

. 160

170.

180.

190.

. 200

21 0.

.

23

22 0.

70.

80. 110 .

0.

12

0.

13

0. 14

70.

0.

290

80.

.

15

.

.

.

160

50

30

60

170.

.

180.

40

50.

190.

200

13

. 40

12 0.

. 160

13 0.

.

30

20.

10.

0.

. 340

260.

0.

0.

.

270.

23

0.

32

40 0.

280.

0.

350.

33

. 31

290 .

.

180.

110 .

170.

12 0.

180.

110 .

100.

190.

100.

90. 0.

90.

70. 80.

.

70.

80.

0.

23

0.

200

190.

.

.

200

.

0.

24

30

.

20.

0.

10.

350.

340

0.

32

0.

33 0.

0.

. 160

.

21

0.

22 0. 0.

0. 14

0. . 160

50

60

0.

.

22

50 60

21

.

.

PKJKP and SKJKS

30

22 0.

40

250

PKS and SKP

21

.

.

24

30

260.

. 250

2

20.

260.

31

0.

270.

.

10.

0. 32

.

280.

270.

0 23

350.

. 340 0. 33

40

.

30

20.

0.

10.

350.

0.

32

. 340 0. 33

31 0.

30 0. 290 .

1

170.

15

PK2IKP

30 .

180.

200

13 0.

.

0.

23

13 0.

12 0.

24

12 0.

190.

70.

. 200

.

260.

280.

0 24

170.

50

60

22 0. 0.

.

280. 270.

60

21

50

30 0. 290 .

PKKSab/PKKSbc and SKKPab/SKKPbc

0.

40 .

0. . 30

30 .

.

250

31

20.

260.

30 0. 290 .

0.

280.

2

10.

0. 0. 32

31 0.

270.

24

. 340

33

40 .

30 .

20.

0.

10.

350.

. 340

0. 32

0. 33

31 0.

30 0. 290 .

180.

P4KPab and P4KPbc

190.

S3KSac

350.

Seismic Phase Nomenclature: The IASPEI Standard, Fig. 6 Seismic rays of multiplereflected and converted core phases

Seismic Properties of Rocks

Cross-References ▶ Body Waves ▶ Earth’s Structure, Core ▶ Seismogram Interpretation ▶ Surface Waves

Bibliography Angenheister GH (1921) Beobachtungen an pazifischen Beben. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse 113–146 Bastings L (1934) Shear waves through the Earth’s core. Nature 134:216–217 Bormann P, Storchak DA, Schweitzer J (2013) The IASPEI standard nomenclature of seismic phases. In: Bormann P (eds) New manual of seismological observatory practice, 2nd edn. GeoForschungsZentrum, Potsdam, IS2.1, pp 1–20. https://doi.org/ 10.2312/GFZ.NMSOP-2_IS_2.1 Bullen KE (1946) A hypothesis on compressibility at pressures of the order of a million atmospheres. Nature 157:405 Chapman CH (1978) A new method for computing synthetic seismograms. Geophys J R Astron Soc 54:481–518 Conrad V (1925) Laufzeitkurven des Tauernbebens vom 28. November, 1923. Mitteilungen der Erdbeben-Kommission der Akademie der Wissenschaften in Wien, Neue Folge 59:23 Dey-Sarkar SK, Chapman CH (1978) A simple method for the computation of body wave seismograms. Bull Seismol Soc Am 68:1577–1593 Geiger L (1909) Seismische Registrierungen in Göttingen im Jahre 1907 mit einem Vorwort über die Bearbeitung der Erdbebendiagramme. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse 107–151 Geiger L, Gutenberg B (1912a) Konstitution des Erdinnern, erschlossen aus der Intensität longitudinaler und transversaler Erdbebenwellen, und einige Beobachtungen an den Vorläufern. Phys Z 13:115–118 Geiger L, Gutenberg B (1912b) Ueber Erdbebenwellen. VI. Konstitution des Erdinnern, erschlossen aus der Intensität longitudinaler und transversaler Erdbebenwellen, und einige Beobachtungen an den Vorläufern. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse 623–675 Gutenberg B (1925) Bearbeitung von Aufzeichnungen einiger Weltbeben. Abh Senckenb Naturforsch Ges 40:57–88 Gutenberg B, Richter CF (1934) On seismic waves (first paper). Gerlands Beitr Geophys 43:56–133 Gutenberg B, Wood HO, Richter CF (1933) Re suggestion by Dr. Harold Jeffreys regarding P and Pg. Gerlands Beitr Geophys 40:97–98 Jeffreys H (1926) On near earthquakes. Mon Not R Astron Soc Geophys Suppl 1:385–402 Jeffreys H, Bullen KE (1940) (reprinted 1948; 1958; 1967; 1970). Seismological Tables. British Association for the Advancement of Science, Gray Milne Trust, London, p 50 Kennett BLN, Engdahl ER, Buland R (1995) Constraints on seismic velocities in the Earth from traveltimes. Geophys J Int 122:108–124 Linehan D (1940) Earthquakes in the West Indian region. Trans Am Geophys Union 30:229–232 Macelwane JB, Brunner GJ, Joliat JS (1933) Re suggestion by Doctor Harold Jeffreys and others regarding P and Pg, Gerlands Beitr Geophys 40:98 Mohorovičić A (1910) Potres od 8. X 1909, God. Izvjeste Zagr. met. Ops. Zag. 1909, Zagreb. (Das Beben vom 8. X 1909, Jahrbuch des meteorologischen Observatoriums in Zagreb für das Jahr 1909 9(4):1–63)

1459 Richter CF (1958) Elementary seismology. W. H. Freeman, San Francisco/London, p 768 Scrase FJ (1931) The reflected waves from deep focus earthquakes. Proc R Soc Lond Ser A 132:213–235 Sohon FW (1932) Seismometry. Part II of Macelwane JB, Sohon FW. Introduction to theoretical seismology. Wiley, New York, 149 pp Somville O (1930) A propos d’une onde longue dans la premiére phase de quelques séismogrammes. Gerlands Beitr Geophys 27:437–442 Somville O (1931) A propos d’une onde longue dans la premiére phase de quelques séismogrammes (II). Gerlands Beitr Geophys 29:247–251 Stechschulte VC (1932) The Japanese earthquake of March 29, 1928. Bull Seismol Soc Am 22:81–137 Storchak DA, Bormann P, Schweitzer J (2002) Standard nomenclature of seismic phases. In: Bormann P (eds) New manual of seismological observatory practice, vol 2. GeoForschungsZentrum, Potsdam, IS2.1, pp 1–18 Storchak DA, Schweitzer J, Bormann P (2003) The IASPEI standard seismic phase list. Seismol Res Lett 74:761–772 Von Dem Borne G (1904) Seismische Registrierungen in Göttingen, Juli bis Dezember 1903. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse:440–464 Willmore PL (1979) Manual of seismological observatory practice, World Data Center A for Solid Earth Geophysics, Report SE-20, September 1979, Boulder, Colorado, 165 pp

Seismic Properties of Rocks Nikolas I. Christensen Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, BC, Canada

Synonyms Rock P and S velocities

Definition Compressional (P) waves Shear (S) waves Poisson’s ratio (s)

Transversely isotropic

Seismic waves in which the rock particles vibrate parallel to the direction of wave propagation. Seismic waves in which the rock particles vibrate perpendicular to the direction of wave propagation. The ratio of the lateral unit strain to the longitudinal unit strain in a body that has been stressed longitudinally within its elastic limit (2s ¼ (R2–2)/(R2–1) where R ¼ Vp/Vs). Anisotropic solids with a single symmetry axis. Rock symmetry axes are usually normal to foliation, cleavage, or layering.

S

1460

Seismic Properties of Rocks

Introduction Although many disciplines have contributed significantly to our knowledge of the Earth’s interior, none has a resolution comparable to seismology. For nearly seven decades, seismic studies have provided geophysicists with worldwide information on crustal and upper mantle compressional (P) and shear (S) wave velocities. Significant data have recently become available on velocity gradients, velocity reversals, compressional and shear wave velocity ratios, and anisotropy in the form of azimuthal variations of compressional wave velocities, as well as shear wave splitting. Reflections within the crust and mantle originate from contrasts of acoustic impedances, defined as products of velocity and density. The interpretation of this seismic data requires detailed knowledge of rock velocities provided by laboratory techniques to a precision at least comparable with that of seismic measurements. In particular, to infer composition of the Earth’s upper 30–50 km, the “crust,” requires studies of the elasticity of rocks at conditions approaching those that exist at these depths. Of fundamental importance is the presence of mean compressive stress and temperature increasing with depth and on the average reaching about 1 GPa and 500
C at the base of the crust. Because of this, the most relevant velocity measurements for identifying probable rock types within the crust have been measurements at elevated pressures and temperatures. These measurements often allow the seismologist to infer mineralogy, porosity, the nature of fluids occupying pore spaces, temperature at depth, and present or paleolithospheric stress based on mineral and crack orientations. Francis Birch (Fig. 1) was the pioneer in the study of rock velocities. In addition to his laboratory work on physical properties of rocks and minerals at high pressures and temperatures, he was well-known for his studies of heat flow and theoretical work on the composition of the Earth’s interior. Two of his benchmark papers on compressional wave velocities in rocks (Birch 1960, 1961) set the stage for modern experimental studies of rock elasticity and have been frequently cited during the past five decades. These papers for the first time provided information on compressional wave velocities for many common rock types, as well as major findings on their anisotropies and relations to density. It is interesting to note that these measurements were carried out to pressures of 1 GPa, a pressure at which even today only a limited number of laboratories generate for modern rock seismic velocity measurements.

Measurement Techniques Rock velocities are usually measured in the laboratory using the pulse transmission technique. The transit time of either a

Seismic Properties of Rocks, Fig. 1 Francis Birch (1903–1992), a pioneer in rock physics research

compressional or shear wave is measured along the axis of a cylindrical rock specimen of known length. The cores are usually taken from rock samples using a 2.54 cm inner diameter diamond coring bit. The cores are trimmed and ground flat and parallel on a diamond grinding disk. The volume of each core is obtained from the length and diameter. The cores are weighed, and densities are calculated from their masses and dimensions. The cores are then fitted with a copper jacket to prevent penetration of high-pressure oil into the rock samples. For measurements at high temperatures, where gas is the pressure medium, the samples are usually encased in stainless steel. Transducers are placed on the ends of the rock core (Fig. 2). Compressional and shear waves are often generated by means of lead zirconate titanate (PZT) and AC cut quartz transducers with resonant frequencies of 1 MHz. The sending transducer converts the input, an electrical pulse of 50–500 V and 0.1–10 ms width, to a mechanical signal, which is transmitted through the rock. The receiving transducer changes the wave to an electrical pulse, which is amplified and displayed on an oscilloscope screen (Fig. 3). Once the system is calibrated for time delays, the travel time through the specimen is determined directly by a computer or with the use of a mercury delay line. The major advantage of the delay line is that it increases the precision, especially for signals with slow rise times, because the gradual onset of the first arrival from the sample is approximated by the delay line. The velocity is the ratio of the length of the specimen to the travel time of the compressional or shear wave. The total error limits for Vp and Vs are estimated to be less than 1%.

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Interfacing the pressure system with a computer for data acquisition and storage permits automatic calculations of velocities as successive readings are taken. Using a leastsquares routine, the computer fits a curve to the data points and calculates velocities for selected pressures. A velocity versus pressure curve is plotted along with recorded data points. Sample length, density, measured pressure velocity pairs, traces of the waveforms at selected pressures, the curve fit equations, and calculated pressure velocity pairs are recorded and stored digitally. Hydrostatic pressure generating systems capable of producing true hydrostatic pressures as high as 3 GPa, equivalent to a depth of approximately 100 km, have been used for rock velocity measurements. Low-viscosity synthetic petroleum and argon for high-temperature measurements are frequently used as pressure media. An alternate technique for obtaining velocities under quasi-hydrostatic conditions has used a

Input pulse

Gum rubber tubing

Electrode

Rock core

Electrode

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Copper jacket with brass shim stock soldered to each end Receiving transducer Gum rubber tubing

Output signal

Seismic Properties of Rocks, Fig. 2 Transducer and rock core assembly for velocity measurements at elevated pressures

Seismic Properties of Rocks, Fig. 3 Electronics for velocity measurements using a mercury delay line

triaxial press with cubic samples. Transducers are placed on the six pistons, and corrections are made for travel times through the pistons. Rock velocities obtained using this technique have provided valuable information on the effect of temperature on velocity but have been limited to pressures of 0.6 GPa. The behavior of a rock’s velocity as a function of pressure is primarily dependent upon mineralogy and porosity. Many igneous and metamorphic rocks have porosities of the order of a few tenths of 1%, which are present as thin openings between grain boundaries. As pressure is applied to the rock, the cracks close and velocities increase. Once the cracks close, any increase in velocity with increasing pressure is related to the intrinsic effects of pressure on the mineral velocities. This is illustrated in Fig. 4 for a garnet granulite. Velocities first increase rapidly over the first 100 MPa as cracks close and then increase slowly as pressure is increased. Also the velocity determined at a given pressure depends upon whether the pressure is approached from lower or higher pressure (Fig. 4). This hysteresis is usually quite small if sufficient time is taken for measurements between pressure increments. A considerable number of investigations have also focused on the influence of temperature on rock velocities. These studies have used either resonance techniques or more frequently the pulse transmission method. Early studies demonstrated that the application of temperature to rock at atmospheric pressure results in the creation of cracks that often permanently damage the rock and dramatically lower velocities. Thus reliable measurements of the temperature derivatives of velocities are obtained only at confining pressures high enough to prevent crack formation. At elevated confining pressures, δVp/δT for common rocks often ranges from 0.3  103 to 0.6  103 km/s/
C, and δVs/δT varies between 0.2  103 and  0.4  103 km/s/
C. At high temperatures, usually between 650
C and 1200
C, rocks undergo partial melting. Several factors influence partial melting, including rock composition, pressure, and the involvement of aqueous and carbonic fluids. For rocks in which velocities are not modified by temperature-induced phase changes, it has been observed

Pulse generator

Input pulse Mercury delay line

Oscilloscope

Amplifier for shear signal

Dual trace plug-in Output signal

Sample assembly

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Seismic Properties of Rocks Seismic Properties of Rocks, Table 1 Average compressional (Vp) and shear (Vs) wave velocities, velocity ratios (Vp/Vs), and Poisson’s ratios (s) at 1 GPa for common rock types (Christensen 1996)

7.30 7.00

Velocity (km/s)

6.70 6.40 6.10 5.80 = Increasing pressure

5.50

= Decreasing pressure

5.20

100

300

500 700 Pressure (Gpa)

900

Seismic Properties of Rocks, Fig. 4 Compressional wave velocity measurements as a function of confining pressure for a mafic granulite

that as temperature increases, velocities decrease linearly below partial melting temperatures and then drop significantly at the onset of partial melting.

Rock Velocities Seismic velocities have been measured for practically all igneous and metamorphic rock types believed to be important constituents of the lithosphere. Because rock classification schemes allow for considerable variations in mineralogy for a given rock type, many rocks have wide ranges in elastic properties. However, some lithologies, such as the monomineralic rocks hornblendite and dunite with little or no alteration, have fairly well-defined velocities. For detailed summaries of rock seismic properties, the reader is referred to the compilations of Birch (1960), Christensen (1982, 1996), Gerbande (1982), Ji et al. (2002), Rudnick and Fountain (1995), and Mavko et al. (1998). Table 1 contains average velocities, in the order of increasing compressional wave velocity, for several common igneous and metamorphic rocks. Volcanic rocks usually have lower velocities than their plutonic equivalents. This is due to the presence of glass, abundant alteration products, and vesicles in volcanic rocks, all of which have lower velocities. In general, for a given composition, velocity increases with increasing metamorphic grade. For example, mica- and quartz-bearing schists have higher velocities than slates and phyllites. Low-grade metamorphosed basalts have lower velocities than higher-grade amphibolite and mafic granulite. Eclogites have the highest velocity of mafic rocks. Note that shear velocities are relatively high in quartzites and low in serpentinites Early attempts to infer crustal composition by comparing laboratory- and field-derived velocities relied primarily on compressional wave velocities. However correlations

Rock Serpentinite Andesite Quartzite Basalt Granitic gneiss Granite-granodiorite Tonalite gneiss Slate Phyllite Mica quartz schist Zeolite facies basalt Diorite Diabase Greenschist facies basalt Marble Mafic granulite Amphibolite Anorthosite Gabbro Pyroxenite Eclogite Dunite

Vp (km/s) 5.607 5.940 6.091 6.118 6.271 6.372 6.366 6.379 6.398 6.523 6.530 6.675 6.814 6.983 6.985 7.000 7.046 7.124 7.299 7.935 8.198 8.399

Vs (km/s) 2.606 3.177 4.054 3.291 3.627 3.726 3.636 3.432 3.608 3.654 3.493 3.756 3.766 3.955 3.794 3.849 3.987 3.717 3.929 4.519 4.594 4.783

Vp/Vs 2.152 1.870 1.502 1.859 1.729 1.710 1.751 1.858 1.774 1.785 1.869 1.777 1.809 1.766 1.841 1.818 1.767 1.917 1.858 1.756 1.785 1.756

s 0.36 0.30 0.10 0.30 0.25 0.24 0.26 0.30 0.27 0.27 0.30 0.27 0.28 0.26 0.29 0.28 0.26 0.31 0.30 0.26 0.27 0.26

between compressional wave velocity and composition are limited due to the similar velocities of many common crustal rock types. Because of this nonuniqueness of compressional wave velocity laboratory and field data comparisons, many recent studies have focused on investigations of crustal composition using both compressional and shear wave velocities. In these studies the ratio Vp/Vs or Poisson’s ratio (s) calculated from Vp/Vs have resolved some of the ambiguities. The values of Vp/Vs and s, assuming isotropic elasticity, are given in Table 1 for several common igneous and metamorphic rocks at 1 GPa. This high pressure eliminates cracks so the values only reflect mineralogy. The relatively low s for quartzites (0.10) agrees well with isotropic aggregate calculations based on the elastic constants of single crystal alpha quartz. Anorthosites, on the other hand, have relatively high Poisson’s ratios (3 years (Nanjo et al. 2016).

Physical Mechanisms Leading to Seismic Quiescence and Activation Let us consider possible physical mechanisms which bring to occurrences of the seismic quiescence and of the foreshock activation. There is no unified explanation at present and it is

Seismic Quiescence and Activation

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48° M = 8.2 M = 8.1

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42°

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Seismic Quiescence and Activation, Fig. 3 Areas of foreshock activation before the Simushir earthquake

expedient to discuss various hypotheses. We would like to note at first that except for the three phases discussed above and the occurrence of foreshocks in the narrow sense, there may appear sometimes also other anomalies. C. Scholz selected additional phases: doughnut pattern was observed in the period of development of the seismic quiescence around its external limits, and seismic silence was noticed just prior to the main event (Scholz 1990). Let us consider in a more detailed manner hypotheses of nature of the stages of a seismic quiescence and of a foreshock activation. Ma et al. investigated the seismic patterns before nine large earthquakes in China (Ma et al. 1989). The authors did not select specially occurrences of a seismic quiescence or activation. They did describe mainly a migration of sources of seismic events for several years before large events in the regions adjacent to their epicenters with the dimensions of about first 100 km. Actual data cited by them are of interest as applied to a problem of physical nature of occurrences discussed in this entry. There was considered a migration of sources mainly with the magnitudes 3–6, while the large earthquakes had the magnitudes of above 8. The regions round the epicenters of large events were separated in three

parts: a hypocenter area A, which included a hypocenter of a large event; area of aftershocks B and external area C. The following main characteristics of migration were selected. During 3–10 years till the moment of the large earthquake there was varied two phases in time. In the first one the events occurred mainly in the area C, while the areas A and B were characterized as relatively quiet. During the second phase which finished in the large earthquake the activity was detected in the areas A and B, while the seismic activity in the area C was reduced. As the second phase developed, the activity in the areas A and B continued to last up to several months or days in relation to the moments of the large earthquake; at the end of such series the activity either accumulated inside the hypocentral area A or continued to last in the area B, while the area A was calm. We should note that the authors did not cite any quantitative evaluations of the seismicity rate, thus conclusions may be made based in a qualitative manner only. Summarizing, there may be made a conclusion that the process of development of a large earthquake may occur according to various scripts, though in described events it related to continental events only.

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The seismic quiescence may be a consequence of increase in the strength of rocks within the seismically active area. In the dilatancy-diffision model the increase in the strength is explained by laboratory experiments, being a consequence of appearance of open microcracks and a relative drying of the rocks. However, it is not explained how such process can develop in the heterogeneous lithosphere in the spatial regions with the linear dimension of above 100 km. The quiescence occurs also when the tectonic stress decreases. This can be a consequence of development of an unstable deformation in the source of a future large earthquake what constitutes one of the corner stones of the model of avalanche-like fracturing (Sobolev 2019). Expenditure of the accumulated potential energy brings to a decrease in stress both in the source and in the surrounding space. This explanation leads to expect an increase in seismic activity in the hypocentral area of a future large earthquake and a simultaneous occurrence of a seismic quiescence in the external area. This cannot always be observed. But the situation that the activation in the source occurs at the level of seismic events with small magnitudes, which are not registered by the available seismological network, cannot be excluded. Decrease in the tectonic stress in some area of lithosphere can be a consequence of the motion of neighboring blocks with a different rate. If one of the blocks stops because of an strong asperity at its tectonic boundaries with any neighboring blocks, then the stress will be accumulated at points of a asperity, while in the middle of the stopped block it will relax gradually. The activation will start at boundaries of the stopped block and then it will migrate to central parts when the asperity is destroyed. As a modification of such assumption is the situation when the stress increases and brings to the seismicity in the stronger blocks, and the block between them is relatively discharged. The quiescence occurring in the discharged block in such situation causes an effect of “false forerunners” as no large earthquake follows after them. Kanamori suggested an explanation of different phases of seismicity based on a model of existence of a strong inclusion at a fault being heterogeneous in its strength (Kanamori 1981). When the stress increases gradually, the less strong inclusions disintegrate sequentially what brings to an accelerated growth of the stress at points being not disintegrated. There is observed a background seismicity at the low stress. As the stress increases, there occurs the doughnut pattern – a mass destruction of rocks around the strong inclusion. The latter has a seismic quiescence. When the stress approaches to a critical level, there occurs a destruction of subunits of the strong inclusion which characterizes the phase of foreshock activation. In this hypothesis, the facts of existence of the seismic quiescence at distances which significantly exceed the rupture sizes when a large earthquake occurs are not explained.

Seismic Quiescence and Activation

The reason of seismic quiescence can be a change of orientation of the tensor of the current stress due to development of a creep at the fault where a future earthquake will occur. The existing fractures, which get into new conditions, require the time for unstable development (kinetics of destruction). It is not proven that this, undoubtedly, existing phenomenon can cause a quiescence at distances which significantly exceed the length of a rupture of a future earthquake. Another reason can be a transformation of the medium surrounding the source to the quasi-plastic state, for example, when the temperature or the hydrostatic compression increases. Then the process of fracturing will occur at a lower scale level which is beyond the registration of earthquakes by the seismic network. There remains a question what are the physical reasons of increase of the temperature or the compression in the lithosphere or the earth’s crust within a relatively short time period (years). It is necessary to mention another effect which influences on the fracturing in the geological medium. Laboratory and field observations prove that an increase or a decrease in the degree of water saturation of rocks results in a significant acceleration or retardation of occurrence of the seismic events (see ▶ “Geomagnetically Induced Currents”). Further, if the rate of the seismic activity increases due to that, then it results in a relative increase in the number of events with the relatively small magnitudes and in a decrease in the number of big events. The seismic quiescence may occur: (a) if a degree of water saturation decreases; (b) if under the influence of the increased water saturation the seismicity transforms to the level of small events being not registered by the seismic network. The activation is a direct consequence of the increased water saturation and may appear also in the form of swarms. It follows from the most available experimental facts that the foreshock activation develops in the epicentral region. Its size is less than the area of seismic quiescence, and the centers of anomalous areas of these two phenomena do not coincide as a rule. There is a ground to suppose that the physics of foreshock activation is connected with the development of unstable deformation which is localized under the laws of mechanics in the zone of mainly two-dimensional extension. The complexity of fault systems, which display a fractal geometry, causes a successive occurrence of several zones of unstable deformation. At the final period of foreshock activation there often occur clusters of seismic events, that is, groups of events, the distances between the hypocenters of which and the times between the successive events are less than mean values of the background seismicity. The occurrence of clusters at the phase of foreshock activation can be explained by two reasons. First, they occur by chance because of the increase in rate of formation of events in the narrow zone of unstable

Seismic Quiescence and Activation

deformation. Second, when the spatial density of accumulated active faults exceeds the critical level, there arise stress interactions among neighboring faults with the formation of the faults of a larger length. In the latter case such effect must appear with the increase of middle magnitudes of seismic events, that is, with the decrease of the b-value. Braun and Peyrard investigated the origin of seismic quiescence with a generalized version of the Burridge–Knopoff model for earthquakes and found that it can be generated by a multipeaked probability distribution of the thresholds at which contacts break. Such a distribution is not assumed a priori but naturally results from the aging of the contacts. It was shown that the model can exhibit quiescence as well as enhanced foreshock activity, depending on the value of some Parameters (Braun and Peyrard 2018). Kawamura et al. applied two different approaches: the pattern informatics (PI) method and the ZMAP method, which is a gridding technique based on the standard deviate (Z-value) test and found that the epicenter of the 2013 ML6.2 Nantou earthquake was surrounded by three main seismic quiescence regions prior to its occurrence. The assumption that this is due to precursory slip (stress drop) on fault plane or its deeper extent of the ML6.2 Nantou earthquake was supported by previous researches based on seismicity data, geodedic data, and numerical simulations using rate- and state-dependent friction laws (Kawamura et al. 2014). Yi-Ying Wen et al. investigated seismicity rate changes associated with the 2010 ML 6.4 Jiashian earthquake applying the RTL and pattern informatics algorithms (Wen et al. 2016). Both temporal and spatial results exhibited signatures of abnormal seismicity change related to the seismic activation and quiescence prior to main shock. Seismicity changes coincided with Coulomb stress change during the same period. The 2004 Sumatra (Mw 9.1) earthquake was preceded by a seismic quiescence that began 13 years before the mainshock (Katsumata 2015). A detailed analysis of the earthquake catalog using a gridding technique (ZMAP) shows the quiescent area is located between 3
and 6
N, which covers the southeastern part of the focal area, including the rupture initiation point of the 2004 Sumatra earthquake. The observed spatial pattern of quiescence can be explained by stress perturbation due to a long-term slow slip located on the deeper edge of the mainshock fault.

Summary It follows from the mentioned hypotheses that there exist various physical mechanisms of occurrence both of the seismic quiescence and of the foreshock activation. For the purpose of more fundamental understanding of such phenomena, the additional laboratory and field works are required. In our

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opinion, special attention should be paid to the following directions: To investigate on the same catalogues of earthquakes both the quiescence and the activation in a complex, paying attention to their distribution in space, time and to magnitudes To compare, where possible, the seismic patterns with the field of deformations being estimated based on the data of satellite geodesy To compare, where possible, the seismic patterns with the data of deep geoelectrical and hydrogeological investigations with the purpose of better understanding of the role of water

Cross-References ▶ Artificial Water Reservoir-Triggered Earthquakes, with Special Emphasis on Koyna, India ▶ Earthquake, Aftershocks ▶ Earthquake, Foreshocks ▶ Earthquake, Magnitude ▶ Earthquakes, Energy ▶ Earthquakes, Intensity ▶ Geomagnetically Induced Currents

Bibliography Braun OM, Peyrard M (2018) Seismic quiescence in a frictional earthquake model. Geophys J Int 213:676–683 Huang QH, Ding X (2012) Spatiotemporal variations of seismic quiescence prior to the 2011 M 9.0 Tohoku earthquake revealed by an improved Region-Time-Length algorithm. Bull Seismol Soc Am 102(4):1878–1883 Kanamori H (1981) The nature of seismicity patterns before large earthquakes. In: Earthquake prediction: an international review. American Geophysical Union, Washington, DC, pp 1–19 Katsumata K (2015) A long-term seismic quiescence before the 2004 Sumatra (Mw 9.1) earthquake. Bull Seismol Soc Am 105(1):167–176 Katsumata K, Sakai S’i (2013) Seismic quiescence and activation anomalies from 2005 to 2008 beneath the Kanto district, central Honshu, Japan. Earth Planets Space 65:1463–1475 Kawamura M, Chen C-c, Wu Y-M (2014) Seismicity change revealed by ETAS, PI, and Z-value methods: a case study of the 2013 Nantou, Taiwan earthquake. Tectonophysics 634:139–155 Kisslinger C (1988) An experiment in earthquake prediction and the 7 May 1986 Andreanof Islands earthquake. Bull Seismol Soc Am 78:218–229 Ma Z, Zhengxiang F, Zhang Y, Wang C, Zhang G, Liu D (1989) Earthquake prediction: nine major earthquake in China. Seismological Press/Springer, Berlin, p 332 Mogi K (1979) Two kinds of seismic gap. Pageoph 117:1172–1186 Nagao T, Takeuchi A, Nakamura K (2011) A new algorithm for the detection of seismic quiescence: introduction of the RTM algorithm, a modified RTL algorithm. Earth Planets Space 63:315–324 Nanjo KZ, Izutsu J, Orihara Y, Furuse N, Togo S, Nitta H, Okada T, Tanaka R, Kamogawa M, Nagao T (2016) Seismicity prior to the 2016 Kumamoto earthquakes. Earth Planets Space 68:187

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Ohtake M, Matumoto T, Latham GV (1977) Seismicity gap near Oaxaca, Southern Mexico, as a probable precursor to a large earthquake. Pageoph 115:375–385 Scholz CH (1990) The mechanics of earthquakes and faulting. Cambridge University Press, New York, p 439 Sobolev G (2001) The examples of earthquake preparation in Kamchatka and Japan. Tectonophysics 338:269–279 Sobolev GA (2011) Seismicity dynamics and earthquake predictability. Nat Hazards Earth Syst Sci 11:1–14 Sobolev GA (2019) Avalanche unstable fracturing formation model. Izv Phys Solid Earth 55(1):1–14 Wen Y-Y, Chen C-C, Wu Y-H, Chan C-H, Wang Y-J, Yeh Y-L (2016) Spatiotemporal investigation of seismicity and Coulomb stress variations prior to the 2010 ML 6.4 Jiashian, Taiwan earthquake. Geophys Res Lett 43:8451–845.7 Wyss M, Habermann RE (1988) Precursory seismic quiescence. Pageoph 126:319–332 Wyss M, Sobolev G, Clippard JD (2004) Seismic quiescence precursors to two M7 earthquakes on Sakhalin Island, measured by two methods. Earth Planets Space 56:725–740

Seismic Ray Theory Vlastislav Červený1 and Ivan Pšenčík1,2 1 Department of Geophysics, Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic 2 Institute of Geophysics, Academy of Sciences of Czech Republic, Praha, Czech Republic

Definition Seismic ray theory

High-frequency asymptotic method of study of seismic wavefields in complex inhomogeneous isotropic or anisotropic media with curved structural interfaces.

Introduction The ray theory belongs to the methods most frequently used in seismology and seismic exploration for forward and inverse modelling of high-frequency seismic body waves. In smoothly varying media with smooth interfaces, it can provide useful approximate solutions of the elastodynamic equation of satisfactory accuracy. Starting from an intuitive description of the propagation of seismic waves along special trajectories – rays, it has developed into a highly sophisticated method, described briefly in this review paper. The ray method has its advantages and limitations. The basic advantages are its applicability to complex, isotropic or anisotropic, laterally varying layered media and its numerical efficiency in such computations. It provides a physical insight into the wave propagation process by separating the wavefield into individual elementary waves and by allowing their

identification. In addition, it makes possible to track the paths (rays) in the medium along which energy of individual waves propagates, an aspect very important in tomography. The ray method also represents an important basis for other related, more sophisticated methods, such as the paraxial ray method, the Gaussian beam summation method, the Maslov method, the asymptotic diffraction theory, various combinations of the ray method with perturbation theory, etc. The ray method also has some limitations. It is approximate and applicable only to smooth media with smooth interfaces, in which the characteristic dimensions of inhomogeneities are considerably larger than the prevailing wavelength of the considered waves. The ray method can yield distorted results and may even fail in some special regions called singular regions. The seismic ray method owes a lot to optics and radiophysics. Although the techniques used in different branches of physics are very similar, there are some substantial differences. The ray method in seismology is usually applied to more complicated structures than in optics or radiophysics. There are also different numbers and types of waves considered in different branches of physics. The first seismological applications of ray concepts date back to the end of the nineteenth century. Then, only kinematics, specifically travel times, were used. Probably the first attempts to use also dynamics (amplitudes and waveforms) were made by Sir H. Jeffreys. The ray series solutions of elastodynamic equation with variable coefficients were first suggested by Babich (1956) and Karal and Keller (1959) for inhomogeneous isotropic media, and by Babich (1961) for inhomogeneous anisotropic media. The Earth’s interior is anisotropic or weakly anisotropic in some of its parts. Seismic anisotropy and its effects on wave propagation play an important role in contemporary seismology and seismic exploration. Consequently, it has also been necessary to develop the ray theory for elastic anisotropic media. It is important to emphasize that, for S waves, the ray theory for anisotropic media does not yield the ray theory for isotropic media in the zero anisotropy limit. For this reason, we describe systematically the ray theory for anisotropic media and also present corresponding formulae for isotropic media, and explain the differences between both of them. S waves require generally a special attention. Wellunderstood phenomenon is propagation of two separate shear waves in anisotropic media. An underestimated phenomenon is shear-wave coupling, which occurs in weakly anisotropic media or in vicinities of shear-wave singularities. In such regions, standard ray theories for anisotropic as well as isotropic media do not work properly. Therefore, we also briefly describe the coupling ray theory for S waves, which fills the gap between ray theories for isotropic and anisotropic media. We give here neither a detailed derivation of raytheoretical expressions nor a relevant systematic

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bibliography. This would extend the text considerably. We refer, however, to several textbooks, in which the ray theory is treated in a considerably greater detail (Červený et al. 1977; Kravtsov and Orlov 1990; Červený 2001; Chapman 2004; Babich and Kiselev 2018). The reader may also find useful information in several review papers devoted to seismic ray theory and its various aspects (Červený et al. 1988, 2007; Virieux 1996; Chapman 2002). Examples of computations based on the ray theory can be found, for example, in Červený et al. (1977) and Gjøystdal et al. (2002). Here we refer only to papers, in which the relevant methods and procedures were first proposed, and/or which give a useful more recent treatment of the subject. We use the following notation. We denote Cartesian coordinates xi and time t. The dots above letters denote partial derivatives with respect to time (üi ¼ @ 2ui/@t2) and the index following the comma in the subscript indicates the partial derivative with respect to the relevant Cartesian coordinate (ui,j ¼ @ui/@xj). We consider high-frequency time-harmonic seismic body waves, with the exponential factor exp(iot), where o is fixed, positive, real-valued circular frequency. The lower-case Roman indices take the values 1, 2, 3, the upper-case indices 1, 2. Hats over bold symbols indicate 3  3 matrices, bold symbols without hats denote 2  2 matrices. The Einstein summation convention over repeating Roman indices is used, with exception of indices in parentheses.

Basic Equations of the Seismic Ray Method For smoothly varying elastic media, the source-free equation of motion reads tij,j  r€ ui ¼ 0:

ð1Þ

Here tij(xn, t), and ui(xn, t) are Cartesian components of stress tensor and displacement vector, respectively, and r is the density. In anisotropic media, the
stress tensor tij and the  infinitesimal strain tensor eij ¼ 12 ui,j þ u j,i are related by Hooke’s law: tij ¼ cijkl ekl ¼ cijkl @uk =@xl :

ð2Þ

cijkl(xn) is a tensor of elastic moduli (stiffness tensor), satisfying symmetry relations cijkl ¼ cjikl ¼ cijlk ¼ cklij. There are, at the most, 21 independent elastic moduli. Inserting Eq. 2 into Eq. 1, we get the elastodynamic equation


 cijkl uk,l , j  r€ ui ¼ 0:

ð3Þ

In the seismic ray method, high-frequency seismic body waves propagating in smoothly varying, isotropic or

anisotropic, media are studied. The formal ray series solution of the elastodynamic equation, see Eq. 3, for the displacement vector u(xn, t) is sought in the form of an asymptotic series in inverse powers of circular frequency o, uðxn , tÞ ¼ exp ½ioðt  T ðxn ÞÞ " # Uð1Þ ðxn Þ Uð2Þ ðxn Þ ð0Þ þ ... : U ðx n Þ þ þ ðioÞ ðioÞ2

ð4Þ

Here T(xn) is the real-valued travel time, U(k), k ¼ 0, 1, 2,. . . are complex-valued vectorial amplitude coefficients. Surfaces T(xi) ¼ const. are called wavefronts (or phase fronts). In perfectly elastic media, functions T(xn), and U(k)(xn) are frequency independent. Also other forms of the ray series have been used in the seismic ray method. For example, Chapman (2004) developed the seismic ray method using the ray series for particle velocity and traction. Such a formal ray series has certain advantages with respect to Eq. 4. Here, however, we consider systematically the traditional ray series for the displacement vector in the form of Eq. 4. Inserting Eq. 4 into elastodynamic equation, see Eq. 3, we obtain a series in inverse powers of o, which equals zero. Consequently, the coefficients of the individual powers of o must also equal zero. This yields a system of equations called the basic recurrence system of equations of the ray method. This system can be used to determine the eikonal equation for travel times T(xn) and, successively, the equations for the amplitude coefficients U(0)(xn), U(1)(xn), U(2)(xn),. . . The equations for U(k)(xn) yield, among others, transport equations. For a detailed derivation of the basic system of equations of the ray method see Červený (2001, Sect. 5.7). The vectorial amplitude coefficients U(k)(xn), k ¼ 1, 2,. . . can be expressed as a sum of the principal component and additional component. The zero-order amplitude coefficient U(0)(xn), which has no additional component, is an exception. Its principal component is determined from the transport equation discussed later. The principal component of U(k)(xn) for k > 0 is the projection of U(k)(xn) into the unit vector parallel to the zero-order amplitude coefficient U(0)(xn), the additional component of U(k)(xn) is the remaining part of U(k)(xn). The complexity of the equations for higherorder amplitude coefficients U(k) increases rapidly with increasing k. Moreover, the higher-order amplitude coefficients are unstable and usually do not increase the accuracy notably. They are very sensitive to fine details of the medium. The instability of the amplitude coefficients increases with increasing k. For these reasons, only the zero-order coefficient U(0)(xn), with the additional component of U(1)(xn), at the most, has been used in seismological applications. In the following, we shall concentrate on the zero-order ray approximation U(0)(xn) only.

S

1474

Seismic Ray Theory

Gm ¼ Gik gi ðmÞ gk ðmÞ ¼ aijkl p j pl gi ðmÞ gk ðmÞ :

The zero-order approximation of the ray method reads: uðxn , tÞ ¼ Uðxn Þ exp ½ioðt  T ðxn ÞÞ:

ð5Þ

In Eq. 5, we have dropped the superscript (0) of U(xn). We call U(xn) the complex-valued vectorial amplitude. In smooth, laterally varying media, containing smooth structural interfaces, the zero-order approximation of the ray method in Eq. 5 usually offers sufficiently accurate results, particularly for travel time T(xn). Its great advantage is that it allows one to work with frequency-independent travel time and amplitude. However, if the medium under consideration becomes more and more complex (less smooth) or if a singularity is close, vectorial amplitude U(xn) becomes less accurate. In structures exceeding certain degree of complexity or close to singularities, the ray method may yield inaccurate results or even fail. The first equation of the basic recurrence system of equations of the ray method reads: ðGik  dik ÞU k ¼ 0,

i ¼ 1, 2, 3:

Gik ¼ aijkl p j pl :

ð7Þ

In Eq. 7, pi are the Cartesian components of the slowness vector p, pi ¼ @T=@xi

ð8Þ

and aijkl ¼ cijkl/r are density-normalized elastic moduli. Note that the classical Christoffel matrix, with elements aijklnjnl, contains components of the real-valued unit vector n (perpendicular to the wavefront) instead of p. For this reason, we call the matrix in Eq. 7 the “generalized” Christoffel matrix. The relation between pi and ni is pi ¼ ni/C , where C is the phase velocity. The generalized 3  3 Christoffel matrix in solid elastic media is symmetric (Γik ¼ Γki), positive definite (Γikaiak > 0, where ai are components of any non-vanishing real-valued vector), and homogeneous function of the second degree in pi (Γik(xn, apj) ¼ a2Γik(xn, pj) for any nonvanishing constant a). It has three real-valued positive eigenvalues Gm(xn, pj), and three corresponding real-valued unit eigenvectors g(m)(xn, pj), m ¼ 1, 2, 3. Gm and g(m) are solutions of the eigenvalue equations ðmÞ

ðGik  dik Gm Þgk

¼ 0,

i ¼ 1, 2, 3:

For isotropic media, elastic moduli cijkl(xn) can be specified in terms of Lamé’s elastic moduli l(xn) and m(xn), describing isotropic media, as follows:
 cijkl ¼ ldij dkl þ m dik djl þ dil djk :

ð9Þ

Eigenvectors g(1), g(2), g(3) are mutually perpendicular. Eigenvalue Gm and the relevant eigenvector g(m) are mutually related as follows:

ð11Þ

Elements of the generalized Christoffel matrix are then given by the relation: Gik ¼

lþm m pi pk þ dik pn pn : r r

ð12Þ

In isotropic media, the expressions for eigenvalues and eigenvectors of the generalized Christoffel matrix can be determined analytically: G1 ¼ G2 ¼ b2 pk pk ,

ð6Þ

Here Γ is the 3  3 generalized Christoffel matrix with elements given by the relation:

ð10Þ

G 3 ¼ a2 pk pk :

ð13Þ

b2 ¼ m=r:

ð14Þ

Here a2 ¼ ðl þ 2mÞ=r,

The eigenvector relevant to the eigenvalue G3 equals n, the unit vector perpendicular to the wavefront. The eigenvectors relevant to coinciding eigenvalues G1 and G2 are mutually perpendicular unit vectors situated arbitrarily in the plane perpendicular to n and tangent to the wavefront.

Eikonal Equation. Polarization Vector The comparison of Eq. 6 with the eigenvalue equation for the 3  3 generalized Christoffel matrix, see Eq. 9, shows that Eq. 6 is satisfied, if the eigenvalue Gm of the generalized Christoffel matrix satisfies the relation
 Gm xi , p j ¼ 1,

ð15Þ

and if the complex-valued vectorial amplitude U of the wave under consideration is related to eigenvector g(m) as follows: U ¼ AgðmÞ :

ð16Þ

Equation 15 is the important eikonal equation. It is a nonlinear, first-order partial differential equation for travel time T(xn). Equation 16 shows that displacement vector U is parallel to the appropriate eigenvector g(m) of the generalized Christoffel matrix. For this reason, we call g(m) the polarization vector. Symbol A(xn) denotes the complex-valued, frequency-independent, scalar amplitude.

Seismic Ray Theory

1475

Taking into account that Gm is a homogeneous function of the second degree in pi, where p ¼ C 1 n, we obtain Gm(xi, pj) ¼ C 2 Gm(xi, nj). This and Eqs. 10 and 15 yield

 C 2 xi , n j ¼ Gm xi , n j ¼ aijkl n j nl gi ðmÞ gk ðmÞ :

ð17Þ

Phase velocity C is the velocity of the propagation of the wavefront in the direction n. The phase-velocity vector C ¼ C(xi, nj)n has the direction of n, i.e., it is perpendicular to the wavefront. It follows from Eq. 17 that the squares of phase velocity C are eigenvalues Gm(xi, nj) of the classical Christoffel matrix with elements aijklnj nl. Generally, eigenvalues Gm, m ¼ 1, 2, 3, of the generalized Christoffel matrix are mutually different. They correspond to three high-frequency body waves propagating in inhomogeneous anisotropic media. Smaller eigenvalues, G1 and G2, belong usually to S1 and S2 waves, largest, G3, to P wave. Such situations are considered here. If the eigenvalues are mutually different, their polarization vectors can be determined uniquely. If two eigenvalues coincide, we speak of the degenerate case of the eigenvalue problem. The corresponding eigenvectors can then be chosen as mutually perpendicular vectors situated arbitrarily in the plane perpendicular to the eigenvector related to the different eigenvalue. Eigenvalues may coincide locally, along certain lines or at certain points, which correspond to the so-called S wave singular directions, or may be close to one another globally in a vicinity of singular directions or in weakly anisotropic media. The approximate but unique determination of polarization vectors in the latter situations is possible using perturbation approach (Jech and Pšenčík 1989). In isotropic media, the S wave eigenvalues G1 and G2 coincide globally, see Eq. 13. Consequently, in isotropic media, the S waves are controlled by a single eikonal equation and we have thus only two different eikonal equations corresponding to P and S waves. As the equations for the eigenvalues in isotropic media can be determined analytically, see Eq. 13, we can express the eikonal equations, see Eq. 15, for P and S waves explicitly: a2 pk pk ¼ 1

for P waves,

ð18Þ

b2 pk pk ¼ 1

for S waves:

ð19Þ

In isotropic media, the generally complex-valued amplitude vector U can be expressed in the simple form, see Eq. 16, only for P waves. In this case the polarization vector g(3) ¼ n, i.e., it is perpendicular to the wavefront. For S waves, U must be considered in the following form: U ¼ Bgð1Þ þ Cgð2Þ :

ð20Þ

Here g(1) and g(2) are two mutually perpendicular unit vectors in the plane tangent to the wavefront, i.e., perpendicular to the vector n. The determination of g(1) and g(2) along the

ray is explained later, see Eq. 38. Symbols B(xn) and C(xn) are the corresponding, generally complex-valued scalar amplitudes. In the seismic ray method, it is common to express the eikonal equation, see Eq. 15, in Hamiltonian form. Hamiltonian ℋ(xi, pj) may be introduced in various ways. We shall consider the Hamiltonian, which is a homogeneous function of the second degree in pi. For inhomogeneous anisotropic media, we can introduce the Hamiltonian expressed in terms of Gm(xi, pj), see Eq. 10:
 1
 1 ðmÞ ðmÞ ℋ xi , p j ¼ Gm xi , p j ¼ aijkl p j pl gi gk : 2 2

ð21Þ

The eikonal equation, see Eq. 15, then reads:
 1 ℋ xi , p j ¼ : 2

ð22Þ

It holds for anisotropic as well as isotropic media. From Eqs. 13 and 21, we get for isotropic inhomogeneous media:
 1 ℋ xi , p j ¼ V 2 ðxi Þpk pk , 2

ð23Þ

where V ¼ α for P waves and V ¼ β for S waves.

Ray Tracing and Travel-Time Computation The eikonal equation in Hamiltonian form as shown in Eq. 22, with pj ¼ @T/@xj, is a nonlinear partial differential equation of the first order for travel time T(xi). It can be solved by the method of characteristics. The characteristics of eikonal equation, see Eq. 22, are spatial trajectories, along which Eq. 22 is satisfied, and along which travel time T can be computed by quadratures. The characteristics of the eikonal equation represent rays. The characteristics of the eikonal equation expressed in general Hamiltonian form are described by a system of nonlinear, ordinary differential equations of the first order: dxi @ℋ ¼ , @pi du

dpi @ℋ ¼ , @xi du

dT @ℋ ¼ pk : du @pk

ð24Þ

Here u is a real-valued parameter along the ray. The relation between parameter u and the travel time along the ray depends on the form of the Hamiltonian used; see the last equation in Eq. 24. For Hamiltonians, which are homogeneous functions of the second degree in pi, the Euler equation for homogeneous functions yields pk@ℋ/@pk ¼ 2ℋ. If we consider Hamiltonian in the form of Eq. 22, we get dT/du ¼ 1 from Eq. 24. For travel time T along the ray, denoted t ¼ T, Eq. 24 simplifies to:

S

1476

Seismic Ray Theory

dxi @ℋ , ¼ dt @pi

dpi @ℋ : ¼ dt @xi

ð25Þ

This system of equations is usually called the ray tracing system. Solution of the ray tracing system, see Eq. 25, with appropriate initial conditions yields xi(t), the coordinates of points along the ray trajectory, and pi(t), the Cartesian components of the slowness vectors along the ray. The travel time T along the ray is obtained automatically, T ¼ t. Inserting Eq. 21 to Eq. 25, we obtain the ray tracing system for m-th wave in the inhomogeneous anisotropic medium: dxi ðmÞ ðmÞ ¼ aijkl pl g j gk , dt

dpi 1 @ajkln ðmÞ ðmÞ ¼ pp g g : dt 2 @xi k n j l

ð26Þ

In the derivation of the first equation of Eq. 26 for @ℋ/@pi, we  ðmÞ ðmÞ took into account that Gik @ gi gk =@pn ¼ 0. An alternative version of ray-tracing equations, see Eq. 26, was derived by Červený (1972), in which the eigenvectors g(m) were not used. The initial conditions for the ray tracing system, see Eq. 26, are xi ¼ x0i, pi ¼ p0i, where x0i and p0i satisfy the eikonal equation, see Eq. 22, corresponding to the wave we wish to compute (P, S1 or S2). Components p0i of the initial slowness vector p0 can be expressed as p0i ¼ n0i/C(x0j), where C is the relevant phase velocity. The eikonal equation, see Eq. 22, is then satisfied along the whole ray. In inhomogeneous isotropic media, the ray tracing system, see Eq. 25 with Eq. 23, yields dxi ¼ V 2 pi , dt

dpi @ ln V : ¼ @xi dt

ð27Þ

The initial conditions for the ray tracing system, see Eq. 27, are again xi ¼ x0i, pi ¼ p0i, where p0i ¼ n0i/V(x0j). Here V ¼ α for P waves, and V ¼ β for S waves. As t is the travel time along the ray, dxi/dt in Eq. 26 represents the Cartesian components U i of the ray-velocity vector U of the m-th wave: U i ¼ aijkl pl g j ðmÞ gk ðmÞ :

In inhomogeneous isotropic media, Eq. 28 for the rayvelocity vector yields U ¼ V2p. For the phase-velocity vector, using Eq. 17, we get C ¼ V2p, where V ¼ α for P waves, and V ¼ β for S waves. Thus, the ray-velocity and phase-velocity vectors are identical in isotropic media. Figure 1 shows mutual orientation of ray-velocity vector U, phase-velocity vector C (parallel to slowness vector p), and polarization vector g of a P wave propagating in an isotropic (left) and anisotropic (right) medium. While U , C, and g are parallel in isotropic media, they generally differ in anisotropic media. For S waves, the vectors U and C have similar orientation as in the case of P waves. The polarization vectors g are, however, perpendicular (isotropic medium) or nearly perpendicular (anisotropic medium) to the ray. Ray tracing systems in Eqs. 26 and 27 can be simply solved if the initial values x0i and p0i are specified at some point S of the ray. We then speak of initial-value ray tracing. The standard numerical procedures of solving the system of ordinary differential equations of the first order with specified initial conditions can then be used (for example, RungeKutta). A very important role in seismology is played by boundary-value ray tracing, in which we seek the ray, satisfying prescribed boundary conditions. The typical boundary value problem is two-point ray tracing, in which we seek the ray connecting two specified points. Mostly, the controlled initial-value ray tracing (controlled shooting method) is used to solve this problem (Červený et al. 2007). Boundary-value ray tracing is considerably more complicated than initialvalue ray tracing. There are four important differences between initial value ray tracing in isotropic and anisotropic media. First: In anisotropic media, we deal with three waves, P, S1, and S2, in isotropic media with two waves, P and S, only. Second: In inhomogeneous anisotropic media, ray tracing system, see Eq. 26, is the same for all three waves. The wave under consideration is specified by the initial conditions, which must satisfy the eikonal equation of the considered wave. In isotropic inhomogeneous media, the ray tracing systems are different for P and S waves, see Eq. 27 with V ¼ α and V ¼ β, p||g|| U

ð28Þ

In elastic anisotropic media, the ray-velocity vector U is also called the group-velocity vector or the energy-velocity vector. As indicated by the name, the energy velocity vector U represents the velocity of the energy propagation. In anisotropic media, the ray-velocity vector U must be strictly distinguished from the phase-velocity vector C . In inhomogeneous anisotropic media, the ray-velocity and phase-velocity vectors U and C are generally different, both in size and direction. Vector U is always greater than C. The two vectors are equal (in size and direction) only in special directions, called longitudinal directions.

Wavefront

U

p

Wavefront

g

Ray Ray P wave in an isotropic medium

P wave in an anisotropic medium

Seismic Ray Theory, Fig. 1 Slowness vector p (perpendicular to the wavefront), ray-velocity vector U (tangent to the ray), and polarization vector g of a P wave propagating in an isotropic (left) and anisotropic (right) medium. For simplicity, the three vectors in the right-hand plot are shown in one plane. In general, this is not the case in anisotropic media

Seismic Ray Theory

1477

respectively. Third: In isotropic media, the initial direction of the slowness vector specifies directly the initial direction of the ray (as the tangents to the ray and the slowness vector have the same directions). In anisotropic media, the direction of the ray is, generally, different from the direction of the slowness vector. Nevertheless, we have to use p0i as the initial values for the ray tracing system. The ray-velocity vector U can be simply calculated from slowness vector p at any point of the ray, including the initial point, however, not vice versa. Fourth: Ray tracing for P and S waves is regular everywhere in inhomogeneous isotropic media. In anisotropic media, problems arise with tracing S-wave rays in vicinities of singular directions, or if medium is nearly isotropic (quasi-isotropic). The problem of ray tracing and travel-time computation in inhomogeneous media has been broadly discussed in the seismological literature, particularly for inhomogeneous isotropic media. Many ray tracing systems and many suitable numerical procedures for performing ray tracing have been proposed. For 1-D isotropic media (vertically inhomogeneous, radially symmetric), the ray tracing systems may be simplified so that they reduce to simple quadratures, wellknown from classical seismological textbooks. For P-wave travel time T and angular distance Δ, they read: ð T ð pÞ ¼

1

p2 a2 r2

1=2

dr , a

 1=2 pa p2 a2 dr: DðpÞ ¼ 2 1  2 r r ð

ð29Þ

Here, α ¼ α(r) is the P wave velocity as the function of the radius r, and p is the ray parameter, p ¼ (r sin i)/α, where i is the angle made by the ray and the radius, see Eq. (9.21) of Aki and Richards (2002), where additional details can be found. Standard programs for ray tracing and travel-time computations in laterally varying isotropic and anisotropic structures are available, see, for example, program packages SEIS (2D isotropic models), CRT, and ANRAY (3D isotropic/ anisotropic models) available at http://sw3d.cz/. Programs for anisotropic media have, however, problems with S wave computations in weakly anisotropic media and in the vicinities of shear-wave singularities. In such cases, the standard ray theory should be replaced by the coupling ray theory. Ray tracing may also serve as a basis for the so-called wavefront construction method (Gjøystdal et al. 2002). In this case, for a selected wave, wavefronts with travel times T ¼ T0 + kΔT are computed successively from the previous wavefronts with travel times T ¼ T0 + (k  1)ΔT. The wavefront construction method has found broad applications in seismic exploration. Let us consider a two-parametric system of rays, call it the ray field, and specify the individual rays in the ray field by ray parameters γ1, γ2. Ray parameters γ1, γ2 may represent,

e.g., the take-off angles at a point source or the curvilinear Gaussian coordinates of initial ray points along the initial surface. The family of rays with ray parameters within the limit [γ1, γ1 + dγ1], [γ2, γ2 + dγ2], is called the elementary ray tube or briefly the ray tube. We further introduce ray coordinates γ1, γ2, γ3 in such a way that γ1, γ2 are ray parameters, and γ3 is some monotonic parameter along a ray (arclength s, travel time t, etc.). Here we consider γ3 ¼ t, but our results may be simply modified for any other monotonic parameter b γ3. We further introduce the 3  3 transformation matrix Q from ray to Cartesian coordinates with elements Qij ¼ @xi/@γj. The Jacobian of transformation from ray to Cartesian coordib can be expressed as follows: nates, det Q, b ðtÞ ¼ ð@xðtÞ=@g1  @xðtÞ=@g2 ÞT U ðtÞ: detQ

ð30Þ

The vectorial product in Eq. 30 has the direction of the normal to the wavefront, specified by n ¼ C p. As p(t) U(t) ¼ 1, see Eqs. 10, 15, and 28, we also obtain b ðtÞ ¼ C ðtÞ j ð@xðtÞ=@g1  @xðtÞ=@g2 Þ j : det Q

ð31Þ

b ðtÞ equals C (t)dΩ(t), where Thus the Jacobian det Q dΩ(t) ¼ |(@x(t)/@γ1  @x(t)/@γ2)| is the scalar surface element cut out of the wavefront by the ray tube. It measures the expansion or contraction of the ray tube, see Fig. 2. For this b ðtÞ is also often called the geometreason, the 3  3 matrix Q rical spreading matrix and various quantities related to b ðtÞ are called geometrical spreading. It plays an impordet Q tant role in the computation of the ray-theory amplitudes.

Transport Equation. Computation of Ray-Theory Amplitudes The second equation of the basic recurrence system of equations of the ray method yields the transport equation for the A0 dΩ B0

S

D0 0

C0

A

D dΩ

B

C

Seismic Ray Theory, Fig. 2 Elementary ray tube. dΩ0 and dΩ are scalar surface elements cut out of the wavefront by the ray tube. This means that in isotropic media, the normals to dΩ0 and dΩ are parallel to rays. In anisotropic media, they are not

1478

Seismic Ray Theory

scalar ray-theory amplitude A(xi) specifying the zero-order vectorial amplitude U(xn), see Eq. 16. The transport equation is a partial differential equation of the first order. It can be expressed in several forms. One of them, valid for isotropic and anisotropic media, and for any wave, reads
 ∇ rA2 U ¼ 0:

ð32Þ

It is common to solve the transport equation along the ray. ∇·U can then be expressed as follows: h  i b =dt ∇ U ¼ d ln detQ

ð33Þ

(Červený 2001, Eq. 3.10.24). Inserting Eq. 33 into Eq. 32 yields the transport equation in the form of the first-order ordinary differential equation along the ray:   b ðtÞ =dt ¼ 0: d rðtÞA2 ðtÞdetQ

ð34Þ

This yields a simple form of the continuation relation for A(t) along the ray: "

b ð t0 Þ rðt0 ÞdetQ Að t Þ ¼ b ð tÞ rðtÞdetQ

#1=2 Aðt0 Þ:

ð35Þ

We obtain another suitable continuation relation for amplitudes along the ray by introducing a special local Cartesian coordinate system y1, y2, y3, varying along the ray. We call it the wavefront orthonormal coordinate system. At any point of the ray specified by γ3 ¼ t, the y3 axis is parallel to slowness vector p, and the y1, y2 axes are confined to the plane tangential to the wavefront at γ3 ¼ t. Axes y1 and y2 are mutually perpendicular. If we denote the 3  3 transformation matrix from ray coordinates to wavefront orthonormal coordinates b ðyÞ, then by Q b ðtÞ ¼ detQ b ðyÞ ðtÞ ¼ C ðtÞdetQðyÞ ðtÞ: detQ

ð36Þ

Here C(t) is the phase velocity, and Q(y)(t) is the 2  2 upperb ðyÞ ðtÞ. Using Eq. 36 in Eq. 35, we obtain left submatrix of Q the continuation relation in an alternative form: "

rðt0 ÞC ðt0 ÞdetQðyÞ ðt0 Þ Að t Þ ¼ rðtÞC ðtÞdetQðyÞ ðtÞ

wavefront, but the axis q3 is taken in a different way than y3, for example, along the ray. This is, e.g., the case of the wellknown ray-centered coordinate system q1, q2, q3. We have det Q(q)(t) ¼ det Q(y)(t), where Q(q) is the 2  2 transformation matrix from ray (γ1, γ2) to ray-centered (q1, q2) coordinates. Transport equations for P and S waves in isotropic media may be also expressed in the form of Eq. 32. The expression is straightforward for P waves. For S waves, transport equations for scalar amplitudes B and C in Eq. 20 are generally coupled. They decouple only if the unit vectors g(1) and g(2) in Eq. 20 satisfy the following relation along the ray:

#1=2 Aðt0 Þ:

ð37Þ

An important property of the continuation relation presented in Eq. 37 is that det Q(y)(t) is uniquely determined by coordinates y1 and y2, confined to the plane tangential to the wavefront at t. Thus, Eq. 37 remains valid for any coordinate systems qi (even nonorthogonal), in which the coordinate axes q1 and q2 are orthogonal in the plane tangential to the

  dgðMÞ =dt ¼ gðMÞ ∇b n,

M ¼ 1, 2:

ð38Þ

In the terminology of the Riemanian geometry, vector g(M) satisfying Eq. 38 is transported parallelly along the ray. If g(1) and g(2) are chosen as mutually perpendicular and perpendicular to n at one point of the ray, Eq. 38 guarantees that they have these properties at any point of the ray. Consequently, g(1) and g(2) are always perpendicular to the ray and do not rotate around it as the S wave progresses. As g(1), g(2), and n are always orthonormal, and n is known at any point of the ray, it is not necessary to use Eq. 38 to compute both vectors g(M). One of them can be determined from the orthonormality condition, once the other has been computed using Eq. 38. b ðtÞ in Eq. 35 may become zero at some Quantity det Q C point t ¼ t . This means that the cross-sectional area of the ray tube shrinks to zero at t ¼ tC. The relevant point t ¼ tC of the ray is called the caustic point. At the caustic point, the ray solution is singular and yields an infinite amplitude there. In passing through the caustic point tC along the ray, the argub (t)]1/2 may change by π/2 or π (Kravtsov ment of [det Q and Orlov 1999). The former case corresponds to the caustic point of the first order, see Fig. 3a, during which the ray tube shrinks to an elementary arc, the latter case corresponds to the caustic point of the second order, see Fig. 3b, during which the ray tube shrinks to a point. If the caustic point tC is situated between t0 and t, it is common to introduce the phase shift TC (t, t0) due to caustic using the relation: "

b ð t0 Þ detQ b ð tÞ detQ

#1=2

1=2

b

 

detQðt0 Þ ¼

exp iT C ðt, t0 Þ : b ð tÞ

detQ

ð39Þ

The phase shift due to the caustic is cumulative. If the ray passes through several caustic points along the ray between t0 and t, the phase shift due to caustics is the sum of the individual phase shifts. For the choice of the exponential factor exp(iot) in Eq. 4, it has the form T C ðt, t0 Þ ¼  12 pk ðt, t0 Þ, where k(t, t0) is an integer, called the KMAH index (to acknowledge the work by Keller, Maslov, Arnold and Hörmander in this field). The continuation relation for raytheory amplitudes given in Eq. 35 can then be modified to read:

Seismic Ray Theory Seismic Ray Theory, Fig. 3 Caustic points of (a) the first order and (b) second order (Fig. 3.13 of Červený 2001)

1479

a

D0

C0

A0

B0

A

b

D0

B

C0 D

A0

C

B0

B

C

" #1=2 b ðt0 Þ j   rðt0 Þ j detQ Að t Þ ¼ exp iT C ðt, t0 Þ Aðt0 Þ: b ðtÞ j rðtÞ j detQ ð40Þ Equation 37 can be transformed to the form analogous to Eq. 40 as the zeros of det Q(y)(t) are situated at the same b ðtÞ. points tC on the ray as the zeros of det Q The KMAH index can be calculated along the ray as a byproduct of dynamic ray tracing. For detailed derivations and discussion see Bakker (1998) and Klimeš (2010). There are some differences between the KMAH indices along the rays in isotropic and anisotropic media. In isotropic media, the KMAH index always increases when the ray passes through a new caustic point, either by one or two. In anisotropic media, however, it may also decrease by one or two at some caustic points. This happens only for S waves as a consequence of the concave form of the slowness surface of the corresponding S wave. Alternative expressions for ray-theory amplitudes and their mutual relations can be found in Klimeš (2019).

Dynamic Ray Tracing. Paraxial Approximations As we can see in Eq. 35, the computation of the ray-theory b where Q b ðtÞ characamplitudes requires knowledge of det Q, terizes the properties of the ray field in the vicinity of the ray b ðtÞ can be computed by the procedure under consideration. Q b ð tÞ called dynamic (or paraxial) ray tracing. In addition to Q

A

D

with elements Qij(t) ¼ @xi/@γj, we also have to introduce a bðtÞ with elements Pij(t) ¼ @pi/@γj. The new 3  3 matrix P equation for Pij must be included to obtain the linear dynamic ray tracing system. Differentiating ray tracing system shown in Eq. 25 with respect to γj, we can easily obtain a system of linear ordinary differential equations of the first order for Qij and Pij, dQij @2ℋ @2ℋ Qkj þ P , ¼ @pi @xk @pi @pk kj dt dPij @2ℋ @2ℋ ¼ Qkj  P , @xi @xk @xi @pk kj dt

ð41Þ

see Červený (1972). This system is usually called the dynamic ray tracing system, and the relevant procedure dynamic ray tracing. It can be solved along a given ray Ω, or together with ray tracing system, see Eq. 25. The dynamic ray tracing system shown in Eq. 41 may be expressed in various forms. Instead of Cartesian coordinates xi, we can use the wavefront orthonormal coordinates yi, or the ray-centered coordinates qi. Then, instead of the 3  3 b and P, b it is sufficient to seek the 2  2 matrices matrices Q (y) (y) (q) Q , P , or Q , P(q). This reduces the number of DRT equations, but complicates their right-hand sides (Červený 2001, Sect. 4.2). As the dynamic ray tracing system presented in Eq. 41 is of the first order and linear, we can compute its fundamental matrix consisting of six linearly independent solutions. The 6  6 fundamental matrix of the dynamic ray tracing system in Eq. 41, specified by the 6  6 identity matrix at an arbitrary

S

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point t ¼ t0 of the ray is called the ray propagator matrix and denoted by P(t, t0). The 6  6 ray propagator matrix P(t, t0) is symplectic:  PT ðt, t0 ÞJPðt, t0 Þ ¼ J,

with J ¼

0

I

I 0

:

ð42Þ

are identical, therefore, they can be expressed in terms of the same 2  2 matrices Q1(t, t0), Q2(t, t0), P1(t, t0), and P2(t, t0). Matrices Q1(t, t0), P1(t, t0) correspond to the plane-wavefront initial conditions at t0, and matrices Q2(t, t0), P2(t, t0) to the point-source initial conditions at t0, see Fig. 4. The 2  2 matrix Q2(t, t0) plays an important role in computing the ray-theory Green function. The quantity

b ðt0 Þ, P bðt0 Þ, we can compute If we know the matrices Q b b QðtÞ , PðtÞ at any point t of the ray by a simple matrix multiplication b ðtÞ Q b ð tÞ P

!

!

¼ Pðt, t0 Þ

b ð t0 Þ Q : b ð t0 Þ P

ð43Þ

The ray propagator matrix P(t, t0) satisfies the chain rule, P(t, t0) ¼ P(t, t1)P(t1, t0), where point t1 is situated arbitrarily on the ray. It is simple to compute the inverse of P(t, t0): P1(t, t0) ¼ P(t0,t). We can express P(t, t0) in the following way: Pðt, t0 Þ ¼

b 1 ðt, t0 Þ Q b1 ðt, t0 Þ P

! b 2 ðt, t0 Þ Q , b2 ðt, t0 Þ P

ð44Þ

b 2 ðt, t0 Þ, P b 1 ðt, t0 Þ, Q b1 ðt, t0 Þ, and P b2 ðt, t0 Þ are 3  3 where Q matrices. Equation 43 can be used to obtain a very important quanb ðtÞ of second derivatives of the tity – the 3  3 matrix M travel-time field with respect to Cartesian coordinates, with elements Mij ¼ @ 2T/@xi@xj:  1 b ð tÞ b ðtÞ ¼ P b ð tÞ Q M :

ð45Þ

b ðtÞ plays an important role in the computation of Matrix M travel time not only along the ray Ω but also in its “quadratic” paraxial vicinity:

ℒðt, t0 Þ ¼ jdetQ2 ðt, t0 Þj1=2

is called the relative geometrical spreading. As in Eq. 45, we can define the 2  2 matrix of the second derivatives of the travel-time field with respect to y1, y2 or q1, q2 as follows: MðtÞ ¼ PðtÞðQðtÞÞ1 :

ð48Þ

Let us briefly summarize several useful ray-theory quantities and applications, which rely fully or partly on dynamic ray tracing. For derivations and more detailed explanations, see Červený (2001, Chap. 4), where also many other applications and references can be found: (1) Paraxial travel times. (2) Paraxial slowness vectors. (3) Paraxial rays. (4) Curvature of the b and the relewavefront. (5) Matrix of geometrical spreading Q b vant matrix P. (6) Continuation relations for ray-theory amplitudes along the ray. (7) Relative geometrical spreading. (8) Phase shift due to caustics. (9) Ray-theory elastodynamic Green function. (10) Higher-order spatial derivatives of the travel-time field. (11) Fresnel volumes and Fresnel zones. (12) Surface-to-surface propagator matrix. (13) Boundary-value problems in fourparametric system of paraxial rays, including two-point ray tracing. (14) Factorization of the geometrical spreading. Dynamic ray tracing is also required in the computation of the two-point paraxial travel times in inhomogeneous, isotropic or anisotropic media (bin Waheed et al. 2013). This approach offers an efficient way of the determination of travel times between sources and receivers situated close to a known



T T 1
b ð tÞ T ðxÞ ¼ T xO þ x  xO pðtÞ þ x  xO M 2
  x  xO : ð46Þ In Eq. 46, x denotes an arbitrary point in the paraxial vicinity of the ray Ω, close to point xΩ ¼ xΩ(t) on the ray Ω; slowness b ðtÞ are given at xΩ. The possivector p(t) and the matrix M bility of computing the travel time in the paraxial vicinity of the ray has many important applications. The properties of the 6  6 ray propagator matrix P(t, t0) described above are also valid for the 4  4 ray propagator matrices P(y)(t, t0) or P(q)(t, t0) expressed in wavefront orthonormal coordinates yi or ray-centered coordinates qi. The ray propagator matrices P(y)(t, t0) and P(q)(t, t0)

ð47Þ

Ray Ω Ray Ω

τ0 τ0 Plane-wavefront initial conditions

Point-source initial conditions

Seismic Ray Theory, Fig. 4 Plane-wavefront and point-source initial conditions for dynamic ray tracing. In anisotropic media, rays are not perpendicular to the wavefront

Seismic Ray Theory

1481

reference ray and specified in Cartesian coordinates. Dynamic ray tracing is also needed in the investigation of chaotic rays and in computations of Lyapunov exponents (Červený et al. 2007), in ray perturbation methods and in modifications and extensions of the ray method such as coupling ray method, Maslov method, Gaussian beam and Gaussian packet summation methods, in Kirchhoff-Helmholtz method, and in various diffraction problems.

Coupling Ray Theory for S Waves in Anisotropic Media In inhomogeneous weakly anisotropic media, the standard ray theory described above yields distorted results since it is unable to describe the coupling of S1 and S2 waves propagating with approximately equal phase velocities. This problem can be removed by using the coupling ray theory. In the coupling ray theory, the amplitudes of the two S waves can be computed along a trajectory called the common ray (Bakker 2002; Klimeš 2006). The closer the common ray approximates actual S wave rays, the more accurate results the coupling ray theory yields. The common rays can be constructed in a reference isotropic medium or in the actual anisotropic medium. A convenient option is to compute common rays using ray-tracing equations given in Eq. 25 with the Hamiltonian given as
 1

 ℋ xi , p j ¼ G1 xi , p j þ G2 xi , p j : 4

ð49Þ

dr1 =dt dr2 =dt

! ¼

0 exp ðio½t2 ðtÞt1 ðtÞÞ d’ dt exp ðio½t1 ðtÞt2 ðtÞÞ 0

!

r1

! ,

r2 ð52Þ

where the angular velocity d’/dt of the rotation of the eigenvectors g(1) and g(2) is given by d’ dgð1Þ dgð2Þ ¼ gð2Þ ¼ gð1Þ : dt dt dt

ð53Þ

For detailed description of the algorithm, see Bulant and Klimeš (2002), for its generalization to the prevailingfrequency approximation of the coupling ray theory, which is effectively independent of frequency, see Klimeš and Bulant (2016). It allows to treat the coupling ray-theory results as standard ray-theory results because it avoids repeated calculations for varying frequencies. There are many possible modifications and approximations of the coupling ray theory. In some of them, the amplitude vector U of coupled S waves is sought along the common ray in the form presented in Eq. 20, in which the amplitude factors B and C can be expressed as BðtÞ ¼ AðtÞℬðtÞ CðtÞ ¼ AðtÞC ðtÞ:

ð54Þ

In Eq. 54, A(t) is again the scalar ray amplitude given in In Eq. 49, G1 and G2 are eigenvalues of the generalized Eqs. 35 or 37 calculated along the common S-wave ray. Christoffel matrix given in Eq. 7, corresponding to S1 and There are many ways how to evaluate factors ℬ and C (Kravtsov 1968; Červený et al. 2007). Here we present a S2 waves. The coupling ray theory solution is sought in the form combination of coupling ray theory and of the first-order ray tracing (Farra and Pšenčík 2010, see also section on “Ray (Coates and Chapman 1990; Bulant and Klimeš 2002): Perturbation Methods”). In the approximation of Farra and h i Pšenčík (2010), the common ray is obtained as the first-order ð1Þ ð2Þ uðt, tÞ ¼ AðtÞ r 1 ðtÞg ðtÞ exp ðiot1 Þ þ r 2 ðtÞg ðtÞ exp ðiot2 Þ ray, see section on “Ray Perturbation Methods”. The vectors exp ðiot Þ: g(K), appearing in Eq. 20, specify the first-order approximað50Þ tion of the S wave polarization plane. The factors ℬ and C in Eq. 54 are then obtained as a solution of two coupled ordinary Here, A(t) is the scalar amplitude given in Eqs. 35 or 37 differential equations, which result from the corresponding calculated along the common ray. The symbols g(1) and g(2) two coupled transport equations: denote the S wave eigenvectors of the generalized Christoffel    dℬ=dt M12 ℬ io M11  1 matrix Γ(xi, pj) calculated along the common ray. The travel ¼ : ð55Þ 2 dC =dt M12 M22  1 C times t1 and t2 are travel times corresponding to the above vectors g(1) and g(2). They can be obtained by quadratures Evaluation of the matrix M with elements MIJ is simple, see along the common ray: Eqs. 7 and 20 of Farra and Pšenčík (2010). h i1=2 h i1=2 The resulting equations reduce to standard ray-theory ð1Þ ð1Þ ð2Þ ð2Þ dt1 =dt ¼ Gik gi gk , dt2 =dt ¼ Gik gi gk : ð51Þ equations in inhomogeneous isotropic media, they describe properly S wave coupling in inhomogeneous weakly anisoThe amplitude factors r1 and r2 are solutions of two coupled tropic media and even yield separate S waves when anisotordinary differential equations (Coates and Chapman 1990): ropy is stronger. Common S wave rays are regular

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Seismic Ray Theory

everywhere. They do not suffer from problems well known from tracing rays of individual S waves in anisotropic media and are suitable for investigating shear-wave splitting, see Pšenčík et al. (2012).

Effects of Structural Interfaces Assume that the ray is incident on a curved structural interface. If we wish to continue the ray computations for the reflected, transmitted, monotypic, or converted waves, see Fig. 5, we have to use relevant transformation relations for the ray tracing system, dynamic ray tracing system, and for the ray theory amplitudes at the interface. The transformation relations for ray tracing and dynamic ray tracing systems at interfaces are well known (Červený 2001). For the sake of brevity, we do not present them here. We shall, however, discuss the transformation of amplitudes. In the zero-order ray approximation, the transformation of ray-theory amplitudes across an interface is described by plane-wave reflection/transmission coefficients. In other words, amplitudes of generated waves do not depend on the curvature of the wavefront of the incident wave and the curvature of the interface at the point of incidence Q. Neither do they depend on the gradients of the density and on gradients of the density-normalized elastic moduli at Q, on both sides of the interface. They depend only on the local values of the density and density-normalized elastic moduli at Q (on both sides of the interface) and on the angle of incidence (the acute angle between the slowness vector of the incident wave and the normal N to the interface at the point of incidence Q).

Incident wave

N S2 S1 P P

Interface Σ S2

S1

Seismic Ray Theory, Fig. 5 Slowness vectors of P, S1, and S2 waves generated at the point of incidence Q of a curved interface separating two inhomogeneous anisotropic media. All slowness vectors at Q are situated in the plane of incidence specified by the slowness vector of the incident wave and the normal to the interface N at Q. Ray-velocity vectors (tangent to rays) of individual waves at Q are, in general, not confined to the plane of incidence. In isotropic media, instead of reflected and transmitted S1 and S2 waves, single reflected and transmitted S waves are generated. Ray-velocity vectors of individual waves at Q are situated in the plane of incidence

Various types of R/T coefficients may be used. The displacement R/T coefficients are used most frequently (Aki and Richards 2002; Červený et al. 1977 for isotropic media; Fedorov 1968 for anisotropic media). Very useful are the energy R/T coefficients, as they are reciprocal. The relation between the energy R/T coefficient R(Q) and the displacement R/T coefficient R(Q) is as follows: 2    31=2 e Un Q e r Q 5 RðQÞ ¼ RðQÞ4 rðQÞU n ðQÞ

ð56Þ

(Červený 2001, Sect. 5.4.3). Here Q is the point of incidence, e the relevant initial point of the R/T wave, both points and Q being, of course, identical. U n is the normal component (perpendicular to the interface) of the ray-velocity vector. We further introduce the complete energy R/T coefficients RC along the ray using the relation RC ¼

N Y

RðQk Þ:

ð57Þ

k¼1

The complete energy R/T coefficient RC corresponds to the ray, which interacts N-times with interfaces (at points of incidence Q1, Q2,. . ., QN) between the initial and end point of the ray. Generalization of the continuation relation presented in Eq. 37 for the ray-theory amplitudes along the ray situated in a laterally varying anisotropic medium containing curved interfaces then reads: "

#1=2 rðt0 ÞCðt0 Þ j detQðyÞ ðt0 Þ j Að t Þ ¼ rðtÞCðtÞ j detQðyÞ ðtÞ j    RC exp iT C ðt, t0 Þ Aðt0 Þ:

ð58Þ

In seismic prospecting, in the technique called amplitude variation with offset (AVO), it is common to work with the so-called weak-contrast R/T coefficients. They are linearized versions of exact R/T displacement coefficients. Linearization is mostly made with respect to the contrasts of the density and elastic moduli across the interface, often also under the assumption of weak anisotropy. There is a great variety of linearized formulae depending on the type of media surrounding the interface (isotropic, anisotropic), strength of anisotropy (weak, strong), etc. The coefficients yield reasonable approximation in the vicinity of normal incidence. For increasing incidence angles, their accuracy decreases. The advantage of the weak-contrast coefficients is their simplicity and the possibility of expressing them in explicit form. The effects of the individual medium parameters on the coefficients can then be easily evaluated.

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1483

Ray-Theory Elastodynamic Green Function The elastodynamic Green function Gin(R, t, S, t0) represents the i-th Cartesian component of the displacement vector at location R and time t, due to a single-force point source situated at location S and oriented along the n-th Cartesian axis, with the time dependence δ(t  t0). We introduce quite analogously the ray-theory elastodynamic Green function, with only two differences. The first difference is that raytheory Green function is defined as a sum of elementary ray-theory Green functions computed along rays of selected elementary waves (direct, multiply reflected/transmitted, etc.). The second difference is that the elementary ray-theory Green functions are not exact, but only zero-order ray approximations. In the frequency domain, the elementary ray-theory elastodynamic Green function Gin(R, S, o) for t0 ¼ 0 reads: Gin ðR, S, oÞ ¼

  gn ðSÞgi ðRÞ exp iT C ðR, SÞ þ ioT ðR, SÞ 4p½rðSÞrðRÞCðSÞCðRÞ1=2 ℒðR, SÞ

RC : ð59Þ

Here ℒ(R, S) is the relative geometrical spreading, given in Eq. 47, gi(R) and gn(S) are the eigenvectors of the generalized Christoffel matrix at R and S (polarization vectors corresponding to the considered elementary wave), T(R, S) is the travel time along the ray from S to R, RC is the complete energy R/T coefficient resulting from interactions of the wave under consideration with interfaces between S and R, and TC(R, S) is the complete phase shift due to caustics along the ray between S and R. The relevant KMAH index in anisotropic media may also include a contribution at a point source S (if the slowness surface of the considered wave is concave at S). In isotropic media, this contribution is always zero. The complete energy R/T coefficient RC, the travel time T(R, S), the relative geometrical spreading ℒ(R, S), and the complete phase shift due to caustics are always reciprocal. Consequently, the elementary ray-theory elastodynamic Green function satisfies a very important property of reciprocity: Gin ðR, S, oÞ ¼ Gni ðS, R, oÞ:

ð60Þ

This relation is valid for any elementary seismic body wave generated by a point source.

Chaotic Rays. Lyapunov Exponents In homogeneous media, geometrical spreading increases linearly with increasing length of the ray. In heterogeneous

media, behavior of geometrical spreading is more complicated and depends considerably on the degree of heterogeneity of the medium. In models, in which the heterogeneity exceeds certain degree, average geometrical spreading increases exponentially with increasing length of the ray. Rays in such a medium often exhibit chaotic behavior, which is characterized by a strong sensitivity of rays to the initial values (for example, to ray parameters). The rays with only slightly differing specification of their initial values tend to diverge exponentially at large distances from the initial point. Consequently, the rays intersect many times and many rays pass through the same point. With such chaotic rays, two-point ray tracing is practically impossible, and the ray tubes are not narrow enough for travel time interpolation. The chaotic behavior of rays increases with increasing length of rays and prevents applicability of the ray theory. The exponential divergence of chaotic rays in the phase space (the space formed by spatial coordinates xi and slowness-vector components pj) can be quantified by the so-called Lyapunov exponents. They may be introduced in several ways. It is common to express them in terms of characteristic values of the ray propagator matrix. The relevant expressions for the Lyapunov exponents and several numerical examples for 2D models without interfaces can be found in Klimeš (2002a). See also Červený et al. (2007), where other references can be found. The estimate of the Lyapunov exponent of a single finite ray depends on its position and direction. The Lyapunov exponents associated with rays of different positions and directions can be used to calculate average Lyapunov exponents for the model. The average Lyapunov exponents play a very important role in smoothing the models to make them suitable for ray tracing (Červený et al. 2007).

Ray Perturbation Methods Ray perturbation methods represent an important part of the ray theory. They can be used for approximate but fast and transparent solutions of forward problems in complicated models. They play even more important role in the inverse problems. Ray perturbation methods are useful everywhere, where we wish to compute the wavefield or its constituents (travel times, amplitudes, and polarization) in complicated models, which deviate only little from reference models, for which computations are simpler. The solutions for complicated models are then sought as perturbations of simple solutions for the reference models. Examples are computations in weakly anisotropic media, which use an isotropic medium as a reference, or in weakly dissipative media, which use a perfectly elastic medium as a reference. Basic role in these approaches is played by reference rays traced in reference

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media. Solutions in perturbed media can be given in the form of a power series in the deviations of the perturbed and reference models. Mostly, the first-order approximation, i.e., the first term of the power series, is used. The most frequent application of ray perturbation methods is in travel-time computations. First-order travel-time perturbation formulae for isotropic media are known and have been used (mostly in tomography) for several decades. Well known and broadly applied are also first-order travel-time formulae for anisotropic media (Hanyga 1982; Červený 2001, Sect. 3.9). Travel-time perturbations are obtained by quadratures along reference rays. As integration parameters, the parameters along reference rays are used. Another application of the perturbation method to the computation of travel times is the use of the so-called weakanisotropy approximation to evaluate approximately reflection moveout. Reflection moveout is commonly calculated as a Taylor series expansion of the squared reflection travel time (for example, Tsvankin and Grechka 2011), which loses rapidly accuracy with increasing offset. Weak-anisotropy approximation of the moveout formulae is free of this limitation. The moveout formulae are available for unconverted or converted waves, for horizontal or dipping reflectors, for anisotropy of arbitrary symmetry. For example, formula for P wave reflection moveout in a homogeneous layer of tilted orthorhombic symmetry can be found in Farra and Pšenčík (2017). Several procedures for computation of higher-order traveltime perturbations for weakly anisotropic media (note that anisotropy of the Earth is mostly weak) also exist. The procedure based on the so-called perturbation Hamiltonians (Klimeš 2002b; Červený et al. 2007) allows computation of highly accurate travel times along a fixed reference ray in a reference medium. Another procedure is based on the so-called first-order ray tracing described briefly below. In the latter method, second-order travel-time perturbations can be calculated along first-order rays. Perturbation methods are also used in first-order ray tracing and first-order dynamic ray tracing for P and S waves in inhomogeneous, weakly anisotropic media (Pšenčík and Farra 2007; Farra and Pšenčík 2010). They allow to compute, approximately, not only rays and travel times but whole wavefields. To derive first-order ray tracing and dynamic ray tracing, the perturbation approach is used, in which deviations of anisotropy from isotropy are considered to be small. Then it is just sufficient to use Eqs. 25 and 41 with Eqs. 21 or 49, in which the exact eigenvalues Gm are replaced by their firstorder approximations. The resulting ray tracing provides first-order rays, first-order travel times, and the first-order geometrical spreading. By simple quadratures along firstorder rays, second-order travel-time corrections can be computed. This approach is applicable to P and S waves. In case of S waves, it can include the computation of coupling effects. First-order ray tracing and dynamic ray tracing are used in this

Seismic Ray Theory

case for computing common S wave rays, first-order travel times and geometrical spreading along them. The wavefield of S waves is obtained by solving second-order coupling equations along the S wave common rays. The procedure yields standard ray-theory results for S waves propagating in isotropic media, and approximate results in anisotropic media when the S waves are coupled or even decoupled.

Ray Perturbation Method for Weakly Dissipative Media Study of wave propagation in dissipative media can be often simplified by the use of the correspondence principle (Borcherdt 2009; Carcione 2014). When applicable, it allows to obtain solutions for viscoelastic waves from solutions for elastic waves by replacing real-valued elastic parameters by their complex-valued, frequency-dependent counterparts. The density-normalized stiffness tensor aijkl(x, o) then reads: aijkl ¼ aRijkl  iaIijkl :

ð61Þ

If aIijkl is small, the dissipative medium can be considered as a perturbation of an elastic medium (Gajewski and Pšenčík 1992; Červený 2001, Sect. 5.5.3), which has a form of the imaginary-valued term iaIijkl . Reference ray in the reference elastic medium and corresponding real-valued travel time T along the reference ray between points S and R can be obtained by standard ray tracing in the elastic medium. The travel-time perturbation ΔT (the perturbation due to the perturbation of the reference elastic medium iaIijkl ), responsible for the exponential amplitude decay along the reference ray, can be obtained by quadratures along this ray: i DT ¼ 2

ðR

Q1 ðtÞdt:

ð62Þ

S

The quantity Q in Eq. 62 is a direction-dependent quality factor for anisotropic media, corresponding to the Hamiltonian given in Eq. 21: Q1 ¼ aIijkl p j pl gi gk :

ð63Þ

For general Hamiltonians, the quality factor Q is given by the relation Q1 ¼ Imℋ(xi, pj). For causal dissipation, the stiffness tensor given in Eq. 61 is frequency dependent. The above-described perturbation approach is then equivalent to the perturbation scheme, in which aIijkl (xn, o) is considered to be of the order of o1 for o ! 1 (Kravtsov and Orlov 1990; Gajewski and Pšenčík 1992).

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1485

In an inhomogeneous isotropic, weakly dissipative medium, the expression shown in Eq. 63 reduces to the well-known formula : Q1 ¼ ImV 2 = Re V 2 ¼  2ImV= Re V,

ð64Þ

in which V is the complex-valued velocity, V ¼ α for P waves and V ¼ β for S waves. Complex-valued quantities α and β are generalizations (to the complex space) of real-valued α and β from Eq. 14. The perturbation method described above uses the real ray tracing in a reference elastic medium. Vavryčuk (2012) shows that more accurate results are obtained by the procedure, which he calls the real viscoelastic ray tracing. Vavryčuk (2012) illustrates it on examples of isotropic dissipative media. Klimeš and Klimeš (2011) extend Vavryčuk’s (2012) work to anisotropic dissipative media. They construct the reference Hamiltonian function using the Hamiltonian function corresponding to a given complex-valued eikonal equation. They show that the reference Hamiltonians can be constructed in various ways leading to travel time perturbations of varying accuracy. Continuing attention is devoted to the study of effects of attenuation on reflection/transmission coefficients (Krebes and Daley 2007; Borcherdt 2009; Ursin et al. 2017). There are still some open questions, mostly concerning the applicability of the correspondence principle to the reflection/transmission coefficients of overcritically reflected waves.

Concluding Remarks. Applications, Modifications, and Extensions of the Ray Method Seismic ray method has found many applications both in global seismology and in seismic exploration. The advantages of the seismic ray method consist in its numerical efficiency, universality, conceptual clarity, and in its ability to treat various seismic body waves independently of other waves. Although its accuracy is only limited, the seismic ray method is the only method which is able to give an approximate answer to many problems of high-frequency seismic body wave propagation in laterally varying, isotropic or anisotropic, perfectly elastic or dissipative, layered and block structures. In classical global seismology, the seismic ray method has been traditionally used to study the internal structure of the whole Earth, assuming that the Earth is radially symmetric. The standard Earth’s model, obtained in this way, is expressed in terms of distribution of elastic parameters varying with depth. At present, the applications of the seismic ray method are considerably broader. It is used to study the 3-D local laterally inhomogeneous structures, their anisotropy and attenuation,

and the form and physical properties of structural interfaces. In addition to forward modelling, the ray perturbation methods are also broadly used in inversions using observed travel times or whole waveforms. In lithospheric structural studies, particularly in crustal seismology, the ray-synthetic seismograms have been also often used for ultimate comparison with observed seismograms. The computation of raysynthetic seismograms requires determination of not only travel times but also ray-theory amplitudes and polarizations of individual waves. Seismic ray method has also found applications in other branches of seismology. Important examples are the localization of seismic sources either in a known structure or simultaneously with structural inversion. In most applications of the ray method in seismic exploration for oil, the use of local 3-D structures with structural curved interfaces is a necessity. Sophisticated algorithms have been developed and used to image the structures under consideration. At present, the most important role is played by migration algorithms. Seismic ray theory and its extensions have found important applications in these algorithms. The ray method is not valid universally. We have briefly described three serious limitations of the ray method: (a) The ray method can be used only for high-frequency signals. (b) In models, in which heterogeneity of the medium exceeds certain degree, the ray field has chaotic character, particularly at large distances from the source. (c) The standard ray method cannot be used for computing S waves propagating in inhomogeneous, weakly anisotropic media. It must be replaced by the coupling ray theory. The coupling ray theory must be used in the vicinity of shear-wave singular directions. The ray method fails, however, even in other singular situations. In smooth isotropic or anisotropic media, the most important types of singularity are caustics. Caustics may attain various forms. Various extensions of the ray method can be used to compute wavefields in caustic regions. These extensions are frequency dependent. See a detailed treatment of wavefields in caustic regions in Kravtsov and Orlov (1999), and also in Stamnes (1986). In models with smooth structural interfaces, other singularities often appear. For edge and vertex points, see Ayzenberg et al. (2007). For critical singular regions, at which head waves separate from reflected waves, see Červený and Ravindra (1971). For the waves, whose rays are tangential to interfaces, see Thomson (1989). Specific methods, designed for different types of singularities may be used for computing wavefields in singular regions. Disadvantage of these methods is that they are different for different singularities. Moreover, singular regions often overlap, and the wavefield in the overlapping region requires again different treatment. It is desirable to have available a more general extension of the ray method, applicable uniformly in any of the mentioned singular regions, or, at least, in most of them. Such an extension

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would simplify ray computations considerably and could even lead to more accurate results. Several such extensions of the ray method have been proposed. We do not describe them here in detail. Instead, we merely present references, in which more details and further references can be found. Let us mention the Maslov asymptotic ray theory introduced to seismology by Chapman and Drummond (1982); see also Thomson and Chapman (1985) and Chapman (2004). Another extension of the ray method is based on the summation of Gaussian beams (Popov 1982; Červený et al. 1982; Klimeš 2018). For the relation of this method with the Maslov method see Klimeš (1984). The Gaussian beam summation method has found applications both in the forward modelling of seismic wavefields and in migrations in seismic exploration. It is closely related to the method of summation of Gaussian packets (Červený et al. 2007). Ray theory can be also used in the Born scattering theory (Chapman and Coates 1994; Chapman 2004). For waves reflected from a smooth structural interface separating two heterogeneous, isotropic or anisotropic media, the Kirchhoff surface integral method can be used. For details and many references see Chapman (2004, Sect. 10.4). Another useful extension of the ray method is the one-way wave equation approach (Thomson 1999).

Cross-References ▶ Energy Partitioning of Seismic Waves ▶ Magnetic Anisotropy ▶ Seismic Diffraction ▶ Seismic Viscoelastic Attenuation ▶ Seismic Waves, Scattering ▶ Seismic, Migration ▶ Seismic, Waveform Modeling and Tomography ▶ Traveltime Tomography Using Controlled-Source Seismic Data Acknowledgments The authors are very grateful to Luděk Klimeš and Ravi Kumar for valuable comments and recommendations. The research was supported by the consortium project Seismic Waves in Complex 3-D Structures, by research projects 205/07/0032, 205/08/0332, and 16-05237S of the Grant Agency of the Czech Republic; and by research project MSM0021620860 of the Ministry of Education of the Czech Republic.

Bibliography Aki K, Richards P (2002) Quantitative seismology, 2nd edn. University Science Books, Sausalito Ayzenberg MA, Aizenberg AM, Helle HB, Klem-Musatov KD, Pajchel J, Ursin B (2007) 3D diffraction modeling of singly scattered acoustic wavefields based on the combination of surface integral propagators and transmission operators. Geophysics 72:SM19–SM34

Seismic Ray Theory Babich VM (1956) Ray method of the computation of the intensity of wave fronts (in Russian). Dokl Akad Nauk SSSR 110:355–357 Babich VM (1961) Ray method of the computation of the intensity of wave fronts in elastic inhomogeneous anisotropic medium. In: Petrashen GI (ed) Problems of the dynamic theory of propagation of seismic waves (in Russian), vol 5. Leningrad University Press, Leningrad, pp 36–46. (Translation to English: Geophys J Int 118:379–383, 1994) Babich VM, Kiselev AP (2018) Elastic waves. High frequency theory. CRC Press, London Bakker PM (1998) Phase shift at caustics along rays in anisotropic media. Geophys J Int 134:515–518 Bakker PM (2002) Coupled anisotropic shear-wave ray tracing in situations where associated slowness sheets are almost tangent. Pure Appl Geophys 159:1403–1417 bin Waheed U, Pšenčík I, Červený V, Iversen E, Alkhalifah T (2013) Two-point paraxial traveltime formula for inhomogeneous isotropic and anisotropic media: tests of accuracy. Geophysics 78:WC65– WC80 Borcherdt RD (2009) Viscoelastic waves in layered media. Cambridge University Press, Cambridge Bulant P, Klimeš L (2002) Numerical algorithm of the coupling ray theory in weakly anisotropic media. Pure Appl Geophys 159:1419–1435 Carcione JM (2014) Wave fields in real media. Wave propagation in anisotropic, anelastic, porous and electromagnetic media. Elsevier, Amsterdam Červený V (1972) Seismic rays and ray intensities in inhomogeneous anisotropic media. Geophys J R Astron Soc 29:1–13 Červený V (2001) Seismic ray theory. Cambridge University Press, Cambridge Červený V, Ravindra R (1971) Theory of seismic head waves. Toronto University Press, Toronto Červený V, Molotkov IA, Pšenčík I (1977) Ray method in seismology. Univerzita Karlova, Praha Červený V, Popov MM, Pšenčík I (1982) Computation of wave fields in inhomogeneous media. Gaussian beam approach. Geophys J R Astron Soc 70:109–128 Červený V, Klimeš L, Pšenčík I (1988) Complete seismic ray tracing in three-dimensional structures. In: Doornbos DJ (ed) Seismological algorithms. Academic Press, New York, pp 89–168 Červený V, Klimeš L, Pšenčík I (2007) Seismic ray method: recent developments. Adv Geophys 48:1–126. http://www.sciencedirect. com/science/bookseries/00652687 Chapman CH (2002) Seismic ray theory and finite frequency extensions. In: Lee WHK, Kanamori H, Jennings PC (eds) International handbook of earthquake and engineering seismology, part A. Academic Press, New York, pp 103–123 Chapman CH (2004) Fundamentals of seismic wave propagation. Cambridge University Press, Cambridge Chapman CH, Coates RT (1994) Generalized Born scattering in anisotropic media. Wave Motion 19:309–341 Chapman CH, Drummond R (1982) Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory. Bull Seismol Soc Am 72:S277–S317 Coates RT, Chapman CH (1990) Quasi-shear wave coupling in weakly anisotropic 3-D media. Geophys J Int 103:301–320 Farra V, Pšenčík I (2010) Coupled S waves in inhomogeneous weakly anisotropic media using first-order ray tracing. Geophys J Int 180:405–417 Farra V, Pšenčík I (2017) Weak-anisotropy moveout approximations for P waves in homogeneous TOR layers. Geophysics 82:WA23–WA32 Fedorov FI (1968) Theory of elastic waves in crystals. Plenum, New York Gajewski D, Pšenčík I (1992) Vector wavefield for weakly attenuating anisotropic media by the ray method. Geophysics 57:27–38

Seismic Reservoir Characterization Gjøystdal H, Iversen E, Laurain R, Lecomte I, Vinje V, Åstebol K (2002) Review of ray theory applications in modelling and imaging of seismic data. Stud Geophys Geod 46:113–164 Hanyga A (1982) The kinematic inverse problem for weakly laterally inhomogeneous anisotropic media. Tectonophysics 90:253–262 Jech J, Pšenčík I (1989) First-order perturbation method for anisotropic media. Geophys J Int 99:369–376 Karal FC, Keller JB (1959) Elastic wave propagation in homogeneous and inhomogeneous media. J Acoust Soc Am 31:694–705 Klimeš L (1984) The relation between Gaussian beams and Maslov asymptotic theory. Stud Geophys Geod 28:237–247 Klimeš L (2002a) Lyapunov exponents for 2-D ray tracing without interfaces. Pure Appl Geophys 159:1465–1485 Klimeš L (2002b) Second-order and higher-order perturbations of travel time in isotropic and anisotropic media. Stud Geophys Geod 46:213–248 Klimeš L (2006) Common-ray tracing and dynamic ray tracing for S waves in a smooth elastic anisotropic medium. Stud Geophys Geod 50:449–461 Klimeš L (2010) Phase shift of the Green tensor due to caustics in anisotropic media. Stud Geophys Geod 54:269–289 Klimeš L (2018) Superpositions of Gaussian beams and column Gaussian packets in heterogeneous anisotropic media. Geophys J Int 215:1739–1746 Klimeš L (2019) Calculation of the amplitudes of elastic waves in anisotropic media in Cartesian or ray-centred coordinates. Stud Geophys Geod 63:229–246 Klimeš L, Bulant P (2016) Prevailing-frequency approximation of the coupling ray theory for electromagnetic waves or elastic waves. Stud Geophys Geod 60:419–450 Klimeš M, Klimeš L (2011) Perturbation expansions of complex-valued traveltime along real-valued reference rays. Geophys J Int 186:751–759 Kravtsov YA (1968) “Quasiisotropic” approximation to geometrical optics. Dokl Akad Nauk SSSR 183(1):74–77. (in Russian) Kravtsov YA, Orlov YI (1990) Geometrical optics of inhomogeneous media. Springer, Heidelberg Kravtsov YA, Orlov YI (1999) Caustics, catastrophes and wave fields. Springer, Heidelberg Krebes ES, Daley PF (2007) Difficulties with computing anelastic planewave reflection and transmission coefficients. Geophys J Int 170:205–216 Popov MM (1982) A new method of computation of wave fields using Gaussian beams. Wave Motion 4:85–97 Pšenčík I, Farra V (2007) First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media. Geophys J Int 170:1243–1252 Pšenčík I, Farra V, Tessmer E (2012) Comparison of the FORT approximation of the coupling ray theory with the Fourier pseudospectral method. Stud Geophys Geod 56:35–64 Stamnes JJ (1986) Waves in focal regions. Adam Hilger, Bristol Thomson CJ (1989) Corrections for grazing rays to 2-D seismic modelling. Geophys J Int 96:415–446 Thomson CJ (1999) The “gap” between seismic ray theory and full wavefield extrapolation. Geophys J Int 137:364–380 Thomson CJ, Chapman CH (1985) An introduction to Maslov’s asymptotic method. Geophys J R Astron Soc 83:143–168 Tsvankin I, Grechka V (2011) Seismology of azimuthally anisotropic media and seismic fracture characterization. Society of Exploration Geophysicists, Tulsa Ursin B, Carcione JM, Gei D (2017) A physical solution for plane SH waves in anelastic media. Geophys J Int 209:661–671 Vavryčuk V (2012) On numerically solving the complex eikonal equation using real ray-tracing methods: a comparison with the exact analytical solution. Geophysics 77:T109–T116 Virieux J (1996) Seismic ray tracing. In: Boschi E, Ekström G, Morelli A (eds) Seismic modelling of Earth structures. Editrice Compositori, Bologna, pp 223–304

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Seismic Reservoir Characterization Dhananjay Kumar Chevron, Houston, TX, USA BP, Houston, TX, USA

Definition Reservoir

Reservoir properties

Reservoir characterization Seismic reservoir characterization Well tie

Seismic amplitude calibration

A petroleum reservoir is a column of layers of sedimentary rocks in the subsurface containing oil and gas. In petroleum exploration, some of the common reservoir properties of interest are thickness, rock type, porosity, permeability, fluid type, and saturation. It is a method to describe the reservoir in terms of its structure and in situ rock and fluid properties in the subsurface. It is the use of seismic data for 3D reservoir characterization away from the well locations. It is the comparison of a measured seismic response with a well log-based synthetic seismic response at a well location. Also, it is a comparison of a seismically derived property with a well log property at well locations. It is a process to relate seismic amplitude to geology (often based on well log synthetics) for quantitative estimation of reservoir properties from seismic.

Introduction In petroleum exploration, geoscientists want to locate and characterize reservoirs that may be a few kilometers deep in the subsurface. Seismic imaging is a noninvasive method that uses seismic data to produce three-dimensional (3D) images of the subsurface in terms of its structure, layered earth stratigraphy, and elastic properties of the rock formations – all the way from the top surface to deeper zones. However, a reservoir characterization study generally focusses on the hydrocarbon-bearing reservoir intervals of interest. To help understand how best to produce hydrocarbons from a reservoir, we want to understand the reservoir in terms of its size, shape, depth, thickness, rock type, rock strength, porosity, permeability, fluid type, saturation, pore pressure, and connectivity. Wells, including well log data, provide abundant information about the reservoir. However, wells usually must be drilled at sparse locations due to cost and safety

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practicalities. Seismic data can be acquired on the surface, and the role of seismic reservoir characterization is to define the structure and interpret reservoir properties away from and between the well locations. Seismic data is a time series representation of the earth’s subsurface that contains information about the reservoir properties filtered with the source wavelet and wave propagation effects. The seismic response is a cumulative effect of wave propagation in the overburden and reservoir. Figure 1 shows a schematic plot of wave energy propagation in a subsurface model highlighting that the wave, represented by ray paths, spends most of the time in the overburden and only a short time in the reservoir. Data processing is carried out to attenuate unwanted seismic noise and enhance spatial and temporal resolution to detect finer details of the reservoir. Processed seismic volumes can be interpreted to map geological boundaries (horizons) and generate a variety of attributes for geological interpretation. For quantitative interpretation, the

seismic amplitudes are first calibrated to well log-based synthetic responses to establish relationships between the seismic waveform response and geology (reservoir properties) at well locations. While seismic reflectivity (amplitude) images have been used for decades in seismic interpretation, seismic inversion is now a standard technique used for quantitative reservoir characterization. Seismic inversion can be undertaken in two ways, either by inverting for elastic rock properties (P-wave velocity Vp, S-wave velocity Vs, impedance), from which reservoir properties (lithology, porosity, fluid saturation) can be estimated using rock physics relationships, or by inverting directly for reservoir properties. The properties estimated from both approaches are used to further estimate reservoir quality (net sand, porosity) and quantity (volumetric hydrocarbon in place), impact well-drilling decisions, and optimize hydrocarbon production and recovery. Figure 2 shows a seismic reservoir characterization workflow.

Seismic Reservoir Characterization, Fig. 1 Schematic subsurface model. An example of seismic wave propagation in the subsurface layers represented by three rays emitting from a source on the surface. Seismic receivers are placed on the top surface and record reflected waveforms in certain time-sampling intervals (i.e., 1 millisecond). Seismic wave propagates in all directions following physics (Snell’s law). At each reflective interface (where contrasts in elastic properties and density are present), part of the energy is reflected, and part of the energy is transmitted, and

with each reflection and transmission, the wave may undergo a change in mode to other wave types. Here, only the primary P-P wave reflection and transmission are shown using three rays: (i) dash-dot red lines, (ii) dotted green lines, and (iii) dashed blue lines. Only the dashed blue color ray is reflecting from the base of the target reservoir, and note that this ray is spending most of the time in the overburden, and therefore its seismic response will be dominated by overburden geology and propagation effects

Seismic Reservoir Characterization

Seismic Data Processing Seismic data are processed, including migration, to attenuate noise and remove 3D wave propagation effects (Yilmaz 2001). After processing, an ideal seismic response is the one responding to P-wave primary reflections only (without converted waves and multiple reflected waves) and without 3D propagation effects. Processing for the P-wave response is common, because our conventional models for interpretation are based on 1D earth models and P-P wave reflections. Advanced methods are being occasionally used (and studied) to consider the 3D and mode conversion effects. In practice, even after careful processing, seismic amplitudes can still be affected by various overburden geologic and wave propagation effects superimposed on those caused by reservoir layer properties (Sheriff 1975). All unwanted seismic responses are termed “seismic noise.” After processing, we use zero-phase seismic data and generate angle gathers, angle stacks, amplitude variation with angle (AVA) attributes, and other attributes to facilitate reservoir characterization (Chopra and Marfurt 2007).

Seismic-Well Tie Calibration of a seismic data with synthetic data (generated using well logs) is referred to as the seismic-well tie process. Figure 3 shows an example. A seismic-well tie offers many benefits: (i) Provides a basis for interpreting seismic events in geologic terms (ii) Helps to establish a time-depth relationship between seismic data and the well depths, respectively (iii) Helps to estimate the phase of the data (iv) Helps to estimate the wavelet for use in synthetic seismogram computation and in seismic inversion (v) Enables general quality control of both seismic and well logs (vi) Enables the understanding of seismic resolution and tuning effects

Seismic Rock Physics Well logs are used to establish relationships between reservoir properties of interest and elastic properties that can be estimated from seismic data (Mavko et al. 1998). For example, relationships can be established to show how seismic impedance (product of velocity and density) can be used to predict reservoir porosity, or how multiple reservoir properties such as lithology and hydrocarbon saturation can be distinguished

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using impedance and Vp/Vs ratio simultaneously. If reservoir properties and elastic properties are strongly related, then we have a better chance of successful reservoir characterization in the presence of good-quality seismic data. But if rock physics relationships are non-favorable, then even in the presence of good-quality seismic data, reservoir characterization will be ambiguous.

Seismic Inversion for Reservoir Properties Seismic reflection data respond to changes in elastic properties and bulk density across interfaces between layered geologic formations. Therefore, the seismic reflectivity response must be inverted for layer properties, as they are better suited for reservoir characterization. Seismic inversion (Menke 1984; Tarantola 1986) is used either to estimate reservoir properties directly or first to estimate elastic properties followed by the application of rock physics models to estimate reservoir properties. Figure 2 shows both methods. The main objective in inversion is to estimate earth model parameters such that the difference between real seismic data and the synthetic seismic data (based on the estimated earth model, the forward modeling method, and the wavelet) is minimum. For the direct estimation of reservoir properties from seismic, rock physics models are included as part of the inversion algorithm and formulate inversion in terms of reservoir properties as model parameters. In this approach, rock physics models are used to compute elastic properties and density for the given reservoir properties (Mavko et al. 1998). Because seismic reflection data is bandlimited in temporal frequency (about 6–60 Hz), inversion using seismic data alone provides relative values of desired properties in seismic frequencies; one such method is colored inversion (Lancaster and Whitcombe 2000). Often, we want to estimate absolute values of reservoir properties, and for that, we use lowfrequency (smooth) models based on available well logs and other geologic understanding including 3D seismic structural maps (Simm and Bacon 2014). Seismic models with relative values will have both negative and positive numbers, and models with absolute values will only have positive numbers. Figure 4 compares porosity predictions using two types of seismic inverted acoustic impedance (Vp times density): (i) relative impedance and (ii) absolute impedance. Various sources of noise in seismic data and our use of various models and approximations give rise to uncertainties in the estimated reservoir properties. Two main areas of research are being undertaken to help deal with uncertainties, as well as the increasing abundance of seismic data. One current focus for research is to estimate uncertainties in seismic predictions, for example, through a Bayesian framework, and consider these in the decision-making for petroleum exploration. Another focus area is machine learning methods.

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Seismic Reservoir Characterization, Fig. 2 Seismic reservoir characterization workflow. Characterization starts with quality control of both seismic and well log data. Seismic data are conditioned, and desired angle gathers, stacks, or AVA attributes are generated. Well logs are used to generate similar synthetic gathers and stacks, followed by well tie analysis (Fig. 3) to calibrate seismic amplitude with synthetics generated using the estimated wavelet. The seismic reflectivity data can be interpreted for reservoir characterization. Also, we can perform seismic inversion to either directly invert for reservoir properties or first invert for

elastic properties and then estimate reservoir properties. These interpretations and inversions require information and calibration from geology (often using well logs). Common practices in terms of seismic inversion for elastic properties includes (i) pre-stack seismic inversion for Vp, Vs, and density; (ii) limited angle stack inversion for elastic impedance (EI); (iii) inversion of AVO intercept and gradient for acoustic impedance (AI) and gradient impedance (GI), respectively; and (iv) inversion of AVO attributes for Vp, Vs, and density. Refer to Whitcombe et al. (2002) for EI and GI

Applications of Machine Learning Methods in Reservoir Characterization

required. It has been used for reservoir property prediction as one way to improve the validity and acceptable geological interpretations (Biswas et al. 2019b). Most ML algorithms are based on a type of statistical learning (Gareth et al. 2014). Note that conventional reservoir characterization is done using the processed seismic data to make sure seismic data contain only the P-wave primary reflection response that can be explained with known physics (e.g., Zoeppritz AVA equation-based convolutional synthetics). One should think about using the raw seismic data in ML methods for reservoir characterization. Although ML methods have potential to provide fast turnaround of a project and reduce human bias, human interactions will be needed to come up with physically interpretable reservoir properties.

Machine learning (ML) methods take a numerical approach to estimate reservoir properties (labels in ML), recognizing the inherent imperfection between physics-based models and seismic amplitudes (data). This is possible because of the advancements in computer algorithms and speed and is done by first deriving abstract numerical relationships between data and desired properties using known data and label combinations, followed by predictions of labels for new datasets. In the absence of the availability of labeled data combination, synthetic models can be used to train the algorithms. This approach of ML is known as supervised learning. ML is becoming an alternative method for geophysical inversion in the presence of a large amount of data and a poor understanding of physics (Russell 2019). Structural (fault, horizon) mapping using ML is becoming a routine (Zheng et al. 2019), but reservoir properties prediction is still not fully developed and accepted. However, physics-guided ML takes the unsupervised approach, where a labeled data is not

Summary Seismic data provides images of the subsurface in terms of the 3D structure, where there are changes in elastic properties and density. Various seismic reflectivity attributes are generated to

Seismic Reservoir Characterization

Seismic Reservoir Characterization, Fig. 3 Seismic-well tie (Biswas et al. 2019a). The first column shows VSH curve to highlight lithology, second column shows Vp (red) and Vs (blue), and third column shows bulk density (red) and porosity (blue). The fourth column evaluates seismic inverted density (blue) with smooth well log density (red), and Seismic Reservoir Characterization, Fig. 4 Seismic-derived porosity section views along a well (Kumar et al. 2014). Log porosity along the wellbore is compared with porosity predicted from two seismic acoustic impedance (AI) models: relative seismic AI (a) and absolute seismic AI (b). The two seismic porosities are comparable in the Eagle Ford (EF) shale, and they also match well with log porosity. The same color scale is used for both seismic and log porosities. Seismic formations (upper EF, lower EF, and Buda) are labelled

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the two compare nicely here. The fifth column shows seismic-well tie as synthetic (blue) and seismic (red) reflectivity comparison, and the last column shows another view of seismic-well tie as seismic section along a well and synthetic (blue) displayed along well trajectory

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enhance seismic sensitivity to reservoir properties. Seismic inversion is utilized further to estimate layer properties for geological interpretation. The two critical ingredients for a successful seismic reservoir characterization are (i) favorable rock physics showing a strong sensitivity of seismic amplitude to reservoir properties and (ii) good-quality seismic data. Uncertainties in estimated reservoir properties exist due to low-quality seismic data, bandlimited resolution of seismic, poor calibration of seismic data to well logs, and approximate seismic physics used in the analysis. Future emphasis is on probabilistic seismic analysis and machine learning methods to estimate reservoir properties from seismic.

Seismic Seiches Tarantola A (1986) A strategy for nonlinear elastic inversion of seismic reflection data. Geophysics 51:1893–1903 Whitcombe DN, Connolly PA, Reagan RL, Redshaw TC (2002) Extended elastic impedance for fluid and lithology prediction. Geophysics 67:63–67 Yilmaz O (2001) Seismic data analysis: processing, inversion, and interpretation of seismic data. Society of Exploration Geophysicists, Tulsa Zheng Y, Zhang Q, Yisifov A, Yunzhi S (2019) Applications of supervised deep learning for seismic interpretation and inversion. The Leading Edge 38(7):528–532. https://doi.org/10.1190/tle38070526.1

Seismic Seiches Cross-References ▶ Earthquake Source Theory ▶ Seismic Data Acquisition and Processing ▶ Seismic Noise ▶ Uncertainties in Geomodelling Related to Geophysical Inversion Acknowledgments The author thanks Stan Davis, Chandan Kumar, Reetam Biswas, Margarita Corzo, John Etgen, Lisa Davies, and Kalachand Sain for reviews and BP for permission to publish.

Bibliography Biswas R, Kumar D, Sen MK, Paul A, Packer K (2019a) Density inversion from seismic using a trans-dimensional approach: a field dataset example. In: 89th Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, pp 539–543 Biswas R, Sen MK, Das V, Mukerji T (2019b) Pre-stack and post-stack inversion using a physics-guided convolutional neural networks. Interpretation 1–76. https://doi.org/10.1190/int-2018-0236.1 Chopra S, Marfurt KJ (2007) Seismic attributes for prospect identification and reservoir characterization. SEG geophysical development series no 11. Society of Exploration Geophysicists, Tulsa Gareth J, Witten D, Hastie T, Tibshirani R (2014) An introduction to statistical learning: with applications in R. Springer Publishing, New York. ISBN 978-1-4614-7137-0 Kumar D, Sugianto H, Li S, Patel H, Land S (2014) Using relative seismic impedance to predict porosity in the Eagle Ford shale. In: 84th Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, pp 2688–2692 Lancaster S, Whitcombe D (2000) Fast-track coloured inversion. In: 70th Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, pp 1572–1575 Mavko G, Mukerji T, Dvorkin J (1998) The rock physics handbook: tools for seismic analysis in porous media. Cambridge University Press, Cambridge Menke W (1984) Geophysical data analysis: discrete inverse theory. Academic, Orlando Russell B (2019) Machine leaning and geophysical inversion: a numerical study. The Leading Edge 38(7):512–519. https://doi.org/10. 1190/tle38070512.1 Sheriff RE (1975) Factors affecting seismic amplitudes. Geophys Prospect 23:125–138 Simm R, Bacon M (2014) Seismic amplitude: an interpreter’s handbook. Cambridge University Press, Cambridge

Art McGarr Earthquake Science Center, U.S. Geological Survey, Menlo Park, CA, USA

Definition Seismic seiche is a term first used by Kvale (1955) to discuss oscillations of lake levels in Norway and England caused by the Assam earthquake of August 15, 1950. This definition has since been generalized to apply to standing waves set up in closed, or partially closed, bodies of water including rivers, shipping channels, lakes, swimming pools, and tanks due to the passage of seismic waves from an earthquake. The first published mention of seismic seiches is thought to be reports of those observed throughout much of Europe due to the great earthquake at Lisbon, Portugal, in 1755 (Wilson 1953; Richter 1958). In addition to the Lisbon and Assam earthquakes, seismic seiches at teleseismic distances have been observed for many other large earthquakes including the 1964 Alaska (McGarr and Vorhis 1968) and the 2002 Denali, Alaska, an earthquake that caused damaging seiches in Lake Union, Seattle, Washington, at an epicentral distance of 2400 km (Barberopoulou et al. 2004). Most recently, Bondevik et al. reported observations of seiches in Norwegian fjords caused by S waves and surface waves from the 2011 M9 Tohoku, Japan, earthquake. Based on evidence from eyewitnesses and surveillance and cell phone cameras, the seiches began about 30 min after the origin time of the earthquake. Kvale (1955) showed that seismic surface waves from the Assam earthquake were the most probable cause of the seiches observed in Norway and England at that time. Moreover, he concluded that the natural period of a basin must be matched by the periods of the passing seismic surface waves. Motivated by observations reported by Donn (1964) of a seiche generated in a channel near Freeport, Texas, at an epicentral distance of about 5040 km from the 1964 Alaska earthquake, McGarr (1965) developed a relation between the ground motion of seismic waves and the resulting seiche. The passing seismic

Seismic Signals in Well Water Observations

wave exerts a horizontal acceleration on a closed, or partially closed, body of water, which can be idealized as a long channel of uniform depth. This causes a seiche, composed of standing water waves whose periods depend on the dimensions of the channel. The amplitude of the seiche depends on channel depth, the amplitudes of the horizontal accelerations of the seismic waves, and the extent to which the periods of the seismic waves match those of standing water waves. The gravest mode of the standing waves has a period given by pffiffiffiffiffiffiffi T ¼ 2L= gH , where T is the period in seconds, L is the channel width in meters, H is the channel depth in meters, and g is gravity. For instance, if L ¼ 100 m and H ¼ 10 m, then the period of the gravest seiche mode is 20 s, which tends to be the period where surface waves show maximum amplitudes at teleseismic distances. Any factor that enhances the amplitudes of seismic waves, such as basins containing lowvelocity sediments, tends to result in greater production of observed seiches from a given earthquake (McGarr and Vorhis 1968; Barberopoulou et al. 2004).

Bibliography Barberopoulou A, Qamar A, Pratt TL, Creager K, Steele WP (2004) Local amplification of seismic waves from the Denali earthquake and damaging seiches in Lake Union, Seattle, Washington. Geophys Res Lett 31:L03607. https://doi.org/10.1029/2003GL018569 Bondevik S, Gjevik B, Sorensen MB (2013) Norwegian seiches from the giant 2011 Tohoku earthquake. Geophys Res Lett 40:3374–3378. https://doi.org/10.1002/grl.50639 Donn WL (1964) Alaska earthquake of 27 March, 1964: remote seiche stimulation. Science 146:261–262 Kvale A (1955) Seismic seiches in Norway and England during the Assam earthquake of August 15, 1950. Bull Seismol Soc Am 45:93–113 McGarr A (1965) Excitation of seiches in channels by seismic waves. J Geophys Res 70:847–854 McGarr A, Vorhis RC (1968) Seismic seiches from the March 1964 Alaska earthquake. U.S. Geological Survey professional paper 544E. pp E1–E43 Richter CF (1958) Elementary seismology. W. H. Freeman and Company, San Francisco, 768 pp Wilson BW (1953) Coastal seiches, pt. 1 of oscillations of the sea and the phenomenon of range. The Dock and Harbour Authority [London], June, pp 41–45

Seismic Signals in Well Water Observations R. K. Chadha CSIR:National Geophysical Research Institute, Hyderabad, India

Synonyms Anomalous water level changes; Groundwater anomalies; Hydrological earthquake precursor

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Definition Seismic signals in well water observations – temporal rise or fall in well water level fluctuations induced by earthquakes.

Introduction Seismic waves generated during an earthquake cause oscillations and permanent offsets in well water levels. Liu et al. (1989) explained the oscillations in terms of expansion and contraction of pore pressure changes induced by seismic waves. Systematic monitoring of well water levels has led to identification of anomalous pre-, co-, post-, and transient changes in well water levels related to earthquakes (Roeloffs 1996). These signals are believed to reflect pore pressure changes related to the redistribution of stress in the near and far fields of dislocation sources. The co-, post-, and transient seismic changes appear as a step or spikelike anomaly in well water oscillations and recover post-earthquake depending on the transmissivity of the medium. While the co- and postseismic signals are attributed to local and regional earthquakes, the transient changes are in response to teleseismic (>1000 km) earthquakes. Several attempts have been made, in hindsight, to interpret preseismic well water levels changes as earthquake precursor.

Co- and Post-seismic Signals The coseismic signals generally coincide with earthquake time and are observed mostly in wells which respond to earth tides. Earth tides are very small deformations in the earth’s surface caused by the gravitational fields of the sun and moon as the earth rotates within their fields. The presence of tidal signals in well level recordings indicates that they are connected to confined, that is, hydraulically locked and fully water-saturated aquifers. This is favorable for the detection of weak pore pressure anomalies induced in crustal stress and hence can be considered as volume strainmeter of resolution of the order of 10 nanostrain (1 in 108) or better. As an alternate mechanism, Rojstaczer et al. (1995) stressed the role of earthquake-induced enhancement of permeability in the shallow crust to explain the co- and post-hydrological changes observed in springs and groundwater flows after the 1989 Loma Prieta earthquake in California. The 1999 Chi-Chi earthquake of Mw7.6 in Taiwan is one of the well-documented events for water level changes in wells. Chia et al. (2001) reported the coseismic changes and compared them to the distances between the earthquake fault and the observations wells. Lee et al. (2002) reported coseismic hydrological changes associated with dislocation caused due to Chi-Chi earthquake, and Lai et al. (2004) compared these changes with the regional geological setting

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and seismic ground motions. They showed that in the nearby Choshui River alluvial fan area, the groundwater levels rose coseismically, and amplitudes increased with increase in ground acceleration and hydraulic conductivity, whereas in the slope regions near the earthquake fault, the water levels dropped with increasing intensity as the ground accelerations increased. In a recent study, Anhua He and Singh (2019) reported coseismic step changes in 138 wells in China associated with the Mw 7.9 Wenchuan earthquake of 2008. Earlier, Zheng et al. (2012) analyzed the coseismic responses of the fluid well pattern system in Jiangsu Province to the 2008 Wenchuan and 2011 Tohoku earthquakes. Shi et al. (2013) studied coseismic response of groundwater level in the three gorges well network and its relationship to aquifer parameters. Akita and Matsumoto (2004) showed coseismic changes induced by M8.0 Tokachi-oki earthquake in 2003 in Japan. These coseismic changes were observed in 29 wells with increases in wells located in southeast part of Hokkaido and decreases in other areas. Matsumoto et al. (2003) studied 28 cases of coseismic changes during 1981 to 1997 at Haibara well in central Japan and obtained an empirical relation between earthquake magnitudes and distances. They attributed these changes to the strong ground motions due to passing of seismic waves. Koizumi et al. (1996) reported groundwater anomalies associated with the 1995 Hyogoken Nanbu earthquake, in Japan. Chadha et al. (2003) reported four cases of coseismic changes in wells in the Koyna region, India. To comprehend the cause-and-effect relationship between local earthquakes and water level changes, they drilled 21 bore holes surrounding a well-constrained seismic source volume of 30  15  10 km3 dimension. Analysis of data from these wells from 1997 to 2003 revealed four cases of coseismic step changes associated with local earthquakes of 5.3 M 4.2 within a 50 km radius. Rudnicki et al. (1993) observed water levels at 17 wells in the Parkfield area along the San Andreas fault during January 1989 to 1990 as a part of the Parkfield earthquake prediction experiment. Using the recovery rates of water level changes due to fault creep or slow slip, they inferred the position of the fault zone. Later, Roeloffs (1998) analyzed persistent water level changes at a well near Parkfield caused by local and distant earthquakes. Gavrinlenko et al. (2000) reported permanent water level drop associated with Spitak earthquake of 1988 at a well located 110 km from the epicenter. This drop occurred postearthquake and might have been influenced by closeness of the fault zone. Following a sequence of nine Mw5.0–5.9 earthquakes in central Italy, changes in water table and spring discharge were reported for several months (Petitta et al. 2018). Data from 22 measurement sites, located within 100 km of the epicentral zones, were analyzed. The intensity of the induced changes was correlated with earthquake magnitude and distance to epicenters.

Seismic Signals in Well Water Observations

Transient Signals Transient changes in well water levels due to teleseismic earthquakes were reported for earthquakes of M > 7.5. Chadha et al. (2008) reported transient and persistent water level changes at few wells in Koyna, India, following Mw 9.3 Sumatra earthquakes in 2004 (Fig. 1). The transient changes were either a spike- or a steplike signal. They attributed these changes to the dynamic strain induced by the passage of seismic waves, most probably long period surface waves. Sil and Freymueller (2006) reported anomalous well water level changes in Fairbanks, Alaska, with the passing of seismic waves due to the Sumatra earthquake of 2004 in the Indian Ocean. Kitagawa et al. (2006) reported well water level changes in 38 observation wells due to Sumatra earthquake at a distance of more than 5000 km in Japan. Ishii-type borehole strainmeters installed in 10 of the observation wells also recorded changes in crustal strains. Sun et al. (2015) showed water level changes at Changping well in China related to Mw 9.0 Tohoku earthquake in Japan. They termed these changes as coseismic induced mostly by the surface waves of this distant earthquake. Similarly, He et al. (2017) reported coseismic response of water levels in the Jingle well in China associated with the Gorkha Nepal (Mw 7.8) earthquake in 2015 in the Himalaya. Several reports of earthquake-induced well water level changes at distant locations came out after the Mw7.9 Denali earthquake in 2002 in Alaska (Cassidy and Rogers 2004). In the Korean Peninsula, Yun et al. (2019) analyzed groundwater level changes in a monitoring well close to Yangsan fault zone, induced by local and offshore earthquakes around the world.

Preseismic Signals During 1970s there was a great optimism to study hydrological and geochemical signals for predicting earthquakes after the success of 1975 Haicheng earthquake in China. Wakita et al. (1988) presented preseismic changes in water level, temperature, radon, and strain associated with the 1978 Izu Oshima earthquake of M 7.0 in Japan. Later, Tsunogai and Wakita (1996) reported preseismic data on radon, chlorine, and strain prior to the 1995 Kobe earthquake of M7.2 in Japan and showed the remarkable similarity in the precursory pattern 1 month prior to the earthquakes. Roeloffs (1996) reported preseismic changes at the observation stations located within 35 km from the epicenter of the 1985 M6.1 Kettleman Hills earthquake along the San Andreas fault system. Two of the four water level sites and two of three volumetric strain sites showed covarying precursory signals beginning 3 days before the event. King et al. (1999) studied the water level drops observed at 16 wells in Tono mines in central Japan, prior to a M 5.8 local earthquake, and suggested

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Seismic Signals in Well Water Observations, Fig. 1 Transient changes in water levels at UKA, TAL, and GOV wells in the Koyna region, western India, induced by Mw9.3 Sumatra earthquake 2000 km away in the Indian Ocean. Arrow shows the arrival of surface waves at the well location

that these changes were premonitory in nature. Chadha et al. (2003) reported preseismic water level drops prior to a coseismic step associated with a M4.4 local earthquake in the Koyna region (Fig. 2). Igarashi et al. (1992) explained the preseismic water level drop as a “rebound anomaly” related to an increase of porosity and permeability due to fracturing, with subsequent recovery attributable either to influx of fluid or to compression.

Mechanisms Three mechanisms have been put forward to explain the seismic signals in the wells. Fault dislocation and static poroelastic strain mechanism states that coseismic water level changes can be explained by the coseismic static strain and the change in pore pressure predicted by the poroelastic theory (Wakita 1975; Ge and Stover 2000). Chadha et al. (2005) computed static volumetric strain change to explain the coseismic steps observed in four cases associated with M 4.2 local earthquakes in the Koyna region, India. They showed a good correlation between steplike increases in wells located in compression and decrease in dilatational zones but reported a misfit in amplitudes of the anomalies. Based on this study, they concluded that coseismic strains may be a function of site effects controlled by local heterogeneity in geological structures. Thus, simple elastic models cannot wholly explain the

amplitudes of the hydrological anomalies. Huang et al. (1995) and Grecksch et al. (1999) have suggested a nonlinear response of water levels to coseismic strain changes due to local heterogeneities. King et al. (1999) computed strain from dislocation of seismic faults in Japan and showed a good quantitative agreement with the strain inferred from the water level changes around the epicenter. Quilty and Roeloffs (1997) showed good agreement between predicted strain and observed water level changes due to 1994 earthquake in Parkfield, California. Lee et al. (2002) found a similar pattern using the 178 auto-recording monitoring wells associated with the 1999 Chi-Chi earthquake. Consolidation hypothesis proposes that seismic shaking causes sediments to consolidate or dilate under undrained conditions (Wang and Chia 2008). The water level will decrease or increase within the near field, and this change depends on the magnitude between the shear strain caused by seismic shaking and the critical threshold (104). However, the water level will increase at regional distances when seismic wave energy exceeds the amount of dissipated energy required to cause undrained consolidation (Yoshimi and Oh-Oka 1975; Hsu and Vucetic 2004; Lai et al. 2004; Wang and Manga 2010). Earthquake-enhanced permeability mechanism suggests enhancement in rock permeability either due to rock fracturing during earthquakes (Rojstaczer et al. 1995) or other processes like clogging and unclogging associated with aquifer deformation (Shi and Wang 2016) and consolidation of

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Seismic Signals in Well Water Observations

Seismic Signals in Well Water Observations, Fig. 2 Records of air pressure, and residual water levels at two wells in the Koyna region, western India, during March-May, 1997. A 23 days precursory drop is clearly seen

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surficial deposits (Xue et al. 2016). Sometimes fractures in the earth are filled up with broken rock material during the normal course of groundwater flow and thus become points of local stress concentration increasing local pore pressure. These local stress points are broken when there is a rapid flow of water during an earthquake (Brodsky et al. 2003) redistributing the pore pressure as the permeability increases that can lead to changes in water levels in wells.

Summary Earthquakes cause dislocations along faults in earth’s crust which leads to redistribution of stresses resulting in alteration of subsurface medium properties and flow of groundwater. Anomalous changes were observed in well water levels after local, regional, and teleseismic earthquakes. These induced anomalous changes termed here as “seismic signals” in well water levels are categorized as pre-, co-, post-, and transient anomalies which manifest as preseismic water level drops, a

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coseismic step- or a transient spikelike change, and sometimes a gradual post-seismic increase. These signals indicate the spatial distribution of seismic energy density that triggers the different responses. Three mechanisms have been put forward to explain these seismic signals in terms of a fault dislocation model and static poroelastic strain hypothesis where predicted strain is found to be in good agreement with the strain predicted by water level changes around the epicentral area. According to static strain hypothesis, the coseismic steplike anomalies can be explained in terms of coseismic static strain and changes in pore pressure predicted by poroelastic theory. The second is the consolidation hypothesis which states that the increase or decrease in well water levels depends on the magnitude of the shear strain caused by seismic shaking and the critical threshold of strain at local distances. However, the water level will increase at regional distance when seismic wave energy exceeds the amount of dissipated energy required to cause undrained consolidation. Finally, the earthquake-enhanced permeability mechanism suggests enhancement in rock permeability either due to

Seismic Signals in Well Water Observations

rock fracturing during earthquakes or other processes like clogging and unclogging associated with aquifer deformation and consolidation of surficial deposits. Seismic signals in well water levels, while clearly are strain indicators, do not have a well-understood relation to strain. They exhibit “response heterogeneity” in the signals which are highly variable in space and time and thus have no straightforward explanation. The mechanisms put forward to explain these signals may not be mutually exclusive but may occur together and may even be causally related. The preseismic changes in well water levels are still far from using them as short-term earthquake precursors. Very often, these precursory changes which represent the pre-earthquake phase are very small and therefore not obvious in the raw data which is also affected by atmospheric pressure, significant precipitation, and earth tides. Innovative filtering techniques have to be developed to decipher the earthquake-related anomalies from non-tectonic effects in the data. Definite patterns of earthquake-related water level anomalies have to be established, both site specific and global, so that predictive capabilities can be developed. We realize that earthquake prediction using short-term precursors is not easy, but sound observations which may lead to a physical model must continue. The earthquake-induced water level changes in groundwater have other implications, too. The supply of groundwater, both in terms of quantity and quality, contaminant transport, and underground waste repositories are some of the important issues which need attention. Plans made to manage these issues are based on pre-earthquake scenario which may drastically change, post event. Thus, there is a need to accelerate these studies by casting wider net-expanding observing programs to capture many more events and their precursors.

Bibliography Akita F, Matsumoto N (2004) Hydrological responses induced by the Tokachi–Oki earthquake in 2003 at hot spring wells in Hokkaido, Japan. Geophys Res Lett 31(16):371–375 Brodsky EE, Roeloffs E, Woodcock D, Gall I, Manga M (2003) A mechanism for sustained groundwater pressure changes induced by distant earthquakes. J Geophys Res 108(B8):503–518 Cassidy JF, Rogers GC (2004) The Mw7.9 Denali fault earthquake of 3 November 2002: felt reports and unusual effects across western Canada. Bull Seismol Soc Am 94:53–57 Chadha RK, Pandey AP, Kuempel HJ (2003) Search for earthquake precursors in well water levels in a localized seismically active area of reservoir triggered earthquakes in India. Geophys Res Lett 30(7):1416. https://doi.org/10.1029/2002GLO016694 Chadha RK, Srivastava K, Kuempel HJ (2005) Earthquake related changes in well water level and their relation to a static deformation model for the seismically active Koyna-Warna region. In: Rummel F (ed) Rock mechanics with emphasis on stress. Oxford & IBH, New Delhi, pp 135–150 Chadha RK, Singh C, Shekar M (2008) Transient changes in well water level in bore wells in western India due to 2004 mw 9.3 Sumatra earthquake. Bull Seismol Soc Am 98(5):2553–2558

1497 Chia Y, Wang YS, Chiu JJ, Liu CW (2001) Changes of groundwater level due to the 1999 ChiChi earthquake in the Choshui River alluvial fan in Taiwan. Bull Seismol Soc Am 91(5):1062–1068 Gavrinlenko P, Melikadze G, Chelidze T, Gilbert D, Kumsiashvili G (2000) Permanent water level drop associated with Spitak earthquake: observation at Lisi borehole (republic of Georgia) and modelling. Geophys J Int 143:83–98 Ge S, Stover SC (2000) Hydrodynamic response to strike- and dip-slip faulting in a half-space. J Geophys Res 105(11):25513–25524 Grecksch G, Roth F, Kuempel HJ (1999) Coseismic well level changes due to the 1992 Roermond earthquake compared to static deformation of half space solutions. Geophys J Int 138:470–478 He A, Singh RP (2019) Groundwater level response to the Wenchuan earthquake of May 2008. Geomat Nat Haz Risk 10(1):336–352 He A, Fan X, Zhao G, Liu Y, Singh RP, Hu Y (2017) Co-seismic response of water level in the jingle well (China) associated with the Gorkha Nepal (mw 7.8) earthquake. Tectonophysics 714/715:82–89 Hsu CC, Vucetic M (2004) Volumetric threshold shear strain for cyclic settlement. J Geotech Geoenviron Eng 130(1):58–70 Huang W, Rojstaczer S, Breau S (1995) Coseismic response of water level to earthquakes in the San Jacinto fault, southern California. EOS Tran Am Geophys Un 76:369 Igarashi G, Wakita H, Sato T (1992) Precursory and coseismic anomalies in well water levels observed for the February 2, 1992 Tokyo Bay earthquake. Geophys Res Lett 19:1583–1586 King CY, Azuma S, Igarashi G, Ohno M, Saito H, Wakita H (1999) Earthquake-related waterlevel changes at 16 closely clustered wells in Tono, Central Japan. J Geophys Res 104(B6):13073–13082 Kitagawa Y, Koizumi N, Takahashi M, Matsumoto N, Sato T (2006) Changes in groundwater levels or pressures associated with the 2004 earthquake off the west coast of northern Sumatra (M9. 0). Earth Planet Space 58(2):173–179 Koizumi N, Kano Y, Kitagawa Y, Sato T, Takahashi M, Nishimura S, Nishida R (1996) Groundwater anomalies associated with the 1995HyogokenNanbu earthquake. J Phys Earth 44:373–380 Lai W-C, Koizumi N, Matsumoto N, Kitagawa Y, Lin C-W, Shieh C-L, Lee Y-P (2004) Effects of seismic ground motion and geological setting on the coseismic groundwater level changes caused by the 1999 Chi-Chi earthquake, Taiwan. Earth Planet Space 56(9):873–880 Lee M, Liu TK, Ma KF, Chang YM (2002) Coseismic hydrological changes associated with dislocation of the September 21, 1999 Chichi earthquake, Taiwan. Geophys Res Lett 29(17):51–54 Liu LB, Roeloffs E, Zheng XY (1989) Seismically induced water level fluctuations in the Wali Well, Beijing, China. J Geophys Res 94(B7):9453–9462 Matsumoto N, Kitagawa G, Roeloffs EA (2003) Hydrological response to earthquakes in the Haibara well, Central Japan–I. Groundwater level changes revealed using state space decomposition of atmospheric pressure, rainfall and tidal responses. Geophys J Int 155(3):885–898 Petitta M, Mastrorillo L, Preziosi E et al (2018) Water-table and discharge changes associated with the 2016–2017 seismic sequence in Central Italy: hydrogeological data and a conceptual model for fractured carbonate aquifers. Hydrogeol J 26(4):1009–1026 Quilty EG, Roeloffs EA (1997) Water-level changes in response to the 20 December 1994 earthquake near Parkfield, California. Bull Seismol Soc Am 87(2):310–317 Roeloffs EA (1996) Poroelastic techniques in the study of earthquakerelated hydrologic phenomena. Adv Geophys 37:135–195 Roeloffs EA (1998) Persistent water level changes in a well near Parkfield, California, due to local and distant earthquakes. J Geophys Res 103(B1):869–889 Rojstaczer S, Wolf S, Micheli R (1995) Permeability enhancement in the shallow crust as a cause of earthquake-induced hydrological changes. Nature 373:237–239

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1498 Rudnicki JW, Yin J, Roeloffs EA (1993) Analysis of water level changes induced by fault creep at Parkfield, California. J Geophys Res 98:8143–8152 Shi Z, Wang G (2016) Aquifers switched from confined to semiconfined by earthquakes. Geophys Res Lett 43(21):11166–11172 Shi ZM, Wang GC, Liu CL, Mei JC, Wang JW, Fang HN (2013) Coseismic response of groundwater level in the three gorges well network and its relationship to aquifer parameters. Chin Sci Bull 58(25):3080–3087 Sil S, Freymueller JT (2006) Well water level changes in Fairbanks, Alaska, due to the great Sumatra-Andaman earthquake. Earth Planet Space 58(2):181–184 Sun X, Wang G, Yang X (2015) Coseismic response of water level in Changping well, China, to the mw 9.0 Tohoku earthquake. J Hydrol 531:1028–1039 Tsunogai U, Wakita H (1996) Anomalous changes in groundwater chemistry. J Phys Earth 44(4):381–390 Wakita H (1975) Water wells as possible indicators of tectonic strain. Science 189(4202):553–555 Wakita H, Nakamura Y, Sano Y (1988) Short-term and intermediateterm geochemical precursors. Pure Appl Geophys 126:267 Wang CY, Chia Y (2008) Mechanism of water level changes during earthquakes: near field versus intermediate field. Geophys Res Lett 35(12):109 Wang CY, Manga M (2010) Earthquakes and water. Lecture notes in earth sciences, vol 114. Springer-Verlag, Berlin Xue L, Brodsky EE, Erskine J, Fulton PM, Carter R (2016) A permeability and compliance contrast measured hydrogeologically on the San Andreas fault. Geochem Geophys Geosyst 17(3):858–871 Yoshimi Y, Oh-Oka H (1975) Influence of degree of shear stress reversal on the liquefaction potential of saturated sand. Soil Found 15(3):27–40 Yun S, Hamm S, Cheong J, Lee C, Seo W, Woo N (2019) Analyzing groundwater level anomalies in a fault zone in Korea caused by local and offshore earthquakes. Geosci J 23(1):137–148 Zheng J, Jiang H, He Z (2012) Analysis of the co-seismic responses of the fluid well pattern system in Jiangsu Province to the Wenchuan and Tohoku earthquakes. Earthq Sci 25(3):263–274

Seismic Stereotomography Gilles Lambaré and Thibaut Allemand CGG, Massy, France

Synonyms Slope tomography

Definition Stereotomography is a special form of reflection tomography proposed by Billette and Lambaré in 1998. It aims to reconstruct smooth, spatially varying properties of the subsurface (in particular the wave velocity) from seismic reflection data. Conventional reflection travel-time tomography relies on the

Seismic Stereotomography

travel-time of continuous reflectors tracked in the prestack dataset over their full extent. On the other hand, stereotomography uses locally coherent reflected events characterized not only by the source and receiver positions and the reflection time, but also by their local slopes (the spatial derivatives of the reflection time with respect to receiver and source positions). Like travel-time tomography methods, stereotomography methods are based on seismic ray theory.

History The roots of stereotomography go back to 1936 with the publication by F. Rieber of a paper in Geophysics describing a new reflection system with controlled directional sensitivity. Soviet scientists recognized the potential of this approach for velocity model building for seismic migration (see ▶ “Seismic, Migration”) and developed a method called CDR (controlleddirectional-reception) in the 1950s and 1960s, principally by Professor Riabinkin (Billette and Lambaré 1998). A few years later, the approach gained interest from western geophysicists and in particular from Stanford’s Charles Sword, who spread it in his community. By 1998, Billette and Lambaré recast CDR and related approaches in a common theoretical framework under the generic term of slope tomography. These approaches have in common their aim to estimate the velocity model, c(x) (x denoting the position in the subsurface), from a set of locally coherent reflected events (events that can be visually followed on a few adjacent traces in a given observation domain) characterized by their reflected travel-time, Tsr, source and receiver positions, s and r, and slopes, that is, the spatial derivative of the travel-time, ps and pr (Fig. 1). These locally coherent events are interpreted within seismic ray theory (see ▶ “Seismic ray theory”) as pairs of ray segments connecting the reflecting point, X, to the shot and receiver positions (Fig. 1). Compared to reflection traveltime tomography, the additional information brought by the slopes allows one to consider the reflection events locally: there is no need to track a reflection event over its full extent in the migrated or unmigrated prestack dataset. The picking (the extraction of the kinematic characteristics of the events, that is, their travel-time and slopes) becomes much easier and can therefore be performed more densely compared with conventional methods. Within this theoretical frame, Billette and Lambaré also proposed a new approach, based on a generalization of the other slope tomography methods and with the expected benefit of improved robustness with respect to complex media involving multipath ray trajectories. They called it “stereotomography” and detailed its numerical implementation based on an efficient local optimization scheme involving paraxial ray tracing (see ▶ “Seismic Ray Theory” and ▶ “Inverse Theory, Linear”). Since this pioneering publication,

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tracing of rays followed by a velocity update. In stereotomography, they are optimized jointly. The model (m) consists of: m¼

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Seismic Stereotomography, Fig. 1 A locally coherent event in the prestack data cube, characterized by its central source and receiver positions s and r, slopes ps and pr, and travel-time, Tsr (Fig. 3 from Lambaré (2008))

stereotomography and, more generally, slope tomography methods have gained in effectiveness and established themselves as powerful approaches for industrial velocity model building. One of their big advantages over the standard tomographic velocity model building methods is the non-linear update of velocity model (see Lambaré et al. (2014))

Data and Model in Slope Tomography All slope tomography methods (and stereotomography in particular) have in common the description of the dataset. It consists of a set of N locally coherent reflected events characterized by their source and receiver positions, source and receiver slopes, and their travel-times,  N d ¼ ðs, r, ps , pr , T sr Þi i¼1 :

ð1Þ

Such a dataset needs to be explained by a model consisting of an associated set of pairs of ray segments and a velocity field (Fig. 1). Various cost functions and strategies have been proposed to estimate the optimum model, with important consequences for the allowed model space and optimization process. In all formulations, the model space contains ray parameters and a velocity model, which can be estimated through either joint or sequential optimization, that is, the

ð3Þ

where CD is the a priori covariance matrix for data parameters (Billette and Lambaré 1998) and Regul(m, d) stands for regularization terms. When ray segments and velocity are optimized sequentially, ray segments can be determined by relaxing partly the data fitting or the reflection conditions (the two ray segments starting from a common point). Once the rays are traced, the velocity is estimated based on fitting the remaining condition (Billette and Lambaré 1998). Among the most popular methods of this kind, the slope tomographic approach initially proposed by Chauris et al. (2002) is widely used in the industry following Guillaume et al. (2008). In this approach, the ray segments are first recovered for a given velocity fitting source and receiver positions, midpoint slopes, and traveltime (this is called kinematic migration). The velocity is then updated considering the misfit of the residual move-out slope in the offset direction in the common image gathers. The numerical scheme used for solving the inverse problem aims at minimizing the cost function (3) through iterative local optimization, using typically a quasi-Newton scheme (see ▶ “Inverse Theory, Linear”). A local optimization offers an efficient solution for ensuring convergence of the optimization process. It requires the computation of the data and Fréchet derivatives (derivatives of the data with respect to the model parameters), which can be estimated by using ray tracing and paraxial ray tracing, respectively (see ▶ “Seismic Ray Theory”), provided the model exhibits proper smoothness. Cubic cardinal B-Splines are a good choice which can be used in gridded or multilayered velocity models (for references, see Lambaré et al. (2014)) not only because of their smoothness but also because they act as a model preconditioner to ease the inversion. Velocity model building from surface reflection data being by nature an ill-posed inverse problem it requires

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regularization methods, such as Tikhonov regularization. More advanced, dip-dependent regularizations have also been proposed, allowing the production of structurallyconformed velocity models. Another strategy requires the introduction of other types of data. For example, in dipconstrained tomography, the velocity is updated considering the misfit between the structural dips of the kinematically remigrated reflected events and an expected dip model (Lambaré et al. 2014). Another example is the incorporation of direct arrival travel times measured from surface sources with receivers located in a well, or first arrival travel times from surface-seismic measurements (see Allemand et al. (2017)). Including this kind of data is helpful, for example, to estimate anisotropy in order to reconcile the kinematics of reflected and diving waves. One of the first and most critical steps in velocity model building is the picking of locally coherent events (see Lambaré (2008) for a review). This can be performed either in the un-migrated data domain or in the (time- or depth-) migrated domain using automatic slope or dip picking tools. If done in the migrated domain, a kinematic demigration step is needed to

recover the so-called kinematic invariants, namely, the kinematic characteristics of the picked locally coherent events in the un-migrated domain (Guillaume et al. 2008). This versatility is a great strength in the approach, which has been extended from surface common-offset migration to any type of prestack depth migration, such as common-angle migration (Montel and Lambaré 2019). After picking, careful editing is needed to remove outliers. This is important as the quality of the picked data has a major influence on the inversion result.

Seismic Stereotomography, Fig. 2 Application of slope tomography to a land dataset. Top right: dip picks (in red) superimposed on the depth migrated stack for acquisition azimuth 0 (black to white). Blue curve indicates the location of top of salt (TOS), green curve the base of salt (BOS). Top left: residual move out curves (in red) superimposed on a

selection of common image gathers (black to white) for azimuth sector 0, located in the yellow box. Bottom: Velocity models obtained by multilayer (left) and high-definition (right) tomography (Hermant et al. 2014). Courtesy of CGG, Petroleum Development Oman, and the Ministry of Oil and Gas of the Sultanate of Oman

Applications of Slope Tomography Since it was first published in 1998, stereotomography and slope tomography have been implemented and applied in various contexts: marine streamer, ocean bottom seismic and land data, in 2D and 3D, isotropic and anisotropic contexts, using PP or PS waves (see Lambaré (2008) and Lambaré et al. (2014)). We show in Fig. 2 an example of such an application (Hermant et al. 2014). The dataset is a high-density full-

Seismic Stereotomography

azimuth seismic land dataset from South of the Sultanate of Oman. The area exhibits complex geological settings with an alternation of fast carbonate layers and slower shale or clastic layers overlying salt bodies. The associated strong lateral velocity variations are particularly challenging for velocity model building. The picking is done in the migrated domain to get the local dip of the events on the migrated stack and the corresponding local slope of the residual move-out curves in offset in the common image gathers (Fig. 2 (Top)). Figure 2 (Bottom left) shows the velocity model obtained by multilayer tomography. This slope tomographic approach updates a layered velocity model with explicit velocity discontinuities iteratively repositioned during the updating process. Figure 2 (Bottom right) shows the velocity model after a further step of high definition tomography (an extension of slope tomography aimed at enhancing vertical resolution).

Challenges and Perspectives Stereotomography and slope tomography methods in general are powerful velocity model building tools widely used in the seismic industry (Lambaré et al. 2014). The picking remains critical, with a major influence on the result. It should be accurate but also as dense as possible. The ability to handle a dense set of picks not limited to a few reflectors, combined with efficient nonlinear solvers and geologically constrained regularizations, allows high-resolution models to be obtained, far exceeding resolution of conventional traveltime tomography. On the other hand suboptimal picking will degrade the result, and a thorough editing should be applied prior to the inversion, for example using machine learning techniques. In terms of tomographic velocity updates, Sambolian et al. (2019) recently investigated replacing ray tracing of individual pairs of ray segments by the use of travel-time maps (computed here by an eikonal solver). They promote the use of a matrix-free approach in order to invert for denser sets of picks. An obvious limitation of these methods is the high frequency approximation, especially when dealing with complex media where waveforms have complicated behaviors with tuning effects. Hence, efforts should be made to extend the approach towards full-wave techniques. The connection between stereotomography and full-wave velocity model building, such as Differential Semblance Optimization, has been pointed out (“differential semblance is essentially dataweighted stereotomography, implicitly computed without event picking” Symes (2008)). Note, however, that the introduction of a picking step allows the extraction of the most significant kinematic information from the data, and the selection of events represents, to our minds, a serious advantage that would be worth preserving in full-wave approaches.

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Conclusions In the last 20 years, stereotomography and its variations (under the generic term of slope tomography) have earned their stripes in terms of industrial applications. This success is due to their versatility and efficiency at dealing with the critical velocity model building challenge (a non-linear and ill-constrained inverse problem). Even if full-wave methods are gaining popularity in complex areas, slope tomography remains and will remain a key tool for industrial velocity model building.

Cross-References ▶ Inverse Theory, Linear ▶ Seismic Data Acquisition and Processing ▶ Seismic Imaging, Overview ▶ Seismic Ray Theory ▶ Seismic Tomography ▶ Seismic, Migration ▶ Seismic, Waveform Modeling and Tomography ▶ Traveltime Tomography Using Controlled-Source Seismic Data

Bibliography Allemand T, Sedova A, Hermant O (2017) Flattening common image gathers after full-waveform inversion: the challenge of anisotropy estimation. In: 87th annual international meeting, SEG, expanded abstract, pp 1410–1415 Billette F, Lambaré G (1998) Velocity macro-model estimation by stereotomography. Geophys J Int 135:671–680 Chauris H, Noble M, Lambaré G, Podvin P (2002) Migration velocity analysis from locally coherent events in 2D laterally heterogeneous media, Part I: theoretical aspects. Geophysics 67:1202–1212 Guillaume P, Lambaré G, Leblanc O, Mitouard P, Le Moigne J, Montel J-P, Prescott A, Siliqi R, Vidal N, Zhang X, Zimine S (2008) Kinematic invariants: an efficient and flexible approach for velocity model building. In: 78th annual international meeting, SEG, expanded abstract, pp 3687–3692 Hermant O, Mansoor K, Baudon H, Al Amri A, Al Maamari A (2014) High-definition tomography: an application to intra-salt velocity update. In: EAGE/SPE workshop on subsalt imaging the challenges of subsalt exploration and imaging in deep water of the Middle East and North Africa, 16–19 February 2014, Limassol, Cyprus, SS21 Lambaré G (2008) Stereotomography. Geophysics 73(5):VE25–VE34 Lambaré G, Guillaume P, Montel JP (2014) Recent advances in raybased tomography. In: 76th annual international conference and exhibition, EAGE, Extended Abstracts, We G103 01 Montel JP, Lambaré G (2019) Kinematics of common-image gathers – Part 1: theory. Geophysics 84(5):S437–S447 Sambolian S, Operto S, Ribodetti A, Tavakoli FB, Virieux J (2019) Parsimonious slope tomography based on eikonal solvers and the adjoint-state method. Geophys J Int 218(1):456–478 Symes W (2008) Migration velocity analysis and waveform inversion. Geophys Prospect 56:765–790

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1502

Seismic Structure at Mid-Ocean Ridges

Definition Lithosphere

Asthenosphere

Anisotropy

The cool, uppermost layer of the solid earth that moves as a unit and has some longterm elastic rigidity. It usually consists of both crust and the uppermost mantle. The mantle portion is typically characterized by high seismic velocities and low attenuation. A more deformable, low-viscosity layer in the mantle underlying the lithosphere. Characterized by low seismic velocities and relatively high attenuation. Physical properties at any one point varying depending on direction; for seismic waves, velocity may depend on direction of propagation and direction of polarization of the wave.

Introduction Mid-ocean ridges are spreading centers where two oceanic, lithospheric plates move apart. The separation of the plates induces upwelling in the underlying astheno-spheric mantle. Because melting temperature decreases with decreasing pressure, as the mantle upwells, it undergoes pressure-release partial melting, producing magma that migrates upward to form new oceanic crust. As the plates move away from the spreading center, heat is lost to the surface conductively by diffusion and convectively by hydrothermal circulation. As the crust and mantle lose heat, the magma solidifies, and the plates thicken and cool with increasing age of the seafloor. Although the general outline of the processes beneath midocean ridges leading to the formation of new seafloor is well known, there are many questions remaining about the details of the mantle flow, melt generation, and melt migration that have not yet been fully answered. For example, along midocean ridges, spreading centers are offset and segmented by transform faults and overlapping spreading centers. Are there distinct centers of mantle upwelling beneath each ridge segment or is the segmentation a shallow manifestation of stresses within the lithosphere with upwelling a more or less continuous phenomenon in the asthenosphere? How broad and deep is the melt production region?

Mantle Structure A cross section of the shear velocity structure beneath the East Pacific Rise spreading center is shown in Fig. 1 in comparison to the velocity variations that would be expected for simple conductive cooling of the plates if shear velocity were sensitive only to temperature and pressure. This tomographic image is based on the propagation of Rayleigh surface waves across two OBS arrays deployed for periods of 6 and 12 months. The expected thickening of the plate with increasing distance from the ridge axis as the plate cools is clearly observed in the form of increasing shear wave velocity near the

Depth (km)

Donald W. Forsyth Department of Geological Sciences, Brown University, Providence, RI, USA

The propagation of seismic waves through the crust and mantle provides one of the most direct ways of probing the structure beneath the ridges associated with plate separation and crust formation. The velocity, attenuation, and anisotropy of the waves are affected by temperature, composition, the presence of melt or cracks, and the crystal fabric. Tomographic images and maps of crustal and uppermost mantle structure are typically created in active source experiments where artificial sound sources, such as air guns, are recorded by ocean bottom seismometers (OBS) and/or long arrays of hydrophones towed behind ships. Deeper mantle structure is probed using signals generated by distant, teleseismic earthquakes recorded on arrays of ocean-bottom and land seismometers over periods of months to years. The logistical difficulty and expense of such experiments means that relatively few mid-ocean ridges have been studied in detail.

4.6 4.4

50 4.2

100 4.2

150 700

600

500

400

300

200

100

0

–100 –200

Distance from Ridge (km) W-E 4.2

Depth (km)

Seismic Structure at Mid-Ocean Ridges

4.1

50 4

100 150 700

4.1 4.2

600

500

400

300

200

100

0

–100 –200

Distance from Ridge (km) W-E

Seismic Structure at Mid-Ocean Ridges, Fig. 1 Tomographic cross section of the East Pacific Rise, comparing predicted shear velocity structure (top) to observed structure (bottom). Contours are labeled in km/s. Velocities are significantly lower than predicted for temperature effects alone, suggesting the presence of partial melt. (After Harmon et al. (2009))

Seismic Structure at Mid-Ocean Ridges

1503

surface. Velocity changes extend substantially deeper than is predicted and the shear velocity is lower than expected for the direct effects of temperature variations. Both of these departures from the predictions are indications that partial melt may be present, which could reduce the shear (S) wave velocity. The lowest velocities are observed at depths of 60–80 km, where petrological models predict maximum melt production. There also are very low shear velocities at shallow depths immediately beneath the ridge axis, which may represent higher melt concentrations in the mantle that accumulated as melt migrates upward and inward to form the new oceanic crust at the ridge axis. Another important feature of the shear velocity structure is the asymmetry across the ridge axis. To the east, beneath the Nazca plate, the high-velocity surface layer increases in thickness more rapidly and the very low-velocity region is absent. This asymmetry is also observed in the degree of shear wave splitting, an indicator of anisotropy, in delays of compressional (P) and S waves, in electrical conductivity, and in the rate of subsidence of the seafloor away from the ridge axis (MELT Seismic Team 1998; Evans et al. 1999). S wave tomography indicates that the asymmetry may extend to depths of 200 km or more (Hammond and Toomey 2003). The asymmetry is probably caused by large-scale mantle flow coming from the hotspot region to the west beneath the Pacific plate, coupled with migration of the spreading center to the west. The fast direction for seismic wave propagation, as indicated by shear wave splitting and Rayleigh wave anisotropy, is perpendicular to the East Pacific Rise, consistent with the alignment of olivine crystals in the mantle expected for plate formation and flow from the west. The East Pacific Rise is one of the fastest spreading ridges, with full spreading rate of about 14 cm/year. Pronounced asymmetry is also observed across the Reykjanes Ridge south of Iceland (Fig. 2), which has a full spreading rate of W

E

50 100 150 200 –400 –300 –200 –100

0

100

200

300

400

Distance (km)

4.0

4.1

4.2

4.3

4.4

4.5

only about 2 cm/year (Delorey et al. 2007). In this case, the tomographic study took advantage of the existence of arrays of stations on Iceland, which straddles the Reykjanes/MidAtlantic Ridge. Like the East Pacific Rise, very low shear velocities (~4.0 km/s) indicative of the presence of melt are found in a broad region beneath the ridge. For both the East Pacific Rise and the Reykjanes Ridge, there is too little attenuation of surface waves to attribute the very low velocities to the effect of high temperature alone. Velocities at depths shallower than 80 km are lower on the west side, beneath the North American plate, perhaps due to the westward migration of the ridge and upwelling in the mantle in the wake of the thicker North American lithosphere farther west. The anisotropy pattern is different than for typical mid-ocean ridges, perhaps indicating that there is along-axis flow away from the Iceland hotspot in the asthenosphere. To date, there have been no experiments that provide good control of along-axis variations in mantle structure at depths of tens of kilometers or more where melt production is expected to occur. In the Gulf of California, a surface wave study showed that there are along-axis variations in shear velocity with minima spaced at intervals of about 250 km, perhaps indicative of discrete upwelling centers (Wang et al. 2009), but that spreading system is flanked by continental crust on both sides and dominated by long transform faults, so it may not be typical. At shallower depths just beneath the Moho (the transition from crust to mantle), there are clearly along-axis variations in P-wave velocity on the northern East Pacific Rise, the boundary between the Pacific and Cocos plates (Toomey et al. 2007). Using long offset arrivals refracting from the Moho (Pn) observed in an active experiment, velocity minima were found spaced about 25 km apart (Fig. 3). P-wave velocities in the minima are 7.4 km/s or less, suggesting the presence of 1–3% melt distributed in films or thin sheets (typical P-wave velocities at the Moho are 7.8–8.4 km/s). However, it is not clear whether these apparent centers of melt lie above centers of upwelling mantle or they represent a scale length associated with melt migration. Most of the centers of melt concentration lie beneath or very close to the spreading center determined from detailed bathymetric surveys, but one at 9
300 N is displaced several kilometers from the axis. Because the fast direction for P-wave propagation is skewed from the spreading direction and is not perpendicular to the strike of the ridge axis, Toomey et al., inferred that upwelling and mantle flow at depth may also be skewed, controlling the location of the velocity minima just below the Moho.

4.6

Vsv (km/s)

Seismic Structure at Mid-Ocean Ridges, Fig. 2 Tomographic cross section of shear wave velocity structure across Reykjanes Ridge south of Iceland. Note strong asymmetry in upper 100 km between North American plate (west) and European plate (east). (After Delorey et al. (2007))

Crustal Structure Basaltic melt migrates upward through the mantle and is focused at the ridge axis. The mechanisms for focusing are

S

1504 Seismic Structure at MidOcean Ridges, Fig. 3 Bathymetry of the East Pacific Rise (left) and tomographic image of the mantle 9 km beneath the seafloor (right). Dashed lines show plate boundary. Solid lines show locations of air gun shots fired at 500-m intervals. Squares and circles are locations of ocean bottom receivers. Contour interval on tomographic image of P-wave velocity is 0.1 km/s. Green lines with double arrowheads indicate fast direction for anisotropic wave propagation. Black lines with arrows indicate direction of relative motion between the Pacific and Cocos plates. Note that some of the slowest region is displaced from the plate boundary. (After Toomey et al. (2007))

Seismic Structure at Mid-Ocean Ridges

10⬚ 00'

9⬚ 30'

9⬚ 00'

8⬚ 30' Nuvel 1A

–104⬚ 30'

2500

–104⬚ 00'

3000

Depth, m

still not well known: there may be melt-rich channels at the base of the lithosphere that guide the melt upward and toward the ridge axis; there may be pressure gradients within the deforming mantle that help push the melt toward the axis; or there may be anisotropic cracks or dunite channels that form easy paths for melt migration. Once the melt reaches crustal levels, there may be redistribution along axis through dikes or a continuous magma chamber. It is clear from seismic studies, however, that there is very little magmatic addition to the crust outside the immediate vicinity of the ridge axis. The crust is essentially full thickness at the spreading center itself (Detrick et al. 1987). The classic model of oceanic crust is a layer of extrusive basalts at the surface in the form of porous pillow basalts and sheet flows, underlain by a region of sheeted dikes that feed the extrusive layer from a magma chamber, and the lower crust consisting of gabbros that solidify from the magma chamber or underlying mush zone. The seismic structure

3500

–104⬚ 30'

–104⬚ 00'

7.2 7.4 7.6 7.8 8.0 8.2 8.4

Vp, km/s

has also commonly been described in terms of layers that have often been equated to the lithological layering: layer 2A is a low-velocity layer at the surface several hundred meters thick (layer 1 is sediments that are deposited on top of the basaltic crust); layer 2B is a transition region in which the P velocity increases rapidly downward; and layer 3 has low vertical velocity gradients and high P-wave velocities of 6.5–7.0 km/s. A number of investigations, however, have demonstrated that the seismologic layering does not correspond exactly to the lithological layering and that seismic structure is primarily controlled by porosity. The middle to lower crust, layer 3, contains both dikes and gabbro bodies. At the axis of fast-spreading ridges, there usually is a lowvelocity region only a few kilometers wide at most that represents a zone of partial melting that extends throughout the crust (Fig. 4). At the top of this low-velocity region is a very low velocity layer, which is imaged as a prominent reflector in seismic reflection profiles at depths of 1.5–2.5 km on the

Seismic Structure at Mid-Ocean Ridges

1505

Seismic Structure at MidOcean Ridges, Fig. 4 Threedimensional view of the anomalous P-wave velocity structure of the East Pacific Rise at 9
300 N (a one-dimensional reference model has been subtracted). (From Dunn et al. (2000).) Overlying bathymetry is shown at top, with red shallow to blue, deep. The red regions below the ridge axis correspond to areas of partial melting. The axial magma chamber is at the top of these anomalously slow regions

4 Moho

8 12

15 10

5

Dis

Depth, km

0

6

0

tan

ce,

km

0

5 6

10 15

12

m

e, k

anc

Dist

–2.2 – 2.0 –1.8 –1.6 –1.4 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 km/s

East Pacific Rise (Detrick et al. 1987). The depth to this reflector tends to increase with decreasing spreading rate or proximity to a fracture zone and it is typically absent at slowspreading ridges (Singh et al. 2006). The width varies from about 250 m to several kilometers. It is interpreted as the top of an axial magma chamber (AMC) or melt sill, with typical thickness less than 100 m. Modeling of P to S conversions in wide-angle reflections indicates that melt content in the AMC varies from nearly 100% to less than 30% along the ridge (Singh et al. 1998; Canales et al. 2006). Estimates of melt content in the deeper crustal mush zone (consisting of a mixture of melt and crystals) are of the order of 2–8% in the lower crust and 3–12% near the Moho (Dunn et al. 2000). The fact that the zone in which melt is present is so narrow even at fastspreading ridges means that hydrothermal circulation must be very efficient in removing heat, as purely conductive cooling of the crust would result in a much wider zone of partial melting. Although there are variations in structure along-axis at fast-spreading ridges, the along-axis variations are much more pronounced at segmented, slow-spreading ridges. Typically, the crust thins approaching transform offsets. Within the fracture zone itself, the basaltic crust may be as thin as 1 km or less, but seismically there may be an altered, fractured layer that is a few kilometers thick and characterized by unusually low velocities, so that it looks like crust. The low-velocity region in the fracture zone, however, probably is mostly mantle that is altered by interaction with water

penetrating down cracks that are repeatedly opened by slip along the transform fault. Near the center of ridge segments between two transform offsets, the crust tends to be thicker and lower in velocity than elsewhere (Fig. 5), suggesting that melt is preferentially delivered to the crust from the mantle at that point. The upper crust at slow spreading ridges is anisotropic, with P-waves traveling faster along axis than perpendicular to it, indicating that faults and fissures are preferentially aligned parallel to the spreading center (Barclay and Toomey 2003).

Summary The seismic velocity structure of mid-ocean ridges is controlled by crustal thickness, cracking or porosity, temperature, melt, and crystal orientation. Low-velocity regions in the mantle that are caused by high temperatures and partial melt indicate that melt is generated in a broad region beneath spreading centers. Asymmetries in the velocity structure show that upwelling and melting beneath ridges are strongly influenced by global mantle circulation and plate motions. Crustal low-velocity regions are concentrated very near the ridge axis, so melt must migrate both vertically and horizontally to the ridge axis from the broad melt production region. The narrowness of the low- velocity zone in the crust requires that hydrothermal circulation must rapidly cool the crust.

S

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Seismic Structure at Mid-Ocean Ridges

South –2

North Oceanographer FZ

OH–1

NTO 1

–4 –6 –8 –10 –12 –14 –50

–40

–30

–20

–10 –0 10 Velocity, km/s

20

30

40

50

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Seismic Structure at Mid-Ocean Ridges, Fig. 5 P-wave velocity structure along the Mid-Atlantic Ridge axis from a non-transform offset on the left to the Oceanographer fracture zone on the right. Crustal thickness from wide-angle reflections off the Moho is shown by thick solid line (open where there are gaps in coverage). The crust is thicker

and the lower crust has lower velocity near the center of the segment. (From Hooft et al. (2000)). The red circles are locations of microearthquakes with uncertainties indicated by error bars. Location of oceanbottom hydrophones indicated by asterisks

Along-axis variations, particularly at slow-spreading ridges, suggest that melt is preferentially delivered to the center of ridge segments.

ridge-hot spot interaction. J Geophys Res 112:B08313. https://doi. org/10.1029/2006JB004785. Detrick RS, Buhl P, Vera E (1987) Multi-channel seismic imaging of a crustal magma chamber along the East Pacific Rise. Nature 326:35–41 Dunn RA, Toomey DR, Solomon SC (2000) Threedimensional seismic structure and physical properties of the crust and shallow mantle beneath the east pacific rise at 9 degrees 30’N. J Geophys Res 105:23537–23555 Evans RL et al (1999) Asymmetric electrical structure in the mantle beneath the east pacific rise at 17
S. Science 286:756–759 Hammond WC, Toomey DR (2003) Seismic velocity anisotropy and heterogeneity beneath the mantle electromagnetic and tomography experiment (MELT) region of the East Pacific Rise from analysis of P and S body waves. J Geophys Res 108. https://doi.org/10.1029/ 2002JB001789 Harmon N, Forsyth DW, Weeraratne DS (2009) Thickening of young Pacific lithosphere from high-resolution Rayleigh wave tomography: a test of the conductive cooling model. Earth Planet Sci Lett 278:96–106 Hooft EEE, Detrick RS, Toomey DR, Collins JA, Lin J (2000) Crustal thickness and structure along three contrasting spreading segments of the Mid-Atlantic Ridge, 33.5
–35
N. J Geophys Res 105:8205–8226 MELT Seismic Team (1998) Imaging the deep seismic structure beneath a Mid-Ocean Ridge: the MELT experiment. Science 280:1215–1218 Singh SC, Kent GM, Collier JS, Harding AJ, Orcutt JA (1998) Melt to mush variations in crustal magma properties along the ridge crest at the southern East Pacific Rise. Nature 394:874–878 Singh SC, Crawford WC, Carton H, Seher T, Combier V, Cannat M, Canales JP, Dusunur D, Escartin J, Miranda JM (2006) Discovery of a magma chamber and faults beneath a Mid-Atlantic Ridge hydrothermal field. Nature 442:1029–1032 Toomey DR, Jousselin D, Dunn RA, Wilcock WSD, Detrick RS (2007) Skew of mantle upwelling beneath the East Pacific Rise governs segmentation. Nature 446:409–414 Wang Y, Forsyth DW, Savage B (2009) Convective upwelling in the mantle beneath the Gulf of California. Nature 462:499–501. https:// doi.org/10.1038/nature08552

Cross-References ▶ Crustal Reflectivity (Oceanic) and Magma Chamber ▶ Earth’s Structure, Upper Mantle ▶ Lithosphere, Mechanical Properties ▶ Lithosphere, Oceanic ▶ Lithosphere, Oceanic: Thermal Structure ▶ Magnetic Anisotropy ▶ Ocean Bottom Seismics ▶ Seafloor Spreading ▶ Seismic Data Acquisition and Processing ▶ Seismic Tomography ▶ Seismic, Velocity, and Density Relationships ▶ Surface Waves ▶ Traveltime Tomography Using Controlled-Source Seismic Data

Bibliography Barclay AH, Toomey DR (2003) Shear wave splitting and crustal anisotropy at the Mid-Atlantic Ridge, 35
N. J Geophys Res 108. https:// doi.org/10.1029/2001JB000918 Canales JP, Singh SC, Detrick RS, Carbotte SM, Harding A, Kent GM, Diebold JB, Babcock J, Nedimovic MR (2006) Seismic evidence for variations in axial magma chamber properties along the southern Juan de Fuca Ridge. Earth Planet Sci Lett 246:353–366 Delorey AA, Dunn RA, Gaherty JB (2007) Surface wave tomography of the upper mantle beneath the Reykjanes ridge with implications for

Seismic Tomography

Seismic Tomography Guust Nolet IRD Geoazur and Université Côte d’Azur, Sophia Antipolis, France

Definition The term tomography derives from the Greek tómoς, or slice. “Seismic tomography” is used for a variety of methods that use transmitted seismic waves to estimate the spatial variations in properties (wave velocity, density, and attenuation) inside the Earth, which are often represented as images of two-dimensional cross-sections or “slices.” It is conceptually different from seismic migration, which uses reflected waves to image sharp discontinuities.

1507

technique of finite-frequency tomography developed by Tony Dahlen and Guust Nolet and their collaborators at Princeton University. For references and a detailed account of the history of seismic tomography, see the reviews by Romanowicz (2003) and Rawlinson et al. (2010). Nolet (2008) provides a general introduction into the methods of seismic tomography, including the theoretical aspects that are here discussed only briefly.

Onset Times Much of seismic tomography is based on estimating the arrival time of a seismic body wave by picking the onset of a phase on the seismogram, and interpreting the travel time T using the infinite-frequency approximation of ray theory: Z T¼

sðrÞ ds

ð1Þ

raypath

History where the raypath is determined by Snel’s law and where s is the inverse velocity n1 of the wave at the location r. The raypaths may be located between two boreholes, e.g., to monitor the exploitation of an oil or gas field, between several explosive sources and an array of seismographs at the surface (see article: Travel Time Tomography Using Controlledsource Seismic Data), between a seismogenic zone and a local array of seismographs, or between an earthquake and the global network of seismic stations. The raypath is often approximated by the path calculated for a spherically symmetric Earth or a horizontally layered background model. Examples of global raypaths are given in Fig. 1. This approximation is permitted because ray trajectories render the travel time stationary, such that small deviations in the true ray path location only cause second order errors in the travel time calculated with Eq. 1, a property of rays known by the name of Fermat’s Principle. In practice, 90˚

˚ 20

0˚

180˚

60˚

˚

15

0˚

1

30

In 1971, P. Bois at the Institut Français de Pétrole was the first to suggest the tomographic method in order to locate the causes of delays in seismic waves between two boreholes. His paper predates many future developments but was initially written in French and remained largely unnoticed. In the mid-1970s, Keiti Aki from MIT applied a linear inversion to locate velocity heterogeneities beneath large nuclear monitoring arrays in Norway and Montana, and Harvard’s Adam Dziewonski began interpreting the time residuals published by the International Seismological Center (ISC) in the UK in terms of global velocity anomalies. In 1982, Guy Masters and his colleagues at the Scripps Institution of Oceanography discovered a strong degree-2 component in the geographical distribution of the slight shifts in the spectral peaks of the Earth’s normal modes. Since then, the Earth’s free oscillations have contributed to constrain the heterogeneity in the Earth at the longest wavelengths and as deep as the inner core. By 1984, John Woodhouse and Adam Dziewonski at Harvard published a first global model for shear velocity in the upper mantle based on long-period surface waves. However, to image smaller scale anomalies, the shorter wavelengths of P and S-waves are indispensable. In particular, Steve Grand at the University of Texas, Rob van der Hilst and Wim Spakman at Utrecht University and Yoshio Fukao and colleagues at the University of Tokyo pioneered highresolution body-wave tomography using iterative solvers for the huge systems of linearized equations and established in the early 1990s that some, but not all, subducting slabs are able to sink well into the depths of the lower mantle. Thermal plumes in the lower mantle were for the first time reliably imaged in 2003 by Raffaella Montelli, using a new

Seismic Tomography, Fig. 1 Examples of raypaths for a P wave in an equatorial cross-section from a hypothetical earthquake at a depth of 248 km located at 0 N,150 W

S

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Seismic Tomography

one usually keeps the raypath fixed to its initial approximation. This keeps the inverse problem linear. One then inverts for the difference between the true Earth and a starting- or background model: δs(r) ¼ s(r)  s0(r), using the difference between observed time T, and its prediction T0 from the background model as datum: δT ¼ T  T0: Z dT ¼

dsðrÞds

ð2Þ

raypath

For strongly heterogeneous regions such as subduction zones, one may need to use three-dimensional ray tracing and an iterative approach using Eq. 2 in which rays are recalculated after each modification of the model s(r).

Model Parameterization and Inversion The change in the model δs(r) can be described by a finite number of parameters if it is developed in terms of a basis of M interpolating or “basis” functions hk(r): dsðrÞ ¼

M X

m k hk ð r Þ

ð3Þ

k¼1

The basis functions may be represented by homogeneous cells, linear interpolators in a tetrahedral or other mesh, spherical harmonics, or 3D wavelets. Substitution of Eq. 3 in Eq. 2 gives a formal system of linearized equations for N estimated travel times δTi, i ¼ 1, . . .N, arranged in a vector dT: dT ¼ Am

ð4Þ

R where Aik ¼ raypathhk(r)ds. Since many raypaths may overlap, this system is usually overdetermined and needs to be solved by a least squares solver that minimizes w2, the length of the misfit vector weighted by the data standard error ei: !2 PM N X A m  dT ik k i k¼1 w ðmÞ ¼ ei i¼1 2

(5)

Mathematically, this is accomplished by first dividing the system of Eq. 4 by the standard error of the data (i.e., multiplying by the inverse square root of the covariance matrix C which is generally assumed to be diagonal), and backprojecting the system using the transpose AT of A: AT C 2 Am ¼ AT C 2 dT 1

1

the following we therefore ignore C. The overdetermined part of the data allows one to obtain objective estimates for the standard errors in the data (Voronin et al. 2014). Usually, Eq. 6 is at the same time overdetermined for some parameters and underdetermined for others (the determinant of ATA being zero): there are in that case infinitely many models that satisfy the data within the measurement uncertainty and one needs to regularize the solution. This can be done by choosing the solution that minimizes a weighted penalty of w2, model norm and model roughness, by strongly truncating a development in spherical harmonics or by choosing the sparsest decomposition in wavelets that still satisfies the data. This generally involves a subjective choice between the degree of detail one allows in a model and the goodness of fit to the data. Generally, one aims for a w2 approximately equal to the number of data N – in other words, one attempts to fit the data at the level of about one standard deviation. Invariably, a trade-off exists between the detail allowed in the model and the statistical precision with which the model parameters can be determined: the sharper the detail, the more uncertain the mk. Regularization can also be done in a fundamentally different way by inverting for local averages in δs(r) with a prior specified variance that can be estimated linearly from the data (Backus-Gilbert theory). This can only be done at the expense of a significant increase in computing time. In this case, the tradeoff is between the size of the averaging volume and the model variance: averages over larger volumes are determined with smaller statistical uncertainty. Though computationally expensive, this is statistically superior to any of the other methods. A stable discretized version of Backus-Gilbert theory was developed by Nolet (1985) and for the first time applied to a large tomographic system by Zaroli (2016). Whatever method is used, the system of Eq. 4 may be very large (e.g., more than 106 data for more than 105 unknown model parameters). Local parameterizations (cells, as opposed to spherical harmonics) render A sparse, and the system can be solved efficiently using iterative solvers that adapt to sparse matrices, such as LSQR (Paige and Saunders 1982).

(6)

One often scales the equations a priori such that the data have unit error ei ¼ 1 and C ¼ I, which has the same effect. In

Normal Modes and Surface Waves The eigenfrequencies n om ‘ of the Earth are characterized by three quantum numbers ‘, m, and n, related to the number of nodal surfaces in the displacement field of the Earth with latitude, longitude, and depth, respectively. For a nonrotating, isotropic, spherically symmetric Earth, the spectrum is degenerate in the sense that the frequency is independent of the azimuthal order m. For the real Earth, a weak dependence on m splits each eigenfrequency into 2‘ + 1 separate frequencies that are too closely spaced to be resolvable except for the

Seismic Tomography

1509

very lowest angular order ‘. Instead, a composite spectral line or “multiplet” is observed with a peak that depends on the location of the seismic station – a direct consequence of the fact that the 2‘ + 1 single peaks have amplitudes that depend on geographical location by virtue of their spherical harmonic dependence on latitude and longitude. Two major strategies exist to exploit the small fluctuations in the spectrum. Decomposing the free oscillation into surface waves traveling in opposite directions, ray theory may be used to establish a linear relationship between the heterogeneity along the great circle between source and receiver and the observed peak shift in the spectral line. At higher frequency, we separate single passages of the surface wave and exploit the linear relationship between fluctuations in the observed phase velocity and the Earth’s heterogeneity. In both cases, the relationship between the Earth’s heterogeneity and the observed datum is a two-dimensional integral along the great circle of the form: do ¼

Z aZ 0

K ðrÞdsðrÞds dr,

ð7Þ

gc

distribution of energy within this window and thus fundamentally different from the delay measured by picking the onset of s1(t). Instead of a “shortest time” satisfying Snell’s law, it represents an integral measure over a time window that is at least as long as the dominant period of s1(t). While Eq. 1 is correct for onset times, it needs to be modified for delays measured by cross-correlation. Heterogeneities inside the Earth may influence the integral by scattering waves in the direction of the recording station that arrive within the time window. The most serious consequence is that energy may diffract around a small heterogeneity, thus masking or dominating the slower or faster arrival that has crossed the heterogeneity. This phenomenon of “wavefront healing” biases observed delay times to zero, especially if the anomaly is located at some distance from the source and receiver. Nolet and Dahlen (2000) give a theoretical analysis of wavefront healing. Taking wave diffraction into account leads to a threedimensional integral constraint of the form: Z dT ¼

K ðrÞ dsðrÞ d3 r,

ð9Þ

V

where the kernel K(r) is computed using first order perturbation theory of the differential equations governing the Earth’s free oscillations. Alternatively, we may exploit the known distribution of amplitudes of single peaks over the surface of the Earth to invert for the location of these peaks (“mode splitting”). The small frequency shifts δom are themselves the eigenvalues of a splitting matrix H of which the elements are linearly related to the variation of density and elastic parameters in the Earth. We can estimate H from the seismic data using autoregressive filtering techniques. This way we avoid any ray-theoretical approximations and obtain a three-dimensional integral constraint that can be used to solve for the large-scale variations in the Earth’s density and elastic properties.

Finite-Frequency Tomography Modern, broadband digital instrumentation allows for a robust estimation of the delay of an observed seismic wave s1(t) with respect to a theoretically predicted (“synthetic”) waveform s2(t) or with respect to the same phase observed elsewhere, by locating the maximum in the cross-correlation C(t) between the two signals: Z dT ¼ arg max CðtÞ, CðtÞ ¼ t  ½t1 , t2 

t2 t1

s1 ðtÞs2 ðt  tÞdt,

ð8Þ

where the integration interval (t1, t2) usually (but not necessarily) extends over the window containing a distinct arrivals such as P or S. The cross-correlation delay is influenced by the

where the kernel K(r) can be efficiently calculated using ray theory for scattered waves. The effective volume of integration V is roughly equivalent to that of the Fresnel zone of the ray, where the sensitivity is largest. Since a positive change in rock velocity gives a negative change in the travel time, the kernel values within the Fresnel zone are negative. There are, however, sidelobes with a positive sign, where a positive velocity anomaly would give a positive change in the observed travel time. This may seem counter-intuitive, but it is possible because cross-correlations involve a finite time window, and it is the energy distribution in this window, rather than the onset of the wave, that defines the delay measured in this way. An example of a finite-frequency kernel is shown in Fig. 2. Remarkably, the cross-correlation travel time is insensitive to perturbations in the Earth’s properties at the center of the kernel, which is at location of the ray path. Because of this hole in the sensitivity and their general shape, finitefrequency kernels are often referred to as “banana-doughnut” kernels. Numerical tests have shown that the finite-frequency kernels accurately model the loss of signal caused by wavefront healing, the gradual reduction of a delay caused by waves diffracting around a small heterogeneity. By estimating delays in different frequency bands, one gains information about the size of the heterogeneity in the Earth. Though finite-frequency kernels were initially proposed to provide a better theoretical modeling of long-period signals affected by wavefront healing, they offer the advantage of being able to model the healing as a function of frequency, and thus obtain independent information on the size of the

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Seismic Tomography

Seismic Tomography, Fig. 2 The sensitivity of the cross-correlation time to the earth structure, for a long-period P wave from a surface source at 0 N, 120 W, recorded at a distance of 60
. The kernel is plotted in an equatorial cross-section of the mantle, the color scale runs from large negative values (red) to positive (blue). The sensitivity is zero (yellow) at the location of the geometrical ray-path, forming a “doughnut hole” in the banana-shaped kernel. The white line denoted the kernel values at 90


derivatives separately for each datum dTi with respect to each model parameter mk. The adjoint approach has some advantages over a direct matrix inversion for large-scale problems, in particular those with few sources and many stations, because it does not require the storage of a large matrix, which is especially important for waveform inversions. Since the gradient is recalculated for every iteration, the method lends itself naturally to tomographic inversions that are strongly nonlinear, such as one expects for imaging of the crust or the D” region just above the Earth’s core. However, Mercerat and Nolet (2013) showed that the inversion of delay times measured by cross-correlation are highly linear in most depth regions of the Earth. For a technical description of the adjoint method (also referred to as “residual back-propagation”) see the article: Seismic, Waveform Modeling and Tomography.

Summary heterogeneity. Such multiple-frequency tomography may significantly increase resolution (Sigloch et al. 2008). Finitefrequency sensitivity can also be formulated for the amplitude perturbations caused by attenuation and by focusing/ defocusing (focusing cannot be handled with ray-theory because ray theory does not model amplitudes correctly at low frequency and is very nonlinear at high frequency). Instead of inverting for a cross-correlation delay or an amplitude perturbation, one can formulate a 3D sensitivity directly for the observed time series, though this has the disadvantage that the observed and predicted waveforms have to be close enough in phase that the phase difference can be adequately modeled by adding a small perturbation to the waveform itself (in fact approximating eif ≈ 1 + if). This limits the inversion to waveforms with small time mismatches, in contrast to the delays estimated through Eq. 8, which remain linear for large anomalies. All types of data – delays, amplitudes, and waveform mismatches – can be iteratively inverted using an adjoint approach, in which best fitting model is sought along the gradient ∇mw2 (Tromp et al. 2005), e.g., for delay times scaled to unit variance: ∇m w2 ¼ AT ðAm  dT Þ

ð10Þ

For waveforms, the matrix A is in this case the matrix representation of a finite-difference or spectral-element algorithm that produces the predicted seismograms and AT is the adjoint operator that projects the seismograms back in time. Since this allows one to backproject all residual seismograms for one earthquake with only one calculation, and since only the product of the adjoint matrix and the residual data vector is needed, this allows us to avoid computing the partial

The equations of seismic tomography are integral equations; the observations are weighted averages of the properties of the Earth (Eq. 9). Though often treated as line integrals, assuming ray theory is valid, a delay of a 1 Hz P wave measured by cross-correlation senses a volume or Fresnel zone inside the Earth that is several hundred km wide. The inversion of integral equations demands care to avoid that noise (errors in the data) are interpreted as small differences in observed delays, which may require large variations in the Earth’s properties. This imposes a fundamental limitation to the resolving power of seismic waves that even finite-frequency theory or waveform modeling cannot completely avoid. The theoretically best available horizontal resolution, using the highest observable seismic frequencies, is of the order of hundred km in the lower mantle. In practice, this lower limit is not yet reached because of the limited coverage of the Earth’s surface with seismic stations. On land, the coverage can be improved markedly using temporary deployments of seismic arrays, but this is much more difficult and expensive to do in the oceanic domain. Major improvements on a global scale are therefore to be expected only if we solve the problem of oceanic seismometry. It is expected that floating seismographs (see article “Mermaids”), and possibly also strain signals extracted from fiber optic cables on the ocean floor, will soon make this possible.

Cross-References ▶ Earth, Density Distribution ▶ Earth’s Structure, Core ▶ Earth’s Structure, Global ▶ Earth’s Structure, Lower Mantle

Seismic Velocity and Temperature Relationships

▶ Earth’s Structure, Upper Mantle ▶ Floating Seismographs (MERMAIDS) ▶ Gravity, Global Models ▶ Inverse Theory, Linear ▶ Mantle Plumes ▶ Plate-Driving Forces ▶ Propagation of Elastic Waves: Fundamentals ▶ Seismic Imaging, Overview ▶ Seismic Ray Theory ▶ Seismic, Waveform Modeling and Tomography ▶ Subduction Zones ▶ Surface Waves ▶ Traveltime Tomography Using Controlled-Source Seismic Data

Bibliography Aki K, Lee HK (1976) Determination of three-dimensional anomalies under a seismic array using first P arrival times from local earthquakes: 1. A homogeneous initial model. J Geophys Res 81:4381–4399 Bois P, la Porte M, Lavergne M, Thomas G (1971) Essai de détermination automatique des vitesses sismiques par mesures entre puits. Geophys Prospect 19:42–81 Dziewonski AM, Hager BH, O’Connell RJ (1977) Large-scale heterogeneities in the lower mantle. J Geophys Res 82:239–255 Masters G, Jordan TH, Silver PG, Gilbert JF (1982) Aspherical earth structure from spheroidal mode oscillations. Nature 298:609–613 Mercerat ED, Nolet G (2013) On the linearity of cross-correlation delay times in finite-frequency tomography. Geophys J Int 192:681–687 Montelli R, Nolet G, Dahlen FA, Masters G, Engdahl ER, Hung SH (2004) Finite-frequency tomography reveals a variety of plumes in the mantle. Science 303:338–343 Nolet G (1985) Solving or resolving inadequate and noisy tomographic systems. J Comput Phys 61:463–482 Nolet G (2008) A breviary of seismic tomography. Cambridge University Press, Cambridge, UK Nolet G, Dahlen FA (2000) Wave front healing and the evolution of seismic delay times. J Geophys Res 105:19043–19054 Paige CC, Saunders M (1982) LSQR: an algorithm for sparse, linear equations and sparse least squares. J ACM Trans Math Softw 8:43–71 Rawlinson N, Pozgay S, Fishwick S (2010) Seismic tomography: a window into deep earth. Phys Earth Planet Inter 178:101–135 Romanowicz B (2003) Global mantle tomography: progress status in the past 10 years. Annu Rev Earth Planet Sci 31:303–328 Sigloch K, McQuarrie N, Nolet G (2008) Two-stage subduction history under North America inferred from multiple-frequency tomography. Nat Geosci 1:458–462 Tromp J, Tape C, Liu Q (2005) Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophys J Int 160:195–216 Voronin S, Mikesell D, Slezak I, Nolet G (2014) Solving large tomographic linear systems: size reduction and error estimation. Geophys J Int 199:276–285 Woodhouse JH, Dziewonski AM (1984) Mapping the upper mantle: three dimensional modelling of the earth structure by inversion of seismic waveforms. J Geophys Res 89:5953–5986 Zaroli C (2016) Global seismic tomography using Backus-Gilbert inversion. Geophys J Int 207:876–888

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Seismic Velocity and Temperature Relationships Kalachand Sain Wadia Institute of Himalayan Geology, Dehradun, India

Definition Seismic velocity is defined as the speed with which a seismic wave travels through a medium and is measured in km/s, whereas temperature is a physical property that quantitatively expresses hot and cold. It is the manifestation of thermal energy, present in all matter, and is expressed in
C. Seismic velocities of subsurface rocks decrease with temperature but experience opposite effects due to increase in pressure (depth). Velocity-temperature equation: On an average, the pressure increases at a rate of 30 MPa/km and the temperature raises at a rate of 25
C/km from surface to deeper parts of the Earth. Since seismic velocity (V) decreases with temperature (T) and increases with pressure (P), we need to know the combined effects of P and T for interpreting seismic velocities at different depths. The generalized relation for the variation of velocity with depth Z can be expressed as dV ¼ dZ

  dV dP dV dT þ dP T dZ dT P dZ

ð1Þ


 where dV with pressure at dP T denotes the change in velocity
dV  constant temperature (isotherm), and dT P is the change in velocity with temperature at constant pressure (isobar). dP dZ and dT are the vertical pressure and temperature gradients dZ respectively. To make petrological inferences, the crustal and lithospheric seismic velocities are to be corrected to the experimental reference values (i.e., constant pressure of 100 MPa and room temperature of 20
C). Rybach and Buntebarth (1984) have defined the correction factor (f) to the P-wave velocity (VP) at a given P and T as V P ð20
C, 100MPaÞ ¼ V P ðP, TÞf   @V P @V P ¼ V P 1 þ DT  DP V P @T V P @P ð2Þ The correction factors for granites, gneisses, amphibolites, gabbros, and ultrabasic rocks of both Precambrian and Phanerozoic period are given in Table 2 of Rybach and Buntebarth (1984). As per eq. (2), the corrections to the field velocities at depths from 1 to 50 km are of the order of 0.1 km/s. As a

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matter of fact, the increase in velocity with pressure is nearly compensated by the decrease in velocity due to temperature increasing with depth. Velocity variation with temperature for some reservoir rocks: The change in seismic velocities with temperature also depends on the saturation of rock. The results of Wang and Nur (1990) in heavy oil sands and hydrocarbon-saturated rocks show that both VP and VS decrease with increasing temperature. Since VS is not affected by fluids, the decrease in VS is due to changes in rock frame and in rock fluid interactions. The VP in heavy oil sands (Tosaya et al. 1987) shows a dramatic decrease. As temperature increases from 25
C to 125
C, VP can drop by 35% to almost 90%. Heavy oils are highly viscous and a strong interfacial force exists between oil and rock grains. The viscosity of oil and interfacial force decrease due to rise in temperature, which decreases the rigidity and bulk modulus leading to reduction in seismic velocities. For temperature up to about 150
C, changes in pore fluid properties play dominant role in changing the velocity. Though there is a negative correlation between seismic velocity and temperature, Ryan and Shalev (2014), from drilling results in Montserrat (West Indies), showed that above ~220
C there is a positive correlation into a geothermal reservoir. They hypothesized that anomalous variation is controlled by the first-order hydrothermal mineral assemblage. Perry et al. (2006) compared temperature and velocity of Moho refracted phase, Pn, for various mineralogical models of the upper mantle from seismic refraction and heat flow studies and demonstrated that the theoretical velocitytemperature conversion used for interpretation of seismic tomographic models is in good agreement with the observations. Schutt et al. (2018) used the velocity-temperature relationship in modeling the deep geotherm within few km of the uppermost mantle in western USA using the Pn velocity. The temperature is a fundamental factor in determining the lithospheric strength, thickness, buoyancy, and mass flux for geoscientific investigation of the subsurface. For example, if temperature of the lower crust is found sufficiently warm, the lower crust may be weak that can decouple the upper crustal and mantle strain fields, affecting the style of deformation. Based on deep seismic studies and gravity modeling in a huge territory of north Eurasia, Yegorova and Pavlenkova (2015) showed that seismic velocities in the upper mantle mainly reflect the temperature regime with no direct velocity-density relationship, as characterized by high-velocity, low-density and low heat flow. Acknowledgments The author is grateful to the Director of Wadia Institute of Himalayan Geology, Dehradun, for his consent to publish this work. This has a Wadia contribution WIHG/0037.

Seismic Viscoelastic Attenuation

Bibliography Perry HKC, Jaupart C, Mareschal JC, Shapiro NM (2006) Upper mantle velocity-temperature conversion and composition determined from seismic refraction and heat flow. J Geophys Res 111(B07301):1–14 Ryan GA, Shalev E (2014) Seismic velocity/temperature correlations and a possible new geothermometer: insights from exploration of a high-temperature geothermal system on Montserrat, West Indies. Energies 7(10):6689–6720 Rybach L, Buntebarth G (1984) The variation of heat generation, density and seismic velocity with rock type in the continental lithosphere. Tectonophysics l03:335–344 Schutt DL, Lowry AR, Buehler JS (2018) Moho temperature and mobility of lower crust in the western United States. Geology 46:1–4 Tosaya C, Nur A, Vo-Thanh D, Da Prat G (1987) Laboratory seismic method for remote monitoring of thermal EOR. SPE Reserv Eng 2:238–242 Wang Z, Nur A (1990) Wave velocities in hydrocarbon saturated rocks: experimental results. Geophysics 55:723–733 Yegorova TP, Pavlenkova GA (2015) Velocity-density models of the Earth’s crust and upper mantle from the quartz, craton, and kimberlite superlong seismic profiles. Phys Solid Earth 51:250–267

Seismic Viscoelastic Attenuation Vernon F. Cormier Physics Department, University of Connecticut, Storrs, CT, USA

Synonyms Seismic intrinsic attenuation

Definition Linear viscoelastic attenuation. The fractional loss of seismic energy in a material in which elastic deformation (strain) induced by one cycle of a seismic wave or mode lags in time the applied stress associated with the wave or mode. Apparent seismic attenuation. The loss of energy in a propagating seismic wave or standing mode due to viscoelasticity combined with the loss of scattered energy redistributed in time and space by heterogeneity.

Introduction The amplitude of seismic waves decreases with increasing distance from earthquake, explosion, and impact sources. How this amplitude decrease occurs and how it depends on frequency of the seismic waves are fundamentally important to the efforts to describe Earth structure and seismic sources.

Seismic Viscoelastic Attenuation

The decay of amplitude of seismic waves with increasing distance of propagation through Earth is known as seismic wave attenuation. The attenuation occurring under hightemperature rheological conditions in the Earth’s interior can be called seismic viscoelastic attenuation. Seismic attenuation and its variation with location within Earth are useful for determining the anelastic properties of Earth as a function of depth. Seismic attenuation also shows large lateral variations that can be related to lateral variations in geological and geophysical properties not as easily detected by measurement of seismic velocities. In addition to providing information on a physical property, research in seismic attenuation has also been strongly motivated by more practical problems. One problem has been the prediction of ground motion due to probable earthquakes in different regions. The frequency content and decay with distance of this strong ground motion is an important input to the design of earthquake-resistant structures and to disaster forecasting (see ▶ “Earthquakes, Strong-Ground Motion”). Another problem has been to estimate the size and detectability of underground nuclear tests (see ▶ “Seismic Monitoring of Nuclear Explosions”).

How Do Seismic Waves Attenuate? The attenuation of seismic waves is due to three effects: geometric spreading, intrinsic attenuation, and scattering. Geometric Spreading Geometric spreading leads to an energy density decrease that occurs as an elastic wave front expands with increasing distance from its source. In a homogeneous Earth of constant velocity and density, the geometric spreading of a seismic body wave is proportional to the reciprocal of the distance between source and receiver. In the real Earth, velocity and density vary strongly with depth and less so laterally. Given a model of this variation, however, the geometric spreading of a body wave can be easily calculated (see ▶ “Seismic Ray Theory”). Intrinsic Viscoelastic Attenuation Intrinsic (viscoelastic) attenuation occurs at high temperatures due to internal friction during the passage of an elastic wave. It is controlled by the thermal and defect properties of the medium in which the wave is propagating. It can result in a phase lag between strain and stress giving rise to strain energy dissipation and associated frequency dependence (dispersion) of the relevant modulus or speed of the propagating elastic wave. The microscopic mechanisms of intrinsic attenuation have been described in several different ways, including the resistive and viscous properties of oscillator models of the atoms in crystalline lattices, the movement of interstitial fluids

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between grain boundaries and cracks (O’Connell and Budiansky 1977), and the frictional sliding of cracks. Jackson (1993, 2007) reviews laboratory experiments that investigate microscopic mechanisms of intrinsic attenuation. This entry concentrates on the measurement of intrinsic attenuation from recordings of seismic waves at great distance. Scattering Attenuation Scattering attenuation occurs when elastic energy is scattered and redistributed into directions away from the receiver or into waves arriving in later time windows at the receiver (see ▶ “Seismic Waves, Scattering”). Scattering takes place by reflection, refraction, and mode conversion of elastic energy by wavelength-scale irregularities in the medium. These irregularities are discontinuous or rapid variations in the velocity and/or density of the medium. In the crust and uppermost mantle, variations in velocity and density can be particularly strong in the lateral as well as the vertical direction.

Linear Viscoelasticity Rheology A stress is a vector force per unit area applied to a solid. A strain is nondimensional measure of the deformation of the solid due to the applied stress, such as the change in a length element divided by the original length. The equation that relates stress and strain is sometimes termed the rheology or the constitutive relation (see ▶ “Mantle Viscosity”). A linear viscoelastic rheology can be described by a linear differential equation: L1 sðtÞ ¼ L2 eðtÞ

ð1Þ

where L1 and LÐ2 are any linear combinations of operators of n the time, dtd n or dtn. This type of equation can describe both the elastic strain of a material over a short time interval of applied stress as well as its viscous behavior and flow over a longer time interval (Gross 1953; Nowick and Berry 1972; Jackson et al. 2005; Kohlstedt 2007). Anelastic Hysteresis Seismic oscillations at distances beyond several fault lengths from an earthquake excite small strains less than 106. These strains are recoverable during a cycle of seismic oscillation and lag the applied stress of the oscillation in time. Because of the time lag, a cycle of increasing and decreasing stress does not produce a perfectly proportional increase and decrease in strain. Instead a hysteresis loop occurs (Fig. 1). The area enclosed by the hysteresis loop is a measure of the energy lost due to heat and internal friction. During the stress cycle associated with the passage of a seismic wave, the

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Seismic Viscoelastic Attenuation

low-frequency limit of the modulus G(0), for the equilibrium or relaxed response, is called the modulus defect ΔG, with Stress σ

DG ¼ Gð1Þ  Gð0Þ

Strain ε

Seismic Viscoelastic Attenuation, Fig. 1 Stress-strain hysteresis curve showing the behavior of strain during a cycle of applied stress induced by a propagating seismic wave

energy lost to this internal friction is not available to deform the adjacent regions of the solid ahead of the wave front, and therefore the amplitude of the wave decreases. From the hysteresis curve, one can see that the stress-strain relation cannot be described by a simple constant of proportionality in the time domain. A more complicated relation involving an integral over time is required to describe strain at any instant of time as a function of the prior time history of the applied stress. By Fourier transforming the rheologic equation, however, and keeping only terms describing the short-term anelastic behavior, the stress-strain relation can be simply expressed either by means of a complex elastic _ _ modulus GðoÞ or by its reciprocal, J ðoÞ the complex elastic compliance: _

_

_

_

_

sðoÞ ¼ GðoÞ e ðoÞ

ð2aÞ

_

e ðoÞ ¼ J ðoÞsðoÞ _

ð2bÞ _

The elastic modulus G and compliance J must be complex numbers to describe the phase lag of strain. They must also be frequency dependent because the phase lag of strain depends on the time history of stress, the shape of the hysteresis curve changing with different load histories. All the usual measures of anelasticity, including the frequency-dependent quality factor Q(o) and the dispersion of the complex phase velocity _ vðoÞ, can be expressed in terms of the complex modulus or complex compliance (Jackson and Anderson 1970).The trend of the frequency dependencies can be inferred from the time lag of strain from applied stress. A feature of the complex modulus is that its real part will be smaller at zero or very low frequency and larger at infinite or very high frequency. That is, there will be an instantaneous response of strain to the applied stress, which is smaller than the eventual equilibrium response after longer time. The difference between the modulus at infinite frequency G(1), representing the instantaneous or unrelaxed response, and the

ð3Þ

The relaxed and unrelaxed moduli are pure real numbers that can be determined by observing a sequence of hysteresis curves for increasing frequencies of monochromatic loads. The frequency dependence of the real part of the modulus G at frequencies between 0 and 1 implies that the propagation of a stress pulse will be dispersive, with higher frequencies traveling faster than lower frequencies. Q and Complex Velocity Since simple mechanical systems, composed of springs and dashpots and simple electric circuits, also obey linear equations of the form of Eqs. 2a and 2b, there are analogies between the quantities describing these systems and quantities in the stress-strain relation. For example, strain behaves like voltage, stress like current, and the complex compliance _ J ðoÞ like the complex impedance of an electric circuit. Similar to the resonance phenomenon in circuits and mechanical systems, a Q can be defined by the average energy W per cycle divided by the energy lost or work done per cycle, ΔW: Q¼

W DW

ð4Þ

Large Q’s imply small energy loss; small Q’s imply large loss. Q is a measure of the area contained in the hysteresis loop of a stress-strain cycle. The inverse of Eq. 4, Q1, is sometimes simply termed the attenuation or internal friction (Knopoff 1964). Plane waves of frequency o and propagatinghinthe + or xi _ direction can be defined by the phasor exp i o t  k x _

_

where k is a complex wave number o _ and c is a complex velocity c _ defined from the local density r and complex modulus G, with sffiffiffiffi _ G _ c¼ r

ð5Þ

From the average energy density and loss per cycle
_of Re G a complex plane wave, it can be shown that Q ¼
_ . Im G

It is often less confusing to report the reciprocal parameter Q1, which represents the usually small perturbations to perfect elasticity. The Q1(v) Relaxation Spectrum _ Since G depends on frequency, Q also depends on frequency. Zener (1960) described the frequency-dependent effects on an elastic modulus of a solid having a single characteristic time t

Seismic Viscoelastic Attenuation

1515

for the relaxation of stress. A distribution of relaxation times can be constructed to give a Q1 having a general dependence on frequency. The function Q1(o) is called the relaxation spectrum. In the Earth and in many solid materials, the relaxation spectrum is observed to be slowly varying and nearly constant over a broadband of frequencies (Lekić et al. 2009). A theoretical requirement is that the attenuation Q1 cannot increase faster than o1 or decrease faster than o1. Figure 2 shows how a continuous distribution of relaxations can produce a Q1 that is nearly constant with a frequency over a broadband. Once the limits of an absorption band are specified, however, it is not possible to have an arbitrarily high Q1 (low viscoelastic Q) over an arbitrarily broad frequency band without making an unrealistically large modulus defect ΔG. Measured modulus defects in shear are typically less than 25%.

disturbance that propagates from its point of initiation as a symmetric narrow Gaussian or triangle-shaped function in time gradually evolves into an asymmetric pulse (Fig. 3). High frequencies traveling faster than low frequencies are preferentially loaded into the front of the pulse (Futterman 1962; Carpenter 1967). Common theories for the physical mechanism of earthquakes as either frictional slip on a plane or a propagating crack triggered by tectonic stress often predict a far-field displacement pulse that has either a different or opposite form of asymmetry than that predicted for the effect of viscoelastic attenuation. These differences can assist in separating the effects of the source-time history from the effects of viscoelastic attenuation.

Effects of Scattering Velocity Dispersion Although the dispersion in elastic moduli had long been known and predicted from the theories of viscoelasticity, it only began to be widely recognized in seismology when velocity models determined in the low-frequency band from the normal modes of the Earth (0.0001–0.01 Hz) were compared with velocity models determined in a high-frequency band (0.1–10 Hz) of body waves (Dziewonski and Anderson 1981). The models were found to differ, and the difference was found to agree with the amount of dispersion predicted from average Q models of the Earth. For example, since the Preliminary Reference Earth Model (PREM) was derived from observations of both the travel times of body waves and the eigenfrequencies of free oscillations, it reports velocities referenced at both 0.001 Hz and 1 Hz. Another more subtle effect of this velocity dispersion can be seen in the propagation of pulses as body waves. A stress

Equivalent Medium At frequencies that are so low that wavelengths are much larger than the characteristic scales of heterogeneity, the attenuative effects of scattering can usually be neglected. At sufficiently low frequency, little energy is lost to scattering, and the medium behaves like an equivalent medium, having properties that are an average of small-scale heterogeneities. Stochastic Dispersion The most complicated domain to calculate the effects of scattering is where the wavelength is of the order of the scale length of the heterogeneity (Fig. 4). In this domain, the presence of heterogeneities can profoundly alter the propagation of the wavefield, affecting both the initial cycle of a body wave pulse and the motion immediately following the initial cycle or coda. At distances less than 2000 km and frequencies

Velocity V/Vo

τlow

Unrelaxed

Source pulse

II

10 s

10−3

S

10−1

Relaxed flow

fhi

Log (1/Q)

101

103

105

Log (Frequency)

Seismic Viscoelastic Attenuation, Fig. 2 Viscoelastic dispersion of seismic velocity (top) and attenuation (bottom) showing a relaxation spectrum constant with frequency between two corner frequencies

1/Q = 0.005

0.001

0.010

0.015

Seismic Viscoelastic Attenuation, Fig. 3 Pulse distortion showing the effects of viscoelastic dispersion for variable low-frequency corner and peak attenuation

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Seismic Viscoelastic Attenuation

higher than 1 Hz, scattered coda waves dominate the signature of S waves propagating in Earth’s heterogeneous crust. Sato et al. (2012) review the theory and measurement of the apparent attenuation Q1 C of these coda waves. They describe strategies to separate the contribution of intrinsic from scattering attenuation to Q1 C based on the shape and frequency content of the energy envelopes of coda waves. For longer range, some of the important effects of scattering can be illustrated by propagating body waves in a one-dimensional medium consisting of thin planar layers in which the velocity in each layer is assigned randomly (O’Doherty and Anstey 1971; Richards and Menke 1983). In fully three-dimensional structure, many of the effects of multiple scattering can be efficiently calculated by a radiative transport modeling of the energy of coda wave envelopes (Margerin 2013). These calculations predict body waves to exhibit a stochastic dispersion in which high-frequency energy is transferred into the coda. Neglect of stochastic dispersion can bias some estimates of intrinsic attenuation. Estimates of the spectral amplitude over a narrow time window can differ depending on the length of window analyzed, with less attenuation of higher frequencies estimated from longer time windows. Pulse measurements such as width and rise time may also be biased because higher-frequency energy has been transferred out of the pulse into the later coda. This behavior is opposite to the effects of intrinsic attenuation on a propagating pulse, in which higher frequencies arrive at the beginning of the pulse. A symmetrically shaped displacement source pulse loses less of its symmetry as it propagates through the heterogeneous medium (Fig. 5). Anisotropy of the scale lengths of heterogeneity can also be important factor (Hong and Wu 2005), attenuation being strongest for paths for which the wavelength is of the order of the characteristic scale length in the medium in that direction.

Anisotropic scale lengths

Isotropic scale lengths

Effects of Anisotropy The existence of general anisotropy in the real part of the elastic modulus has the potential to bias some estimates of anelastic attenuation from either shear wave pulses or surface waves. In a medium having general anisotropy, the decompositions of shear wave motion into SH and SV motion will each contain the interference of two orthogonal shear wave polarizations that are neither SH nor SV (see ▶ “Shear-Wave Splitting: New Geophysics and Earthquake StressForecasting”). The broadening of the SH component due to the interference of two quasi-S waves arriving close in time can be mistaken for the broadening due to anelastic attenuation. The regions of the deep Earth characterized by the strongest elastic anisotropy are the upper 400 km of the mantle (Silver 1996) and the lowermost 400 km of the mantle near the core-mantle boundary (Panning and Romanowiz, Panning and Romanowicz 2006). The effects of elastic anisotropy must be removed by combined analysis of SV and SH components of motion, resolving the polarizations of two quasi-S waves, before viscoelastic attenuation can be properly measured.

Measurement and Modeling Attenuation Measurements of amplitude of seismic waves may be taken directly from seismograms or from their frequency spectra. To measure the attenuation, we must predict its effects from a model and vary the parameters of the model to fit the observed amplitude, amplitude ratio, or waveform. The effects of intrinsic attenuation in any modeling algorithm operating in

Source pulse

Scale length (km) II

10 s

0.10

0.80

6.40

51.2 Pulse attenuation: max to stratification

Pulse attenuation: all directions equal

Seismic Viscoelastic Attenuation, Fig. 4 Example heterogeneity in the Earth and the directional dependence of attenuation of a body wave pulse for wavelengths that are approximately equal to either the dominant scale length (isotropic heterogeneity) or the dominant scale length in the direction of propagation (anisotropic heterogeneity)

ΔV/V = 4.6%

9.7 %

14.8 %

20.0

Seismic Viscoelastic Attenuation, Fig. 5 Pulse distortion showing the effects of scattering attenuation for variable scale lengths and velocity perturbation calculated by Cormier and Li (2002) using the Dynamic Composite Elastic Modulus (DYCEM) theory of Kaelin and Johnson (1998)

Seismic Viscoelastic Attenuation

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the frequency domain can be simply obtained by allowing elastic moduli and/propagation velocities to become complex. Elastic boundary conditions, reflection and transmission at boundaries, travel times, and amplitudes are calculated exactly as in a non-attenuating solid but with elastic moduli and associated velocities analytically continued to complex values. This step of analytic continuation of real moduli to complex moduli is the same whether one wishes to predict the waveform of a body wave or surface wave or spectrum of free oscillations. The size of the imaginary part of the elastic moduli, parameterized by the value of Q1 as a function of depth and frequency, is chosen to match an observed waveform, spectrum, amplitude ratio, or spectral ratio. The Attenuation Operator for Body Waves As an example of these procedures, consider an experiment with body waves. The effects on a body wave of source radiation, geometric spreading, reflection-transmission, and intrinsic attenuation are most conveniently expressed in the frequency domain   by a product of complex functions. The _ ! complex O x , o spectrum of a body wave propagation !

!

from a point x o to a receiver at x is    _ _ _ *! ! ! O x , o ¼ B x o , x , o SðoÞ AðoÞ

ð6Þ

_

The function S_ðoÞ is the Fourier transform of the source! ! time function. B x o , x , o incorporates a product of reflection-transmission coefficients, reverberations at source and receiver, geometric spreading, and source radiation pat_ tern. In the frequency domain, an attenuation operator AðoÞ is defined by h _ i _ AðoÞ ¼ exp io T ðoÞ

ð7Þ

_

where T ðoÞis the complex travel time obtained by integrating the reciprocal of complex velocity along a ray or normal to the wave front of the body wave: _

ð

T ðoÞ ¼

ds c ðo Þ

_

ray

ð8Þ

The effects of attenuation on reflection-transmission coefficients and geometric spreading through the complex velocity are much smaller and can be neglected unless the attenuation is very large (Q is very small). For Q >> 1, _ AðoÞ can be rewritten as     _ ot  ðoÞ H ½ t  ðo Þ exp io Re T ð1Þ  AðoÞ ¼ exp 2 2 _



ð9Þ

where ð t  ðo Þ ¼

Q1 ds c ðoÞ

_

ray

ð10Þ

InhEq. 9, ithe attenuation effect is contained in the factor ðoÞ exp o t , and the dispersive effect is in the factor n h2 _ io exp io Re T ð1Þ  H½t2ðoÞ . The operator H is a Hilbert transform. In a band of frequencies in which Q and t are nearly constant H½t  ðoÞ=2 ¼

ln ðo=o0 Þ t p

ð11Þ

where o0 is a reference frequency contained in the band (Liu et al. 1976). The value of T(1) need not be known and can be replaced by some reference time or predicted from an Earth model for the phase being analyzed. The Hilbert transform _ relation in Eq. 11 for the dispersive phase of AðoÞ says that _ AðoÞ must be a minimum phase filter in the frequency domain. In general, the Fourier transform of the source-time _ function, SðoÞ, is not a minimum phase filter, which can help in the separation and discrimination of the source spectrum _ from the effects of A ð o Þ in the total expression for the far-field   _ ! spectrum O x , o . The phase given by Eq. 11 will be accurate only between and far from the low- and high-frequency corners of the _ relaxation spectrum. Accurate representations of AðoÞ across a broad frequency band can be obtained for general relaxation _ spectra by substituting expressions for complex velocity c ðoÞ in Eq. 8 obtained by superposing multiple Zener relaxations centered on single relaxation times whose strength is varied to achieve a desired shape for the relaxation spectrum. A useful _ expression for c ðoÞ that is accurate for all frequencies across a relaxation spectrum, which is flat between two corner frequencies, can be derived from formulae for complex modulus given by Minster (1978) and is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2p Q1 ln ½cðoÞ  _ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ðoÞ ¼ cref ðo0 Þ Re 1 þ 2 p Q1 ln ½cðo0 Þ

ð12aÞ

where cðoÞ ¼

i o þ 1=t1 i o þ 1=t2

ð12bÞ

with t1 and t2 the relaxation times corresponding to the lowand high-frequency corners, respectively. cref(o0) is a real velocity at the reference frequency o0.

S

1518

Seismic Viscoelastic Attenuation

Most measurements of attenuation attempt to measure only the amplitude effect of attenuation through the term exp[o t  (o)/2] from the spectral shape of body waves. There are basically two types of experiments commonly reported: (1) matching of spectral decay rates and (2) spectral ratios. In experiment (1), a shape for the displacement source _ spectrum SðoÞ is assumed usually to be a flat level followed by a decay of o2 above a corner frequency. The additional decay observed at high frequencies in data spectra is taken as a measure of t in exp[o t  (o)/2]. In experiment (2), a ratio of two different seismic phases from the same source is observed in which the source spectrum is assumed to approximately cancel and the factor related to ratios of geometric spreading and near source and receiver crustal reverberations can be assumed to contribute a simple constant scalar factor. If the phases analyzed are recorded at the same receiver and are incident at nearly the same angles, then crustal reverberations at the source and receiver will approximately cancel. Both types of experiments usually apply some type of smoothing to the spectra to remove biasing effects of spectral holes caused by interfering crustal multiples, source complexities, scattering, and multipathing that are not included in the simple propagation model. Figure 6 illustrates an attenuation experiment of this type. Since t measures only the path-integrated effect of attenuation, many such experiments for different ray paths, bottoming at a range of different depths, are needed to construct a model of Q as a function of depth. Serious

t∗P = ln(APP) − ln(AP) πf

Natural log amplitude

Slo

pe =

−π t∗ Sl P op e =

−π

t∗

PP

AP APP Frequency

Seismic Viscoelastic Attenuation, Fig. 6 The steps (top) to measure the path-integrated attenuation t of P waves in the mantle from a log-log plot (bottom) of stacked PP and P spectra (APP and AP). The distances of observed P and PP spectra are chosen such that each turning ray path of PP is identical in shape and length to that of the single turning ray path of P in the mantle. (Adapted from figures in Warren and Shearer 2000)

breakdowns in this approach, however, exist for cases in which the factorization of the observed spectrum into a product of a geometric spreading, source spectrum, and crustal effects is no longer accurate. One such case is when the body waves in question experience frequency-dependent effects of diffraction near caustics or grazing incidence to discontinuities. The spectral ratios of PKnKP waves, for example, are dominated by the effects of frequency-dependent reflection and transmission coefficients at grazing incidence to the coremantle boundary. Instead of decreasing linearly with increasing frequency, an observed spectral ratio increases with frequency and exhibits a curvature in a log-log plot, which is consistent with a Q near infinity (Q1 ¼ 0) in the other core (Cormier and Richards 1976). It is becoming more common to model and invert for viscoelastic attenuation parameters in the time domain, including not only the magnitude of the viscoelastic attenuation parameter Q1 but also its frequency dependence. Examples of such a study are the inversions for Q1 in the inner core assuming either a viscoelastic (Li and Cormier 2002) or a scattering origin of attenuation (Cormier and Li 2002). In these studies, the combined effects of mantle attenuation and source-time function were first modeled by fitting P waves observed in the great circle range
30–90 . Attenuation in the liquid outer core was assumed to be zero. Parameters defining a viscoelastic relaxation spectrum in the inner core were then varied to match the observed PKIKP waveforms. Care must be taken to examine a broad range of attenuation parameters because waveform inversions of this type are very nonlinear. Equally good fits between observed and predicted waveforms can sometimes be achieved with quite different domains of parameters. Free Oscillations and Surface Waves Measurements of attenuation in the low-frequency band of the free oscillations of the Earth are conducted in the frequency domain by observing the width of the individual resonance peaks associated with each mode. These measurements face special problems associated with the broadening produced by lateral heterogeneity of elastic Earth structure. This heterogeneity splits the degenerate modes of a radially symmetric Earth, making a set of modes that would have the same frequency have slightly different frequencies. The slightly different frequencies of the split modes may not be easily resolved in the data spectra and can be confused with the broadening of a single resonance peak of a mode caused by attenuation. Lateral heterogeneity also complicates the measurement of viscoelastic attenuation of surface waves. Heterogeneity introduces focusing, defocusing, and multipathing, all of which must be accurately modeled to understand the separate attenuative effects of viscoelasticity.

Seismic Viscoelastic Attenuation

1519

The frequency band of free oscillation and surface waves (0.001–0.1 Hz), however, offers the best hope of obtaining radially symmetric whole-Earth models of viscoelastic attenuation in this frequency band. This is because lateral variations in attenuation structure are averaged by the gravest modes of oscillation and surface waves that make multiple circuits around the Earth. Computational advances have made the division between free oscillation and surface wave studies fuzzier, with common approaches now amounting to timedomain modeling of complete low-frequency (