Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation [1st ed.] 9783030589677, 9783030589691

Table of contents :
Front Matter ....Pages i-xiv
Structure of the Earth’s Magnetic Field (Alexei Gvishiani, Anatoly Soloviev)....Pages 1-27
EMF Observations and Data Processing (Alexei Gvishiani, Anatoly Soloviev)....Pages 29-96
Mathematical Models of the EMF (Alexei Gvishiani, Anatoly Soloviev)....Pages 97-146
Electrodynamic Processes in the Earth’s Ionosphere (Alexei Gvishiani, Anatoly Soloviev)....Pages 147-205
Extreme Events and Reconstruction of the Solar Activity Parameters Based on Geomagnetic Measurements (Alexei Gvishiani, Anatoly Soloviev)....Pages 207-231
Geomagnetic Information and Big Data (Alexei Gvishiani, Anatoly Soloviev)....Pages 233-268
Applications of the Geomagnetic Field in Technological Problems (Alexei Gvishiani, Anatoly Soloviev)....Pages 269-310
Back Matter ....Pages 311-311

Citation preview

Alexei Gvishiani Anatoly Soloviev

Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation

Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation

Alexei Gvishiani Anatoly Soloviev •

Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation

123

Alexei Gvishiani Geophysical Center of the Russian Academy of Sciences Moscow, Russia

Anatoly Soloviev Geophysical Center of the Russian Academy of Sciences Moscow, Russia

ISBN 978-3-030-58967-7 ISBN 978-3-030-58969-1 https://doi.org/10.1007/978-3-030-58969-1

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

Considering the long and august history of the Earth’s magnetic field, it might well be asked “Why a new book on geomagnetism is needed now?” I believe, however, that this book answers the current need to better understand the geomagnetic field. Over the last decades, new observations, numerical simulations, and experiments have offered more information about the geomagnetic field, which is currently changing dramatically. The dipole moment decreased by about 10% since the geomagnetic field vector is continuously measured. Besides, the intensity of the magnetic field in the center of the South Atlantic Anomaly, as well as the extension of this specific feature, have dramatically changed over the last decades. It is then crucial to understand the complexity of the physical processes behind the changes of the Earth’s natural magnetic fields, whereas it is well-known that the geomagnetic field is characterized by a multiscale spatial structure and a wide range of temporal variations, due to its various sources. The geomagnetic field has its sources inside the Earth (internal contributions) and outside it (external contributions). The most important internal source is the core field, originating in the external fluid core of the Earth, together with the lithospheric field caused by magnetic minerals in the crust and subordinately the upper mantle. The external sources have their origins in the ionosphere, the magnetosphere and from electrical currents coupling the ionosphere and magnetosphere, the field aligned currents. These sources external to the solid Planet induce secondary fields in the Earth. The need to understand the multitude of temporal variations is then essential when, for example, the space weather is considered. The core magnetic field acts like a shield to the solar wind that the Sun continually emits. Nevertheless, geomagnetic disturbances in the ionosphere appear and can affect the high-tech infrastructures, from the electric and railway networks to aviation, telecommunications, and satellite navigation. It is then necessary to know and understand well the nature of the solar-terrestrial relations, to increase our ability to forecast the space weather and mitigate its negative effects. Nowadays, an avalanche-like increase in the data volumes related to geosciences in general, and to geomagnetism in particular takes place. Due to toughening of the requirements to the quality of the collected data, as well as due to the perspectives v

vi

Foreword

of using the coordinated measurements in different geographical locations at the Earth’s surface and in space, it becomes crucial to develop new technological solutions for collecting, processing, storage, and distribution of the data. Development of the new mathematical methods for the data analysis using the modern geoinformatics approaches is equally important. These approaches should, on the one hand, provide automated intelligent control of the quality of incoming information with the possibility to classify the disturbances of the anthropogenic origin and natural magnetic field variations. On the other hand, it is necessary to develop techniques allowing to structure the information and distinguish some certain indicators that simplify tracking the development of the physical processes. Thus, a large variety of the interrelated problems arises, which requires the use of the methods of systems analysis, geoinformatics, mathematical geophysics, and data science for their solution. And the present monograph moves in this direction. This book represents the result of many years of fundamental and applied research carried out by the two authors, members of the Geophysical Center of the Russian Academy of Sciences (GC RAS), with specific support of the Russian Academy of Sciences, the Ministry of Education and Science of the Russian Federation and a number of industries. These researches are mainly aimed to: i) develop new methods for collecting, processing, systematizing, and intellectual analysis of geomagnetic data recorded on ground and in space; ii) coordinate the operation of the magnetic observatory network in Russia and contribute to its expansion; iii) develop new scientific and technical principles for effective functioning of intellectual systems and mathematical methods for system analysis of large data volumes from the Earth’s magnetic field observations; iv) create hardware-software system for collecting, processing, storing, disseminating, and analyzing a wide variety of geomagnetic information; v) build modern high-precision multi-scale models of the Earth’s magnetic field. This monograph combines both classical concepts about geomagnetism with its applications and authors’ recent results. The book well-made contributions are many, though the first to be noted is the MAGNUS (Monitoring and Аnalysis of Geomagnetic aNomalies in Unified System) hardware-software system. This system is the core of the analytical center for geomagnetic data in the authors’ institution, residing in high-quality observations collected by the network of magnetic observatories in Russia and near-abroad countries. The MAGNUS system largely involves artificial intelligence component for online quality control, improved baseline calculation, geomagnetic activity estimation, extraction of core field signal, as well as interactive software for online modeling of ionospheric electromagnetic parameters in polar and low-latitude regions. Another interesting contribution of the authors is a novel mathematical approach, DMA (Discrete Mathematical Analysis) for dealing with time series and multidimensional arrays and vast applications to geomagnetic data analysis. In particular, DMA-based regression derivatives and gravitational smoothing are capable to steadily detect geomagnetic jerks and secular acceleration pulses in the noisy observatory records of the magnetic field recorded over a long period of time. On the applicative side of the geomagnetic field, the authors provide an exciting investigation of the geomagnetic data role in the

Foreword

vii

directional drilling of wells in the Arctic region. The authors present the estimations of the geomagnetic disturbance impact on the typical parameters of the directional drilling path, which are constantly monitored at oil and gas fields. Strong geomagnetic disturbances and their localization are crucial for a highly accurate drill positioning employed by high-tech enterprises. Finally, I would like to stress that this monograph is drawing on first-hand observations and analysis of available data. The authors are both outstanding researchers in the field and are well-known and respected international scientific experts. The book is comprehensive, far-reaching and is at the state-of-the-art level. The arrangement of the material and chapters is logical and follows a natural progression to introduce the various geomagnetic observations, sources, and Sun-Earth interactions. The conducted experimental investigations are combined with development and implementation of methods of geoinformatics, artificial intelligence, systems analysis, and data science to better understand the geomagnetic field. Paris, France

Prof. Mioara Mandea President of the International Association of Geomagnetism and Aeronomy

About This Book

Geomagnetic field penetrates through all shells of the solid Earth, hydrosphere and atmosphere, spreading into space. The Earth Magnetic Field plays a key-role in major natural processes. Geomagnetic field variations in time and space provide important information about the state of the solid Earth, as well as the solar-terrestrial relationships and space weather conditions. The monograph presents a set of fundamental and, at the same time, urgent scientific problems of modern geomagnetic studies, as well as describes the results of the authors’ developments. The new technique introduced in the book can be applied far beyond the limits of Earth sciences. Requirements to corresponding data models are formulated. The conducted experimental investigations are combined with development and implementation of new methods of mathematical modeling, artificial intelligence, systems analysis and data science to solve the fundamental problems of geomagnetism. At that, formalism of Big Data and its application to Earth sciences is presented as essential part of systems analysis. The book is intended for research scientists, tutors, students, postgraduate students and engineers working in geomagnetism and Earth sciences in general, as well as in other relevant scientific disciplines. Best wishes, Prof. Alexei Gvishiani Chief Scientist and Chair of the Scientific Council Geophysical Center RAS, Full Member of RAS, M.A.E. Dr. Anatoly Soloviev Director of the Geophysical Center RAS Сorresponding Member of RAS

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2 EMF Observations and Data Processing . . . . . . . . . . . . . . . . . . 2.1 EMF Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Ground Based Observatories and Stations . . . . . . . . 2.1.2 Aerial and Marine Magnetic Surveys . . . . . . . . . . . 2.1.3 Satellite Based Measurements . . . . . . . . . . . . . . . . . 2.2 Indices of Geomagnetic Activity . . . . . . . . . . . . . . . . . . . . 2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 K and Kp Indices: Background, Calculation and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Collecting, Storing and Processing the Geomagnetic Data in MAGNUS System . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Some Recent Improvements in Routine Baseline Production

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1 Structure of the Earth’s Magnetic Field . . . . . . . . . . . . . . . 1.1 Main Sources of the EMF . . . . . . . . . . . . . . . . . . . . . . 1.2 Internal EMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Main Magnetic Field and Structure of the Earth 1.2.2 World Magnetic Anomalies . . . . . . . . . . . . . . . 1.2.3 Magnetic Field of the Lithospheric Anomalies . . 1.2.4 Dynamics of the Internal Magnetic Field . . . . . . 1.3 External EMF: Sources and Dynamics . . . . . . . . . . . . . 1.3.1 Magnetosphere . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Reconnection of Magnetic Field Lines . . . . . . . 1.3.4 Magnetospheric Substorms . . . . . . . . . . . . . . . . 1.3.5 Magnetic Storms . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Magnetic Pulsations . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.5

Mathematical Methods of Analysis of the Geomagnetic Data Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Methods of the Discrete Mathematical Analysis . . . . . . 2.5.2 Use of the DMA Methods for Tracking the Geomagnetic Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 DMA-Based Regression Derivatives . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Mathematical Models of the EMF . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dipole Model of the Main EMF . . . . . . . . . . . . . . . . . . . . . 3.2 Models of the Main Field Based on the Spherical Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Model Design Principles . . . . . . . . . . . . . . . . . . . . . 3.2.2 Model Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Studying Rapid Core Magnetic Field Dynamics Based on Magnetic Observatory Data . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Data Selection Methodology . . . . . . . . . . . . . . . . . . . 3.3.2 Detection of Secular Acceleration Pulses from Observatory Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Studying Geomagnetic Jerks Using Regression Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Models of the Crustal Magnetic Field . . . . . . . . . . . . . . . . . 3.5 Magnetosphere Field Models . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Electrodynamic Processes in the Earth’s Ionosphere . . . . . 4.1 Field-Aligned Currents According to the Satellite Geomagnetic Measurements . . . . . . . . . . . . . . . . . . . . 4.2 Modeling the Global Distribution of the Ionospheric Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Electric Fields in the Polar Ionosphere Controlled by the FACs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Large Scale Convection Patterns . . . . . . . . . . . . 4.3.2 Localized Current Vortices . . . . . . . . . . . . . . . . 4.4 Penetration of the Electric Fields from the Polar Cap to the Mid-Latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Asymmetric Structures of the FACs and Convection of the Ionospheric Plasma Controlled by the Azimuthal Component of the IMF and Season . . . . . . . . . . . . . . . 4.6 Low Latitude Electric Currents and Their Response at the Earth’s Surface . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Determination of Sq Field from Ground Based Magnetic Data . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Regional Features of the Sq Field . . . . . . . . . . .

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Contents

4.6.3 4.6.4 4.6.5 4.6.6 References .

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Global Seasonal Distribution of Sq Field . . . . EEJ Contribution Near Magnetic Equator . . . Latitudinal Dependence of Sq(X) Amplitudes Equivalent Sq Current System Modeling . . . .

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5 Extreme Events and Reconstruction of the Solar Activity Parameters Based on Geomagnetic Measurements . . . . . . . . . . . 5.1 Solar Dynamo, Solar Cycles and Geoeffective Manifestations of Solar Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Reconstruction of the Solar Wind Speed Based on the High-Latitude Geomagnetic Data . . . . . . . . . . . . . . . . 5.3 Reconstruction of the Monthly Average and Seasonal Values of the Solar Wind Speed and Definition of Extrema . . . . . . . 5.4 Geomagnetic Activity, High-Speed Solar Wind and Evolution of the Magnetic Field of the Sun . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Geomagnetic Information and Big Data . . . . . . . . . . . . . 6.1 General Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Big Data Formalism . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Examples of Big Data Sources . . . . . . . . . . . . . . . . 6.4 Big Data of the Meteorological Observation Systems 6.5 Geo-ecological Observation Network SMEAR . . . . . 6.6 Geo-ecological Big Data . . . . . . . . . . . . . . . . . . . . . 6.7 History and Methods of the Earth Remote Sensing . . 6.8 Big Data of the Earth Remote Sensing . . . . . . . . . . . 6.9 Data Flows of Geomagnetic Observations . . . . . . . . 6.10 Importance of Geomagnetic Data . . . . . . . . . . . . . . . 6.11 Geomagnetic Information Acquisition, Storage and Distribution Networks . . . . . . . . . . . . . . . . . . . . 6.12 Geomagnetic Information and Big Data Formalism . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Applications of the Geomagnetic Field in Technological Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Space Weather and Geomagnetic Activity . . . . . . . . 7.2 Geomagnetic-Based Orientation and Navigation . . . . 7.2.1 Magnetic Compass . . . . . . . . . . . . . . . . . . . . 7.2.2 Navigation and Detection Systems Using the EMF Measurements . . . . . . . . . . . . . . . . 7.3 Geomagnetically Induced Electric Currents . . . . . . .

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7.4

Geomagnetic Support of Directional Drilling of Wells in the Arctic Region . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Drilling Trajectory Control . . . . . . . . . . . . . . . 7.4.2 Assessment of the Geomagnetic Disturbance Impact on the Directional Drilling Path . . . . . . 7.5 Radiation Hazard for the Space Technologies . . . . . . . 7.6 Prevention of Radio Communication Failures . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Chapter 1

Structure of the Earth’s Magnetic Field

1.1 Main Sources of the EMF The geomagnetic field is characterized by a heterogeneous spatial structure and a wide range of temporal variations. This is explained by the fact that the field is created due to the sources of different nature located inside the Earth, as well as in the near-Earth space—in the magnetosphere and ionosphere. The main sources of the geomagnetic field are in the liquid core, mantle and crust of the Earth. These sources create the internal EMF, which contribution to the total vector of the magnetic induction B, observed on the Earth’s surface is the most essential (Parkinson 1983; Hulot et al. 2010). The main EMF (or core field) arising as a result of the dynamo processes in the outer part of the liquid core, has, generally speaking, a multipole structure, but the symmetric dipole plays the dominant role. The main field changes within the time scales of decades, forming a so-called “secular variation”. The satellite and groundbased measurements show that, in general, the geomagnetic field strength decreases, moreover, unevenly. On average, it has decreased by 1.5–2% over the last decades, and in some regions, for example, in the southern part of the Atlantic Ocean, by 10%. In some places, the field strength has increased slightly contrary to the general trend. The observations of the evolution of the main EMF are necessary to update the global models used in navigation, aerospace and other innovative technologies. The main field changes gradually, in the time scales from a year to millennia. The inversions, jerks, which are sharp changes in the behavior of the main EMF, and formation of the large-scale spatial inhomogeneities, such as South Atlantic anomaly, can be considered as the extreme events in the evolution of the main magnetic field of the Earth.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Gvishiani and A. Soloviev, Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation, https://doi.org/10.1007/978-3-030-58969-1_1

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1 Structure of the Earth’s Magnetic Field

The various geological formations in the upper mantle and crust are the sources of the lithospheric part of the internal magnetic field, also called anomalous field. Such local magnetic anomalies are quasi-constant in time, but their geographical distribution and spatial scales are very diverse. The magnitude of the magnetic anomalies varies within a wide range from 1 nanotesla (nT) to several thousand nT. In general, the lithospheric input makes about 1% of the total internal field value. The source of the external magnetic field is a complex and variable threedimensional system of electric currents flowing in the Earth’s magnetosphere and ionosphere. The magnetic effect of these currents is observed on the Earth in the form of regular variations (diurnal, seasonal and 11-year related to the solar activity cycle) and sporadic fluctuations within the time scales from one second to many-hour deviations from the quiet level. The short-term variations related with the powerful (up to tens of millions Amperes) and very variable current systems of magnetic storms, substorms and other unsteady phenomena in the near-Earth space, form the geomagnetic activity. Due to the geometry of the magnetosphere the geomagnetic activity is maximal at the latitudes 65–75° in the both hemispheres. The disturbances of the external geomagnetic field of high amplitude are classified as the extreme events of the space weather (Geomagnetism, Aeronomy and Space Weather… 2019). Thus, the geomagnetic field observed on the Earth’s surface is the sum of the three fields (the first two form the internal EMF) with sources of different physical formation mechanisms: – the main magnetic field with the sources located in the Earth’s core; – the anomalous magnetic field with the sources located in the Earth’s crust; – the external magnetic field with the sources located in the ionosphere and magnetosphere. The main part of the EMF is the field of the Earth’s magnetic dipole (>80%), and about 15% is provided by the field of the world magnetic anomalies. Thus, the contribution of the main magnetic field to the value of induction observed on the Earth’s surface is more than 95%, the anomalous field contributes about 4% and the external field is less than 1%. But this ratio is not constant in space; for example, in the magnetosphere, the external sources can contribute more than 50%. At certain times, even on the Earth’s surface, the contribution of the external sources can increase by an order. It is often necessary to separate the signal of the internal and external fields in the geomagnetic records, which is nontrivial problem (Olsen et al. 2010a). The geomagnetic field induction vector B = µ0 H (where H is the magnetic field strength, µ0 is magnetic permeability) is completely described by three independent components (Yanovskiy 1963). As applied to the preferred coordinate system (Cartesian, cylindrical or spherical), the components are defined as follows: X is the north component (positive direction to the geographic north pole), Y is the east component (positive east along the geographical meridian), Z is the vertical component (positive down to the center of the Earth), H is the horizontal component, D is the angle of magnetic declination (positive if the field line deviates eastward from the north direction), I is the angle of inclination of the magnetic field (positive below the horizontal), F is the total intensity of the geomagnetic field (modulus of the geomagnetic

1.1 Main Sources of the EMF

3

Fig. 1.1 Decomposition of the magnetic field vector in the Cartesian (X, Y, Z), cylindrical (D, H, Z) and spherical (D, I, F) coordinate systems

field vector, F = |B|). The representation of the components of the geomagnetic field vector is given in Fig. 1.1. The relationship between the two commonly used systems (X, Y, Z) and (H, D, Z) is determined as X = H·cosD, Y = H·sinD, Z is without change.

1.2 Internal EMF 1.2.1 Main Magnetic Field and Structure of the Earth Analyzing the distribution of the magnetic field over the Earth surface, English scientist William Gilbert showed that the Earth’s magnetic field represents a field of the giant permanent magnet located in the center of the Earth with axis slightly (by 11.5°) deviated from the Earth’s rotation axis (Fig. 1.2). In 1600, Gilbert stated the results in his book “On the Loadstone and Magnetic Bodies, and on That Great Magnet the Earth” and thereby laid the foundation for the science of geomagnetism. The first hypothesis of the magnetic field source was an assumption on the presence of the large magnetic core of the Earth, but soon it became clear that the temperature inside the Earth was so high that none of the magnets could exist there. This is related with the loss of magnetic properties of magnetized materials upon their excess of the temperature of a certain value, called the Curie point. This temperature is different for the different materials, for example, it has the value about 750–800 °C for ferrum, while the temperature in the Earth’s core reaches 5,000–6,000 °C. All modern theories of the Earth’s structure indicate the presence of the melted material area with high electrical conductivity inside the Earth (Aldridge et al. 1990). Models of the Earth’s structure (Fig. 1.3) indicate lithosphere, which is an external

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1 Structure of the Earth’s Magnetic Field

Fig. 1.2 Configuration of the magnetic field lines of the dipole main field of the Earth (NASA) Fig. 1.3 Configuration of the internal structure of the earth: 1—continental crust, 2—oceanic crust, 3—upper mantle, 4—lower mantle, 5—outer core, 6—inner core, A—Mohoroviˇci´c discontinuity, B—Gutenberg discontinuity (core–mantle boundary), C—Lehmann-Bullen discontinuity

1.2 Internal EMF

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layer of the Earth’s solid component including the crust and upper part of the mantle to the depth of approximately 70 km, and the asthenosphere, being the lower part of the upper mantle. The lithosphere is split approximately into 10 large tectonic plates, along the boundaries of which the vast majority of the earthquake focuses is located. Under the rigid lithosphere a layer of high flowage, entitled the Earth’s asthenosphere, is located, where the temperatures of the mantle substance come most closely to the melting points. Due to a low viscosity of the asthenosphere the lithosphere plates “swim” in the asthenospheric “ocean”. This explains phenomenon called the continental drift. Deeper, the Earth’s core consisting of two parts, inner and outer, is located. The outer core represents the layer 2,100–2,200 km thick and has the properties of electroconductive viscous liquid, which makes about 30% of the Earth’s mass. On the mantle—outer core boundary the density, temperature (from 2,000 to 4,000 °C) and, respectively, the pressure increase sharply. The inner core has the radius of approximately 1,250 km and has the properties of the solid body. The temperature in its center reaches 5,000 °C. The inner core consists of ferrum by 80% and nickel by 20%. The theory of hydromagnetic dynamo was proposed to explain the mechanism of the EMF genesis. This theory is currently well developed and is the main explanation the origin of the EMF (Braginskiy 1978). According to the dynamo, if a liquid conductor during its relative motion crosses the initial (seed) field lines, the electric current is generated in it, creating a magnetic field, which is directed such that it increases the external seed field, and this, in its turn, increases the electric current and so on. At the substance temperature of several thousand degrees, its conductivity is high enough so that the convective movements occurring even in a weakly magnetized medium can excite varying electric currents (Fig. 1.4). According to the electromagnetic induction laws, these currents create new magnetic fields, which leads to an increase in the currents and fields, i.e. a system with positive feedback emerges. The growth of these fields is limited by an increase of the losses for heating the substance by currents flowing in it (Bloxham and Roberts 2017). Thus, the dynamo effect is self-excitation and maintenance of the magnetic fields in the stationary conditions due to the movement of the conducting liquid or gas plasma. Its mechanism is similar to generation of the electric current and magnetic field in the direct-current generator with self-excitation. The origin of the selfmagnetic fields of the Sun, Earth and planets is associated with the dynamo effect, which also relates to their local fields, for example, the fields of the Sun active regions. The dynamo effect creates the main part of the magnetic field observed on the Earth’s surface, which spreads from the depth of the globe outward through the mantle and crust beyond the surface of our planet. It is often approximated as a displaced inclined dipole magnetic field with the magnetic moment of 7.94 × 1022 A m2 . At the Earth’s magnetic poles, where the field vector is oriented perpendicularly to the surface, the magnitude of the magnetic induction is about 60,000 nT, and about 30,000 nT near the equator, where the vector is horizontally directed. Between the pole and equator, the EMF vector is oriented at an angle to the Earth’s surface, i.e. it has horizontal and vertical components. For example, at the latitude of 65° N, the magnitude of the total vector is approximately 52,000 nT, while the azimuthal

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1 Structure of the Earth’s Magnetic Field

Fig. 1.4 EMF internal sources and geodynamo

(horizontal) component of this vector is about 11,000 nT. At a latitude of 70° N, the total vector is 53,000 nT, and the azimuthal component is approximately 8,000 nT.

1.2.2 World Magnetic Anomalies In fact, the EMF configuration is much more complicated than can be assumed from a simple dipole model. Its non-dipole part results in the world anomalies. These are deviations of up to 15% from the Earth’s dipole field in several regions with the typical spatial scales of around 10,000 km. It is considered that the sources of these anomalies are the inhomogeneities of the structure of the core-mantle boundary, as well as the magma flows in the upper layers of the Earth’s mantle. The largest anomalies are the South Atlantic (or Brazilian) and the East Siberian ones (Fig. 1.5). The anomalies show up mainly in the vertical (Z) component of the magnetic vector. The positive anomalies are considered those with the direction of the Z component deviation coinciding with the direction of the dipole field, and the negative ones are those with the opposite direction. The world anomalies extend far into the space, their intensity decreases slowly with the height, which indicates a deep, close to the core-mantle boundary location of its sources. Due to the presence of the world anomalies in the internal EMF, a multipole component becomes apparent; thus the

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Fig. 1.5 Modern EMF with the generally dipole configuration, revealing clear large-scale anomalies. In the figure, the areas with the higher intensity of the magnetic field are red, and the areas with the lower intensity are purple

EMF can be considered as a quasi-dipole field. The field of the Earth’s dipole coupled with the world anomalies makes the main EMF.

1.2.3 Magnetic Field of the Lithospheric Anomalies One more internal source of the magnetic field is the local magnetized areas in the Earth’s crust spreading from several units to hundreds of kilometers, which create the field of the lithospheric magnetic anomalies (Langel and Hinze 1998). These magnetic anomalies are mainly related to the magmatic and metamorphic nature of the rocks. They are developed by the various geological formations in the upper mantle and crust being the sources of the lithospheric part of the internal magnetic field. The growth of the temperature with depth leads to the fact that the rocks located at a certain depth (the depth of the Curie point) no longer have any magnetic properties. The Curie depth varies from ~20 km in the continental areas to ~2 km in the oceans, i.e. the lithospheric field is connected with the layers located relatively close to the Earth’s surface. The magnetic field generated in a geologically heterogeneous lithosphere remains almost unchanged in time. The anomalies are ununiformly distributed across the globe. The map of the relatively large-scale magnetic anomalies is given in Fig. 1.6a. One of the most intense magnetic anomalies is the Kursk Magnetic Anomaly. To create the detailed maps of the local magnetic anomalies of the lithospheric nature, in addition to the ground-based and spaceborne observations

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1 Structure of the Earth’s Magnetic Field

Fig. 1.6 Maps of the large-scale magnetic anomalies of total intensity at geoid altitude (https:// geomag.us/models/MF5.html) (Maus et al. 2007) (a) and magnetic signal of oceanic tides (https:// www.esa.int/Applications/Observing_the_Earth/Swarm/) (b)

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the field magnetic and aeromagnetic surveys are carried out. The latter involves the specially equipped aircrafts cruising over the selected regions along the routes and measuring the magnetic field (usually, total field). The Earth’s dipole field slightly distorted by the field of world magnetic anomalies together with the lithospheric field form the internal EMF. The field induced by the ocean currents as a result of the dynamo effect arising during motion of the conductive fluid (salt ocean water) in the geomagnetic field can also be referred to the internal field. Although the contribution of this field is small and makes not more than ten nT, this signal can be distinguished in the data of the low-orbit satellite measurements (Fig. 1.6b).

1.2.4 Dynamics of the Internal Magnetic Field The dynamics of the geomagnetic field on the Earth’s surface and its spatiotemporal structure are the reflection of the dynamics of the system of electric currents in the liquid core of the planet, specified by the physical processes in its deep interior (Korte and Muscheler 2012). Several distinctive processes in the dynamics of the internal magnetic field are specified. The long-term observations of the geomagnetic field from the ground-based magnetic observatories show that the value of all three orthogonal components of its vector varies slightly from year to year. The secular variability for each component can be of different shapes and achieve several percent of the total measured value (Finlay 2008). The long-term changes of the EMF are determined based on the average annual values of the field elements that each geomagnetic observatory provides. Such diagrams show a slow smooth change at the particular location. The differences between the successive annual values of an element are called the secular variation (SV) of the element. Figure 1.7 shows the values and SV of the horizontal components X and Y at two observatories Chambon-la-Foret (France) and Gnangara (Australia) from both hemispheres over the last 80 years. It can be seen that the amplitude might exceed 1000 nT, i.e. more than 2%. In general, the secular change has a period of 60–80 years. Despite the continuous improvement of the quality of the main EMF models, the magnitudes of the observed geomagnetic field at given geographical locations can significantly differ from the predicted ones. Figure 1.8 shows the annual values and SV of the geomagnetic total intensity (F) for the two Russian high-latitude stations Dikson and Chelyuskin, according to the IGRF model and the observations. It can be seen that the differences achieve 300 nT, the interannual SV has a long-period wave-like component with a period of about 60 years, and the interannual differences between the simulated and measured SV show a linear trend, negative for one station and positive for another. Figure 1.9 shows the global change of F for a century as the difference between the values in 2008 and 1908. The contours show that, in general, F decreased in

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1 Structure of the Earth’s Magnetic Field

Fig. 1.7 Values (a) and SV (b) of the horizontal X (black) and Y (red) geomagnetic components at the observatories Chambon-la-Foret (France, 50° N geomagnetic latitude, solid line) and Gnangara (Australia, 42° S geomagnetic latitude, dashed line) over 1938–2015. Vertical axis for Y values is on the right

the southern hemisphere and in the American sector, while in the eastern part of the northern hemisphere it remained unchanged. The secular variation is common for both the dipole and non-dipole components of the EMF. So, during the last century, the dipole field decreased by about 0.05% per year. The relative magnitude of the annual rate of change of the non-dipole field is, in average, higher, but varies from region to region, where the field intensity can either decrease or increase. The total magnetic moment of the Earth’s dipole decreases systematically. Figure 1.10 shows the change of the magnetic moment value over the last 450 years (Finlay 2008). It is seen that during this period the magnetic moment of the Earth decreased by 20%, i.e. approximately by one thousandth of its value for each year. If this trend keeps, the dipole moment will become zero in 1500–2500 years. It is also found that the world anomalies move slowly westward at the speed of 0.2° longitude per year. This phenomenon, discovered in 1951, was called the westward drift of the Earth’s magnetic field. The westward drift becomes especially evident at the low latitudes.

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Fig. 1.8 Left column: annual F values for the stations Dikson (69° N geomagnetic latitude, upper row) and Chelyuskin (78° N geomagnetic latitude, lower row), derived from the IGRF model (black dots) and the direct measurements (red dots). Middle column: interannual SV of F according to the measurements (black broken line) and 5-point smoothed curve (blue line). Right column: difference between the modeled and measured interannual SV and fitted linear trend

Fig. 1.9 Contours of the difference between the total intensity of the geomagnetic field in 2008 and in 1908

The modern maps of annual SV in the EMF components show that there are several areas where these components increase or decrease much more intensively than in the other regions. In the southern hemisphere, the South Atlantic magnetic anomaly, a zone of an anomalously weak magnetic field, behaves quite dynamically. The anomaly shifts by 0.3° to the west and 0.1° to the north and deepens up to 100 nT per year. The central parts of the regions where the maximum changes are observed are called the secular variation focuses (SVF). Studying the maps of the modern SV,

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1 Structure of the Earth’s Magnetic Field

Fig. 1.10 Change of the Earth’s magnetic moment in time

one can specify several such regions: the South Atlantic SVF with the maximum rate of change −200 nT/year, the Ceylonese SVF with the rate +100 nT/year, the Indonesian SVF with the rate −40 nT/year, the Pacific SVF with the rate +60 nT/year and some smaller ones. The movement of magma in the Earth’s mantle as well as the emergence of reverse magnetic flux patches at the core surface are among the possible causes of the SVF occurrence. The Earth’s magnetic poles also shift noticeably. In the north, the magnetic pole drifts from the Canadian sector of the Arctic towards the East Siberian coast. In the south, the pole shifted from the continental part of Antarctica to the Southern Ocean, towards New Zealand (Fig. 1.11). The magnetic poles are the locations where the field is perpendicular to the Earth’s surface. The geomagnetic poles are those predicted by the dipole field models. The ground and satellite based measurements show that the drift rate of the north magnetic pole towards the geographical pole increased sharply in the 1990s and achieved 50 km/year, while at the beginning of the 20th century it

Fig. 1.11 Shift of the magnetic (black dots) and geomagnetic (red dots) poles in the northern and southern hemispheres (https://www.gfz-potsdam.de/en/section/geomagnetism)

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was only 10 km/year. If the pole keeps its speed and direction, then in about 50 years it could reach the Northern Siberian islands. Upon achievement of the value of about 60 km/year in 2003, the drift rate of the north magnetic pole then began to slow down and in 2009 decreased down to about 45 km/year. At that, the pole began to turn slightly towards Canada, still moving in the northwest direction. The drift of the south magnetic pole, on the contrary, is directed away from the geographical South pole. At that, the shift rate of the South pole, which used to be approximately the same as that of the North pole (about 10 km/year), has even decreased recently. The shift of the magnetic poles results in a change in the geographical position of the regions that are related with the certain plasma domains in the Earth’s magnetosphere. If the motion of the north magnetic pole from the northern islands of the Canadian archipelago towards East Siberia continues, then the aurora zone surrounding it will shift in the same direction. In turn, auroras are associated with the maximum precipitations of energetic particles from the magnetosphere into the upper atmosphere and the most intense and variable ionospheric electric currents. Now, this zone is located above the continental territory of Canada, and the Canadian power systems are mostly sensitive to the attacks of the geomagnetic storms, during which the parasitic induced currents increase catastrophically, disabling the transformers and intensifying the corrosion processes in the pipeline systems. If the pole continues moving in the same direction as it moves now, the zone of the maximum geomagnetic variations, dangerous for the technological systems, will gradually cover the northern territories of Russia. These are exactly those territories where it is supposed to develop new energy projects. Thus, acceleration of the magnetic pole drift and change in the trajectory of their motion can significantly affect development of many natural and technological processes and lead to the extreme changes in the geomagnetic conditions. A decrease in the dipole moment and drift of the magnetic poles indicate an increase in the probability of geomagnetic reversal, which is the most dramatic global change in the EMF. In the course of geomagnetic reversal, the dipole component of the field weakens, and the quadrupole and tetrapole components play an increasingly significant role. This leads to a complex, multipolar field topology and eventually to the changeover of the North and South magnetic poles. The paleomagnetic studies indicate that such a configuration always preceded a geomagnetic reversal. In fact, during this period, which can last for thousands of years, the Earth loses gradually its strong magnetic shield. This leads to the increasing inflow of charged particles of the galactic cosmic rays from the outer space, affecting the atmosphere chemistry. In particular, the concentration of ozone in the atmosphere can significantly decrease due to an increase in the amount of nitrogen oxides. The inversion dating is based on the paleomagnetic information obtained by studying the igneous and sedimentary rocks of the relevant geological age (Amit et al. 2011). Extensive paleomagnetic and paleointensity data are available for the last 80 million years; however some reliable samples date back to 1.86 billion years (Veselovskiy et al. 2019). One of the results of the paleomagnetic studies carried out at the beginning of the 20th century was the discovery that the rocks of the different geological age have a natural remnant magnetization oriented both in and against the

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1 Structure of the Earth’s Magnetic Field

Fig. 1.12 Direction of the Earth’s magnetic moment: black segments match today’s normal polarity, white ones denote periods of reversed polarity, time scale indicates million years to the past

direction of the modern main EMF. Magnetization of the rock consists of two components: induced magnetization, which is always directed along the modern field, and remanent magnetization, emerging during the rock formation. Its direction reflects the direction of the magnetic field that existed at that geological time when the rock was formed. In many rocks, the remanent field does not coincide in direction with the modern field. This enables studying the history of the geomagnetic field during the past geological periods. In the course of the completely different experimental studies that began in the middle of the 19th century, the evidence appeared that the magnetized rocks could contain the fossilized records of the Earth’s magnetic field during their formation. This idea was extended to the archaeological materials at the beginning of the 20th century. In 1905, the baked clays were discovered in France, magnetized in the direction opposite to the modern field, and the conclusion was that the Earth’s magnetic field had reversed its polarity in the geological past (Brunhes 1905). This discovery was confirmed in the rock samples around the world and became an important evidence leading to the revolution in understanding the mechanism of our planet. Studying the magnetic properties of the geological sections of the igneous and sedimentary rocks showed that in past, the north and south magnetic poles indeed reversed—the sign of the magnetic field inverted, while at some time point the intensity of the main EMF approached zero. The inversion duration makes 2000– 6000 years. Based on the inversion age dating using the radioactive methods, a world magnetochronological scale of inversions was made (Fig. 1.12). A quite short period of time from the geological perspective is sufficient to change completely the total pattern of the geomagnetic field, including its zero crossing and polarity change. In the past, the inversion occurred on average approximately every 250,000 years. However, already 780,000 years have passed since the last one. There is no explanation for such a long period of stability. Some small-scale changes occur comparatively often in the internal EMF, including the so called geomagnetic jerks, which are relatively sharp changes in the rate of change (SV) of one (usually eastern) of the EMF components (Bloxham et al. 2002). Upon the jerk, the sign of the first time derivative of the field component changes to the opposite one (Fig. 1.13). At present, the physical processes underlying the jerk appearance do not have a generally accepted explanation, although it is clear that they are most likely specified by the extra dynamics in the upper layer of the liquid Earth’s core. As their origin has not been established yet, the jerks are a serious obstacle for predicting the behavior of the geomagnetic field forward for years and decades. The recent events observed in the satellite geomagnetic data have been related with the short-time and equatorially localized secular acceleration (second

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Fig. 1.13 The rate of change (SV) of the EMF east component recorded by European observatories. Geomagnetic jerk epochs are marked by arrows (Macmillan 2007)

time derivative) pulses, which is likely connected with the rapidly alternating flows on the surface of the Earth’s core. The numerical simulation of the geodynamo has shown that the jerks can occur due to the interaction between slow convection of the core and fast hydromagnetic waves and can be caused by the localized Alfven waves generated upon the sudden increases in buoyancy inside the core (Aubert and Finlay 2019). When they achieve the core surface, the waves focus their energy in the direction of the equatorial plane along the Earth’s magnetic field lines, creating sharp interannual changes in the core flow. Various methods including regression, wavelet and discrete mathematical analysis made it possible to establish the exact dates of appearance of the jerks at various observatories without any additional assumptions. One of the most noticeable geomagnetic jerks was observed during several months of 1969–1970 independently at many observatories of the world. Other events were observed in 1901, 1913, 1925, 1978, 1991, 1999 (Jacobs 1994) and later on. The most recent jerk occurred in 2017. However, the jerks are not always apparent at all observatories, and the observed ones are not always simultaneous. For example, the specific structure of the 1969 jerk was registered in the southern hemisphere 2 years later than in the northern one.

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1 Structure of the Earth’s Magnetic Field

1.3 External EMF: Sources and Dynamics The magnetic field sources are not only inside the Earth, but also in the near-Earth space. The source of the external magnetic field is a complex and variable threedimensional system of electric currents flowing in the Earth’s magnetosphere and ionosphere (Akasofu and Chapman 1972; Kivelson and Russell 1995). The magnetic effect of these currents is observed on the Earth in the form of both regular variations and sporadic fluctuations within the time scales from one second to days. As the electric currents amplify in the ionosphere-magnetosphere system, the observed magnetic field deviates from the quiet level.

1.3.1 Magnetosphere A supersonic stream of energized particles, mainly protons, is constantly flowing from the Sun, forming the solar wind (SW). The plasma flow of the solar wind transfers the interplanetary magnetic field (IMF) of the solar origin “frozen” into it. Encountering an obstacle in the form of the EMF in its way, the magnetized plasma of the SW slows down and flows around the dipole. As a result, a cavity is formed inside the solar wind flux, filled with the properly magnetic field of the Earth - the magnetosphere. Striking of SW against the Earth’s dipole leads to the fact that on the side facing the Sun, the magnetic field lines of the dipole compress and on the opposite night side, on the contrary, they stretch away (Fig. 1.14). As a result, the shape of the magnetosphere deforms; its outer boundary from the sunlit side is approximately ten Earth radii (R E ≈ 6371 km) away from the Earth’s surface, whereas on the

Fig. 1.14 Earth’s magnetic field under the action of the solar wind stream (not to scale, image source—NASA)

1.3 External EMF: Sources and Dynamics

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opposite side, the elongated geomagnetic field lines form the cylindrical tail of the magnetosphere (magnetotail) up to 30 RE in diameter. The geomagnetic field in the tail decreases gradually and becomes vanishingly small at a distance of about 100 RE . The magnetotail, especially its boundary layers, is filled with the cosmic plasma with a wide energy spectrum. The magnetosphere and ionosphere are coupled with each other and the SW through various electrodynamic and plasma processes, which are determined by the configuration of the geomagnetic field and the structure of the interplanetary magnetic field. The magnetospheric and ionospheric plasma motion in the geomagnetic field leads to generation of the electric fields and currents, thus making the electric currents in the magnetosphere-ionosphere system closely interrelated. The magnetic effect of these currents is observed both on the Earth and in the near-Earth space. Configuration of the Earth’s magnetosphere and the currents flowing in it are shown in Fig. 1.15. It includes the following basic elements: • The current on the day side of the magnetopause. This current provides the balance of the dynamic pressure of the solar wind and the magnetic pressure of the Earth’s dipole. The current closes on the night side of the magnetopause in the northern and southern halves of the magnetotail. The magnetopause current magnitude is about 108 A. This current provides a positive (directed to the north) disturbance of the magnetic field on the Earth’s surface near the equator. • The currents in the magnetotail. The current sheet across the tail separates the northern and southern parts of the tail, in which the dipole magnetic field has

Fig. 1.15 Configuration of the Earth’s magnetosphere and its currents (thick arrows—electric currents, thin arrows—geomagnetic field lines) (Russell et al. 1995)

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1 Structure of the Earth’s Magnetic Field

an opposite direction. In the current system of the magnetotail, the transverse current closes at the night magnetopause, forming figuratively two solenoids. On the Earth’s surface, the tail current appears as a negative disturbance of the horizontal component of the magnetic field. • The ring current developing during the global magnetic storms in the equatorial plane of the magnetosphere at a distance of 2–3 RE . The current is connected with injection of the energetic particles from the plasma sheet and the circular drift of protons and electrons along the orbits around the Earth in the dipole magnetic field. The ring current magnitude is of the order 106 A. This current also provides a negative disturbance of the magnetic field on the Earth’s surface near the equator. • The field-aligned currents flowing along the field lines of the Earth’s dipole and connecting the low-latitude boundary layer of the magnetosphere and the highlatitude ionosphere. The field-aligned currents form a system of layers of the inflowing and outflowing currents from the ionosphere, extended along the latitude circles. The total current makes ~1 × 106 A at the quiet times and ~3 × 106 A during the periods of geomagnetic disturbances. The field-aligned currents are perpendicular to the ionospheric sheath, and their magnetic signal is almost not detected on the surface of our planet. The internal magnetosphere is the region of a relatively stable quasidipole field, specified by the internal sources of the Earth. In the equatorial plane, this region extends from the Earth to the distance of about 6–8 RE , which approximately corresponds to the geomagnetic latitude of 65°. The inner magnetosphere is a relatively stable formation and is included in the disturbance processes only during the periods of the global magnetic storms. The outer magnetosphere is the region adjacent to the magnetopause and contacting directly with the solar wind in one or another way. The magnetic field in this region is mainly specified by the electric currents flowing at the boundary of the magnetosphere, and thus it is sensitive to all changes in the SW conditions.

1.3.2 Ionosphere Being a layer with high electric conductivity due to ionization of the air molecules by the high-frequency part of the solar electromagnetic radiation, the ionosphere is formed at the altitudes of 60–500 km. The most intensive electric currents flow at the altitudes of 100–120 km (in the E-layer of the ionosphere). The ionosphere can be considered as the base of the region occupied by the magnetosphere. The magnetosphere is electrodynamically coupled with the ionosphere through the fieldaligned currents flowing along the highly conductive geomagnetic field lines from the boundary layers of the magnetosphere into the high-latitude ionosphere. The ionosphere parameters are quite variable and depend on time of day, season, phase of the solar cycle, interaction with the magnetospheric plasma and electric field.

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The ionospheric electric currents, which can be considered horizontal, spread over the spherical ionospheric sheath, generating the geomagnetic variations at all latitudes. The most intense and localized ionospheric currents (electrojets) are located along the auroral oval at the magnetic latitude of 65–75° in both hemispheres. The auroral electrojets are related with the electric circuit, closed by the field-aligned currents of the magnetospheric origin. The ionospheric current system includes the following basic elements: • Solar diurnal variation (Sq). A regular change in the geomagnetic field is related with rotation of the Earth around its axis, a daily change in illumination, and, as a consequence, of the ionospheric conductivity. As a result, the circular currents appear in the ionosphere at the altitude of 100–120 km, which move above the Earth’s surface following the Sun (Fig. 1.16). This current system causes a spatiotemporal distribution of the disturbance vectors with a maximum at the local noon. The amplitude of the Sq-variation depends on the latitude of the observation point and varies from 5 to 100 nT. The maximum amplitude is observed at the magnetic equator and the mid-latitudes. There is a dependence of intensity on the season of the year. The maximum amplitude of the Sq-variation is observed in summer, when the Sun is above the horizon and the minimum one during the winter solstice. • The eastward and westward auroral electrojets. The field-aligned currents of the magnetospheric origin are closed in the auroral ionosphere, where the eastward and westward electrojets are formed between the field-aligned currents flowing into the ionosphere and those flowing out from it (Fig. 1.17). The electrojets +20 = 20 000 Ampere 0

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Fig. 1.16 Contours of the electric current density corresponding to the solar-diurnal variations (equinox)

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1 Structure of the Earth’s Magnetic Field

Fig. 1.17 Eastward and westward auroral electrojets and return ionospheric currents closing them. View from the north magnetic pole in the “magnetic local time (MLT)—magnetic latitude” coordinates

make the greatest contribution to the geomagnetic activity, and their intensity depends on the parameters of the solar wind and magnetosphere. A substorm auroral electrojet developing on the night side in the auroral zone is typically a continuation of the westward electrojet. The substorm electrojet is connected with the current sheet of the magnetotail through the field-aligned currents. Figure 1.18 shows the hourly average values of the geomagnetic variations at the Thule polar observatory (Greenland, 88° N geomagnetic latitude) for the H, D, and Z components of the field vector for 2001, as well as their smoothed values. Here, the secular trend curve (annual averages) is deducted. It can be seen that the amplitude of the variations in summer is significantly higher than in winter. This is due to illumination of the ionosphere by the solar radiation, which results in the increase of the ionization rate, ionospheric conductivity and, respectively, the ionospheric currents generating the geomagnetic disturbances. Figure 1.19 shows minute values of the geomagnetic variations of the horizontal (X) geomagnetic component recorded at the same observatory during the different seasons, in January and July. The figure also shows the diurnal quiet variation. Usually, such curve is calculated according to the data during the quietest days of the month, when the variations are minimal. The five so-called international quietest days of the month are calculated globally basing on a fixed set of observatories. This level can be considered as the background for the magnetic variations in the hour, minute and second time scales. For the longer period variations, the secular trend can be considered as the background conditions. It can be seen from the figure that in summer the amplitude of the diurnal variation is almost twice higher than in winter, which is related with the increased ionospheric conductivity, and also partially with

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Fig. 1.18 Hourly mean values of the geomagnetic variations at the Thule observatory (88° N geomagnetic latitude) for the H, D and Z components (blue line) and smoothed average values (red line) for 2001 (Stauning et al. 2007)

the system of the thermospheric neutral winds depending on solar heating. The disturbances of the external geomagnetic field are considered as the deviations from the quiet diurnal curve. The figure shows that in 2000, which was the year of the very active Sun, all days were notably disturbed, and the deviations from the quiet diurnal curve exceeded 200 nT.

1.3.3 Reconnection of Magnetic Field Lines When the direction of the IMF incorporated into the plasma cloud streaming from the Sun becomes opposite to the direction of the geomagnetic field on the day side, the process of the so-called reconnection begins. When the oppositely directed field lines approach each other, the IMF vanishes and the “freezing-in” principle is violated. From a “closed” geomagnetic field line and “free” interplanetary field line, two “open” field lines are formed, which at one end begin in the Earth’s polar caps with

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1 Structure of the Earth’s Magnetic Field

Fig. 1.19 Minute values of the geomagnetic variations of the northern X component at the Thule observatory in January (top) and July (bottom) 2000. The quiet diurnal variation is shown in red (Stauning et al. 2007)

the other end stretching into the interplanetary space (Fig. 1.20a). The reconnection is “beneficial” from an energy point of view, since the total length of the field lines is reduced. The SW stream blows the “open” lines off to the night side, where the oppositely directed field lines come together again. The magnetosphere and ionosphere become involved in the cycle of global convection. The reconnection rate on the night side is lower than on the day side, therefore, in the magnetotail an accumulation of the “open” field lines and, consequently, magnetic energy takes place. The size of the polar caps grows, and the auroral oval zone shifts closer to the equator by several degrees. After some time (several hours) of the continuous reconnection, the magnetotail, “overloaded” with a magnetic field, loses stability, the reconnection process on the night side takes on an explosive character, and in a few minutes the excessive force lines discharge. This cyclic process is called the magnetospheric substorm and is accompanied by significant disturbance of the entire outer magnetosphere of the Earth. In fact, a part of the magnetotail breaks off, and its remainder is compressed to the Earth (Fig. 1.20b). At this moment, a part of the plasma of the outer magnetosphere becomes “superfluous” and is discharged along the field lines into the auroral zone of the ionosphere.

1.3 External EMF: Sources and Dynamics

23

(a)

(b)

Fig. 1.20 Reconnection mechanism caused by the interaction between negative IMF and geomagnetic field leading to emergence of the “open” field lines (a); evolution of a separate field line and the breakage of a magnetotail (Hargreaves 1979) (b)

When the magnetotail is broken off, the electric (tail) current, under normal conditions, flowing across the magnetotail (see Fig. 1.15), is forced to bypass this gap through the nearest conductor, which is the ionosphere, using the “backup circuit” along the field lines to the Earth, then through the ionosphere in the region of the night part of the polar oval and back to the tail (Petrukovich and Zeleny 2001). The newly formed current is called the electrojet. The strength of such an ionospheric current sometimes exceeds one million amperes, and the induced magnetic field on the Earth’s surface in the auroral zone introduces significant variations in the geomagnetic field. The most impressive manifestation of substorms is aurora borealis resulting from the bombardment of neutral atoms of the atmosphere by plasma flows from the magnetotail, accelerated along magnetic field lines. Neutral atoms in the collision with energetic ions and electrons emit photons. The magnetosphere

24

1 Structure of the Earth’s Magnetic Field

can for a long time dump excess energy into the polar regions of both hemispheres of the Earth in the form of substorms with a frequency of about 3 h (Yermolaev and Yermolaev 2010).

1.3.4 Magnetospheric Substorms A substorm is related with a current break in the magnetotail and its closure to the night ionosphere through the field-aligned currents with formation of a substorm ionospheric westward electrojet in the midnight sector of the local time. A substorm is accompanied by sharp increase in luminosity of the auroras and violent geomagnetic disturbances. The substorms can occur one after another during the high solar activity periods. Since the ionospheric substorm electrojet is concentrated in the midnight sector of the local time and in a quite narrow latitudinal interval, the highest magnetic disturbances are usually observed at the geomagnetic observatories and stations located at the latitudes of the auroral oval (60–70°) during the hours when the station is on the night side of the Earth. In the magnetogram, the northern (X or H) component shows a “negative bay” lasting 0.5–1 h with amplitudes varying from tens to thousands of nT in the minute time scale. With an increase in the substorm intensity, not only the current density in the electrojet increases, but it also expands and moves to the lower latitudes. To monitor the electrojets, the data of 12 magnetometers located at the auroral latitudes are used to derive 1 min indices of the geomagnetic activity AL and AU, which represent the maximum negative and positive deviations of the horizontal, north-directed geomagnetic component of the magnetic field from the quiet level.

1.3.5 Magnetic Storms The magnetic storms reflect the global disturbances of the external EMF. The storms are caused by a solar flare and arrival of the interplanetary magnetic cloud front to the Earth with a sharp change in the SW parameters. During a storm, a rapid amplification followed by slow (during a day or more) attenuation of the magnetospheric ring current takes place; the latter is formed from the charged particles coming from the magnetosphere. The initial compression of the magnetosphere when colliding with a solar plasma cloud and current amplification at the magnetopause lead to the geomagnetic response on the Earth, expressed in appearance of a jump in the horizontal field component, which is called the storm sudden commencement (SC). The geomagnetic effect of the ring current is expressed in a decrease in the Hcomponent of the field at the equator by the value from tens to hundreds of nT. An increased geomagnetic activity is also observed at other latitudes. The substorm activity also intensifies, substorms follow one after another, and the auroral zone

1.3 External EMF: Sources and Dynamics

25

expands up to midlatitudes. The heavy storms with a sharp SC and duration of 1– 2 days usually occur during the years of the maximum of the 11-year solar cycle, when the number of solar flares increases. For the phase of the solar cycle decay, appearance of the long-lived coronal holes on the Sun and high-velocity SW streams are common; they cause the longstanding, multi-day storms of a relatively low intensity. A geomagnetic storm belongs to the category of the extreme natural processes, when the geomagnetic field is most disturbed. Figure 1.21 shows the variations of the auroral AL and AU indices, which quantitatively estimate the westward and eastward ionospheric electrojets, respectively, as well as the low-latitude SYM-H index, which quantitatively estimate the magnetospheric ring current. It can be seen that the auroral disturbances in absolute magnitude exceed by an order the disturbances at the low latitudes. Simultaneous amplification of the ring current, convection of the magnetospheric plasma, electric currents in the magnetotail and in the ionosphere and substorm electrojets leads to the propagation of the storm-time geomagnetic disturbances all over the globe. The rapid changes in the external magnetic field cause the induced (telluric) electric currents in the Earth’s conducting layers, which, in turn, generate an additional localized magnetic field. The telluric currents achieve their maximum magnitude during the magnetic storms. Thus, the observed variations of the geomagnetic field represent the sum of the variations caused by the currents in the ionosphere and near-Earth space, as well as by intraterrestrial electric currents. At the low solar activity, the contribution of the external field to the total observed magnetic field does not exceed fractions of percent. During the magnetic storms, the amplitude of

Fig. 1.21 1 min values of the auroral AU and AL indices and low-latitude SYM-H index of the geomagnetic activity during the magnetic storm of July 15–17, 2000

26 Table 1.1 Types and periods of the magnetic pulsations

1 Structure of the Earth’s Magnetic Field Regular

Irregular

Type

Period (s)

Type

Period (s)

Pc1

0.2–5.0

Pi1

1–40

Pc2

5–10

Pi2

40–150

Pc3

10–45

Pc4

45–150

Pc5

150–600

fluctuations increases sharply and, depending on the field component and latitude, the contribution might amount tens of percent. So, in the auroral zone during the geomagnetic disturbances (substorms), the magnitudes of the horizontal H component of the external and internal fields can become comparable.

1.3.6 Magnetic Pulsations In response to the non-stationary motion of the SW in the magnetosphere, the plasma waves are excited, which is identified by the magnetometers as magnetic pulsations. The plasma waves in the ultralow frequency range (ULF) with the wavelengths comparable to typical scale lengths of the entire magnetosphere have the frequencies in the range from 1 MHz to 10 Hz and play a fundamental role in energy transfer in the magnetosphere and ionosphere. The ULF waves can also interact with the charged particles in the magnetosphere, causing their acceleration, transfer and loss from the ring current and radiation belt. The magnetic pulsations are divided into regular and irregular (Table 1.1).

References Akasofu SI, Chapman S (1972) Solar-terrestrial physics. Oxford University Press, Ch. 2 Aldridge KD, Bloxham J, Dehant V, Gubbins D, Hide R et al (1990) Core-mantle interactions. Surv Geophys 11(4):329–353. https://doi.org/10.1007/BF01902965 Amit H, Korte M, Aubert J, Constable C, Hulot G (2011) The time-dependence of intense archeomagnetic flux patches. J Geophys Res Solid Earth 116–600 Article Number B12106. https://doi. org/10.1029/2011jb008538 Aubert J, Finlay C (2019) Geomagnetic jerks and rapid hydromagnetic waves focusing at Earth’s core surface. Nat Geosci 12:393–398. https://doi.org/10.1038/s41561-019-0355-1 Bloxham J, Roberts PH (2017) The geomagnetic main field and the geodynamo. Rev Geophys 29(S1):428–432. https://doi.org/10.1002/rog.1991.29.s1.428 Bloxham J, Zatman S, Dumberry M (2002) The origin of geomagnetic jerks. Nature 420(6911):65– 68. https://doi.org/10.1038/nature01134 Braginskiy SI (1978) The geomagnetic dynamo, Izvestiya, Acad. Sci. USSR. Phys Solid Earth 14:659–668

References

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Brunhes B (1905) Sur la direction de l’aimantation permanente dans une argile de Pontfarein, C. R. Acad Sci Paris 141:567–568 Finlay CC (2008) Historical variation of the geomagnetic axial dipole. Phys Earth Planet Inter 170(1–2):1–14. https://doi.org/10.1016/j.pepi.2008.06.029 Hargreaves JK (1979) The upper atmosphere and solar-terrestrial relations: an introduction to the aerospace environment. Van Nostrand Reinhold, New York, 298 pp. Hulot G, Finlay CC, Constable CG, Olsen N, Mandea M (2010) The magnetic field of planet earth. Space Sci Rev 152(1–4):159–222. https://doi.org/10.1007/s11214-010-9644-0 Jacobs JA (1994) Reversals of the earth’s magnetic field. Cambridge University Press, Cambridge 358 pp. Kivelson MG, Russell CT (1995) Introduction to space physics. Cambridge University Press, 588 pp. Korte M, Muscheler R (2012) Centennial to millennial geomagnetic field variations. J Space Weather Space Climate 2:A08. https://doi.org/10.1051/swsc/2012006 Langel RA, Hinze WJ (1998) The magnetic field of the Earth’s lithosphere: the satellite perspective. Cambridge University Press, 429 pp. https://doi.org/10.1017/cbo9780511629549 Macmillan S (2007) Geomagnetic jerks. In: Gubbins D, Herrero-Bervera E (eds) Encyclopedia of geomagnetism and paleomagnetism. Springer, Dordrecht Mandea M, Korte M, Yau A, Petrovsky E (eds.) (2019) Geomagnetism, aeronomy and space weather: a Journey from the earth’s core to the sun. Cambridge University Press, 350 pp. Maus S, Lühr H, Rother M, Hemant K, Balasis G, Ritter P, Stolle C (2007) Fifth-generation lithospheric magnetic field model from CHAMP satellite measurements. Geochem, Geophys, Geosyst 8(5). https://doi.org/10.1029/2006gc001521 Olsen N, Glassmeier KH, Jia X (2010a) Separation of the magnetic field into external and internal parts. Space Sci Rev 152(1–4):135–157. https://doi.org/10.1007/s11214-009-9563-0 Olsen N, Mandea M, Sabaka TJ, Toffner-Clausen L (2010b) The CHAOS-3 geomagnetic field model and candidates for the 11th generation IGRF. Earth, Planets Space 62(1). https://doi.org/ 10.5047/eps.2010.07.003 Parkinson WD (1983) Introduction to geomagnetism. Scottish Academic Press, Edinburgh, 446 pp. Petrukovich A, Zeleny L (2001) In the arms of the Sun. Sci Life N7 (in Russian) Russell CT, Snare RC, Means JD, Pierce D, Dearborn D, Larson M, Barr G, Le G (1995) The GGS/POLAR magnetic fields investigation. Space Sc 71:563–582 Stauning P, Troshichev O, Janzhura A (2007) Polar Cap (PC) index. Unified PC-N (North) index procedures and quality, DMI Scientific Report, SR-06-04 Veselovskiy RV, Samsonov AV, Stepanova AV, Salnikova EB, Larionova YO, Travin AV, Arzamastsev AA, Egorova SV, Erofeeva KG, Stifeeva MV, Shcherbakova VV, Shcherbakov VP, Zhidkov GV, Zakharov VS (2019) 1.86 Ga key paleomagnetic pole from the Murmansk craton intrusions—Eastern Murman Sill Province, NE Fennoscandia: Multidisciplinary approach and paleotectonic applications. Precambr Res 324:126–145. https://doi.org/10.1016/j.precamres. 2019.01.017 Yanovskiy BM (1963) Terrestrial magnetism, part II (Zemnoy magnetism, ch. II). Izd. Leningr. Gos. Univ., 461 pp (in Russian) Yermolaev YI, Yermolaev MY (2010) Solar and interplanetary sources of geomagnetic storms: space weather aspects. Izv Atmos Ocean Phys 46:799–819. https://doi.org/10.1134/S00014338 10070017

Chapter 2

EMF Observations and Data Processing

2.1 EMF Measurements The continuous and diverse time and space variations of the EMF lead to the necessity of deployment of the stationary observation sites for monitoring and recording these variations. The regular and continuous geomagnetic observations are carried out in order to solve a wide range of problems of the physics of the Earth and solarterrestrial relations, radiophysics and ecology, geology, geodesy, mineral exploration, etc. The basis of the Earth’s magnetic field studies are the measurements of the three components of the geomagnetic field vector B and its modulus (total intensity) at spatially distributed sites. The high-precision measurements of the geomagnetic field are carried out using the ground-based (magnetic observatories and stations) and spaceborne (low-orbit satellites) magnetometric instrumentation.

2.1.1 Ground Based Observatories and Stations A full-scale geomagnetic observatory provides automatic recording of the total intensity of the geomagnetic field and relative variations of its components, as well as regular absolute observations of the geomagnetic field. The observatory type measurements are characterized by the following fundamental features. Firstly, the observatory ensures the maximum possible measurement accuracy. Secondly, it ensures the continuity of recording of the EMF variations at the various time scales, from days to decades. The data gaps are irrecoverable and can lead to the loss of significant information in case of unexpected geophysical events or anthropogenic disasters. Thirdly, the data are collected and distributed with the minimal time delay.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Gvishiani and A. Soloviev, Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation, https://doi.org/10.1007/978-3-030-58969-1_2

29

30

2 EMF Observations and Data Processing

Fig. 2.1 Photographic recording (analog magnetogram) of the geomagnetic field components dated 1859

Finally, the primary task of the observatory is to control the secular variation of the EMF using the absolute measurements. For a long time, the main media of the observatory recordings was the magnetograms—the photographic records of the geomagnetic field components (Fig. 2.1). In the recent decades, the observatories have switched to the digital data acquisition and processing systems, producing most commonly daily files of the second and minute values of the geomagnetic field variations adjusted to the absolute measurements. Because of the large magnitudes of the total EMF at the Earth’s surface (40,000– 70,000 nT) and the necessity to measure its variations with high accuracy (up to 0.1 nT), it is almost impossible to carry out measurements with a single instrument. The modern observatory protocol specifies the carrying out of the measurements using two different methods and different instruments, which are being improved constantly (Jankowsky and Sucksdorf 1996; Rasson 2007). The absolute measurements of the basic EMF elements are carried out mainly by the scalar magnetometer and a fluxgate inclinometer/declinometer (DIflux). These instruments provide the required accuracy; however, DIflux instruments are still operated manually, which requires about 30–40 min per one measurement series. Such measurements are normally carried out once or twice a week by a highly qualified observer. The absolute measurements are an essential part of observatory operations, whereas the world magnetic observatories total roughly 50% of all stationary sites of the geomagnetic field monitoring. Most modern ground- and satellite-based instruments for measuring the geomagnetic field vector are divided into two broad categories: (a) magnetometric sensors measuring the field along one or several axes using the ferromagnetic properties of some materials—fluxgate vector magnetometers, and (b) magnetometric sensors based on the quantum-mechanical properties of liquid and gas for measuring the magnetic field modulus—scalar magnetometers. The magnetometer family also includes the induction coil magnetometers that record the field variations at the higher time frequencies, and cryogenic magnetometers with a superconducting quantum interference device (SQID), mainly used in the gradiometry and paleomagnetic studies.

2.1 EMF Measurements

31

In a vector magnetometer, each direction is determined by a sensor consisting of highly permeable ferromagnetic core, an excitation coil, and a measuring (more sensitive) coil. When the alternating current is passed through the excitation coil, the core is alternately magnetized in the opposite directions, which, in turn, causes an electromotive force in the receiving coil. Since the magnetization curve of the ferromagnetic material is non-linear, the output signal doesn’t have exactly the same frequency characteristics as the input field when adding a nonzero ambient field along the sensor axis. This property is used to determine the magnitude of the ambient field. The long-term stability is an important requirement for using the vector magnetometers at the magnetic observatories. At present, the global observatory network INTERMAGNET (http://intermagnet.org) requires the vector magnetometer drift to be not greater than 5 nT/year at all observatories. The drift affects the sensitivity of the instrument, the misplacement (i.e. the output if a zero field exists) and the direction of the sensor axis. It can be caused by various small changes in the electronics including analog-digital converter, coil system, or even the sensor material itself. Some of these changes are related with the environment temperature, since all sensor materials (core, coils, etc.) are subject to thermal expansion, and the temperature affects the electric conductivity. The temperature dependence can be mitigated to some extent by adding the temperature feedback circuit. The best observatory fluxgates for today have the temperature coefficients of about 0.2 nT/o C. Another external source of instability is a slow motion of the instrument itself, for example, caused by a pillar inclining. The noise level is another important feature of the magnetometer, since it sets a limit for the smallest and fastest geomagnetic signal that can be registered. Before, the requirements in this respect were less strict for the main magnetic field observations and stricter for the space observations. However, since more and more observatories have√already started recording 1-second data, the noise level should not exceed 10 pT/ Hz in order to measure accurately the geomagnetic activity in this part of the spectrum. The scalar measurements of the geomagnetic field are usually made using a proton precession magnetometer, Overhauser magnetometer, or optically pumped magnetometer. These three types of the magnetometers combine the use of the quantummechanical properties of liquids or gas and measurement of the frequency linearly related to the module of the ambient field. The scalar magnetometers can provide very precise measurements (with the uncertainty less than 0.5 nT), and they are not exposed to the long-term drifts. For this reason, they are used as the absolute reference at the magnetic observatories and on board the geomagnetic satellites. They are also widely used in the aeromagnetic and ground surveys. The low sampling rate is a serious limitation for the proton precession magnetometers. In the magnetometers based on the Overhauser effect, magnetization is enhanced by adding free electrons (for example, in the free radicals) to a liquid and applying the suitable radio frequency radiation that saturates the selected energy levels. This leads to a higher sampling rate, lower power consumption and fewer scattered fields in the immediate vicinity of the instrument. A modern device can achieve the sensitivity of 10–20 pT in the second measurements. Due to their good

32

2 EMF Observations and Data Processing

characteristics and relative availability, the Overhauser magnetometers have become the preferable absolute instruments at the observatories, and reference instruments on the satellites. In the optically pumped magnetometers, the sensory fluid is replaced by the gas, helium or alkaline vapor, such as cesium, rubidium, or potassium. The optical pumping is used to enhance the effects of electron magnetic resonance by raising the electrons to the higher energy levels. A resonance occurs between the sublevels created by the ambient magnetic field and is induced by the radio frequency radiation. Since the pump light is transmitted through a sensor cell containing gas, its intensity changes are used to detect a resonance. The optically pumped magnetometers, in general, provide higher frequency measurements and lower noise level than the Overhauser magnetometers, which makes them suitable for measuring on the fast-moving platforms, such as airplanes and satellites. The absolute scalar magnetometer of this type is installed on the modern Swarm geomagnetic mission satellites, √ which are currently operational. This instrument achieves a resolution of 1 pT/ Hz over a DC to 100 Hz bandwidth, and its final corrected scalar accuracy has been demonstrated to be better than 50 pT (Léger et al. 2009). The absolute values and secular variation of the EMF are controlled using a DIflux magnetometer. It represents a combination of the one-axial fluxgate magnetometer with a demagnetized theodolite. The measurements are carried out manually by a trained observer. They consist in determining the magnetic declination D, i.e. the angle between the geographic north direction and horizontal component of the geomagnetic field, and the magnetic inclination I, i.e. the angle between the horizontal plane and the geomagnetic field vector. D is measured using the reference azimuthal mark (mira) with the precisely determined azimuth angle and is obtained by determining the angle when the field is zero in the horizontal plane. I is obtained by determining zero field in the previously determined plane of the magnetic meridian. Here, two methods can be used: null and offset. Although there are many sources of errors, the experienced observers can carry out the consecutive measurements over a long time period and achieve accuracy better than 5 arc seconds, which corresponds to the intensity accuracy of 1 nT at the mid-latitudes. At the magnetic observatories, all three types of the magnetic measurements are available to ensure the most accurate observations in a wide range of the time scales. In the remote areas, the magnetic stations are deployed; they are quipped only with the automatically operating vector magnetometers for recording the variations of the external field. In the marine and airborne magnetic studies, the scalar magnetometers are towed or aerotowed above the selected oceanic and land areas to reconstruct the small spatial scales of the lithospheric field that cannot be detected from the satellites. For each of these observational methods, the observation standards and data distribution are agreed at the national or global level and are constantly developed due to the new and improved technologies.

2.1 EMF Measurements

33

The modern geomagnetic observatories represent the globally distributed complex facilities providing accurate monitoring of the geomagnetic field at the fixed sites over significant time intervals. The final data product represents 1-minute (1-second at a growing number of modern observatories) and hourly complete values of the geomagnetic field components. The frequency range of the measured geomagnetic variations is typically from 1 to 10−5 Hz, and the amplitude varies from the fractions of nT to several thousand nT. Thus, the absolute values of the magnetic field are essential for studying the internal EMF dynamics. The view of the magnetic observatory pavilion and measuring instruments are shown in Fig. 2.2. Magnetic stations represent another class of the geomagnetic measuring facilities. The can operate for a long time in the automatic mode, however stations measure only the deviations of the EMF vector components from some fixed, and, in general, unknown value. If at a certain time there are simultaneous absolute and variation

Fig. 2.2 Non-magnetic pavilion for the absolute measurements of the observatory (above). The small opening in the right window is for observing the azimuth mark (mira), and the pipe in the upper right corner is for observing the Polar star. The DIflux instrument, vector and scalar magnetometers are given below

34

2 EMF Observations and Data Processing

measurements, then subtracting the variation measurements from the absolute values gives the complete values of the geomagnetic components corresponding to the zero values of the vector magnetometer. These values are called the baseline values of the relevant components. As long as these baseline values remain unchanged, it is possible to obtain the complete values of the magnetic field components by adding baseline values to the values, constantly recorded by the vector magnetometer. In practice, it is very difficult to ensure the stability of the station baseline values, so they have to be periodically controlled by carrying out the absolute measurements of the EMF components and introducing some corrections. The variation values themselves, even not knowing the baseline values, are still essential for solving various problems of space physics and industrial sector, including the studies of the external ionospheric and magnetospheric sources of the magnetic field and geomagnetically induced telluric currents. Thus, the geomagnetic measurements are carried out at the observation sites of two classes - full-scale observatories with a complete set of instruments for measuring the magnetic variations and absolute values of the magnetic field, having trained staff to perform these measurements, and the magnetic stations, providing geomagnetic variation measurements only. The latter can be attended and unattended, capable to operate autonomously for up to a year and more. The introduction of the new digital technologies allowed to increase the accuracy of measurements and automate a part of the processes of the data acquisition and processing. If several decades ago, the measurement error within 5–10 nT (about 0.015% of the total field and about 1% of typical strong disturbances) was considered satisfactory, now the error makes 0.1–0.5 nT (about 0.001% of the total field and about 0.1% of typical strong disturbances). The magnetic observatories and stations should be located in places with a small spatial gradient of the magnetic field, which indicates the absence of the local magnetic anomalies, and a low level of the industrial interferences. The most strong source of interferences in the magnetic field is the electrified transport, so the interferences from the direct current railway is noticeable at the distance of up to 20–30 km from it, the tram produces electromagnetic noise within a radius of 5–10 km. At that, the interferences are generated mainly by the spreading currents in the upper layers of the Earth, and their distribution depends on the conductivity structure of the soil. Thus, the value of the interferences at the site of proposed construction of a new observatory is specified experimentally. One of the main problems the observatories encounter worldwide is an increase of the electromagnetic pollution of the environment, in particular, because of urbanization and emerging anthropogenic interferences, which leads either to observatory shutdown or its transfer to a new site disrupting the continuity of the observation series. According to the modern requirements, the level of the industrial noise at the site of measuring the EMF variations should not exceed 0.1 nT, and in the near future, taking into account the quality improvement of the magnetometers, this threshold will be reduced to 0.01 nT.

2.1 EMF Measurements

35

INTERMAGNET. The largest coordinated network, which includes the geomagnetic observatories performing a full cycle of the measurements according to the highest international quality standards, is the International Real-time Magnetic Observatory Network, INTERMAGNET (Kerridge 2001; Love 2008; Love and Chulliat 2013). INTERMAGNET project (http://www.intermagnet.org) was founded in the late 1980s as a voluntary international association of the observatories being a part of the national networks and meeting the agreed criteria in terms of the measurements, data transfer and processing. A necessary condition of fulfillment of INTERMAGNET standards by the observatories is the quality, stability and continuity of the measurements, as well as transmission of geomagnetic data in quasi real-time mode to the dedicated data centers (Geomagnetic Information Nodes, GIN). At present, INTERMAGNET unites almost 60 institutions from 40 countries supporting more than 130 observatories, and their number grows constantly (Fig. 2.3). Nevertheless, a long-standing constraint of the magnetic observatories is their uneven geographical distribution. While some regions (for example, Europe, North America, Australia, parts of Asia) are covered by the dense networks of the observatories, the other ones (for example, Africa, Pacific Ocean) have extensive gaps. In the recent decades, the obstinate efforts have been made to upgrade old and install new observatories in the remote places. Nevertheless, despite these efforts, the global geographic coverage of the observatories remains almost constant since the International Geophysical Year, held in 1957–1958. One way to improve the geographic coverage would be designing and installing the fully automated observatories in remote locations, as well as at the seafloor.

Fig. 2.3 Map of the INTERMAGNET observatories and their IAGA-codes as of 2020

36

2 EMF Observations and Data Processing

Table 2.1 Russian geomagnetic observatories Observatory name Code

Latitude Longitude INTERMAGNET GIN certificate

Data transmission interval

Arti

ARS

56.4

58.6

+

BGS

30 min

Borok

BOX

58.1

38.2

+

IPGP Day

Cape Schmidt

CPS

68.9

−179.4

−

−

Day

Irkutsk

IRT

52.2

104.5

+

BGS

Day

Kazan

KZN

55.9

48.7

−

−

15 min

Khabarovsk

KHB

47.6

134.7

+

BGS

Day

Klimovskaya

KLI

60.9

39.5

−

−

10 min

Magadan

MGD 60.1

150.7

+

BGS

3h

Mikhnevo

MHV 54.9

37.7

−

−

10 min

Novosibirsk

NVS

54.9

83.2

+

BGS

Day

Paratunka

PET

53.0

158.3

+

BGS

Day

Saint-Petersburg

SPG

60.5

29.7

+

IPGP Day

Vostok (Antarctic) VOS

78.3

106.5

+

KYO 30 min

White Sea

WSE

66.6

33.1

–

–

30 min

Yakutsk

YAK

62.0

129.7

+

BGS

Day

Monitoring of the EMF is especially important in the high-latitude areas where the external part of the magnetic field is very variable. In the Russian sector of the Arctic region, the great efforts have been and are being made to develop an observational network, arrange the information collection centers and online data analysis (Gvishiani et al. 2014). At present, there are 15 geomagnetic observatories operating in Russia (Table 2.1), equipped with the instrumentation meeting the INTERMAGNET requirements, as well as having the capability of online data transmission. The Russian INTERMAGNET segment includes 10 certified magnetic observatories. Each observatory transmits the geomagnetic data to one of five Geomagnetic Information Nodes, which are INTERMAGNET data collection centers in real and nearreal time. The GINs communicates with the magnetic observatories via the satellite, computer or cellular networks. The preliminary (real-time) data of the geomagnetic observations are transmitted from the observatories to the GINs within 72 h. These are the unprocessed magnetograms that may contain the baseline inaccuracies. They may also contain spikes and jumps (artificial disturbances) or missing values. The preliminary data are available to the users with a very short delay. A new type of the geomagnetic data, quasi-definitive data, was introduced ten years ago (Peltier and Chulliat 2010). Such magnetograms are corrected using the temporary baselines and contain no artificial disturbances and gaps. The difference between the definitive and quasi-definitive magnetograms should be within 5 nT. The definitive data are cleared of the interferences and represent continuous time series of the complete values of the three orthogonal components of geomagnetic field. These data sets are produced

2.1 EMF Measurements

37

Fig. 2.4 Map of the geomagnetic information nodes of the international real-time magnetic observatory network INTERMAGNET. The oval areas present the coverage area of the geostationary communication satellites for transmitting the data to the GINs from the observatories (http://interm agnet.org/)

by the employees of the observatories and GINs on a yearly basis and sometimes less often, which makes substantial delay in their availability. An independent group of INTERMAGNET experts checks the data annually. Both, preliminary and definitive data are available on the official INTERMAGNET web-site. At present, the INTERMAGNET GINs are located in the following cities (Fig. 2.4): – – – – –

Golden (Colorado, USA), operated by US Geological Survey; Kyoto (Japan), operated by Kyoto University; Ottawa (Canada), operated by Canadian Geological Survey; Paris (France), operated by Paris Institute of Earth Physics; Edinburg (Great Britain), operated by British Geological Survey.

Magnetometer network. Due to the global nature of the Earth’s magnetic field, it is crucial to ensure the most uniform and dense geographical coverage of the globe with the observation sites, along with data collection, storage, access and exchange. There are several formal and informal associations in the world for storing the geomagnetic data obtained at the magnetic observatories and stations. Many stations are united into the networks and chains of different scales and forms depending on the target scientific applications. Despite the unofficial status, these associations develop in a coordinated manner. A chain of the magnetometers along a magnetic meridian is a common configuration. It enables studying various magnetosphericionospheric processes, such as, geomagnetic storms, substorms, auroral and equatorial ionospheric electrojets, geomagnetic pulsations, and other manifestations of the space weather. The examples of such projects are IMAGE, SAMBA, WAMNET, MACCS, AMBER. The magnetometer arrays in both hemispheres can be used to

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2 EMF Observations and Data Processing

study the geomagnetic activity at the conjugate points of the opposite hemispheres (for example, MACCS, GMA, PENGUIn, BAS). Until recently, each magnetometer network project distributed its data by its own, for example, via a website. Around ten years ago, the SuperMAG joint project was started (Gjerloev 2009, 2012), which began distributing the data from the different arrays in a uniform manner via a single website (http://supermag.jhuapl.edu). Now, SuperMAG includes the data of about 300 magnetometer stations and observatories, including all INTERMAGNET observatories. After receiving the data, SuperMAG provides anthropogenic noise removal, data conversion to the uniform local coordinate system (local magnetic north, east and vertical direction) and time resolution (one minute) and baseline subtraction, including the long-term trends and daily variations. The whole procedure is aimed at facilitating the use of the data from the various arrays of the magnetometers for studying the ionospheric and magnetospheric current systems. This is a markedly different approach than the one used by INTERMAGNET, which does not require any data conversion from its member observatories, but, instead, focuses on assisting the observatories to meet the strict data quality standards and provide the data in the near-real time mode. During the International Geophysical Year in 1957–1958, an agreement was reached on development of the World Data Centers (WDC), which became the first depositories for geomagnetism. Particularly, two WDCs for Solar-Terrestrial Physics and Solid Earth Physics were established in Moscow (http://wdcb.ru/), where the data from the geomagnetic observatories and stations located on the territory of the USSR, as well as the data from many foreign observation sides, are being accumulated. These data depositories also continue developing worldwide.

2.1.2 Aerial and Marine Magnetic Surveys The magnetized rocks in the Earth’s lithosphere generate the magnetic fields with a wide range of spatial scales, from thousands of kilometers to several meters. With the scales exceeding 3,000 km, the lithosphere field is overlapped with a more intense core field and is mainly unknown. Within the intermediate scales (from 3,000 to 250–300 km), the lithospheric field dominates and can be mapped at the global level according to the data of the low-orbit satellites. At the scales less than 250 km, the lithospheric field becomes too small to be detected at the satellite altitude using the existing instruments and field modeling technology. The measurements on the surface are necessary to recover this part of the spectrum. There are two types of such measurements: the marine magnetic survey, when the magnetometer is towed or attached to a ship or submarine, and the aeromagnetic survey, when the magnetometer is attached to a flying object, such as an airplane, helicopter, balloon or unmanned aerial vehicle (UAV). The scales of only few hundred meters can be recovered by the aerial surveys at the low altitude. Even smaller scales can be recovered using other means, such as cycling or walking surveys, but the geographical coverage of such measurements is very limited.

2.1 EMF Measurements

39

Usually, the scalar magnetometers ensuring the accuracy of up to 1 nT are used. The survey accuracy also depends on how well the position of the magnetometer is determined. The appearance of the Global Navigation Satellite System (GNSS) led to a significant improvement in the positioning and navigation systems with the errors less than a few meters horizontally and vertically when using the singlefrequency receivers. The aerial surveys usually performed along the regular flight profiles with the interval of 50–500 m and perpendicular connection lines with the ten times larger distance. A base station (for example, a nearby observatory or a temporary station specially installed for this purpose) is often used to provide a reference record of temporal changes of the magnetic field in the survey area, which allows to accurately take into account the external magnetic field signal and correct the survey data accordingly. A scalar magnetometer towed by a ship is used for the marine magnetic surveys. Usually a tow rope of up to 200 m long is required to avoid contamination of the survey data with the ship’s magnetic field. A large contribution to marine geomagnetism was made by the measurements of the Soviet non-magnetic schooner “Zarya”, which are still used at present. At that time, the main source of the field measurement errors was a positioning error, which could achieve 100 m during the surveys preceding the appearance of GNSS. Recently, some surveys involve the vector magnetic measurements combined with the inertial measurements of orientation to improve interpretation of the magnetic anomalies, as well as the deep-submergence vehicles.

2.1.3 Satellite Based Measurements Besides the ground based facilities, the artificial Earth satellites are used for studying the geomagnetic field, having a low (400–800 km) orbit lying in the meridional plane and crossing the polar regions of the globe. The satellite observations of the Earth’s magnetic field began in the 1960s. The world’s first satellite magnetic survey was carried out by the USSR in 1964– 1965 (“Cosmos-36” and “Cosmos-49” missions) and in 1970 (“Cosmos-321”) (Krasnoperov et al. 2020). At different times there were several satellites equipped with the magnetometers on the polar orbit: Triad (1972, USA), Magsat (1978– 1979, USA), DMSP (a series of satellites since 1979 up to the present day, USA), Interkosmos ICB-1300 (1981–1983, USSR), ST5 (2006, USA), SAC-C (2000–2010, Argentina). They carried out measurements of the magnetic field along particular pass trajectories, and these measurements were fragmental. At the turn of the century, the European satellites Oersted and CHAMP of new generation were launched; they were designed for regular spatiotemporal measurements of the geomagnetic field and study of the internal and external field sources. The constellations of the low-orbit spacecrafts measuring the geomagnetic field (past and current projects) are schematically shown in Fig. 2.5, and the characteristics of magnetometric instrumentation are given in Table 2.2.

40

2 EMF Observations and Data Processing

Fig. 2.5 Constellations of low-orbit spacecrafts measuring the geomagnetic field (past and current projects). The numbers indicate the amount of the spacecrafts involved in each project

Table 2.2 Characteristics of the satellite magnetometers Name of Orbit the altitude mission (km)

Weight Magnetometer Measurement Measurement Number of (kg) type range (nT) accuracy (nT) measurements per minute

Magsat

350–550

182

Scalar, vector

±64,000

0.5 (sc) 3 (v)

10

SAC-C

707

485

Scalar, vector

±65,500

1 (sc) 2 (v)

100

DMSP

830

1220

Vector

±10,000

2

10

ST5

300–4500

25

Vector

±10,000

0.1

16

Iridium

780

680

Vector

±10,000

48

0.005

522

Scalar, vector

±64,000

0.5 (sc) 2 (v)

32 (64)

61

Scalar, vector

±65,500

0.5 (sc) 1 (v)

20 (100)

CHAMP 350–450 Oersted

655

The Oersted satellite (Denmark), equipped with a triaxial magnetometer for absolute measurements of the magnetic field with an accuracy of 5 nT and instruments for accurate orientation of the magnetometer in space, was launched in 1999 into the polar orbit with the perigee of 650 km and apogee of 850 km. The satellite worked actively until 2001, and then continued carrying out observations in a limited mode. In the meridional plane, its orbit shifted gradually in local time at the speed of ~1 min

2.1 EMF Measurements

41

per day. Thus, the magnetic data of many thousands of passes above all longitudinal sectors of both hemispheres were obtained over the satellite lifetime. The European satellite CHAMP, launched in 2000 and worked for more than ten years, also made a great contribution to the Earth’s magnetic field studies. The low orbital height of the CHAMP satellite (~200 km) allowed, in addition to measuring the large-scale main field and geomagnetic variations, to monitor efficiently the lithospheric field inhomogeneities. Besides, the CHAMP satellite was equipped with an accelerometer to measure the motion speed of the neutral gas component in the thermosphere, which made it possible to evaluate the variations of the thermosphere density. Processing of the obtained data sets of the magnetic field measurements above the ionosphere carried out by CHAMP and Oersted made a valuable contribution to the improvement of the models of the Earth’s main magnetic field and magnetic anomalies. The magnetic field variations caused by the external sources were also specified and analyzed, which led to the appearance of the qualitatively new models of distribution of the field-aligned currents flowing in the magnetosphere-ionosphere system. Since the late 1990s, the spacecrafts of the Iridium satellite communication system, covering 100% of the Earth’s surface, including the polar regions, have been used to measure the magnetic field variations above the ionosphere. The Iridium orbital constellation, launched by Japan for the Motorola commercial telecommunication project and then transferred to the United States, includes 66 satellites revolving round the Earth along 11 orbits at the altitude of 700–800 km; one complete revolution takes about 100 min. Although the magnetic measurements are not the main task of Iridium, all satellites are equipped with the low-sensitive engineering magnetometers with the resolution of about 50 nT. In the framework of the data usage for the scientific purposes, the United States arranged monitoring, although with low accuracy, of the magnetic field variations above the ionosphere. It is assumed that the new Iridium satellites will be equipped with more sensitive magnetometers. Modern constellation of Swarm geomagnetic satellites. At present, the unique conditions have formed for further development of the geomagnetism research. This is specified by the fact that the ground-based observations are supplemented by the space observations performed by three satellites, which are combined into the Swarm constellation and equipped with the high-precision magnetometers and other scientific instruments. The Swarm project of the European Space Agency (ESA) is designed for mapping the EMF and studying the mechanisms of its formation (FriisChristensen et al. 2006, 2009; Swarm (Geomagnetic LEO Constellation) 2019). The three identical satellites of the Swarm constellation operate at a polar orbit (Fig. 2.6). The satellites were launched in November 2013 as part of the Earth exploration program. The Swarm satellites operate in two different near-polar orbital planes that provide global coverage of the Earth. Two satellites (Swarm A and C) were launched into a circular orbit with the inclination of 87.4° at the altitude of 450 km. The third satellite (Swarm B) has an orbit with the inclination of 88° and the altitude of 530 km. Four years after the start of the mission, this resulted in a difference of nine hours local time between the Swarm A/C and B satellites (Fig. 2.7). Gradually shifting along the longitude, the orbits of the satellites cover all local time sectors of the

42

2 EMF Observations and Data Processing

Fig. 2.6 SWARM satellites on the orbit (ESA)

Fig. 2.7 Evolution of the orbits of the Swarm constellation satellites (A and B are the spacecrafts with the orbit altitude of 450 km, C is the spacecraft with the orbit altitude of 500 km) and the estimated change in the longitude (local time) of the orbits over 5 years of operation

globe. Originally, the estimated duration of Swarm operation was 4.5 years, but this period has been significantly exceeded. The payload of the three satellites is completely identical. It includes the vector field magnetometer (VFM), absolute scalar magnetometer (ASM), electric field instrument (EFI) and auxiliary equipment (accelerometer, GPS receivers, star laser trackers). The measurements of the geomagnetic field vector with the frequencies of 1 and 50 Hz are supplemented with the accurate observations of plasma and electric field using navigation and accelerometer (Olsen et al. 2013). The characteristics of the satellite magnetometers are given in Table 2.3. The continuous geomagnetic recordings made by three Swarm satellites include the values of three orthogonal components of the geomagnetic field vector in the VFM coordinate reference system and NEC (North-East-Center) reference system, values of the total intensity of the geomagnetic field, UTC time stamps and geographical coordinate reference of the recorded values with the sampling frequency of 1 and 50 Hz. They are stored in the online archive in the binary data format (Common Data

2.1 EMF Measurements

43

Table 2.3 Magnetometers of the Swarm satellites Instrument name

Measured values

Measurement accuracy and dynamic range

Fluxgate vector magnetometer (VFM)

X, Y, Z components of the geomagnetic field

0.5 nT, ±65,536.0–0.0625 nT

Proton absolute scalar magnetometer (ASM)

Total field intensity F

0.3 nT, 15,000–65,000 nT

Format, CDF). Access to the 1st level data (calibrated and verified instrument data) and 2nd level data (data products) is provided to the users via the FTP server. The primary task of processing spaceborne geomagnetic data is to separate the magnetic field from the sources inside the Earth and the magnetic field of the electric currents flowing in the ionosphere-magnetosphere system. Figure 2.8 presents an example of recordings of the satellite vector magnetometer with a flight trajectory passing through the polar zone of the northern hemisphere from the dawn to the dusk side. Here, three orthogonal components of the magnetic induction B are shown: δBx is measured along the east-west direction (i.e., approximately across the flight trajectory), δBy is measured along the meridian (along the trajectory), and δBz is measured along the normal to the surface of the ionospheric shell. In the δBX component, the large-scale variations with a change in the sign of the field gradient can be seen, indicating the current sheet cross-over, as well as the small-scale fluctuations superimposed on them. The variations of smaller amplitude are also observed in the meridional δBY component of the measured field.

Fig. 2.8 Example of the satellite flight trajectory through the polar region of the northern hemisphere (left) and the variations of three orthogonal components of the geomagnetic field δBX , δBY and δBZ measured by the satellite magnetometer (right)

44

2 EMF Observations and Data Processing

2.2 Indices of Geomagnetic Activity 2.2.1 Overview Regular diurnal variations of the magnetic field are caused mainly by the Earth’s ionospheric currents that emerge due to illumination of the ionosphere by the Sun. Irregular variations in the magnetic field are induced due to the impact of the unstable solar plasma flow (solar wind) on the Earth’s magnetosphere, further processes within the magnetosphere, and the magnetosphere and ionosphere coupling. Indices of geomagnetic activity are a quantitative measure of geomagnetic activity caused by irregular sources (Lincoln 1967). All indices are derived from the ground-based measurements of the magnetic field and published in UT time format. The book of (Mayaud 1980) is one of the most comprehensive monographs on the subject, which provides a description of all geomagnetic activity indices suggested by that time. Index calculation involves data from unevenly situated networks of magnetic stations (Love and Remick 2007), thus the responses of various magnetosphericionospheric current systems are taken into account. Figure 2.9 depicts allocation of five networks of ground stations that provide data for calculating different indices. The existing geomagnetic activity indices (http://www.ngdc.noaa.gov/IAGA/ vdat/) are divided into three groups: 1. regional indices giving local estimation of geomagnetic disturbance over a territory, such as C, K, aK , r H or Q; 2. planetary indices reflecting geomagnetic activity all over the Earth, such as C i , C p , C 9 , K p , K m , K s , K n , ap , Ap , am , Am , aa and Aa; 3. source driven indices describing magnetic disturbance intensity of a well-defined origin, for instance: – Dst (disturbance storm time) index characterizes the intensity of the symmetric part of the equatorial ring current, – AE (auroral electrojet) index estimates the disturbance in the aurora oval, – PC (polar cap) index is derived from the geomagnetic data of the diametrically opposite Vostok or Thule observatories and satellite data and characterizes the perturbation in polar caps.

Fig. 2.9 Observatories and stations that provide data for calculation of AE, K p , am , Dst and aa indices (http://isgi.unistra.fr/)

2.2 Indices of Geomagnetic Activity

45

Fig. 2.10 Zoning in the magnetic coordinate system: northern zone of auroras (gray color), subauroral zone (green color), low latitude zone (pink color with hatching). The map shows the observatories used for calculating the indices: K p (green squares), AU, AL, AE, AO (yellow circles) and Dst (red triangles)

There are more than 20 indexes in total, some of them are officially accepted by IAGA (International Association of Geomagnetism and Aeronomy). Each geomagnetic activity index is based on data recordings from a defined set of observatories (Figs. 2.9, 2.10). Traditionally, each index should fulfill a number of requirements: 1. 2. 3. 4. 5. 6.

clear physical meaning; clarity and ease of use; detailed definition of disturbance by amplitude; detailed definition of disturbance by time; timeliness of becoming available; long duration of the calculated time series.

Currently, there is not a single index that would simultaneously satisfy all these requirements to the same extent; each index has some advantages over others. When using classical geomagnetic activity indices, it is worth remembering that originally the input data were analog magnetograms recorded at a tape drive speed of about 20 mm/h. Thus, the magnetic field disturbances in this case are in the frequency range between 5 × 10−3 and 10−8 Hz, which corresponds to periods from several minutes to a year.

46

2 EMF Observations and Data Processing

It might be assumed that with sufficient statistics of magnetic storms there should be a correlation between the extreme values of various indices. Such an analysis was carried out for 1085 magnetic storms over 1957–1993 (Loewe and Prolss 1997). On the other hand, such results can lead to the illusion that magnetospheric indices are interchangeable. However, the very first attempts to analyze real data demonstrated the non-identical behavior of different indices during the same event. For example, in the time interval 15–23 UT on 24 October 2003 (Veselovsky et al. 2004), with the noted high values of the Kp index, the Dst index remained at the usual level. Dependence of the extreme values of the K p index on the extreme value of the Dst index for 611 magnetic storms (–300 < Dst < –60 nT) over 1976–2000 is presented in Fig. 2.11 (Yermolaev and Yermolaev 2003). The large scatter of the data is explained by the fact that the K p and Dst indices are measured at different geomagnetic latitudes and are sensitive to different current systems: auroral electrojet (magnetic substorms) and ring current (magnetic storms). Thus, in order to study the relationship of magnetic storms with various phenomena and to exclude auroral phenomena from the analysis, the Dst index should be used. On the contrary, the dedicated AE index is used for studying the influence of the auroral electrojet on various systems. The K p index is sensitive to both phenomena and does not allow to study separately the influence of each current system (Yermolaev and Yermolaev 2010). For the correct use of geomagnetic activity indices as applied to related disciplines, it is necessary to generally understand the principles of their construction and physical meaning, their interdependence and value ranges that correspond to different levels of geomagnetic activity (Yermolaev and Yermolaev 2010). More information can be found in (Mayaud 1980; Loewe and Prolss 1997). Often, the most in-depth and detailed studies of the morphological features of the geomagnetic disturbance distribution around the globe are related to the analysis of the entire set of initial observational data, and not to the analysis of indices. M. Bobrov, studying the records of 72 stations during the International Geophysical Year (1957–1958), found that almost all geomagnetic disturbances are of a planetary Fig. 2.11 The relation of the extreme values of the K p and Dst indices for 611 magnetic storms with—300 < Dst < –60 nT in the period 1976–2000. The solid line approximates the data presented (Yermolaev and Yermolaev 2003)

2.2 Indices of Geomagnetic Activity

47

nature (Bobrov 1961). Consequently, the study of magnetic disturbances from the materials of a small number of observatories, covering limited intervals of latitudes and longitudes, cannot give a complete picture of the phenomenon. Geomagnetic activity is the most direct expression of the impact of solar corpuscular streams on the Earth’s atmosphere. Therefore, the structure of the corpuscular flows should be reflected in certain typical features of geomagnetic disturbances. The researcher should have an easily visible, general planetary picture of the disturbance under study using data recorded at various latitudes and longitudes over a given period (Bobrov 1961). Below, we give a more detailed description of the K and K p indices, which are most commonly used by the researchers to quantify geomagnetic activity at the regional and planetary scales, respectively.

2.2.2 K and Kp Indices: Background, Calculation and Applications The first and the simplest C index was introduced in 1906 to estimate magnetic field activity over a day on a three-point scale (0, 1 and 2). For each day, an expert at each observatory carried out the estimation visually: the quietest daily records were evaluated by 0 (Fig. 2.12a), the most disturbed ones by 2 (Fig. 2.12b). In order to avoid the subjective nature of the estimation, J. Bartels proposed planetary C i index to replace C index. It is calculated as the arithmetic mean of the C indices of approximately 30 geomagnetic observatories, and therefore it is more objective as compared to the C index. C i index values are calculated with an accuracy of one tenth and they can take values from 0.0 to 2.0. Thus, the C i index enabled estimation of the magnetic activity globally. However, both C and C i indices had a very low daily time resolution, whereas many studies require more frequently defined geomagnetic activity measure, devoid of any subjectivity. Therefore, in 1939 IAGA decided to introduce K index of a tenpoint scale in magnetic observatory practice, which was also proposed by Bartels (Bartels et al. 1939). It characterizes the variability of geomagnetic activity at a given observatory over 3-hour intervals starting from 00:00 UT. The variability is understood as the amplitude of the observed horizontal field relative to the quiet daily variation (Sq field). The K index takes values from 0 to 9, where 9 points correspond to the strongest geomagnetic disturbance. Considering that the amplitude of the geomagnetic disturbance depends on the latitude of the observation site (the maximum amplitude is observed in the aurora zone), a value of K = 9 corresponds to a disturbance greater than 2500 nT in the aurora zone and about 300 nT at low latitudes (excluding the equator). For all other observatories located in the polar cap and middle latitudes, a value of K = 9 corresponds to disturbances smaller than 2500 nT but larger than 300 nT (see Table 2.4). Specifically, for each observatory, a correspondence between the value K = 9 and the disturbance amplitude was obtained

48

2 EMF Observations and Data Processing

Fig. 2.12 Quiet daily variations on 17 March 2002 (C = 0) (a) and strong magnetic storm on 6–7 April 2000 (C = 2) (b) at Irkutsk magnetic observatory

by considering the extremely strong geomagnetic disturbance that was observed on 16 April 1938. It was agreed that on that day between 06:00 and 09:00 UT the K index at all observatories was 9 points, and the maximum value of the disturbance amplitude over the considered interval was taken as the lower limit of the amplitude corresponding to K = 9. Originally, when only analog magnetograms were in place, geomagnetists were confined to determine geomagnetic disturbance amplitudes and consequently K index values by hand (Fig. 2.13). As the digital registration of the magnetic field was introduced, the necessity for automated K calculation immediately arose. Currently, there are several officially accepted algorithms for derivation of K index: 1. FMI-method for K indices computation, provided by Finnish Meteorolical Institute, Finland; 2. Hermanus algorithm, using the linear-phase robust non-linear smoothing method, provided by Hermanus Magnetic Observatory of the Council for Scientific and Industrial Research, South Africa (Hattingh et al. 1989); 3. KASM method for K indices computation using the adaptative smoothing method, provided by the Institute of Geophysics, Polish Academy of Science, Poland (Nowo˙zy´nski et al. 1991);

0−25

0−20

0−18

0−15

0−12

0−10

0−8

0−6

0−5

0−5

0−4

0−3

2

3

4

5

6

7

8

9

10

11

12

0

K index

1

Scale

3−6

4−8

5−10

5−10

6−12

8−15

10−20

12−25

15−30

18−36

20−40

25−50

1

6−12

8−16

10−20

10−20

12−24

15−30

20−40

25−50

30−60

36−72

40−80

50−100

2

12−24

16−30

20−40

20−40

24−48

30−60

40−80

50−100

60−120

72−144

80−160

100−200

3

24−40

30−50

40−70

40−70

48−85

60−105

80−140

100−175

120−210

144−252

160−280

200−350

4

40−70

50−85

70−120

70−120

85−145

105−180

140−240

175−300

210−360

252−432

280−480

350−600

5

70−120

85−140

120−200

120−200

145−240

180−300

240−400

300−500

360−600

432−720

480−800

600−1000

6

120−220

140−230

200−330

200−330

240−400

300−500

400−660

500−825

600−1000

720−1188

800−1300

1000–1650

7

200−300

230−350

330−550

330−550

400−600

500−750

660−1000

825−1250

1000−1500

1188−1800

1300−2000

1650−2500

8

Table 2.4 Correspondence of geomagnetic disturbance amplitudes to K index values depending on the latitude of observation 9

>300

>350

>500

>550

>600

>750

>1000

>1250

>1500

>1800

>2000

>2500

0−40

31−47

36−57

48−54

55−58

58−62

60−83

62

60−83

80

65−80

64−90

Lat

2.2 Indices of Geomagnetic Activity 49

50

2 EMF Observations and Data Processing

Fig. 2.13 Determination of K (purple bars) and K p values by hand

4. USGS method for K indices computation, provided by US Geological Survey, USA (Wilson 1987). The comparison of the K index computer production can be found in (Menvielle et al. 1995). Herein, we give a brief description of the KASM algorithm (Nowo˙zy´nski et al. 1991), as the one widely used by the members of the worldwide magnetic observatory network INTERMAGNET (International Real-time Magnetic Observatory Network) (Love and Chulliat 2013). It is designed to deal with 1-minute digital data. For the determination of the K indices, it is necessary to determine and remove the regular S q part from the record (see Fig. 2.13). For the S q estimation, no any physical assumptions are used. The general philosophy of the method is the simulation of the hand-scaling procedure. The simple theory, based on the least-squares method with additional limitation of the second derivatives, is applied. For a quiet day, the smoothing coefficient, which is introduced to achieve a smooth S q curve, is chosen to give the best approximation of the hand-made S q curve. When treating a partly quiet and partly disturbed day, the weighting factors are introduced for making the influence of the quiet period larger than that of the disturbed one. Thus, the errors

2.2 Indices of Geomagnetic Activity

51

connected with large variations are avoided. These factors being inversely proportional to the variations are defined recursively assuming different smoothing factor values. Removing the determined S q curve from the daily data series under consideration, the final K indices are obtained. The three free parameters of the algorithm must be determined individually for each observatory to achieve the best agreement with the hand-scaling method. Because the procedure has two steps, and the weighting factors depend on the data, the method is called the Adaptive Smoothing (AS). In the course of the automated method comparison with the results from the hand-scaled method, the percentage of the differences larger than one unit was less than 0.3 per cent. As mentioned above, for estimating the geomagnetic activity in the planetary scale on a daily basis C i index was introduced. As the need for more detailed information on the planetary geomagnetic activity arose, in 1949 J. Bartels suggested new planetary K p index (http://www.ngdc.noaa.gov/IAGA/vdat/) for estimating the disturbance level over 3-hour intervals (see Fig. 2.13). Mainly data from subauroral observatories were used for calculating planetary indices of geomagnetic activity. 20° wide, the subauroral zone surrounds aurora oval from the outside; it is remote from any particular magnetospheric or ionospheric source and at the same time is sensitive to any of them. All these contributed to the use of data from only observatories of the subauroral zone for estimating planetary geomagnetic disturbances. K p calculation is based on K index values from 13 observatories situated between 63°N and 46°S geomagnetic latitudes. These observatories ensure prompt calculation and timely data transmission to International Service of Geomagnetic Indices (ISGI, http://isgi.unistra.fr/), where Kp is calculated. They include Meanook (Canada), Ottawa (Canada), Sitka (USA), Fredericksburg (USA), Lerwick (United Kingdom), Eskdalemuir (United Kingdom), Uppsala (Sweden), Brorfelde (Denmark), Wingst (Germany), Niemegk (Germany), Hartland (United Kingdom), Canberra (Australia) i Eyrewell (New Zealand) (Fig. 2.14, Table 2.5). Planetary K p index is calculated as the average of the K indices from these 13 observatories and determined with an accuracy of 1/3 (see Fig. 2.15, Table 2.6). Similar to K p , ap index represents 3-hour equivalent amplitude and its values are derived according to Table 2.6. Ap index is an equivalent daily mean planetary amplitude and represents an average from 8 ap indices over a day. Figure 2.15 depicts some examples of K p index graphical representation over a year. Some other planetary indices based on K, K p and ap were introduced later. They include K n (an ) and K s (as ) for estimating activity separately in the Northern and Southern hemispheres respectively, K m (am ) that gives mean activity level over the globe using data from more observatories as compared to K p index, and some others. K p index underlies the determination of international most quiet (10 Q-days) and most disturbed (5 D-days) days of each month, officially endorsed by IAGA. The three criteria used for each day are as follows: 1. the sum of the eight 3-hour K p values; 2. the sum of squares of the eight K p values; 3. the maximum of the eight K p values.

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2 EMF Observations and Data Processing

Fig. 2.14 Map of magnetic observatories providing data for K p index calculation (ISGI, http://isgi. unistra.fr/)

According to each of these criteria, a relative order number is assigned to each day of the month, the three order numbers are averaged and the days with the lowest and the highest mean order numbers are selected as the ten (respectively five) quietest and most disturbed days. It should be noted that these selection criteria give only a relative indication of the character of the selected days with respect to the other days of the same month. As the general disturbance level may be quite different for different years and also for different months of the same year, the selected quietest days of a month may sometimes be rather disturbed or vice versa. International Q-days are commonly used in many geomagnetic studies. These days are considered when the data are selected to construct the core magnetic field models, obtain and analyze Sq variations in the magnetic field components and in many other applications.

2.2 Indices of Geomagnetic Activity

53

Table 2.5 Coordinates of observatories contributing to K p index Code

Observatory

Corrected geomagnetic latitude

Geographic latitude

Longitude

62.5°

54.62

246.67

Northern hemisphere MEA

Meanook

SIT

Sitka

60.0°

57.07

224.67

LER

Lerwick

58.9°

60.13

358.82

OTT

Ottawa

58.9°

45.4

284.45

UPS

Uppsala

56.44°

59.9

ESK

Eskdalemuir

54.3°

55.32

17.35

BFE

Brorfelde

52.7°

55.62

11.67

FRD

Fredericksburg

51.8°

38.2

282.63

WNG

Wingst

50.9°

53.75

9.07

HAD

Hartland

50.0°

50.98

355.52

NGK

Niemegk

48.8°

52.07

12.68

356.8

Southern hemisphere EYR

Eyrewell

50.2°

43.41

172.35

CNB

Canberra

45.2°

35.30

149

Fig. 2.15 Traditional representation of 3-hour planetary K p index, 1996 (left) and 1999 (right) (https://www.gfz-potsdam.de/kp-index/)

54

2 EMF Observations and Data Processing

Table 2.6 Correspondence between K p and ap indices Kp

0o

0+ 1− 1o

1+ 2− 2o

ap

0

2

5

2+

3−

3o

3+

4−

4o

4+

4o 4+

6

7

9

12

15

18

22

27

32

Kp

27 32

6− 6o

6+

7−

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2.3 Collecting, Storing and Processing the Geomagnetic Data in MAGNUS System At present, all existing geomagnetic data centers are divided into categories: depositories (for example, geomagnetic data centers of the World Data System of the International Council for Science), depositories with the additional functions for data sampling and processing (for example, the interactive resource SuperMag, which contains the data of more than 300 variometers), centers for on-line geomagnetic data collecting (for example, INTERMAGNET). The hardware and software complex MAGNUS (Monitoring and Analysis of Geomagnetic aNomalies in Unified System) for collecting, processing, storage and analysis of the geomagnetic information was created at the Geophysical Center of the Russian Academy of Sciences (Gvishiani et al. 2016). MAGNUS underlies the analytical Geomagnetic Data Center of the Russian INTERMAGNET segment (http://geomag.gcras.ru) (Gvishiani et al. 2018; Pilipenko et al. 2019). The system is designed to perform the following tasks: search and systematization of initial ground-based and satellite (Swarm) observations of the magnetic field, automated filtering of the observational data from the artificial disturbances, its verification according to the INTERMAGNET standards, recognition, classification and coding of the data on the extreme geomagnetic phenomena that are dangerous for various technological infrastructures, model calculations, provision of the interactive access to the original data and information about the extreme phenomena, visualization of the geomagnetic data. Storage of the ground and satellite geomagnetic data (original and processed) in MAGNUS is arranged in the relational database. This is a significant advantage over the existing INTERMAGNET information nodes, which use mainly the conventional file databases. The innovative analytical component of the MAGNUS system is based on formalized integration of the knowledge and experience accumulated by the data experts in recognition and study of the natural extreme phenomena and anthropogenic disturbances in the magnetograms. The corresponding algorithms included in the MAGNUS system involve artificial intelligence elements. MAGNUS is characterized by the unified information and telecommunication modular infrastructure providing data transfer, storage and multicriteria processing. The infrastructure remains open for introducing additional software and hardware components. The data flow and main functions of the MAGNUS system are schematically shown in Fig. 2.16.

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Fig. 2.16 Flowchart of the MAGNUS system

The input magnetograms are divided into three groups: measurements, which are automatically and continuously recorded and transmitted by the observatories in the near real-time mode, absolute measurements of the geomagnetic field, carried out manually by the observatory operators, and operational satellite based magnetic observations. The operational data recorded at the magnetic observatories are the continuous time series of variations of the three orthogonal components and total intensity of the geomagnetic field vector with UTC time synchronization, recorded with the sampling rate from 1 s to 1 min. The data are transmitted in one of three formats: IMF1.23, IAGA-2002 or Mingeo. Depending on the telecommunication capabilities, the data are transmitted to MAGNUS from the observatories by e-mail or via the FTP-protocol with a certain delay (from 10 min to 2 days). The absolute observations are made several times a month and are provided in the form of the text files or using the web-interface. After loading the measurement results, the baseline values of the vector three-component magnetometer (variometer) and the absolute values of the magnetic field components are automatically calculated and stored in the database. The output parameters provided to the end user include the unprocessed data (preliminary magnetograms) and the results of its automated processing: recognized anthropogenic events, absolute values of the geomagnetic field, baselines, quasi-definitive, definitive magnetograms, derived geomagnetic activity indicators, etc. The end user is also provided with the results of model calculations based on both observatory and satellite data. All data and results of its processing are available to the users online. The requested data are provided in both digital and graphic formats.

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The MAGNUS software components are divided into six blocks: “Data acquisition”, “Data export to database”, “Data analysis”, “Model calculations”, “Data access”, “Visualization”. The “Data acquisition” block includes the software modules transmitting the operational data and absolute measurements from the observatories, as well as the satellite data. “Data export to database” consists of the software modules for loading the received data into the geomagnetic database. The “Data analysis” software modules are designed for operational processing and analysis of the data transmitted continuously to the MAGNUS system including automatic recognition of the anthropogenic disturbances in the magnetograms, calculation of quasi-definitive values, recognition of the anomalous geomagnetic events using the measure of anomalousness, estimation of the rate of change and amplitude of the geomagnetic variations and online calculation of the K index of the geomagnetic activity. The “Model calculations” block provides digital maps of the hourly distribution of the total intensity deviations for the territory of Russia, online modeling of the diurnal Sq variations for selected observatories, modeling of the spectral characteristics of geomagnetic time series, modeling of the dynamic electromagnetic parameters of the polar ionosphere and others. The “Data access” software modules provide online access to all the geomagnetic data stored in the relational database. The “Data visualization” module is aimed at visualization of the selected data online, displaying real-time data on the video board and visualization of the spatial characteristics of the magnetic field on the spherical display. The MAGNUS features are being constantly extended. The MAGNUS hardware components include four servers (mail server, FTPserver, DB-server, and web-server), video controller for visualizing the geomagnetic data on multiple displays, spherical visualization system and the system administrator’s computer. Configuration and interaction of the MAGNUS hardware components are shown in Fig. 2.17. The mail server is required to receive e-mail messages from the observatories containing the fragments of the continuous records, with their subsequent transmission to the FTP-server. The latter is designed for storing the geomagnetic data files in their original format and contains the software for searching the observatory and satellite data via the FTP-protocol. The FTP-server also contains a backup copy of the complete geomagnetic database, which is regularly synchronized with the main copy of the database in the automatic mode. The database server (DB server) uses a relational database management system and is designed for storing the observatory data (preliminary magnetograms) and satellite based observations, as well as all processing results. The software modules “Data analysis” and “Data export to database” are located on the DB server as well. The complete configuration of the database tables is shown in Fig. 2.18. The web-server contains all web content and software modules that provide users with interactive access to MAGNUS and its feature set. This includes the web applications for the database queries, server based software for modeling and online visualization, auxiliary PHP scripts providing interaction between the server applications and web interface. The procedure of collection of the source data is controlled by

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Fig. 2.17 MAGNUS components, their functions and interaction

the software modules of the “Data acquisition” block located on the FTP and webservers. These modules allocate the newly transmitted data in its original format in the file storage on the FTP server, from where it is automatically converted to the geomagnetic database elements. The data access modules providing end users with access to the MAGNUS features are located on the web-server, and the data visualization modules are distributed between the web-server, video controller and spherical visualization system. When a user requests the modeling results, the module “Model calculations” is executed using the PHP-script on the web-server, and the results are sent back to the user using the “Data access” module. An important feature of the MAGNUS system is automated production of the quasi-definitive data from preliminary magnetograms streaming from magnetic observatories. Quasi-definitive (verified) data represent corrected vector measurements of the field, which are adjusted to temporary baseline values immediately after registration and very close to definitive data. Today, only a quarter of the INTERMAGNET network observatories are capable of producing satisfactory quasidefinitive data that diverge from the definitive data by less than 5 nT. Quasi- definitive data are most in demand in such geophysical domains as constructing timely models and studying rapid variations of the core magnetic field (Soloviev et al. 2017a), as well as calculating geomagnetic activity indices (Peltier and Chulliat 2010). According to INTERMAGNET regulations, the delay in calculating such data should not exceed 3 months, but in fact it does not exceed 1 month. The production of quasi-definitive data can be divided into two main stages: quality control of continuous vector measurements of the magnetic field by recognizing and removing

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Fig. 2.18 Configuration of the database tables for storing ground and satellite based geomagnetic data and the results of its processing. The tables with orange filling refer to the initial data of the observatories; blue refers to the absolute data of the observatories; purple refers to the initial satellite data; green refers to the modeling results; yellow refers to the geomagnetic activity indicators; white refers to auxiliary information (e.g., observatory metadata)

anthropogenic disturbances in magnetograms; determination of dynamic baseline values according to the results of absolute measurements and consequent calculation of time series of the full values of field elements. In the MAGNUS system, an automated method consisting of two stages is used to detect and remove spikes from 1-sec and 1-min vector measurements: application of algorithms for recognition of anthropogenic anomalies and verification of the final result by comparing recognition results obtained from data from multiple observatories. At the first stage, the developed recognition algorithms based on the

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principles of artificial intelligence are executed (Bogoutdinov et al. 2010; Soloviev et al. 2012a, b, 2013; Sidorov et al. 2012). The algorithms have several free parameters determined for each observatory individually in the process of the algorithm learning. The second stage is necessary to eliminate false alarms of these algorithms. For instance, if the disturbance is detected at two or more observatories at the same time, most likely it is not of an anthropogenic but natural origin (for example, geomagnetic pulsation). In addition, to reduce the likelihood of false positives of the algorithms, values of geomagnetic activity indicators are involved (Gvishiani et al. 2016, 2018), derived continuously in the MAGNUS system as data become available. They include measure of anomalousness (Soloviev et al. 2016a; Agayan et al. 2016), peak amplitude, rate of change of the magnetic field and the operational K index of geomagnetic activity. They are quantitative indicators of geomagnetic activity, simplifying the determination of the time boundaries of natural geomagnetic events (such as magnetic storms and substorms) and making possible to study their fine structure. If a recognized anomaly falls into the period of increased geomagnetic activity observed at several observatories, it ceases to be classified as anthropogenic. Upon completion of the recognition of the anthropogenic anomalies in the incoming magnetograms, the MAGNUS system performs their baseline correction and further calculation of the quasi-definite data. By means of the web-interface of the MAGNUS system the observatory operators transmit the results of the absolute measurements carried out manually using absolute (DIFlux) magnetometer. Upon the receipt of these data the DB is requested to retrieve cleared variational data over the considered period, and information on the orientation of the vector magnetometer. Then, the calculation of the so-called observed (spot) baseline values for the three components at the times of absolute measurements is executed. The procedure for calculating spot baseline values for the orientation of a vector magnetometer along both geographic and geomagnetic meridian is described in detail in (Jankowsky and Sucksdorf 1996). Subsequently, an SQL query to the database is generated and the results along with their time references are imported into the database. Due to the tightening of requirements for the observatory operations in the INTERMAGNET network and the growing need of scientific community for the observatory data of higher quality, it became necessary to constantly control the values of the difference in the geomagnetic total intensity between the pillars in the absolute pavilion of the observatory. The indicated difference (F0) is used in the calculation of quasi-definitive and definitive data and, in fact, is another baseline value. For this reason, a special service and a data processing algorithm to automatically derive F0 was introduced to the MAGNUS system. The observed baseline values are an irregular time series, because absolute measurements themselves are irregular (usually 1–2 measurements per week). To recalculate continuously recorded components into the full values of field elements with the same sampling rate, a regularization of the baseline values is required. The procedure for deriving regular baseline series (baseline curves) is described in detail in the next Chapter. In practice, the baseline curve is not calculated from all available values. Sometimes, some observed baseline values are outliers against the background of the general set, for example, due to a mistake made in the course of

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absolute measurements. Another common failure is jumps in the observed baseline values. Their cause may be a change in the variometer settings, its movement, as well as the appearance or removal of static noise at the observatory. Such events are monitored by the observatory staff, who enter information about it into the database using a special web form. In the MAGNUS system, the processing of these events is automated; both kinds of interference are automatically removed before constructing regular baseline curves. Next, the full values of the field components are calculated using baseline curves and variational values for the period from 1 December of the previous year to the current time. When storing the resulting quasi-definitive data set in the database, it is cleared from the anthropogenic disturbances that were recognized by the method described earlier. In the MAGNUS system, recalculation of the baseline approximated curve and subsequent recalculation of quasi-definitive data are executed automatically when new absolute measurement results arrive, which ensures a minimum time delay. The block diagram of the developed technique is presented in Fig. 2.19. In addition to acceleration of the quasi-definitive and definitive data production, such automation also improves data quality. Let us demonstrate it by the example of the data recorded at Saint-Petersburg observatory (IAGA code SPG) (Sidorov et al. 2017). To assess the quality of the obtained quasi-definitive data, we adduce their comparison with the definitive data of 2015–2016 approved by INTERMAGNET and published on the website http://intermagnet.org (Soloviev et al. 2016b, 2017b). As the first criterion for the quality assessment, the component-wise differences with 1-min definitive data were evaluated. As a result of the calculation, it was found that the difference did not exceed 5 nT for 99.97% of X component values, 96.16% of Y component values and 100% of Z component values. As the second criterion for the

Fig. 2.19 Automated calculation of the quasi-definitive values of the magnetic field (“Data analysis” block)

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quality assessment, the differences between temporary baseline values used to derive quasi-definitive data and definitive baseline values were evaluated. The former were obtained for the period from 1 December 2015 and then monthly until 31 December 2016, the latter covered the period from 1 December 2015 to 31 January 2017. The analysis showed that for the Y 0 and Z 0 baseline the difference did not exceed 5 nT, and for the X 0 baseline it did not exceed 10 nT. The latter was associated with the outliers, which were filtered out during the calculation of the definitive baseline curve (Fig. 2.20). The presented comparison argues that the obtained quality indicators of the quasi-definitive data completely satisfy the INTERMAGNET standards, and even exceed the required characteristics in some indicators. Also, a comparison of SPG quasi-definitive data and spherical harmonic model predictions over the period of 2016 proved that the secular variation of the core magnetic field at the observatory, derived from (WMM Chulliat et al. 2019, IGRF Thébault et al. 2015) models, corresponds to the processed observatory data (Fig. 2.21). For magnetically quiet days, the mean-square difference between quasi-definitive daily means and modeled data did not exceed 5 nT (Kudin et al. 2020). Thus, the MAGNUS system is an effective tool that helps to solve the problems of collecting, storing, managing, processing and analyzing the magnetic data. The routine operations with the data flows, adopted around the world, were transformed from the manual to automatic analytical mode using the mathematical elements of

Fig. 2.20 The differences of the baseline values for the quasi-definitive and definitive data, which were obtained from the results of absolute measurements at the SPG observatory in 2016: X (top), Y (middle) and Z (bottom) components

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Fig. 2.21 The centered daily means of the magnetic field components at the SPG observatory in 2016: quasi-definitive data (continuous line), IGRF model data (dash-dotted line), and WMM model data (dash line)

the artificial intelligence. This leads to an increase in the data accuracy and decrease in the delays in definitive data production as compared to other data centers. In addition to the fundamental studies, the unhindered and reliable provision of the quasi-definitive data is also important in the industrial sector. MAGNUS integrates online tools for processing and analyzing data sets, including those uploaded by the user. This makes the MAGNUS system an efficient and flexible tool for the geomagnetism researchers, in particular facilitating the timely modeling of the magnetic field. All automated operations are carried out in the near real-time mode enabling multi-criteria evaluation of the geomagnetic activity and provision of the subsequent forecasts with a minimum time delay. The MAGNUS functionality can be used by the experts and decision makers who need this information to assess and reduce the risks caused by the extreme geomagnetic events.

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2.4 Some Recent Improvements in Routine Baseline Production The data on the geomagnetic field’s state and its temporal variations are registered using modern magnetometers installed on special satellites and ground-based magnetic observatories. Despite the advantages of satellite measurements, there are a number of limitations that do not allow us to abandon the classical magnetic field measurements of stationary observatories. The measuring satellites are relatively recent and inevitably have a limited lifetime. The typical duration of the continuous satellite data time series does not exceed 10 years, whereas the oldest geomagnetic observatories provide continuous series of observations lasting more than a 100 years. Such records are of exceptional value for the fundamental researches in the field of geomagnetism assuming the study of the evolution of the Earth’s magnetic field and the associated dynamic processes in the outer core over longtime intervals. The development of science in the study of the magnetic fields of the Earth and the Sun requires a permanent improvement of the quality of the data provided by the observatories. Despite the development of methods for measuring the magnetic field, the present high-precision vector magnetometers do not allow automatic measurements of the total values of the magnetic field components in an autonomous mode. Temperature changes, pillar movements, aging of electronics and many other factors inevitably influence the magnetic field vector measurements. To ensure the precision needed for contemporary researches, a periodic calibration of the variation magnetometer should be carried out at every observatory. Such calibration is provided as a result of observation of the absolute magnetic field values (absolute observations), performed by a trained specialist using the non-magnetic theodolite, on which a single-axis fluxgate magnetometer is mounted. The corresponding instrument is called an absolute magnetometer or a declinometer/inclinometer. However, even with the strict fulfillment of all the requirements, the resulting measurements can be still burdened with inaccuracies due to increased geomagnetic activity or weather conditions during the absolute observations. The resulting measurements make it possible to calculate the calibration corrections, the so-called baseline values, for each component of the vector magnetometer (Jankowski and Sucksdorff 1996). Spot (or observed) baseline values are calculated using the results of absolute observations, as well as spot values of the component variations and the field modulus (total intensity) registered with two independent instruments at times of absolute observations. Further, they are used to obtain a regular series of baseline values applied for the correction of 1-min (or 1-s) data, which are derived from continuous variometer recordings. According to the INTERMAGNET rules for the data processing, such regular series should represent daily values resulting from the interpolation of the observed baseline values. The interpolation algorithm is selected by each observatory independently, ensuring the smoothness of the resulting baseline. Most observatories solve this problem by using cubic smoothing splines or the approximation with polynomials.

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Nevertheless, the described approach has some serious disadvantages. First of all, application of interpolation to obtain the baseline values for the periods of the absence of the absolute observations does not take into account the physical effects caused by the behavior of the magnetic field during the corresponding time intervals. That is, the common agreement that the baseline variations should be described by smooth functions (quadratic polynomials, splines, etc.) based on the absolute observation values generally has no rigorous physical justification. At the same time, the field behavior between the time moments of absolute observations can be fully taken into account, as observatories normally provide continuous and independent registration of both relative variations of the field components and its modulus values. In the current observatory practice, when calculating the regular baselines, continuous records of the components and the modulus of the magnetic field are ignored. Taking into account the detected geomagnetic signals with the amplitude of several nT with a spatial scale of some hundred meters and less than 1-day duration (Lesur et al. 2017), this circumstance becomes especially critical. Also, the selection of baseline values at each time moment is carried out for each component independently, which does not allow to take into account the accuracy of orientation and orthogonality of the vector magnetometer sensors. Another disadvantage of the widely accepted method is that it does not admit baseline value variations within a day, which may well occur in actual practice due to an abrupt temperature change in the pavilion or small-scale geomagnetic signals described above. The mentioned disadvantages necessarily lead to the deterioration of the quality of the definitive data, which in turn introduces errors in model predictions built using observatory data, especially when studying rapid (shorter than a year) core field variations. In Soloviev et al. (2018) a new approach to the calculation of regular baselines is proposed, in which both of the mentioned disadvantages are eliminated. The proposed approach is based on the simultaneous analysis of the results of absolute observations and the values of the regular time series F = F(t) which is widely used for estimating the data quality (Reda et al. 2011). The latter represents a series of differences between the field modulus recorded with a scalar magnetometer and calculated from the full component values, which are derived from the vector magnetometer data. In the proposed method, the systematic F analysis allows to take into account all available information about the operation of observatory instruments in the intervals between the moments of absolute observations and enables continuous monitoring of the quality of work of the devices. Of course, it implies a continuous recording of the absolute field modulus at observatory. To establish a connection with the observed baseline values, a function for approximate evaluation of the intermediate baseline values is introduced. The informal essence of the algorithm is in the selection of such a combination of the three baseline values for every time moment that (1) the resulting F tends to zero, (2) their discrepancy with the adjacent observed baseline values is minimal. The first criterion is formalized as a function G tending to zero, and the second one— as the minimization of the function A. Then, the target function ϕ is introduced, which is a linear combination of the G and A functions connected with a weighting factor λ. To optimize the computational process, while calculating the ϕ function,

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the time step is set equal  to 1 h. The brute force search is performed on the 3D grid j i k i, j,k  = X b , Yb , Z b with a typical step around 0.1–0.25 nT for each component. S Hence, f  de = (X b , Yb , Z b ) = arg min ϕ(t), ∀t ∈ T, B(t) Si, j,k

 where B(t) is the desired baseline vector for the time moment t on a regular time interval T with a step of 1 h and “argmin” returns the value of Si, j,k , which minimizes ϕ(t) over the set of candidates. The resulting series of the found minima of the ϕ function (Qϕ ) represents a quantitative estimation of confidence in the obtained baseline values for each time moment. The method was first validated using synthetic data set. Assuming predefined regular baselines (sine functions with a phase and frequency shift between the components and an offset with respect to zero), variations of the three components and synthetic total intensity and the irregularly pseudo-observed baseline values, the method was tested to recover hourly baseline values. An important indicator of the quality of the calculated baseline values is the difference function G (Fig. 2.22). As it is seen, its values are within the [−0.5 0.5] interval, which indicates the high definitive data quality according to the INTERMAGNET regulations. The Qϕ time series is also present in Fig. 2.22. It is seen that during the periods of increasing variability of the baseline curve and at the moments, which are remote in time from the observed baseline values, the minimum of the ϕ function increases. On the contrary, in between the two first observed baseline values Qϕ does not show any significant peaks, as (1) these two values are very close for each component, and (2) vector/scalar measurements are in good agreement. We recall that the value of the weighting factor λ = 0.5 means equal “confidence” of the algorithm both in the observed baseline values and in the coherence of vector and scalar measurements of the magnetic field. The algorithm validation process consisted of 100 computational experiments based on the synthesized data. In every computational experiment, an additive Gaussian white noise with a RMS σ = 0.5 and a mean μ = 0 was added to the pseudoobserved baseline values. During every computational experiment, a baseline calculation was performed consequentially for λ = 0.1, 0.5 and 0.9. Such values were selected in order to explore the computational side effects of the algorithm when a confidence factor is selected incorrectly. When λ = 0.1 the increased confidence of the algorithm in the observed baseline values reduces the dispersion of the computed baseline values; however, the dispersion of the G function values increases with that (STD ≈ 0.19 nT). On the other hand, when λ = 0.9, representing the excessive confidence of the algorithm in the quality of the work of the scalar and vector magnetometers of the observatory, the decrease in the dispersion of the G function (STD ≈ 0.03 nT) is seen. Accordingly, the lack of confidence in the observed baseline values decreases smoothness and adds a noise component to the calculated baselines, which can adversely affect the quality of the definitive data.

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Fig. 2.22 Calculated hourly baseline values (blue) with λ = 0.5, pseudo-observed values (red) and functionally defined hourly baselines (green) for the X (first plot), Y (second plot) and Z (third plot) components of the geomagnetic field. Fourth and fifth plots represent the corresponding Qϕ and G functions

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Fig. 2.23 Visualization of obtained function ϕ minima (Qϕ series) for λ = 0.1 (a), λ = 0.5 (b), λ = 0.9 (c) resulting from 100 experiments. The horizontal axis stands for the experiment number, the vertical axis stands for the numerical order of the synthesized data hour, and the color represents the Qϕ values. Red lines indicate the mean time moments corresponding to the values of the Qϕ local maxima for all the computational experiments with λ = 0.5

The hourly series Qϕ is a main characteristic function for definitive data obtained using the proposed algorithm. The local Qϕ maxima can indicate excessive variability of the observed baseline values over the corresponding time interval or the problems with synchronous data registration by the scalar and the vector magnetometers. The latter may include wrong scale factors, time desynchronization, sensor temperature effects and others. In Fig. 2.23, the Qϕ time series for all the computational experiments are plotted. As it is seen from the figure, the alternation of the confidence factor λ does not have a significant impact on the distribution of the local maxima of the Qϕ functions. However, with increasing the confidence in the function G (λ → 1) the contribution of the noise component of the variometer values increases, which leads to the occurrence of noise effects in the Qϕ values (Fig. 2.23c). With this, the order of magnitude of the total noise effect becomes comparable to the one of the noise rate of the observed baseline values, which almost completely masks possible problems with the baseline values on the Qϕ data. For λ = 0.5, the local maxima of the function Qϕ are clearly seen and strictly localized in time on all random sets of test data. This indicates that the change in noise characteristics in the initial data does not significantly affect the result of the algorithm. The found local maxima of Qϕ correspond to the intervals of low confidence of the algorithm in the obtained definitive data quality with respect to its general level. On the contrary, local Qϕ minima indicate a higher quality of the resulting data. An introduction of extra pseudo-observed values to input data leads to the amplitude decrease or disappearance of local Qϕ maxima, whereas removal of pseudo-observed values leads to amplification or appearance of extra local Qϕ maxima. Thus, the function Qϕ represents a quantitative measure of the quality of the initial data and, as a consequence, reflects the level of confidence in the resulting data. The latter is one of the most important features of the developed algorithm.

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In the course of the method validation in real conditions, an initial data from the Saint Petersburg geomagnetic observatory (IAGA code SPG) over 2015 was used. The observatory is located 100 km north from the city of Saint Petersburg, Russia (60.542 N, 29.716E) and based at the site of the former Krasnoe Ozero (Red lake) observatory, where continuous observations were carried out from 1960 to 1990. Through the efforts of the GC RAS in 2014, it was equipped with the modern magnetometric instrumentation, and in 2016 the first definitive data set for 2015 was produced (Soloviev et al. 2016b). These data were accepted by INTERMAGNET, and the observatory was officially included into the network (Sidorov et al. 2017). To demonstrate the advantages of the described method, its comparison with the traditional method used by the INTERMAGNET community by the example of the SPG data over 20 February–1 October, 2015 is given. The traditional method produced F amplitude below 1 nT, baseline amplitudes for X, Y and Z components below 5, 10 and 2 nT during the whole year, respectively. As the input data for the developed algorithm, the 1-min variations of the components and the geomagnetic field modulus record cleaned from anthropogenic disturbances using the algorithms (Bogoutdinov et al. 2010; Soloviev et al. 2012a, b; Sidorov et al. 2012), as well as the absolute observations after outlier removal were used. The comparative analysis of the results displays much more variability of the baseline values calculated by the new method both for the hourly and for the daily cases (see Fig. 2.24). The main indicator for the proposed method efficiency is the resulting curve F, which is widely used in INTERMAGNET practice as the data quality measure. The general difference from the F series, derived from the approved definitive data of the SPG observatory (Gsrc ), is the significant decrease in the root mean square deviation from σ = 0.226 nT to σ = 0.041 nT when using the hourlybaseline  values  n  n Ghour and σ = 0.088 nT when using the daily baseline values Gday , and the decrease of the mean to 0. Figure 2.25 shows histograms of their value distributions in percentage. Also, the Qϕ series was plotted (Fig. 2.26) for the whole interval under consideration, characterizing the level of confidence in the definitive data for each value. The data with the highest confidence level (the lowest Qϕ values) are obtained in

Fig. 2.24 The comparison between the baseline values for the X component, obtained by the classical and the proposed methods over 20 February–1 October, 2015 at SPG observatory. The baseline according to the classical method with the use of the smoothing spline interpolation is marked with  red (X0 ), the hourly baseline values obtained by the described algorithm are marked with blue nXh0 , the daily values calculated by averaging the hourly values nXh0 are marked with   green nXd0 , and the observed baseline values are marked with light-blue circles (spot)

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Fig. 2.25 Histograms of the value distributions for Gsrc (left), Gnday (middle) and Gnhour (right) in percentage of the total number of values over 20 February–1 October, 2015

Fig. 2.26 Hourly Qϕ values for the period from 20 February–1 October, 2015

July and August. High-frequency low-amplitude spikes around July–August are due to electric works that took place nearby and consequently affected G function. Low confidence level (the highest Qϕ values) corresponds to the time intervals of the most intense baseline variations, for example, at the end of April, at the end of May and at the end of June. Single spikes in the Qϕ values correspond to the intervals of technical gaps in the initial data.

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The comparative analysis shows that the definitive data obtained using the developed algorithm meet the INTERMAGNET requirements. Due to the new method, the data quality indicators are essentially improved, comparing to the classical approach, and also a relevant estimation of the measurement quality during the analyzed time interval is provided. Taking this estimation into account, it becomes possible to set the minimal quality level required for the final data and, when processing the observations from several observatories, to rank the individual data fragments according to their quality. It also allows to build a baseline in a semiautomatic mode using all the available data from the observatory. The obtained results prove that the baseline variability in time should not necessarily be smooth. In particular, this may be due to the distance between the variation and absolute pavilions: the more this distance, the less smooth is the baseline (Lesur et al. 2017). The assumption of the baseline smoothness might lead to the loss of the information about the geomagnetic signals of small spatial scale (100–200 m) but quite lasting (~1 day long), with the amplitude of several nanoteslas. In turn, this will add significant distortion into the models of rapid core field variations built using observatory data.

2.5 Mathematical Methods of Analysis of the Geomagnetic Data Time Series An intense increase in the volume of the geomagnetic measurement data and the need of its analysis requires to use the effective modern geoinformatics technologies, develop new mathematical approaches and methods of searching the anomalies of the physical fields. In the history of development of the approaches to recognition of the anomalies in the time series, three stages can be specified characterized by the dominant concept and the capabilities of the computer equipment. At the first stage, a deterministic approach to data processing was applied on the assumption that the solution to the problem of searching for the anomalies is completely determined by the system of the initial values of the measured physical field (Agayan et al. 2010). At the second stage, the methods of analytical extension of the fields to the upper and lower half-spaces are used, the frequency analysis methods are developed, a new probabilistic-statistical approach to processing the geophysical data based on the representation of the fields in the form of the random functions is formed. At the third stage, the probabilistic-statistical methods of analysis are actively improved and applied in the various fields. Actually, such a representation is largely approximated to the real situations when the sources of the natural physical fields vary within a sufficiently wide range, and the signals that are the object of the determination (useful signals) are complicated by the interferences and observation errors, i.e., they are a combination of the constant and random components. Upon the probabilisticstatistical approach, the result of solving the problem is the probability distribution specified over the possible values of the result. The problem of signal extraction

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includes several simultaneously solved problems: detection, which is understood as determination of the signal availability, properly selection or measurement consisting in evaluating the signal parameters, in particular, its location, and recognition of several signals. The algorithms for extracting the anomalies of the geophysical fields can be based on the spectral, asymptotic and functional methods, for example, of autoregressive description, maximum likelihood, recognition with learning, neural networks. The filtration-based methods of searching for the anomalies in the frames of the probabilistic-statistical approach are usually implemented if the a priori information about the shape of the anomalies and their correlation characteristics (field distribution law, dispersion) is available and effecent when extracting the strong anomalies. One of the simplest filtration-based methods of detection of the anomalies is averaging of the signal and further subtraction of the averaged field from the source one. The problem of detecting the weak anomalies is more complicated and can be effectively solved based on the mathematical apparatus of the theory of statistical solutions, which provides the maximum extraction of useful information by calculating the posterior probabilities of availability of the anomalies. In the practice of analyzing the observatory magnetograms, the problem of recognizing the disturbances has its own specific features. Processing of the magnetograms includes primarily identification of the disturbances of the artificial (anthropogenic) origin that need to be removed, which is usually done manually by the experts. At that, the characteristics of a particular disturbance are determined only based on the general ideas about this kind of disturbances. There is very limited a priori information about the desired disturbances, and there is no data on the shape of the anomaly and its correlation properties. Therefore, a magnetic field analysis algorithm is required that would allow to solve the problems of detecting the anomalies in the most general case. Here, self-adjusting (adaptive) processing procedures, in which the processing algorithm can “adapt” to a change in the properties of the anomalies and interferences (dispersion, correlation properties) directly during processing, should be considered as perspective for detecting and classifying the anomalies.

2.5.1 Methods of the Discrete Mathematical Analysis The analysis of a geophysical process is a complex and ambiguous concept. Such complexity results from both the objective complexity of the process itself and the subjective nature of expert analysis. Therefore, an adequate formalization of this process requires an entirely new approach, one that is capable of overcoming both the objective complexity of geophysical data (irregularity and inaccuracy), and the intrinsically ambiguous nature of expert judgments. The development of Discrete Mathematical Analysis (DMA) (e.g., Gvishiani et al. 2010; Agayan et al. 2018) is an important step in this direction. DMA is essentially an expert-oriented approach that occupies an intermediate position in data analysis between straightforward mathematical methods (statistical analysis, spectral-temporal analysis, etc.) and “soft”

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combinatorial methods (imitational modeling, neural networks, etc.). The concept of DMA states, first, that the expert’s advantage over formal methods in data analysis is a result of the expert’s flexible, adaptive perception of the fundamental properties of limits, proximity, continuity, connectivity, trend, and others. All DMA based data analysis algorithms are assembled on the basis of these properties, as if from a constructor. Thus, it is necessary to find discrete analogs of the aforementioned properties and then (the second part of the concept) to connect them in the scenario of classical continuous mathematics and thus to improve the data analysis. In the implementation of the concept, it was immediately taken into account that the expert thinks and operates not with numbers but with concepts. Therefore, in addition to classical mathematics, fuzzy mathematics and, through it, artificial intelligence, were involved in the DMA modeling. The DMA results in a series of the algorithms aimed at solving the main data analysis problems: clustering and tracing in multidimensional arrays, studying discrete time series (both one- and multidimensional, noisy, irregular, …) and identifying trends, extrema, etc. based on the functional scenario resembling the classical analysis of smooth functions. All DMA algorithms are universal and are joined by a single formal basis (e.g., Agayan et al. 2010; Gvishiani et al. 2002, 2003, 2004, 2008a, b, 2010). The decisive element of the study are the measures of activity of a discrete function characterizing its local activity from an expert’s point of view. The points, in which the local activity is high, are considered locally anomalous. The anomalies are obtained from the locally anomalous points using DMA-clustering and studied by the DMA methods, in particular, are ranked using the DMA indices. Besides, the intellectual analysis of the finite time series includes monitoring its measures of activity at the mentioned levels. Configuration of finite time series analysis is shown in Fig. 2.27. Let us list the definitions associated with it. Observation period T is a finite, in the general case, irregular set of nodes ti :   T = {t1 < t2 < . . . < t N } = ti |1N . Function f is the time series defined on T:

Fig. 2.27 Configuration of the finite time series analysis

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f : T → R;

73

  f : f i = f (ti )|1N .

Dynamic index D is the functional on T, parametrized by T: D : F(T ) × T → R, F(T ) is the function space on T. The values of D( f, t) are denoted D f (t) and understood as the quantitative estimates of the function f behavior in the node t ∈ T according to the approach D to its dynamics. The examples of dynamic readings: let δi ( j) be a measure of the proximity of the node t j to the node ti in T. 1. Energy N j=1 f j − M f (ti |δ) δi ( j) E f (t|δ) = , N j=1 δi ( j) where N j=1

M f (ti |δ) =  N

f j δi ( j)

j=1 δi ( j)

.

2. Jaggedness N  j=1 f δ (ti ) δi ( j) , L f (ti |δ) = N j=1 δi ( j) where f δ (ti ) is the DMA-derivative of function f with respect to the proximity δ: N N δ j (k) f k tk k=1 δ j (k)tk k=1  N N δ j (k) f k k=1 δ j (k) . f δ (ti ) = k=1 N  N 2 kN δ j (k)tk kN δ j (k)tk k δ j (k)tk k δ j (k) 3. Scatterness: let p > 0; assume

O f (ti |δ, p) = M f (ti |δ, p) − M f (ti |δ, − p), where

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N M f (ti |δ, p) =

p j=1 f j δi ( j) N j=1 δi ( j)

N M f (ti |δ, − p) =

1/ p

−p j=1 f j δi ( j) N j=1 δi ( j)

, −1/ p .

The dynamic activity measure μD f (ti ) is a measure of the maximality of dynamic index D f (ti ) ≡ the membership function to the fuzzy concept “activity of f in the node ti from the index D point of view”: μD f (ti ) = mesmax(Im D f ,δi ) D f (ti ),   where Im D f , δi is an image of D f , centered in the node ti using the proximity δ:         Im D f , δi = Im D f t j , δi t j | Nj=1 . There are four implementations of the structure mesmax: “fuzzy comparisons” n, “Kolmogorov averages” K, “fuzzy averages” S and “fuzzy boundaries” G:   ⎧  n Im D f , δi , D f (ti )  ⎪ ⎪ ⎨ K Im D f , δi , D f (ti ) μD f (ti ) = . ⎪ S Im D f , δi , D f (ti )  ⎪ ⎩  G Im D f , δi , D f (ti ) “Fuzzy comparisons” structure examples: 1. binary structure ⎞⎛ ⎞−1   N N   D f (ti ) − D f t j     δi t j ⎠⎝ μD f (ti ) = ⎝ δi ( j)⎠ ; t D + D (t ) f i f j j=1 j=1 ⎛

2. gravity structure   D f (ti ) − D f t j μD f (ti ) =  , D f (ti ) + D f t j where

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  D f tj =

N j=1

  D f t j δi ( j)

N

j=1 δi ( j)

75

;

3. sigma structure     σ r D f (ti ) − σ l D f (ti )   , μD f (ti ) = r  σ D f (ti ) + σ l D f (ti ) where N         D f (ti ) − D f t j δi ( j) : D f t j < D f (ti ) σ l D f (ti ) =

 r



σ D f (ti ) =

j=1 N  

.     D f (ti ) − D f t j δi ( j) : D f t j > D f (ti )

j=1

“Kolmogorov averages” structure μD f (ti ) =

4 ar ctg p ∗ − 1, π

where p ∗ is the equation solution

N

p∗ j=1 f j δi ( j) N j=1 δi ( j)

1/ p∗ = D f (ti ).

Complex activity measures The transfer D f → μD f translates the function f analysis into the language of fuzzy logic and fuzzy mathematics. Initially, the dynamic indices D f for various D are not comparable to each other and cannot be combined for joint study of f . On the contrary, the activity measures μD f take the values within a single scale of the segment [−1; 1] and can be combined in any configurations and quantities using the numerous operations of fuzzy logic, which are denoted by *. It becomes ¯ possible to give a sense to the activity of f per totality of the dynamic indices D:   μ D¯ f (∗) = ∗ D∈ D¯ μD f (∗) . Statistical analysis based on the activity measures The expert’s opinion E on the function f is generally modeled by a complex activity ¯ measure μ D¯ f = μ D(E) f . The analysis by experts E 1 , . . . , E M of the function f by the activity measures μ D¯ 1f , . . . , μ D¯ M f includes:

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1. local analysis of f in point t ∈ T ≡ construction and comparison of the measures μ D¯ mf (t) in the point t for m = 1, . . . , M to monitor the mutual relations of the expert dynamics for f in t. Besides, it includes calculation of the total integral activity of f in t by all μ D¯ mf (t)|1M . 2. global analysis of f ≡ average and covariance characteristics μ D¯ mf and   Cov μ D¯ mf 1 , μ D¯ mf 2 . The result is coding of f and the possibility of comparing f through such coding with another f¯ and, generally speaking, within another observation period T¯ . Algorithm DPS Let X be the finite set, and A, B, . . . and x, y, . . . be the subsets and points in it, respectively. Density P on the set X is mapping from 2 X × X into the segment [0, 1], increasing by the first argument: P( A, x) = PA (x) . ∀x ∈ X, A ⊂ B ⇒ PA (x) ≤ PB (x) The value PA (x) is the density of the subset A in the point x. The subset A is called α-perfect in X for α ∈ [0, 1], if A = {x ∈ X :

PA (x) ≥ α}.

The process of construction of the maximum α-perfect subset X (α) in X is called the algorithm DPS (Discrete Perfect Sets): X (α) = DPS(X |P, α). The subset X (α) is obtained as a result of intersection k X (α) = ∩∞ k=1 X (α),

where   X k+1 (α) = x ∈ X k (α) : PX k (α) (x) ≥ α . The intersection ∩∞ k=1 is necessarily achieved, because it is always finite due to the finiteness of X and nesting of X k+1 (α) in X k (α) at all k = 0, 1, . . . Search for anomalies in the time series using the DPS algorithm 1. Regular case. The set of the nodes T = {t1 < . . . < t N } is regular, if ti+1 − ti = h is for all i = 1, . . . , N − 1.

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1.1. Functional clustering. Let us determine the density P on T according to dynamic activity measure μD f : let  > h be the parameter of the local observation, A be the subset in T, t be the node from T, then

PA (t) =

  μD f t¯ + 1 : t¯ ∈ [t − , t + ] ∩ A. 2N

The result of the algorithm DPS operation with such density on T with the fixed α represents the first part of the nodes anomalous for the function f within the observation period T from the dynamic index D point of view: T (α) = DPS(T |P, α, μD f ). The set T (α) is represented as a disjoint union of the segments [ak , bk ], k = 1, . . . , m. Every [ak , bk ] lies in T and has no gaps inside it (Figs. 2.28, 2.29): T (α) = ∨m k=1 [ak , bk ].   Interval clustering. The set of intervals [ak , bk ]|m 1 is denoted by X (α) and we introduce a measure of proximity ρ between two intervals from X (α):   ak , bk ]|+|[ak , bk   1 1 2 2     . ρ ak1 , bk1 ], [ak2 , bk2 = min ak , ak , max bk , bk 1

Fig. 2.28 Time series and the activity corresponding to it

2

1

2

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Fig. 2.29 Results of detection of the anomalies on the activity index

The proximity ρ allows to construct the density P on X (α): if S ⊂ X (α), [ak , bk ] ∈ X (α), then PS =

max

[ak¯ ,bk¯ ]∈S\[ak ,bk ]

  ρ ak , bk ], [ak¯ , bk¯ .

The result of the DPS algorithm operation with such density on X (α) with the fixed β represents the second part of the nodes anomalous for the function f within the observation period T from the point of view of the dynamic index D (Figs. 2.30, 2.31):

Fig. 2.30 Interval clustering

Fig. 2.31 Reduction to the initial function

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X (α)(β) = DPS(X (α)|P, β, μD f ). The algorithm DPS at the second stage shows what segments from X (α) should be additionally joined into one anomaly—these are precisely the segments from X (α)(β). Thus, the isolated segments from X (α) that were not included in X (α)(β), as well as the new segments obtained after combining the segments from X (α)(β) into groups shall be finally anomalous for the function f from the dynamic index D point of view.   Let us denote the set of all anomalies as A = A μD f :  L   . A = Al = Al μD f |l=1 2. Irregular case. Let I be the minimal natural number, for which

h=

N −1 t N − t1 ≤ min(ti+1 − ti ). i=1 I −1

Let us specify by T¯ the regular discrete segment [t1 , t N ] with the nodes t¯i = t1 + (i − 1)h, i = 1, . . . , I so that t¯1 = t1 and t¯I = t N . Let f¯ be the linear extension of f on T¯ :

  If A¯ = A μD f¯ are the anomalies for f¯ on T¯ found according to the regular   scenario, then the anomalies A = A μD f for f on T are the traces of A¯ on T: A = A¯ ∪ T =



L  . A¯ l ∩ T |l=1

Anomalies ranking. Each anomaly A ∈ A has four indicators: 1. 2. 3. 4.

massiveness m A =  |A|;  activity μA = M μD  f (A) ; dynamics D A = M D f (A) ; value f A = M( f (A)). Four sets appear:         L L L L ; μA = μAl |l=1 ; DA = D Al |l=1 ; f A = f Al |l=1 . mA = m Al |l=1

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The comparison of the indicators of the anomaly A with the relevant indicators of the remaining anomalies from A gives four new global indicators indm A,indμ A,ind D A,ind f A for the anomaly A, characterizing its role and place, i.e. ranking it in a full series of the anomalies from A: indm A = mesmaxmA m A indμ A = mesmaxμA m A . ind D A = mesmax DA m A ind f A = mesmax f A m A

2.5.2 Use of the DMA Methods for Tracking the Geomagnetic Variations Let us denote St the network N of the measurement points st i : St = {st i }|T , yi = yi (t)= y(st i )(t) is the record of variations. T is the period of observation and analysis of all records yi (t). F is the algorithm of search for the anomalies in the records yi (t) using the DMA methods. F(yi |T ) = Ai is the anomaly detected in the record yi by the algorithm F with the support Supp Ai ⊂ T : Ai = yi |Supp Ai . The search for the anomalies Ai in the record yi (t) can be interpreted as an estimation by the positive numbers Φ yi (t) of the activity of its small fragments yi |[t −Δ, t+Δ] . Thus, we can pass from the source record to the nonnegative function t → Φ yi , which is a straightening of yi (t), since the larger values of Φ yi (t) correspond to the more active points of yi (t). The search for the anomalies in the record is reduced to the search for the elevations in its straightening, corresponding to the most active sections of the records. The DMA algorithms find the elevation Bi in Φ yi and associate their bottoms suppBi with the anomalies Ai on yi : Ai = yi |Supp Bi . The algorithm F estimates the strength f (Ai ) of the anomaly Ai , the boundaries of the st i anomaly Ai , and performs their morphological examination in the form of initial, central, and final stages with separation of the strong and weak phases in the central stage. The calculation of the value of the observatory anomaly for estimating the geomagnetic disturbances. The disturbance is estimated using the anomaly strength in each magnetogram by constructing two structures. MaxF structure. This algorithm divides the record into the anomalous and background parts: T = Ai ∪ (T − A). Let us specify the average background value of yi on T by the si :  si =

yi (t) : t ∈ T − Ai . |T| − |Ai |

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Let us specify Max(Ai ) as the maximum modulus deviation of the anomaly Ai from si : Max(Ai ) = max(|yi (t) − si | : t ∈ Supp Ai ). MaxDispF structure. Let d i be the background dispersion of yi on T, then d i is the average modulus deviation from si on T − Ai :  di =

|yi (t) − si | : t ∈ T − Ai . |T| − |Ai |

Let us specify MaxDispF of Ai as the ratio to d i of one or another variant of deviation of yi from si in the process of anomaly Ai : Max Disp F(Ai ) = [max(|yi (t) − si | : t ∈ Supp Ai )]/di , Max Disp F(Ai ) =



 |yi (t) − si | : t ∈ Supp Ai /di |Ai |.

In Fig. 2.32, an example of the algorithm application to data from Magadan magnetic observatory (MGD, Russia) for the geomagnetic storm on December 19– 21, 2015 is given (see second plot). Four degrees of anomalousness are marked by color according to the selected ranges. All numerical values of the parameter f (Ai ) are distributed in the interval [− 1; 1] and it is assumed that at f (Ai )∈ [0.7, 1] there is an extreme anomaly, at f (Ai ) ∈ [0.55, 0.7) there is a strong anomaly, at f (Ai )∈ [0.4, 0.55) it is average and at f (Ai )∈ [− 1; 0.4) there is no anomaly. Construction of the anomaly distribution maps at INTERMAGNET observatories. The anomaly Ai shall be certainly considered as a manifestation of the geomagnetic event A at the station st i , and the anomaly strength f (Ai ) as the quantitative indicator of such a manifestation. Based on fuzzy comparison, the structure f (Ai ) provides a possibility to answer quantitatively the question about the station anomalousness of Ai , i.e. characterize the anomaly strength f (Ai ) in the scale of the interval [− 1; 1]. Using the MaxF structure, it is possible to construct the disturbance distribution maps not only for a separate observatory, but also for the selected region or globally (Soloviev et al. 2013, 2016a; Gvishiani et al. 2016a, b). Figure 2.33 shows an example of construction of the disturbance parameter (station anomalousness) distribution map over the whole INTERMAGNET observatory network for the maximum of the storm on September 8, 2017.

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Fig. 2.32 Automated multi-criteria estimation of geomagnetic activity in MAGNUS system. The initial record (H component recorded during the December 2015 geomagnetic storm at Magadan observatory, upper plot) is followed by four geomagnetic activity indicators calculated in real-time: 1-min measure of anomalousness, 1-h dB/dt, 1-h amplitude and 3-h K index (Gvishiani et al. 2016a, b, 2018). The segments of an extremely high level of disturbance (strongly anomalous) are marked in red, segments of high level are purple, medium level is green and background level is blue (http:// geomag.gcras.ru)

2.5.3 DMA-Based Regression Derivatives DMA is widely applied to problems of the morphological analysis of time series, including the construction of geometric measures (Agayan et al. 2005) and dynamic corridors (Kagan et al. 2009) and the identification of trends. Within the latter DMA branch, new mathematical constructions of regression derivatives for discrete time series generally defined on an irregular grid were proposed by (Agayan et al. 2019). Like other DMA concepts, they are based on the modeling of expert’s logic: the expert’s view on the geophysical series y at node t is modeled by a fuzzy δt structure, which reflects the proximity of other nodes to t; the tangent to y at t is a R y,δ (t) linear regression of y weighted by δt , the angular coefficient of which R’y(t) = R’y(t, δ) would be a regression derivative of y at t (relative to δ). The expected relationship between the y trends and the areas of the sign constancy of R’y, with similar trends and derivatives obtained via classical mathematical analysis, in fact exceeded all expectations and was a pleasant surprise for the authors.

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Fig. 2.33 Distribution of the disturbance parameter (station anomalousness) for the magnetically quiet (top) and disturbed (bottom) periods. The parameter value is represented in the form of circles of different color from the high (red) to the low (blue) anomalousness

Theoretical Background. Let function f be defined around zero on the interval [a, b], (ab < 0) and be integrable on it. Let us denote the f restriction as f  for 0min(|a|,|b|) on the segment [− ,]: f  = f |[−,] and calculate the projection prf  of the function f  in the space L 2 [–, ] onto the two-dimensional space of the linear functions Lin2 [–, ].

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Let us denote the orthonormal basis in Lin2 [–, ], which is obtained from the natural basis (1, x) by Gramm–Schmidt orthogonalization (Kolmogorov and Fomin 1976) as e1 = e1 (), e2 = e2 (). Then, pr f  = ( f  , e1 ) e1 + ( f  , e2 ) e2 . Let us assume e1 = c, e2 = ax + b. The following three restrictions are then valid for a, b, c: 

e1  = 1 ⇔ ∫ c2 d x = 1 ⇔ c2 = −

1 , 2

 (e1 , e2 ) = 0 ⇔

c(ax + b)d x = 0 ⇔ b = 0, − 

e2  = 1 ⇔ ∫ a 2 x 2 d x = 1 ⇔ a 2 = −

3 . 23

Thus, 3 1  ∫ f (x)d x + ( pr f  )(x) = 2 − 23







∫ x f (x)d x x.

−

In addition, the function f should be differentiable in 0: f (x) = f (0) + f  (0)x + α(x)x, where α(x) → 0 when x → 0. Then, the tendency 1  ∫ f (x)d x → f (0) 2 − in the constant term of prf  projection is explained by the mean value theorem (Fikhtengoltz 1969). Let us analyze the prf  expansion coefficient of x:  



   3 3  (0)x + α(x)x d x ∫ ∫ x f x = x f + f (x)d (0) 23 − 23 − 3  3  2  3  ∫ x f (0)d x + ∫ x f (0)d x + ∫ α(x)x 2 d x 23 − 23 − 23 − 3  ∫ α(x)x 2 d x. = 0 + f  (0) + 23 −

=

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85

The latter integral tends to zero when  → 0: ∀ε > 0∃(ε) : ∀ < (ε) |α||[−,] < ε ⇒ 3  3  2 2 ∫ α(x)x d x ≤ ∫ εx ≡ ε. 23 − 23 − Thus, the following is proved: Statement. If the function f is differentiable at zero, then, if  → 0, the linear projection prf  tends to its tangent at zero. Replacement of the tangent to f by the projection prf  for small  makes it possible to determine the tangent for discrete functions, since prf  is nothing more than the linear regression of f on [–, ], which allows generalization at the discrete case. Discrete Regression Tangent. Let T be the observation period, as represented by a finite, irregular (in the general case) set of nodes t: T = {t},|T| < ∞. The function y is a time series defined on T: y : T → R, y = {yt = y(t), t ∈ T }. The limit transition  → 0 in the discrete case is replaced by the measure of  proximity δt t¯ —a fuzzy binary relation on T, which shows the extent to which node t¯ is close to node t in T.  y (δt ) =  regression  is constructed according to weighted graph  The linear t¯, yt¯ , δt t¯ , t¯ ∈ T . It is considered to be the tangent R y,t t¯ = at t¯ + bt to the function y at node t ∈ T. Omitting the standard things associated with linear regressions, we present the expressions for at and bt :      ¯ t¯δt t¯ yt¯ ¯ t¯δt t¯ t ∈T   t ∈T   δt t¯ yt¯ δt t¯ ¯ ¯ at = t ∈T 2    t ∈T   , ¯ t¯ δt t¯ ¯ ¯ t ∈T   t¯∈T t δt t ¯ ¯ ¯ t¯∈T t δt t t¯∈T δt t    ¯ t¯2 δt t¯ ¯δt t¯ yt¯ t ¯ t ∈T t ∈T    ¯ ¯ ¯ t¯∈T t δt t t¯∈T δt t yt¯ bt =     . ¯ t¯2 δt t¯ ¯ ¯ t¯∈T t δt t t ∈T    ¯ ¯ ¯ t¯∈T t δt t t¯∈T δt t Definition 1. The angular coefficient at of the regression tangent Ry,t is called the regression derivative of function y at node t and is denoted as (R’y)(t). Definition 2. The value Ry, t (t) of the tangent Ry, t at node t is called the regression value of y in t and denoted as (Ry)(t).

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Thus, 

 R  y (t) = at ,

(Ry)(t) = at t + bt .

Examples and Properties of Regression Derivatives and Values. In constructing regression derivatives and values, we use two modifications (global and local) of the same proximity construction, depending on two nonnegative parameters r and p: (A) Global proximity measure (Fig. 2.34):

  δt (r, p) t¯ = δt t¯ = 1 −

t¯ − t max(maxT − t, t − minT ) + r

(B) Local proximity measure (Fig. 2.35):

  δt (r, p) t¯ = δt t¯ =

  |t¯−t | p 1− r , i f t¯ − t ≤ r . 0, i f t¯ − t > r

Fig. 2.34 Global proximity measure: dependence on p with fixed r

Fig. 2.35 Local proximity measure: dependence on p with fixed r

p .

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87

Fig. 2.36 Stability of R operation to disturbances (the initial series is shown in gray, regression smoothing is black): a R operation applied to a smooth function, b R operation applied to a noisy function

It should be stressed that the proximity measure is one of the basic concepts of DMA and the corresponding time series processing algorithms (Agayan et al. 2016; Bogoutdinov et al. 2010; Gvishiani et al. 2008a, b; Soloviev et al. 2012a; Zelinskiy et al. 2014; Kulchinsky et al. 2010; Zlotnicki et al. 2005). Let us state the main properties of regression derivatives and values: 1. As follows from definitions 1 and 2, the regression derivative R  , y → R  y and the regression value R, y → Ry are linear operators in the function space F(T ) on T. 2. Operations R’ and R are based on regression and are therefore resistant to disturbances of the original functions (Figs. 2.36, 2.37). 3. In contrast to the classical continuous case, the discrete tangent Ry, t does not have to take the value y(t) at point t; thus, R can be regarded as a new type of smoothing operation that is not only not inferior in terms of its versatility but is

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Fig. 2.37 Stability of R ’ operation to disturbances: a R’ operation applied to a smooth function, b R’ operation applied to a noisy function. Grey and black colors denote the increasing and decreasing zones of the function y(t), respectively

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Fig. 2.38 Regression smoothing (black) and classical averaging (gray) of the straight line y = x with respect to the same proximity measure δ

Fig. 2.39 Regression smoothing based on local proximity measure: dependence on p with fixed r

superior to conventional averaging in its results (Fig. 2.38). 4. Operation R ’ is closely related to stochastic trends: the areas of positive (negative) sign constancy of R’y correspond to increasing (decreasing) trends of y, and the boundaries between them correspond to the y extrema (Fig. 2.37). 5. The relationship between the operations R’ and R and the proximity measure δ on T makes it possible to solve the problems of smoothing (Fig. 2.39) and to determine the trends and extrema for function y at different scales. This is why the regression derivatives are applied to the detection of geomagnetic jerks, as reported in the next Chapter, since jerks represent the extrema of the geomagnetic field secular variation on a certain scale.

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Fig. 2.40 Regression derivative of a smooth function: dependence on p with fixed r. The upper plot shows the original curve, and the lower plot shows the sign of the regression derivative for different p values. The increase and decrease of the analyzed curve are marked in dark gray and black, respectively; extrema are distinguished by light gray color

Figures 2.40 and 2.41 demonstrate the possibilities of the use of regression derivative to find the extrema and monotonic intervals of different levels. The upper plot of Fig. 2.40 shows the original curve under consideration. In the lower plot, dark gray and black colors denote the ascending and descending zones of the original curve for each p value, respectively. The extrema zones of four levels are marked by light gray color. 6. There are currently numerous ways to determine trends and extrema in stochastic time series (Gumbel 1958; Leadbetter et al. 1983; Lyubushin 2007; Mallat 1999). Using the function f shown in Fig. 2.40 as an example, we compare the use of regression derivatives with that of the WTMM (Wavelet Transform Modulus Maxima) algorithm to find trends and extrema using points of the maximum of the wavelet transform module (Lyubushin 2007; Mallat 1999). The WTMM algorithm was borrowed from the program Spectra_Analyzer (http://www.ifz.ru/ applied/analiz-dannykh-monitoringa/programmnoe-obespechenie/). For convenience, Figs. 2.42a, b duplicate Fig. 2.40, and the results obtained with the WTMM algorithm are shown in Fig. 2.42c, where the gray solid and black solid

2.5 Mathematical Methods of Analysis of the Geomagnetic …

91

Fig. 2.41 Regression derivative of the noisy time series: dependence on p with fixed r. The legend is the same as in Fig. 2.40

chains show a different-scale picture of the maxima and minima of the f function, respectively. Gray dotted (black dotted) chains characterize the strongest points on increasing (decreasing) f trends of different scales. The comparison should be made at the upper level (p ≥ 15): as follows from Figs. 2.42b, c, the regression derivatives yielded two more extrema (at the edges of the graph) than the WTMM algorithm, whereas the trends in the internal areas almost coincided.

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Fig. 2.42 Comparison of the performance of regression derivatives and the WTMM algorithm: a the original function, b result of the application of regression derivatives, and c result of the application of the WTMM algorithm. Legends of figures (a) and (b) are similar to Fig. 2.40. Figure (c) represents the structure of long chains of skeletons of modules of the wavelet transform with the Gaussian zero derivative. According to the program Spectra_Analyzer, the notations are: a is the scale of the observation (a ∈ [γ × (T /6), T /6]), γ = 0.12 is the method parameter, and T = 201 is the number of elements in the set. Black solid lines show the minima of the averaged curves, gray solid lines show the maxima of the averaged curves, black dotted lines show the maxima of the negative trends, and gray dotted lines show the maxima of the positive trends

References Agayan SM, Bogoutdinov SR, Gvishiani AD, Graeva EM, Zlotniki Z, Rodkin MV (2005) Study of the signal morphology on the basis of fuzzy logic algorithms. Geofiz Issled Moscow: IFZ RAN 1:143–155 (in Russian) Agayan SM, Gvishiani AD, Bogoutdinov SR, Kagan AI (2010) Sglazhivaniye vremennyh ryadov metodami diskretnogo matematicheskogo analiza (smoothing time series using methods of discrete mathematical analysis). Russ J Earth Sci 11(RE4001). https://doi.org/10.2205/2009es 000436 (in Russian) Agayan S, Bogoutdinov S, Soloviev A, Sidorov R (2016) The study of time series using the DMA methods and geophysical applications. Data Sci J 15:16. https://doi.org/10.5334/dsj-2016-016 Agayan SM, Bogoutdinov SR, Krasnoperov RI (2018) Short introduction into DMA. Russ J Earth Sci 18:ES2001. https://doi.org/10.2205/2018ES000618 Agayan SM, Soloviev AA, Bogoutdinov SR, Nikolova YI (2019) Regression derivatives and their application to the study of geomagnetic jerks. Geomag Aeron 59(3):359–367. https://doi.org/10. 1134/S0016793219030022

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and the near abroad. Vestn Otd nauk Zemle 10:NZ4001. https://doi.org/10.2205/2018nz000357 (in Russian) Hattingh M, Loubser L, Nagtegaal D (1989) Computer K-index estimation by a new linear-phase, robust, non-linear smoothing method. Geophys J Int 99:533–547 Jankowsky J, Sucksdorf C (1996) Guide for magnetic measurements and observatory practice. IAGA, Warsaw, p 238 Kagan AI, Agayan SM, Bogoutdinov SR (2009) Definition of stochastic continuity by fuzzy logic methods and geophysical applications. In: Nechitailenko V (ed) Materials of the international conference “Electronic geophysical year: state of the art and results”. GC RAS, Moscow, p 117. https://doi.org/10.2205/2009-regyconf Kerridge DJ (2001) INTERMAGNET: Worldwide near-real-time geomagnetic observatory data [Electronic source]. Proceedings of ESA space weather workshop. ESTEC, Noordwijk, Netherlands. http://www.intermagnet.org/publications/IM_ESTEC.pdf. Date of access 01 April 2019 Kolmogorov AN, Fomin SV (1976) Elementy teorii funktsii i funktsional’nogo analiza (Elements of the theory of functions and functional analysis). Nauka, Moscow (in Russian) Krasnoperov R, Peregoudov D, Lukianova R, Soloviev A, Dzeboev B (2020) Early Soviet satellite magnetic field measurements in the years 1964 and 1970. Earth Syst Sci Data 12:555–561. https:// doi.org/10.5194/essd-12-555-2020 Kudin DV, Soloviev AA, Sidorov RV, Starostenko VI, Sumaruk YP, Legostayeva OV (2020) System of accelerated production of quasi-definitive data of INTERMAGNET standard. Geomagnetism Aeronomy Kulchinsky RG, Kharin EP, Shestopalov IP, Gvishiani AD, Agayan SM, Bogoutdinov SR (2010) Fuzzy logic methods for geomagnetic events detections and analysis. Russ J Earth Sci 11:RE4003. https://doi.org/10.2205/2009ES000371 Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, New York, p 336 Léger JM, Bertrand F, Jager T, Le Prado M, Fratter I, Lalaurie JC (2009) Swarm absolute scalar and vector magnetometer based on helium 4 optical pumping. Procedia Chem 1:634–637. https://doi. org/10.1016/j.proche.2009.07.158 Lesur V, Heumez B, Telali A, Lalanne X, Soloviev A (2017) Estimating error statistics for Chambonla-Forêt observatory definitive data. Ann Geophys 35(4):939–952. https://doi.org/10.5194/angeo35-939-2017 Lincoln JV (1967) Geomagnetic indices. In: Matsushita S, Campbell WH (eds) Physics of geomagnetic phenomena, vol 1. Academic, New York, pp 67–100 Loewe CA, Prolss GW (1997) Classification and mean behavior of magnetic storms. J Geophys Res 102:14209 Love JJ (2008) Magnetic monitoring of Earth and space. Phys Today 61(2):31–37. https://doi.org/ 10.1063/1.2883907 Love JJ, Remick KJ (2007) Magnetic indices. In: Gubbins D, Herrero-Bervera E (eds) Encyclopedia of geomagnetism and paleomagnetism. Springer, New York, pp 509–512 Love JJ, Chulliat A (2013) An international network of magnetic observatories. Eos Trans Am Geophys Union 94(42):373–374. https://doi.org/10.1002/2013eo420001. 2013 Lyubushin AA (2007) Analiz dannyh sistem geofizicheskogo i ekologicheskogo monitoringa (Analysis of the data from geophysical and ecological monitoring systems). Nauka, Moscow (in Russian) Mallat S (1999) A wavelet tour of signal processing. Elsevier, Academic Press, Amsterdam, p 620 Mayaud PN (1980) Derivation, meaning and use of geomagnetic indices. AGU Geophys Monograph Ser no. 22. AGU, Washington Menvielle M, Papitashvili N, Hakkinen L, Sucksdorff C (1995) Computer production of K indices: review and comparison of methods. Geophys J Int 123:866–886 Nowo˙zy´nski K, Ernst T, Jankowski J (1991) Adaptive smoothing method for computer derivation of K-indices. Geophys J Int 104:85–93

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Yermolaev YI, Yermolaev MY (2003) Statistical relationships between solar, interplanetary, and geomagnetic disturbances, 1976–2000: 3. Cosm Res 41:539–549. https://doi.org/10.1023/B: COSM.0000007952.09069.b8 Yermolaev YI, Yermolaev MY (2010) Solar and interplanetary sources of geomagnetic storms: space weather aspects. Izv Atmos Ocean Phys 46:799–819. https://doi.org/10.1134/S00014338 10070017 Zelinskiy NR, Kleimenova NG, Kozyreva OV, Agayan SM, Bogoutdinov SR, Soloviev AA (2014) Algorithm for recognizing Pc3 geomagnetic pulsations in 1-s data from INTERMAGNET equatorial observatories. Izv Phys Solid Earth 50(2):240–248 Zlotnicki J, Le Mouël J-L, Gvishiani A, Agayan S, Mikhailov V, Bogoutdinov Sh, Kanwar R, Yvetot P (2005) Automatic fuzzy-logic recognition of anomalous activity on long geophysical records: application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island). Earth Planet Sci Lett 234(1–2):261–278

Chapter 3

Mathematical Models of the EMF

The source of the data on the internal geomagnetic field is the ground-based magnetic observations, aeromagnetic surveys, as well as the spaceborne measurements. The main EMF is determined by the temporal (over a year) and spatial (over the area of at least 106 km2 ) averaging of the field vector by each component. Carrying out measurements at each point of the Earth’s surface is almost impossible, and, actually, it is not a necessary problem. To determine the values of the magnetic elements at any location at a given time moment, the EMF maps are created based on the mathematical models assimilating the data from the ground and space observations. There are a great number of the geomagnetic models created by different geophysical organizations and teams worldwide for different purposes. In general, they can be divided into two large categories: global and regional models. The regional geomagnetic models describe the main Earth’s field magnetic only for particular, sometimes very vast, geographical areas. The additional measurements are required at the secular variation repeat stations to build the regional models, which allows to achieve the resolution of about 350 km. The global models are used to represent the main EMF over the entire globe and have a resolution of about 3,000 km. The global models of the main EMF are built for both particular epochs and a certain time interval. In the latter case, a model is constructed for a series of successive epochs and further interpolated between them. The historical models are not very accurate, but cover long time intervals. The other models are highly accurate, but require constant updating. Among numerous global models, those that are positively evaluated as the most reliable by the international geophysical community, are generally accepted and widely used. Such models include International Geomagnetic Reference Field, IGRF (Thébault et al. 2015), and World Magnetic Model, WMM (Chulliat et al. 2015). Each of them is based on the magnetic field expansion into the spherical harmonics with determination of the Gaussian coefficients according to the satellite and observatory measurements. The maps of the large-scale magnetic field and

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Gvishiani and A. Soloviev, Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation, https://doi.org/10.1007/978-3-030-58969-1_3

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those of magnetic anomalies based on the longstanding international efforts towards global magnetic survey are collected in the Atlas of the Earth’s Magnetic Field for 1500–2010 (Zhalkovsky et al. 2009; Berezko et al. 2011; Soloviev 2013). By now, a large amount of the vector and scalar data has been accumulated, obtained at different altitudes from the Earth’s surface to the upper ionosphere, with the different densities covering the different territories. The significant progress in the satellite data processing and the methods of their compilation and integration with the ground data allowed expanding the capabilities of global and regional modeling of the EMF.

3.1 Dipole Model of the Main EMF The dipole model of the main EMF can be considered as the first and the simpliest approach to constructing its global mathematical model, which allows to obtain the predictions well-agreed with the direct measurements. The measurements of the geomagnetic field demonstrate that on the Earth’s surface it can be generally represented as a field of a strip magnet placed in the center of the planet. The configuration of this field changes slowly; according to the modern concepts, it is probably due to the movement of the molten material in the outer liquid core of the Earth at the depths of more than 3,000 km. Thus, the main magnetic field is specified by the sources located in the deep Earth’s interior, and the dipole model describes the configuration of the geomagnetic field from these particular sources (Fig. 3.1). Since the Earth’s surface has a spherical shape, it is more convenient to describe the magnetic field in the polar coordinate system. Let us introduce the designations: r is the radial distance from the center of the Earth, θ is the angle measured from the

Fig. 3.1 Configuration of the Earth’s geocentric dipole (Walt 2005)

3.1 Dipole Model of the Main EMF

99

dipole axis. Then, the radial and azimuthal components of the geomagnetic field can be represented by the following equations (Thomson et al. 2013):  H = −B0  Z = −2B0

RE r RE r

3 sinθ,

(3.1)

3 cosθ,

(3.2)

where B0 is the equatorial value of the radial component of the magnetic field B0 ≈ 30000 nT, R E is the radius of the Earth taken to be equal to 6,371 km. The value of the total of intensity of the magnetic field is specified as F=



   RE 3 H 2 + Z 2 = B0 1 + 3cos2 θ r

(3.3)

It can be seen from the Eq. (3.3) that the values of the geomagnetic field at the poles (θ = 0◦ ) are two times higher than the values at the equator (θ = 90◦ ), F p ≈ 60000 nT and Fe ≈ 30000 nT respectively. The magnetic field is perpendicular to the Earth’s surface at the poles, while at the equator it is parallel. The direction of the field in space changes with the magnetic latitude. The horizontal component of the magnetic field is very weak to the north and south of 75° magnetic latitude, so there’s  3 no use to rely on the compass at the polar latitudes. The value RrE is equal to 1 on the Earth’s surface. But with an increase in r, the dipole component of the magnetic field decreases inversely as the cube of the distance, that is, for the distance r = 2R E , the field intensity will be only 12.5% of the value of the field on the surface. The importance of the dipole model of the Earth’s magnetic field is given by the fact that it allows to establish a coordinate system related with the geocentric dipole. It is called a “geomagnetic coordinate system”. This coordinate system is most often used in studying the solar-terrestrial relations and magnetosphere, since the non-dipole component of the main EMF, which appears especially intensely at the Earth’s surface and below, does not play any role for such studies.

3.2 Models of the Main Field Based on the Spherical Harmonic Analysis 3.2.1 Model Design Principles A fundamental approach to modeling the EMF by expanding into the spherical functions was proposed by C. F. Gauss. This approach is still widely applied nowadays. It is based on the assumption of the field potentiality, i.e. in the region free from the electric currents the following is true:

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V = 0,

(3.4)

where V is the potential. Then, the magnetic field induction vector B is a negative gradient of V, i.e. B = −∇V

(3.5)

To solve the Eq. (3.5), the potential V can be represented as a spherical functions series (Backus et al. 1996). At that, it should be born in mind that the sources of the EMF have a different nature. Some of them are located below the Earth’s surface in the liquid core and crust, hereinafter referred as the internal sources (int), and some sources are in the ionosphere and magnetosphere, hereinafter referred as the external sources (ext). The potential V is represented as the sum of two components. N  n

 a n+1  int

V =V

int

+V

ext

=a

n=1 m=0

r

 m gnm cos(mφ + h m n sin(mφ))Pn (cos θ )

N  n

  a n+1  m qn cos(mφ + snm sin(mφ))Pnm (cos θ), +a r n=1 m=0 ext

(3.6)

where a = 6371.2 is the mean Earth’s radius, (r, θ, φ) are the spherical coordinates, Pnm are the associated Legendre polynomials, N int is the maximum degree and order of the Gaussian coefficients gnm i h m n describing the fields from the internal sources, and N ext , qnm i snm are same for the external sources. For the sphere with the radius r = a, for which all sources are located either inside or outside the sphere, the expression for the field components can be obtained by differentiation (3.6) by the spherical coordinates: Bθ = −

1 ∂V −1 ∂ V ∂V , Bφ = , Br = − . r ∂θ r sin θ ∂φ ∂r

Then, e.g., the Br component specified by the internal sources, will be expressed as N  n  int

Br (r, θ, φ) =

n=1 m=0

(n + 1)

a n+2  r

 m gnm cos(mφ + h m n sin(mφ))Pn (cos θ ) (3.7)

In the Cartesian coordinate system, in which geomagnetic observations are usually made, the field components would be X = −Bθ ; Y = Bφ ; Z = −Br .

3.2 Models of the Main Field Based on the Spherical Harmonic Analysis

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If there were a set of values Br (ri , θi , φi ), where i = 1 … K, then the Eq. (3.7) would turn into a system of equations; its solution would give the Gaussian coefficients and thereby the construction of an analytical model of the EMF. However, the problem is much more complicated, as the EMF sources are of the different origins and differ in their location, while all of them contribute to the measured field values. The sources referred to the internal ones can be divided into two types. First, these are magnetized rocks, which represent mainly the residual magnetization with the partial presence of the magnetization induced by the modern magnetic field. The field of these sources can be considered constant. And second, these are current systems in the Earth’s liquid core determining the main field and its secular variation. All other sources of the observed magnetic field are external and represent the current systems in the ionosphere and magnetosphere. Besides, the time-varying components of the EMF induce the currents in the crust and oceans. The analysis of the properties of the generated fields can contribute to separation of the contributions from the corresponding sources. The EMF sources differ not only by location but also by the amplitude of the created field. The main EMF is the most intense on the Earth’s surface. Its changes are the longest wavelengths both in the temporal and spatial scales. The lithosphere field or the field of the magnetized rocks is much weaker, constant in time, but changes significantly in space. Due to the spatial variability of this part of the field, it contributes to the spatial spectrum of the EMF almost along the total wavelength range of less than 2,000 km on the Earth’s surface. The contribution of the external sources to the observed EMF is mainly determined by the current systems in the ionosphere and magnetosphere. The induced currents in the surface layer of the Earth are primarily determined by the rapidly changing field of the external sources. Their magnitude depends strongly on the electric conductivity of the medium where they are induced, that’s why it changes not only during the daytime, but also in space. They are referred to the field from the external sources. While the data from the observatories, repeat stations of secular variation, and the results of the marine and aeromagnetic surveys are used to build a model, the problem does not go beyond the potential field and Eq. (3.6). But at that, it is necessary to put up with the highly uneven distribution of the observation sites over the globe. The use of the satellite data removes the problem of the “white spots”, but creates new problems related with the flight altitude. At the altitude of 400 km, the measured field values correspond approximately to averaging over a region of the same size, which can be useful to specify the contribution of the main EMF, but complicates construction of the high-resolution anomaly field models. Besides, due to a high motion speed, it is not always possible to distinguish a spatial change of the field from its time variations. And finally, the low-orbit satellite trajectories intersect the region with the flowing currents, connecting the magnetosphere and ionosphere. In the expression for the field values measured at the satellite flight altitude, a term should be included that describes the toroidal component generated by these currents. As a result, the satellite measurements are described by the equation

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B(r, θ, φ, t) = B cor e (r, θ, φ, t) + B cr ust (r, θ, φ) + B ext (r, θ, φ, t) + B tor (r, θ, φ). (3.8) It can be seen from the Eq. (3.8) that the satellite measurements contain abundant information about a wide variety of the magnetic field manifestations, which allows to use them in studying the processes occurring in all geospheres. To build the analytical models, each of the summands can be represented as a series expansion in basic functions with subsequent determination of the relevant coefficients. By including all sources of the observed EMF into consideration, the expression for the potential (3.6) should be rewritten in the form

V =a

⎧ int N n   a n+1 m   ⎪ ⎪ ⎪ gnm cos mφ + h m Pn (cosθ ) ⎪ n sin mφ r ⎪ ⎪ ⎪ n=1 m=0 sv ⎪ N n  ⎪    ⎪ ⎪ +   g˙ m cos mφ + h˙ m sin mφ (t − t0 ) a n+1 P m (cosθ ) ⎨ n n n r

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

n=1 m=0

n  2   n  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ qnm cos mTd + snm sin mTd ar Pnm (cosθd ) + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n=1 m=0 ⎪ ⎪ ⎪ ⎪   1  ⎪ ⎪       2 ⎪ ⎪ r a m m m ⎪ q˜1 cosTd + s˜1 sinTd P1 (cos θd ) ⎪ ⎭ ⎩ + E st (t) a + Ist (t) r m=0

(3.9) m Here a = 6371.2 km, (r, θ, φ) are the geocentric spherical coordinates, )  m m  Pn (cosθ are the Schmidt normalized associated Legendre polynomials, gn , h n and qnm , snm are theGaussian  coefficients describing the internal and external sources, respectively, g˙ nm , h˙ m n are the coefficients of the linear part of the secular variation (SV) relative to the epoch t 0 , θd is geomagnetic co-latitude and Td is geomagnetic local time. The last two terms in the expression of the potential (3.9) are related with the contribution from the magnetospheric current; in a first approximation, it is assumed that the spherical harmonics with n = 1 and m = 0 are sufficient for its spatial description, taking into account separation of the Dst index, which estimates the magnetospheric ring current flowing in the equatorial plane perpendicular to the dipole axis, into the main and induced parts: Dst = E st + I st . Such representation is related with the fact that the ring current and the geomagnetic coordinates are eminently suitable to describe the magnetic contribution of this current. Besides, such approach is in agreement with the magnetosphere models. Actually, all methods of constructing the EMF models are based on the Eq. (3.9), and differ by the number of the coefficients-parameters, approach to solving the inverse problem, and taking into account the additional terms not included in (3.9). Choosing the maximum degree of a polynomial. In the case of construction of the EMF model for a single epoch the time dependence is not considered, and the contribution of the external sources is considered as an interference. The problem  is limited by determining the coefficients gnm , h m n . If only the results of the vector measurements are used when constructing the model, the problem of solving the truncated Eq. (3.9) is linear with respect to the coefficients and can be solved by

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the least squares method, or using the weighted least squares. When a set of scalar (module) data is included in the initial data, the problem becomes non-linear and the iterative least squares method is applied. The number of the unknowns is determined by the maximum degree N of the polynomials. When choosing the value N, two factors are essential. First, what part of the field of the internal sources is modeled, i.e. whether it is a model of the main or anomalous field. And second, what was the satellite flight altitude during the measurements, used to build a model. Actually, as the modeling practice has shown, the rate of decrease of the coefficients with an increase of the degree N depends significantly on the flight altitude. Figure 3.2 shows the dependence of change in the spectrum of particular components on the satellite flight altitude (figure from (Olsen et al. 2014)). Considering that all measurement conditions are ideal and allow the resolution of the magnetic field structures with an amplitude square higher than 0.1 nT2 , according to the Ørsted satellite, it makes sense to increase the degree of polynomials only to N = 20. While for the CHAMP satellite at the height (h) of 450 km, N can be increased to 40, at h = 300 km it can be increased to N = 60 and at h = 250 km to N = 80. It must be kept in mind that the criterion 0.1 nT2 is selected as an example and, in practice it should be correlated with the accuracy of the satellite data. Approaches to solving the inverse problem. As for the problem of constructing the model of the main Earth’s magnetic field, two approaches are possible. In the first case, the spherical function series is limited  n < N and the inverse problem is solved with respect to the coefficients gnm , h m n . At that, the contributions of all other sources of the EMF are considered as noise. The problem is that this noise

Fig. 3.2 Spectra of the geomagnetic field depending of the flight altitude of the satellite (Olsen et al. 2014). The spectra of the geomagnetic field on the Earth’s surface are shown in black, for different altitudes of the CHAMP satellite are in blue and for the Ørsted satellite are in red; h is the corresponding flight altitude

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cannot be considered as uncorrelated; therefore, some a priori information about the covariance interference matrix is required. The approach to solving this problem was proposed by Gillet et al. (2013). The main point of the proposed approach is to consider a set of different EMF models as an ensemble of several implementations of the stochastic process for each of the coefficients. At that, the coefficients as the time functions are assumed to have a zero mean, zero covariance between the different coefficients and the same autocorrelation function for all coefficients of n degree. The model extension is inclusion of the time dependence of the spherical coefficients, usually represented as B-splines of different orders. This increases the number of the unknowns in the model, but allows to generate a model regularized in time, and further to forecast the secular variation. With the second approach, the inverse problem is solved for a long series. At that, the contribution of the external sources is considered as noise. Then, the obtained coefficients are divided into two intervals with respect to a certain value of nbd . At that, it is assumed that the terms of the series with n ≤ nbd describe the model of the main EMF, and the rest terms describe the model of the anomalous field. As Fig. 3.2 shows, a change in the nature of the dependence of Rn on the polynomial degree occurs in the region n ≈ 16. Generally, it is assumed that already after n = 13–14, the long-wavelength components of the lithosphere magnetic field begin to contribute to the EMF spectrum. However, some authors believe that the maximum degree of polynomials referred to the main field can be increased up to 20 and even to 30. At that, including the contribution of the long-wavelength components of the lithospheric sources to the main field will not have significant effect on the spatial structure of this part of the EMF, and excluding this contribution from the anomalous field increases the resolution of the smaller scale anomalies, which are of the maximum interest for practical use. A completely new approach, a so-called complex inversion, is currently used in construction of the most modern models (Sabaka et al. 2004). This approach assumes including all coefficients being a part of the equation for the potential (3.9) in the solution of the inverse problem. The second term in the Eq. (3.9) reflects the linear dependence of the spherical coefficients on time. In fact, this can be compared with expansion of gnm (t) and h m n (t) in the Taylor series with conservation of only the first order terms. Expansion in the Taylor series including the first and second orders, as a rule, describes well the changes within a short time interval. It is preferable to use the splines for longer periods. A description of this dependence by the B-splines of the different orders is also used for interpolation within the short time intervals: gnm (t) =

L 

m gn,l Ml (t),

(3.10)

l=1

where M(t) is the basis B-spline functions. The similar expression can be written for h m n . Depending on the degree of the basis splines, the number of the coefficients determined when solving the inverse

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problem increases. At that, it should be kept in mind that using the splines of higher orders may lead to a boundary effect. Two last terms in (3.9) express the potential of the external sources, and, first of all, describe the large-scale magnetospheric sources. It is generally accepted that n ≤ 2 is sufficient to describe their contribution. The last term describes the time dependence of the contribution of the magnetospheric sources. Further complication of parameterization of the model is related with introduction of two more coefficients corresponding to the contribution of the remote magnetospheric sources. It should be noted that the basic level of the magnetosphere field is considered constant in this case. When constructing some models, more coefficients are added to describe the time variation of the basic level and include the influence of the ionospheric and magnetospheric currents (Laundal et al. 2018). In addition to accounting the contribution of the external sources, an important factor is correction of the orientation of the satellite coordinate system, i.e. determination of the Euler angles, which adds parameters to the problem. The contribution of the magnetic field of the ocean tides, which makes about 3 nT at the satellite flight altitude, can be also taken into account (Kuvshinov and Olsen 2005).

3.2.2 Model Families International Geomagnetic Reference Field (IGRF). The global models of the magnetic field of the internal sources are divided into four types: analytical models of the main EMF, analytical models of the sum of the main and anomalous EMF, analytical models of the anomalous EMF and digital models of the anomalous EMF. At that, in the analytical models of the main Earth’s magnetic field either the dependence of the coefficients on time is included or this dependence is assumed to be linear. The latter includes the IGRF model built as a weighted average of the candidate models developed by the different research teams. Its construction principle consists in averaging the coefficients of the models of different authors with some weights. The first digital model was built as a result of the World Magnetic Survey program, which started in the International Geophysical Year (1957–1958) and lasted 12 years. As a result, the data from the marine, aeromagnetic and satellite profiles were collected together. The first POGO geophysical exploration satellites, which operated in the orbit between 1965 and 1971, made it possible to improve significantly the quality of the main EMF models, despite the fact that only the magnetic field module was measured on the POGO board. Based on all compiled data, the first generally accessible global spherical model of the main EMF with the maximum degree of polynomials N = 8 was built by 1969. The widespread practical use of the model was facilitated by the rapid development of the computer engineering and creation of the specially configured software. During the subsequent years, new satellite projects helped forward intensive development in constructing the global models of the main field. In fact, after operation

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of each new satellite, the new models based on the obtained new data appeared. The next step in the modeling development is related with the launch of the Magsat satellite in 1979. Both scalar and vector magnetic field measurements were performed on this satellite for the first time. It resulted in the first global catalog of the high quality vector data. Previously, the information on the spatiotemporal structure of the field vector components was available only from the data obtained by the magnetic observatories and during the expeditions of the nonmagnetic schooner “Zarya” (Kasyanenko and Pushkov 1987). The magnetometric equipment was improved with each new satellite, and in parallel, the methods for creating the global EMF models were developed and enhanced. The geophysical satellites launch program was continued, and a Working Group was created under the International Association for Geomagnetism and Aeronomy (IAGA) to deal with the integration of the efforts of the IGRF model developers. Until now, this model is still being updated thanks to the contributions of the modelers all over the world. Based on the high-precision geomagnetic measurements made on the satellites equipped with the vector magnetometers, all other global models were significantly improved. The IGRF model is an international, confirmed mathematical model of the geomagnetic field of the internal sources. The model is generally accepted, accessible and widely used in practice. It describes the long-wavelength part of the main EMF and allows to calculate the components of the EMF vector in the three-dimensional Cartesian coordinate system, as well as the values of magnetic declination and inclination at any geographical point. It is assumed that the contribution of ionospheric and lithospheric sources is excluded. The strict initial data selection is carried out to reduce the influence of the external sources of the EMF. This selection is mainly based on the Kp index of the global geomagnetic activity and the Dst index of the magnetospheric ring current intensity. These two parameters should have the minimum deviations from the quiet level. The observatories providing data to create the models of the main EMF are shown in Fig. 3.3. Each analytical model is based on the procedure of approximating the model field to the main EMF by its expansion into the spherical harmonics. In this procedure, the main component of the geomagnetic field is approximated by a double series of the normalized associated Legendre functions Pnm (x) with a pair of the Gaussian coefficients for each function—gnm and h m n . In the spherical coordinates, the geomagnetic scalar potential can be written by the following equation: V (r, θ, φ, t) = R E

 n  k   R E n+1

(gnm (t) cos(mφ) r n=1 m=0  m +h n (t)sin(mφ) Pnm (cosθ )

(3.11)

where V is the scalar potential of the geomagnetic field, R E is the Earth’s radius, r is the distance from the center of the Earth, θ is the colatitude, i.e. the polar angle, φ is longitude, Pnm are the Schmidt quasi-normalized associated Legendre functions of degree n and order m, gnm (t) and h m n (t) are the time-dependent Gaussian coefficients, published by the IAGA for a particular epoch.

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Fig. 3.3 Map of magnetic observatories providing data for constructing the internal field models since 2000 (http://www.wdc.bgs.ac.uk/dataportal/)

The double series of functions, being infinite, cannot be used as a formula for practical calculations; therefore, to be used in the model, it is cut off on a certain term of degree N. In this case, only the terms of degree n ≤ N and order m ≤ n participate in the sum. The choice of the limit expansion degree, as well as appropriate determination of the coefficients of spherical harmonics gnm and h m n determine the accuracy of the built model. In 1969, the first IGRF model was elaborated, adopted by the IAGA and recommended for use. Since the main field is gradually changing, the Gaussian coefficients used in the models also change accordingly. The IGRF model is updated every 5 years; besides, since the 7th generation, it is extended into the past until 1900. Each new generation represents a set of spherical harmonic coefficients for the current epoch. The coefficients of the earlier generations are simultaneously reviewed. The IAGA Working Group approves a set of the Gaussian coefficients which is best agreed with the available measurements from the global network of geomagnetic observatories, as well as the artificial Earth satellites. In addition to the spherical coefficients for deriving the spatial structure of the EMF components, the model includes the forecast coefficients describing the geomagnetic field change over the next 5 years. The latest

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Fig. 3.4 Distribution of the horizontal intensity of the main EMF (left) and its annual rate of change (right) according to the latest generation model IGRF-13 (http://www.geomag.bgs.ac.uk/research/ modelling/IGRF.html)

coefficient generation published so far is the IGRF-13. The data from the magnetic observatories and satellite missions, including CHAMP (2000–2010), Ørsted (1999– 2010), SAS-C (2001–2003) and, which is especially important for the 2015–2020 model, Swarm constellation (since 2013) were used to build the model. Up until 2000, the IGRF model, in its full form, used expansion into spherical harmonics with a depth up to the Legendre functions of the 10th degree and order (total 120 Gaussian coefficients). In the latest versions of the model, the depth of expansion into spherical harmonics was increased to N = 13 (195 Gaussian coefficients). This value was obtained empirically, as a compromise between the intention to achieve higher certainty of the main field models and to avoid the influence of the interferences generated by the crustal fields. The main field coefficients are rounded to tenths of nT, which corresponds to the resolution limit of the observations. The predictive model of the secular variation is limited to N = 8 (80 coefficients), the coefficients are also rounded to tenths of nT/year. The IGRF model accuracy is within 30 arcminutes for the magnetic declination and inclination and within 0.2 μT for intensity. Figure 3.4 shows the distribution of the horizontal component intensity of the main EMF and its annual rate of change according to the latest 13th generation of the IGRF model. When the coefficients for some epoch do not require any further refinement, the corresponding model obtains the status of definitive (Definitive Geomagnetic Reference Field, DGRF). DGRF is a refined model of the geomagnetic field that comes out upon expiration of a five-year period. Each IGRF model will be eventually replaced by a definitive model in its future revision. IGRF is the model of the main EMF most commonly used in practical applications and is available online at http://www.ngdc.noaa.gov/IAGA/vmod/. On the website, the coefficients are available for download in various formats, along with the online calculator for computing values for the geomagnetic field and secular variation for a given set of coordinates and date. World Magnetic Model (WMM). The WMM comprises joint efforts of the US National Centers for Environmental Information (NCEI) and British Geological Survey (BGS). Similar to IGRF, the WMM model is updated every 5 years. The

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data from the Ørsted, CHAMP and Swarm satellites, as well as the ground magnetic observatories, are used for modeling. The selected observatories are the same as shown in Fig. 3.3. The WMM allows to derive the values of the EMF orthogonal components, as well as the values of the magnetic declination and inclination with the same accuracy as the IGRF model. The WMM fully utilizes the expansion into spherical harmonics with the depth up to the Legendre functions of the 12th degree and order (total 168 Gauss coefficients). To consider the magnetic field variations in between the model updates, a table of 168 corrections of the main model coefficients (up to the 12th degree and order) is published additionally during the next 5 years after the model adoption. It is assumed that this model describes well all components of the EMF in a wide range of the altitudes, from 1 km below and up to 850 km above the Earth’s surface. The model resolution is estimated as 3200 km on the Earth’s surface. Based on the coefficients of the model, all field components can be calculated at any point, at any time, and at a given altitude range relative to the ellipsoid of the World Geodetic System (WGS-84) reference frame. Besides, the model contains the secular variation coefficients. Due to the singularity in the direction of the geomagnetic field vector at the geographic poles, an additional angle is introduced to describe the declination D, which is defined as GV = D − λ, when ϕ > 55◦ and ϕ < −55◦ , where ϕ is the geodetic latitude. This angle can be understood as the angle in the polar stereographic projection. As for the altitude of interest, the user sets the altitude above the mean sea level to calculate the field components, which is later converted to the altitude above the ellipsoid. Although the difference in the calculation of the field components is very small, it is important in some practical problems. The model accuracy is estimated by the authors as 0.1 nT. The model coefficients, necessary software and maps are available on the website http://www.ngdc.noaa.gov/geomag/WMM/. As the IGRF, the WMM model is also widely used in practical applications including navigation and direction calculation systems, as well as by the oil and gas companies when drilling wells. Also, the WMM is used as a standard geomagnetic model by the ministries of defense of the USA and other NATO countries, as well as in the civil navigation systems. The distribution of the total intensity of the main EMF according to the WMM model is shown in Fig. 3.5. The enhanced version of the WMM model (Enhanced Magnetic Model, EMM) is built using the satellite data, land, sea and aeromagnetic survey data and the latest digital anomalous field model EMAG2-v3 developed by the NCEI. Until 2016, the EMM included the spherical coefficients up to the degree N = 15. In 2017, this model was developed further. As the WMM, the EMM model allows to calculate the values of the EMF components at any point, but at that, in the new EMM version, the internal sources are not divided into the lithospheric and liquid core sources. If the WMM model gives the expansion of the magnetic potential to degree and order 12, providing the resolution of about 3,000 km at the Earth’s surface, in the new EMM model the degree of polynomials is increased up to N = 790, which corresponds to the resolution of 51 km. A map of the declination values for a particular region

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Fig. 3.5 WMM model for the epoch of 2020.0 showing the contours of the magnetic declination of the main EMF (https://www.ngdc.noaa.gov/geomag/WMM/). Asterisks denote magnetic poles

derived from the WMM and EMM models is given in Fig. 3.6. The EMM model also includes the forecast coefficients of the secular variation of the order and degree N = 15. As a rule, the forecast is limited by N = 8. Another EMM extension, the High Definition Geomagnetic Model (HDGM), describes cumulatively the main EMF and the field of the lithospheric sources. As compared to the EMM, its development involved a huge number of the marine and aeromagnetic survey profiles (more than 50 million points). In this model, the order of the polynomials is N = 720; the main EMF and its secular variation coefficients are extended back to 1900. Also, it includes the averaged model of the external sources. As expected, this model will be annually updated for refining the time variations of the geomagnetic field. The model resolution makes 28 km over the significant territory. HDGM is a commercial model; therefore its coefficients are not freely available. For this reason, an independent comparison of its predictions with other models’ predictions is difficult. British Global Geomagnetic Model (BGGM). The BGGM model is developed by the British Geological Survey based on the satellite and observatory data (http://geomag.bgs.ac.uk/data_service/directionaldrilling/bggm.html). This is a global model of the EMF and its temporal variations, which involves the spherical polynomials of the degree N = 13; the time dependence is described by 6-order B-splines. The BGGM model is annually updated, and interpolation of the spherical coefficients is carried out only within the interval of 1 year. When building the

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Fig. 3.6 Geomagnetic declination derived from the WMM (dashed line) and EMM (solid line) (https://www.ngdc.noaa.gov/geomag/EMM/)

model, the ground-based data from the same set of observatories, shown in Fig. 3.3 are used. Both satellite and observatory data selection is based on the strict criteria and the weight coefficients are introduced. Figure 3.7 shows the applied weights for the satellite and observatory data depending on time. Unlike the IGRF and WMM, the BGGM model is commercial and widely used by the oil and gas companies as well

Fig. 3.7 Values of the time-dependent weight coefficients for the satellite and observatory data, introduced in the BGGM model

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as other industries dealing with the systems for accurate positioning and navigation with the geomagnetic field reference. CHAOS. The models of the CHAOS family have been developed at the National Space Institute of the Technical University of Denmark (DTU Space) since 2005 (Finlay et al. 2016). The model is primarily built on the basis of the satellite (Oersted, CHAMP, SAC-C and Swarm) and observatory (160 observatories) monthly average data and represents a solution of the complete inverse problem. This means that the parameters determined in the model describe all possible sources contributing to the measured values of the EMF, including the sources located inside the Earth in the liquid core and lithosphere, as well as the ionospheric and magnetospheric sources. The maximum degree of the polynomials describing the large-scale internal sources is N = 110. The model describes the EMF within the period of time starting from 1997. For the time-dependent internal field, the polynomials with the degrees n from 1 to 20 are used, and the relevant Gauss coefficients are determined with a time step of 6 months. The time-dependent coefficients are further expanded in a basis of sixth-order B-splines. To describe the crustal field, a statistical model of the field of the small-scale internal sources is introduced; it comprises the coefficients for the polynomials of degrees n from 21 up to 110. As a candidate for the new generation International Geomagnetic Reference Field, a truncated model of degree/order N = 13 is submitted. The model coefficients and the necessary software can be found on the website https://www.space.dtu.dk/english/Research/Scient ific_data_and_models/Magnetic_Field_Models.

3.3 Studying Rapid Core Magnetic Field Dynamics Based on Magnetic Observatory Data Variations of the core magnetic field at the Earth’s surface and above encompass time scales from typically one year to millions of years. These variations include various manifestations of the geomagnetic secular variation (SV), such as the westward drift, the gradual decay of the geomagnetic dipole, the enlargement of the abnormally low intensity area in the South Atlantic region and the North magnetic pole drift. The SV is generated by convective flows, magnetohydrodynamic waves and diffusion processes in the Earth’s outer core (e.g., (Finlay et al. 2010)). On shorter time intervals, up to ten years, less evident SV variations can be detected by calculating the second time derivative of the field, i.e. the secular acceleration (SA). Sudden changes of SA polarity happening within a year or so can be seen in observatory data and models derived from observatory and satellite data. Such events, called geomagnetic jerks, take place at the junction of two time intervals characterized by practically linear change of SV (e.g., (Mandea et al. 2010)). Despite extensive studies of geomagnetic jerks over the past 30 years (see, e.g., (Brown et al. 2013) for a recent compilation), their origin is still unknown. Jerks can be both regional and global in their spatial extent, and the time shift between manifestations

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at different observatories can reach up to 3 years. Recent geomagnetic jerks were detected in 2003 (Olsen and Mandea 2007), 2007 (Chulliat et al. 2010; Kotze 2011), 2011 (Chulliat and Maus 2014) and 2014 (Torta et al. 2015). One of the ways to investigate outer core dynamics is to construct time variant, spherical harmonic expansions of the core field using observations obtained at or above the Earth’s surface. Since late ‘90s, almost continuous, low-Earth orbit satellite observations of the Earth’s magnetic field (e.g., (Chulliat et al. 2017)) have made it possible to build spherical harmonic models describing the time-varying part of the core field at much higher resolution (e.g., (Finlay et al. 2016)). Assuming the mantle is an electrical insulator, these models enable charting the SV and SA not only at the Earth’s surface, but also at the core-mantle boundary (CMB) (e.g., (Bloxham and Gubbins 1985)). However, downward continuation of the SA is only possible for models derived from recent, high accuracy satellite observations, providing a geographically homogeneous coverage. On the other hand, the most accurate information about the time variations of the Earth’s magnetic field on much longer time scales is provided by geomagnetic observatories. For now, the highest quality of the recorded data is provided by the INTERMAGNET network (http://intermagnet.org). The observatory data are crucial for the calibration of satellite measurements, models reflecting the time variability of the Earth’s magnetic field, estimating the geomagnetic activity of different origins, etc. The observatory data are continuous and cover long time intervals, which make it possible to study the evolution of the core field on time intervals of up to a few hundred years. Herein, we investigate the possibilities to derive accurate signals from the observatory data reflecting fine structure of the core field dynamics.

3.3.1 Data Selection Methodology Today, the main method for evaluating the degree of the geomagnetic field disturbance caused by the magnetic activity of the Sun involves the geomagnetic activity indices. Since the beginning of continuous measurements of the magnetic field variations, a plethora of indices has been developed for the complex monitoring of the geomagnetic disturbances. There are three main and most widely used geomagnetic indices (Menvielle and Berthelier 1991): the Dst-index for describing the ring current based on the data of four subequatorial observatories; AU/AL/AE index for estimating the maximal intensity of the auroral electrojets (Davis and Sugiura 1966) based on the data of high-latitude observatories; and the set of K-indices reflecting the maximal disturbances in the horizontal component of the geomagnetic field on a scale from 0 to 9 (Bartels 1957a, b; Rangarajan 1989; Rangarajan and Iyemori 1997; McPherron 1995) at individual observatories. The K-index is calculated over the successive three-hour intervals (Chapman and Bartels 1940, 1962) under the assumption that this time span is, on the one hand, sufficiently long for the adequate estimation of various disturbances with a duration of 1–2 h, including the so-called substorm bays and, on the other hand, sufficiently short for distinguishing between the neighboring

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intervals with different levels of activity caused by the same event. The dependence of the K-index on solar activity is demonstrated, e.g., in (Mariska and Oster 1972); the detailed procedure for calculating the K index is described in Chap. 2.2. The planetary Kp-index (Rangarajan 1989) is the mean of the K indices obtained at 13 particular observatories. It is worth noting that, with the continuous development of the observational networks, the Kp index does not currently provide objective information about the activity on the planetary scale (Menvielle and Berthelier 1991; Love and Remick 2007). Specifically, this is due to the highly non-uniform distribution of the 13 mentioned observatories across the globe, with most of them located in North America and Western Europe and only two observatories in the Southern hemisphere. Besides, considering the progress in digital data recording and mathematical methods for data processing, it appears obsolete to use a unit measure for estimating the magnetic activity from an extremely limited subset of the available data. Nevertheless, the Kp-index is still widely used for the detection of low magnetic activity caused by the external magnetic field and, hence, for selecting the data that reflect variations of the internal magnetic field of the Earth. Based on the values of the Kp-index, the International Association for Geomagnetism and Aeronomy determines ten magnetically quiet and five magnetically disturbed days of every month. Thus, the data for constructing the core magnetic field models are commonly selected depending on the Kp-index for the respective days (e.g., in the set of the CHAOS models (Olsen et al. 2006, 2009, 2010, 2014; Finlay et al. 2016) and SIFM models (Kotsiaros et al. 2014; Olsen et al. 2015). A fairly new method for estimating geomagnetic activity from observatory data with the use of the measure of anomalousness (MA) was recently developed in (Soloviev et al. 2016). The method is based on the principles of the Discrete mathematical analysis (DMA) (e.g., (Gvishiani et al. 2008a, b, 2010, 2014; Soloviev et al. 2013; Agayan et al. 2016)) and allows estimating the geomagnetic activity with the maximal time resolution equal to the discretization (sampling rate) of the initial time series. Applications of the MA to the recognition of the increased geomagnetic activity periods are considered in (Soloviev et al. 2016; Agayan et al. 2016; Gvishiani et al. 2014, 2016a, b). (Soloviev and Smirnov 2018) employed the dual property of the measure for determining the diurnal intervals of the minimal magnetic activity of external origin separately for each observatory and each component. Apparently, the data selected in this way reflect to a great extent the core field dynamics. MA derived from each original time series represents a new time series of the mapping with the range of values from (–1, 1). This mapping assigns the larger values (those close to 1) to the anomalous variations of the initial record, and the smaller values (those close to –1) to the background measurements. An important property of the MA is that it has the same discretization as the initial record. The graduated scale of the MA values (Table 3.1) for evaluating the level of geomagnetic activity based on the observatory data was empirically developed in Soloviev et al. (2016). Soloviev and Smirnov (2018) suggested a new approach for determining magnetically quiet days with the use of the MA for a given month, component of the magnetic field, and observatory. The magnetically quiet days are defined as days in which as

3.3 Studying Rapid Core Magnetic Field … Table 3.1 Empirical scale of correspondence between MA values and level of geomagnetic activity

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MA value

Geomagnetic activity level

μ ∈ [−1, 0.4]

Background

μ ∈ (0.4, 0.55]

Weakly anomalous

μ ∈ (0.55, 0.75]

Anomalous

μ ∈ (0.75, 1]

Strongly anomalous

many measurements (measured values) as possible have the respective MA values below 0.4 (Table 3.1). At the same time, in case when most of the measured values in a certain day correspond to μ < 0.4 but still there is a short period activity characterized by the MA values close to 1, this day, clearly, cannot be considered as completely quiet. Therefore, when identifying the magnetically quiet days, the days containing at least one MA value above 0.55 are rejected. For further ranging of the days in terms of the degree of quietness, the coefficient of quietness K D is introduced: KD =

 |μ(t∈D) A among the available ones; black polyline represents percentage of available observatories per month. Characteristic function peaks near previously detected 2006, 2009 and 2012 SA pulses; peaks near 1996.5, 1999.5, 2002.5 and 2014.5 suggest newly detected SA pulses

the suggested method, even though intermediate scales, especially from degree 4 to 6 of the spherical harmonic expansion, are known to dominate the SA pulse signal at the CMB (Chulliat and Maus 2014). Interestingly, the characteristic function for the Y component displays peaks that are shifted by a few months (2012) to two years (2006) with respect to the pulse epochs. The reason for these shifts is unclear; it could be related to the upward continuation of the core field from the CMB to the Earth’s surface, which places extrema for the X and Y components at different locations at the Earth’s surface and therefore leads to different characteristic functions when using a geographically inhomogeneous set of observatories. The characteristic functions for X and Z change sign from one pulse to the next, not only in 2005–2012, but also over the whole period under consideration. Therefore, the suggested method correctly detects changes of pulse polarity at the core surface at the global level, information that is not contained in SA power spectra and can only be recovered by plotting the SA at the core surface. Increasing the characteristic function threshold from 2 to 5 nT/yr2 makes some peaks sharper (e.g., in 2009 and 2012 on X) while smoothing out some others (e.g., in 2012 on Z). This indicates that the method is very sensitive to threshold values, and that it is necessary to scan a range of threshold values to detect pulses.

3.3 Studying Rapid Core Magnetic Field …

127

A more detailed picture emerges when considering regional subsets of observatories in the Atlantic and Southeast Asian regions. These are the two areas where the 2006, 2009 and 2012 pulses were found to be the strongest at the CMB. All negative, respectively positive, characteristic functions for the Atlantic region contain conspicuous maxima around 2006 and 2012, respectively 2009. When increasing A, these peaks become sharper but are still present. This provides evidence that the considered SA pulses are particularly strong in the Atlantic sector, as previously shown by using spherical harmonic models downward continued to the CMB. The polarity of the peaking characteristic function changes from one pulse to the next in the Atlantic sector, reflecting the alternating dominance of negative and positive SA patches in this area at the CMB. In the South Asian region, characteristic functions robustly detect peaks in 2006, 2009 and 2012 for all values of the threshold A, even the largest ones, thus showing that the pulses are strong in this area too. However, the characteristic function for the Y component has the same polarity in 2006 and 2009. This is likely due to geographical distribution of the selected observatories, which are all in the Northern hemisphere and therefore cannot detect the effect on the Y component of the most intense patch at the core-mantle boundary in the South Asian region, which is located in the Southern hemisphere (see, e.g., Fig. 1 of (Chulliat et al. 2015)). This effect is largest to the East and West of the patch, as indicated by the Green function relating the radial field at the CMB and the Y component at the Earth’s surface (e.g., (Constable et al. 1993; Chulliat et al 2010)). Figure 3.13a, X component, shows that the same pair of characteristic functions that peak near 2006, 2009 and 2012, also displays maxima around 1997, 1999–2000, 2002 and 2014, and that the polarity of the peaking function is alternately positive and negative for the whole series, from 1997 to 2014. Specifically, over a total of 56 observatories, the SA was larger than 2 nT/yr2 at 40 observatories in 2000 and 36 observatories in 2002, which amounts to percentages of total of 65–70%. Such percentages are comparable to the percentages obtained for SA pulses between 2006 and 2012, which fall within the 65 to 80% range. Maxima in 1997 and 2014 are smaller, but the number of available observatories is also smaller; as a result, the percentage of observatories with SA smaller than −2 nT/yr2 is larger than 65% for both epochs. There is an additional maximum of the positive characteristic function in 1994, but the percentage of observatories is significantly smaller than that for other peaks and therefore it is not further considered. Characteristic function maxima are also found in Z component, around the same times as in X component. The 1997 and 2002 maxima are nearly as large as the maxima between 2006 and 2012, with about 65 and 57% of available observatories, respectively. The 1999 and 2014 maxima are smaller, below 50% of available observatories. Besides, the 1997 and 1999 maxima are shifted in time by about one year. As for the X component, the polarity of the peaking characteristic function is alternately positive and negative from 1997 (or 1996) to 2014. As already noted at the validation stage, characteristic functions for the Y component are not indicative of possible SA pulses, since pulses between 2006 and 2012 do not lead to clearly identifiable maxima or lead to maxima that are shifted in time. Similarly, there are no clear maxima for that component around 1997, 2002 and 2014; there is a strong maximum near 1999, which looks like an exception.

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3 Mathematical Models of the EMF

The four new maxima identified in the X component characteristic functions are robust when increasing the threshold value to ±5 nT/yr2 . Although smaller than maxima caused by the 2009 and 2012 pulses, they are comparable or larger than the 2006 maximum. Although the picture is less clear for the Z component because of smaller values at all epochs, maxima are conspicuous near 1997, 1999, 2002; no maximum is visible in 2014, nor in 2012, which suggests that geographical areas with the largest SA on the Z component for these two epochs are outside areas covered by selected observatories. Like for the smaller threshold of ±2 nT/yr2 , the polarity of peaking characteristic functions is alternately positive and negative from 1997 to 2014, both for the X and Z component. We conclude from this analysis that the SA peaked around 1997, 1999, 2002 and 2014 in large areas of the Earth’s surface, and that the signature of these events on observatory-based characteristic functions is very similar to that of SA pulses in 2006, 2009 and 2012. Moreover, the polarity of the peaking SA was alternately positive and negative from 1997 to 2014, both on the X and Z components, and for small (±2 nT/yr2 ) and large (±5 nT/yr2 ) SA thresholds. Although the method doesn’t allow for downward continuation at the core-mantle boundary, these results strongly suggest that (a) large-scale SA pulses occurred at the core surface in 1997, 1999, 2002 and 2014, and (b) SA pulse polarity changed from one pulse to the next from 1997 to 2014. In order to determine the regional distribution of the newly discovered SA pulses, (Soloviev et al. 2017) analyzed characteristic functions separately calculated for each of the following regions: Pacific, North America, Atlantic, Europe, North Asia, South Asia and Australia. The results are given in Table 3.2. It provides information on the presence of the most stable SA pulses, i.e. seen with different thresholds in characteristic functions, and their polarity (‘+’ for positive and ‘−’ for negative). In the Atlantic sector, all SA pulses are robustly detected from 1997 to 2014 on the X component, and their polarities alternate from one pulse to the next. This is not the case on the Z component, as the 2014 pulse has no signature on that component and some successive pulses have the same polarities. All pulses are also detected with alternating polarities on the X component in the European and South Asian regions. This is conspicuous for example in Fig. 3.13b, top, where characteristic functions for X with a threshold of ±3 nT/yr2 are plotted for the European subset of observatories. On the contrary, a more complicated picture emerges in the Pacific, Australian, North Asian and North American regions, with some pulses having no signature and/or successive pulses having strong signatures of the same polarity. As mentioned above, successive SA pulses of opposite polarity are expected to be separated by ‘V-shaped’ geomagnetic jerks at the Earth’s surface. The geographical extent of such jerks is of course related to that of the preceding and following SA pulses. Such was the case between the 2006 and 2009 pulses, and between the 2009 and 2012 pulses, with jerks identified in 2007 and 2011. Results from (Soloviev et al. 2017) suggest the existence of jerks between (a) 1997 and 1999, (b) 1999 and 2002, (c) 2002 and 2006, (d) 2012 and 2014. Here we’d like to point out that the date of SA pulses, and therefore of associated jerks, comes with an uncertainty of about 1 year due to the geographically sparse distribution of observatories (which also leads to timing differences between components) and the filtering effect of the

d 2 Z/yr 2

–

+

–

+

–

+

–

1996–1997

1999–2000

2002–2003

2005–2007

2009–2010

2012–2013

2014–2015

–

+

–

+

–

+

+

+

–

+

–

–

–

+

+

–

d 2 Y/yr 2

d 2 X/yr 2

d 2 X/yr 2

d 2 Y/yr 2

North Asian (4 obs)

European (18 obs)

–

+

+

+

+

–

d 2 Z/yr 2

–

+

–

+

–

+

–

d 2 X/yr 2

–

–

d 2 Y/yr 2

South Asian (6 obs)

+

–

Region/Year

–

+

–

d 2 Z/yr 2

–

+

2014–2015

+

+

–

–

+

2012–2013

–

+

–

–

+

+

–

–

+

+

–

2009–2010

–

2002–2003

–

–

+

–

+

–

–

d 2 X/yr 2

+

–

–

d 2 Y/yr 2

Australian (7 obs)

–

–

+

–

d 2 Y/yr 2

Atlantic (13 obs) d 2 X/yr 2

+

+

1999–2000

+

d 2 Z/yr 2

2005–2007

–

d 2 Y/yr 2

North American (6 obs) d 2 Z/yr 2

d 2 X/yr 2

d 2 Y/yr 2

Pacific (4 obs)

d 2 X/yr 2

1996–1997

Region/Year

–

+

–

–

–

+

–

–

–

+

d 2 Z/yr 2

d 2 Z/yr 2

Table 3.2 Detection of SA pulses in characteristic functions associated with regional subsets of observatories, and polarities of these characteristic functions

3.3 Studying Rapid Core Magnetic Field … 129

130

3 Mathematical Models of the EMF

weakly conducting mantle. Also, SA pulses extend over at least 1–2 years and the pulse dates adopted so far are the dates for which the number of observatories having a GrS-based SA larger than a given threshold is significant. The existence of a jerk near 2014 was recently reported by (Torta et al. 2015). On the Y component, this jerk is strongest in the Atlantic sector observatories. It also has clear signatures on the X and Z components in the European and Australian regions. This picture is consistent with the findings using the GrS-based method—the 2014 SA pulse starts at the beginning of 2014 in Europe (see Fig. 3.13b), a bit earlier in Australia. Jerks were also previously detected in 2003 (Olsen and Mandea 2007) and 1999 (Mandea et al. 2000; De Michelis and Tozzi 2005), mostly in Europe. It is clear from Fig. 3.13 that the 2003 jerk occurred between the 2002 and 2005 SA pulses, and that its effect was indeed strong in Europe (Fig. 3.13b) and also Australia. It is less straightforward to relate the 1999 jerk reported in the literature with the SA pulses detected by our analysis, because both cited studies focus on the Y component and SA pulses are less conspicuous on that component. The number of observatories with a large SA on Y peaks in 1998, just before the reported 1999 jerk. However, when looking at the X and Z components, we see ‘V-shaped’ jerks near 1998, for example at Guam (GUA, Western Pacific), Memambetsu (MMB, Japan), Niemegk (NGK, Germany), San Juan (SJG, Puerto Rico) and TAM (Fig. 3.14), which are scattered all over the world. This is in agreement with (Brown et al. 2013), who, using a method for systematically determining jerk-like features in worldwide observatory data, found a peak in the number of detected jerks in 1995–1998. We conclude that the 1997 and 1999 SA pulses are indeed separated by ‘V-shaped’ jerks, mostly on X and Z, and sometimes Y, in many parts of the world. Although no jerk was previously reported between the 1999 and 2002 SA pulses, V-shaped variations of the SV can be seen at several observatories around 2001–2002 (Fig. 3.14). These changes in the polarity of the SA were likely not reported as jerks until now because they are less sudden than typical jerks. However, we choose to qualify them as jerks, following the definition proposed by Chulliat and Maus (2014), as they occur between two successive SA pulses. In this way, being applied to the data from 56 observatories worldwide recorded during 25 years, the method by (Soloviev et al. 2017) successfully recognized the satellite-inferred SA pulses of 2006, 2009 and 2012, demonstrating its reliability. Moreover, four new SA pulses were detected in 1996, 1999, 2002 and 2014, each between a pair of successive geomagnetic jerks and each with an opposite polarity to that of the next pulse, thus extending to twenty years the validity of the observed jerk/pulse relationship and the 6-year periodicity of the pulse phenomenon.

3.3 Studying Rapid Core Magnetic Field …

131

Fig. 3.14 GrS-based SV (purple) superposed over SV computed in accordance with formula (3.19) using monthly means (black) at several observatories. The years are given in the horizontal axes and vertical axes represent nT/yr for X (left), Y (center) and Z (right) components

3.3.3 Studying Geomagnetic Jerks Using Regression Derivatives Conventional geomagnetic jerk analysis is based on observatory data, which are largely noisy, mainly due to the influence of external fields. The noise magnitude is often comparable to the magnitude of the signal of internal origin, but the signal and

132

3 Mathematical Models of the EMF

the noise always have different frequency characteristics. This justifies the use of various mathematical methods to detect jerks in automatic or semiautomatic mode (e.g., (De Michelis and Tozzi 2005; Mandea et al. 2010)). Such methods may include DMA-based regression derivatives (Agayan et al. 2019) as well, since they allow one to search for trends and local extrema in noisy time series. Herein, we study these properties of regression derivatives, which are crucial for the detection of geomagnetic jerks from noisy data recorded at several observatories of the INTERMAGNET standard (Kerridge 2001). The time interval of 1991.0–2015.3, which encompasses a widely known series of jerks (see the previous Section), was chosen to assess the efficiency of the jerk detection method based on regression derivatives. The source data of the present study are the secular variations of the three components of the magnetic field (dX/dt, dY /dt, dZ/dt) obtained from the following observatories (the IAGA codes are listed): GUA (Guam Island), MMB (Memambetsu, Japan), NGK (Niemegk, Germany), SJG (San Juan, Puerto Rico), and TAM (Tamanrasset, Algeria). The secular variations were calculated based on observatory measurements of “definitive” status (Love and Chulliat 2013), which ensures their high quality over the entire considered period. The method to calculate the secular variations is described in detail by (Soloviev et al. 2017) (see the previous Section). The choice of observatories is determined by their wide geographical coverage. The algorithm parameters are the r and p values. They are used to determine the local measure of proximity δ(r, p). The parameter p is assumed to vary from 0 to 50, whereas r takes a value from 50 to 100. Within this range, the regressionderivative graph characterizes the initial variations quite accurately. Since the method is sensitive to the slightest changes of the sign of the derivative, we do not consider p and r values exceeding the specified range. Otherwise, the resulting graph would represent a set of sharp changes in the sign of the derivative (jumps), which are primarily noise (especially for smaller r values). If r > 100, the resulting graph would not contain such small jumps; however, such r values allow actual targets to be missed (the transition from increasing zone to decreasing zone (and vice versa), which is clearly visible in the initial function, would not be marked in the graph of regression derivatives). Two graphs were plotted for each observatory/component: the graph of the initial secular variations and that displaying the result of the application of regression derivatives (Figs. 3.15, 3.16 and 3.17). There are three colors in the graphs of the regression derivative: dark gray, black, and light gray. Dark gray indicates the function predominantly increases in the indicated ranges. Black indicates a decrease of the function, and light gray means that the original function is quasi-constant. With increasing p, the appearance of the resulting function changes, since its values reflect more local consideration of the original function (and therefore more local extrema are found). The gray and white vertical lines in the graphs indicate the epochs of the previously determined SA pulses. They correspond to the time limits of the intervals within which the jerks took place. We studied the results of the algorithm obtained with different constructions of parameter δ. For each fixed value of r i , we looked for such pi  range within 0 < p < 50, where its lower limit ensures the detection of the most known jerks, and its upper limit

3.3 Studying Rapid Core Magnetic Field …

133

Fig. 3.15 Application of the method to the search for trends in the secular variation of the X component recorded at GUA observatory data for the period 1991.0–2015.3. Top: the initial dX/dt graph (nT/year). Bottom: the regression derivative graph for p values from 0 to 50. On this graph, positive trends are marked in dark gray, and negative trends are in black. The range 25 < p < 35 is marked with horizontal lines. The previously studied SA pulses of 1997, 1999, 2002, 2006, 2009, 2012, and 2015 are marked with vertical lines

Fig. 3.16 Application of the method to the search for trends in the secular variation of the Y component recorded at NGK observatory for the period 1991.0–2015.3. The legend is the same as in Fig. 3.15

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3 Mathematical Models of the EMF

Fig. 3.17 Application of the method to the search for trends in the secular variation of the Z component recorded at TAM observatory for the period 1991.0–2015.3. The legend is the same as in Fig. 3.15

provides the minimum number of false signals (noise). The search was conducted empirically with numerous experiments. In this way, we obtained the single p’ range for each r i for all observatories. The borders of this range are indicated in Figs. 3.15, 3.16 and 3.17 by horizontal white lines on the lower graphs. Tables 3.3, 3.4 and 3.5 give the results for several constructions δ(r i , pi  ), where r 1 = 50, r 2 = 80, r 3 = 100. From these results, it follows that increased r expectedly leads to increase in the number of observatories having target misses; the number of false positives, however, decreases. When the parameter value is small, all events are recognized, including insignificant ones, while minor events can be ignored with increasing value. Since our primary goal is to establish a jerk presence between a pair of neighboring SA Table 3.3 r 1 = 50, 25 < p1  < 35 Component\interval 1997–1999 1999–2002 2002–2006 2006–2009 2009–2012 2012–2015 % of observatories without missed target X

100

100

100

100

100

Y

80

100

100

80

80

100 100

Z

100

100

100

100

100

100 100

% of observatories with ≥1 false signals X

80

100

100

100

100

Y

60

80

100

60

60

80

Z

60

80

100

100

60

80

3.3 Studying Rapid Core Magnetic Field …

135

Table 3.4 r 2 = 80, 25 < p2  < 40 Component\interval 1997–1999 1999–2002 2002–2006 2006–2009 2009–2012 2012–2015 % of observatories without missed target X

100

100

100

100

100

100

Y

80

100

100

40

80

100

Z

100

60

100

80

80

100

% of observatories with ≥1 false signals X

60

100

100

100

100

100

Y

20

80

100

20

60

60

Z

40

40

100

80

60

60

Table 3.5 r 3 = 100, 30 < p3  < 45 Component\interval 1997–1999 1999–2002 2002–2006 2006–2009 2009–2012 2012–2015 % of observatories without missed target X

100

80

100

100

100

100

Y

80

100

100

40

60

100

Z

100

60

100

80

80

100

100

100

100

80

% of observatories with ≥ 1 false signals X

40

60

Y

20

60

80

0

40

60

Z

40

40

100

60

60

60

pulses, maximizing the percentage of observatories without missing the target is our higher priority. In other words, the parameter values presented in the Table 3.3 (r 1 = 50, 25 < p1  < 35) are preferable. With these parameters, the target was only missed in the dY/dt component record and only at one observatory within each of the three considered intervals: for 1997–1999 at MMB observatory, for 2006–2009 at TAM observatory, and 2009–2012 at SJG observatory. This result can be considered quite satisfactory given (1) the rather small number of observatories involved, (2) the spatial localization of geomagnetic jerks, and (3) the manifestations of jerks only on individual components. Due to their versatility, regression smoothing and derivatives could significantly simplify and speed up the calculations related to noisy time series. The second time regression derivative might be an efficient tool in searching for SA pulses and subsequent jerk detection between them. Also, a similar algorithm can be applicable to studying the spatiotemporal migration of jerks on the basis of the data from the entire global network of INTERMAGNET observatories. However, these directions are still under development.

136

3 Mathematical Models of the EMF

3.4 Models of the Crustal Magnetic Field The magnetic field of the lithospheric sources is an interference for constructing the global models of the main EMF, and the attempts are made to exclude it from the measured values by various ways, or, at least, reduce its contribution. However, recently the CHAOS-like EMF models have become widespread, which are based on the satellite data and describe both the time-varying core magnetic field and the static crustal field. Although a strict selection and correction of the satellite data are carried out to construct the models of the lithospheric anomalies, due to the fact that measurements are performed at high altitudes, a blurring effect arises for the lithospheric sources. To achieve a higher spatial resolution, the degree of the polynomials is constantly increasing. But even for the harmonics of the degree n = 130, the horizontal resolution makes approximately 350 km. World Digital Magnetic Anomaly Map (WDMAM). As compared to the satellite data, the measurements on the Earth’s surface provide a spatial resolution of about 1 km. However, the geographical uneven distribution of the observatories is the significant drawback. That’s why the spherical coefficients based on these data cannot be estimated without interpolation, which leads inevitably to the errors. Constructing the high resolution global maps of the anomalous magnetic field of the Earth became the subject of the WDMAM international project. The WDMAM project is aimed at combining all results of the ground, sea, aeromagnetic and spaceborne geomagnetic surveys accumulated during 50 years of the studies to create the first global map of the anomalous EMF. The synthetic data derived from the lithospheric field models based on satellite data were used for the regions of the globe, which were not surveyed by marine or aeromagnetic facilities. As a result, the first edition of the WDMAM digital map (Korhonen et al. 2007) and afterwards the latest WDMAM 2.0 map (Lesur et al. 2016) were compiled. The magnetic anomaly map is defined on a global geographical grid with 3 × 3 arcminute (i.e. roughly 5 × 5 km) cell size at the altitude of 5 km above the WGS84 ellipsoid, except for the marine data and model where altitude is the sea level (Fig. 3.18).

Fig. 3.18 World Digital Magnetic Anomaly Map WDMAM 2.0 (http://www.wdmam.org)

3.4 Models of the Crustal Magnetic Field

137

A global magnetic field model of the lithosphere contribution, parameterized by spherical harmonics, has been derived up to degree and order 800. The grids of the anomalous EMF are available for use on the website http://www.wdmam.org. Regional models of the magnetic anomalies. In many cases, for practical purposes, such as geological exploration or navigation, the lithospheric field models of higher resolution are required. The concept of the regional modeling is based on a higher density of the data underlying such models. As a rule, the regional EMF models are constructed to increase the resolution of the anomalous magnetic field. A large number of techniques have been developed to construct the regional models, but the increased density of the source data points remains the primary condition. A widespread network of repeat stations of the secular variation serves to fulfill this requirement. These stations are not operated continuously but used to carry out regular observations of the geomagnetic field in the course of several-day expeditions, which have a many year history. So far, a set of requirements have been developed and adopted for the long-term establishment of the stations in fixed locations, instrumentation, measurement methodology and processing of the results. Figure 3.19 shows the locations of the repeat stations operating at different times since 1975. As it can be seen from the figure, the repeat station density, although uneven, is significantly higher than the density of the magnetic observatory distribution in some regions (Fig. 3.3). With the beginning of the satellite era, the impression appeared that there was no need in the repeat stations anymore, since the density of the satellite measurements was sufficient to fully describe the EMF. However, the practice has shown that the satellite flight altitude does not allow tracking adequately the regional features of

Fig. 3.19 Repeat stations of secular variation around the globe

138

3 Mathematical Models of the EMF

the EMF. The currents flowing in the Earth’s ionosphere are below the satellite, and their separation is quite complicated; it is also difficult to eliminate completely the errors related with the variations of the magnetospheric current systems. Now it is assumed that the average error of determining the magnetic declination using the modern satellite models has a value of about 0.5° at the altitude of about 400 km, and at the ground level in presence of the local magnetic anomalies it can achieve 3 and more degrees. So the observations at the existing repeat stations continued, and their network only expanded. The regional models make the greatest contribution to the description of the spatial structure of the secular variation. This is especially evident when comparing the average annual values of the field components obtained at the magnetic observatories and predicted by the global and regional models. The maximum differences between the lithospheric field models and the actual values of the EMF anomalies obtained along the profiles fall on the ocean region. It is mainly related with the different nature of the continental and oceanic crust, linearity and arrangement of the magnetic field of the latter. One of the preferred alternative regional modeling techniques over the last decades is the Spherical Cap Harmonic Analysis (SCHA), introduced by (Haines 1985). Unlike the global models based on classical spherical harmonic analysis, SCHA represents a very similar segmental spherical harmonic analysis. The difference is that the segmental spherical harmonics are applied only to part of the sphere, whereas the conventional spherical harmonics are applied to the whole sphere. In this case, a local expansion of a potential within a spherical cap rests upon associated Legendre functions with integer order m but in general real (not necessarily integer) degree nk (m). In addition to lithospheric magnetic field modeling, is has been widely applied to modeling main field and its secular variation as well as studying ionospheric and magnetospheric–ionospheric coupling electrodynamics (Torta 2020). The Canadian Geomagnetic Reference Field (CGRF) is one of the most common SCHA based regional models for the practical applications (e.g., underground navigation while drilling wells) (Haines and Newitt 1986). In addition to Canada, the CGRF regional model also describes the geomagnetic field for its adjacent territories. The maximum expansion depth for the CGRF model achieves 16. The model is updated every five years. The Gaussian coefficients gnm and h m n are time-dependent and a correction table is issued for them. Unlike the global models, which are mainly based on the satellite data when determining the Gaussian coefficients, the CGRF model uses widely the geomagnetic data obtained on the Earth’s surface and in the sea. Along with the fact that the considered area is much smaller as compared to the global models, this results in a higher accuracy of the CGRF model, which enables revealing small spatial variations of the magnetic field invisible for the global models. It is recommended to conduct a general magnetic survey of the area every 10 years to obtain the reliable models of the anomalous magnetic field and provide the accurate data on the magnetic declination values. It should be noted that about 30 years have passed since the last magnetic survey of the total territory of Russia. A high-precision aeromagnetic component survey at the altitudes of 5–8 km using GPS-GLONASS satellite navigation systems along with the appropriate ground-based support by the magnetic stations should be carried out to construct a modern reliable model of the

3.4 Models of the Crustal Magnetic Field

139

EMF vector distribution over the Russian territory. In the regions with the higher requirements for the model accuracy and its spatial resolution, the survey at the altitudes of 5–8 km can be supplemented by a detailed survey of the magnetic field modulus at the altitudes of 0.5–1 km using unmanned aerial vehicles. It is reasonable to make a resurvey of the territory in two-three years to refine the rate of change of the magnetic field, which enables recalculating more accurately the field values for the required epoch.

3.5 Magnetosphere Field Models To solve a number of problems, it is necessary to estimate the magnetic field intensity and direction in the near-Earth space at the different spots of the magnetosphere, for example, to determine the shape of the magnetic field lines, the position of the boundaries of magnetospheric domains and the dependence of these values on the level and type of magnetic activity. The configuration of the large-scale magnetospheric field changes significantly during the magnetic storms and substorms. The dynamic digital models of the magnetosphere have been developed to describe such variations. Two of them are most commonly used in practice. In the models, the magnetic field in the Earth’s magnetosphere is presented as a superposition of contributions of the large-scale current systems: B = Bint + BCF + BRC + BT + BFAC ,

(3.21)

where Bint is the internal EMF, usually represented by the IGRF model, BCF is the magnetic field produced by the currents on the dayside magnetopause (ChapmanFerraro currents), screening the field of the internal sources, BRC is the ring current field, BT is the field of the magnetotail currents including the currents across the tail and currents closing them on the magnetopause, BFAC is the field of the field-aligned currents forming the three-dimensional current system together with the currents closing them in the ionosphere and magnetosphere. The family of the empirical models by Tsyganenko (2002) most widely used by the scientific community is based on the totality of the direct measurements of the magnetic field performed at the various magnetospheric satellites and their comparison with the conditions in the solar wind. The approach is implemented by approximating the measurements by the family of the basic functions obtained when solving numerically the Laplace equation for the scalar magnetic potential inside the magnetopause of a given shape. The expressions for the magnetic field of the magnetospheric currents are represented in the form of expansions into the harmonic functions, and the expansion coefficients are derived from the boundary conditions. Parameterization of the solar wind effect on the magnetosphere has the form: K 0 + K 1 · (Pd /Pd0 )α + K 2 · S + K 3 · Dst ,

(3.22)

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where K are the weight coefficients calculated for each source of the magnetic field by minimizing the standard deviation from the values derived according to the measurement database from several satellites (Polar, Geotail, CRRES, etc.), Pd /Pd0 is the current dynamic pressure of the solar wind to the average one ratio, S is the parameter describing the previous conditions of the solar wind, Dst is the geomagnetic index estimating the intensity of the magnetospheric ring current. In the first models, it was possible to select several modifications according to the level of magnetic activity. It was enough for a user knowing the Fortran programming language to enter the date, initial coordinates and value of the Kp index and, depending on a problem, under consideration obtain the configuration of the magnetic field lines, the value of the magnetic field along the line or the coordinates of the field line projection from the ionosphere to the magnetosphere. This model, T2001, is still actively used nowadays. However, this version has two limitations. The first one is its applicability within the distance of approximately 10 RE , i.e. the quasicapture region of the charged particles. The second limitation is the difference of this averaged model of the magnetosphere from its rapidly changing actual condition. This difference is especially significant during the magnetic storms, for which the database is rather small. For this reason the T2001 model and its earlier version are used under the quiet and moderately disturbed conditions. A new version T2004 of the model was developed later, which is specifically designed to describe the variations in the magnetosphere during the magnetic storms (Tsyganenko and Sitnov 2005). A new database of the satellite measurements during 37 magnetic storms over 1996–2000 was used in it. The general mathematical formalization has remained the same, but the developers of the new version have come up with the complex multi-parameter constructions taking into account not only the current condition of the solar wind, but also its time evolvement during a particular storm. Each of the six current systems including the ring current, partial ring current, near and far currents of the plasma sheet, external and field-aligned currents between the magnetosphere and ionosphere is given for a specific moment of the magnetic storm as a function of speed, pressure and magnetic field of the solar wind. Each source varies in its own time scale and depends on a particular set of parameters. At that, not only the current values are considered, but also the solar wind conditions within the preceding period with a 5-min time step. The influence attenuation rate of the preceding activity is taken into account; for example, if the preceding influence decays quickly in the auroral currents, the ring current decreases slowly. The intensity of each field source is parameterized according to the following expression:   K 0 + K 1 · (Pd /Pd0 )α + K 2 · (t)/ 1 + ((t)/0 )2 ,

(3.23)

where t is the time, F describes the nonlinear dynamics of the current system during its development and attenuation as a function of speed and density of the solar wind and the magnitude of the south (geoeffective) component of the interplanetary magnetic field. Figure 3.20 shows the configuration of the magnetosphere during

3.5 Magnetosphere Field Models

141

Fig. 3.20 Magnetosphere configuration on October 29, 2003 in the Z-XY plane (R is the Earth’s radius) according to the T2004 model. Dashed and solid lines depict the day side and the night side of the magnetosphere, respectively. The McIlwain parameter for the field lines is specified by the numbers

the main phase of the magnetic storm on October 29, 2003 according to the T2004 model. The models of the “T” family are being constantly upgraded. In particular, the latest version is capable to resolve the magnetospheric structure details not available for the previous models. The mathematical apparatus is also being improved, and the satellite data set is replenished with the results of the new space missions. The version description and the relevant software are available online at http://geo.phys. spbu.ru/magmodel/empiric.html. The paraboloid model of the magnetosphere (Alexeev et al. 2001) is another popular model. Its specificity is that the internal field and field of currents are initially introduced analytically and then adjusted to real observations using the best fit principle. The source superposition approach is also applied (3.21). With a specified spatial distribution of the main magnetospheric current systems, the magnetic field generated by each system is determined as a solution of the magnetostatic problem, and the magnetopause shape is specified by a paraboloid of revolution and cylinder. Using the parabolic coordinates enables in some cases to separate the variables in the Laplace equation and obtain an analytical solution based on the expansion in the harmonic and Bessel functions. The expansion coefficients are determined from the boundary condition of the zero normal field component at the magnetopause for each magnetospheric current system, which depends on its own set of parameters. The following is a description of the input parameters used in the paraboloid model. First, the inclination angle of the geomagnetic dipole to the Z axis of the solarmagnetospheric coordinate system, which is uniquely determined by the UTC and characterizes the seasonal and diurnal variations of the magnetospheric field. Then follows the magnetic flux in the fractions of the magnetotail , distance from the Earth to the subsolar point at the magnetopause r 1 and distance to the current sheet in the magnetotail r 2 , magnetic field of the ring current bR , which according to the Dessler-Parker-Sckopke equation, is proportional to the total energy of the captured particles in the ring current region, and the intensity of the field-aligned currents J. The magnetosphere parameters can be determined from the measurement data or directly specified. For this purpose, a specific set of submodels is provided, which

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Fig. 3.21 Magnetosphere configuration on October 29, 2003 in the Z-XY plane (Re is the Earth’s radius) according to the paraboloid model. Dashed and solid lines depict the day side and the night side of the magnetosphere, respectively

enables calculating the distance from the Earth to the subsolar point at the magnetopause, the magnetic flux in the magnetotail, the magnetic field of the ring current and more. The magnetic field at the specified magnetosphere point is determined by the system of currents, each of which is controlled by its own set of the parameters: Bm = Bint + BRC ( , b R ) + BT ( , r1 , r2 , 0 ) + BSD ( , r1 ) + BST ( , r1 , b R ) (3.24) + BFAC (J ), where BSD and BST are a part of the current field at the magnetopause, which screens the field of the Earth’s dipole, and a part of the field, which screens the ring current field, respectively. Figure 3.21 shows the configuration of the magnetosphere during the main phase of the magnetic storm on October 29, 2003 according to the paraboloid model.

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Chapter 4

Electrodynamic Processes in the Earth’s Ionosphere

In the upper atmosphere, with an increase of the altitude and ionization intensification under the influence of solar radiation, the electrodynamic processes controlled by the magnetic field and plasma flows emanating from the Sun forming the solar wind (SW) begin to play an increasing role. The presence of the magnetic fields in a conducting medium of the moving plasma determines the determinant role of the electrodynamic processes. During interaction of the SW and the interplanetary magnetic field (IMF) with the Earth’s magnetosphere, about 1012 W of energy enters the near-Earth space resulting in the occurrence of the electric fields and currents, acceleration of the charged particles, wave exciting and many other complex and variable processes in the magnetospheric and ionospheric plasma. The electrodynamic interaction between the SW energy and magnetosphere occurs mainly in the boundary layers and magnetotail, which are conjugated along the geomagnetic field lines with high-latitude regions, such as the auroral oval, cusp, and polar cap. The electrodynamics of the polar ionosphere is directly related with the regular and sporadic geomagnetic variations observed on the Earth, because it is exactly the ionospheric electric fields and currents that create an external magnetic field, measured by the ground and satellite based magnetometers.

4.1 Field-Aligned Currents According to the Satellite Geomagnetic Measurements The field-aligned electric currents (FAC) are generated in the boundary layers of the magnetosphere, which are mapped to the high-latitude ionosphere. The existence of these currents was predicted by the Norwegian physicist K. Birkeland at the beginning of the 19th century, and only in the 70ties of the 20th century they

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Gvishiani and A. Soloviev, Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation, https://doi.org/10.1007/978-3-030-58969-1_4

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were first truly discovered in space. The FACs are the sources of excitation of the ionospheric electric fields, ionospheric plasma convection and determine largely the distribution of electron concentration in the high-latitude ionosphere. In the ionosphere, the FACs are closed by the Pedersen currents. The horizontal ionospheric Hall currents perpendicular to the Pedersen currents cause geomagnetic variations on the surface of the Earth (Fig. 4.1). The magnetic effect of the FACs is almost not registered by the magnetometers on the Earth’s surface; however, the magnetic field created by these currents can be measured above the ionosphere. The problem of determining the FACs is solved by measuring the magnetic variations using the low-orbiting satellites with a polar orbit. For the first time the problem of detecting and measuring the FACs in the space above the ionosphere was solved by the Triad satellite in 1972. For a long time, the approximate FAC schemes were mainly reconstructed based on the separate satellite passes. The physical concepts were introduced with respect to three principal FAC zones concentrated in the high-latitude regions (Iijima and Potemra 1976). The FACs flowing into the ionosphere and outflowing from it have the shape of quasi-circular zones.

Fig. 4.1 Configuration of the current circuit of the ionosphere-magnetosphere coupling: the fieldaligned currents are closed by the ionospheric Pedersen and Hall currents (Le et al. 2010)

4.1 Field-Aligned Currents According to the Satellite Geomagnetic Measurements

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The circumpolar shell is referred to as Region 1 (R1); here the currents flow into the ionosphere in the dawn half of the latitudinal zone and outflow from the ionosphere in the dusk half of the same zone. In Region 2 (R2), located closer to the equator, the direction of currents is reverse. These two current sheets are caused by different physical mechanisms, but they are connected through the ionosphere and form a single circuit. The total current of Region 1 makes ~1 million A during quiet perids and ~3 million A during geomagnetically disturbed periods. The current of Region 2 is approximately one third less. In the midday sector, an additional current system is located poleward from Region 1, termed the Region 0 (R0); it is controlled by the azimuthal component (By) of the IMF. The distribution of FACs in Regions 0, 1, and 2 can be approximated by the sinusoidal functions in the first approximation. Later, the other satellites (Oreol, Cosmos, Magsat, DMSP and others) carried out measurements of the magnetic field along particular flight trajectories, however, these measurements were fragmentary and allowed to obtain only a one-dimensional profile of the FACs, which provided only an approximate quasi-two-dimensional FAC representation. The more detailed two-dimensional FAC images were calculated using the magnetic measurements of the DE2 satellite as a function of the SW parameters and the inclination angle of the Earth’s dipole (Weimer 2001). At that, there was no data separation for the northern and southern hemispheres; to consider the interhemispheric and intrahemispheric (relative to the midday-midnight meridian) asymmetry, the mirror transformation was applied. In 1999–2000, the European satellites of new generation Orsted and CHAMP were launched, which were designed to study both the main geomagnetic field and FAC magnetic field. These satellites equipped with the highly sensitive magnetometers, performed a vast amount of measurements above the ionosphere in all sectors of the magnetic local time (MLT). The Orsted satellite equipped with a magnetometer with the accuracy of up to 5 nT was launched to the polar orbit with the perigee of 650 km and apogee of 860 km. In the meridional plane, the orbit shifted gradually in local time at the rate of ~0.9 min per day. All satellite trajectories cross the polar region, passing near the pole, and the orbit shifts gradually so that the trajectory passes through the various local time sectors. The magnetic data of many thousands of passes above all longitudinal sections of the high-latitude regions of both hemispheres were obtained during several years of the satellite lifetime. The CHAMP satellite (the orbit altitude is about 200 km) was operational for more than ten years and also made a great contribution to the study of the EMF. The Orsted, CHAMP satellites and the earlier Magsat project provided a huge database of the magnetic field measurements above the ionosphere. At the beginning of the 2000s, the descriptions of the FAC distribution based on statistical processing of a large amount of new satellite data became available. The analysis and interpretation of these measurements allowed to determine the two-dimensional structure of the FACs. Figure 4.2 shows two pass trajectories of the Orsted satellite through the high latitude region MLat > 60° of the northern hemisphere in the geomagnetic latitude (MLat)—MLT coordinate system. Figure 4.2a shows an example of a pass under the disturbed geomagnetic conditions on June 8, 2000, and Fig. 4.2b is obtained under the quiet conditions on June 2, 2000. Along the trajectory, the variation vectors of

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Fig. 4.2 Variations of the vector of the horizontal component of the magnetic field along the Orsted satellite pass trajectory (a) during the magnetic storm on June 8, 2000 and (b) under the quiet geomagnetic conditions on June 2, 2000

the horizontal (i.e., lying in the X-Y plane, where the X axis is directed along the pass trajectory and the Y axis is perpendicular to the trajectory) component of the geomagnetic field are shown. In each point of the trajectory, the X and Y values of the components represent the differences between the measured field and the IGRF model of the main field. The variation of the vertical (Z) component after subtracting the IGRF value is close to zero, whereas the variations of the horizontal component are specified by the magnetic effect of the FACs. These currents at the high latitudes are directed almost perpendicularly to the ionospheric sheath, so the generated magnetic field lies in the horizontal plane. Thus, for example, if a satellite crosses a single FAC sheet, then a positive or negative jump is observed in the X component of magnetic field, depending on the current direction. If a satellite crosses two adjacent to each other, elongated along the latitude current sheets of the opposite polarity, the measured variation of the X component becomes V- or -shaped. At the same time, the Y component increases near the edges of the current sheets. The simplest algorithm for calculating the FAC density from the magnetic signal is based on the calculation of the gradient of the magnetic field component directed across the pass trajectory. At that, it is considered that the FACs have a layered structure and are elongated along the latitude. Then, the second component can be neglected. The FAC density value jZ is proportional to the spatial gradient of the magnetic induction B. Provided that the satellite crosses the infinitely elongated current sheet, it is calculated according to the formula: jz =

d δ By . dx

(4.1)

The shape of the zonal component allows to determine the number of FAC sheets intersected by the trajectory, their polarity and current density. A change in the sign of

4.1 Field-Aligned Currents According to the Satellite Geomagnetic Measurements

151

Fig. 4.3 (Top) variation of δBY and (bottom) FAC density jZ . The axis x is directed along the satellite pass trajectory through the polar region of the northern hemisphere, the magnetic pole is in the center

the field gradient indicates the crossing of the FAC sheets of different directions, i.e. flowing into and outflowing from the ionosphere. An example of the calculated FAC density based on the corresponding magnetic variation measured along the trajectory is shown in Fig. 4.3. With a large number of passes, the whole high-latitude region can be divided into the cells with a certain latitudinal and longitudinal size. Superposition of all available passes falling into a certain cell enables construction of statistical twodimensional distribution of magnetic variations, which in turn makes it possible to calculate distribution of the FACs in the projection onto the ionosphere. As a result of the extensive data processing, a statistical FAC model was created for both hemispheres and different seasons, parameterized by the IMF magnitude and direction (Papitashvili et al. 2002). The distribution of the FACs in the northern and southern hemispheres is divided into 3-month seasonal intervals and corresponds to eight main IMF orientations in the Y-Z plane of the Geocentric Solar Magnetospheric (GSM) coordinate  system with a step of 45° given the magnitude of the full IMF vector BT = B Z2 + BY2 = 5 nT. The FAC maps were built on the spatial grid with a cell size 1°MLat × 1 h MLT. Later, using the interpolation methods, the FAC maps were developed with a higher temporal resolution and a resolution of 1 nT for the BZ and BY IMF components with the upper limit BT = ±12 nT (Lukianova et al. 2008). Recently, the model was supplemented with the data from the modern geomagnetic Swarm satellites (Lukianova and Bogoutdinov 2018). Figure 4.4 presents an example of the FAC maps of the northern and southern hemispheres for the winter and summer solstices (December 21/June 21) and for the equinox (March 21) with the northern (BZ = +8 nT) and southern (BZ = −8 nT) direction of the IMF. Basic FAC structures. The two-dimensional distribution of the field-aligned currents in the high-latitude ionosphere can be represented as a superposition of

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Fig. 4.4 Distribution of the FACs for the northern spring equinox (middle), northern winter (left) and northern summer (right) solstices in the northern (top row) and southern (bottom row) hemispheres with the northern BZ = +8 nT (a) and the southern BZ = −8 nT (b) direction of the IMF. The field-aligned currents flowing into and outflowing from the ionosphere are marked by the red and blue contours respectively

several current systems. On the dawn and dusk sides, the R1 (~70° MLat) and R2 (~60° MLat) currents elongated along the parallels dominate. The R1 currents flow in on the dawn side and flow out on the dusk side, and the R2 currents have the opposite polarity. These currents are mainly controlled by the vertical IMF component, intensifying when the IMF turns south, i.e. with the negative values of BZ (BZ < 0). If the IMF is oriented northward (BZ > 0), the so-called NBZ system develops with the currents located above 80° MLat, shifted to the day side, flowing into the ionosphere on the dusk side and flowing out on the dawn side. In the R0 current system controlled by the IMF BY component, the inflowing or outflowing currents are located in the dayside cusp area. The direction, intensity and localization of these currents are determined by the sign and value of the BY component. So, at BY > 0 in the northern (southern) polar cap, the outflowing (inflowing) FAC intensifies. At BY < 0, the direction of the currents changes to opposite.

4.2 Modeling the Global Distribution of the Ionospheric Electric Potential

153

Fig. 4.4 (continued)

4.2 Modeling the Global Distribution of the Ionospheric Electric Potential The plasma convection in the magnetosphere results from the reconnection of the IMF field lines with the EMF lines at the magnetopause, in the magnetotail, as well as the processes of viscous friction at the magnetopause. In the collisionless plasma, a presence of the electric field transverse to the magnetic field shows as the convective plasma motion at the electric drift speed. The electric field is mapped from the magnetosphere to the ionosphere along the highly conductive lines of the geomagnetic field almost without distortion. The distribution of the electric fields in the ionosphere can be represented in the form of a system with the convection lines identical to the isolines of the electric field potential. The field-aligned currents of the magnetospheric origin determine the distribution of electric potential and the structure of the convective drift of the magnetic tubes, which enclose the ionospheric plasma. In the high-latitude ionosphere, convection together with precipitation of the auroral particles leads to formation of the large-scale inhomogeneities of the electron concentration. The structure of the convection lines is defined by the magnitude and orientation of the IMF and the solar zenith angle, which varies depending on the day of the year and the time of day. The statistical FAC models, describing the distributions of the field-aligned currents in both hemispheres for various combinations of BZ and BY components

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of the IMF, have revealed the new prospects for numerical simulations of the global distribution of the electric potential in the ionosphere. In these simulations, the FACs are used as the input parameters (Lukianova and Christiansen 2006). Such problem statement is physically correct, since exactly the FACs of the magnetospheric origin excite the electric fields in the polar ionosphere of the northern and southern hemispheres, while opposite circumpolar regions are electrodynamically interconnected through electric fields penetrating from high to middle latitudes. The importance of setting problems of ionospheric electrodynamics in the twohemisphere approximation is specified by the following main reasons: (a) the inclination of the Earth’s axis and discrepancy of the geographical and geomagnetic poles lead to seasonal and daily asymmetry between the conducting ionospheric shells of the northern and southern hemispheres; (b) the topology of interaction of the geomagnetic field and IMF, especially when its azimuthal component is increased, generates interhemispheric asymmetry in distribution of the electrodynamic parameters and creates a potential difference between the opposite polar caps; (c) to determine the total amount of energy entering the upper atmosphere of the Earth from the SW, the proportion of each hemisphere must be known; mirroring the parameter distributions from the northern to the southern hemisphere is not correct; (d) interaction processes in the SW-magnetosphere-ionosphere system occur mainly at the high latitudes. The electric fields propagate from both polar caps to the middle and low-latitude regions of the ionosphere; at that, in many cases their contributions are not equal. The structure of the electric currents providing the magnetosphere-ionosphere coupling can be determined through the current continuity equation: divJ = j  · sin χ ,

(4.2)

where J is the integral horizontal currents flowing in the ionosphere, j is the FAC density and χ is the magnetic inclination. The FAC distribution at the upper boundary of the ionosphere approximated by a thin spherical surface is set as a source. The electrodynamic connection of the current-carrying ionospheric shells of the opposite hemispheres is the key matter. The problem statement considers this conjugation of the ionospheres. Namely, inside the northern and southern polar caps (open lines of the geomagnetic field), the process of spreading of the integral currents is described by its continuity equation, with the relevant distribution of conductivity and FACs for each cap. A the same time, outside the polar cap region the closed magnetic field lines of the Earth’s dipole equalize effectively the electric potential at the conjugate points of the opposite hemispheres. Setting the boundary conditions, representing the continuity of the whole current circuit and equalizing the potential at the boundaries of the caps, leads to an interdependence of distribution of the electric fields inside the polar caps and influence of both caps on potential distribution in the mid-latitude region. When designing the computational scheme, the following approximation was used. The conducting ionospheric shell is split into three regions: the northern and southern polar regions and the rest part of the sphere. There is no direct potential relation between the caps, while in the mid-latitude region (closed field lines) the

4.2 Modeling the Global Distribution of the Ionospheric Electric Potential

155

conjugate points of the opposite hemispheres are equipotential. The two-dimensional equation of continuity of the integral ionospheric current in the spherical geomagnetic coordinates (θ is the co-latitude, ϕ is the longitude) is solved separately for the northern, southern, and mid-latitude regions, and the relevant boundary conditions connect these three regions in a unified system. In the mid-latitude region, the equation is solved in a half of the region (for definiteness, the northern) with a low-latitude boundary spaced away from the equator not less than 20° (at the subequatorial latitudes the Cowling conductivity component and the vertical component of the electric field become too large). In the mid-latitude region, the sum of conductivities and the sum of the sources at the conjugate points of both hemispheres are used as an integrated conductivity and source, respectively. The boundary problem is given by divJl = j1 at θ ≤ θl ,

(4.3)

divJ2 = j2 at π − θ1 ≤ θ < π,

(4.4)

divJ3 = j3 at θ1 ≤ θ ≤ θ3 .

(4.5)

The boundary conditions are as follows: Ul (θl , ϕ) = U3 (θl , ϕ) = U2 (θ2 , ϕ),

(4.6)

Jl (θl , ϕ) − J3 (θl , ϕ) = J2 (θ2 , ϕ),

(4.7)

J3 (θ3 , ϕ) = 0,

(4.8)

where U α , J α and jα are the electric potential, integral ionospheric current and source in the form of the radial FAC component in the relevant regions α = 1, 2, 3. Index 1 corresponds to the northern polar cap with the equatorial boundary at θ = θ 1 ; index 2 denotes the southern polar cap with a boundary at θ = θ 2 ; index 3 is the mid-latitude region with the boundaries at θ = θ 1 and θ = θ 3 . The computational scheme for the ionospheric shell is schematically represented in Fig. 4.5. The boundary condition (4.6) means the absence of a potential jump across the boundary of the given cap and between the boundaries of the opposite caps at each point of the boundaries. The condition (4.7) means that the possible discontinuities of the normal current component in the (θ , ϕ) plane at the boundaries of the northern and southern caps compensate each other due to the FAC flow across these boundaries. The condition (4.8) corresponds to the absence of the current through the equatorial boundary. The boundaries of the northern and southern polar caps are combined with each other and with the boundary of the mid-latitude region in such a way that in this configuration the opposite polar caps having relevant FAC and conductivity distributions appear as if nested inside each other and resting on the common boundary of

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Fig. 4.5 Configuration of the ionosphere regions in the computational scheme. The circumference shows a thin ionospheric shell with integral conductivity. The dotted lines indicate the geomagnetic field lines closed in the mid-latitude region and open in the polar caps. The heavy gray arrows in the northern and southern polar caps indicate the FACs flowing in and out of the ionosphere. The solid horizontal lines show the boundaries of the polar caps and subequatorial boundaries of the mid-latitude regions; the dotted horizontal line indicates the magnetic equator. The computational domains (α) 1, 2 and 3 are shown in gray hatching (only the northern one among two mid-latitude regions)

the mid-latitude region. The boundary problem (4.3)–(4.5) is solved using an iterative method, consisting in regularization of the differential equations, Fourier series expansion, and sweep for solving the system of the linear algebraic equations for the Fourier coefficients. The parameters U α iJ α are related with the Ohm’s law: Ja = Σa · (−∇Ua ),

(4.9)

where α is the tensor of the altitude integral ionospheric conductivity, i.e.  Σθθ Σθϕ , Σ= −Σθϕ Σϕϕ  Σθθ = Σ P sin2 χ ,  Σθϕ = Σ H sin2 χ , 

Σϕϕ = Σ P ,

 sin χ = 2 · cos θ (1 + 3 · cos2 θ )1/2 ,

where P and H are the ionospheric Pederson and Hall conductivities.

(4.10)

4.2 Modeling the Global Distribution of the Ionospheric Electric Potential

157

Fig. 4.6 Distribution of the conductivity in the high-latitude regions of the northern (upper) and southern (lower) hemispheres during the December solstice (DOY = 355), UT = 10.6, solar activity level F10.7 = 150, geomagnetic activity level Kp = 1 (left) and Kp = 3 (on the right). The intensity scale is given in the S units

The distribution of conductivity in the ionosphere depends on both solar ultraviolet radiation and high-latitude precipitations of energetic particles. Both contributions were considered through the sum of the squares of the relevant values at each point of the computational grid covering the ionosphere. The conductivity is approximated according to the models (Robinson and Vondrak 1984; Hardy et al. 1987). The algorithm for calculating the conductivity maps with any temporal resolution and input parameters F10.7, Kp index, DOY (day of the year) and UT was implemented as a subroutine included in the software package of the convection model. Figure 4.6 shows the examples of distribution of the Pedersen conductivity in the high latitude regions of the northern and southern hemispheres during the December solstice given the geomagnetic activity index Kp = 1 and 3. As a result of solving the problem (4.3)–(4.5), depending on distribution of the conductivity and FACs in the northern and southern polar caps and in the region of the closed field lines, the reproduced potential distributions can be significantly different in the northern and southern caps. However, these distributions are not independent of each other and together form the potential distribution in the equatorward region from the cap boundary. Setting the distribution of the magnetospheric fieldaligned currents, exciting convection in the ionosphere, at the outer boundary of the ionosphere corresponds to the terms “magnetosphere is a current source”. This is opposite to the approach “magnetosphere is a voltage source”, which suggests the distribution of the electric potential is set at the polar cap boundary. The boundaries separating the northern (boundary θ 1 ) and southern (boundary θ 2 ) regions of the open field lines from the region of the closed lines should be located in the northern and southern hemispheres at the same latitudinal circles. When setting the boundaries for calculating the potential distribution, it is to be expected that inside the northern and southern polar regions (from the poles to the boundaries θ 1 and θ 2 ),

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the distributions of the input parameters (FACs and conductivities) can be specified independently, while in the mid-latitude region of both hemispheres the distribution of the input parameters is the same. Thus, modeling the global distribution of electric potential in the Earth’s ionosphere is based on the solution of the two-dimensional equation of continuity of the ionosphere-magnetosphere current circuit. The potential distribution is described by a boundary problem for the system of the partial differential equations with separation of the spherical shell approximating the ionosphere into three sub-regions with the nonlocal boundary conditions specified at their boundaries. Fulfillment of the boundary conditions, representing the continuity of the general current circuit and equalizing the potential at the boundaries of the polar caps, provides an interdependence of the potential distribution inside the northern and southern polar caps and influence of both caps on the potential distribution in the mid-latitude region.

4.3 Electric Fields in the Polar Ionosphere Controlled by the FACs 4.3.1 Large Scale Convection Patterns The distribution of the ionospheric electric potential in the high latitude regions of the Earth is controlled by the field-aligned currents (FAC), which depend primarily on the orientation and magnitude of the IMF, as well as on the solar zenith angle. The structure of the potential contours determines the drift (convection) trajectories of the plasma in crossed electric and magnetic fields. The convection patterns in the northern and southern hemispheres for eight main IMF orientations in the Y-Z GSM plane (Fig. 4.7) are given in Fig. 4.8. Figure 4.8 shows the modeled distributions of the contours of the ionospheric electric potential in the northern and southern hemispheres in the conditions of the June solstice, i.e. with the maximum seasonal interhemispheric differences in the magnitude of the solar zenith angle. In June, the illumination of the northern polar cap, and therefore the value of the ionospheric conductivity is maximal, and the illumination of the south cap is minimal. The vertical (BZ ), the most geoeffective IMF component changes the direction from the north (minimum geoefficiency) to the south (maximum geoefficiency). The azimuthal (BY ) component of the IMF determining the asymmetry of the solar wind energy entering the magnetosphere on the magnetosphere flanks changes √ the direction from the negative to the positive. The total IMF intensity is B = (B2Z + B2Y ) = 5 nT. Main convection structures. The configuration of the plasma convection in the ionosphere can also be subdivided into the subsystems controlled by the BZ and BY components of the IMF. Thus, with BZ < 0, two powerful vortices develop on the dawn and dusk sides, the plasma moves in the antisolar direction through the polar cap. When BZ turns northward, they are replaced by weaker vortices of the opposite

4.3 Electric Fields in the Polar Ionosphere Controlled by the FACs

159

Fig. 4.7 Main IMF orientations in the Y-Z GSM plane

direction shifted to the day side. The effect of the BY component is expressed in the dawn (dusk) vortex expansion and domination in the northern hemisphere, if BY < 0 (BY > 0). In the southern hemisphere, the situation is close to mirroring of the northern structure. It can be seen that the reproduction of the basic properties and structures of the double-vortex convection is determined by the nature of the solar wind energy entering the magnetosphere at the different IMF orientations, namely, an increase in the potential in the vortex foci when the IMF rotates to the south, the extension of the vortices to the dawn or dusk sides depending on the BY sign, and some prevalence of the dusk vortex over the dawn one. With the constant value of BZ , but with the opposite signs of BY inside one hemisphere, the structure of the vortices is not a mirroring with respect to the noon-midnight meridian. Also, there is no mirroring between the opposite hemispheres: with the specified BY sign, the convection patterns in the northern and southern polar regions differ not only in the intensity, but also in the shape of the contours. With the completely zero IMF, as well as at BZ = 0, the double-vortex convection develops in both hemispheres, but in the summer northern hemisphere (Fig. 4.8a), the dusk vortex propagates further through the noon meridian to the dawn side, rather than the vortex in the winter southern hemisphere (Fig. 4.8b). For BZ > 0 and BY = 0, the circular round-pole plasma flow is noticeable, moreover, in summer it makes a part of the main dawn or dusk vortex, and in winter a separate circumpolar cell is distinguished at the latitude above 80° magnetic latitude. At BZ < 0, the dawn and dusk vortices are more symmetrical in winter (southern hemisphere) than in summer (northern hemisphere), where the dusk vortex prevails. In the conditions of IMF BY > 0 and north (BY < 0, south), the dusk (dawn) vortex spreads more efficiently to the opposite side than with the opposite BY sign. Thus, in the solstice conditions, the differences in conductivity amplify not only the quantitative, but also the structural differences of the convection systems in the opposite hemispheres. It can be also seen that the equipotentials extend to the latitudes located

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Fig. 4.8 Contours of the ionospheric electric potential in the northern (a) and southern (b) hemispheres in the MLT-geomagnetic latitude coordinates, day of the year 173. The positive (negative) contours are shown by the solid (dashed) lines. The values of the potential in the vortex foci are given at the bottom of each picture. The outer boundary is located at the magnetic latitude of 50°

much lower than the polar cap boundary, and setting the boundary condition U = 0 at the boundary θ ≈ 50–60° MLat (which is typical for most of the older models) is not quite correct. The potential difference cross wise the polar cap ( F), corresponding to the difference between the positive and negative potential extrema at the vortex foci of the opposite direction, is a representative parameter of the convection intensity. This parameter expresses both the seasonal differences and the effect of the IMF orientation. At the solstice, F in the winter hemisphere exceeds F in the summer hemisphere on average 1.1–1.2 times. The studies of the seasonal changes in the FACs carried out using a large array of the satellite data, showed that, in general, the current

4.3 Electric Fields in the Polar Ionosphere Controlled by the FACs

161

Fig. 4.8 (continued)

density in the most lit (summer) high-latitude ionosphere is 1.5–1.8 times higher than the density in the unlit (winter) ionosphere (Christiansen et al. 2002). The IMF orientation modulates the structure of the contours of the potential and affects the value F. In the northern hemisphere, F is higher at BY < 0 than at BY > 0, and in the southern hemisphere, vice versa. The combination of the conditions BY > 0/northern winter and BY < 0/southern winter provides the maximum seasonal hemisphere differences in F. At that, F in the summer hemisphere is almost the same for both BY signs, and in the winter hemisphere it depends on the BY sign. In the northern winter, F is higher at BY > 0, and in the southern winter—at BY < 0. In both cases, BY component contributes to development of the dusk vortex and its extension through the midday meridian to the dawn side, and in general, such a configuration provides higher F values. Thus, the global maps show that it is necessary to take into account the combined influence of both IMF orientation and

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solar zenith angle on distribution of the ionospheric potential in both hemispheres for appropriate characterizing the structure of the high-latitude convection (Lukianova 2005; Lukianova et al. 2010). The seasonal FAC variations are mainly specified by the ionosphere conductivity depending on the solar zenith angle δ, which magnitude is affected not only by the seasonal factor, but also by the factor of mismatch of the geomagnetic and geographical axes of the Earth, which causes the regular daily UT changes in the position of the terminator in the geomagnetic coordinate system. The terminator shift to the day or night side occurs, especially during the equinox season, near the FAC density maxima associated with the dawn-dusk meridian, that’s why, even a small change in the conductivity can lead to a significant change in the value of the potential difference cross wise the polar cap F. At certain time points of the day, the differences in illumination of the northern and southern ionospheres are maximum, which leads to additional interhemispheric asymmetry (Lukianova and Christiansen 2008). It is important that the southern and northern hemispheres are electrodynamically related with each other at the ionosphere level and through the magnetosphere of our planet. The inclination of the Earth’s axis, the mismatch of the geographical and geomagnetic poles and the differences in the structure of the internal magnetic field lead to differences in the ionospheric conductivity of the hemispheres. The topology and efficiency of interaction of the magnetosphere with the solar wind, which are determined mainly by the IMF orientation, can also differ slightly in the northern and southern hemispheres. The geomagnetic disturbances arising in the course of such interaction in the high-latitude ionosphere propagate through the conducting ionospheric shell to the lower latitudes, and along the closed field lines of the Earth’s dipole—to the opposite hemisphere. As the patterns of the contours in Fig. 4.8 show, the electric field from the polar caps, where the main FACs are concentrated, can propagate to the mid-latitude region. A distinctive feature of the mid-latitude region is the closeness of the geomagnetic field lines and the equipotentiality of the geomagnetically conjugate points of the opposite hemispheres. The effect of the propagation of the electric field from one polar cap to the region of the closed field lines and to the polar cap of the opposite hemisphere is illustrated in Fig. 4.9. In this configuration, the FAC is set in the form of a sinusoidal function only in the northern polar region (α = 1) at the co-latitude circles θ = 5–10°. The boundaries of the polar regions are located at θ = 20°. There is no source of potential excitation in the southern polar region (α = 2). The conductivity in both hemispheres is constant and correlates as NH : SH = 1:2. It can be seen that two convective vortices develop in the northern polar region. The potential difference F between the centers of these vortices is taken as unity. Also, two convection cells are formed in the southern hemisphere, which centers are associated with the boundary between the regions of the open and closed field lines, and F = 0.1, i.e. makes 10% of the value in the northern region. The potential in the southern hemisphere appears due to the overflow of the currents between the boundaries of the polar caps (regions α = 1, 2, 3). The current approaching the common boundary from the region α = 1, where there is an external source of the potential (i.e. the FAC) flows over to the regions α = 2 and 3, being divided

4.3 Electric Fields in the Polar Ionosphere Controlled by the FACs

163

Fig. 4.9 Distribution of the electric potential in the northern (NH) and southern (SH) hemispheres. The source is specified only in the north polar cap. The potential difference in the centers of convective cells F and the step between the contours of the potential dU are expressed in the relative units

in proportion to the conductivity of these regions. Figure 4.10 shows the relations between F in the opposite hemispheres with the change in the conductivity ratio. The nonlinear dependence shows generally that the higher the conductivity of the hemisphere without a current source is, the smaller the interhemispheric difference F, which indicates the possibility of overflow from the hemisphere with a lower conductivity to the hemisphere with a higher one, under the certain conditions of FAC. The simulation results show that the interhemispheric asymmetry in distribution of the FACs and ionosphere conductivity is almost always observed. Fig. 4.10 Change in the ratio between F in the opposite hemispheres upon the change in the ratio of conductivities in the NH and SH

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4.3.2 Localized Current Vortices The energy transfer of electromagnetic disturbances from the Earth’s magnetosphere to the ionosphere occurs mainly due to field-aligned currents (FACs). At high latitudes, where the geomagnetic field B is almost vertical, a magnetic disturbance at the Earth’s surface originates from a system of vortex Hall currents excited by FACs. Vortex ionospheric structures can have various scales, depending on the type of magnetospheric disturbance: from planetary scales under the influence of an interplanetary shock wave (Fujita et al. 2005) to small kinetic scales in auroral structures (Chmyrev et al. 1988). In localized structures associated with vortex currents, the main energy of non stationary magnetospheric-ionospheric disturbances is concentrated. It is they that are responsible for bursts of intensity of geomagnetically induced currents (GICs) in long power lines (Ngwira et al. 2015; Belakhovsky et al. 2019). Thus, the ability to automatically detect localized vortex structures and determine their dynamic parameters from ground-based magnetometers is a crucial task. As a rule, the presence of vortex structures in geomagnetic disturbances is determined visually from the chart of equivalent ionospheric currents derived from magnetic data (e.g., (Engebretson et al. 2013)). (Chinkin et al. 2020) proposed a new method, which not only makes it possible to automatically recognize several simultaneous vortex structures in the ionosphere using the data of the 2D magnetometer network, but also to determine their specific parameters. To validate the method of recognizing vortex structures, we consider convective Hall vortices, referred as traveling convection vortices (TCV) (Glassmeier 1992). TCVs in the ionosphere are a well-known and visible manifestation of the pulse impact of the solar wind on the magnetosphere (Kivelson and Southwood 1991; Sibeck and Korotova 2000; Kataoka et al. 2002). The ground response of such events are isolated impulse geomagnetic disturbances with typical time scales of ~10 min and amplitudes up to the first hundreds of nT, referred as magnetic impulse events (MIE) (Luhr and Brawert 1994; Lanzerotti et al. 1990, 1992). Sporadic MIE/TCV pulses are one of the non-stationary phenomena typical for the daytime high-latitude region. As an example, herein we consider a typical daytime convective vortex based on data from arctic magnetic stations. Model of ionosphere convective vortex. FAC j|| , flowing along the magnetospheric field lines between the magnetosphere and the ionosphere are closed in the conducting E-layer of the ionosphere to a system of transverse Pedersen JP and Hall JH currents: J = Σ P E + Σ H [e × E]. Here, J is a horizontal ionospheric current integrated over the thickness of the conducting layer of the ionosphere; E is a horizontal electric field in the ionosphere; e is the normal to the ionosphere. The ratio between the Pedersen and Hall currents

4.3 Electric Fields in the Polar Ionosphere Controlled by the FACs

165

is determined by the ratio of the integral (over the thickness of the E-layer of the ionosphere) Pedersen P and Hall H conductivities. From the mathematical point of view, Pedersen JP and Hall JH currents are curl-free (divergent) and divergentfree (curl) components of the horizontal current vector J. Using two-dimensional operators RotA = rotz A and DivA = ∂Ax /∂x + ∂Ay /∂y, these features on a plane of the ionosphere can be represented as RotJP = 0 and DivJH = 0. At higher latitudes where the geomagnetic field is almost vertical and j|| ≈ jz , the condition of the current continuity is fulfilled: jz = −DivJ P . This relation shows that the FAC jz is closed only by the Pedersen current in a horizontally homogeneous ionosphere. In the vertical geomagnetic field, the total magnetic effect under the ionosphere from the FAC and Pedersen currents is 0, so the magnetic disturbance at the Earth’s surface is caused only by a system of ionospheric Hall currents. Thus, the equivalent currents in a horizontally homogeneous ionosphere, determined from ground-based magnetic data, are Hall currents. In the region of the ionosphere with horizontally homogeneous integral conductivities j Z = (Σ P /Σ H )RotJ H .

(4.11)

The ratio H / P ~1.5–2.0 varies slightly even with strong variations of the ionospheric conductivity. A source of vortex systems that generates a TCV pulse at the Earth’s surface is usually represented as either (a) an azimuthally symmetric current tube or (b) a pair of oppositely directed currents (Glassmeier 1992). (Chinkin et al. 2020) use the model (a) shown in Fig. 4.11. Figure 4.12 shows qualitatively the spatial structure diagrams of the horizontal Br (R) and vertical Bz (R) components of the magnetic field generated by a homogeneous current tube, which would be observed at the ground based stations under a horizontally homogeneous ionosphere at high latitudes. Pedersen currents flow symmetrically from the center of the incoming FAC, but they do not excite the magnetic response on Earth, since it is compensated by the field created by the FAC (Bϕ = 0). We qualitatively imagine that at a distance of the order of radius Rc , the Pedersen currents drop to a negligible value. In turn, Hall currents fill the region from r = 0 to r = Rc . The magnetic field Br created by them has a bipolar shape (turns to 0 under the center of the vortex current system). In the Bz component, the modeled TCV pulse represents a unipolar magnetic disturbance. Qualitatively, the magnetic disturbance presented in Fig. 4.11 corresponds to the numerical model of TCV by (McHenry and Clauer 1987). Below we briefly describe a technique by (Chinkin et al. 2020) for approximate determination of the position of centers of such localized current

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Fig. 4.11 Schematic representation of the FACs (J|| ) flowing in and out of the ionosphere and their closure to the transverse Pedersen (JP ) and Hall (JH ) currents in the conducting E-layer of the ionosphere for a magnetospheric disturbance in the form of an azimuthally symmetric current tube

Fig. 4.12 The dimensionless radial functions Br (R) and Bz (R) of the magnetic field components and the derivative ∂Bz /∂z (R) for the vortex model shown in Fig. 4.11. The ratio of the height of the ionosphere to the radius of the contour is h0 /Rc = 3/16. The origin coincides with the center of the vortex; dashed lines indicate its boundaries

structures. As an estimate of the typical size of the current circuit and demonstration of the physical meaning of such TCV model, the line ∂Bz /∂z = 0 is taken. Constructing vector field of the equivalent ionospheric current. The 2D magnetic network data are used as initial data, which are components of the horizontal magnetic induction vector Bi (k) = (X i (k),Y i (k))T , i = 1,…,N st , where k is the number of the minute processed, registered at N st stations. The data are

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pre-processed in order to interpolate short-time gaps, eliminate the constant component, daily variation, and long-term variations using a high-pass filter with a finite impulse response. A magnetic record with eliminated low-frequency trend is shown in Fig. 4.13. As an example, we consider the TCV pulse on 31 January 1997. Its signature on the magnetograms of the magnetic stations along the meridional profile is shown in Fig. 4.14. The unevenly distributed station data are further spatially interpolated onto a regular geographic grid using cubic polynomials. The resulting matrices Bk (i, j), where k is a time mark and i, j are grid node coordinates, contain a discrete 2D vector field of geomagnetic variations of the magnetic field B⊥ (t, x, y) at the Earth’s surface. The equivalent ionospheric current field can be represented by rotating the ground field vectors B⊥ by an angle π/2 around the basis vector ez (see Fig. 4.15). Since such vector field may not be divergence-free, the flat curl-free component of the magnetic field is reconstructed in order to derive the divergence-free electric field J associated with Hall currents:

Fig. 4.13 Examples of filtering and removing the trend in magnetograms for the three components recorded at PBQ station (Poste-de-la-Baleine, Canada) on 31 January 1997: “Orig” is the original record, “Filt” is the filtered record

168 Fig. 4.14 TCV impulse recorded along the meridional profile of magnetic stations (geomagnetic components X, Y and Z). Codes of the stations and their geomagnetic coordinates are given on the right

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Fig. 4.15 Plane vector field of the equivalent ionospheric current J, constructed for the event of 31 January 1997 at 14:53 UT before correction. The direction to the Sun is on the left, the length of the arrows depends on the intensity of the magnetic disturbance. Bold arrows represent real equivalent current vectors at station locations plotted with their IAGA codes

J ≈ μ0 J H .

(4.12)

To do this, the analyzed discrete magnetic field B⊥ (x,y) is represented as the gradient of a potential function and this function is determined. The Gauss smoothing filter is applied afterwards to eliminate noise arising from differentiation. An example of the corrected divergence-free field of the equivalent ionospheric current J is shown in Fig. 4.16. Determination of the vortex structure parameters. When automatically determining the centers of the vortices, the center of the magnetic disturbance is estimated from the extrema of the following function:

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Fig. 4.16 An example of the recognition of vortex structures for the event of 31 January 1997 at 14:53 UT: the flat corrected vector field of the equivalent ionospheric current J and the values of the FAC modulus density |J z | (marked in gray-scale color). The circle marker in the vortex center corresponds to its clockwise direction and the inflowing current jz > 0; the asterisk marker in the vortex center corresponds to its counterclockwise direction and the outflowing current jz < 0

G(x, y) = Rot(J/|J|).

(4.13)

The extrema of the function (4.13) correspond to the maxima of the FAC flowing into or out of the ionosphere, according to (4.11). Moreover, from the relationship of the curl and circulation, it follows that it also represents the maximum circulation of the minimum loops. Normalization is introduced so that relatively small vortices are not lost against the background of large structures. Therefore, each extremum of

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the function (4.13) can be recognized as a distinctive internal point of a current loop in the ionosphere. The numerical calculation of the function G is associated with the calculation of the divergence of the field B⊥ /|B⊥ |, which is performed by the standard Matlabfunction divergence. To exclude random maxima, the standard deviation criterion is applied:  |Rot(J)| > χ M[Rot(J)2 ],

(4.14)

where M[Rot(J)2 ] is the mean square of the curls throughout the field and χ is a configurable parameter, empirically set to 2. An example of this technique can be seen in Fig. 4.16, where 4 vortex structures were recognized. Circles and asterisks mark the centers with inflowing and outflowing FACs, respectively. Let the point r0 = (x 0 , y0 )T be the found center of the vortex, the contour (R) be the circle of the radius R centered at the point r0 , and S be the surface formed by this contour. If we consider the circulation of the vector J along the contour (R) 

 Jdl =

C( (R)) = Γ

 RotJd S ∼

S

 Div B ⊥ d S = −

S

S

∂ Bz d S, ∂z

then for the flat approximation it corresponds to the partial derivative of the flux  along the direction perpendicular to the Earth through the surface S: C( (R)) = −

∂ . ∂z

Considering these formulas to be a physical interpretation of the method, we emphasize that  is not calculated, and the equations presented just indicate the connection with the numerical model shown in Fig. 4.12. The scale of the vortex structure can be estimated by finding the extremum of the circulation: ∂C( (R)) = 0. ∂R The value of the maximum equivalent current circulation corresponds to the scale of the maximum change in the flux of the magnetic induction in the vertical direction. Figure 4.17 shows the graphs C(G(R)) for the vortices, which were determined at the previous step (Fig. 4.16). They show extrema by which the scales of these vortices are estimated. The FAC density jz , which characterizes the intensity of the vortex under consideration at a given moment in time, is numerically calculated at the grid nodes based on expressions (4.11) and (4.12): jz (x, y) = (1/μ0 ) Rot J.

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Fig. 4.17 The dependence of the normalized circulation on the radius for each recognized vortex in Fig. 4.16. The vortex coordinates are given in the colatitude–colongitude system (ψ, φ)

Since the conductivity ratio varies slightly during the day, P / H is set to 1. An example of this technique for the vortex scale estimation can be seen in Figs. 4.16 and 4.17. Each vortex trajectory is determined by introducing the special metrics between the recognized structures, which connects the consecutive frames of observations. The connected points reflecting the trajectory of the motion of each vortex are then identified. The implementation of this procedure can be observed in Fig. 4.18. Thus, the method proposed by (Chinkin et al. 2020) enables tracking each vortex from its nucleation until its disintegration or passage beyond the limits of the magnetometer network. The threshold level, when one can speak of the vortex nucleation/disintegration, is determined on the basis of Eq. (4.14): it is unique for each moment of time and depends on the general geomagnetic conditions and the empirically selected parameter χ. Dynamic parameters of the vortices during the event on 31 January 1997. Let us study more in detail the daytime TCV vortices on 31 January 1997. The corresponding impulse (see Fig. 4.14) is a typical TCV event in the noon hours. The source data are geomagnetic observations from stations of the CARISMA and MACCS networks obtained through the SuperMAG portal. Figure 4.18 shows the trajectories of the centers of the two longest disturbances that took place from 14:50 to 14:58 UT, indicating the time of each vortex nucleation, duration, and distance traveled during its existence. The typical lifetime of the analyzed vortices is ~10 min. It can be seen from the figure that the vortices during their existence shift to the night side and to the Earth’s pole. The anti-solar direction of propagation is caused by their “blowing off” from the day side by a stream of solar wind. The typical displacement velocities of the vortex centers are approximately 3.9 km/s and 1.3 km/s. For comparison, the linear speed of the Earth’s rotation at the considered latitudes (~70 °N) is of the order of 0.16 km/s, which indicates that the vortices drifting in the westward direction on the morning side move relative to the Earth’s surface.

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Fig. 4.18 An example of determining the motion trajectories of the vortex structures during the event of 31 January 1997. The lines indicate the displacement of the centers of two vortices (with numbers 2 and 4 in Fig. 4.16) over the observation period, the beginning of the movement is marked with a circle. The figure shows the time of nucleation (Time), duration in minutes (Dur) and the path length in kilometers (L) of each vortex. The position of the vortices is shown at 14:55 UT, the designations are similar to Fig. 4.16

The plots in Fig. 4.19 show the time variations of the smoothed characteristics of the vortex disturbances: the FAC value J z at the center of each vortex, the estimated radius of the vortex, and the current circulation along this radius circuit. The typical size of the vortex is ~500 km, the order of magnitude of the FAC density is up to ~0.2 A/km2 , given the values of geomagnetic disturbance are ~150 nT. The presented plots clearly demonstrate a direct relationship between the magnitude of the FAC J z and the circulation along the circuit (Fig. 4.19, first and third plots). The determination of the vortex parameters often becomes complicated in the last minutes of its existence because of the low FAC values. In addition, sometimes the vortex disappears from the observation zone and enters the blind zone of the station network. This is the case

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Fig. 4.19 Evolution of the parameters of the two continuous vortices (with numbers 2 and 4 in Fig. 4.16) on 31 January 1997, including the FAC intensity J z in the center of the vortex, its radius and the current circulation along this radius circuit. The abscissa indicates the time in minutes from the nucleation of each vortex. The graph line types correspond to those of vortex trajectories in Fig. 4.18

for the negative vortex (dash dotted lines in Figs. 4.18 and 4.19), which disappears almost at the peak of the FAC values. While estimating the relative position of the recognized vortices (Fig. 4.16), one can observe pairs of alternating inflowing (circle marker) and outflowing (asterisk marker) FACs forming adjacent vortices. This observation is consistent with the generally accepted idea of the closure of FACs coming from the magnetosphere in the conducting E-layer of the ionosphere, forming pairs of oppositely directed currents. Thus, the proposed technique allows, in principle, to distinguish the hierarchy of vortices with different spatial scales. Observations on regional magnetometer networks show the presence of horizontal propagation effects of TCV along the ionosphere. Typically, to evaluate the propagation effects of geomagnetic disturbances, magnetograms of stations located either along the geomagnetic meridian or parallel are compared. In the longitudinal direction, the observed time shifts between pulses at spaced stations, as a rule, correspond to phase velocities of ~10 km/s in the anti-solar direction. In the projection onto the equatorial plane of the magnetosphere, this value approximately corresponds to the typical velocity of the magnetosheath plasma flow around the flanks of the magnetopause. In the latitudinal direction, the observed time shifts indicate a specious propagation to the pole with a phase velocity of ~1 km/s. Latitudinal propagation is associated with the fact that the signal propagates along magnetic field lines from

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the equatorial region of the magnetosphere to the high-latitude ionosphere. Since the propagation time increases with increasing latitude , a delay corresponding to the specious propagation to the pole will be observed at the Earth’s surface. The described technique determines the motion of the vortex center, i.e. estimates the group velocity of the disturbance. For the analyzed event, this speed along the geographic latitude is 4.1 km/s, and along the geographic longitude it is 0.3 km/s, so it turns out to be less than typical phase velocities.

4.4 Penetration of the Electric Fields from the Polar Cap to the Mid-Latitudes The currents and magnetic field variations at the latitudes below the auroral zone are relatively small as compared to the auroral ones, and the empirical models are hardly applicable in this area. At the same time, the knowledge of distribution of the electric field, for example, at the subauroral latitudes, is highly important, in particular, for studying the processes near the plasmapause boundary, where polarizing electrojets can develop. The disturbances of the mid-latitude electric fields occur mainly due to direct penetration of the magnetospheric convection fields to the equator through the closure of currents in the ionosphere. In this area, the potential distribution is controlled by the field-aligned currents located in both polar caps. The interhemispheric asymmetry in the electric field distribution specified by the seasonal differences becomes more complex when the IMF azimuthal component evolves. With a non-zero value of BY , the two-vortex convection pattern modifies so that the dawn or dusk vortex, which can cover the entire polar cap, begins to dominate in the circumpolar zone depending on the sign of BY . In the southern hemisphere, the pattern is almost mirrored, so that in the opposite polar caps there is an oppositely directed zonal component of the plasma drift velocity, connected with the meridional component of the electric field directed towards the pole or equator. Under the solstice conditions, the electric field in the area of the closed field lines of the Earth’s dipole is determined to a greater degree by the contribution of the source located in the summer cap. If the round-pole plasma flow, caused by the IMF BY component, dominates in the summer cap, then it may happen that the meridional component of the electric field of a certain sign dominates in the mid-latitude region of both hemispheres. This scenario is shown in Fig. 4.20, with the MLT profiles of the meridional (E θ ) and zonal (E ϕ ) components of the electric field given at 45° magnetic latitude, derived from the convection patterns for the conditions BY < 0 and BY > 0 (|BY | |BZ |) at the December solstice. It can be seen that the zonal component E ϕ is almost identical for both BY signs, while the meridional component E θ has a predominant direction controlled by the BY sign. At BY > 0, in the middle latitudes at all hours of the local time, E θ is directed equatorward in the northern hemisphere, and against the equator in the southern hemisphere. At BY < 0 the direction of E θ is opposite. The average value of E θ is about 0.1 mV/m. From the predominant direction of the meridional E θ

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Fig. 4.20 Components of the electric field E θ and E ϕ (positive eastward direction) at the magnetic latitude θ = 45° during the December solstice season given the IMF BY < 0 (top) and BY > 0 (bottom), | BY | | BZ |

component in the mid-latitude region it follows that the predominant direction of the zonal plasma drift takes place there as well. At BY > 0, in the entire area extending from the north pole through the equator almost to the southern polar cap boundary, the zonal component of the drift velocity is directed mainly eastward, and at BY < 0, westward. This result suggests that the mutual influence of the opposite hemispheres can modify the plasma drift across the entire globe.

4.5 Asymmetric Structures of the FACs and Convection of the Ionospheric Plasma Controlled by the Azimuthal Component of the IMF and Season The problem of determining the degree of influence of the azimuthal (BY ) component of the IMF on the electrodynamic parameters distribution in the near-Earth space has a long history, starting from the study carried out by (Mansurov 1969), when the geomagnetic field variations related to BY sign were first revealed at the high latitudes. The basic conception of the role of the IMF azimuthal component in the electrodynamics of the magnetosphere-ionosphere system was enunciated in (Nishida 1971; Leontyev and Lyatsky 1974). This conception consists in the following: if the solar wind plasma and the magnetic field B frozen into it propagate in the anti-solar direction with the velocity V, then in the coordinate system fixed relating to the Earth, this magnetic field generates an electric field of the solar wind E = −V × B. In particular, BY component of the IMF (positive direction from the dawn to dusk side) generates

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an electric field directed from south to north, at that creating a potential difference between the parts of the magnetosphere. An electric field is effectively projected to the ionosphere of the polar caps along the highly conductive geomagnetic field lines and excites E × B plasma convection. The result of the potential difference between two poles is the radial electric field in the caps and the azimuthal round-poles motion of plasma. At that, the flow directions in the south and north are opposite. Thus, for example, with the negative BY values of the IMF, in the northern (southern) polar cap plasma flows counterclockwise (clockwise). Penetration of the electric field of the solar wind into the magnetosphere depends on the configuration of the magnetic field lines, which determines the area position of reconnection of the geomagnetic field and IMF and the mapping of the electric field into the ionosphere. As it is shown in (Nishida 1971), in both caps BY of the IMF can cause an electric field directed from the dawn to the dusk side and a potential difference between the polar caps of the opposite hemispheres. (Lyatsky 1978) has schematically shown that the convection pattern excited by the IMF BY component can be represented as the sum of two structures, namely, a homogeneous flow through the polar cap in the anti-solar direction (symmetric part DY 0 ) and nearpole vortex (asymmetric part δDY ± ), depending on the BY sign of the IMF. Figure 4.21 illustrates this concept; the direction of the vortices δDY + and δDY – is shown in the northern hemisphere for positive and negative BY of the IMF. The flow direction will be opposite in the southern hemisphere. In general terms, the bidirectional plasma

Fig. 4.21 Configuration of ionospheric convection, excited by the BY component of the IMF, as a superposition of two structures: a homogeneous flow in the anti-solar direction (symmetric part DY 0 ) and a vortex in the polar cap (asymmetric part δBY + or δBY – ) (Lyatsky 1978)

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motion in the opposite hemispheres in the region of the closed geomagnetic field lines having an ideal conductivity, cannot exist under the stationary conditions; it is assumed that the asymmetric flow is localized in the polar caps in the region of the open magnetic field lines. However, the recent observations have shown that a simultaneous bidirectional flow in the northern and southern hemispheres is also possible in the region of the closed lines. This may be connected with the appearance of the anomalous resistance during the flow of the field-aligned currents between the two hemispheres (Kozlovsky et al. 2003). Also, as it follows from the topological model (Watanabe et al. 2007), the interhemispheric asymmetry in distribution of the electrodynamic parameters can occur due to the specific configuration of the reconnecting field lines and the corresponding mapping of the electric field into the ionosphere. An asymmetric part of the electric field controlled by the azimuthal component of the IMF appears mainly at the very high latitudes near the geomagnetic poles, i.e. in the regions where the number and density of measurements are still relatively low, so the problem to obtain accurate quantitative estimates remains rather difficult. Constellation of the European low-orbit satellites such as Magsat, Oersted, CHAMP and Swarm became one of the modern techniques that pushed forward significantly the modeling of the electrodynamics of the high-latitude ionosphere. These satellites, equipped with the highly sensitive magnetometers, performed a vast amount of the magnetic measurements above the ionosphere F-layer. As a result of these data processing, a statistical model of the FACs was created for both hemispheres and different seasons, parameterized by the IMF magnitude and orientation (see Sect. 4.1). The sufficiently intense field-aligned currents in the polar caps were obtained in this model, with the distribution mainly determined by the IMF BY sign. Later, the statistical FAC maps were used as the input parameters for a numerical model of global distribution of the ionospheric electric potential (convection model), which considers the mutual influence of electrodynamically connected opposite hemispheres. The asymmetric structures can be detected by constructing the charts representing the differences between the corresponding maps for the opposite signs of the IMF BY (Lukianova et al. 2010). This allows considering the effects of the simultaneous impact of two factors responsible for intra- and interhemispheric asymmetry, namely, the orientation of the IMF and zenith angle of the Sun (season of the year). In order to determine the structures associated exactly with the impact of the IMF BY component during the different seasons, the difference between the FAC values for the opposite BY signs was calculated at each point of the computational grid; the total IMF intensity, magnitude and sign of the BZ component remained constant. The statistical FAC maps were used according to the data of the Magsat, Oersted and CHAMP satellites (see Sect. 4.1). The same differences were obtained for distribution of the electric potential derived from the convection model (see Sect. 4.2). The subtraction procedure allows to compensate significantly the impact of the BZ component and quasi-viscous interaction, keeping only the “asymmetric” structures controlled by BY . Difference charts “BY > 0 minus BY < 0” for the FACs. Figure 4.22 shows the maps for the equinox, winter and summer in the northern and southern hemispheres

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Fig. 4.22 Difference charts “BY > 0 minus BY < 0” for the field-aligned currents in the northern (upper row) and southern (lower row) hemispheres for the equinox (left), winter/summer (middle) and summer/winter (right) solstices at BZ > 0 (a) and BZ < 0 (b). The value of the maximum density of the inflowing/outflowing current is given in the lower right corner of each map. Inflowing and outflowing currents are shown with solid and dashed lines, respectively; outer and inner latitudinal circles are given at 50° and 70° of the magnetic latitude; magnetic local noon is at the top

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for the positive (Fig. 4.22a) and negative (Fig. 4.22b) IMF BZ sign, which allows to compare manifestations of the BY effect during different seasons for different general orientation of the IMF vector. In the figures, the upper charts refer to the northern and the lower ones to the southern hemisphere. The structures controlled by BY are clearly distinguished in the circumpolar region near the noon meridian in each difference chart. Indeed, when subtracting the FAC distribution typical for a given BY sign from the FAC distribution for the opposite BY sign, all else being equal, the currents specified by the IMF BZ impact, quasi-viscous interaction and the effect of the BY component where it causes a transpolar plasma flow in the anti-solar direction (DY 0 in Fig. 4.21) are compensated; at the same time, the “asymmetric” currents specified by the potential introduced by the BY component into the circumpolar region are summed up. When BZ > 0 (Fig. 4.22a), the difference charts at the equinox and in the summer hemisphere reveal the circular cross-section current localized near the pole at the noon meridian. In the northern cap the current outflows from the ionosphere, and in the southern cap the current flows in. The second, weaker horseshoe-shaped current is localized on the dayside at the lower latitudes. The obtained configuration is quite symmetrical as to the noon-midnight meridian. In summer, the current density is about three times higher than at the equinox. In the winter cap at the noon meridian, the circumpolar current can also be identified. The low latitude current of the opposite direction is stretched and shifted to the dawn side. In Fig. 4.22b, the similar difference charts are shown, but at BZ < 0. It can be seen that in summer and at the equinox the circumpolar currents slightly stretch along the latitude, and the lowlatitude horseshoe-shaped currents also stretch, becoming ring-like. In the winter hemisphere, it is difficult to specify the clear structures, but it is seen that the most intense current is on the dawn side. Difference charts for the electric potential. The configuration of equipotential contours along which the ionospheric plasma moves is closely related to the distribution of the field-aligned currents. Similarly to Figs. 4.22 and 4.23 shows the difference charts depicting the potential distribution for the equinox, winter and summer in the northern and southern hemispheres for positive (Fig. 4.23a) and negative (Fig. 4.23b) IMF BZ component. When subtracting the convection pattern for BY < 0 from the one for BY > 0, the two-vortex convection system with a transpolar flow along the noon-midnight meridian related with the BZ component, reconnection, and quasiviscous interaction becomes more or less efficiently compensated and disappears. Figure 4.23a shows that during the summer season and at the equinox a circular cell remains in the difference charts with a focus almost coinciding with the geomagnetic pole in the summer and slightly shifted to the dayside at the equinox. The shape of the cells is almost the same in the northern and southern hemispheres, and the direction of the plasma flow is opposite. In winter, a two-vortex structure with the stretched cells is revealed, having one focus near the pole at the noon meridian, and another one in the post-midnight sector. The obtained configuration most likely suggests a more effective displacement of the polar cap under the influence of the BY sign exactly in winter, and also that the convection subsystems associated with BY differ not only quantitatively, but also qualitatively in the opposite hemispheres during the

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Fig. 4.23 Difference charts “BY > 0 minus BY < 0” for the electric potential at BZ > 0 (a) and BZ < 0 (b). Spacing between the isolines is 5 kV. The positive and negative potentials are shown by the solid and dashed lines, respectively

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solstice period. Figure 4.23b, showing the difference charts for BZ < 0, demonstrates the convection subsystems similar in structure, but larger in area. In summer and at the equinox, in addition to the prominent circumpolar vortex, the weak vortices on the night side appear, which is probably related with the more intense precipitation with the IMF directed southward. A two-vortex subsystem with focuses located in the afternoon and after-midnight sectors is revealed in winter. Convection increases in the dawn (dusk) sector at BY > 0 (BY < 0) in the northern hemisphere. Further, the more detailed studies revealed the more complex relations between the IMF orientation/season of the year and distribution of the large-scale electric fields and currents. It turned out that as a result of the various factor combination, the distribution of the fields and currents in the opposite hemispheres is not a mirror-like pattern with the given magnitude and sign of the BY component. The configuration of the convection systems depends on the combination of the BY sign and season. Thus, with a combination of BY > 0/summer i BY < 0/winter, a more circular convective round-pole vortex arises than with a combination of BY < 0/summer and BY > 0/winter. However, the question of how asymmetry arises and what additional factors affect it, requires a more detailed consideration. The asymmetric effect of BY can be distinguished by applying the superposition principle to the new statistical maps of the field-aligned currents created according to the above-ionospheric satellite measurements of the magnetic field, as well as to convection systems derived from these maps. The resulting difference charts “BY > 0 minus BY < 0” with the removed symmetric part, mainly related to the impact of the BZ component, allow to reveal the following structures. Field-aligned currents. For the summer and equinox conditions in both hemispheres, one almost circular field-aligned current is concentrated at the pole, and the other one, with the opposite sign and horseshoe-stretched along the latitude, is located on the dayside at the lower latitudes. At BZ > 0, the currents are limited in the near-noon sector, and at BZ < 0, they stretch to the dawn and dusk sides. In winter conditions, a circular circumpolar current can be revealed, at that, an intense current of a low-latitude reverse branch remains on the dawn side. Such configuration is typical for both BZ > 0 and BZ < 0 and might result from displacement of the polar cap as a whole to the dawn or dusk side when the BY sign changes. Convection systems. In the summer and at the equinox, an intense round-pole vortex develops with some shift to the 6–12 MLT sector, controlled by the IMF BY . The plasma rotates in opposite directions in the northern and southern caps, and the potential value in the vortex foci is approximately the same during the corresponding season. However, in winter, two vortices of the comparable intensity appear in the difference charts. Their foci fall on the afternoon and after-midnight hours of the local time. During the solstice seasons, the structure of the electric potential isolines controlled by the BY component differs significantly in the opposite hemispheres. The asymmetric FACs controlled by the BY component, arising due to the potential difference between the polar caps, can flow between the hemispheres. Figure 4.24 shows schematically the current system at BY > 0 (Fig. 4.24a) and BY < 0 (Fig. 4.24b). Thus, at BY > 0, the current flows along the open field lines of the geomagnetic field

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Fig. 4.24 Diagram of the field-aligned current flow generated by the IMF BY of the positive (a) and negative (b) signs, and the relevant convective vortices in the northern hemisphere (c, d)

into the southern cap, then flows through the ionosphere (Pedersen current) to the border of the area of the closed field lines, then flows into the northern hemisphere, then through the ionosphere to the northern polar cap and, finally, flows away back to the solar wind. At BY < 0, the currents are of the reverse direction. The convective vortices excited by these field-aligned currents are shown in Fig. 4.24c, d. In the caps, the more intense currents (indicated by the thicker lines in Fig. 4.24a, b flow in/out on the dawn or dusk side, i.e. they shift as to the noon meridian following the BY sign. In this scenario, the appearance of two current maxima or some other structure stretched along the dawn–dusk meridian might be expected in the difference charts. However, almost circular distribution follows from the statistical FAC model, at least for summer and equinox, which suggests an insignificant displacement of the current from the pole when the BY sign changes. Figure 4.25 shows a qualitative model of the field-aligned currents depending on the IMF BY in one (in this case, northern) hemisphere. The idealized FAC configuration “BY > 0 minus BY < 0” represents a circular current in the center of the cap related with the solar wind and surrounded by the oppositely directed current flowing

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Fig. 4.25 Evolution of the idealized (a) distribution of the field-aligned currents when the IMF turns from the northward direction (b) southwards and the ionospheric conductivity increases (c). The dark and light-gray colors indicate the FACs of different directions

between the hemispheres (Fig. 4.25a). Under the real conditions, both currents shift to the daytime hours due to a higher ionospheric conductivity and more intense reconnection on the dayside, as it is shown for the case BZ > 0 in Fig. 4.25b. When BZ turns southward, the cap area increases and currents shift to the low latitudes and stretch along the parallels (Fig. 4.25c). At the equinox, under the conditions of the highest interhemispheric symmetry, as well as in summer, the statistics of the aboveionospheric FAC measurements demonstrates exactly such evolution of the current shape: from a circular shape to a stretched one (see Fig. 4.22). In winter, upon the low conductivity, the current closure path through the ionosphere becomes complicated, and the FACs flowing between the hemispheres are unevenly distributed along the equatorial boundary of the cap.

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4.6 Low Latitude Electric Currents and Their Response at the Earth’s Surface Regular diurnal geomagnetic variations at the Earth’s surface, at middle and low latitudes, are caused by currents of ionospheric origin (Campbell 1989). The influence of regular solar daily variations on geomagnetic measurements was first observed in the 18th century by Graham. Later, in the 19th century there were some attempts to quantify these variations and to figure out their origin (e.g. (Stewart 1882)), however unsuccessful, as by that time the ionosphere had not been discovered. The most regular of all daily geomagnetic variations are the solar quiet daily (Sq) variations (Elemo and Rabiu 2014). The first serious discussions and analysis of solar diurnal variations emerged during the 1940s, after (Chapman and Bartels 1940) finally worked out and explained the Sq current system. Since then, a large and growing body of literature has investigated solar variations, attributed to the Sq current system (e.g. (Chapman 1951; Hasegawa 1960; Mayaud 1965; Matsushita and Maeda 1965; Stening et al. 2005a, b, 2007)). These variations are attributed to the two currents, situated in E region of the ionosphere on altitudes from 90 to 130 km on the day-side in each Earth’s hemisphere. The current vortex in the northern hemisphere is counterclockwise and the vortex in the southern hemisphere is clockwise (Richmond and Matsushita 1976; Mazaudier and Blanc 1982). The centers of the both vortices, Sq foci, are situated around 30° of the northern and southern magnetic latitudes (MLat), respectively (Fig. 4.26).

Fig. 4.26 The ideal Sq current schema (taken from (Amory-Mazaudier 1983; Anad et al. 2016)), degrees are referred to magnetic latitude

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Various studies were devoted to analyzing the shape of Sq current system (e.g., (Mayaud 1965; Mazaudier and Blanc 1982; Stening 1971, 1991, 2008; Stening et al. 2005a, b; Chen et al. 2007; Stening and Winch 2013; Vichare et al. 2016; Yamazaki and Maute 2017)). Sq current vortices’ position, as well as Sq foci position, are highly variable, as shown, for instance, in (Stening et al. 2005a, b; Chen et al. 2007). (Vichare et al. 2016) pointed out that in winter, when the amount of solar illumination in the northern hemisphere is the smallest, a disintegrated Sq current system can be observed. (Kirchhoff and Carpenter 1976) assumed that variability of Sq current system is associated with changing ionospheric conductivities. There is an unambiguous relationship between Sq amplitude and latitude, local time (Chapman and Bartels 1940), solar activity (Rastogi and Iyer 1976), as well as seasonal effects (Sq amplitudes are weak in local winters and strong in local summers) (Takeda 2002a). (Takeda 2002a) argued that some other factors can cause fluctuation in Sq amplitudes and used spherical harmonic analysis to figure out global features of Sq fields. Furthermore, as the Sq current system complies with the Ohm’s law, Sq amplitude depends on the strength of the main geomagnetic field (B), neutral wind (V) and electrical conductivity (σ). Their effects on Sq amplitude were discussed in (Takeda 1996, 2013a, b). It was demonstrated that albeit the main geomagnetic field’s strength may reduce, an increase in ionospheric Pedersen and Hall conductivities might be sufficient enough to overcome the effect of the main field reduction (see also (Elias et al. 2010)). (Soloviev et al. 2019) demonstrated the enhanced seasonal Sq(X) amplitudes in the region of the South Atlantic magnetic anomaly (SAA), which is known to be the largest depression in the Earth’s main magnetic field (Matzka et al. 2009).

4.6.1 Determination of Sq Field from Ground Based Magnetic Data In order to obtain and analyze both monthly and seasonal Sq variations in the three orthogonal magnetic field components (X, Y, Z), it is necessary to accurately select magnetically quiet days within each month. Most studies are based on the selection of the low planetary Kp-index values (e.g., (Bello et al. 2014; Takeda 2013a, b; Owolabi et al. 2014; Elemo and Rabiu 2014; Anad et al. 2016)). (Vichare et al. 2016) used Ap-index and selected days with Ap < 5. (Soloviev et al. 2019) proposed to use “Measure of anomalousness” (MA) algorithm, based on the fuzzy logic, to select quiet days for each observatory. This method was described in (Soloviev et al. 2016a; Agayan et al. 2016). (Soloviev and Smirnov 2018) applied this algorithm to detection of low magnetic activity periods; they convincingly demonstrated its advantages over the traditional approach based on Kp index estimation (International Q-days). For instance, it was shown that some of the planetary quiet days selected according to the Kp index may contain intense but short-period (less than 1.5 h) magnetic events of comparable with the Sq amplitude; such magnetic disturbances, ignored by the

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accepted methodology, were detected using the MA-based approach and those days were reasonably classified as magnetically disturbed. To analyze the local and global features of the Sq three-component field, let us consider the 1 min data from 121 geomagnetic observatories and stations (75 INTERMAGNET observatories and 46 SuperMAG stations (Love and Chulliat 2013; Gjerloev 2012)) for the year 2008, which was the minimum of the solar cycle 23/24. The year 2008 had generally low Kp-index values and low solar activity index F10.7 ≈ 70, and therefore large portions of magnetically quiet conditions can be selected to study the Sq-variations. The locations of the observatories are shown in Fig. 4.27. The chosen observatories cover geomagnetic latitudes from 60 °S to 62 °N Mlat to avoid the prevalence of magnetospheric current effects over the ionospheric ones at polar latitudes. For each observatory, the time-series of the three magnetic field components (X, Y and Z) are analyzed in the local time (LT) frame. We first select magnetically quiet days using the “Measure of anomalousness” algorithm as described in (Soloviev and Smirnov 2018) and Sect. 3.3.1. In order to evaluate local features of Sq fields, we selected the quiet days for each observatory individually instead of choosing planetary quiet days. In order to obtain close-to-ideal Sq-variation curves with 1-h time step over a given month (hereafter referred to as “monthly Sq-variation”) for each component, we first derive 1-min time series over a given month by the same-UT-minute averaging among the selected quiet days. Then, the shift from UT to LT is performed and the LT-hourly means are calculated. There are different approaches to Sq amplitude calculation. (Takeda 2002a, b, 2013a, b; Takeda et al. 2003) calculated Sq amplitudes as the difference between the maximum and minimum values for each component in the daytime (06-18 LT). Another Sq

Fig. 4.27 Locations of the INTERMAGNET observatories and SuperMAG stations used in the study and their IAGA codes

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calculation method defines the baseline value, which is afterwards subtracted from the values of the magnetic field’s component. There are also two possible ways of calculating the baseline value: the early studies determined the baseline value as the mean value of Sq variation, while the later papers determined the baseline value as the average of the nighttime values (Yamazaki and Maute 2017). Nowadays, it is evident that the latter method of baseline calculation is superior, as the Sq current vortices are fixed in their position with respect to the Sun, thus, such vortices flow on the sunlit side of the Earth, while in the nighttime they have practically zero effect. Such calculation technique was used, for instance, by (Rabiu et al. 2007). The authors calculated Sq-variations for each component by defining the baseline value of the component as the average of the nighttime values (23-02 LT) and then subtracted this value from the hourly values of the magnetic field’s components. We will stick to the latter Sq calculation method, as it gives us an opportunity to analyze hour-to-hour variability of Sq fields. We will briefly describe the chosen calculation technique. We define the baseline value as follows: M0 =

M01 + M02 + M23 + M24 , 4

(4.15)

where M—is the component of the magnetic field, M 01 , M 02 , M 23 , M 24 —averaged hourly values for 01, 02, 23 and 24 h LT, respectively. After that, we calculate the hourly departures as the differences between each hourly value of the component and the baseline value, which can be expressed as follows:

Mi = Mi − M0 ,

(4.16)

where i = 1…24 h LT. (Rabiu et al. 2007) described the so-called non-cyclic variation as a difference between the values of the magnetic field’s components during preceding and succeeding local midnights (i.e., during 01 and 24 LT). We further correct the hourly departures for non-cyclic variations by making a linear adjustment, calculated as a difference between 01 and 24 hourly values, and divided by 23. The final expression can be written in the following form:

M01 − M24 , Sq Mti = Mi + (ti − 1) 23

(4.17)

where Mti is an averaged hourly value of the component, t i is hour (i = 1…24). Further, to get Sq-variation curves with 1-h time step over a given season (hereafter referred as “seasonal Sq-variation”) for each component, we perform the sameUT-hour averaging among the considered monthly Sq-variation curves. We divided months into seasons following Lloyd’s seasons: D-season (November, December, January and February), J-season (May, June, July, August), spring (March and April) and autumn (September and October). Spring and autumn may be combined together into E-season, as they have a tiny difference in terms of solar effect and Sq behavior.

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Fig. 4.28 Monthly (a) and seasonal (b) Sq(Y) variations at CLF observatory in local time (LT) frame during 2008

Such classification is based on the fact that it enables June and December Solstices to be approximately in the middle of J- and D-seasons, whereas March and September Equinoxes are approximately in the middle of the E-seasons, respectively. Figure 4.28 demonstrates monthly and seasonal curves of the Sq(Y) for CLF (Chambon-la-Foret, France, 49.82 °N MLat) observatory derived according to the described methodology.

4.6.2 Regional Features of the Sq Field Figure 4.28 demonstrates clear dependence of Sq(Y) on solar activity: during Lloyd’s D-season, the amount of solar illumination in the northern hemisphere is minimal, which leads to the smallest amplitude of Sq(Y); solar activity during the J-season is the greatest, which leads to enhanced Sq(Y) amplitude. The amounts of solar illumination during spring and autumn are approximately equal; hence, the amplitudes of Sq(Y) are practically the same for both E-seasons. All Sq(Y) curves show the same, “northern-type”, behavior: the variations are negative in the afternoon, having minimums from 12 to 15 LT. In the morning, the variations are mainly positive during E- and J-seasons, while during D-season we can also see the negative values during 0–8 LT decreasing before the corresponding minimums (−5 nT) at 4–5 LT and then increasing with the sign change at 8 LT, showing maximums at 9–10 LT. The Sq(Y) reversal time is practically the same for all seasons, as all seasonal Sq(Y) curves change their sign during 10–11 LT. Figure 4.29 shows averaged monthly Sq(Y) variations at a set of European observatories. In order to demonstrate the ideal Sq current schema proposed by (AmoryMazaudier 1983) and presented in Fig. 4.26, we analyze the shape of Sq-variations of the three components during J-season for the latitudinal chain of European and

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Fig. 4.29 Schema of Sq(Y) month-to-month variability for European observatories. The observatories are presented in the descending order according to their geomagnetic latitudes

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Fig. 4.30 Sq diurnal variations for J-season 2008 observed at European and African observatories in the LT frame. Horizontal lines represent zero amplitude. Quasi-dipole latitudes are shown on the right side

African observatories (see Fig. 4.30). The chosen network of observatories covers all segments of the ideal Sq current schema (Fig. 4.26). According to the magnetic latitude of observatory, Sq(Y) can be divided into “northern-type” and “southern-type” variations. The typical behavior of the “northern-type” is a positive variation in the morning and negative variation in the afternoon (Anad et al. 2016), with the typical Sq(Y) reverse between 10.00 and 11.00 LT. Typical “southern-type” Sq(Y) variations are negative in the morning and positive in the afternoon. From the Fig. 4.30 it is evident that Sq(Y) curves are in good agreement with the ideal Sq current schema (Fig. 4.26). The Northern hemisphere observatories from ESK to MBO exhibit the “northern-type” behavior, showing positive values in the morning and negative values in the afternoon. The two Southern hemisphere observatories HER and CZT show “southern-type” behavior, showing negative Sq(Y) values in the morning and positive values in the afternoon with reversal times between 12.00 and 12.30 LT. Sq(X) behavior can be divided into three groups according to the observatory magnetic latitude: higher than 30 °N MLat, between 30 °N and 30 °S MLat, and below 30 °S MLat (see Fig. 4.26). Behavior of Sq(X) curves shown in Fig. 4.30 also

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corroborates the ideal Sq current schema suggested by (Amory-Mazaudier 1983). Observatories from ESK to PAG show negative Sq(X) values (as they are located at geomagnetic latitudes higher than 40°), which matches the ideal Sq current schema. However, Sq(X) curve at SFS (39.36 °N MLat) and GUI (33.29 °N MLat) show positive values, which according to the ideal schema must be the case for observatories located to the south from the Sq focus. Such Sq(X) behavior at SFS and GUI is due to the northern Sq focus shift during J-season 2008 (Sq focus shift is discussed later). In general, observatories from SFS to ASC exhibit positive Sq(X) values, as they are situated between the northern and the southern Sq foci. Observatories HER and CZT are located below the southern Sq focus, hence, these Sq(X) curves show negative values, which concurs fairly well with the ideal schema, shown in Fig. 4.26.

4.6.3 Global Seasonal Distribution of Sq Field Worldwide estimated seasonal distributions of Sq(X), Sq(Y) and Sq(Z) amplitudes are charted in Figs. 4.31, 4.32 and 4.33, respectively. The amplitude values were interpolated between the points of observatories using the Kriging method (Cressie 1990). The global distribution of the ground-based geomagnetic stations is uneven, having a gap in the South Pacific region; to avoid the edge effects of the numerical interpolation this area was shaded in Figs. 4.31, 4.32 and 4.33.

Fig. 4.31 Global seasonal distribution of Sq(X) amplitudes. The bold black line represents the magnetic (dip) equator for 2008. Large black dots show the location of geomagnetic stations; the small dots mark the magnetic coordinates. The area with no magnetometers available in the South Pacific region is shaded

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Fig. 4.32 Global seasonal distribution of Sq(Y) amplitudes. The notations are identical to Fig. 4.31

Fig. 4.33 Global seasonal distribution of Sq(Z) amplitudes. The notations are identical to Fig. 4.31

In the Fig. 4.31, we can clearly observe that the Sq(X) amplitudes are increased in the equatorial region, due to the influence of the equatorial electrojet (EEJ). For instance, during D-season, Sq(X) amplitudes at middle latitudes (10 °N–60 °N and 10 °S–60 °S MLat) are between 5 and 20 nT, while at the equatorial latitudes (10 °S– 10 °N MLat) there is an increment in amplitudes up to 25–30 nT. During J-season, the values in the southern hemisphere are approximately the same, while amplitudes

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in the northern hemisphere, at the latitudes 40 °N–60 °N MLat, for the most part increase up to 40–45 nT (most likely due to the higher density of observatories). In the geographical latitudinal band from 10° to 40° MLat in both hemispheres, Sq(X) amplitudes remain similar to those during D-season (5–20 nT). Sq(X) latitudinal dependence will is discussed more in detail later. (Cnossen and Richmond 2013) analyzed the global distribution of the threecomponent Sq amplitudes using CMIT (Coupled Magnetosphere-IonosphereThermosphere) model simulations during E-Season (spring) (see Fig. 9 in (Cnossen and Richmond 2013)). The results in Fig. 4.31 coincide well with those obtained by (Cnossen and Richmond 2013) using CMIT model. Both approaches suggest (1) ~20 nT amplitudes at middle latitudes, and (2) enhanced Sq(X) amplitudes over the region of South Atlantic magnetic anomaly (SAA) with amplitudes ∼ 80 nT during March-April (E-season Spring). During D-season, one can observe gradual latitudinal increase in Sq(Y) amplitudes from ∼6–8 nT at higher latitudes (50°–60 °N MLat) in the northern hemisphere up to ∼50 nT over the Australian mainland (35°–50 °S MLat) (Fig. 4.32). Conversely, during J-Season, the biggest Sq(Y) amplitudes (> 50 nT) occur at higher latitudes (30°–60 °N MLat) in the northern hemisphere; Sq(Y) amplitudes decrease with latitudes, exhibiting moderate values (25–30 nT) at equatorial zone and the lowest values of ∼2–6 nT at higher latitudes in the southern hemisphere (55°–60 °S). This result can be explained by the fact that Sq amplitudes are strongly influenced by the amount of solar illumination. During two E-seasons, one can observe practically similar distributions of Sq(Y) amplitudes. In the northern hemisphere, amplitudes at the middle latitudes (10°–60 °N MLat) are around 30–40 nT. In the southern hemisphere at the middle latitudes (10°–60 °S MLat) Sq(Y) amplitudes are also within 30–40 nT with a slight increment over the Australian mainland. During both Eseasons, one can also observe that Sq(Y) amplitudes over the equatorial observatories are smaller in comparison to those over the observatories at middle and high latitudes, which was also observed by (Cnossen and Richmond 2013). Using CMIT data, the authors showed that Sq(Y) amplitudes during E-season (Spring) exhibit lower values (∼20 nT) at low latitudes, gradually increasing up to ∼60 nT at higher latitudes, which is in line with the results demonstrated in the Fig. 4.32. (Anad et al. 2016) studied solar quiet daily variations at Medea observatory (Algeria, 27.98 °N MLat) for 2008– 2011; the authors reported that Sq(Y) variations were subjected to an equinoctial asymmetry, as the amplitude of Sq(Y) in Autumn was always higher than in Spring. It is crucial to note that (Anad et al. 2016) were correct to argue that at Medea observatory there was an equinoctial asymmetry, however, their calculations referred to a single observatory, while here it is typical for observatories situated at magnetic latitudes of ∼30° to have practically the same Sq(Y) amplitudes during both Eseasons. As shown in Fig. 4.32, in some cases Sq(Y) amplitudes at low and middle latitudes during E-season (autumn) are lower than during E-season (spring) (e.g., 180°–150 °W). Figure 4.33 shows worldwide seasonal distribution of Sq(Z) amplitudes, ranging between ∼5 nT and ∼30 nT during all seasons. Also, across all longitudes Sq(Z) amplitudes vary between ∼5 and 15 nT, except for some observatories, where the

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amplitudes slightly increase up to ∼20 nT. The local enhancements of Sq(Z) amplitudes can be attributed to the differences in lateral conductivity of the solid Earth, the conducitivity contrasts between the coasts and continents, and the oceanic induction. The local variations of Sq(Z) amplitudes, as well as the influence of oceanic induction, have been investigated by many studies. For instance, (Srivastava and Habiba Abbas 1980) showed that the coastal effect (i.e., the effect of induced electric currents along the coastline) can be strong enough to cause 1.6–1.8 times increased Sq(Z) amplitudes. This conclusion was based on the comparison of two datasets for 1964– 1965, one of ABG (Alibag, India) observatory, situated within the coastal influence, and the other of HYB (Hyderabad, India), the nearest observatory situated in the central inland.

4.6.4 EEJ Contribution Near Magnetic Equator From the Fig. 4.31, one can also see enhanced Sq(X) amplitudes over SAA region, which makes this area the one of special interest. There are two possible explanations for such an increment in Sq amplitudes. Firstly, there is one observatory HUA (Huancayo, Peru), which during all seasons exhibited increased values of Sq(X) amplitudes. The observatory is located within the latitudinal belt of ± 6° from the magnetic equator, which is known to be affected by the equatorial electrojet (EEJ). The EEJ equivalent current system has a shape of ribbon flowing alongside the magnetic equator. In fact, it might be quite difficult to quantify the real influence of EEJ and to separate it from the total amplitude, in order to obtain the amplitude of Sq-variation. Interactions between Sq variations and EEJ have been studied extensively (e.g., (Yamazaki et al. 2010; Abdul Hamid et al. 2014; Yamazaki and Maute 2017)), and there are several techniques of calculating daily EEJ curves. One of the methods, usually called a “two-station method” (e.g., (Yamazaki and Maute 2017)), defines EEJ variations as the difference between the daily horizontal component curves of equatorial and off-equatorial stations. As the off-equatorial station, we choose PUT, which has approximately the same longitude as HUA. In order to analyze them individually, we define the total amplitude of diurnal X variation as the sum of effects of the EEJ and Sq current system (Soloviev et al. 2019). Total amplitude of diurnal X variation at HUA, Sq(X) at PUT and EEJ amplitudes over HUA are shown in Fig. 4.34. In Fig. 4.31, maximum amplitudes at HUA refer to E-seasons, with the maximum value (> 100 nT) in Spring. The same result for HUA can be seen in Fig. 4.34: there are two maxima during E-seasons, and the Spring maximum is greater. In fact, equinoctial maximums in total amplitude of diurnal variation at HUA are a typical sign of the EEJ influence. At PUT, there are no equinoctial maximums; hence, such equinoctial maximums at HUA are attributed to the influence of EEJ. Examples of the increased solar diurnal variations’ amplitudes at equatorial stations during E-seasons are shown, for instance, in (Doumouya et al. 1998). It is worth mentioning that Sq(Y) amplitudes at HUA (Fig. 4.32) are the same as at consecutive observatories, which argues that X amplitudes are more influenced by

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Fig. 4.34 Total amplitude of diurnal X variation at HUA observatory (Ta(X)), Sq(X) at PUT and EEJ strength over HUA, which is Ta(X)–Sq(X)

EEJ than Y amplitudes. According to (Rabiu et al. 2017), the longitudinal inequality in the amplitudes of Sq(X) along the equatorial region may be explicable in terms of the effects of local winds, dynamics of migratory tides, propagating diurnal tide, and meridional winds. Another possible explanation for an enhancement in Sq(X) amplitudes is that HUA observatory is close to the area of the SAA. Due to the depression in the main magnetic field of the Earth, ionospheric Pedersen and Hall conductivities can experience a sufficient increment. In this case the amplitudes of ground observatory measurements will increase due to higher ionospheric conductivities (see e.g., (Takeda 2013a, b)).

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4.6.5 Latitudinal Dependence of Sq(X) Amplitudes To estimate dependence of daily Sq(X) amplitudes from latitude, we used regression analysis and obtained regression trend curves of different orders (Soloviev et al. 2019). Figure 4.35 shows daily Sq(X) amplitudes and regression trend curves of different orders. According to the ideal Sq current schema shown in Fig. 4.26, Sq(X) behavior can be divided into three groups: (1) at magnetic latitudes higher than 30 °N, (2) at magnetic latitudes between 30 °N and 30 °S, and (3) at magnetic latitudes below 30 °S. The first group shows negative Sq(X) values in the afternoon (due to the effect of westward electric current in the ionosphere); the second group presents positive values (due to the effect of eastward ionospheric current), while in between of them, at the northern vortex focus (∼30 °N), Sq(X) amplitudes are close to zero. The third group exhibits the same behavior as the first group, and at the focus of the southern vortex (∼30 °S), Sq(X) amplitudes are also minimal. Hence, the minimal order of the polynomial trend curve, showing two minimums approximately at 30 °N and 30 °S,

Fig. 4.35 Regression trend curves of different orders, showing Sq(X) latitudinal dependence

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a maximum between them and two maximums around ± 60° latitude, is six. The two minimums of the sixth order trend curve between ± 30° and ± 40° MLat, as well as three maximums around ± 60° and 0° MLat (Fig. 4.35) are in good agreement with the ideal current schema, shown in Fig. 4.26. The obtained equation of the trend curve of the sixth-order is x = − 2−9 y 6 + 2 · 10−8 y 5 + 2 · 10−5 y 4 − 3 · 10−4 y 3 − 0.0345 · y 2 − 0.0355 · y + 36.619, where Sq(X) amplitude and quasi-dipole latitude are denoted as x and y, respectively. Although the actual coefficients before the corresponding power of y are small, they are necessary for the more precise approximation of Sq(X) latitudinal dependence.

4.6.6 Equivalent Sq Current System Modeling To visualize the equivalent two-dimensional Sq current system, we use hourly values of seasonal Sq(X) and Sq(Y) variations. Sq(X) and Sq(Y) can be combined into timedependent vector with coordinates (Sq(Y), Sq(X)), showing the effect of the ionospheric currents on geomagnetic measurements. The rotation of this vector clockwise by 90° gives the equivalent vector, showing the direction and strength of the ionospheric current. This method of the equivalent current system modeling has been used since the early works by Akasofu (e.g., (Akasofu et al. 1980)). Such calculations are based on the current-sheet assumption and do not take into account the field aligned currents (FACs) due to the fact that the Sq cells correspond to the low-and middle latitudes and the influence of FACs at these latitudes is small. (Yamazaki and Maute 2017) provide the theoretical justification of the method and show that under the above mentioned assumptions the horizontal magnetic disturbance of 1 nT corresponds to the equivalent current density of ∼1 mA/m. In order to analyze both northern and southern current loops, we chose two segments: a narrow latitudinal chain of 10 European and African INTERMAGNET observatories covering latitudes from 62 ° N to 10 °N MLat, and the cluster of 6 INTERMAGNET observatories located in the Australian mainland between 20 °S and 60 °S MLat. In order to exclude the EEJ contribution and derive pure Sq equivalent current system, the observatories located within ± 3° degrees from the dipequator are not considered (Soloviev et al. 2019). We assume that although Australian observatories are situated in several time zones, the morphology of the Sq current system within the region during each season and along each latitude is practically the same. Furthermore, the equivalent current system modeling is based on the assumption that ionospheric conductivities remain practically constant. Figures 4.36 and 4.37 show averaged seasonal equivalent Sq current systems simulated for the Southern and Northern hemispheres. Sq current vortices show clear dependence on solar activity, as current loops practically disintegrate during the local winter seasons (J-season in

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Fig. 4.36 Averaged seasonal equivalent Sq current systems over the Australian mainland (Southern hemisphere). Southern hemisphere Sq focus is given with red asterisk. Geomagnetic latitudes of the observatories are shown on the right y-axis. Spacing on y-axis is proportional to geomagnetic latitudes

the southern hemisphere and D-season in the northern hemisphere). In general, both southern and northern vortices exhibit the same oval shape as shown in the ideal Sq current schema (Fig. 4.26). There is a considerable number of studies reporting the absence of Sq current vortices’ formation during local winter seasons (e.g. (Campbell and Matsushita 1982; Vichare et al. 2016)). In the northern hemisphere (Fig. 4.37), there is no trace of Sq current loops’ formation during D-season; in the southern hemisphere, a disintegrated current vortex can be observed during J-season (Fig. 4.36). In the northern hemisphere, Sq current vortices start to emerge in February, and practically vanish by December. The vectors’ magnitudes exhibit the greatest values during J-season; the magnitudes during E-seasons are approximately equal. The behavior and strength of the Sq current system over the Australian observatories during D- and two E-seasons are approximately similar in terms of the current vectors’ magnitudes. To estimate the local time (LT) and the magnetic latitude of the Sq focus, we use the method described by (Vichare et al. 2016). The authors defined magnetic

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Fig. 4.37 Averaged seasonal equivalent Sq current systems over European-African sector (Northern hemisphere). Northern hemisphere designations are the same as in Fig. 4.36

latitude of Sq focus as the central point between two neighboring stations showing opposite current directions; in case the station shows zero current, its location is considered to be the location of Sq focus. Local time of the Sq focus is defined as the middle point of the LTs, with the corresponding current vectors showing northward and southward directions. The defined latitude and the local time of the southern Sq focus (red asterisks in Fig. 4.36) remain roughly the same during all seasons, except for J-season when it is impossible to identify these two parameters. Sq focus is located always between GNA and ASP observatories, having the geomagnetic latitude ∼36 °S. The local time of the focus is always around noon (12.00 LT). As opposed to the southern hemisphere, the latitude and the local time of the northern Sq focus (red asterisks in Fig. 4.37) show seasonal variability. Figure 4.38 shows month-to-month variability of Sq focus latitude and LT from February to November 2008, as it is impossible to estimate the latitude and the local time of the Sq focus during January and December. Sq current focus latitude is subjected to a significant seasonal difference, varying between 45 and 48 °N MLat during D-season and Eseason (Spring), and 28–37 °N MLat during J-season and E-season (Autumn). A number of studies (Campbell and Schiffmacher 1985; Yamazaki et al. 2011; Vichare et al. 2016) have found that in the northern hemisphere during J-season and E-season

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Fig. 4.38 Monthly average latitude (top) and local time (bottom) of the Sq focus for 2008 in the Northern hemisphere

(Autumn) the Sq focus moves to lower latitudes leading to asymmetry between Sq focus latitude during Spring and Autumn. The authors related such an asymmetry to the difference in the winds in the dynamo region during two equinoctial seasons. The results presented here are in line with those obtained previously.

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Chapter 5

Extreme Events and Reconstruction of the Solar Activity Parameters Based on Geomagnetic Measurements

5.1 Solar Dynamo, Solar Cycles and Geoeffective Manifestations of Solar Activity The dominating part of the solar energy falls on the electromagnetic radiation, the power of which is about 1017 W. With the temperature of approximately 106 K, the solar corona represents the plasma, with magnetic field prevailing in its’ dynamics. The strongest magnetic fields are concentrated in the sunspots. The solar corona extends at a supersonic speed, filling the heliosphere, as was first proposed by Parker (1958). Parker’s equations assume the knowledge of the magnetic field vector of the source on the surface close to the Sun, which allows to extrapolate the equations to the heliosphere. Now, it is clear that the surface magnetic field of the source is a complex function depending on location and time, which complicates Parker’s model for plain field expansion to the heliosphere. The plasma flows ascend constantly from the Sun, forming the solar wind (SW) and the interplanetary magnetic field (IMF)—the part of the solar magnetic field that is carried away into the interplanetary space. Due to the high plasma conductivity the magnetic field moves along with it. The condition of “freezing-in” of the field in the solar wind is fulfilled due to a large track length of the particles and high conductivity of the plasma. Although the energy inflow from radiation exceeds significantly the energy input from the solar wind, complex interaction of the solar wind with the Earth’s magnetosphere is the source of the majority of geomagnetic phenomena. The SW and IMF parameters are defined by the evolution of the solar magnetic field which changes cyclically. The visible cycle of the solar activity is traditionally related with formation of the sunspots and their migration to the solar equator. This cycle has a wellknown periodicity of about 11 years. The spots represent the areas with the reduced temperature and intense magnetic fields. Some certain spots or spot groups may

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Gvishiani and A. Soloviev, Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation, https://doi.org/10.1007/978-3-030-58969-1_5

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Fig. 5.1 Solar cycles since 1600. The blue color indicates the period of the instrumental measurements, and the red represents reconstruction during the pre-instrumental period. The Maunder and Dalton minima and the modern Grand maximum are specified

exist during several solar revolutions, thereby providing the recurrent manifestations of the activity. A common measure of the solar activity is the Wolf’s number measuring the number and total area of the sunspots. During 11-year cycle, the number of spots varies from almost 0 at the minimum to 100 or more at the maximum. Figure 5.1 shows the average annual sunspot numbers (SSN) over 400 years. Apart from 11-year cycle, the variation of the solar activity has a longer-period component of approximately 100 years. During the time period shown in the figure, two prolonged decreases in the solar activity occurred, the Maunder and Dalton minima in 1600–1700 and in 1800s, respectively. In the twentieth century, on the contrary, an increase in the solar activity peaks was observed, and the absolute peak took place in the 1950s. This period constitutes the Grand Solar Minimum (Usoskin et al. 2014). In the 11-year variation, the spots of the new cycle are formed several years before the minimum at relatively high heliolatitudes ~35°–40°. The spot formation zone moves to ever lower latitudes over time, approaching the solar equator. The extension and shift of the sunspots along the heliolatitude along with the evolution of the activity cycles was first identified by Maunder (1904). The parameter often used instead of the sunspot numbers is the flux of the solar radio emission on the frequency of 2800 MHz or at the wavelength of 10.7 cm. This parameter is known as the solar activity index F10.7 ; it has been constantly measured by the ground-based solar observatories since 1947 (Tapping 2013). The physical mechanisms underlying the measurements of the solar radio flux F10.7 are thermal bremsstrahlung emission in the corona and radiation of gyromagnetic resonance in the strong magnetic fields above the sunspots (Schonfeld et al. 2015). The gyroresonance component (about 90% of the total intensity for the F10.7 index) is related with the magnetic fields of the sunspots. This provides a close relation of F10.7 with the Wolf numbers. However, approximately since 2000, an excess of the F10.7 radio flux has been noted as compared to the sunspot numbers using the earlier correlations (Lukianova and Mursula 2011). The reason of this deviation from the earlier correlation has not yet been exactly determined, although it is probably related with a decrease in the magnetic field strength in the sunspots, decrease of the contrast with the surrounding photosphere, so that the smaller spots may not be taken into account, but the magnetic

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field strength is still sufficient for generating the gyroresonance interactions leading to radio emission (Livingston et al. 2012). A complete cycle with a period of 22 years is related with a change in distribution and polarity of the global solar magnetic field and is called the solar dynamo cycle. The 22-year magnetic cycle is fundamental in the dynamics of the internal structure of the Sun. The physical mechanism of the solar dynamo is still not completely understood, despite the progress made in studying the internal structure of the Sun using helioseismology, as well as the surface magnetic observations. The model initially proposed in the 1960s by Babcock and Leighton (1961, 1969) is still the basic approach for modern developments in the solar dynamo theory. Many improvements have been made for this model; among the most important ones is the meridional circulation mechanism observed in the solar cycle (Wang and Sheeley 1991). The continuing accumulation of helioseismic observations, their increasing resolution and quality and analysis of these data during the last three solar cycles have confirmed the complexity of the solar dynamo, including its multipolar nature; for the time being, the latter cannot be adequately described by the modern models. The asymmetry in the activity between the hemispheres, possible existence of the second dynamo, quasi-two-year modulation of the solar activity and some other effects have been also identified. The global solar magnetic field is changing in the antiphase with the sunspot number. Its structure is close to the dipole at the minimum of the 11-year activity cycle. The appearance of the sunspots with the beginning of the cycle is related with a complication of the dynamics of the magnetic flux tubes below the photosphere. The formation of spots follows the movement of tubes with a strong magnetic flux from the convection zone through the photosphere. The spots appear in pairs of the opposite magnetic polarity, and the pairs are usually oriented from east to west. The polarities in the bipolar structures are always ordered in such a way that the previous (western) spot has one polarity in the southern hemisphere and the opposite polarity in the northern hemisphere (Hale et al. 1919). The following (eastern) spots also have the opposite polarities in two hemispheres. The field strength at the poles gradually decreases as the cycle is approaching its’ maximum. The general structure of the solar magnetic field is already multipole at the maximum. Then, the solar dipole intensity increases again, but this time it has the opposite polarity. Thus, the complete cycle of the solar magnetic field variation, taking into account the sign change, is equal to the double duration of the 11-year cycle of the solar activity, which makes approximately 22 years (Fig. 5.2). The ampitude and shape of the 11-year cycles change significantly from cycle to cycle reflecting the long-term variations of the solar activity. The solar cycle forecasts are based on the historical correlations between the sunspot numbers and parameters that may be physically related with the sunspot cycles. Determination of the functional relationships remains probably the most important topic of studies that tie the solar variability with the processes occurring on the Earth. It is remarkable that some ground-based indices of the geomagnetic activity turned out to be the most reliable predictors of the solar activity cycles. A large number of the geomagnetic indices were brought into correlation with the solar activity in order to verify their potential use for predicting the main parameters of

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Fig. 5.2 11-year cycle of the solar magnetic field reversal (above) and 11-year cycle of the sunspots (below)

the future cycles. In particular, it has been established that the minimum level of the geomagnetic activity, as measured by the aa index near the solar minimum, is a good predictor of the next solar maximum power (Wang and Sheeley 2009). The minimum geomagnetic activity near the solar cycle minimum is related with the axial-dipole component of the solar magnetic field, and is the best predictor of the activity in the next cycle. The dynamic processes occurring inside the Sun and in the solar atmosphere propagate in the interplanetary space, reach the orbit of the Earth and determine the space weather conditions. The space weather disturbances originate from the solar energy events and are transferred by the solar wind. The direct satellite measurements of the IMF and SW parameters have been carried out since the mid-1960s by several generations of the satellites. The average annual values of the sunspots, SW speed and vertical (BZ ) and azimuthal (BY ) components of the IMF are given in Fig. 5.3 (http://omniweb.gsfc.nasa.gov/). These parameters are the most geoeffective. It can be seen from Fig. 5.3 that the SW speed with the average value of 440 km/s peaks in 1974, 1983, 1994 and 2003 exceeding 500 km/s, which falls on the decrease phases of the solar cycles. The BZ and BY components of the IMF change irregularly, and their average values for the considered period are close to zero. The solar flares and coronal mass ejections (CMEs) are the powerful sporadic events causing the disturbances of the geomagnetic field. The flare is a response of the solar atmosphere to a sudden rapid process of the magnetic-originated energy release, being the product of instabilities in the multiscale dynamic magnetic structures.

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Fig. 5.3 Average annual values of the sunspot numbers, SW speed and BY , BZ components of the IMF for the period of the satellite measurements since 1966

During the flare, a large amount of plasma is ejected into the heliosphere at the speed of about 1000 km/s and higher. The interplanetary magnetic clouds, representing the structures with an intensified magnetic field and increased values of the main plasma parameters (speed, density and temperature), are related with the flares. Inside the cloud, the vector of the intensified magnetic field rotates smoothly. The main factor affecting the CME efficiency in terms of the disturbances in the near-Earth space environment is a large-scale, often very fast, shock wave that is formed in the front of the magnetic cloud while it propagates from the Sun. Both the frequency of the solar flares and CMEs vary depending on the solar activity cycle and, as a rule,

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correlate with the sunspot numbers. The flares and coronal mass ejections more often occur at the maximum of the solar cycle. Depending on the speed, the SW near the Earth’s orbit is conventionally divided into two classes: slow (280–400 km/s) and fast (800–2000 km/s). By detecting the coronal holes in the 1970s and revealing their relation with the high-speed streams (HSS) of the SW, it became clear what was the source of the fast solar wind. The coronal holes, and the polar coronal holes in particular, provided almost a uniform fast SW with the speeds of about 700–800 km/s. The other areas of the Sun and corona were the source of a more variable and much slower component of the SW. The fast SW speed is related with the expansion of the unipolar magnetic flux tubes resting on the corona base in the coronal hole where the wind originates (Wang and Sheeley 1990). The characteristic feature of the slow SW is its high variability in all parameters: speed, density, temperature. The slow wind arises from the areas of the closed magnetic field lines of the corona. The recurrent fast SW streams are emitted by the Sun for several months, usually during the decrease of the 11-year cycle, and their periodicity makes 27 days when observing from the Earth (the period of the Sun rotation). The highest values of the SW speed observed at the decrease phase of the 11-year cycle are caused by the SW HSS emanating from the equatorial coronal holes located in the ecliptic plane. As HSS spread in the outer space, they compress a slower background SW. This is how the stream interaction region (SIR) is formed, where the plasma becomes denser, magnetic field at the front edge of the interaction region is intensified, and SW speed gradually increases to the fast flow speed. When the coronal structure is stable during several solar revolutions, the interaction pattern is successively repeated, which leads to the Corotating Interaction Region (CIR). The CMEs and CIRs cause an increase in the geomagnetic activity, magnetic storms and substorms, at that, the CMEs cause the most intense and shortest magnetic storms, while the CIRs cause less intense storms, but lasting much longer. With the HSS, the magnetic storm begins gradually and without a sudden pulse. The HSS periods are characterized by the fast repeating oscillations of the IMF with relatively moderate amplitude, but the amount of energy entering the magnetosphere can be comparable to or even exceed the amount of energy during the storms caused by the CMEs (Tsurutani et al. 2006). Forecasting the CIR evolution is an important problem in the space weather studies.

5.2 Reconstruction of the Solar Wind Speed Based on the High-Latitude Geomagnetic Data The SW speed is one of the main factors in the interaction of the SW with the Earth’s magnetosphere. Construction of a long time series of the values of the SW parameters constitutes a great interest, but reconstruction of these values in the presatellite period is possible only using the indirect data. Such type of data is represented

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by the geomagnetic activity records, which is varying under the effect of the solar activity. The reconstruction attempts have been made several times, mainly, according to the mid-latitude global geomagnetic index aa, which has the longest series of observations, since 1864 (Lockwood 2013). The results demonstrated the general trends in the SW parameters, but turned to be very approximate in dealing with extreme HSS values. This could be observed in making comparison of model values and speeds measured by the satellite. The average annual SW speed in 2003, when a series of powerful eruptive events and large-area coronal holes were observed, was the highest ever observed by the satellites. The impact of the SW HSS on the magnetosphere in 2003 was so strong that it resulted in a visible outlier corresponding to 2003 in the normally smooth curve of the geomagnetic field evolution represented by a series of the annual means of the vertical component Z recorded at the circumpolar observatories. Figure 5.4 demonstrates the annual means of the Z component (hereinafter referred to as AM_Z) at Godhavn (GDH, Greenland) observatory located in the northern polar cap. The AM_Z values were calculated for five magnetically quietest and five most disturbed days of each month denoted by AM_Zq and AM_Zd, respectively (the list of five international quietest and most disturbed days is available at the Kyoto World Data Center for Geomagnetism website http://wdc.kugi.kyoto-u.ac.jp). The geomagnetic field intensity level obtained for the quietest days is commonly used as a contribution of the main EMF and as a zero level to estimate the development of geomagnetic disturbances of the external magnetosphere-ionosphere origin against the background. However, in this case, it can be seen that in 2003 the quietest level was also disturbed.

Fig. 5.4 Annual means of the vertical Z component of the geomagnetic field at Godhavn observatory (GDH, Greenland) calculated for five quietest (blue curve) and most disturbed (red curve) days of each month over 1926–2010

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The impaired smoothness of the secular trend in 2003 was observed both in the north and in the south polar caps. The southern polar observatories possess much shorter series of continuous observations than the northern ones. The data of two Antarctic observatories Dumont d’Urville (DRV) and Casey (CSY) are available since the 1990s. The long-term observations were also carried out at the Vostok observatory (VOS), however, in the very 2003 the observatory was not operational because of the fire that occurred there. Figure 5.5 demonstrates the annual means of the Z component for DRV and CSY. Here, the long-term trend of the AM_Z is almost the same as for the northern observatory but with the opposite sign. The gradual increase is interrupted in 2003, when a jump in the Z component by about 20–30 nT is observed at both observatories; such deviation is quite significant as compared to the overall secular trend. Thus, the coherent data from the northern and southern polar observatories suggest that in 2003 a global increase (more positive Z values in the north and more negative Z values in the south) of the geomagnetic field occurred. The analysis of the geomagnetic data shows that the observed anomaly is specified by a substorm westward auroral electrojet (Westward Electrojet, WEJ). Indeed, at the polar latitudes, the contribution to the Z component can be made by those currents of the ionosphere-magnetosphere system that flow along the round-pole trajectory. In Fig. 5.5 Annual means of the Z component at the DRV and CSY observatories located in the southern polar cap

5.2 Reconstruction of the Solar Wind Speed Based …

215

general, the magnetic effect of the following currents can enhance the Z component in the polar caps of both northern and southern hemispheres: (1) (2) (3) (4)

WEJ (Hall ionospheric current); ring current (westward drift of the positive ions); the round-pole plasma flow specified by the IMF BY < 0; dawn-dusk current in the magnetotail.

The magnetic fields from other currents, on the contrary, weaken Z component in both polar caps: (5) Chapman-Ferraro current at the magnetopause; (6) eastward electrojet; (7) the round-pole plasma flow at the IMF BY > 0. The influence of the currents (2)–(4), listed in the first group, excluding the first one (WEJ) can be considered as insignificant. Indeed, a weak ring current is typical for the geomagnetic storms driven by CIR as the solar source. Thus, in 2003 the average Dst value was around −25 nT, which is far from being an extreme value. The annual average value of BY component of the IMF was negative (Fig. 5.3), and in the northern polar cap the system of the ionospheric currents specified by BY component could have technically created an observed increase in the geomagnetic Z component (Lukianova et al. 2010; Lukianova and Kozlovsky 2011). However, in this case, at the Antarctic observatories the weakening instead of increase of the Z intensity in modulus would have been observed. Thus, the effect of intensification of the vertical magnetic field observed in the polar caps of both hemispheres can be explained exactly by the intensification of the WEJ flowing in the ionosphere along a semicircular round-pole trajectory. The substorm WEJ occupies the night and early dawn sectors of the magnetic local time (MLT), and the less intense eastward electrojet is concentrated in the noon and dusk MLT sectors. The WEJ geomagnetic effect is as follows: the magnetic field generated by WEJ reduces the horizontal (H) component in the auroral zone in the northern hemisphere, strengthens the Z component in the poleward region from the WEJ in both hemispheres and weakens the Z component in the equatorward region from the WEJ (Lyatsky et al. 2006). The geomagnetic variations at the auroral latitudes, especially where the currents of the westward and eastward electrojets flow in the ionosphere, confirm that the SW HSS cause the most intense substorms. The conventional WEJ intensity indicator is the AL index (Davis and Sugiura 1966). In order to identify the interannual variability of the quiet and disturbed WEJ, similarly to the Z component the annual means of AL were calculated for five most disturbed and quietest days of each month, i.e. the indices ALd and ALq, respectively. Figure 5.6 represents the variations of the ALd and ALq indices for the total period of AL availability, starting since 1966 (for some years the index values are absent). The average level of ALq makes about −40 nT, and for ALd it is approximately −300 nT. It can be seen that the ALd minima fall on the decrease phases of the solar cycles in the years of the SW HSS 1974, 1982/1984, 1991/1994 and 2003 with the deepest minimum being

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5 Extreme Events and Reconstruction of the Solar …

Fig. 5.6 Annual means of the AL index calculated for five quietest (blue line) and most disturbed (red line) days of each month

observed exactly in 2003. The latter is followed by a strong shift of the index in the positive direction, which corresponds to an exceptionally quiet Sun over the period of the last minimum of the solar activity. In the time series of the ALq index, the variation of the solar cycle is less evident. However, it is clearly seen that ALq has the minima in the same years as ALd, repeating even their relative depth. Both indices indicate the deepest minimum in 2003, when ALq ≈ −75 nT, i.e. almost two times less than the average index value. The second largest minimum in ALq is observed during the decrease of the previous 22-year solar cycle, in 1982. In Fig. 5.7, the differences between the most disturbed and quietest levels for the AL index and Z component recorded at THL, RES, DRV and CSY observatories (their names and coordinates are given in Table 5.1), as well as the SW speed are given, which makes it possible to correlate the change in the WEJ intensity and geomagnetic parameters in the polar cap and in the auroral zone. The monthly means of the considered parameters and their sliding average are shown in the left column, and the annual–means are given in the right column. It can be seen that for each observatory the difference Z between the most disturbed and quietest level, as a whole, shows the same variation peaking in the decrease phase of the 11-year solar cycle, when the SW speed is maximal. This is especially clearly seen in RES observatory data in 1974, 1983, 1994 and 2003. Two Antarctic observatories DRV and CSY show the minima (the signs in the southern and northern hemispheres are opposite) of the difference Z also during the years of the SW HSS, in 1994 and 2003. A further study of the “2003 effect” argued that SW HSS evoke substorms of the maximum intensity with the substorm WEJ dominating in the high-latitude current system (Lukianova et al. 2012; Mursula et al. 2015). For the high-latitude stations, the geomagnetic parameter representing a difference between the most disturbed and quietest levels, is the appropriate indicator of the WEJ intensity and has the highest

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217

Fig. 5.7 Difference values between the most disturbed and quietest levels for the AL index (top row) and for the Z component recorded at THL, RES, DRV and CSY observatories (consecutive rows). Plots on the left show the monthly means of the considered parameters and their sliding average; plots on the right show their annual means. The annual means of the SW speed are shown in the bottom row

218 Table 5.1 List of observatories

5 Extreme Events and Reconstruction of the Solar … Name

IAGA code

Geographical coordinates

Geomagnetic coordinates

Thule/Qaanaaq

THL

78°N 69°W

88°N 14°E

Resolute bay

RES

75°N 95°W

83°N 64°W

Godhavn

GDH

69°N 54°W

79°N 35°E

Sodankyla

SOD

67°N 27°E

64°N 121°E

Dumont D’Urville

DRV

67°S 140°E

75°S 128°W

Casey

CSY

66°S 111°E

76°S 176°E

correlation with the SW speed. As mentioned above, the geomagnetic effect of the extreme event of solar activity in 2003 could be observed in the secular trend of the annual means of the Z component at the polar observatories of both hemispheres. At THL and RES observatories located in the northern polar cap (see Table 5.1), this effect reproduced the only visible jump in the annual means series for the total history of measurements, as the regular monitoring at these observatories was started in the late 1950s. Hence, it can be reliably assumed that in 2003 we observed the most intense HSS over the last half a century. However, there are two unique observatories with even longer series of observations in the high latitudes of the northern hemisphere. Godhavn Polar Observatory (GDH) and Sodankylä Auroral Observatory (SOD) (see Table 5.1) began their work in 1926 and 1914, respectively. The analysis of the records of two oldest high-latitude observatories allowed detecting other extreme events of the SW HSS similar to the 2003 event (see Fig. 5.4). Figure 5.8 shows the vertical component difference Z at the GDH observatory between the maximum disturbances (the average of five international most disturbed days) and the quiet level (the average of five international quietest days) for each year since 1926. The Z value is correlated with SSN within the 11-year solar cycle. It can be seen that Z peaks in the decrease phase of each cycle. Two highest peaks observed in 1952 and 2003 stand out in the plot. The annual means of the horizontal component differences H at SOD observatory are given in Fig. 5.9. While Z at GDH observatory were derived using all hourly means for each day, H at SOD observatory were based on the data from the 23–02 MLT sector only. This is specified by the fact that in the auroral zone the signal of substorm WEJ appears in the horizontal component only on the night side, and in the polar cap the WEJ contribution to the disturbance of the vertical component has the weak dependence on the local time. It is seen from the figure that two maxima in the H at SOD observatory are also distinguished, both in 1953 and 2003. Based on the obtained results, it is possible to reconstruct the SW speed in the past and to specify the annual peaks with a high reliability. As the comparisons with the data of direct measurements over the last 50 years show, the method adequately reproduces the extreme values of the SW speed in each solar cycle. For the period of the direct satellite measurements since 1965, the correlation coefficient between the measured SW speed and the Z/H parameters is 0.78 for both observatories,

5.2 Reconstruction of the Solar Wind Speed Based …

219

Fig. 5.8 Difference between the most disturbed and quietest levels of the vertical geomagnetic component (Z) for each year at GDH observatory since 1926 (black vertical lines, left axis), and variation of the 11-year solar cycles in terms of SSN (grey curve, right axis)

Fig. 5.9 Annual means of the horizontal geomagnetic component differences (H) derived from the SOD observatory data recorded in the 23–02 MLT sector since 1914

and the linear regression is described by the equations Vsw = 0.95·H – 258.8 and Vsw = 3.2·Z + 367.8, respectively. These relationships allow not only to reconstruct the values of the SW speed for each year in the past, but also to specify the extreme values. The fact that it was possible to distinguish the SW HSS in 1952 (see Fig. 5.4) is important in confirming the modern solar dynamo theory. In fact, the extreme SW HSS in

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5 Extreme Events and Reconstruction of the Solar …

1952 in the decrease phase of the solar cycle 18 was followed by an exceptionally highly active solar cycle 19, which was the peak of the modern centennial grand maximum. The change in the toroidal field of the Sun occurs in the phase with a change of the sunspot number, and its evolution can be traced over a long time interval, since the SSN are well known. The poloidal field changes approximately in the antiphase with the toroidal field, and exactly its intensity specifies the SW HSS coming from the coronal holes. The SW speed is an indicator of the poloidal field value, but the direct measurements of this indicator were started only in the mid-60 s. Previously, it was not possible to adequately recover the differences in the peak values of the SW speed because the mid-latitude instead of the high-latitude magnetic variations were considered (Lockwood et al. 2009). The obtained results for the first time provide an experimental confirmation of the modern theory of the solar dynamo, which suggests the interpretation of the solar activity cyclicity as a successive transformation of the poloidal and toroidal configuration of the solar magnetic field (Babcock 1961; Leighton 1969; Charbonneau 2014). The HSS peak in 1952 shows that an exceptionally powerful toroidal field of the 19th solar cycle has been preceded by an increased poloidal field of the 18th cycle.

5.3 Reconstruction of the Monthly Average and Seasonal Values of the Solar Wind Speed and Definition of Extrema The data of the Sodankylä observatory (SOD), which provides the series of the hourly means of the geomagnetic field vector measurements, were also used to reconstruct the monthly means of the SW speed. The observatory is located near the equatorial boundary of the auroral oval and is well suited for the substorm WEJ monitoring. According to the substorm development scenario, the three-dimensional system of the magnetospheric-ionospheric currents is formed when the current across the magnetotail breaks, which in turn is closed on the night ionosphere through the field-aligned currents. With a substorm, the maximum negative deviations of H from the quiet level are observed in the 22–01 MLT sector, which corresponds to 20–23 UT for SOD observatory. The positive deviations are observed in the afternoon sector and related with the eastward electrojet. Similarly to the above, for each month of the year, the parameter H is determined as a difference between the average value of the H component and the quiet level calculated in the interval 20–23 UT for five quietest, but now locally quietest days. The linear regressions were constructed for the H values and SW speed for each month. In Fig. 5.10, the correlation for all Januaries in 1964–2014 is given. The correlation is statistically significant (the correlation coefficient is 0.76, the zero correlation probability is p = 0.0002 when using the first-order autoregressive model). The deviation of the values from the regression line is symmetric (the lower plot in Fig. 5.10) for the total range, which indicates the constancy of the average

5.3 Reconstruction of the Monthly Average and Seasonal …

221

Fig. 5.10 Correlation between H at SOD observatory and SW speed for all Januaries in 1964– 2014 (top) and deviation of the values from the regression line (bottom) (Lukianova et al. 2017)

variance and applicability of the least squares method approximation. In Table 5.2, the coefficient values of the regression equation Vsw = a·H + b and the correlation coefficients for each month are given. The coefficients a and b for each month in the form of a diagram are given in Fig. 5.11. It can be seen that the slope of the regression line varies significantly from month to month, while the intercept varies slightly. The maximum slope is observed in winter and summer, and the minimum is on the equinox; therefore, the H response is higher on the equinox than on the solstice for the given value of the SW speed. In Fig. 5.12, the dependences between the SW speed measured by the satellite and reconstructed speed according to the data of Table 5.2 are given for the period of 1964–2014, when the direct measurements were available. Also, the parameter δ = V meas – V recon depending on the speed value is given. It can be seen that the average variance is evenly distributed almost in the total range, which indicates the adequacy of the model. Table 5.2 Regression and correlation coefficients Coefficient/Month 1

2

3

4

5

6

7

8

9

10

11

12

a [km/s/nT]

1.87 1.62 0.99 0.97 1.26 1.69 1.34 1.00 0.85 0.72 1.05 1.79

b [km/s]

367

363

Correlation coefficient

0.7

0.74 0.72 0.77 0.82 0.73 0.74 0.64 0.68 0.54 0.73 0.77

377

373

359

360

372

386

373

379

374

363

222

5 Extreme Events and Reconstruction of the Solar …

Fig. 5.11 Regression coefficients a (left) and b (right) for each month (Lukianova et al. 2017)

Fig. 5.12 Dependences between the SW speed measured by the satellite and reconstructed one for the period 1964–2014 (top) and parameter δ = V meas – V recon depending on the SW speed value (bottom) (Lukianova et al. 2017)

5.3 Reconstruction of the Monthly Average and Seasonal …

223

The obtained relationships were further used to reconstruct the monthly means of the SW speed for a hundred-year period, since 1914 (Lukianova et al. 2017). In Fig. 5.13, the values of the reconstructed speeds for each month and the standard deviation ±σ, as well as the measured speeds since 1964 are presented. It can be seen that the change in H reflects the SW speed change quite precisely, except for some peaks that go beyond ±1σ. This may be related with the absence of the measurements for some periods. Figure 5.14 shows the measured and reconstructed values over 1964–2014, including the months when more than 30% of the measurements was

Fig. 5.13 Values of the reconstructed SW speeds for each month (blue) and standard deviation ± σ (grey) over 1914–2014, as well as the measured values since 1964 (red) (Lukianova et al. 2017)

224

5 Extreme Events and Reconstruction of the Solar …

Fig. 5.14 Measured (red) and reconstructed (blue) values of the SW speed in 1964–2014, including the months when more than 30% of the measurements were available (top), and the share of the gaps in the measurements (bottom) (Lukianova et al. 2017)

available; the lower plot of the figure shows the share of the gaps in the measurements. Until 1995, the number of gaps was quite large, which certainly affects the accuracy of reconstruction. A large number of the gaps in the measurements refer to the period when the SW was monitored by only one IMP-8 satellite. Since 1995, when the ACE satellite for measuring SW parameters came up, there are almost no gaps. The standard deviation of the measured and reconstructed values in 1964–1994 makes 39 km/s, and in 1995–2014 it is 31 km/s. Figure 5.15 shows the reconstructed and measured values of the monthly means of the SW speed as two superimposed time series. It can be seen that the peaks are dated for the decrease phase of each solar cycle (the lower plot in Fig. 5.15). In cycles 15–23, the peak values are observed in 1919, 1930, 1941, 1952, 1959, 1973, 1982, 1994 and 2003. By comparing the years of the largest monthly and annual means (see (Mursula et al. 2015)) of the HSS, one can observe their good coincidence. This argues that during the centennial grand maximum of the solar activity, the coronal holes were long-lived, at least for several months, i.e. existed over several revolutions of the Sun. The comparison of the HSS distribution and sunspot number indicates that although the SW speed increases on average with the long-term evolution of the solar activity, the frequency of HSS occurrence (i.e., the coronal holes) remains constant.

5.3 Reconstruction of the Monthly Average and Seasonal …

225

Fig. 5.15 Reconstructed (blue, 1914–2014) and measured (red) values of the monthly means of the SW speed (top), monthly sunspot number and solar cycle numbers (bottom) (Lukianova et al. 2017)

To reveal the seasonal component, the H monthly means were averaged over a 3-month seasonal period for spring (February–April), summer (May–July), autumn (August–October) and winter (November–January). The season was determined based on the heliographic latitude of the Earth’s orbit. In Fig. 5.16, the correlations between H and SW speed values measured in 1964–2014 are given separately for the solstice (winter, summer) and equinox (autumn, spring); as also, the deviations of the values from the regression line are presented. The regression and correlation coefficients are given in Table 5.3. The deviations are symmetrically distributed as to the regression line for the total range of H, i.e. the variance is evenly distributed, and the reconstruction is robust. Figure 5.17 shows the values of the reconstructed and measured SW speed by the seasons (Mursula et al. 2017). It can be seen that the small values ( 500 keV), characterized by the sharp flux increase in the distance range of the whole outer radiation belt of the Earth, are considered as one of the most dangerous space weather factors for the AES equipment. Such sharp increases in the flux of the relativistic magnetospheric electrons cause serious anomalies in the spacecraft operation, so these electrons are even called ‘killer electrons’. For example, in May 1994, several geostationary satellites (ANIK-1, ANIK-2 and INTELSAT-K) were lost, and in May 1998, some serious problems arose with operation of the electronics on board of Equator-S, Polar and Galaxy-4 spacecrafts as a result of increased flux intensity of relativistic electrons. The failure rate at the spacecrafts increases drastically during the periods of the maximum flows of relativistic electrons in the magnetosphere against the background of the high-speed streams of the solar wind near the minimum of the solar activity (Wilkinson 1991; Wrenn et al. 2002). In addition to the high-energy electrons, the radiation belts are filled with the electron fluxes with the energies of tens to hundreds of thousands of electron-volts. The fluxes of these particles can increase dramatically (by 2–5 orders of magnitude) during the geomagnetic storms and substorms. The electrons with the energies of ~ 10–100 keV penetrate into the dielectric material (spacecraft skin or elements located on its surface) to the depth up to 10–20 microns, creating a potential difference between the spacecraft parts with a subsequent breakdown. The most severe events on the Sun lead to a sharp increase in the proton fluxes with the energies of several MeV inside the magnetosphere, which can keep for a long time, leading to formation of the additional proton belts (Lorentzen et al. 2002). Another pernicious factor is the cosmic rays. The solar cosmic rays (SCR), consisting mainly of protons and helium ions, appear during the solar flares and coronal mass ejections. Usually, the energy of the accelerated particles does not exceed 50 MeV for protons and 1 MeV for electrons. Solar flares occur quite often, once a week during the years of the increased solar activity. Approximately once a month, solar flares originate the particles with the energy of 100 MeV. And statistically once a year, the solar flares generating the particles with the energy of 1 GeV appear. Two-three times during the 11-year solar cycle, the events accompanied by the particle flows with the energy of 10 GeV and higher take place. The maximum magnitude of the SCR flux at the level of the Earth’s orbit is observed from 1 to 15 h after the solar flare. The magnitude of the SCR flux varies from 105 to 1011 particles/cm2 . The SCR injection can be of both pulsed and long-term nature; some

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7 Applications of the Geomagnetic Field in Technological Problems

cases are known when the significant SCR fluxes were recorded within 24 h. After the intense solar flares, the proton and heavy nuclei fluxes increase by several orders of magnitude and cause “blinding” and loss of orientation of the interplanetary and geostationary spacecrafts (Kuznetsov and Nymmik 1997). Besides the high-energy protons and electrons, the failures in the onboard systems of the spacecrafts can be also caused by the galactic cosmic rays (GCR). The GCR penetrate through a significant part of the magnetosphere and cause single event failures in the spacecraft electronics (Berezinskii et al. 1990). The intensity of the GCR fluxes is in anti-correlation with the solar activity. Along with the long-term GCR variations related with 11- and 22-year solar cycles, their fluxes undergo some shorter-period changes. First of all, they include the 27-day variations of the cosmic rays specified by the Sun rotation. The geomagnetic storms and substorms represent a response of the solar activity manifestation. The geomagnetic activity by itself is not capable to affect the operation of the spacecraft onboard equipment, but during the geomagnetic disturbances the conditions are created in the magnetosphere that are favorable for penetration of the high-energy particles in the regions where the satellite orbits are located. The response of the magnetosphere to almost identical external disturbances can differ significantly. The geomagnetic activity can often serve as a proxy for the change of the flux intensity of the various charged particles and consequent increase in the probability of the satellite failures. The wave activity of the magnetosphere can be formalized by introducing some special wave indexes (ULF index) (Kozyreva et al. 2007). The index characterizes the hourly average power of the variations of the magnetospheric and interplanetary magnetic fields in the Pc5 range (f ~ 2–6 MHz); it is calculated based on the data of the world network of the magnetic stations, geostationary and interplanetary satellites. The ULF index shows that the fluxes of relativistic electrons increase up to the extreme energies exactly against the background of the increased wave activity. The database on ULF wave index is available online at http://ulf.gcras.ru/.

7.6 Prevention of Radio Communication Failures For many years, the study of the ionospheric inhomogeneities has been among the fundamental problems of geophysics and is still relevant owing to the continued use of the ionosphere as an information transmission path and deployment of the global telecommunication systems in it. It is exactly the heterogeneous structure of the ionosphere that causes instability of the engineering system operations used for radio communications. The main processes generating the ionospheric inhomogeneities occur in the equatorial and auroral regions, where the conditions are created for development of the ionospheric plasma instabilities. In the equatorial region, this is specified by the complex dynamics of the South Atlantic anomaly and equatorial electrojet, and in the auroral regions by the particle precipitation and

7.6 Prevention of Radio Communication Failures

307

auroral electrojet. With the appearance of the modern telecommunication infrastructures, including the dense networks of the ground-based GNSS receivers, a more detailed study of the ionospheric plasma inhomogeneities has become available at all latitudes. The modern telecommunication systems operate mainly in the VHF (very high frequency) and UHF (ultra high frequency) ranges. The UHF range extends from 300 to 3,000 MHz, which includes, in particular, the GPS frequencies (f 1 = 1575.25, f 2 = 1227.2 MHz). The US military satellites operate at the high-frequency edge of the VHF band (250 MHz). The intense small-scale ionospheric inhomogeneities may cause strong amplitude and phase flickers of the GPS signal, leading to a loss of the signal phase and making it impossible to carry out the high-precision navigation. The spatial scale of such inhomogeneities is of the order of the first Fresnel zone, which makes 150– 300 m for GPS frequencies f 1 and f 2 . It was confirmed that the main cause of the malfunctions is the signal scattering during propagation through the area with intense small-scale ionospheric inhomogeneities, and it was shown that phase malfunctions at the auxiliary frequency are recorded more often than at the main frequency. At that, the phase failures accompany the equatorial boundary of the auroral oval when it widens during geomagnetic storms. It has been established that at the fronts of the intense travelling ionospheric disturbances (TID) of the auroral origin, the smallscale inhomogeneities can be generated, causing the phase measurement failures. The interrelation between the failures in the GPS system and intensity of the solar radio emission has been revealed, and it has been shown that the amplitude of the broadband radio emission during the powerful solar flares can be comparable with the amplitude of the signals from the satellite navigation systems. This leads to numerous failures in operation of the receivers, up to the complete loss of the signal from the satellites. It has been argued that the probability of recording the phase failures depends on the angle between the ‘satellite-receiver’ radio beam and the Earth’s magnetic field. This is related with the presence of the large-scale magnetically oriented inhomogeneities (Ledvina et al. 2004). The fact that the magnetically oriented inhomogeneities (so called plasma bubbles) affect seriously the radio communication not only at the equator, where their development is most favorable because of the Rayleigh-Taylor instability, but also at the mid-latitudes, was demonstrated when studying the failure causes of the military operation “Anaconda” in March 2002 (Kelly et al. 2014). Indeed, due to the strong anisotropy represented by elongation along the geomagnetic field, the impact of the plasma bubbles affects the latitudes up to 30°. In the early morning on March 4, 2002, the military command at the Bagram base tried unsuccessfully to contact the crew of the Chinook helicopter, which was holding a course for the snowy peak of Takur Ghar. There was a detachment of 20 soldiers onboard flying to rescue the SEALs crew blocked by a Taliban detachment. It turned out that the crew was initially given the wrong coordinates, but the pilots failed to receive the corrected data because of the communication malfunctions. As a result, the helicopter was fired from the ground and crashed onto the mountain slope occupied by the enemy. During the 17 h battle, seven American soldiers died, which was one of the most severe losses for

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7 Applications of the Geomagnetic Field in Technological Problems

the United States in the course of the Afghan campaign. The reason was the plasma bubbles formed in the upper atmosphere, which deflected the signals of the military satellite providing the communication. During the operation in the mountains, the NASA TIMED (Thermosphere Ionosphere Mesosphere Energetics and Dynamics) spacecraft passed over the battlefield. The data from its ultraviolet camera confirmed that a plasma bubble inhibiting a contact actually appeared between the military and communication satellite at the moment of the operation. Since the plasma bubble formation affects seriously the radio communications, a challenge to prevent such scenarios or predict the conditions of their appearance arises. In general, their appearance depends on the season and solar activity cycle; however, the online forecast is based on the dependence of the plasma bubble appearance on the geomagnetic activity. So, during the continuous positive BZ of the IMF, the eastward component of the electric field penetrates from the magnetosphere into the ionosphere within several hours. This electric field leads to a rise of the layer F to the higher altitudes, where the frequency of ion-neutral collisions decreases so that the instability development, and therefore, generation of the plasma bubbles becomes possible. At that, the most favorable conditions for the plasma bubble evolution are formed during the main phase of the geomagnetic storm. The above mentioned examples show that knowledge of the geomagnetic field characteristics can contribute to solving a number of problems of ensuring stable operation of spacecrafts and radio communications.

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Acknowledgments

The authors wish to thank Dr. Renata Lukianova for her contribution to preparing the materials in Russian. The authors are grateful to member of the Russian Academy of Sciences, space pilot Yuri Baturin. He provided tangible contribution to the part of the book, devoted to the Earth remote sensing as a source of Big Data. In the Fig. 6.10, the reader can see Yuri in space while he makes remote sensing operations. The authors also thank Dr. Mikhail Dobrovolsky for his efficient assistance in the work on the Chap. 6.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Gvishiani and A. Soloviev, Observations, Modeling and Systems Analysis in Geomagnetic Data Interpretation, https://doi.org/10.1007/978-3-030-58969-1

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