Seismic data interpretation using digital image processing 9781118881798, 1118881796, 9781119125594, 1119125596

Table of contents :
Content: Cover
Title Page
Copyright
Contents
Foreword
Preface
Chapter 1 Introduction
1.1 Image Processing of Exploration Seismic Data
1.2 Exploration Seismic Data: From Acquisition to Interpretation
1.2.1 Seismic Data Acquisition
1.2.2 Seismic Data Processing
1.2.3 Seismic Data Interpretation
1.3 The Seismic Convolution Model
1.4 Summary
Chapter 2 Seismic Data Interpretation
2.1 Introduction
2.2 Structural Features
2.2.1 Faults
2.2.2 Folds
2.2.3 Diapirs
2.3 Stratigraphic Features
2.3.1 Channels
2.3.2 Reefs
2.3.3 Truncation
2.4 Seismic Interpretation Tools 2.4.1 Seismic Sequence Stratigraphy2.4.2 Seismic Facies Analysis
2.4.3 Direct Hydrocarbon Indicators
2.4.4 Tying Seismic and Well Data
2.4.5 Seismic Modeling
2.4.6 Time-to-Depth Conversion
2.4.7 Seismic Attributes
2.5 Pitfalls in Seismic Interpretation
2.6 Summary
2.7 Problems and Computer Assignments
Chapter 3 Seismic Image Enhancement in the Spatial Domain
3.1 Introduction
3.1.1 The Mean (Running-Average) Filter
3.2 The Median Filter
3.3 The Edge-Preserving Smoothing Algorithm
3.3.1 Two-Dimensional Structure-Preserving Smoothing
3.4 Wavelet-Based Smoothing
3.4.1 Method 3.4.2 Sharpening Filter3.5 Summary
3.6 Problems and Computer Assignments
Chapter 4 Seismic Image Enhancement in the Spectral Domain
4.1 Introduction
4.2 The Fourier Transform
4.3 Filtering in the Spectral Domain
4.4 Spectral Attributes
4.5 Summary
4.6 Problems and Computer Assignments
Chapter 5 Seismic Attributes
5.1 Introduction
5.2 Detection of Interesting Regions from Time or Depth Three-Dimensional Slices using Seismic Attributes
5.3 Two-Dimensional Numerical Gradient Edge-Detector Operators
5.4 Application to Real Seismic Data 5.5 Two-Dimensional Second-Order Derivative Operator5.5.1 The Coherence Attribute
5.5.2 The Dip Attribute
5.6 The Curvature Attribute
5.7 Curvature of the Surface
5.7.1 Curve, Velocity, and Curvature
5.7.2 Surface, Tangent Plane, and Norm
5.8 Shape Operator, Normal Curvature, and Principal Curvature
5.8.1 Normal Curvature
5.8.2 Shape Operator
5.8.3 The Principal Curvatures
5.8.4 Calculation of the Principal Curvatures
5.8.5 Summary of Calculation of Principal Curvature for a Surface
5.9 The Randomness Attribute
5.10 Technique for Two-Dimensional Images 5.10.1 Problem Statement and Preliminaries5.10.2 Review of Fast Noise Variance Estimation Algorithm
5.10.3 Design Mask by Constrained Optimization
5.11 The Spectral Decomposition Attribute
5.12 Summary
5.13 Problems and Computer Assignments
Chapter 6 Color Display of Seismic Images
6.1 Introduction
6.2 Color Models and Useful Color Bars
6.2.1 The RGB Model
6.2.2 The CMY Model
6.2.3 The HSI Model
6.2.4 Useful Color Bars
6.3 Overlay and Mixed Displays of Seismic Attribute Images
6.4 Summary
6.5 Problems and Computer Assignments
Chapter 7 Seismic Image Segmentation

Citation preview

Seismic Data Interpretation using Digital Image Processing

Seismic Data Interpretation using Digital Image Processing Abdullatif A. Al-Shuhail King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia

Saleh A. Al-Dossary Saudi Arabian Oil Company Dhahran, Saudi Arabia

Wail A. Mousa King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia

This edition first published 2017 © 2017 John Wiley & Sons Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa to be identified as the author(s) of this work have been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Al-Shuhail, Abdullatif Abdulrahman, 1962- author. | Al-Dossary, Saleh A., 1965- author. | Mousa, Wail Abdul-Hakim, author. Title: Seismic data interpretation using digital image processing / Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary, Wail A. Mousa. Description: Hoboken, NJ, USA : John Wiley & Sons Inc., 2017. | Includes bibliographical references and index. Identifiers: LCCN 2017005390 (print) | LCCN 2017008974 (ebook) | ISBN 9781118881781 (hardback) | ISBN 9781118881804 (Adobe PDF) | ISBN 9781118881798 (ePub) Subjects: LCSH: Seismic reflection method. | Image processing–Digital techniques. | Petroleum–Prospecting–Data processing. Classification: LCC TN269.84 .A425 2017 (print) | LCC TN269.84 (ebook) | DDC 551.22028/7–dc23 LC record available at https://lccn.loc.gov/2017005390

Cover Design: Wiley Cover Image: (Background) © Artpilot/Gettyimages; (Inset Images) Courtesy of authors Set in 10/12pt WarnockPro by SPi Global, Chennai, India 10 9 8 7 6 5 4 3 2 1

To my dear parents and family. AAS To anyone who has ever taught me something. SAD To my beloved father, mother, wife, and children. To KFUPM and my country. WAM

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Contents Foreword xi Preface xv 1

Introduction 1

1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.4

Image Processing of Exploration Seismic Data 1 Exploration Seismic Data: From Acquisition to Interpretation 1 Seismic Data Acquisition 2 Seismic Data Processing 2 Seismic Data Interpretation 2 The Seismic Convolution Model 3 Summary 6

2

Seismic Data Interpretation 7

2.1 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 2.4.7 2.5 2.6 2.7

Introduction 7 Structural Features 11 Faults 11 Folds 18 Diapirs 21 Stratigraphic Features 22 Channels 24 Reefs 26 Truncation 27 Seismic Interpretation Tools 27 Seismic Sequence Stratigraphy 29 Seismic Facies Analysis 33 Direct Hydrocarbon Indicators 35 Tying Seismic and Well Data 35 Seismic Modeling 35 Time-to-Depth Conversion 41 Seismic Attributes 44 Pitfalls in Seismic Interpretation 44 Summary 48 Problems and Computer Assignments 49

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Contents

3

Seismic Image Enhancement in the Spatial Domain 59

3.1 3.1.1 3.2 3.3 3.3.1 3.4 3.4.1 3.4.2 3.5 3.6

Introduction 59 The Mean (Running-Average) Filter 60 The Median Filter 63 The Edge-Preserving Smoothing Algorithm 66 Two-Dimensional Structure-Preserving Smoothing Wavelet-Based Smoothing 70 Method 70 Sharpening Filter 71 Summary 72 Problems and Computer Assignments 73

4

Seismic Image Enhancement in the Spectral Domain 77

4.1 4.2 4.3 4.4 4.5 4.6

Introduction 77 The Fourier Transform 77 Filtering in the Spectral Domain 80 Spectral Attributes 83 Summary 85 Problems and Computer Assignments 85

5

87 Introduction 87 Detection of Interesting Regions from Time or Depth Three-Dimensional Slices using Seismic Attributes 87 Two-Dimensional Numerical Gradient Edge-Detector Operators 89 Application to Real Seismic Data 91 Two-Dimensional Second-Order Derivative Operator 96 The Coherence Attribute 96 The Dip Attribute 100 The Curvature Attribute 101 Curvature of the Surface 103 Curve, Velocity, and Curvature 103 Surface, Tangent Plane, and Norm 104 Shape Operator, Normal Curvature, and Principal Curvature 105 Normal Curvature 105 Shape Operator 105 The Principal Curvatures 106 Calculation of the Principal Curvatures 106 Summary of Calculation of Principal Curvature for a Surface 107 The Randomness Attribute 108 Technique for Two-Dimensional Images 109 Problem Statement and Preliminaries 109 Review of Fast Noise Variance Estimation Algorithm 110 Design Mask by Constrained Optimization 111 The Spectral Decomposition Attribute 113 Summary 115 Problems and Computer Assignments 116

5.1 5.2 5.3 5.4 5.5 5.5.1 5.5.2 5.6 5.7 5.7.1 5.7.2 5.8 5.8.1 5.8.2 5.8.3 5.8.4 5.8.5 5.9 5.10 5.10.1 5.10.2 5.10.3 5.11 5.12 5.13

Seismic Attributes

67

Contents

6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.3 6.4 6.5

123 Introduction 123 Color Models and Useful Color Bars 124 The RGB Model 125 The CMY Model 125 The HSI Model 126 Useful Color Bars 127 Overlay and Mixed Displays of Seismic Attribute Images 127 Summary 130 Problems and Computer Assignments 130

7

Seismic Image Segmentation 133

7.1 7.2 7.3 7.3.1 7.3.1.1 7.3.2 7.4 7.5

Introduction 133 Basic Seismic Image Segmentation 134 Advanced Seismic Image Segmentation 136 Color-Based Segmentation 136 The Imposed Constraints for the POCS Color Segmentation Method 137 Graph-Based Segmentation 139 Automatic Fault Extraction 140 Summary 143

6

Color Display of Seismic Images

Glossary

145

References 151 Index 157

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Foreword Human beings are experts at pattern recognition. We are well equipped to find a missing set of keys in a cluttered drawer, to find a family member in a crowd, to find the image of a cow in the clouds, or to find a mythological character in a constellation of stars. Such pattern recognition is key to seismic data analysis, where a human interpreter identifies and integrates the amplitude, spectral content, waveform, and geometric configuration of the seismic response to the underlying geology with an appropriate tectonic, depositional, or diagenetic model. For a seasoned interpreter, much of this pattern recognition is done subconsciously, requiring little conscious thought, much like pedaling and maintaining your balance on a bicycle. However, advances in seismic acquisition and processing technology are increasing more rapidly than the number of seismic interpreters. Some modern seismic surveys may be 100 Gbytes in size, while merged surveys may be larger still. It has become increasingly intractable for an interpreter to examine each and every seismic voxel. Seismic attributes attempt to reduce the amount of data the interpreter needs to examine by capturing the same key components used in the conventional interpretation of vertical seismic amplitude sections. All attributes except the input seismic amplitude data itself implicitly or explicitly define an analysis window. Instantaneous attributes are, in reality, not instantaneous, but rather integrate the information of neighboring samples through the use of a Hilbert transform. Spectral magnitude and phase components use the information of neighboring samples on a seismic trace, while post-stack impedance inversion uses the information content of all the overlying samples as well. Geometric attributes such as coherence, curvature, and texture analysis operate in a three-dimensional (3D) window, including neighboring seismic samples and traces, often oriented along structural dip. Amplitude versus offset, inversion for Pand S-impedances, and estimates of amplitude versus azimuth increase the size of the data to be analyzed further. By extracting such key components as an auxiliary attribute volume, an experienced interpreter can now rapidly animate through time or depth slices to identify subtle, and otherwise easily overlooked, channels, mounds, collapse, slumps, faults, and folds, as well as zones of anomalous porosity or anisotropy. Equally important, other professionals, including stratigraphers, structural geologists, drilling engineers, and completion engineers, now have access to images that have removed much of the overprint of the seismic wavelet and begin to mimic interpreted geologic cross-sections and maps.

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Foreword

While geoscientists call such resulting images seismic attribute volumes, most of the scientific world refers to them as the results of image processing. Recent advances in medical X-ray analysis, positron emission tomography–computed tomography scans, video images, and meteorological data have not yet been applied to 3D seismic data volumes. For example, can one apply techniques to map the corkscrew veins seen in 3D images of a human kidney to mapping crosscutting channels in a fluvial deltaic complex? Can one apply techniques used to monitor changes in land use of a suspected terrorist or narcotics site to monitor changes due to reservoir completion? Can one use techniques in Doppler radar to map tornadoes to identify potential drilling hazards seen in seismic anisotropy? While most seismic interpreters have a rudimentary understanding of human anatomy, land use, and weather, few radiologists, security analysts, and meteorologists, let alone algorithm developers with a traditional electric engineering background, have a good understanding of geology and geologic processes, let alone geology convolved with a seismic wavelet. The authors of Seismic Data Interpretation Using Digital Image Processing, Al-Shuhail, Al-Dossary, and Mousa, are among the few who have bridged the gap between modern image processing practiced by the scientific community at large and the world of geology and reflection seismology. This book bridges the gap in both ways, providing a path for non-geoscience image processors to better understand the seismic interpretation process through real data examples, and a path for geoscientists by presenting modern image-processing algorithms in the context of filters and convolutional operators routinely used in seismic data analysis. The authors begin in Chapter 1 with a succinct synopsis of the seismic experiment from acquisition to processing, emphasizing the alternative formats of seismic data display. In Chapter 2 the authors define and illustrate geologic concepts of structure and stratigraphy through their seismic expression. They draw heavily upon online examples published in the Virtual Seismic Atlas, showing slices through the data with and without interpretation of horizons, faults, unconformities, channels, slumps, gas hydrates, and carbonate buildups. In this manner, the reader sees the seismic amplitude data by itself and is able to mentally reconstruct the process by which human interpreters make their analysis. The more algorithmic reader can mentally apply their favorite image-processing method to determine if it might be able to reproduce such an interpretation. The chapter continues with a summary of the more common seismic interpretation tools, including concepts of seismic sequence stratigraphy, seismic facies analysis, direct hydrocarbon indicators, seismic ties to well logs, and the value of seismic modeling. The chapter concludes by summarizing some of the more common interpretation pitfalls to both human interpretation and imaging processing, including velocity pull-up and push-down and over- or undermigrated data. In contrast to Chapter 2, which focuses on building a bridge for the non-geoscience algorithm developer into the world of geoscience, Chapters 3, 4, and 5 focus on building a bridge for the geoscientist to cross into the world of image processing. Chapters 3 and 4 introduce concepts of spatial seismic image enhancement, an area geoscientists call data conditioning. These filters, and indeed most filters in the book, are summarized as simple convolutional stencils that are easily grasped by a seismic interpreter. In contrast to

Foreword

Chapter 2, where the authors show the rich diversity of the seismic expression of geology, Chapters 3, 4, and 5 present a single example from the NW Arabian Peninsula to illustrate the rich diversity of image-processing algorithms. The mathematics are presented simply and clearly, with the important concepts of nonlinear versus linear filters accurately addressed. Chapter 3 uses spatial filters oriented along geologic structure, while Chapter 4 addresses the same problem with two-dimensional spectral filters applied to time or depth slices. The interpretive value of alternative data-conditioning techniques described in Chapters 3 and 4 is demonstrated using a single image-processing tool, the Sobel filter edge detector, allowing easy comparison of alternative workflows. The input and output of the algorithms described in Chapters 3 and 4 are the original and an enhanced version of the seismic amplitude data. The input of the algorithms described in Chapter 5 is, in general, the enhanced version of the seismic amplitude data, while the output is a quantitative measure of the lateral change in seismic waveform and amplitude. This collection of edge detectors, orientation and shape estimators, and texture analysis image-processing tools are called seismic attributes by geoscientists. Good attributes enhance and delineate subtle features in the geology that might otherwise be overlooked. The various edge-detection algorithms described in Chapter 5 can be used to map faults, joints, channel edges, and slumps. As in Chapter 3, the edge detection and, later in the chapter, noise estimation algorithms of Chapter 5 are presented as simple stencils. The dip and curvature algorithms map folds, flexures, and faults that fall below seismic resolution and appear to be faults. Chapter 6 begins with multiattribute display using HSI, RGB, and CMYK color gamuts. Such coloring is an interpreter-driven form of clustering and forms the basis for subsequent discussions of image segmentation using projections onto convex sets and graph-based algorithms. In the final chapter, Chapter 7, the authors give an overview of important image segmentation techniques and show how to adapt them for seismic data interpretation using real seismic data examples. The inputs to these techniques are preferably seismic attribute images, while the outputs are subsets of these images that constitute target geologic elements, including horizons, channels, and diapirs. The chapter concludes with a summary of a popular automatic fault extraction algorithm. Seismic Data Interpretation Using Digital Image Processing is an excellent book for both undergraduate and graduate students in signal processing, graduate students in geology and geophysics, and experienced seismic interpreters who want to look “under the hood” to learn how these different algorithms work. Most of the chapters are followed by a carefully constructed suite of questions and exercises to solidify mastery of the key concepts. Since most prefer to “learn by doing”, the Matlab exercises at the end of each chapter, the Matlab and C codes and seismic data sets available on the book’s website are particularly effective parts of the book. I have spent 25 years of my career working in the area of image processing. As a geophysicist, I found many of the published papers to be both terse and too mathematical. Most important, almost all applications seemed to be applied to images of the beloved Lena, and not to 3D seismic data. Seismic data analysis has experienced several revolutions from digital recording and processing, through improved migration using supercomputers, to interactive 3D visualization on the desktop. I strongly believe the

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Foreword

next revolution will be in seismic image processing and pattern recognition. Seismic Data Interpretation Using Digital Image Processing provides an excellent vehicle for scientists in both communities to share their problem objectives and solution techniques. March 2016

Kurt J. Marfurt Frank and Henrietta Schultz Professor of Geophysics The University of Oklahoma, Norman, OK, USA

xv

Preface Oil and gas are extremely important natural resources for human beings. During 2008–2009, for example, the world used 150,000 L/s of oil, the highest when compared with previous years. According to Cambridge Research Associates, 3.7 trillion barrels of oil are left, including unconventional resources, in addition to 1.2 trillion barrels of oil if enhanced oil recovery factors increase from 35% to 50%. Today’s global challenges require new and innovative approaches. Oil and gas resources are buried in deep land or marine subsurface geologic structures. In order to produce oil, we need first to determine the subsurface structure; that is, get a clear and accurate image of the subsurface through large-scale data acquisition. Then oil wells can be drilled, and further data acquisitions, which are considered small-scale data, may be taken and used to confirm the large-scale maps of the subsurface structures. Both large- and small-scale geo-signals can be acquired by various geophysical exploration methods that include seismic, magnetic, and gravity surveys. Furthermore, other small-scale data acquisitions are measured within boreholes, such as resistivity measurements, nuclear magnetic resonance measurements, gamma ray measurements, and so on, all of which require careful analysis as well as efficient processing. Another important category of geo-data includes lab-scale acquisitions and measurements. Core measurements such as dielectric measurements, petrographic thin sections, computed tomography and magnetic resonance imaging scans of core plugs or slabbed cores are just a few examples of data that require careful analysis and processing. In all the aforementioned three categories, the process generally requires acquisition, processing, and interpretation of data in order to ultimately deduce useful analyses, depending on the objective in which we are interested: oil/gas exploration, well optimization, or enhanced oil recovery. For example, acquired seismic data alone do not reveal an accurate image of the subsurface unless we use appropriate processing techniques from amplitude gain to migration. Using artificial intelligence for steering horizontal well drilling requires accurate real-time processing of real-time acquired data. Thin-section petrographic images reveal useful porosity and grain matrix rock information if we use robust signal or image analysis techniques, which assist in accurate estimations. It is well known that signal processing, in general, plays an important role in many applications of science and engineering disciplines, including seismology, sonar, radar, medicine, communications, and so on. In the case of seismology, the application of signal processing theory began with the work of the Geophysical Analysis Group at the Massachusetts Institute of Technology between 1960 and 1965, where it was one of the great historic milestones in seismic data processing. Another example

xvi

Preface

is the use of Fullbore Formation MicroImager (FMITM ) technology, which provides micro-resistivity formation images in water-based mud, which was first initiated by an image-processing scientist at Schlumberger Riboud Product Center, Clamart, France, and introduced as a tool to the industry in 1992. With the increased demand for oil/gas production, innovative technologies that rely heavily on signal analysis and processing will be required for geo-signal acquisition, processing, and interpretation. And seismic signals are at the top of the list in terms of the requirements for careful acquisition, processing, and interpretation. This book serves as a textbook on the interpretation of seismic data using digital image processing. Seismic interpretation has been using algorithms from the field of digital image processing over the past few decades, owing to the recent advancement in computing and the field of image processing. This textbook is suitable for advanced undergraduates and graduate students in electrical engineering and graduate students in Earth sciences, who are primarily interested in understanding the processing methods in order to devise algorithms, software, and hardware for interpreting seismic data. This textbook is also suitable for image-processing researchers interested in entering the field of exploration seismology. Likewise, it should be interesting to seismic data interpreters to learn more advanced techniques used from digital image processing in order to expand their knowledge and provide them with opportunities to advance such useful image-processing algorithms. We want to teach students the basics of seismic exploration with more emphasis on seismic interpretation, where seismic textures and attributes are explained in detail, as a necessary background for any electrical engineering student. Geoscience students may want to skip this part. We also aim to allow students to learn through this textbook about multi-attribute analysis, which basically uses pseudo-coloring and various color models. Through this textbook, we aim to show them how to process two-dimensional (2D) and three-dimensional (3D) seismic images in different domains and extract useful and important geologic features using image-processing techniques and algorithms. This book is written as a textbook. The focus of the text is on analysis and processing of 2D seismic images. Many real data examples are provided after each theory, concept, or algorithm to provide readers with a real practical understanding of the image-processing algorithms used. Most chapters end with a few problems for readers to solve and entertain. Some Matlab and C codes are provided on the book’s web site along with some data sets to allow instructors, students, and researchers to see for themselves the results and play with the data and algorithms. The book is divided into seven chapters and a glossary at its end. Chapter 1 mainly motivates readers regarding the subject of exploration seismology and its important role in coping with global energy demands. It also provides a general overview of seismic data acquisition, processing, and interpretation. This is a very good start for non-geoscientists to learn about the field of exploration seismology and is prerequisite knowledge for them to proceed with reading the remaining book chapters. Chapter 2 focuses more on seismic interpretation. It introduces details on its tools, such as seismic sequence stratigraphy, seismic facies analysis, and direct hydrocarbon indicators. Chapter 3 highlights conventional techniques used to enhance the resolution of seismic images and their signal-to-noise ratios in the time–distance domain, all at the seismic interpretation stage. Chapter 4 surveys important methods employed to enhance seismic images, at the seismic interpretation stage, using various spectral

Preface

domains, such as the frequency–distance and frequency–wavenumber. In Chapter 5 we define seismic attributes and their various classification systems, and we highlight the important ones that are mainly used in the industry. Chapter 6 describes the different color models used to display and interpret seismic images. These models are currently used in practice, where interesting geological features can be revealed using, for example, pseudo-coloring of seismic images. Chapter 7 is devoted to segmentation of seismic images, which can be used to locate interesting geological objects and boundaries (channels, faults, etc.), where it is intended in this chapter to show different image segmentation techniques used for the sake of interpreting seismic images. Finally, the glossary provides definitions of terms that may be new to electrical engineers and those that might be new to geoscientists. We take this opportunity to thank H.E. Prof. Khaled S. Al-Sultan, Rector, King Fahd University of Petroleum & Minerals (KFUPM) and the university for their support. We also thank engineers Arbab Latif and Umair Yaqub from KFUPM, who helped very much in preparing many figures and codes for this book, as well as for their efforts in the final compilation of the chapters. Abdullatif Al-Shuhail (KFUPM) Saleh Al-Dossary (Saudi Aramco) Wail A. Mousa (KFUPM)

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1

1 Introduction 1.1 Image Processing of Exploration Seismic Data The exploration seismic method is the most widely used and well-known geophysical technique among many exploration geophysical surveying methods. Such data can be processed to reveal details of geologic structures on scales from the top tens of meters of the Earth’s crust to its inner core [1, 2]. Part of its success lies in the fact that raw seismic data can be processed to produce seismic sections (images) of the subsurface structure. A geologist and/or a geophysicist can then make an informed interpretation by understanding how such images were created [1, 3]. In the past few decades, advances in signal- and image-processing algorithms have greatly contributed to the geologic interpretation of exploration seismic data sets. Large areas of petroleum-bearing sedimentary basins are currently imaged via advanced three-dimensional (3D) seismic processing techniques. Such advances and research contributions from the image-processing community have enabled seismic interpreters to better assess possible oil/gas reservoirs.

1.2 Exploration Seismic Data: From Acquisition to Interpretation The exploration seismic method involves three main stages: seismic data acquisition, seismic data processing, and seismic data interpretation. The main aim of the exploration seismic method is knowledge of the distribution of wave impedance that reflects the subsurface geology. Exploration seismic waves typically penetrate the subsurface up to a few kilometers in depth. Basically, one records (a) the amplitudes of the reflected seismic waves and (b) the time at which such amplitudes arrived at the receiver. Then the data can be processed and interpreted using signal- and image-processing algorithms, which ultimately influence the decision on where to drill wells that will eventually produce hydrocarbons.

Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

2

Seismic Data Interpretation using Digital Image Processing

1.2.1 Seismic Data Acquisition

The first stage in exploration seismic surveys is the acquisition stage, where the seismic data are collected by an array of receivers (geophones for land and hydrophones for marine), transmitted over a narrow-band channel, and stored for processing. The acquisition can be done to acquire two-dimensional (2D) or 3D seismic data. Both 2D and 3D acquisition receivers can be single component (recording compressional vertical ground motion) or multicomponent (recording additional horizontal ground motions). The acquisition is commonly based on the so-called common midpoint (CMP) technique, a recording–processing method where each subsurface point is covered using many source-receiver pairs. After performing some data correction, the data are combined in a way which provides a CMP section that approximates the traces that would be recorded by a coincident source and receiver at each location, but with improved signal-to-noise ratio (SNR). The objective is to attenuate random effects and events whose dependence on offset is different from that of primary reflections. Other acquisition methods include narrow azimuth. More advanced acquisition techniques such as multi-azimuth, wide-azimuth, and rich-azimuth are recent innovations that aim to address the illumination problems inherent in the traditional narrow azimuth seismic technique. These types of advanced reflection seismic acquisition methods improve the SNR and illumination in complex geology. 1.2.2 Seismic Data Processing

Next comes the processing of the acquired seismic data sets. Seismic data processing involves the use of many sequential mathematical and signal processing techniques, which are blended with a somewhat subjective interpretation by an experienced geophysicist. This includes seismic data processing steps like geometric spreading correction, frequency filtering, deconvolution, velocity analysis, static correction, seismic migration and imaging, and so on [1]. Finally, the processed data go to the interpretation stage, which mainly aims to derive a simple, plausible geologic model that is compatible with the observed data. It can utilize subjective and objective ways to interpret exploration seismic images. 1.2.3 Seismic Data Interpretation

Seismic data interpretation aims to extract all available geologic information from the processed and imaged seismic data, such as structure, stratigraphy, and rock properties. Interpretation involves sophisticated image processing algorithms, thanks to advances in computers and image processing techniques. The interpretation is done based on the reservoir types: structural or stratigraphic. Considering the rock structures, it is very important to understand the regional tectonic settings and similar ones existing around the world. Additionally, the interpreter has to know the style of the structure – such as a fault, a salt dome, an anticline, and so on – and identify the complexity of the structures in the imaged seismic data. When considering the rocks’ stratigraphy, the interpreter has to understand the age relationships and depositional environment of important geologic

Introduction

formations of oil/gas exploration interest. Also, one must know the time, space, and production relationship between the structure and stratigraphy in the area of interest. Finally, the interpretation process should consider the rock properties and lithology by knowing the rock types (e.g., sandstone, carbonate) and the existence of salt, anhydrite, limestone or dolomite in the shallow section of the data, where they can distort the seismic signals. One must also know the weathering layer lithology and how variable it is with respect to the survey area.

1.3 The Seismic Convolution Model The convolutional model of seismic data is essential to understand how a seismic trace is formed. The trace represents a combined response of layered ground and a recording system to a seismic source wavelet. Any display of a collection of one or more seismic traces is termed a seismogram. Assuming that the wavelet shape and amplitude do not change as the wavelet propagates through such layered ground, the resultant seismic trace may be regarded as the convolution of the input wavelet with a time series known as a reflectivity function, which is composed of unit sample functions. Each unit sample function has an amplitude related to the reflection coefficient at a layer boundary and a traveltime equivalent to the two-way reflection time from the surface to that boundary.1 Furthermore, the reflectivity function represents the impulse response of the layered ground, which is basically the output for a unit sample input. Since the source wavelet has a finite length, individual reflections from closely spaced boundaries may overlap in time on the resultant seismogram (seismic section). Figure 1.1 represents a typical seismic convolution model. Once the data are processed and a seismic image is obtained, several ways exist of displaying seismic images. One of the most commonly used displays is, for example, the wiggle display, as seen in Figure 1.2a. It vertically plots seismic trace amplitudes as functions of time (with time increasing downward), where the positive peaks are to the right of every trace and the negative ones are to the left of every trace. Another common method of displaying a seismic images is the so-called variable-area display, which shades the area under the wiggle trace to make coherent seismic events (see Figure 1.2b), rendering the positive peaks evident.2 The combination of variable area and wiggle is known as wiggle–variable-area display (Figure 1.2c). Additionally, the variable-density display represents amplitude values by the intensity of shades of gray (black is for the maximum amplitude value and white is for the minimum amplitude value) and in colors (e.g., blue is for the maximum amplitude value, white is for the zero amplitude value, and red is for the minimum amplitude value) like the ones shown in Figure 1.2d and e, respectively. These can be used in different ways with different seismic attributes and color maps, as will be discussed later in the book. 1 Traveltime is the time difference between zero time and the arrival time of a seismic event. It can be a one-way time, such as for direct waves, or two-way time, such as for reflected waves. 2 A seismic event is the arrival of a new seismic wave, usually indicated by a phase change and an increase in amplitude on a seismic record. It may be a reflection, refraction, diffraction, surface wave, or random noise.

3

Exploration seismic data acquisition

Exploration seismic data processing

Geological Reflectivity section function

A noisy seismogram Noise

+ Input source wavelet

Exploration seismic data interpretation Figure 1.1 During the acquisition stage, a seismic wavelet is convolved with the reflectivity function of the rocks and recorded via many receivers to obtain a seismogram. The seismogram is processed to yield a seismic image. This image is then analyzed by interpreters. Note that the reflectivity function is related to the geologic section of the subsurface through the reflection coefficient of each geologic boundary and the two-way traveltime.

Introduction

(a)

(b)

(c)

(d)

(e)

Figure 1.2 Various displays for a seismic section image: (a) wiggle display, (b) variable area display, (c) wiggle-variable area display, (d) gray-scaled variable density display, and (e) colored variable density display. Note that the vertical axis represents the two-way traveltime increasing downward, while the horizontal axis represents the distance from left to right. (See color plate section for the color representation of this figure.)

5

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Seismic Data Interpretation using Digital Image Processing

1.4 Summary This chapter introduced the exploration seismic method. It also described the basic convolution model by which seismic data are acquired and recorded. There are various means to display seismic images, depending on the interpreter’s desire as well as the attributes one is looking for after applying a suitable image-processing algorithm, as will be seen later throughout the book.

7

2 Seismic Data Interpretation 2.1 Introduction Seismic interpretation refers to the extraction of geologic information from seismic data. Extracted geologic information includes structure, stratigraphy, and pore-fluid content. It is performed often on migrated and stacked seismic data. However, it can be performed also on unmigrated or prestacked data. It is usually augmented by other nonseismic data, such as gravity, magnetic, well-log, and petrographic data. It is mainly used for prospect evaluation and reservoir development. Although seismic data interpretation comes after acquisition and processing, it is important for acquisition, processing, and interpretation professionals to communicate continuously (Figure 2.1). Occurrence of a commercial petroleum prospect requires essential factors [4]. These factors include a source rock that is porous, impermeable, and rich in organic matter. After it undergoes a suitable combination of time and temperature, organic matter will generate petroleum in commercial amounts. Petroleum will eventually migrate from the source rock to a porous and permeable reservoir, where it will accumulate when barred from further migration by an impermeable cap rock (seal). These factors have to be timed appropriately to trap petroleum in commercial amounts. Next, we discuss some of these factors in more detail. Petroleum forms in the source rock from the transformation of organic matter, which initially produces a substance called kerogen that matures eventually to petroleum under appropriate temperature and pressure conditions. Kerogen maturation to petroleum is a function of temperature and depth (or geologic time). Oil typically forms in temperature and depth windows of 60–120 ∘ C and 1.5–3 km respectively, while gas forms at 120–225 ∘ C and 3–6 km respectively (Figure 2.2). A source rock has generally a high porosity at deposition time in order to allow for the accumulation of large amounts of organic matter; but it has a low permeability in order to preserve the organic matter from oxidation. These properties are commonly associated with shales, which form the most common petroleum source rock. Eventually, petroleum migrates from the source rock to the reservoir rock, where it is usually found and produced. It is generally not feasible to produce petroleum from source rocks due to their low permeability. However, recent advancements in drilling and production technologies

Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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Seismic Data Interpretation using Digital Image Processing

Figure 2.1 Relationship among the requirements of interpretation, processing and acquisition. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_ Analysis. Used under CC-BY-SA 3.0 https:// creativecommons.org/licenses/by-sa/3.0/deed .en [1].

Acquisition area Partial-fold, Prestacked, Unmigrated Processing area Full-fold, Stacked, Unmigrated Interpretation area Full-fold Stacked Migrated

Oil/gas window

Depth (km)

Temp (°C)

Spore Colour Index

Vitrinite Reflection

Subsurface process

1

1

30

Kerogen

2

Diagenesis

3 2

60

3

90

4

0.5

5 Oil

6

4

Gas

Wet

120

7

Katagenesis 1.2

8 5

150

2.0 9

Dry

Metagenesis 6

250

10

5.0 Metamorphic

Figure 2.2 Temperature and depth generation windows of petroleum (intechopen.com). Source: Adapted from Tissot and Welte [5].

made it commercially feasible through extensive fracturing of the source rock. A new boom in petroleum exploration and production has been triggered under many names, including unconventional resources, shale gas, tight gas, and so on. A petroleum reservoir is a sedimentary rock with a high porosity and a high permeability for better storage and movement of petroleum. Porosity in a reservoir rock

Seismic Data Interpretation

(a)

(b)

Figure 2.3 (a) Micrograph showing primary intergranular porosity. (b) Micrograph showing primary intragranular porosity. Source: Selley and Sonnenberg [4]. Reproduced with permission of Elsevier.

can be classified into two main categories: primary and secondary. Primary porosity is formed during sediment deposition and can be intergranular, where pore space occurs between sediment grains (such as in clastic rocks – Figure 2.3a) or intragranular (where it occurs within sediment grains, such as in carbonates – Figure 2.3b). Secondary porosity is formed after sediment deposition (diagenesis). Diagenesis can form porosity via many mechanisms, including: • grain solution, where porosity enhancement occurs through cavities commonly encountered in carbonates (Figure 2.4a); • tectonic stresses, where porosity enhancement occurs through fracturing (Figure 2.4b); • dolomitization, where porosity enhancement occurs through mineral changes from calcite to dolomite in carbonates. Although diagenesis can enhance rock porosity through these mechanisms, it can also destroy it through cementation by quartz, calcite, and clay minerals.

9

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Seismic Data Interpretation using Digital Image Processing

(a)

(b)

Figure 2.4 (a) A rock core exhibiting vuggy porosity. (b) A rock core exhibiting fracture porosity. Source: Selley and Sonnenberg [4]. Reproduced with permission of Elsevier. Table 2.1 Types of structural traps. Diapiric

Mud

Salt

Fold

Compressional anticlines

Compactional anticlines

Fault

Normal

Reverse

Strike-slip

The petroleum reservoir is a part of the trap which also includes the cap rock. A trap is a place where petroleum is prevented from further movement (migration). Traps can be grossly divided into structural or stratigraphic [4]. Structural traps are caused by tectonic processes that result in compression and extension of sedimentary layers, resulting in their folding and faulting. Table 2.1 shows common types of structural traps. On the other hand, stratigraphic traps are caused by depositional morphology or diagenesis. As an example of depositional morphology, a coral reef that is a good reservoir can be covered by a shale layer that is a good cap rock, forming a reef trap. An example of diagenesis is the weathering of a dipping reservoir rock that is later covered by a cap rock, forming a truncation trap. Table 2.2 shows common types of stratigraphic traps. These traps are important to identify at the interpretation stage in order to better mark possible locations for petroleum production. In the coming sections, we will further explain each trap type in detail. It is thus important for the image processing interpreter to be able to identify such traps.

Seismic Data Interpretation

Table 2.2 Main types of stratigraphic traps. Associated with unconformity Supra-unconformity

Sub-unconformity

Channel

Truncation

Onlap

Unassociated with unconformity Depositional

Pinchout

Bar

Diagenetic

Reef

Porosity and/or permeability change

2.2 Structural Features Structural features are generally easy to identify in seismic sections. Such structural features that are of interest to petroleum exploration include faults, folds, and diapirs. We discuss relevant aspects of these features next. 2.2.1 Faults

Faults are ruptures along which subsurface rocks move past each other. This movement can juxtapose reservoir and nonreservoir rocks in a way that can trap petroleum from further upward movement. The fact that fault surfaces can be sealing or nonsealing to fluid movement is an important factor in the formation of a fault trap. In terms of movement direction, faults are divided into three types: • Normal faults, in which the hanging wall of the fault moved down the fault plane. • Reverse (thrust) faults, in which the foot wall of the fault moved down the fault plane. • Strike-slip (wrench) faults, in which the walls of the fault moved horizontally across the fault plane. These are usually characterized by an associated flower structure. Figure 2.5 presents typical configurations of normal, reverse and strike-slip faults. Figure 2.6 shows examples of trapping mechanisms associated with normal and reverse faults. Figures 2.7, 2.8, and 2.9 show seismic sections of normal, reverse, and strike-slip faults respectively. Faults are relatively easy to define on stacked seismic data using one or more of the following lines of evidence [6]: 1 2 3 4 5 6

reflection termination against the fault plane (Figure 2.10); offset (vertical and horizontal) reflections across the fault plane (Figure 2.11); differential reflection dip across the fault plane (Figure 2.12); diffractions along the fault plane on unmigrated sections (Figure 2.13); parts of the fault plane might produce reflections (Figure 2.14); some fault planes are associated with fracture zones causing scattering and loss of seismic events (Figure 2.15).

Once a fault is defined on a seismic depth section, its throw and apparent dip angle can be calculated with good accuracy. Normal faults commonly exhibit 50–60 ∘ , reverse faults 30–40 ∘ and strike-slip faults almost vertical dip angles on seismic depth sections. It should be noted here that measurement of a fault throw or dip angle on a seismic

11

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Seismic Data Interpretation using Digital Image Processing

(a)

(b)

(c)

(d)

Figure 2.5 Main types of faults; (a) normal fault; (b) reverse (thrust) fault; (c) strike-slip (wrench) fault; (d) oblique (combined) fault. Source: https://en.wikibooks.org/wiki/Historical_Geology/Faults#/media/ File:Faults6.png. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/.

Figure 2.6 Normal faults forming unlimited (a) and limited (b) petroleum traps; reverse faults forming unlimited (c) and limited (d) petroleum traps. Source: Selley and Sonnenberg [4]. Reproduced with permission of Elsevier.

Spill point

(a)

(b)

Spill point

(c)

(d)

500 ms TWT

500 ms TWT 1 km

1 km (a)

(b)

Figure 2.7 (a) Seismic section showing normal faults; (b) interpreted section (VSA author: Butler, 2015; data courtesy of Fugro N.V.). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

(a)

(b)

Figure 2.8 (a) Seismic section showing a reverse fault; (b) interpreted section (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

2 km

2 km (a)

(b)

Figure 2.9 (a) Seismic section showing a flower structure associated with a strike-slip fault; (b) interpreted section (VSA author: Stewart, 2015). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

15 km (a)

500 ms TWT

on-lapping basin fill

500 ms TWT

dip decreases up section

15 km (b)

Figure 2.10 (a) Seismic section showing reflection termination at faults; (b) interpreted section (VSA author: Butler, 2015; data courtesy of Fugro N.V.). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

Seismic Data Interpretation

500 ms TWT

Figure 2.11 Seismic section showing reflection offset across faults (VSA author: Butler, VSA, 2015; data courtesy of Fugro N.V.). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

1 km

500 ms TWT

(a)

1 km

(b)

Figure 2.12 (a) Stacked seismic section showing differential reflection dip across faults; (b) interpretation showing fault locations (VSA author: Butler, 2015; data courtesy of Fugro N.V.). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

17

18

Seismic Data Interpretation using Digital Image Processing CMP 140

340

540

740

940

1140

1340

1540

1740

1940

1340

1540

1740

1940

s

1

2

3

4

5

(a) CMP 140

340

540

740

940

1140

s

1

2

3

4

5

(b)

Figure 2.13 (a) Diffractions on an unmigrated seismic section; (b) diffractions removed after seismic migration. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

time section is inaccurate because both decrease with time due to the general increase of velocity with time (or depth). 2.2.2 Folds

Folding of sedimentary layers is associated with many environments, including excessive horizontal compressive stresses commonly associated with convergent tectonic plate boundaries producing prolific anticlinal petroleum traps (Figure 2.16). The giant fields

Seismic Data Interpretation

Agbada Formation

(a)

high amplitude on fold crest = Direct Hydrocarbon Indicator

detachment at top of Akata Shale

(b)

Figure 2.14 (a) Seismic section showing a fault producing a reflection; (b) interpreted section. (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

of the Arabian Gulf are examples of this type of trap. Another common environment of folding is doming by salt and mud diapirs as well as magmatic intrusions that are emplaced forcefully beneath sedimentary layers causing the overlying layers to arch and dome (Figure 2.17). The Gulf of Mexico fields are examples of this type of trap. In addition, differential compaction can produce folding with no apparent tectonic activity. This

19

20

Seismic Data Interpretation using Digital Image Processing

500 ms 2 km

(a)

artefacts in forelimb

single thrust strand

500 ms 2 km

(b)

Figure 2.15 (a) Seismic section showing seismic fault zone exhibiting amplitude loss; (b) interpreted section (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

Seismic Data Interpretation

(a) channel complex

fold located between strike-slip zones

channel complex

5 km thrust anticline located by basement step?

(b)

Figure 2.16 (a) Seismic section showing compressional folds; (b) interpreted section (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

happens when layers are deposited on a preexisting high and undergo less compaction magnitude on the high than on the lows, creating a mild fold (Figure 2.18). The Forties field in the UK is an example of this type of trap. Important characteristics commonly associated with anticlinal folding on seismic sections include normal faulting at the top of the anticline, which produces grabens in some parts of the anticline that might form good traps. Another characteristic is the abundance of fractures at the top of the anticline, which may result in signal deterioration below the anticline. Folding tends to disappear if associated with faulting due to the release of compressive stress through these faults. Furthermore, it is common for domes and arches to be associated with radial and concentric normal faults. 2.2.3 Diapirs

Some salt and mud can act as a fluid under certain temperature and pressure conditions. The fluid-like salt or mud moves up if its density is less than that of the overlying layers due to buoyancy, thus forming structural forms known as diapirs. Salt and mud diapirs move upward, pushing the overlying layers to form domes and many related

21

22

Seismic Data Interpretation using Digital Image Processing

V = 2H (a)

V = 2H (b)

Figure 2.17 (a) Seismic section showing a fold generated by a salt diapir; (b) interpreted section (VSA author: Stewart, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

traps. Diapirs are probably the easiest among structural features to identify in seismic sections owing to their prominent textural contrast from surrounding sediments. The Dammam and Awali fields of the Arabian Gulf region are examples of salt dome traps, while the Niger Delta basin forms an example of a mud dome trap (Figure 2.19).

2.3 Stratigraphic Features Stratigraphic traps are those in which the main trapping mechanism is not formed due to structural causes but due to spatial variations in lithology, porosity, and permeability. Stratigraphic traps are generally subtler and difficult to recognize on seismic sections,

200 ms (TWTT)

N

S

10 km

(a)

200 ms (TWTT)

N

S

10 km

(b)

Figure 2.18 (a) Seismic section showing a compactional folds; (b) interpreted section (VSA author: Jackson, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

24

Seismic Data Interpretation using Digital Image Processing

(a) ridge split by faults

frontal anticline

pitch out against flank

sand pitches out against ridge 5 km

(b)

strike-slip fault zone transpressional ridge

Figure 2.19 (a) Seismic section showing a mud diapir; (b) interpreted section (VSA author: Jackson, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www .seismicatlas.org). (See color plate section for the color representation of this figure.)

as opposed to structural traps. Next, we discuss in more detail some of the stratigraphic features relevant to petroleum exploration. 2.3.1 Channels

Channels are rivers or water streams that have been filled with sediments. They can form good and prolific reservoirs if filled by clean sand, but they can also act as seals if filled by impermeable sediments. Since they follow the river bed, it is generally difficult to follow them for long distances using drilling or sparse 2D seismic lines. They are best recognized on time slices or horizon slices of 3D seismic data, where the meandering nature of channels can be observed (Figure 2.20). Nevertheless, wide and deep channels may be recognized easily as separate structures on 2D seismic sections (Figure 2.21).

(a)

(b)

Figure 2.20 (a) A 3D seismic horizon slice showing channels; (b) interpreted slice (VSA author: Sylvester, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

26

Seismic Data Interpretation using Digital Image Processing

A

Eroded anticline

Channel valleys

Seafloor

NW

SE

100 ms (twtt)

Base of channels sequence 5 km

Normal faults

Figure 2.21 A 2D seismic section showing channels (VSA author: Torvela, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

Cretaceous Devonian Figure 2.22 A 2D seismic section showing channels (indicated by arrows) with false synclines below them due to low velocity of channel fill material (VSA author: Posamentier, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

Occasionally, a false-positive or -negative folding may develop under the channel due to the channel’s velocity being different from that of surrounding sediments (Figure 2.22). 2.3.2 Reefs

Reefs are carbonate depositional buildups that develop in tropical areas. They are formed by corals and algae, which build these reefs with originally high porosity. These reefs are often enclosed later by shales that form the source rock as well as the seal for these traps. The Intisar (Idris) field of Libya is a good example of this type of trap [4]. Reefs are generally difficult to observe on 2D seismic sections. However, they can be recognized on seismic data provided some characteristics are present. For example, pinnacle and patch reefs can be distinguished easily by their distinctive round shapes on time slices of 3D seismic data. Often, the reef top boundary may produce a prominent reflection. Another characteristic feature of reefs is the absence of reflections inside them due to their heterogeneous nature. Diffractions may develop at the reef’s edges on

Seismic Data Interpretation

Isolated Patch Reefs

1 km

100 ms

Horizon “C” Horizon “B” Abundant Patch Reefs

Figure 2.23 A 2D seismic section showing patch reefs with strong reflections from reef tops due to high contrast with overlying layers (VSA author: Posamentier, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

unmigrated stacked sections due to abrupt termination of surrounding beds at the reef. Differential sedimentation (i.e., changes in layer dip and thickness) may occur on opposite sides of a barrier reef. In addition, overlying reflections may show signs of differential compaction, as explained previously (Section 2.2.2). Similar to channels, a false-positive or -negative folding may develop under the reef due to the reef’s velocity being different from that of surrounding sediments. Figure 2.23 shows a typical reef expression on 2D seismic sections exhibiting several of its characteristics, while Figure 2.24 shows various 3D views of two patch reefs in 3D seismic data. 2.3.3 Truncation

Truncation traps are commonly associated with angular unconformities. These traps can form very prolific petroleum reservoirs, such as the East Texas oil field. A prominent characteristic of angular unconformities on seismic sections is that the overlying and underlying sediments (reflections) across it have different dips. Occasionally, there might be a reflection from the unconformity surface because the overlying and underlying sediments differ in acoustic impedance. In the case of an angular unconformity, the reflection may change its amplitude and polarity laterally due to lateral changes in the underlying sediments. Figure 2.25 shows a 2D seismic section of an angular unconformity.

2.4 Seismic Interpretation Tools Important tools and concepts commonly employed in seismic data interpretation include seismic sequence stratigraphy, seismic facies analysis, and direct hydrocarbon indicators (DHIs).

27

Moat

~100 m

Debris aprons

1 km

5 km

Figure 2.24 A 3D seismic volume showing a patch reef (VSA author: Posamentier, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas .org). (See color plate section for the color representation of this figure.)

Seismic Data Interpretation

(a)

ESE

WNW

3 km (b)

Figure 2.25 (a) A 2D seismic section of a truncation trap formed by an angular unconformity; (b) interpreted section (VSA author: Hunt, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

2.4.1 Seismic Sequence Stratigraphy

Seismic sequence stratigraphy refers to separating time-depositional units in seismic data based on unconformities [7]. The central concept of sequence stratigraphy is that sediment deposition is controlled by crustal subsidence (which creates accommodation space), sediment supply (which fills the accommodation space), rises and falls of the absolute sea level, and climate that determines the sediment nature (e.g., carbonate deposition dominates in warm climates). A consequence of this concept is that seismic reflections become geologic-time rather than lithologic surfaces. These reflections are

29

Seismic Data Interpretation using Digital Image Processing

the result of either sediment deposition or erosion that occurred during the same time period. Depositional models of sequence stratigraphy include lower-boundary and upper-boundary reflections. Lower-boundary reflections include onlaps, downlaps, and concordances. In an onlap, reflections terminate against an originally dipping boundary, indicating proximal deposition (i.e., close to sediment source). In a downlap, reflections terminate down onto an originally horizontal boundary, indicating distal deposition (i.e., far from sediment source). In contrast, reflections in a concordance do not terminate but are parallel to a boundary. Concordances can occur with lower-boundary and upper-boundary reflections. Similarly, upper-boundary reflections include toplaps and erosional truncations. In a toplap, reflections terminate from above, parallel to a boundary of nondeposition. In contrast, reflections in an erosional truncation are truncated from above by an unconformity. Figure 2.26 shows examples of downlap and toplap, Figure 2.27 shows an onlap, Figure 2.28 shows an erosional truncation, and Figure 2.29 shows concordance on 2D seismic sections. To map a seismic sequence, we first mark reflection angularities followed by drawing unconformities using onlaps, downlaps, toplaps, and truncations, and finally continue unconformities to their correlative conformities. A system tract is a part of a sequence deposited during a single sea-level change. It can be a lowstand system tract (LST)

500 ms

5 km

(a) 5 km 500 ms

30

(b)

Figure 2.26 (a) Seismic section showing downlaps and toplaps; (b) interpreted section (VSA Source: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

S

500 ms TWTT

5 km

N

S

(a) fan

500 ms TWTT

faulted topography associated with the Middle Jurassic-Early Cretaceous rift event

5 km

N

S

(b)

Figure 2.27 (a) Seismic section showing onlaps; (b) interpreted section (VSA source: Jackson, 2015; data courtesy of CGG Veritas) Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

500 msTWT

Seismic Data Interpretation using Digital Image Processing

5 km

500 msTWT

(a)

5 km

(b)

+

Figure 2.28 (a) Seismic section showing erosional truncation; (b) interpreted section (VSA source: Stewart, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

–

Amplitude

50 ms(TWT)

32

1 km

Figure 2.29 Seismic section showing concordance (VSA author: Calves, 2015; data courtesy of PeruPetro). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

Seismic Data Interpretation

that is deposited during a rapid sea-level fall. Downlaps and toplaps are formed during a lowstand system tract. In contrast, a transgressive system tract is deposited during a rapid sea-level rise, usually forming onlaps. A highstand system tract is deposited during a period spanning late sea-level rise, stillstand, and early fall, resulting in downlaps with no accompanying toplaps. 2.4.2 Seismic Facies Analysis

The term “facies” refers in geology to all features characterizing the depositional environment of a sedimentary unit. Therefore, seismic facies analysis is the study of the internal and external forms of a group of reflections within a sequence in order to establish the depositional environment of this group of reflections. Internal seismic forms include the configuration of reflections. Examples of these configurations are the parallel (Figure 2.30), divergent, chaotic (Figure 2.31), reflection free, and prograding (Figure 2.32). Other internal forms of seismic reflections include their amplitude, continuity, frequency, and interval velocity. External seismic forms include sheets, drapes, wedges, banks, lenses, mounds, and fills [7]. Table 2.3 shows associations of some seismic forms with their corresponding depositional environments.

C H8 (~2 Ma)

B

C’

Pockmark

H7 (~5 Ma)

H6 (8.2 Ma)

H5 (~10 Ma)

H4 (16.5 Ma)

Figure 2.30 Seismic section showing parallel reflection configuration (VSA author: Jobe, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

33

1s TWT

Seismic Data Interpretation using Digital Image Processing

20 km

Figure 2.31 Seismic section showing chaotic reflection configuration (VSA author: Butler, 2015; data courtesy of Fugro N.V.). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

5 km 500 ms

34

Figure 2.32 Seismic section showing prograding (sigmoidal) reflection configuration (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www .seismicatlas.org). (See color plate section for the color representation of this figure.)

Table 2.3 Associations of some seismic forms with their corresponding depositional environments. Seismic facies

Depositional environment

Sheet of high continuity and amplitude

Marine

Sheet of low continuity and variable amplitude

Non-marine

Mound of local reflection void

Reef

Oblique internal reflections

High depositional energy and sediment supply

Sigmoid internal reflections

Low depositional energy and sediment supply

Sheet drape

Low-energy deep marine

Source: Mitchum et al. [7].

Seismic Data Interpretation

2.4.3 Direct Hydrocarbon Indicators

A direct hydrocarbon indicator (DHI) is a change in the amplitude and/or polarity of seismic reflections related to changes in the pore-fluid phase (i.e., liquid versus gas). Its basis stems from the relation between the rock and pore-fluid velocities, such as the time-average equation. Important DHIs include bright spots (Figure 2.33), dim spots (Figure 2.34), polarity reversals (Figure 2.34), flat spots (Figure 2.35), gas chimneys (Figure 2.36), P-wave without S-wave anomalies (Figure 2.37) and amplitude variation with offset (AVO) anomalies (Figure 2.38). 2.4.4 Tying Seismic and Well Data

Another tool of seismic interpretation involves the direct comparison of well-log and seismic data. Well-log information commonly used for tying includes formation tops (i.e., depth to layer tops), lithology (i.e., rock matrix, fluid properties, and porosity), and location of faults and unconformities to determine missing sections. Disagreement between the well-log and seismic data may arise after tying for various reasons. For example, time-to-depth conversion errors may be caused by incorrect seismic velocities used to convert seismic two-way traveltime to depth. Another reason might be due to dip effects, because points on a dipping seismic reflector are not vertically below their surface recording stations. On the other hand, points along a vertical well-log are assumed to be vertically below the well location. Seismic migration can correct for the dip effects in seismic data so that points on the stacked seismic section correspond to their real subsurface locations. Another source of error is vertical resolution, since wavelengths of well-logs are very short (10 cm), while those of seismic data are very long (100 m). This means that some well-log details will not be resolved by surface seismic data. Similarly, lateral resolution can be another source of error because seismic data sample a large (Fresnel) zone, whereas logs sample a very small annulus around the borehole. This means that the seismic data average the rock properties of a very large zone around the borehole, missing out details within the annular zone of the well-log. Furthermore, problems arising from seismic data processing can contribute to the disagreement between seismic and well-log data. For example, static shifts might erroneously shift seismic events vertically from their true positions, leading to misties. Furthermore, multiples in seismic data might be confused with primaries, leading to the interpreter tying a primary event on the well-log with a multiple on the seismic data, which generates a mistie between the two. Figure 2.39 shows an example of tying various well-logs with seismic data. 2.4.5 Seismic Modeling

Owing to the complexity of the subsurface, it is necessary to use a simplified model that includes only important elements of the subsurface. Forward modeling involves computing seismic data using a specific subsurface model. Results of forward modeling are compared with observed seismic data and changed iteratively until errors between model and observed data sets are “acceptably small.” Forward modeling may be achieved using physical modeling, which involves building a miniature simplified model of the subsurface with appropriate scaling parameters. This model is used to generate the corresponding synthetic seismic data, which can be compared with the observed seismic data acquired over the real reservoir after properly scaling the synthetic seismic data.

35

(a)

mass transport complexes

examples of channel complexes

Bottom Simulating Reflector gas hydrates

?sand pinch-outs onto flank of anticline North

1 s TWT

high amplitude on fold crest = Direct Hydrocarbon Indicator

Agbada Formation

slide scars

South

complex forelimb fold-thrust zone

5 km detachment at top of Akata Shale

(b)

Figure 2.33 (a) Seismic section showing bright spot; (b) interpreted section (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

Seismic Data Interpretation

dim out and phase reversal 1 km

200 ms TWT

200 ms TWT

−

− Amplitude

Amplitude

+

+

1 km

(a)

(b)

Figure 2.34 (a) Seismic section showing a dim spot and phase reversal; (b) interpreted section (VSA author: Calves, 2015; data courtesy of PeruPetro). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). 1.8

Top Sand OWC

2.2

Figure 2.35 Seismic section showing a flat spot due to an oil–water contact at about 2 s, as confirmed by drilling and logging results. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1]. (See color plate section for the color representation of this figure.)

200 ms TWT

200 ms TWT

− Amplitude

− Amplitude

+

+

BSR and base of chimneys

(a)

(b)

Figure 2.36 (a) Seismic section showing a gas chimney; (b) interpreted section (VSA author: Calves, 2015; data courtesy of PeruPetro). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

37

1090

542 1000

500 900

450 800

400 700

350 600 2000

2248

2496

2744

2992

3240

3480

3704

Figure 2.37 Seismic volume showing P effect (left) but no S effect (right) due to fluid effects. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1]. (See color plate section for the color representation of this figure.)

Seismic Data Interpretation

Figure 2.38 Seismic CMP gather showing AVO effect at 1.25 s. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/ deed.en [1].

Offset (m)

Time (s)

200

1500

1.0

1.0

1.2

1.2

1.4

1.4

A more preferred modeling technique is computer modeling, which involves calculating synthetic seismograms (traces) from a subsurface model using one of many modeling algorithms. For example, ray-tracing (forward convolutional modeling) employs well data to generate a synthetic seismogram through the following procedure: 1 Obtain depth, density log, and sonic log in a well close to the surface location of a CMP from surface seismic data. 2 Extract the seismic source wavelet from an observed stacked seismic trace close to the well location (we will discuss some wavelet extraction techniques later in this section). 3 Use sonic velocity and depth to compute the two-way traveltime. 4 Use density and sonic velocity to compute the reflectivity series. 5 Convolve the extracted wavelet with the reflectivity series to generate the synthetic trace. 6 Compare the synthetic with the stacked trace closest to the well. 7 If needed, stretch/compress the synthetic trace to get the “best match” between synthetic and stacked traces. Synthetic and stacked traces may not match perfectly for the same reasons mentioned in Section 2.4.4. Another computer seismic modeling algorithm is finite difference, which uses the finite-difference method to solve the wave equation for the pressure wavefront and calculate the corresponding synthetic seismic data. There are several versions of the finite-difference method and wave equation depending on the modeling accuracy and model complexity. Other methods for solving the wave equation involve finite elements and finite volumes. Figure 2.40 shows an example of finite-difference seismic modeling.

39

Seismic Data Interpretation using Digital Image Processing 0

travel time

integrated travel time

0.0

impedance

reflectivity

seismic

synthetic

500 0.5 1000

1500

1.0

two-way time [s]

measured depth [m]

40

2000

2500

3000

1.5

2.0

3500 2.5 4000

4500 slowness [μs/m]

two-way time [s]

3.0 0.0 0.5 1.0 1.5 2.0 kg/m2 s2 1e7

Figure 2.39 Example of tying well and seismic data sets. Source: Yilmaz, http://wiki.seg.org/wiki/ Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/ deed.en [1].

Inverse modeling (or inversion) involves the direct calculation of a possible subsurface model from observed seismic data. It should be noted here that different subsurface models can produce “acceptably small” errors between the calculated and observed seismic data sets, making inversion inherently nonunique. This is due to the simplification of models used for inversion and/or noise present in the observed data that is not accounted for in the inversion process. As we mentioned earlier, a wavelet is required to generate synthetic seismograms. The wavelet can be assumed using analytical wavelets, such as Ricker or Klauder wavelets. However, it is better to estimate a wavelet from the observed seismic data. If the source is a Vibroseis, the corresponding Klauder wavelet can be calculated using information about the sweep. If an impulsive source (e.g., dynamite) is used, one of many methods can be used to estimate the minimum-phase equivalent of the real seismic wavelet. For example, the double-inverse method assumes that the seismic wavelet is the inverse operator of the best spiking-deconvolution operator of the observed seismic data. In addition, least-squares fitting can be used to calculate a wavelet that minimizes the error between the synthetic and seismic traces. Furthermore, deterministic deconvolution of the seismic trace near a well can be used to calculate the seismic wavelet provided the reflectivity series from that well is known.

Seismic Data Interpretation

s H1 0.5

1.0

H2

1.5

H3 H4 H5

2.0

H6 (a) km

0.5 1.0 1.5 2.0 2.5

(b)

Figure 2.40 (a) Synthetic seismic data generated from the velocity model in (b) using finite-difference modeling. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1]. (See color plate section for the color representation of this figure.)

2.4.6 Time-to-Depth Conversion

Seismic data are recorded in the time domain. However, many interpretation decisions require a depth image of the subsurface. Time-to-depth conversion refers to converting the vertical axis of the stacked seismic data from two-way traveltime to depth using appropriate subsurface velocities. These velocities can come from seismic data through the velocity analysis step of the data processing workflow or from well data. Next, we briefly discuss some of these methods.

41

42

Seismic Data Interpretation using Digital Image Processing

The Dix method uses velocities calculated from surface seismic data. The stacking velocities and total two-way traveltimes are used to calculate the Dix (interval) velocities of each layer, which are used to calculate the thickness and depth of each layer. This method gives inaccurate depths for thin or deep layers. Furthermore, different reflectors are usually used by the processor and interpreter, which leads to errors in assignment of depths to reflectors. In addition, layer dip, anisotropy, and curvature might not be accounted for while using this method, which leads to depth errors. Figures 2.41 and 2.42 show an example of time-to-depth conversion using the Dix method. s

0.5

1.0

1.5

2.0

(a) s

0.5

1.0

1.5

2.0

(b)

Figure 2.41 (a) Synthetic seismic stacked section derived from the velocity–depth model in Figure 2.40b. (b) Stacking velocity field derived from (a), where colored curves indicate picked primary reflections. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1]. (See color plate section for the color representation of this figure.)

Seismic Data Interpretation

km

0.5 1.0 1.5 2.0 2.5

(a) km

0.5 1.0 1.5 2.0 2.5

(b)

Figure 2.42 (a) Velocity–depth model derived from the time and stacking velocities in Figure 2.41b using the Dix method for time-to-depth conversion. (b) True velocity–depth model. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons .org/licenses/by-sa/3.0/deed.en [1]. (See color plate section for the color representation of this figure.)

Another method involves building velocity–depth functions. For example, discrete velocity functions can be derived from a dense distribution of wells in the area. These functions are then interpolated and/or extrapolated vertically and spatially to form a highly dense 3D velocity volume that can be used for time-to-depth conversion across the whole area of the seismic survey. Analytical velocity functions can also be estimated along each well in the area and later used to convert time to depth along seismic traces nearby each well. For example, linear velocity functions are used commonly: V (Z) = V0 + kZ, where V (Z) is the interval velocity of the P-wave in the rock at depth Z, V0 is the P-wave velocity at the surface (Z = 0), and k is a velocity gradient with units of s−1 . When using this method, it is important to observe that the calculated depths from the seismic and well data must tie exactly at the wells.

43

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Seismic Data Interpretation using Digital Image Processing

Other sources of velocity information include vertical seismic profiling, check-shot, and sonic-log surveys. The interpreter combines all sources of velocity information using plausible geologic interpretation to produce a velocity volume to be used for time-to-depth conversion across the whole survey area. 2.4.7 Seismic Attributes

One of the major tasks of an image-processing interpreter is to extract useful information from seismic data, such as seismic attributes. A seismic attribute is any quantity derived from seismic data. Seismic attributes are used to enhance the visibility of structural, stratigraphic, or pore-fluid features on seismic data. They are extracted usually from the times, amplitudes, frequencies, or attenuation of seismic data. Examples of attributes include acoustic impedance, instantaneous frequency, coherence, and curvature. Any successful seismic attribute inversion project must use correct seismic-well ties and properly interpreted seismic data. In addition, the reservoir property derived from seismic attributes must honor the values of that property at wells within high statistical significance provided that there is a physical link between the seismic attributes and the inverted reservoir property [8]. More details on seismic attributes will be given in Chapter 3.

2.5 Pitfalls in Seismic Interpretation Seismic stacked migrated sections resemble geologic depth sections only in areas of simple geology. Therefore, direct interpretation of seismic sections in complex areas may result in serious errors in interpretation due to false effects, or simply pitfalls. Interpretation pitfalls can be produced due to geometric, shallow, deep, acquisition, or processing reasons [9]. Geometric effects occur on unmigrated seismic time sections and can produce pitfalls such as gentler and wider convex-up structures such as domes and anticlines. The opposite is true for convex-down structures such as synclines (Figure 2.43). Another common geometric pitfall is a bowtie over a syncline (Figure 2.44). Non-conformable dip sets due to sideswipes (offline reflections) on 2D sections are also the result of such geometric pitfalls (Figure 2.45). To avoid these geometric effects, it is recommended to migrate seismic sections (using 3D migration, if possible) before interpretation. Shallow effects are produced due to lateral velocity and/or thickness changes in near-surface layers. They can produce false lows below low-velocity and/or thick near-surface zones as well as false highs below high-velocity and/or thin near-surface zones (Figure 2.46). A good validity check for all these shallow effects is to see if the anomaly follows near-surface velocity and/or thickness profiles. Deep effects are produced due to velocity changes in overlying layers caused by structural and/or stratigraphic features. They can produce many false effects, such as downdip thinning of reflections due to velocity increase with depth. A good validity check for this effect is to see if the thinning effect increases with burial depth. Similarly, thinning of reflections on the downthrown side of normal faults may occur due to velocity increase with depth. A good validity check for this effect is to see if the

Seismic Data Interpretation

S

2

4

Figure 2.43 (Left) Unmigrated seismic section showing a gentler and wider dome; (right) migrated section with sharper and narrower dome. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_ Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

(a)

(b)

Figure 2.44 (a) Unmigrated seismic section showing bowties. (b) Migrated section with bowties removed. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

45

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Seismic Data Interpretation using Digital Image Processing

T B

Figure 2.45 An inline (top left) and a crossline (top right) stacked section from a land 3D survey. Also shown are the 2D (center row) and 3D (bottom row) migrations of the two stacked sections. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https:// creativecommons.org/licenses/by-sa/3.0/deed.en [1].

Seismic Data Interpretation

km

0.5

1.0

1.5

2.0

2.5

(a) s

0.5

1.0

1.5

2.0

(b)

Figure 2.46 (a) Velocity–depth model showing lateral velocity variation in the near-surface layer; (b) synthetic stacked seismic section derived from this model. Note the false structural high under the near-surface lateral velocity increase. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1]. (See color plate section for the color representation of this figure.)

47

48

Seismic Data Interpretation using Digital Image Processing

S

1 B

2

Figure 2.47 Stacked seismic section showing a false structural high of horizon B under a salt diapir due to the velocity increase of the diapir relative to surrounding rocks. Source: Yilmaz, http://wiki.seg .org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/bysa/3.0/deed.en [1].

thinning is confined to the downthrown side of the normal fault. Another deep effect is lows beneath normal faults and highs beneath reverse faults due to juxtaposition of layers with highly different velocities. A good validity check for this effect is to see whether the anomaly only exists beneath the fault. As mentioned in Section 2.3, in stratigraphic traps it is common to observe false highs (or lows) beneath reefs or channels due to the high (or low) velocity of reef or channel relative to surrounding rocks. A good validity check for this effect is to see if the anomaly coincides with the reef or channel. Similarly, lows beneath shale diapirs may occur due to the low velocity of shale, and highs beneath salt diapirs due to the high velocity of salt are other examples of deep pitfalls (Figure 2.47). A good validity check for this effect is to see whether the anomaly only exists beneath the diapir. Finally, acquisition and processing effects are produced due to improper selection of acquisition and/or processing parameters. An example of a processing pitfall is conformable and nonconformable dip sets due to multiples generated in a shallow layer (Figure 2.48). A good validity check for this effect is to see whether the velocity of the suspected multiple is the same as that of a shallow primary reflection. An example of an acquisition pitfall is bedding over unconformities due to a multicycle wavelet, such as the case produced by the bubble effect commonly associated with marine sources. A good validity check for this effect is to estimate the source wavelet from a clean part of the seismic section.

2.6 Summary This chapter provides an overview of important structural and stratigraphic features commonly observed in exploration seismic images. It also introduces tools and concepts

Seismic Data Interpretation

1

2

3

4

5

6 (a) 1

2

3

4

5

6 (b)

Figure 2.48 (a) Marine stacked seismic section contaminated with water-bottom multiple. (b) Section after multiple removal. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

required for seismic data interpretation, such as seismic sequence stratigraphy. Finally, it describes a suite of interpretation pitfalls that any interpreter of seismic data should be aware of before interpreting seismic images.

2.7 Problems and Computer Assignments 2.1

Given Figure 2.49 of a migrated stacked section: (a) Indicate a normal fault. (b) Indicate the relative motion direction across the fault.

49

Seismic Data Interpretation using Digital Image Processing

Figure 2.49 Image for Problem 2.1.

(c) What criteria did you use to identify the fault? (d) Indicate a good location of a good petroleum trap. 2.2 Given Figure 2.50 of a migrated stacked section: (a) Indicate a reverse fault. (b) Indicate the relative motion direction across the fault. (c) What criteria did you use to identify the fault? (d) Indicate a location of a good petroleum trap. WEST 0.0

900

EAST 850

800

750

700

650

600

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5 TWT (s)

TWT (s)

50

c. 4x vertical exaggeration

Figure 2.50 Image for Problem 2.2.

Seismic Data Interpretation

4000.0

5000.0

6000.0

Figure 2.51 Image for Problem 2.3. (See color plate section for the color representation of this figure.)

2.3

Given Figure 2.51 of a migrated stacked section: (a) Indicate a strike-slip fault. (b) Indicate the relative motion direction across the fault. (c) What criteria did you use to identify the fault? (d) Indicate a location of a good petroleum trap.

2.4

Given Figure 2.52 of a migrated stacked section: (a) Indicate a fold. (b) What criteria did you use to identify the fold? (c) Indicate a location of a good petroleum trap.

2.5

Given Figure 2.53 of a migrated stacked section: (a) Indicate a fold. (b) What criteria did you use to identify the fold? (c) Indicate a location of a good petroleum trap.

51

52

Seismic Data Interpretation using Digital Image Processing

2

3

+

TWT (sec)

4

Amplitude

5

−

200 ms (twtt)

Figure 2.52 Image for Problem 2.4. (See color plate section for the color representation of this figure.)

NW

Figure 2.53 Image for Problem 2.5. (See color plate section for the color representation of this figure.)

2.6 Given Figure 2.54 of a migrated stacked section: (a) Indicate a diapir. (b) What criteria did you use to identify the diapir? (c) Indicate a location of a good petroleum trap. 2.7 Given Figure 2.55 of a 2D migrated stacked seismic section: (a) Indicate a reef. (b) What criteria did you use to identify this reef? (c) Suggest a location of a good petroleum trap formed by the reef.

SE

Seismic Data Interpretation

Figure 2.54 Image for Problem 2.6. (See color plate section for the color representation of this figure.) 1 km

1 Sec

2 Sec

3 Sec

4 Sec

4 Sec

Figure 2.55 Image for Problem 2.7. (See color plate section for the color representation of this figure.)

2.8

Given Figure 2.56 of a 2D migrated stacked seismic section: (a) Indicate an unconformity. (b) What criteria did you use to identify it? (c) Suggest a location of a good petroleum trap formed by the unconformity.

2.9

Given Figure 2.57 of a 2D migrated stacked seismic section: (a) Indicate a channel. (b) What criteria did you use to identify it? (c) Suggest a location of a good trap formed by the channel.

53

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Seismic Data Interpretation using Digital Image Processing

Figure 2.56 Image for Problem 2.8.

Figure 2.57 Image for Problem 2.9.

2.10

Given Figure 2.58 of a 3D migrated seismic time slice: (a) Indicate a channel. (b) What criteria did you use to identify it? (c) Suggest a location of a good trap formed by the channel.

2.11

Given Figure 2.59 of a 2D seismic stacked section: (a) Indicate an onlap.

Seismic Data Interpretation

Figure 2.58 Image for Problem 2.10.

(b) Indicate a toplap. (c) Indicate a downlap. (d) Indicate a system tract. (i) What type is it? (ii) What is its depositional environment? (iii) Indicate the sediment deposition direction.

500 ms

5 km

Figure 2.59 Image for Problem 2.11. (See color plate section for the color representation of this figure.)

55

56

Seismic Data Interpretation using Digital Image Processing

Figure 2.60 Image for Problem 2.12. (See color plate section for the color representation of this figure.)

2.12

Given Figure 2.60 of a 2D seismic stacked section: (a) Indicate a DHI. (b) What type is it? (c) Why does it occur?

2.13

A 2D seismic line was acquired over a sabkha in the Rub Al-Khali desert of southern Saudi Arabia. The line traverses a rectangular sand-filled channel with a width of 300 m, a height of 50 m, and a sand P-wave velocity of 1000 m/s. There is originally a horizontal reflector at a depth of 2000 m whose root-mean-square velocity is 3000 m/s. The reflector has a hydrocarbon trap with the shape of a horst structure extending below the whole channel with a throw of 25 m. Figure 2.61 shows the depth model. (a) Draw the time section showing the shape of the structure below the channel before static correction. You should show the exact two-way traveltime at various points along the reflector and the structure. Use the sabkha as the reference surface (i.e., depth = elevation = 0 on the sabkha surface). (b) Draw the time section showing the shape of the structure below the channel after static correction. You should show the exact two-way traveltime at various points along the reflector and the structure. Use the sabkha as the reference surface (i.e., depth = elevation = 0 on the sabkha surface). (c) Answer the following questions: (i) Were you able to see the trap before static correction?

Seismic Data Interpretation

300 m 50 m

57

Sabkha

Vs = 1000 m/s

VRMS = 3000 m/s

Depth = 2000 m Reservoir top 25 m

Reservoir top

300 m Reservoir top

Figure 2.61 Image for Problem 2.13.

(ii) Were you able to see the trap after static correction? (iii) Was it important for the interpretation to do the static correction?

59

3 Seismic Image Enhancement in the Spatial Domain 3.1 Introduction In seismic exploration, seismic noise refers to a noninterpretable or unwanted component of recorded seismic signals due to a multitude of causes. These unwanted features may actually be somebody else’s signal, such as converted waves in what we think of as “P-wave” data, but more commonly these unwanted “noise” features provide little or no information about the subsurface and are referred to as random noise and coherent noise (pseudorandom noise). Examples of random noise include wave action in a marine environment, wind, vehicle traffic, and human activities in a land environment, and electronic instrument noise in both environments. Examples of coherent noise are noise not generated by the seismic experiment, such as 60 Hz powerline noise and pump jack noise and noise that is generated by the seismic experiment, such as ground roll, reverberating refractions, and multiples. Other types of noise that look random in time but may be highly organized in space include the acquisition footprint, which is highly correlated to the acquisition geometry. Whatever their cause, all these types of seismic noise can result in significant artifacts that may negatively impact subsequent interpretation products, from simple structural and spectral attributes to advanced seismic imaging and analysis. Seismic attributes are among the processes that can be affected severely since they are sensitive to subtle changes in signal and noise. To solve this challenge, random-noise-suppression algorithms, such as mean and median filters, have been developed to suppress this random seismic noise by removing noise along reflectors, while striving to preserve major structural and stratigraphic discontinuities. Unfortunately, these mean and median filters may also smear lateral discontinuities, such as fault boundaries. Thus, this will negatively impact the location of fractures and identification of their orientations, which in turn hinders production of hydrocarbons. Recently published structure-oriented filtering algorithms, such as the multi-window Kuwahara (or edge-preserving filtering) algorithm (1976) work well when the structures appears in blocky format, such as smoothing data between fault boundaries [10]. Unfortunately, this algorithm blurs small linear features narrower than the analysis window that may be associated with joints and fractures. The quality of obtaining reliable seismic attributes such as edges is directly related to the effectiveness of the noise-reduction

Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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Seismic Data Interpretation using Digital Image Processing

filters applied prior to the calculation. In this chapter, we explore some of the commonly used methods to enhance seismic images in the spatial domain. 3.1.1 The Mean (Running-Average) Filter

The mean filter is the simplest and most familiar random-noise-suppression filter. The mean filter is a low-pass filter, which is typically implemented as a running window that outputs the average of all the samples that fall within an analysis window at its center. The window size is usually an odd number, such as 3 × 3 or 5 × 5, and may be either rectangular or elliptical. The definition of the mean filter is 1∑ d, J j=1 j J

dmean =

(3.1)

where d denotes the jth of J traces falling within the analysis window at time t. Figure 3.1 illustrates how to apply a 3 × 3 window to a matrix of size 5 × 5. From Figure 3.1a, we start by choosing the first 3 × 3 matrix and then calculate the mean and assign the value to X22 , the central element in that window. In the next step we move the window to the right and keep moving until we reach the end of the matrix, as displayed in Figure 3.1b and c. Similarly, in the next iteration the window will move back to the first column but slide one row below, and the same procedure is repeated until we reach the end of the matrix. In Figure 3.2, we apply the mean-filtering concept to a 5 × 5 matrix using a window of size 3 × 3. As you can see from Figure 3.2a and b, we take the original matrix in (a) and filter it using a 3 × 3 window. The border encapsulates the first window with the set elements {5, 0, 2, 1, 3, 3, 5, 1, 2}, which equates to a mean of 2.44, with the shaded cell representing the assigning of the mean value to the central element. X11

X12

X13

X14

X15

X11

X12

X13

X14

X15

X11

X12

X13

X14

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Seismic Image Enhancement in the Spatial Domain

As can be seen in Figure 3.2b, this method of mean filtering skips the first and last rows as well as the first and last columns. In general, for a W1 × W2 window, the mean filter will not affect the top and bottom (W1 − 1)∕2 rows and the left- and rightmost (W2 − 1)∕2 columns. So, for a 3 × 3 window, the first and last rows and columns will not be filtered. To deal with this issue, we can apply (W1 − 1)∕2 rows of zeros to the top and bottom of the matrix and (W2 − 1)∕2 columns to the left and right. Then we can apply the mean filter and discard the extra rows and columns. So, for a 3 × 3 window we will have a zero padding of one row and column in each direction and two rows and columns for a 5 × 5 window, and so on. Figure 3.2c is actually the mean-filtered matrix after zero padding. As can clearly be observed, all of the elements have been filtered. An example of zero padding for a 3 × 3 window is illustrated in Figure 3.3a and b for the general 5 × 5 case and for the matrix in the example respectively. Averaging is actually a linear operation. Consequently, the mean filter is a linear filter. This means that it can be replaced by the convolution operation; a 2D filter can be replaced by 2D convolution. Mean filtering with a W1 × W2 window is the same as convolving the image with a W1 × W2 matrix with all the elements set to a value of 1∕(W1 × W2 ). So, for a 3 × 3 filter the convolution matrix will be 3 × 3 with all elements set to 1∕9, as illustrated in Figure 3.4. In general, for W1 and W2 greater than one, the convolution of an N × M image will result in an N + W1 − 1 × M + W2 − 1 matrix. Taking the central matrix by discarding (W1 − 1)∕2 rows and (W2 − 1)∕2 columns in each direction will retrieve the original mean-filtered matrix. Historically, linear mean and nonlinear median filters have been widely used to improve the interpretability of the seismic data. Figure 3.5 shows the application of the mean filter on a 2D seismic image. Unfortunately, the mean filter can also severely blur 0

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Figure 3.5 (a) Original image. (b) Image after the application of the mean filter. (c) Sobel edge detection of the original image without filtering. (d) Sobel edge detection of the image after mean filtering.

coherence and other edge-sensitive attributes. For example, Marfurt published work on a survey from the Fort Worth Basin in Texas and illustrated that the mean filter can, on the one hand, significantly improve the geologically interesting long-wavelength characteristics of the data but, on the other hand, smear the short-wavelength fracture patterns, making them difficult to see [11]. Although a median filter can preserve edges by separating fault blocks and stratigraphic features that are several traces in width, they will, in general, obliterate narrow curvilinear features associated with joints and fractures that are only a single trace wide. Nevertheless, the edge-preserving and impulse-removing properties are the most desirable features of the median filter [12]. In recent image-processing and synthetic aperture radar applications, simple mean

Seismic Image Enhancement in the Spatial Domain

filters have been supplanted by more advanced Lee and Frost filters [13, 14]. The more recently published speckle-reducing anisotropic-diffusion filters are also effective in reducing speckle noise [15]. Although these filters were developed to smooth images with speckle noise, some of them have been modified to smooth images with additive noise. Others have been developed to take care of both kinds of noise.

3.2 The Median Filter The median filter is one of the most widely used nonlinear techniques in signal and image processing [12]. In the seismic world, the median filter is routinely used in velocity filtering of vertical seismic profile data to distinguish between downgoing and upgoing events using the differences in their apparent velocities. The median filter works by replacing each sample in a window of a seismic trace by the median of the samples falling within the analysis window. The window size is typically an odd number (e.g., 3 × 3 or 5 × 5). One way to calculate the median is simply to order all of the J samples in the analysis window using an ordering index k: dj(1) ≤ dj(2) ≤ · · · ≤ dj(k) ≤ dj(k+1) ≤ · · · ≤ dj(J) .

(3.2)

The median is then given by dmedian = dj[k=(J+1)∕2] .

(3.3)

Using the matrix given in the example for the mean filter, we demonstrate how to apply the median filter. As we did for the mean filter, we choose a 3 × 3 window and apply it in a similar manner. Figure 3.6a illustrates the selection of the elements (covered by the green border) after the first application of the window. The elements are then converted to a one-column vector and sorted in ascending order so that the central element is the median, as shown in Figure 3.6b and c by the shaded cell. The median-filtered matrix is then displayed in Figure 3.6d. Figure 3.7 shows the application of the median filter on a 2D seismic image. There is also a filter that can act in between the mean and the median filter. This filter is called the alpha-trimmed mean. The alpha-trimmed mean is given by d𝛼 =

(1−𝛼)J ∑ 1 d , (1 − 2𝛼)J k=𝛼J+1 j(k)

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where 0 =< 𝛼 < 0.5. If 𝛼 = 0.5, we replace Equation 3.4 with the median filter. If 𝛼 = 0.0, we obtain the conventional mean filter. Figure 3.8 shows the application of the alpha-trimmed mean filter for 𝛼 = 0.25 on a 2D seismic image. The median filter is well known for preserving sharp discontinuities and removing impulse noise in the signal. The median and alpha-trimmed mean filters perform better than the mean filter in suppressing noise and preserving details. However, neither of these two filters is capable of preserving the thin lineaments. In recent image-processing and synthetic aperture radar applications, simple median filters have also been improved upon. Al-Dossary and Marfurt showcased a multistage median-based modified trim-mean filter, which was superior in filtering volumetric dip

Seismic Image Enhancement in the Spatial Domain

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Figure 3.8 (a) Original image. (b) Image after the application of the alpha-trimmed mean filter for 𝛼 = 0.25. (c) Sobel edge detection of the original image without filtering. (d) Sobel edge detection of the image after the application of the alpha-trimmed mean filter for 𝛼 = 0.25.

estimates, which form the basis of a majority of attribute analysis, including coherence, curvature, and lateral changes in reflectivity [16]. The filter worked by having the multistage median component of the filter search for and preserve constant-value lineaments running through the analysis window. Then the range trimmed-mean component of the filter improved signal-to-noise by smoothing data. They proved that the multistage median-based modified trim-mean filter worked well on attributes with Gaussian statistics, such as dip, unnormalized amplitude gradients, and many of the curvature attributes, and was thus able to identify faults and fractures on short-wavelength curvature images that were otherwise difficult to see. As you can see, developing novel image-processing techniques for seismic data interpretation(especially for low SNR

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data) is an ongoing effort, and many petroleum companies have assigned their subject matter experts constantly towards production of better imaging for now and the future.

3.3 The Edge-Preserving Smoothing Algorithm Smoothing is an effective means of reducing random noise in 3D seismic data. Much work has been done regarding this attribute, and many explorationists have developed their own variants for use in seismic interpretation. Filters used in the past, such as the Gaussian and mean filters, were structure indistinguishable, and thus smeared the edges and texture boundaries. After filtering, the resolution of horizons, faults, and unconformities was degraded or even lost. Inspired by the multiwindow “Kuwahara” filter, Luo et al. [17] developed an edge-preserving smoothing (EPS) algorithm that was able to keep edges in 2D seismic data, by searching for the most homogeneous block around each location within an input cube and assigning the average/median value of the block to that location [10]. In two dimensions, EPS can be computed using nine masks (Figure 3.9) originally developed by Nagao and Matsuyama [18]. EPS can be computed using the following steps: 1 2 3 4 5 6 7

for pixel at location (x, y) loop for each mask in Figure 3.9 compute the variance of each mask find the smallest variance of all the nine masks compute the average of the smallest variance of all the nine masks replace the pixel value at (x, y) by the average computed in step 5 goto next pixel.

Figure 3.10 shows the application of the EPS filter on a 2D seismic image, and Figure 3.11 compares the results of the EPS filter with the mean and the median filters on seismic data after Sobel edge detection. However, its 3D counterpart could not keep planar structures, such as faults [19]. Structure-oriented filtering solved these challenges by first computing the structural orientation, followed by applying a

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Seismic Image Enhancement in the Spatial Domain

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Figure 3.10 (a) Original image. (b) Image after the application of the EPS algorithm. (c) Sobel edge detection of the original image without filtering. (d) Sobel edge detection of the image after applying the EPS algorithm.

diffusion scheme along the known orientation [20]. However, it was computationally costly and was not feasible for nonstructured areas, which are common in coherence or curvature data. 3.3.1 Two-Dimensional Structure-Preserving Smoothing

Further advancement of smoothing was achieved when Al-Dossary and Wang proposed a data-adaptive smoothing method called structure-preserving smoothing (SPS), which did not require extensive computations of structural orientation for preserving existing structures [21]. SPS and EPS work in similar ways. In EPS, a set of predefined neighborhood sub-windows with size of 3 × 3 is used in Figure 3.12, and the best result (usually

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Seismic Image Enhancement in the Spatial Domain

the one with minimum deviation) is selected as the smoothened output. Likewise, in SPS, a set of predefined orientations is used for smoothing, and the best result is selected. If structures (planar or linear features) exist, the selected result is likely to be the best in alignment with the true structure. The selection rule of SPS can be based on minimum deviation (cf. EPS). For polar data containing both positive and negative numbers (i.e., seismic amplitude), the selection rule can also be an absolute maximum. For mono-polar data containing only positive numbers (i.e., coherence or curvatures), the selection rule can be either maximum or minimum summation, depending on which end the structure resides. Figure 3.13 shows the application of the SPS filter on a 2D seismic image.

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Figure 3.13 (a) Original image. (b) Image after the application of the SPS algorithm. (c) Sobel edge detection of the original image without filtering. (d) Sobel edge detection of the image after applying the SPS algorithm.

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Their results showed that SPS significantly removed random noise and seismic footprints, and greatly improved coherency of seismic attributes. In comparison with structure-oriented filtering [20], Al-Dossary and Wang’s SPS method was faster and more robust for both structured and nonstructured areas, but was not as accurate when orientations were available [21].

3.4 Wavelet-Based Smoothing In this case study we will illustrate the performance of EPS based on the redundant wavelet transform (RWT) and demonstrate its usefulness before running an edge-detection algorithm to reveal channel patterns in seismic data. Our examples demonstrate that RWT can successfully preserve, enhance, and delineate channel edges that are otherwise not readily visible on conventional seismic amplitude displays. Al-Dossary and Ananos demonstrated the usefulness of RWT before edge detection to enhance channel patterns in seismic data [22], applying the concept to data collected from Saudi Arabia. The RWT is a type of discrete wavelet transform (DWT). However, it differs from the standard DWTs in that it does not carry out decimation or subsampling at successive resolution levels; rather, RWT decomposes the data into low-frequency information (approximation) and high-frequency information (wavelet coefficients) to obtain a projective decomposition of the data into different scales. RWT can be used for noise reduction in image processing, texture classification, and image fusion. RWT’s advantage in feature characterization lies in its pixel-wise analysis of images without performing image decimation. After application of RWT, locally linear features appear at adjacent scale levels, whereas nonsignificant features such as random noise slowly decrease as the scale level increases. 3.4.1 Method

For a discrete implementation of the RWT one can use the “à trous” algorithm, which avoids the decimation of the typical DWT transform [23]. A series of the original image I0 (i.e., amplitude time slice) in terms of the wavelet coefficients 𝑤i can be written as I0 = IP +

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where M is a 2D convolution operator. Here, 𝑤i defines the wavelet plane “scale level” and is computed as the difference between two consecutive approximations of I0 : 𝑤i = Ii−1 − Ii .

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The à trous wavelet transform is a nonorthogonal multiresolution decomposition that separates the input data into a set of low-frequency components Ip and high-frequency components 𝑤i [24].

Seismic Image Enhancement in the Spatial Domain

The associated wavelet function 𝜓(x) is defined by 1 1 𝜓(x) = 𝜙(x) − 𝜙(x). 2 2

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Edge features on the 𝑤i wavelet planes are typically preserved as i increases from 1 to P, whereas random noise is not preserved among these levels. Therefore, it is possible to enhance the edge details by summing together the wavelet planes. The output of this superposition is called the enhanced edge and is given by IEI =

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in two dimensions. The implementation of the algorithm is done by convolving with the image once. The result is shown in Figure 3.14. 3.4.2 Sharpening Filter

Sharpening kernels can be helpful in the image processing of seismic data. They are second-order derivative kernels. The main goal of sharpening is to make the image crisper by emphasizing the high frequency in the image. A common way of sharpening seismic images is to convolve the image with a 2D second-order derivative kernel. One example is as follows: ⎡− k ⎢ 9 ⎢ k ⎢− ⎢ 9 ⎢− k ⎣ 9

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3.5 Summary In this chapter we present tools to reduce the seismic random noise that can contaminate seismic data. The application of noise reduction prior to edge detection (or similar attributes, such as coherency and curvature) is critical to preserve subtle geologic features, such as faults, fractures, and channels, which are required for accurate subsurface models. There are many different methods that range from simple to complex algorithms. In this part of the book we reviewed the following methods: mean filter, median filter, EPS algorithm, 2D SPS, and wavelet-based smoothing. When applied prior to edge detection, these filters show promising results for enhancing channel and fault patterns

Seismic Image Enhancement in the Spatial Domain

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Figure 3.15 Effect of sharpening kernel on seismic coherence data (k = 9). (a) Original image (b) Sharpened image; edges are much thinner and contrast is increased. A slight ringing effect appears. Noise in the initial data set is amplified as well. (c) Sobel edge detection of the original image. (d) Sobel edge detection of the image after applying the sharpening filter.

in seismic data. Testing demonstrates these filters improve seismic reflection continuity and increase resolution in subsequent seismic attribute calculations.

3.6 Problems and Computer Assignments For Problems 3.1–3.4, use the 5 × 5 matrix in Figure 3.16. 3.1

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(a) Apply a 3 × 3 mean filter without zero padding. (b) Apply a 3 × 3 mean filter with zero padding. (c) Perform 2D convolution with a 3 × 3 matrix with all the elements set to 1∕9. Does the result match the answer to (b)? (d) Apply a 3 × 3 median filter. Explain how it differs to the results for parts (a)–(c). 3.2

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In this problem we will use computer programs to apply the mean filter. (a) Write a program to solve Problems 3.1(a) and 3.1(b). (b) Modify this program such that it accepts any matrix as an input, performs zero padding, and applies a 3 × 3 mean filter. (c) Modify the program of part (b) such that the filter can be of any window size W 1 × W2 . (d) Finally, modify the program to accept an image as an input and output the filtered image.

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(b) Replace the matrix in Figure 3.17 with the operator for a 3 × 3 mean filter. Compare your answers with Problem 3.1(c). (c) Now apply the operators for the wavelet smoothing algorithm and the sharpening filters; then compare your results with Problems 3.3 and 3.4 respectively.

Seismic Image Enhancement in the Spatial Domain

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4 Seismic Image Enhancement in the Spectral Domain 4.1 Introduction Most seismic image enhancement processes can be performed in the spatial or spectral domain with, ideally, identical quality. However, some of these processes perform more efficiently and often more robustly in the spectral domain. In this chapter, important spectral processes that are frequently used by seismic data processors to enhance the quality of the seismic image will be discussed. The main goals of most of these processes are image enhancement and noise suppression. The chapter will discuss the 2D Fourier transform of seismic images. Examples of each process will be discussed in detail. Finally, spectral attribute images will be presented with real data examples.

4.2 The Fourier Transform The Fourier transform provides several mathematical tools that are useful in the analysis of seismic images. Such representations basically involve the decomposition of the signals in terms of complex exponential/sinusoidal components. The signal with such decomposition is represented in the frequency and/or wavenumber domain. Seismic signals are examples of signals that are heavily analyzed and processed in the frequency and/or wavenumber domain. Frequency (and/or wavenumber) analysis of a signal (which is a function of time and/or space) involves the resolution of the signal into its frequency/wavenumber (sinusoidal) components. Frequency (and/or wavenumber) synthesis involves the recombination of the sinusoidal components to reconstruct the original signal. Spectrum is used when referring to the frequency and/or wavenumber content of an image. The Fourier transform of one variable (e.g., time), continuous function g(t) can be given by ∞

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Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

(4.1)

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where Ω is the analog angular frequency and j = G(Ω) is given by

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For example, the 2D Fourier transform of the boxcar function { 1 if 0.4 < t ≤ 0.6, 2000 < x ≤ 3000 g(t, x) = 0 otherwise is calculated using Equation 4.3, yielding ( ) −4e−j[(Ω∕2)−2500Kx ] sin Ω2 sin(2500Kx ) , G(Ω, Kx ) = ΩKx

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(4.5)

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which is a typical sinc function in Ω and Kx . Figure 4.1 shows both g(x, t) of Equation 4.5 and the magnitude spectrum of G(Ω, Kx ) given by Equation 4.6. Seismic images are already stored digitally, so the interest will rather be on the 2D discrete Fourier transform (DFT). In this case, consider, without loss of generality, a seismic image section g(nt , nx ) of size Nt × Nx . Its 2D DFT G(n𝜔 , nkx ) is given by [ ( )] n𝜔 nt − nkx nx g(nt , nx ) exp −j2π Nt Nx n =0 n =0 ∑ ∑

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x

for n𝜔 = 0, … , Nt and nkx = 0, … , Nx . At the same time, the inverse 2D DFT g(nt , nx ) of size Nt × Nx can be obtained using G(n𝜔 , nkx ) and g(nt , nx ) =

[ ( )] Nt −1 Nx −1 n𝜔 nt − nkx nx 1 ∑ ∑ G(n𝜔 , nkx ) exp −j2π Nt Nx n =0 n =0 Nt Nx t

(4.8)

x

for nt = 0, … , Nt and nx = 0, … , Nx . Note that both Equations 4.7 and 4.8 comprise the 2D DFT pair, where nt and nx represent the temporal and spatial variables of the seismic image respectively. Also, n𝜔 is the frequency variable and nkx is the wavenumber variable. The magnitude spectrum of G(n𝜔 , nkx ) is given by √ (4.9) |G(n𝜔 , nkx )| = ℜ{G(n𝜔 , nkx )}2 + ℑ{G(n𝜔 , nkx )}2 .

0

5000

–3

4500

0.1 –2

0.2

–1

Ω (rad/s)

0.4

Time (s)

4000 3500

0.3

0.5

3000 2500

0

2000

0.6 1

1500

0.7 0.8

1000

2

500

0.9 1

3 0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0 –3

–2

–1

0

1

Space (m)

Kx (rad/m)

(a)

(b)

Figure 4.1 (a) g(t, x) in Equation 4.5. (b) its magnitude spectrum G(Ω, Kx ) given by Equation 4.6.

2

3

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Seismic Data Interpretation using Digital Image Processing

where ℜ stands for the real part and ℑ is the imaginary part of a given complex function. In addition, the phase spectrum of G(n𝜔 , nkx ) is ∠G(n𝜔 , nkx ) = arctan

ℑ{G(n𝜔 , nkx )} ℜ{G(n𝜔 , nkx )}

.

(4.10)

while the power spectrum P(n𝜔 , nkx ) of G(n𝜔 , nkx ) is the square of Equation 4.9, as follows: P(n𝜔 , nkx ) = |G(n𝜔 , nkx )|2 .

(4.11)

It is worth mentioning here that the exponential kernels in the 2D DFT in Equation 4.7 (or its inverse in Equation 4.8) are separable. This means that in order to compute the 2D DFT, or its inverse, one can apply a one-dimensional (1D) DFT in the nt (or n𝜔 ) direction followed by another 1D DFT in the nx (or nkx ) direction to obtain G(n𝜔 , nkx ) (or g(nt , nx )). We recommend the reader refer to the Handbook of Image & Video Processing [25] for more information on the 2D DFT and its properties of images.

4.3 Filtering in the Spectral Domain Linear filtering to enhance seismic images, such as smoothing, can be performed in the spectrum domain (n𝜔 , nkx ) instead of performing convolution in the time–space domain, as was seen earlier in the book. This requires transforming the seismic images g(nt , nx ) into G(n𝜔 , nkx ) and then multiplying G(n𝜔 , nkx ) by the 2D DFT of the filtering masks h(n, m), say, H(n𝜔 , nkx ). This will result in a function F(n𝜔 , nkx ), where its 2D inverse DFT f (nt , nx ) is the filtered (enhanced) seismic image. For example, consider the 5 × 5 smoothing mask: ⎛1 ⎜ ⎜1 1 ⎜ 1 h(n, m) = 25 ⎜ ⎜1 ⎜ ⎝1

1 1

1 1

1 1

1 1

1 1

1 1

1

1

1

1⎞ ⎟ 1⎟ 1⎟ . ⎟ 1⎟ ⎟ 1⎠

(4.12)

Figure 4.2 shows smoothing of the seismic image in (a) via Equation 4.12 but in the spectrum domain. This required performing two DFT computations in addition to multiplying spectra G(n𝜔 , nkx ) and H(n𝜔 , nkx ). Another example is the application of sharpening filters (see Section 3.4.2). Consider the sharpening filter that detects lines in the +45∘ direction: ⎛−1 ⎜ h(n, m) = ⎜−1 ⎜2 ⎝

−1 2 −1

2⎞ ⎟ −1⎟ . −1⎟⎠

(4.13)

The result of applying such a filter to detect the channel in the +45∘ direction in the spectrum domain is shown in Figure 4.3. It is interesting to note that all linear filtering that has been shown previously in Chapter 3 can be performed in the spectrum domain, instead of using 2D convolution.

Seismic Image Enhancement in the Spectral Domain

(a)

(b)

(c)

(d)

(e)

Figure 4.2 (a) Smoothing of the seismic image. (b) The magnitude spectrum of (a) using the 2D DFT. The smoothing operator filter in Equation 4.12 magnitude spectrum is shown in (c). (d) The filtered seismic image in the spectrum domain. (e) The smoothed image after applying the inverse 2D DFT on (d).

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(a)

(b)

(c)

(d)

(e)

Figure 4.3 (a) Sharpening of the seismic image. (b) The magnitude spectrum of (a) using the 2D DFT. The sharpening operator filter in Equation 4.13 magnitude spectrum is shown in (c). (d) The filtered seismic image in the spectrum domain. (e) The sharpened image after applying the inverse 2D DFT on (d).

Seismic Image Enhancement in the Spectral Domain

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 (a)

(b) 0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4

(c)

(d)

Figure 4.4 (a) A seismic image, (b) its instantaneous amplitude, (c) its instantaneous phase, and (d) its instantaneous frequency. (See color plate section for the color representation of this figure.)

4.4 Spectral Attributes Among the commonly used seismic attributes are the envelope, instantaneous frequency, and instantaneous phase. They are obtained based on the notion of a complex trace. Given a seismic trace g(t), one can use it to calculate a complex seismic trace z(t) as follows: { 2G(Ω) if Ω > 0 Z(Ω) = . (4.14) 0 if Ω ≤ 0 Then, z(t) can be given by the inverse Fourier transform in Equation 4.2 to yield the following: z(t) = g(t) + jy(t),

(4.15)

where y(t) happens to be the quadrature of g(t). y(t) can be calculated by taking the Hilbert transform of g(t) as follows: 1 (4.16) y(t) = g(t) . πt

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60 50 40 30 20 10

(a)

(b) 0.5 0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4 –0.5

(c)

(d)

Figure 4.5 (a) A seismic image, (b) its instantaneous amplitude, (c) its instantaneous phase, and (d) its instantaneous frequency. (See color plate section for the color representation of this figure.)

Another way to calculate the complex seismic trace via the use of 1D Fourier transform is stated as follows [26]: 1 2 3 4

Calculate the Fourier transform of the real trace. Zero the amplitude spectrum for negative frequencies. Double the amplitude spectrum for positive frequencies. Calculate the inverse Fourier transform using the resulting amplitude spectrum.

The complex seismic trace z(t) is used to calculate many seismic attributes. For example, the instantaneous amplitude (i.e., trace envelope) a(t) is calculated from z(t) as follows [27]: √ a(t) = g(t)2 + y(t)2 , (4.17) while the instantaneous phase Φ(t) is calculated as [ ] y(t) Φ(t) = arctan . g(t)

(4.18)

Seismic Image Enhancement in the Spectral Domain

The instantaneous frequency f (t) is the derivative of the instantaneous phase, as follows: 1 dΦ(t) f (t) = . (4.19) 2π dt To obtain such spectral attributes from seismic images, one should compute first the complex trace of each seismic trace in the seismic image. Then, for each complex trace, apply Equations 4.17, 4.18, and 4.19 to obtain the seismic envelope attribute image, instantaneous phase attribute image, and instantaneous frequency attribute image respectively. Figures 4.4 and 4.5 show two examples of seismic images and their instantaneous amplitude, phase, and frequency attribute images.

4.5 Summary In this chapter we introduce spectral domains calculated through various forms of the Fourier transform. For the purpose of illustration, we apply some of these processes on simple images. As an application of spectral processing to seismic images, we define spectral attributes and show examples of them on real seismic images.

4.6 Problems and Computer Assignments 4.1

Consider two continuous functions x(t) and y(t), with Fourier transform X(Ω) and Y (Ω) respectively. Show that  {a1 x(t) + a2 y(t)} = a1 X(Ω) + a2 Y (Ω), where a1 and a2 are constant.

4.2

For a continuous function g(t), show that ( ) 1 Ω  {g(ct)} = G , c c where  {g(t)} = G(Ω) is the Fourier transform of g(t).

4.3

Show that Equation 4.7 (and 4.8) is separable. Provide an explicit form for your answer.

4.4

Compute the 3 × 3 2D DFT of the following image: ⎡ 1 0 0⎤ g(nt , nx ) = ⎢0 1 0⎥ . ⎢ ⎥ ⎣ 0 0 1⎦

4.5

Compute the 3 × 3 2D inverse DFT of the following image: ⎡1 0 2⎤ G(n𝜔 , nkx ) = ⎢ 1 0 2 ⎥ . ⎢ ⎥ ⎣−1 0 −1⎦

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4.6

Recall that a complex seismic trace is defined as z(t) = x(t) + jy(t). Assume that x(t) = cos 𝜔o t and y(t) = sin 𝜔o t where 𝜔o is a constant angular frequency and j = following for this complex seismic trace: (a) the instantaneous amplitude (envelope); (b) the instantaneous phase; (c) the instantaneous frequency.

√

−1. Calculate and sketch the

4.7

It is well known that Equations 4.7 and 4.8 are implemented via the fast Fourier transform (FFT). Compute the number of multiplications and additions required to implement Equations 4.7 and 4.8 in a computer program.

4.8

Using Matlab, compute the 2D DFT via FFT of the seismic image shown in Figure 4.4. Also, perform the same on Figure 4.5. Swap the phase spectrum of both results and then apply a 2D inverse DFT via an inverse FFT on both images. Display both seismic images and comment on your findings.

4.9

Using Matlab, compute the average trace of the seismic image shown in Figure 4.4. Plot its instantaneous amplitude, frequency, and phase. Comment on your plots.

87

5 Seismic Attributes 5.1 Introduction Advances in image-processing algorithms and computer technology have greatly contributed to the geologic interpretation of seismic dataset. An interpreter’s computer workstation can now access integrated databases containing complete seismic and well log data needed for quantitative analysis, thus allowing better assessment and measurement of oil and gas reservoirs. Large areas of petroleum-bearing sedimentary basins are now imageable via advanced 3D seismic techniques combined with parallel processing. In particular, the challenges of accurate fault detection and the tracking of ancient river channels can now be solved via imaging algorithms and attributes, such as coherency and curvature analysis [28]. Additionally, removal of random noise and artifacts, such as processing footprints, can be solved via image-processing techniques like EPS/SPS algorithms [29]. Picking of seismic traveltimes of low signal-to-noise data is also now possible with the aid of edge-preserving filters. Separation of seismic arrivals can be achieved via image segmentation in the time-scale domain. Thus, parallel processing workstations synergize well with these advanced algorithms in achieving high-fidelity results. The integration of image-processing algorithms and advanced computing techniques now makes quantitative interpretation possible [30]. In this chapter we will discuss some of these advanced imaging algorithms and seismic attributes and their roles in processing and interpreting seismic data. The focus will be on 2D seismic images, such as seismic slices and seismic sections.

5.2 Detection of Interesting Regions from Time or Depth Three-Dimensional Slices using Seismic Attributes As stated in Chapter 2, a seismic attribute is a quantitative derivative of Earth’s reflection information observed in time or depth slices of 3D seismic data volume. Analysis of attributes has been integral to interpreting seismic reflections since the 1930s [28]. This is when geophysicists started picking traveltimes of the reflections of seismic field records. The objective of seismic attributes is to enhance the visibility of static and dynamic characteristics of subsurface structures by quantifying both amplitude and phase features observed in seismic data, through multiple image-processing algorithms that are performed on a computer. For example, algorithms commonly used to delineate Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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body shapes and skeletons of medical pictures can be used to detect faults and ancient riverbeds of the Earth’s subsurface [31]. Algorithms used to enhance continuity of forensic fingerprints are now optimized for removal of processing artifacts of seismic data. These algorithms are now capable of processing large volumes of geophysical and well information in producing seismic attributes, which are routinely used by interpreters in seeking geologic and reservoir information. Edge-detection attributes are among the major seismic attributes in use by petroleum explorationists today. In principle, edge detection is one of the fundamental steps in image processing, image analysis, image pattern recognition, motion estimation and object tracking in digital video, and computer vision techniques. It is a set of mathematical image segmentation techniques that determines the presence of an edge or a line in a digital image and outlines them in an appropriate way [32], by identifying points in an image at which the image brightness changes sharply or has discontinuities. Specifically, an edge is defined as the boundary pixels (sample amplitude in our case) that connect two separate regions with changing image amplitude attributes, such as changes in reflectance, changes in surface orientation (within the object), changes in luminance, and tristimulus values in an image [33, 34]. The main purpose of edge detection is to simplify the image data in order to minimize the amount of data to be processed [35]. Ideally, the result of applying an edge detector to an image may lead to a set of connected curves that indicate the boundaries of objects, the boundaries of surface markings, and curves that correspond to discontinuities in surface orientation. Thus, applying an edge-detection algorithm to an image may significantly reduce the amount of data to be processed and may, therefore, filter out information that may be regarded as less relevant while preserving the important structural properties of an image. Edge detection begins with the examination of the local discontinuity at each pixel element in an image. Amplitude, orientation, and location of a particular subarea in the image are essential characteristics of potential edges [32]. Based on these characteristics, the detector has to decide whether each of the pixels examined is an edge or not. The process of edge detection consists of three main steps: 1 Noise reduction Owing to an edge detector’s great sensitivity to noise, the use of image-smoothing techniques before applying the edge-detection operator is necessary. 2 Detection of edge points Local operators that respond strongly to edges and weakly elsewhere are applied to a given image, resulting in an output image whose highlighted pixels can be selected to become edge points. 3 Edge localization Edge detection results are then postprocessed, where expendable pixels are removed and broken edges are turned into meaningful lines and boundaries, via techniques such as the Hough transform. In the past few decades, advances in image-processing algorithms have greatly contributed to the geologic interpretation of seismic data sets. Several edge-detection algorithms are commonly used in image processing of 2D seismic data to extract important but subtle geologic structural features, such as faults and channels, which often appear as sharp edges and are not so easily detected. One such algorithm is the Sobel filter, which is a discrete differentiation operator that approximates the local gradient by combining derivatives of the amplitude between neighboring traces

Seismic Attributes

along the x, y, and z directions. For example, this conversional Sobel filter provides a quantitative measure of the changes in seismic waveform across a discontinuity, which is important in seismic data processing and detection of seismic lineaments, faults, and fractures [36, 37]. In the following, we shall go into detail on some of the edge-detector operators in use in the petroleum industry today.

5.3 Two-Dimensional Numerical Gradient Edge-Detector Operators Some traditional 2D gradient operators that are used to detect edges in images in general and 2D seismic data in particular are now examined [38]. Simple finite difference operator This is the simplest gradient operator. Its three-point representation in terms of matrices Dx and Dy along the x and y directions respectively is Dx =

⎡0 1⎢ −1 2⎢0 ⎣

0 0 0

0⎤ 1⎥ , ⎥ 0⎦

Dy =

⎡0 1⎢ 0 2 ⎢0 ⎣

−1 0⎤ 0 0⎥ . ⎥ 1 0⎦

(5.1)

Because of similarity, the operator along the y direction is simply the transpose of that along the x direction, a property that holds for other operators. Roberts operator This is a 2 × 2 approximation of a first-order derivative. Its expression in matrix form along the x and y directions is given by [ ] [ ] 1 0 0 1 , Dy = . (5.2) Dx = 0 −1 −1 0 The Roberts operator can also be represented as a 3 × 3 convolution mask in the form ⎡0 Dx = ⎢0 ⎢ ⎣0

0 1 0

0⎤ 0 ⎥, ⎥ −1⎦

⎡0 Dy = ⎢ 0 ⎢ ⎣−1

0 1 0

0⎤ 0⎥ . ⎥ 0⎦

(5.3)

At each point of an image we calculate an approximation of the gradient at that point by combining both the aforementioned results: √ (5.4) D = D2x + D2y . Equation 5.3 can be approximated by D = |Dx | + |Dy |.

(5.5)

The angle of orientation of the edge that produces the spatial gradient is given by ( ) Dy 3π 𝜃 = arctan − . (5.6) Dx 4 Sobel operator This operator is mostly used in edge detection. Its expression in matrix form along the x and y directions is given by Dx =

⎡−1 1⎢ −2 2 ⎢−1 ⎣

0 0 0

1⎤ 2⎥ , ⎥ 1⎦

Dy =

⎡−1 1⎢ 0 2⎢1 ⎣

−2 0 2

−1⎤ 0 ⎥. ⎥ 1⎦

(5.7)

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At each point of an image we calculate an approximation of the gradient at that point by combining both the aforementioned results using Equation 5.4 , while the angle of orientation of the edge that produces the spatial gradient is given by ( ) Dy 𝜃 = arctan . (5.8) Dx Prewitt operator The gradient operator here is similar to the Sobel, however different weighting factors {1, 1, 1} are used in the smoothing. Its expression is ⎡−1 Dx = ⎢−1 ⎢ ⎣−1

0 0 0

1⎤ ⎡−1 −1 −1⎤ 1⎥ , Dy = ⎢ 0 0 0 ⎥. (5.9) ⎢ ⎥ ⎥ 1⎦ 1 1⎦ ⎣1 It is obvious that the Sobel and Prewitt operators are orthogonally smoothed versions of the simple finite difference operator, in which the gradient in the x direction is smoothed along the y direction. The smoothing operator is {1, 2, 1} for Sobel and {1, 1, 1} for Prewitt. Laplacian of Gaussian (LOG) edge operator This is a linear and time-variant operator. It uses a discrete convolution kernel that approximates the second-order derivative. It works by searching for zero crossings in the second derivative of an image to find edges. Its expression in matrix form along the x and y directions is given by ⎡0 Dx = ⎢−1 ⎢ ⎣0

−1 4 −1

0⎤ −1⎥ , ⎥ 0⎦

⎡−1 Dy = ⎢−1 ⎢ ⎣−1

−1⎤ −1⎥ . (5.10) ⎥ −1⎦ Canny operator The Canny edge operator [35] is one of the most powerful and optimal edge-detection operators. Its implementation is not straightforward. It is based on the image gradient computations, but in addition tries to maximize the SNR, find a good localization of the edge points, and minimize the number of positive responses around a single edge. Its algorithm involves the following steps: 1 Gaussian filtering The input image I is smoothed using a Gaussian low-pass filter with a specified standard deviation 𝜎. 2 Spatial gradient estimation The local gradient is computed for each point in the smoothed image to identify regions with high spatial derivative. The first derivative of a Gaussian filter is utilized here. Sometimes Sobel or Prewitt are used. 3 Non-maximum suppression The edge points at the output result in wide ridges. The algorithm thins those ridges leaving only the pixels at the top of each ridge. 4 Hysteresis The ridge pixels are then thresholded using two thresholds T1 and T2 . Ridge pixels with values greater than T2 are considered strong edge, weak edge will be contained in the output. The theory basis is shown in Equations 5.11, 5.12, 5.13, 5.14, and 5.15: [ ] (x2 + y2 ) D(x, y) = exp − (5.11) 2𝜎 2 −1 8 −1

𝜕D , (5.12) 𝜕n where Dn is the first derivative operator of a 2D Gaussian D in some direction n. n should be oriented normal to the direction of an edge to be detected and is given as follows: Dn (x, y) = −

Seismic Attributes

• edge normal (5.13)

n = Δ(D ∗ I)∕|Δ(D ∗ I)| • maximal strength 𝜕 D ∗ I = 0. 𝜕n n

(5.14)

Substituting for Dn yields 𝜕2 Dn ∗ I = 0. 𝜕n2

(5.15)

Compass masks Finding horizontal and vertical masks used by the Prewitt and Sobel operators can be extended to include all eight compass directions: north, northeast, east, southeast, south, southwest, west, and northwest. Examples of these include Robinson, Kirsch, Frei–Chen edge detectors, and so on. Kirsch and Robinson are both nonlinear convolution kernels that find the maximum edge strength in a predetermined direction. These operators take a single kernel and rotate it in 45∘ increment in all eight compass directions. The Kirsch operator detects edges using eight compass filters. All eight filters are applied to the image, and the maximum one is retained for the final image. The discrete convolution kernels of the Kirsch operator are of the following form: ⎡−3 D0 = ⎢−3 ⎢ ⎣−3

−3 0 −3

5⎤ 5⎥ , ⎥ 5⎦

⎡−3 D1 = ⎢ 3 ⎢ ⎣−3

5 0 −3

5⎤ 5⎥, ⎥ −3⎦

⎡ 5 D2 = ⎢−3 ⎢ ⎣−3

5 0 −3

5⎤ −3⎥ , ⎥ −3⎦

⎡ 5 D3 = ⎢ 5 ⎢ ⎣−3

5 0 −3

−3⎤ −3⎥ , ⎥ −3⎦

⎡−3 D5 = ⎢ 5 ⎢ ⎣ 5

−3 0 5

−3⎤ −3⎥ , ⎥ −3⎦

⎡−3 D7 = ⎢−3 ⎢ ⎣−3

−3 −3⎤ 0 5⎥. ⎥ 5 5⎦

⎡5 −3 D4 = ⎢5 0 ⎢ ⎣5 −3 ⎡−3 D6 = ⎢−3 ⎢ ⎣ 5

−3 0 5

−3⎤ −3⎥ , ⎥ −3⎦ −3⎤ −3⎥ , ⎥ 5⎦

(5.16)

5.4 Application to Real Seismic Data To verify the effectiveness of these different gradient operators, they are applied separately to the same input data: a 2D real seismic image. The input image is a time slice extracted from a 3D seismic survey in central Saudi Arabia. The input image Figure 5.1a shows two seismic (geologic) subsurface channels. White arrows indicate the visible part of the main channel. Black arrows indicate the visible part of the secondary channel. Our goal is to detect the edges of these two channels. The input image is run through the following operators: simple finite difference (Figure 5.1), Sobel (Figure 5.2), Prewitt (Figure 5.3), Canny (Figure 5.4), LOG (Figure 5.5), and Kirsch (Figure 5.6).

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(a)

(c)

(b)

(d)

Figure 5.1 Edge detection example: (a) original image, (b) result of simple finite difference horizontal kernel, (c) result of simple finite difference vertical kernel, and (d) gradient magnitude.

(a)

(b)

Figure 5.2 Edge detection example: (a) original image, (b) result of Sobel horizontal kernel, (c) result of Sobel vertical kernel, and (d) gradient magnitude. (See color plate section for the color representation of this figure.)

Seismic Attributes

(c)

(d)

Figure 5.2 (Continued)

(a)

(b)

(c)

(d)

Figure 5.3 Edge detection example: (a) original image, (b) result of Prewitt horizontal kernel, (c) result of Prewitt vertical kernel, and (d) gradient magnitude. (See color plate section for the color representation of this figure.)

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(a)

(c)

(b)

(d)

Figure 5.4 Edge detection using the Canny edge detector: (a) original image, (b) default values, 𝝈 = 2, T1 = 0.01, T2 = 0.1, (c) 𝝈 = 1, T1 = 0.01, T2 = 0.1, and (d) 𝝈 = 0.5, T1 = 0.11, T2 = 0.2. (See color plate section for the color representation of this figure.)

(a)

(b)

Figure 5.5 Edge detection using the LOG edge detector: (a) original image, (b) default value (𝝈 = 0.0034), (c) 𝝈 = 0.005, and (d) 𝝈 = 0.0078. (See color plate section for the color representation of this figure.)

Seismic Attributes

(c)

(d)

Figure 5.5 (Continued)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 5.6 Edge detection using the Kirsch compass mask detector. Results in (a) north, (b) northeast, (c) east, (d) southeast, (e) south, (f ) southwest, (g) west, and (h) northwest directions. (i) Resultant edge.

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All the aforementioned operators show good results. It should be noted that the results could be better if the input image were preprocessed (like noise reduction) before the application of edge detection.

5.5 Two-Dimensional Second-Order Derivative Operator In Section 5.4 the edge detector was approximated using the first-order derivative of the pixel value of an image. In this sense it is also possible to use the second-order derivative for detection of edges. One of the most popular operators for this purpose is the Laplacian operator. By definition the Laplace operator is defined by Δf (x, y) =

d2 f (x, y) d2 f (x, y) + . dx2 dy2

(5.17)

The second-order derivative in two dimensions (Laplacian) can be approximated using the five-point stencil finite-difference method: f (x − h, h) + f (x + h, y) + f (x, y − h) + f (x, y + h) − 4f (x, y) . h2 Implementation in image processing: Δf (x, y) =

(5.18)

• 1D filter D2 x = [1 − 2 1]

(5.19)

• 2D filter (see Figure 5.7) ⎡0 D2 xy = ⎢1 ⎢ ⎣0

1 −4 1

0⎤ 1⎥ ⎥ 0⎦

(5.20)

• 2D filter including diagonal D2 xy =

2 ⎡1 1⎢ 2 −12 2 ⎢1 2 ⎣

1⎤ 2⎥ . ⎥ 1⎦

(5.21)

We should note that the first-derivative operator exaggerates the effect of noise. As a consequence, the second-order derivative will exaggerate the noise twice as much. The other disadvantage of this method is the absence of directional information about the edge. So, by using Gaussian smoothing in conjunction with the Laplacian operator, it is possible to detect edges. 5.5.1 The Coherence Attribute

The coherence attribute is an edge-detection method that is widely used for interpreting faults on 3D seismic time slices. One of the most important steps in seismic data interpretation is the ability to map structural faults reliably. It is common practice to interpret faults using constant or horizon time/depth slices extracted from 3D coherence volumes, with coherence being defined as a measure of similarity between reflector waveforms.

Seismic Attributes

(a)

(b)

(c)

Figure 5.7 (a) Input of time slice seismic data; (b) after applying the Laplacian operator (Equation 5.20); (c) after applying the Laplacian operator (Equation 5.21). (See color plate section for the color representation of this figure.)

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Different concepts of coherence are described by researchers such as Drecun and Lucas [39] and Claerbout [40]. One of the earliest documented examples of the use of coherence for fault mapping was the application of a three-trace cross-correlation coherence algorithm to a large 3D seismic survey acquired over South Marsh Island, Louisiana, USA [36]. In a 3D coherence time slice, the coherence along a fault plane decreases because the similarity between the traces across the fault is low (i.e., incoherent). As a result, coherence can reveal significant structural and stratigraphic boundaries because these features generate sharp discontinuities that are reflected on the seismic traces. Additional research on coherence was developed for production, such as with estimates of seismic coherence [36, 37, 41, 42], which provided a quantitative measure of the changes in waveform across a discontinuity. Then the development of estimates of apparent dip was published, which provided a measure of change in reflector dip/azimuth across a discontinuity [29, 42–44]. Estimates of amplitude gradients provided a measure of changes in reflectivity across a discontinuity. More recently [37, 43], Al-Dossary and Marfurt used spectrally limited volumetric curvature to help predict fractures [45]. Skirius et al. used seismic coherence in carbonates in North America and the Arabian Gulf to detect faults and fractures [46]. Faults and channels could manifest themselves as abrupt or gradual changes of amplitude in 3D seismic volumes. When changes are abrupt, detecting them has been addressed via several methods – such as a coherence cube [36], an eigenvalue coherence cube [47], and a derivative method [43]. However, when amplitude changes associated with faults and channels are gentle and gradual, illuminating possible faults and channels with computer-based algorithms has been a challenge. In detail, the coherence cube method developed by Bahorich and Farmer consisted of a volume, or cube, of coherence coefficients within which faults were revealed as numerically separated surfaces [36]. In this volume, the 3D seismic data were binned into a regular grid. By calculating localized waveform similarity in both in-line and cross-line directions, estimates of 3D seismic coherence were obtained. Small regions of seismic traces cut by a fault surface generally had a different seismic character than the corresponding regions of neighboring traces. This resulted in a sharp discontinuity in local trace-to-trace coherence. Calculating coherence for each grid point along a time slice resulted in lineaments of low coherence along faults. When this process was repeated for a series of time slices, these lineaments became fault surfaces, even though fault plane reflections had not been recorded. Stratigraphic boundaries generate similar discontinuities. The technique may be employed to produce coherence horizon slice maps, or to transform a reflection amplitude 3D data volume into an entirely new volume or “cube” of coherence coefficients. Additionally, in the process of calculating coherence for a 3D data volume, faults became numerically separated from the data that surrounded them. In other words, faults generate surfaces of low coherence. Not only were these surfaces distinctly separated from neighboring data, they were also numerically separated, enabling them to be picked using horizon autotracing software. This fault autopicking technique enabled the interpreter to have a fault pick on every trace, which added significant detail to the fault interpretation. Luo et al. [43] then developed the “difference method” of processing seismic data that was able to identify faults and stratigraphic boundaries (such as sand channels) early in the interpretation process and without interpretational bias (see Figure 5.8). The fundamental concept of the difference method is to measure the difference between

Seismic Attributes

(a)

(b)

Figure 5.8 (a) Input time slice; (b) after applying the difference method of calculating coherence using Luo et al. algorithm. Source: [43]. (See color plate section for the color representation of this figure.)

seismic signals. Luo et al. assumed they had a signal A on the target trace and signal B on an adjacent trace; they calculated the difference between the two traces in the following way [43]: d=

∥A − B∥ . (∥A∥ + ∥B∥)

(5.22)

In addition to the difference method, they also developed a technique that calculates the spatial derivative of the instantaneous phase called the “derivative method,” which was a high-resolution edge-detection technique that identified the location of rapid lateral changes in instantaneous phase. These technologies enabled rapid

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and high-resolution evaluation of 3D seismic data at all stages of exploration and exploitation of reservoirs. 5.5.2 The Dip Attribute

One of the most challenging tasks in interpretation of seismic data is delineating seismic anomalies related to subsurface unconformities and noise interference, both of which may exist in a data volume. Developed to solve these challenges, seismic attributes are one of many tools used by petroleum explorationists to delineate faults and stratigraphic features that are difficult to map using standard amplitude seismic data. Some of the most important seismic attributes in interpretative work consist of quantitative estimates of dip and azimuth throughout the seismic volume. Pioneered by Dalley et al. [44], these estimates of reflector dip and azimuth (or vector dip) are the basic building blocks for all seismic attributes, as well as for structurally oriented filtering. Accurate dip and azimuth maps can thus highlight subtle faults as well as stratigraphic features that reveal themselves through differential compaction or subtle changes in the seismic waveform. Today, petroleum explorationists are able to calculate 3D data cubes of reflector dip and azimuth without explicitly picking a given horizon, as variability in reflector waveform and seismic noise can cause difficulties with attribute extractions made along picked horizons [48]. Among the earliest pioneers in estimating dip directly from seismic data were Picou and Utzmann [49]. In their work, they used a 2D cross-correlation scan over candidate dips on 2D seismic lines. Continuing work with dip and azimuth included papers published by Luo et al. [43] and Barnes [27], who described an alternative method of estimating vector dip based on a 3D extension of the analytic trace (or complex-trace) attributes. Further extension of this method was published by Barnes [50], who worked on another alternative approach based on the 3D complex-trace analysis that Scheuer and Oldenburg [51] originally had applied to velocity analysis, which improved stability and generated a smoother, more realistic image at the expense of some loss of lateral resolution. Additionally, Gersztenkorn and Marfurt [41] developed a true 3D scan by generalizing the semblance-based scan established by Finn and Backus [52] in order to generate a more robust means of estimating reflector dip. Alternatively, Bakker et al. [53] presented an estimate based on the gradient structure tensor by calculating the first eigenvector of the gradient structure tensor. However, this method sometimes produced erratic and uninterpretable orientations when noisy data were encountered. To overcome this difficulty, Luo et al. [17] incorporated a data-adaptive weighting function to reformulate the gradient structural tensor. They adopted the squared instantaneous power as the weight factor, which simplified the computation when the instantaneous phase was used as input. Their weighted structural tensor method effectively eliminated erratic dips produced by unweighted methods and allowed the use of either the instantaneous phase or amplitude as input for computing orientations, thus producing better results that were suitable for further deriving of curvature or other seismic attributes. No matter how dip and azimuth data volumes are calculated, they are valuable interpretation tools. Currently, the most important function of dip and azimuth volumes is to define a local reflector surface upon which petroleum explorationists can estimate a measure of discontinuity or, conversely, along which they filter the data to

Seismic Attributes

Figure 5.9 Pixel operation to find the dip angle.

I(i, j)

I(i, j+1) α

I(i+1, j+1)

extract their continuous components. Previous papers published with work containing dip and azimuth volumes include the various coherence and edge-detection measures published by Marfurt and co-workers [41, 42] and Luo and co-workers [29, 43], and conventional f –x–y deconvolution and structurally ordered filtering by Höcker and Fehmers [54] (also known as edge-preserving smoothing). To simplify the dip computation, Figure 5.9 shows how to find the dip angle 𝛼 using the geometric concept by evaluating the trigonometric tangent function: tan(𝛼) =

I(i + 1, j + 1) − I(i + 1, j) , I(i, j + 1) − I(i, j)

(5.23)

where 𝛼 is the dip angle, which is the arctan of 𝛼. Figure 5.10 is a simple demonstration of the dip attribute.

5.6 The Curvature Attribute While coherence and amplitude gradients can often detect lineaments, reflector curvature is more directly linked to fracture distribution [55–57]. Spectral analysis of curvature provides a powerful tool to detect the edges of features such as faults, fractures, and channels, while also providing alternate perspectives of the same geology by analyzing different wavelengths. One of the most accepted geologic models is the relation between reflector curvature and the presence of open and closed fractures. Such fractures, as well as other small discontinuities, are relatively small and below the imaging range of conventional seismic data. Depending on the tectonic regime, structural geologists link open fractures to either Gaussian curvature or to curvature in the dip or strike directions. Reflector curvature is fractal in nature, with different tectonic and lithologic effects being illuminated at the 50 m and 1000 m scales [58]. Hart et al. [59] and Melville et al. [60] used horizon-based attributes, including various curvature attributes, to identify structural features that may be associated with fracture-swarm sweet spots. Stewart and Wynn [61] pointed out that it may be necessary to examine curvature at various scales to account for different wavelengths, which was later initiated by Wynn and Stewart [62] and Bergbauer et al. [57] on interpreted horizons. While his paper also dealt with curvature computed from interpreted horizons, Roberts [56] anticipated that volumetric estimation of reflector curvature should be possible. In 2006, Al-Dossary and Marfurt [58] developed the first volumetric spectral estimates of reflector curvature. They discovered that the most positive and negative curvatures

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(a)

(b)

Figure 5.10 (a) Input time slice. (b) Dip attribute slice computed using Equation 5.23. (See color plate section for the color representation of this figure.)

are the most valuable in the conventional mapping of lineations – including faults, folds, and flexures. Curvature is mathematically independent of, and interpretatively complementary to, the well-established coherence geometric attribute. They also concluded that the long spectral wavelength curvature estimates were of particular value in extracting subtle, broad features in the seismic data, such as folds, flexures, collapse features, fault drags, and under- and overmigrated fault terminations. In addition, domes appeared as positive anomalies on most negative curvature, while bowls appeared as negative anomalies on most positive curvature. The shape index, when co-rendered with curvedness, allowed one to visualize 3D reflector morphology on simple time slices. Finally, they inferred that reflector rotation was a good indicator of data quality, as well as scissor movements along faults. Through their experiments, Al-Dossary and Marfurt had developed a means by which powerful seismic attributes

Seismic Attributes

that could previously be applied only to interpreted horizons could now be applied to the entire uninterpreted volume of seismic data [58].

5.7 Curvature of the Surface In this section we review the basic concepts that are related to the geometry of a surface. If we consider an image z = f (x, y) as a surface in 3D space, then the geometry of this surface may be utilized to drive the curvature of the surface itself. We begin with a short review of quantities related to the geometry of curves. Then we focus on the geometry of a curved surface, such as tangent planes, normal curvature, and the principal curvatures. The study of the geometric shapes of curves and surfaces is the subject of differential geometry. For a more comprehensive introduction to differential geometry in 3D Euclidean space, please refer to Differential Geometry and Its Applications by Oprea [63] and Elementary Differential Geometry by Pressley [64]. 5.7.1 Curve, Velocity, and Curvature

We start from a curve in 3D space. A curve is essentially a 1D entity that maps a “time” variable to a position in 3D space. The instantaneous direction of a point on a curve can be represented by the velocity (or tangent) vector. Curvature measures the deviation of a curve from being a line. Curvature measures one of the two important features of a curve. The other quantity is torsion, which measures the deviation of a curve from lying in a plane. Torsion is not related to our current project, so we omit it here. The calculation of curvature is related to the second-order derivatives of the curve with respect to the parameter “time.” Indeed, if we reparameterize the curve function using the arc length as the “time” parameter, then the curvature is exactly the second-order derivative. For illustration of the curvature of a curve, we use a demo project from Mathematica. The curvatures of a curve at different points are different. If we use the inverse of the curvature as the radius of an osculating circle, then we can visualize the curvature in terms of this osculating circle. Please see Figure 5.11. k1 < k2 k1 =

1 R1

k2 = R1 A

1 R2 R2 B

Figure 5.11 Illustration of curvature of a curve. At point A on the left side the curvature is small; hence, the curvature radius is large. At point B on the right side the curvature is large; hence, the curvature radius is small.

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5.7.2 Surface, Tangent Plane, and Norm

A surface is essentially a 2D entity in 3D space. It is usually represented as a mapping from 2D to 3D; that is, from (u, 𝑣) ∈ R2 to (x, y, z) ∈ R3 . One such surface can be seen in Figure 5.12. The geometry (or shape) of the surface can be described by the change of its local direction. While the local direction of a curve can be represented by the tangent (or velocity) vector, the local direction of a surface can only be represented by the direction of the tangent plane. tangent plane P

Figure 5.12 A surface along with a tangent plane at a given point P.

κη

N Xμ

μ

Figure 5.13 Illustration of the idea of normal curvature.

Seismic Attributes

For a given point on a surface, we may have different curves passing through this point. See, for example, the point marked on the surface in Figure 5.12. Two curves passing through the same point obviously have different local directions. So each such curve may have its own tangent vector and these tangent vectors may be different. Hence, using the tangent vectors to describe the direction of a surface is not possible. Actually, these tangent vectors form a plane, the tangent plane. One tangent plane is shown in Figure 5.12. For a plane, its direction is measured by the norm direction, because this is unique. The norm vector that is perpendicular to the given tangent plane is shown in Figure 5.13. If we ask how the local surface changes along different directions, then the equivalent question is: How does the norm change along different directions? So the shape of a surface can be described by describing how this normal vector changes along different directions. This is done by the so-called shape operator.

5.8 Shape Operator, Normal Curvature, and Principal Curvature 5.8.1 Normal Curvature

Now, the picture in our mind about the surface should be the following. For a given point on the surface, we may have different curves passing through this point; hence, we may have different tangent vectors, and all these tangent vectors form a plane. This plane is characterized by its direction, the norm direction. Next, we may be interested in how the surface bends along different directions. How to find the bend? Referring to Figure 5.13, given a direction, say 𝜇, we can find the tangent vector along this direction, say X𝜇 . This tangent vector and the norm of the tangent plane form a plane that is perpendicular to the tangent plane (see the vertical plane in Figure 5.13). Then the intersection of this “normal” plane with the surface will be a curve. The curvature of this curve is what we call the normal curvature along the direction u. In Figure 5.13, the osculating circle is also drawn and the radius of this circle is also illustrated. 5.8.2 Shape Operator

You can think about the shape operator  as a little machine, to which you can feed a vector v, and it outputs another vector w. The input vector is the direction you are interested in. For example, if you are interested in the change of surface shape along the x-direction, then the vector v is the unit vector pointing to the positive x-direction. The output w is another vector, which tells you the change of the norm vector along the given direction 𝑣. Fortunately, these two vectors reside in the same tangent plane  . If we choose a basis for this tangent plane, then we can use a matrix S to represent the shape operator. We call it the shape matrix here. For example, if we choose two vectors corresponding to the tangent vectors of the curves along the u axis and the 𝑣 axis – that is, xu and x𝑣 – then the first column of the matrix S is the column vector (xu ). This (xu ) tells us how the shape of the surface changes along the direction of the u axis. Similarly, the second column of the matrix S is the column vector (x𝑣 ). This (x𝑣 ) tells us how the shape of the surface changes along the direction of the 𝑣 axis. Similar to the case of

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calculating the gradient of a scalar field, if you know the change of the field along the x-direction and along the y-direction, then you can tell the change along any direction; that is, the directional derivative. Here, if we know the shape of the surface along the two directions, then we can tell the shape along any direction. So the shape matrix S tells us everything we need about the shape of the surface: [ ] S(xu ) S= . (5.24) S(x𝑣 ) The connection between the shape operator and the normal curvature is the following. The dot product between the output of the shape operator (u) and the direction u is the normal curvature. We may think about this as projection of the output of the shape operator (u) onto the direction u. This connection coincides with our intuition: the larger the change of the norm direction when you travel along the direction u, the larger the curvature. 5.8.3 The Principal Curvatures

Since, along different directions on the surface, the normal curvatures are different, we may need to know along which direction the bend of the surface is the maximum and along which direction the bend of the surface is the minimum. The two extreme values of curvature are called the principal curvatures, and the two corresponding directions are called the principal directions (Figure 5.14). Then the question is how to find the principal curvatures and the principal directions. If the shape operator tells us the curvature along the two basis directions, then we can rotate the basis vectors (or the coordinate system) to find out the maximum curvature direction and the minimum curvature direction. Rotating the coordinate system can be done by eigenvalue decomposition (EVD). It turns out that the two principal curvatures are the two eigenvalues of the shape operator, and the two principal directions correspond to the two eigenvectors of the shape operator. So, in principle, we can find the principal curvatures by calculating the EVD of the shape matrix. Even though the principal curvatures can be found by EVD in principle, we have to have the shape operator matrix first. This matrix depends on the basis we choose for the tangent plane. In fact, we do not need to do this. If an image is given, and the first-order and second-order partial derivatives can be estimated, then we can calculate the principal curvatures from these partial derivatives. 5.8.4 Calculation of the Principal Curvatures

In this section we focus on the calculation of the principal curvatures from a given image. From the previous discussion, we know that the principal curvatures are the eigenvalues of the shape operator. But the calculation of principal curvatures can be done directly from the derivatives of the image intensity with respect to the axes. First, we summarize the process of calculation of the principal curvature for a surface with analytic expression, and then we study the estimation of principal curvatures from a real image.

Seismic Attributes

Figure 5.14 The principal curvatures and the principal directions. For (a), along the direction of vertical plane P1, the curvature is the maximum. For (b), following the direction along the vertical plane P2, the curvature is the minimum.

vertical plane P1

maximum curvature

(a) vertical plane P2

minimum curvature

(b)

5.8.5 Summary of Calculation of Principal Curvature for a Surface

If we have a surface z = f (x, y) in 3D space, then the surface can be represented as a level surface g(x, y, z) = z − f (x, y) = 0. From this surface, we can obtain the partial derivatives of f (x, y): fx , fy . Then the principal curvatures can be calculated using the algorithm listed in Table 5.1 [64, 65]. The coefficients for the first fundamental forms are essentially the inner products between the tangent vectors along the axes. The coefficients for the second fundamental forms are essentially the second-order partials projected in the direction of the surface norm.

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Table 5.1 Algorithm for calculating the principal curvatures, Gaussian, and the mean curvatures. Inputs:

Surface z = f (x, y)

Outputs:

Principal curvatures: c1 and c2

1

Calculate the coefficients for the first fundamental form: E = 1 + fx2 ,

2

fxx

1+fx2 +fy2

,

M= √

fxy

1+fx2 +fy2

,

N= √

fyy

1+fx2 +fy2

Calculate the mean curvature: H=

4

G = 1 + fy2

Calculate the coefficients for the second fundamental form: L= √

3

F = fx fy ,

fxx (1+fy2 )−2fxy fx fy +fyy (1+fx2 ) 2(1+fx2 +fy2 )3∕2

Calculate the Gaussian curvature: K=

fxx fyy −fxy2 (1+fx2 +fy2 )2

5

Calculate the two principal curvatures from the mean curvature H and Gaussian curvature K: √ √ k1 = H + H 2 − K, k2 = H − H 2 − K

6

Output the curvatures: k1 , k2 , K and H

From the algorithm in Table 5.1, we see that even though the principal curvatures are eigenvalues conceptually, in the calculation we do not need any explicit eigenvalue decomposition computations.

5.9 The Randomness Attribute Estimates of the randomness or chaos attribute in seismic data can be beneficial to grade the reliability of seismic interpretations and, therefore, to assess the risk in new drilling proposals. Determining seismic quality is particularly important in locations where well costs are substantial, such as deep marine. In general, the chaos attribute used in seismic interpretations represents a measure of the chaotic or disordered quality from statistical analysis of local seismic responses; that is, abruptly changing signals are chaotic, but smoothly varying signals are not. Furthermore, it does not vary with seismic amplitude or dip orientation, meaning that the chaos value will be the same whether in a low- or high-amplitude or dipping or flat region of the seismic data volume [66]. The chaos attribute can also be tuned specifically for fault detection by taking the elongated, more or less vertical nature of faults into consideration, by estimating the chaos attribute within elongated vertical windows, or dip guided windows orthogonal to the dominating orientation [67]. Previously, Immerkær [68] proposed a fast noise variance estimation algorithm that averaged noise estimates at each pixel location to create a single (global) measure of image quality. In this section we will modify Immerkær’s technique to measure the spatial variation of chaos and to output an estimate of the randomness attribute at each pixel location (similar to other seismic attributes).

Seismic Attributes

The randomness attribute is a well-known concept in image processing, and it is known as noise variance. Since noise variance estimation (randomness attribute) is an inseparable part of many image-processing algorithms, most of the noise variance estimation algorithms are presented as only part of larger tasks. A comprehensive comparison of the noise variance estimation methods can be found in Olsen [69]. Olsen surveyed six methods for estimating the variance of additive white noise in an image, and concluded that the more effective way for noise variance estimation is to suppress the image structure first and then estimate the noise variance. Most algorithms developed from then roughly follow this line. There are usually three basic steps in image noise variance estimation: (1) image structure suppression, (2) local variance estimation, and (3) global variance estimation. The most important step among them is the image structure suppression. Rank et al. [70] use a simple two-tap difference filter in both horizontal and vertical directions to suppress the image structure. It is found that the filters should be cascaded for better results. However, this simple two-tap filtering also leaves a lot of edge information in the filtered images, so special postprocessing is essential in the global variance estimation stage to correct the noise variance histogram. In order to better suppress the image structure, Immerkær used the difference of two Laplacian filters as the mask to filter the image [68]. However, some thin lines and edges in man-made scenes are not well suppressed by this filtering technique. To overcome this, Corner et al. [71] used both the “difference Laplacian” mask and a gradient operator to suppress the influence of the image structure. The Immerkær mask is used first to suppress the image structure, and then the gradient operator is used to detect the edge in the original image. When calculating the local variance, the edge points are excluded. In this way, the influence of the image structure can be further reduced [68]. We can easily apply these image-processing algorithms to seismic data processing.

5.10 Technique for Two-Dimensional Images This section presents the proposed mask design technique for an image. We will also prove the optimality of the original mask designed by Immerkær [68]. 5.10.1 Problem Statement and Preliminaries

The signal model of a noisy image contaminated by additive Gaussian noise is depicted in Figure 5.15. The observed noisy image y(m, n) is the additive combination of the clean image x(m, n) and the additive noise z(m, n), where the index range 1 ≤ m ≤ M and 1 ≤ n ≤ N. M and N are the number of rows and columns of an image respectively. The additive noise is assumed to be uncorrelated with the clean image x(m, n). In most cases, the additive noise is white, which means that the noise samples are uncorrelated with one another. Here, we are mainly interested in Gaussian noise. The reason is that this kind of noise distribution is common in various images. So algorithms developed Figure 5.15 Noisy image model.

z(m, n) x(m, n)

y(m, n)

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for Gaussian noise will be more general than those for other noise distributions. This form of noise is also found to be the worst noise from a communication perspective [72]. The mean of the noise is assumed to be zero. For situations with nonzero mean, the mean can be subtracted from the noisy image as long as the mean is a known parameter. We will denote the distribution of the noise as z(m, n) ∼ N(0, 𝜎z2 ). So the only parameter needed to specify the noise is the noise variance 𝜎z2 . The clean image contains various structures, such as flat regions, edges, and textures. For local structures, the ramp edges and step edges are the two main structures that will affect the accuracy of noise variance estimation. So, in the following development, we will consider the clean image as a combination of flat regions, ramp edges, and step edges. We will devise an algorithm to suppress the influence of these structures on the accuracy of noise variance estimation. The estimation of image noise variance belongs to the larger research field of statistical parameter estimation. So, it is important to know the best performance we can achieve in this task. Since the influence of the image structure can only deteriorate the estimation accuracy (i.e., it will not increase our estimation accuracy by introducing more noise), the best performance can be achieved when the influence of the structure is totally eliminated. Although this is not possible, it will give us an indication as to the upper bound of the best performance. We also want the estimator to be the class of unbiased estimator, which means that the expected value of our estimate will be the true parameter. With these considerations, the best estimation performance will occur if we consider the estimation of the variance of the pure noise z(m, n). The best performance of an unbiased estimator in terms of the variance of the estimator is given by the Cramer–Rao lower bound [73]: if the observed signal is of length N, and the samples of signal are independent and identically distributed as a Gaussian distribution with mean zero and variance 𝜎 2 , then the variance of the best unbiased estimator 𝜎̂ 2 of the signal variance is lower bounded by var(𝜎̂ 2 ) ≥

2𝜎 4 . N

(5.25)

Any unbiased estimator that achieves this lower bound is said to be efficient. The usual estimator ( )2 N N 1 ∑ 1 ∑ 2 z − 𝜎̂ = z (5.26) N − 1 n=1 n N m=1 m is the asymptotically efficient estimator for noise variance as N goes to infinity. Note that we have reindexed the noise samples z(m, n) as a 1D signal z(n) for simplicity. This estimator will be used herein for variance estimation. 5.10.2 Review of Fast Noise Variance Estimation Algorithm

The fast noise variance estimation (FNVE) algorithm proposed by Immerkær uses the difference of two Laplacian filters to suppress the influence of image structure [68]. The general form of the Laplacian filter is L𝛼 =

⎡ 𝛼 1 ⎢ 1−𝛼 1+𝛼 ⎢ 𝛼 ⎣

1−𝛼 −4 1−𝛼

𝛼 ⎤ 1 − 𝛼⎥ , ⎥ 𝛼 ⎦

(5.27)

Seismic Attributes

where 𝛼 is between 0 and 1. It can be verified easily that the Immerkær mask is the difference between L1 and L0 . Although the design of a 2D mask is easy, the extension of this mask to three dimensions is not straightforward. So we are interested in a systematic approach in designing the 3 × 3 mask of this kind for image structure suppression. In order to design the mask, we need to explore the good properties inherent in the Immerkær mask: 1 When filtering a constant gray level with the Immerkær mask, the output is zero. This is ensured by the zero sum property of the Immerkær mask entries. 2 When filtering the constant slope ramp with the Immerkær mask, the output is zero. This property is also guaranteed by the zero sum property of the Immerkær mask entries. 3 When the image has horizontal or vertical step edges, the output of the filtering is also zero. 4 When the image has diagonal step edges, the output of the filtering is not zero. Considering these properties, the first three are good properties and can be used to suppress image structures like constant gray level, ramp edges, and step edges. The fourth property is actually the main drawback of the Immerkær scheme. In Immerkær’s original paper, his experiments report that the Immerkær mask cannot suppress some thin lines. We found from our experiments that these thin lines are all diagonal lines. The simple 3 × 3 mask restricted the simultaneous suppression of both nondiagonal edges and diagonal edges. 5.10.3 Design Mask by Constrained Optimization

Instead of designing the mask for FNVE using the difference of two Laplacian masks, we will design the mask directly based on the desired properties. We first list the desired properties of the mask and then represent these requirements using algebraic equations. Based on the discussion in Section 5.10.2, the desired properties are: 1 The mask should be isotropic, which means that it should work equally well no matter what the direction of the image structure is. For example, it should work equally well for both horizontal and vertical ramp edges. This requirement suggests that the mask coefficients should be symmetric around the center block. 2 The mask coefficients should be zero sum. This property will guarantee that the mask will suppress a constant gray level and constant-slope ramp edge. 3 When the range of the mask contains a horizontal edge or vertical edge, the output of the filtering should be zero. The next step is to formulate the aforementioned requirements as an optimization problem. According to requirement (1), the mask should have the following structure: ⎛ x2 ⎜ x3 ⎜ ⎝ x2

x3 x1 x3

x2 ⎞ x3 ⎟ . ⎟ x2 ⎠

The center coefficient is marked by a square. The other coefficients are symmetric with respect to the horizontal, vertical, and diagonal lines through the center coefficient. So

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we need to determine three coefficients: x1 , x2 and x3 . The zero-sum requirement can be formulated as x1 + 4x2 + 4x3 = 0.

(5.28)

For ease of final noise variance estimation, we impose a fourth requirement that is not present in the Immerkær mask. 4 Noise variance conservation. The image noise is assumed to be independent Gaussian noise. When passing the Gaussian noise through the designed mask, the noise is no longer spatially independent but correlated noise. It is expected that the noise variance will not change. The requirement indicates that the sum of squares of the mask coefficients should be equal to one; that is: x21 + 4x22 + 4x23 = 1.

(5.29)

In order to show this, let us consider a 1D signal and filtering mask. The input noise nk is passed through a filtering mask with impulse response hk , where 1 ≤ K ≤ N. The noise samples are independent of each other; that is, E(nk nl ) = 𝜎n2 𝛿kl , where 𝜎n2 is the variance of the noise, 𝛿kl is the delta function which equals one if k = l and zero otherwise. The filtering mask is symmetric, so that convolution can be replaced by correlation. The output signal variance can be expressed as ) (N N ∑ ∑ nm+k hk nl+k hl var(ym ) = E ( =E

k=1

l=1

N N ∑ ∑

)

nl+k hl nm+k hk

k=1 l=1

=

N N ∑ ∑

E(nl+k hl nm+k hk )

k=1 l=1

=

N N ∑ ∑

hk hl E(nl+k nm+k )

k=1 l=1

=

N N ∑ ∑

hk hl 𝜎n2 𝛿kl

k=1 l=1

=

𝜎n2

N ∑

h2k .

k=1

So, in order to maintain the noise variance after filtering, we must design the filter mask so that N ∑

h2k = 1.

(5.30)

k=1

With these constraints, our goal now is to minimize the response of the mask to horizontal and vertical edges. We now illustrate the two cases shown in Figure 5.16. This figure only considers an edge along the vertical direction. Since the mask we are designing is isotropic, this situation is the same as when the edge is along the horizontal

Seismic Attributes

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direction. In the first case, the center of the mask is on the “1” side, as illustrated by the left image in Figure 5.16. The filter output in this case is y1 = x1 + 2x2 + 3x3 .

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By this formulation, the goal is to minimize the sum of squares of y1 and y2 subject to the equality constraints listed earlier: min f (x1 , x2 , x3 ) = (x1 + 2x2 + 3x23 + (2x2 + x3 )2 subject to x1 + 4x2 + 4x3 = 0 x21 + 4x22 + 4x23 = 1 This optimization problem can be solved by a Lagrange multiplier or by numerical minimization searching [74]. The solution1 is found to be ) ( 2 1 1 . (5.33) x = − ,− , 3 6 3 Comparing this result with the Immerkær mask, this mask is actually the normalized version of the Immerkær mask. So, we can conclude that the Immerkær mask is the optimal mask under the criterion presented in this section [68]. The randomness estimation algorithm based on the aforementioned mask can be used to assess data quality and visualize the spatial distribution of noise, which helps identify drilling complications such as gas chimneys, reservoir-controlling features such as fracture zones, and the boundaries of merged seismic surveys; see Figure 5.17.

5.11 The Spectral Decomposition Attribute Spectral decomposition is the process of decomposing a seismic signal into its Fourier components. 1 Exercise on computing the randomness attribute.

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Figure 5.17 (a) Input time slice. (b) Randomness attribute slice computed using Immerkær mask. Note that in image (b) the noise is more emphasized than the channels. (See color plate section for the color representation of this figure.)

Spectral decomposition can be a powerful aid to the imaging and mapping of bed thickness and geologic discontinuities [75]. Partyka et al. [75] developed a spectral decomposition algorithm that utilized seismic data and the DFT for imaging and mapping of temporal bed thickness and geologic discontinuities over large 3D seismic surveys. By transforming the seismic data into the frequency domain via the DFT, the amplitude spectra delineated temporal bed thickness variability, while the phase spectra indicated lateral geologic discontinuities. This technology has delineated stratigraphic settings (such as channel sands and structural settings involving complex fault systems) in 3D surveys.

Seismic Attributes

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Figure 5.18 (a) Seismic time slice. (b–d) Single-frequency time slices of (b) 30 Hz, (c) 50 Hz, and (d) 100 Hz. The channels are clearly visible on the 100 Hz time slice. (See color plate section for the color representation of this figure.)

Figure 5.18a shows the time slice with the channels. Figure 5.18b, c, and d show the single-frequency time slices generated at 30 Hz, 50 Hz, and 100 Hz respectively using DFT.

5.12 Summary Certain image-processing techniques can be used to extract a specific quantity derived from seismic data. These extracted pieces of information are called seismic attributes. These can be analyzed in order to enhance information for better geologic or geophysical interpretation of the data. A seismic attribute can help to identify the presence or absence of subsurface geologic features, such as faults and channels, that can be important for hydrocarbon passage and accumulation. Actually, there are many distinct seismic attributes, but in this chapter we have covered the following: edge detection, coherence, dip, curvature, randomness, and spectral decomposition. For example, edge detection can identify faults and channels. Based on this concept, certain image-processing techniques can be used to enhance certain features in the data, and they can provide the basis for the automation of certain interpretation tasks.

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5.13 Problems and Computer Assignments 5.1

Apply the simple finite-difference method to the matrix given in Figure 5.19a. 4 5 0 5 3

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Figure 5.19 (a) Matrix for Problems 5.1–5.5. (b) Matrix for Problem 5.6.

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Apply the following edge-detection algorithms to the matrix given in Figure 5.19a. Give the results in (i) the horizontal direction, (ii) the vertical direction, and (iii) both directions combined using: (a) Roberts’s method; (b) Prewitt’s method; (c) the Sobel edge detector.

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Use the Kirsch compass masks to calculate the edges in all eight directions and then compute the resultant edge.

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Seismic Attributes

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Estimate the noise variance of the matrix of Figure 5.19b using: (a) the unbiased estimator in Equation 5.26; (b) the FNVE algorithm given in Equation 5.27.

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In this part we will tackle first-order edge detection using computer programs. (a) Solve Problems 5.1 and 5.2 using a computer program such as Matlab. (b) Modify the programs such that they accept any matrix as an input and output the resultant edges for each of the four methods (simple finite difference, Roberts’s, Prewitt’s, and Sobel). (c) Finally, develop functions that will accept any image as an input and output the edges.

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Write a computer program that will accept any image as an input and output the Kirsch compass masks for all directions as well as the resultant image.

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(a) From your experience, identify for each of the seismic sections in Figure 5.21 which of the aforementioned Sobel filters has been applied to the seismic section in Figure 5.20 to generate them. Why? (b) Which Sobel filters have been used to obtain edge attributes in Figure 5.22? Which of them provides better edge attributes? 5.10

Consider Figure 5.23 and answer the following. (a) Which is the most blurry seismic slice: Figure 5.23a, b, c, or d? (b) Match each seismic time slice in part (a) with its corresponding wavenumber magnitude spectrum from the numbered images in Figure 5.24. Seismic time slice

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6 Color Display of Seismic Images 6.1 Introduction Several ways exist of displaying seismic images, as explained earlier in the book. Some of the most commonly used displays are (a) the wiggle, (b) the variable-area display, (c) the wiggle–variable-area display, (d) the gray variable-density display, and (e) the color variable-density display. These can be used in different ways with different color maps to display not only images of seismic amplitudes but also attribute images (e.g., edges, complex trace envelope, and instantaneous frequency). Displaying colored seismic images and seismic attribute images along with color bars is extremely useful for interpreters because colors will reveal as much detail as contained in such images. This has been known in the image-processing community for many years, where two principle factors motivate the use of colors in image processing: (a) color is a powerful descriptor that often simplifies object identification and extraction from a scene, and (b) humans can discern thousands of color shades and intensities versus only two dozen shades of gray [76, 77]. Color image processing can be done as a full-color processing, where images are acquired with a full-color sensor (e.g., color digital cameras, TV cameras). There is also pseudo-color processing, where, similar to the case of seismic images and seismic attribute images, a color is assigned to a particular monochrome intensity or range of intensities. Humans and animals perceive colors in an object based on the nature of the light reflected from the object. A body that reflects light that is balanced in all wavelengths appears white to the observer, while a body that favors reflectance in a limited range of the visible spectrum exhibits some shades of color (e.g., green objects reflect light with wavelengths in the range of 500–570 nm), while absorbing most of the energy at other wavelengths. In general, light can be achromatic, where its only attribute is intensity. In terms of images, these are gray-scaled, where a gray level is a scalar measure of intensity that ranges from black to grays, and, finally, to white. Light also can be chromatic and spans the electromagnetic spectrum in the rage of 400–700 nm (Figure 6.1). Three quantities exist that describe the quality of a chromatic light source. The first quantity is the radiance, which is defined as the total amount of energy that flows from the light source. The second is luminance, which gives a measure of the amount of energy an observer perceives from a light source. Finally, brightness, the third quantity, embodies the achromatic notion of intensity and is considered a key factor in describing color sensation. At the same time, brightness is a subjective descriptor that is practically impossible to measure. It is fascinating to know that the human eye has 6–7 million color Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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Figure 6.1 Color wavelengths comprising the visible range of the electromagnetic spectrum. Source: https://9-4fordham.wikispaces.com/Electro+Magnetic+Spectrum+and+light. (See color plate section for the color representation of this figure.)

sensors called cones. Such cones can be divided into three principle sensing categories. Interestingly, 65% of the eye cones roughly correspond to the color red (700 nm), and about 33% of them correspond to the color green (546.1 nm). The remaining 2% of cones, which are considered to be the most sensitive ones, sense the color blue (435.8 nm). Hence, owing to the human eye absorption characteristics, colors can be seen as variable combinations of the so-called primary colors red, green, and blue. At the same time, no single color exists that is called red, green, or blue. Also, the three primary color wavelengths for the purpose of standardization do not mean that red, green, and blue components acting alone can generate all spectrum colors. The primary colors can be added to produce secondary colors of light known as magenta, cyan, and yellow. In this chapter, readers will be exploring color models and useful color bars used for seismic image display. Also, they will see various ways to color displays of seismic amplitude and attribute images. This includes overlying, mixing, and blending different displays.

6.2 Color Models and Useful Color Bars Three color models are commonly used in computer displays of seismic images and seismic attribute images. They are: (a) the red, green, and blue (RGB) model; (b) the cyan, magenta, and yellow (CMY), or sometimes known as the CMYK, model (K stands for black); and, finally, (c) the hue, saturation, intensity (HSI) model. Each color appears in its primary spectral components of red, green, and blue. The model is based on a Cartesian coordinate system, as shown in the color cube of the 3D subspace in Figure 6.2.

Color Display of Seismic Images

The Color Space

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Figure 6.2 A 3D subspace normalized color cube of the color space, where red, green, blue, cyan, magenta, yellow, black, and white are at the corners of the cube. Note also that the gray scale is only a diagonal linking the black corner to the white one. By coloring seismic images and seismic image attributes, one will be able to see more details than with gray-scaled ones. (See color plate section for the color representation of this figure.)

Notice the RGB values in the three corners, while black is at the origin and white is at the farthest corner from the origin. The cube is a normalized cube, so all the values of R, G, and B are in the range [0, 1]. The different colors in this model are points inside or on the cube. Colors, for convenience, can be defined by vectors extending from the origin, where each color vector is of three components. 6.2.1 The RGB Model

The RGB model is used on computer and television screens. Three component images (red, green, and blue) are fed into an RGB monitor. The number of bits used to represent each amplitude sample value or picture element (known in image processing as a pixel) in RGB space is called the pixel depth. Most ordinary monitors define values of R, G, and B in the range of [0, 255] for 8-bit integer values. Hence, each RGB color pixel is said to have a depth of 24 bits. The total number of colors in a 24-bit RGB image is (28 )3 = 16,777, 216 (see the example of an RGB color cube shown in Figure 6.2). 6.2.2 The CMY Model

Cyan, magenta, and yellow are secondary colors of light or the primary colors of pigments. When a surface coated with a cyan pigment is illuminated with white light, no red light is reflected from the surface. The cyan subtracts red light from reflected white light, which is itself composed of equal amounts of red, green, and blue light. Most devices that deposit colored pigments on paper, such as color printers, require CMY data input or perform RGB to CMY conversion internally. Assuming that the colors are normalized to be in the [0, 1] range, RGB to CMY can be performed using the following relation: ⎛ C ⎞ ⎛1⎞ ⎛ R ⎞ ⎜M⎟ = ⎜1⎟ − ⎜G⎟ . (6.1) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Y 1 B ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

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Equal mixtures of cyan, magenta, and yellow pigment generate a dark brown (muddy) color. This muddy color is commonly known to be the black color (K), which is added to form the CMYK color model. The range of human perception spanned by CMY is unfortunately smaller that that covered by RGB. Hence, printed copies of (seismic) images are sometimes inferior to what is displayed on computer monitors. 6.2.3 The HSI Model

It is interesting to note that humans describe a colored object by its hue, saturation, and brightness. Hue is a color attribute that describes a pure color (e.g., pure orange, pure red), while saturation provides a measure of the degree to which a pure color is diluted by white light. Brightness is a subjective descriptor that is practically impossible to measure, where it embodies the achromatic notion of intensity and is one of the key factors in describing color sensation. It is well known that intensity (gray level) is a most useful descriptor of monochromatic images, where it is measurable and easily interpretable. The HSI model decouples the intensity component from the color-carrying information (hue and saturation) in a color image. It is considered an ideal tool for developing image-processing algorithms based on color descriptions that are natural and intuitive to humans. Sometimes the HSI model is called the hue, lightness, and saturation model. The HSI model does not directly map to either computer monitors or printer hardware. The HSI model needs to be converted to the RGB (or CMY) model before the final display. The RGB model and CMY model are not well suited for describing colors in terms that are practical for human interpretation. Typically, the HSI model is defined as a double-sided cone, and seismic interpreters find it more useful to use the HSI model like the one shown in Figure 6.3. For example, the hue axis rotates through various colors at various angles. It is useful for mapping seismic attributes like the phase and the azimuth. The saturation axis (along the radius) is useful for dip–azimuth attribute images. Given a (seismic or attribute) image in RGB color format (with normalized values in [0, 1]), one can convert the colors from RGB to HSI using the following relations. The White (W)

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Figure 6.3 The HSI model defined by a double-sided cone. The hue axis rotates through various colors at various angles ranging from 0 to 360∘ so it can be used to map seismic attributes such as azimuth. The saturation axis is useful for dip–azimuth attribute images. The intensity is perpendicular to saturation. (See color plate section for the color representation of this figure.)

Color Display of Seismic Images

hue component of each RGB pixel is obtained using { 𝜃 if B ≤ G H= , 360∘ − 𝜃 if B > G where

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6.2.4 Useful Color Bars

Interpreters usually use either single- or dual-gradational color bars to map seismic data and their attributes, despite the fact that color maps used in seismic data interpretation vary from one software to another. Color bars can help the interpreter to intuitively remember which amplitudes represent high, low, zero, or intermediate values. Note that the addition of more colors might improve the resolution of the seismic (attribute) image but does not modify the overall appearance of a given attribute. The extra colors are merely interpolated between the original colors. Deviation from these simple color-display models can result in overly busy and difficult-to-interpret seismic images. One can classify the color bars into three main categories. The first is single-polarity color bars. These are used for various attributes, such as coherence, dip magnitude, amplitude, envelope, time, and structure. The second is dual-polarity (dual-gradational models) color bars that are routinely used for seismic display. They are very effective in displaying conventional seismic data, particularly when overplotting (overlaying) colored horizon and fault interpretations. The last category represents specialized color bars like the one used for contour appearance, such as the time–structure attributes or the one used for cyclic seismic attributes; for example, the phase, azimuth, and strike. Figure 6.4 shows a few examples of the three categories. The use of such color maps is shown on real seismic images in Figure 6.5, where variable-density displays using various color maps are shown for different amplitude and attribute images with suitable color maps.

6.3 Overlay and Mixed Displays of Seismic Attribute Images A seismic overlay is used with some color displays comprising attributes represented with the original seismic image. That is, the attribute of interest is plotted in color to form a background. Then the data are plotted in variable-area format and overlaid on the background attribute. This method is widely used in attribute displays. There are many variations of seismic overlays. A limitation in the overlay method is that, traditionally,

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Color Display of Seismic Images

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in wiggle–trace displays, the negative lobe of the seismic amplitudes is not displayed. So half of the data are not displayed. Plotting the negative lobes in gray is one solution. Figure 6.6 shows an example of both overlaying the positive peaks variable area and overlay of positive peaks and negative lobes in gray along with the instantaneous phase attribute of a seismic image. Mixed (sometimes called blended) displays attempt to render two seismic attributes simultaneously at all points. One technique is known as alpha blending. Assume that an interpreter is interested in blending two seismic attribute images, namely, SA1 and SA2 , which are represented in the RGB domain. The blended seismic image SAb is simply the weighted average of the two input seismic attribute images. Mathematically, this is given by Ab = 𝜆A1 + (1 − 𝜆)A2 .

(6.6)

where 0 < 𝜆 < 1. Figure 6.7 shows various examples of seismic attribute images mixed/blended with other seismic images and/or other seismic attribute images.

6.4 Summary Color displays, along with typical seismic displays, are important in seismic interpretation. The use of color bars with color maps reveals much more detail from seismic displays compared with gray-scaled, wiggle or variable-area displays. There are many ways to utilize colors and color maps for displaying seismic images and seismic attribute images. This chapter provided some of the commonly used and basic ones.

6.5 Problems and Computer Assignments 6.1

Indicate which of the following statements are true and which are false. (a) Color is a powerful descriptor that often simplifies object identification and extraction from a scene. (b) Humans and animals perceive colors in an object based on the nature of the light observed by an object. (c) Cyan, magenta, and yellow are secondary colors of light. (d) Humans describe a colored object by its hue, saturation, and brightness. (e) In the hue model, the intensity component is given by I = 1 + [(R + G + B)∕3]. (f ) Two colors with identical RGB values will always have the same spectra. (g) Two colors that produce identical cone responses will always have the same spectra. (h) Two colors with identical HSI values will always have the same spectra.

6.2

Assume that one is to replace the RGB color cube in Figure 6.2 by CMY color. What are the colors that would appear on the vertices of the new CMY color cube when displayed in a computer monitor?

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Based on the RGB image on the left in Figure 6.8, sketch the CMY and HSI images.

Color Display of Seismic Images

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Figure 6.8 Images for Problem 6.3.

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Using Matlab, select and display a seismic image in a variable density and investigate the use of various color maps such as single-polarity attributes as well as dual-polarity attributes. Discuss the main differences.

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Using Matlab, select a seismic image and compute its instantaneous amplitude. Overlay part (say 10% of the seismic traces) of the black positive peaks and gray negative lobes with the instantaneous amplitude. Repeat the same exercise but with the instantaneous frequency instead of the amplitude.

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Using Matlab, select a seismic image and display its variable density. Compute its instantaneous phase. Blend-mix parts from both the seismic images with the instantaneous phase at the same locations. Investigate the effect of varying 𝜆. Comment on your result. Repeat this exercise but with the dip magnitude instead of the instantaneous phase.

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7 Seismic Image Segmentation 7.1 Introduction Image segmentation can be defined as the process of dividing an image into different segments, regions, or clusters such that each region is nearly homogeneous, whereas the union of any two regions is not (see Figure 7.1). In image analysis, it serves as a fundamental step toward low-level vision, which is significant for object recognition such as seismic attributes and geologic structures and objects. In image segmentation, one is usually concerned with object boundaries, where once the boundaries are identified, the next step is to segment the object of interest. Various segmentation algorithms exist [25, 77–79]. When considering intensity (gray-scaled) images, such algorithms are generally based on (a) discontinuity, where the image segments are identified based on abrupt changes in intensity values (e.g., points, lines, edges), and (b) similarity, where the image segments are identified based on similar regions according to a set of criteria [25]. Image segmentation algorithms can also be classified into three major categories: (a) spatial segmentation, also referred to as region-based methods (similar to intensity images, discontinuity and similarity just described); (b) feature-space-based clustering; and (c) graph-based approaches. Feature-space-based clustering approaches capture the global characteristics of an image through the selection of the image features. Such features are usually based on the color or texture. Graph-based approaches, on the other hand, can be regarded as image perceptual grouping and organization methods based on the fusion of the feature and spatial information. Such approaches are based on several factors, like similarity, proximity, and continuation. A graph is partitioned into multiple components that minimize some cost function of the vertices in the components and/or the boundaries between those components. So far, several graph cut-based methods have been developed for image segmentation, such as the normalized cuts method [78]. As discussed earlier in the book, seismic attribute images provide another important means by which an interpreter can better analyze seismic images. Segmentation of seismic objects, such as salt diapirs or channels, is better performed on attribute images or a combination of attribute images. So, in this chapter, some of the basic and advanced image segmentation techniques will be used for seismic images. Such techniques will be applied on real seismic data.

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Figure 7.1 The basic principle of image segmentation. (See color plate section for the color representation of this figure.)

7.2 Basic Seismic Image Segmentation Several well-known and basic segmentation algorithms exist, such as the use of discontinuity information based on detection of points, lines, and edges. This is usually followed by thresholding, where two or more segments can be separated. In the case of seismic images, in many instances it is difficult to directly segment objects of interest such as salt diapirs, which are useful in velocity analysis, using the seismic image itself. The detection and thresholding can better be performed on attribute images. Depending on the application, several types of point, line, and/or edge detectors can be used (see Chapter 5). Consider the case of the seismic image containing a channel seen in Figure 7.2a. After applying some preprocessing (namely, high-boost filtering), the channel has been enhanced with the Laplacian spatial enhancement method, as seen in Figure 7.2b. The segmented channel is shown in Figure 7.2c, after performing thresholding that distinguishes the channel from the background. It is clear that there are some other sample values that were segmented along with the channel and that the channel itself was not perfectly segmented.

Seismic Image Segmentation

(a)

(b)

(c)

Figure 7.2 (a) A seismic slice image showing a channel, (b) the slice after applying some preprocessing enhancement, and (c) the channel after segmentation.

(a)

(b)

(c)

(d)

Figure 7.3 (a) A seismic image showing a salt diapir, (b) the eigenvector of the amplitude envelope attribute, (c) the boundary detected between the salt diapir and the layers above it, and (d) the segmented salt diapir. (See color plate section for the color representation of this figure.)

Another example of seismic image segmentation is the detection of the salt diapir seen in Figure 7.3a. The segmentation is performed based on the eigenvectors derived from the amplitude envelope attribute of the image, as seen in Figure 7.3b. After applying thresholding based on the amplitude envelope attribute image, the boundary (shown in red in Figure 7.3c) between the salt diapir and the layers above it is identified, allowing

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the salt diapir to be segmented, as shown in Figure 7.3d. Note the missing part near the top of the diapir, which requires better segmentation. Also note the spike that is apparent just above the right side of the salt diapir boundary.

7.3 Advanced Seismic Image Segmentation This section provides two examples of feature-based segmentation and graph-based segmentation. 7.3.1 Color-Based Segmentation

Color images carry much more information than gray-level ones (as stated in detail in Chapter 6) [76, 77]. It is well known that the use of color information is more advantageous than the use of only gray-scaled information due to the fact that colors are powerful descriptors that simplify object identification and extraction from a scene. As a result, great efforts have been made in recent years to investigate segmentation of color images due to demanding needs [76, 79–83]. By using a specific distance measure that ignores the spatial information, the feature samples are handled as vectors, and the objective is to group them into compact but well-separated clusters. Many color-based segmentation schemes exist, and most of them rely on clustering methods such as the k-means [25]. Here, the color segmentation scheme based on projection onto convex sets (POCS) will be discussed. Developed by Bregman [84] and Gubin et al. [85], POCS is one of the mathematical techniques where we can easily, in this case, express our desired color-based segmentation characteristics. In general, POCS can handle any number of constraints, including linear and convex types. It is additionally very flexible, since crucial requirements can be incorporated very conveniently. Finally, the resulting operators will satisfy all predefined constraint sets as long as the imposed constraints are not contradictory and mutually exclusive [86]. Owing to its aforementioned attractive properties, POCS has been used for many applications, including filter design [87–91], image recovery and restoration [92, 93], deconvolution [86, 94, 95], kernel synthesis for time–frequency analysis [96], inversion of cross-borehole imaging [97], image compression [86], and so on. The basic idea of POCS is as follows (see [98]). Given a reference color vector r, we wish to find all color vectors h ∈ I ⊆ H, where I is an image that is a subset of a given Hilbert space H, subject to certain constraints Ci ∈ H. We assume explicitly that the constraints Ci form closed convex sets in the Hilbert space. For m known properties there are m closed convex sets C1 , C2 , … , Cm . The segmentation requirement for h is that h ∈ C0 , where C0 denotes the intersection of all sets Ci . That is, C0 = C1 ∩ C2 ∩ · · · ∩ Cm . Given the sets Ci , we derive the associated projection operators PCi for each i = 1, 2, … , m based on the nearest neighbor rule. Now, the objective is to find a solution h within C0 that satisfies all constraint sets and therefore represents an acceptable solution. The solution can generally be found iteratively using the simultaneous POCS formula hk+1 =

m ∑ i=1

𝑤i PCi hk ,

(7.1)

Seismic Image Segmentation

where we start with an arbitrary h0 for the given image in this case. Note that the 𝑤i are ∑m positive constants such that i=1 𝑤i = 1. Iterations (if required) are stopped once the convergence rate drops below a predefined threshold. The final solution h will converge to a point in C0 , thereby satisfying all imposed image segmentation constraints as long as the constraints Ci are not mutually exclusive [88]. 7.3.1.1 The Imposed Constraints for the POCS Segmentation Method

The desired properties for obtaining colored segmented images are based on finding all statistically similar as well as close color vectors h to a given reference color vector r with respect to a given digital image. Moreover, we can model pixels in color images as 3D vectors in ℝ3 . Hence, given a digital image color vector in ℝ3 , the proposed constraint sets that describe the aforementioned characteristics can be defined as follows. 1 C1 is the set of all color vectors h that are close to a given reference color vector r with respect to the Euclidean norm (refer to Figure 7.4a): C1 = {h ∈ I ⊆ ℝ3 ∶ ||h − r|| ≤ 𝜖},

(7.2) B

B

r

θ

ε

r

ρ = cosθ

G

G

R

R (b)

(a) B

r C 0

G

R (c)

Figure 7.4 Assuming that we are in the RGB space, our proposed constraint sets are (a) C1 , which describes the set of all color vectors that are close to the given reference color vector r within a sphere of radius 𝜖, (b) C2 , which describes the set of all color vectors that are statistically similar to the given reference color vector r by a cone, where its correlation coefficient 𝜌 = cos 𝜃 and 𝜃 is the angle between any vector h and r, and (c) the solution set C0 = C1 ∩ C2 , which includes all color vectors that are close and similar to the given reference color vector.

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where 𝜖 is the required maximum distance between h and r. 2 C2 is the set of all color vectors h that are statistically similar to a reference color vector r; that is, C2 is the set of all correlated color vectors h to r with a correlation coefficient 𝜌 (see Figure 7.4b): { } ⟨h, r⟩ 3 ≥𝜌 , (7.3) C2 = h ∈ I ⊆ ℝ ∶ ||r|| ⋅ ||h|| where ⟨, ⟩ denotes the dot product between two given vectors in the defined Hilbert space ℝ3 . Note that 0 ≤ |𝜌| ≤ 1. Clearly, C1 and C2 are closed sets. Also, C1 represents a sphere in ℝ3 and, therefore, is convex. Similarly, C2 formulates a cone in ℝ3 and is also convex. By considering this case of digital image segmentation based on color vectors in a given colored image, the two sets should always be intersecting; therefore, we have a solution set, as seen in Figure 7.4c. Taking into account that we are only interested in color vectors which are similar and close to a given reference color, we can easily derive the associated projection operators of an arbitrary color vector h ∈ ℝ3 , where h ∉ Ci for i = 1, 2 by the use of the nearest neighbor rule. This is based on a Lagrangian formulation for each of the constraint sets defined in Equations 7.2 and 7.3 . We can derive the associated operators with each of the aforementioned constraint sets to give { h, if h ∈ C1 PC1 h = (7.4) 𝟎, otherwise. Note that, within a given image, PC1 finds the closest color vector h to the reference color vector r within a sphere of radius 𝜖 and ignores any other color vectors (refer to Figure 7.4a). In addition, the second projection is { h, if h ∈ C2 PC2 h = (7.5) 𝟎, otherwise. In this case, PC2 identifies the correlated color vectors h to r with respect to the correlation coefficient 𝜌 and ignores the other noncorrelated color vectors (refer to Figure 7.4b). Given a seismic image or a seismic attribute image, the amplitudes are converted to variable-density colors using indexed colors, which assign each amplitude an indexed color. After that, those amplitude/attribute values indexed colors are transferred into a 3D color domain of pixels such as the RGB domain. In this case, the POCS color segmentation algorithm will be as follows: 1 Regroup the pixels of the colored seismic (attribute) image into 3D color vectors h, where each vector component represents a given color component depending on which color space we are using. 2 Select the POCS 𝑤1 and 𝑤2 positive constants (where their sum is equal to one) for 𝜖 and 𝜌. Note that these parameters are data dependent and some testing must be done to select the specific values of those parameters. 3 Select from the image the reference color vector r. 4 Apply Equation 7.4 on all possible image color vectors h. 5 Apply Equation 7.5 on all possible image color vectors h.

Seismic Image Segmentation

(a)

(b)

(c)

(d)

Figure 7.5 (a) A seismic image showing a salt diapir, (b) and (c) the two selected color classes, and (d) the segmented salt diapir. Note that the POCS parameters were 𝜌 = 0.8, 𝜖 = 0.35, and 𝑤1 = 𝑤2 = 0.5. (See color plate section for the color representation of this figure.)

6 Apply Equation 7.1 , which yields the output vector h that satisfies both constraints in Equations 7.2 and 7.3 . 7 If the mean-square error between hk+1 and hk (Equation 7.1 ) is less than or equal to a certain threshold 𝛽, stop. Otherwise, repeat steps 2–6. After that, the interpreter can segment the object of interest. An example of segmenting the salt diapir from other layers (similar to the one in Figure 7.3) using the POCS color segmentation scheme is shown in Figure 7.5. 7.3.2 Graph-Based Segmentation

Over the years, many researchers have proposed various approaches to the generalized image segmentation problem. Among the famous approaches is the method of normalized cuts (Ncut ). This method aims at extracting the global impression of an image, where image segmentation is treated as a graph partitioning problem. The normalized cut criterion measures both the total dissimilarity between the different groups (image segments) and the total similarity within the groups. Generally, the normalized cut method

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relies on the following. The set of points in an arbitrary seismic image (feature) space is presented as a weighted, undirected graph, say G = (V , E), where V is a vertex set and E is a function of the similarity between the nodes of the image. Nodes of the graph are the points in the feature space, and an edge is formed between every pair of nodes. The weight on each edge 𝑤(i, j) is a function of the similarity between nodes i and j. A graph G = (V , E) is partitioned into two disjoint complementary sets A and B, B = V − A, by removing the edges connecting the two parts. The degree of dissimilarity between the two sets can be computed as a total weight of removed edges. The mathematical formulation of a cut is as follows: ∑ 𝑤(u, 𝑣). (7.6) cut(A, B) = u∈A,𝑣∈B

The problem now is to find an optimal partitioning of the graph G. If N is the cardinality of V , then 2N will be the number of possible partitions. There exists an efficient algorithm for finding a minimum cut [78]. The normalized cut computes the cut cost as a fraction of the total edge connections to all nodes as follows [78]: Ncut (A, B) = where, asso(A, V ) =

cut(A, B) cut(A, B) + , asso(A, V ) asso(B, V ) ∑

𝑤(u, 𝑣).

(7.7)

(7.8)

u∈A,𝑣∈B

The association value asso(A, V ) is the total connection from nodes A to all nodes in the graph. Note that the Ncut value will not be small for the cut that partitions isolating points, since the cut value will be a large percentage of the total connection from that set to the others. Equation 7.7 is minimized to partition a given image by considering the eigenvectors of the seismic image or its attribute. The eigenvector corresponding to the second smallest eigenvalue is the real-valued solution that optimally subpartitions the entire graph, the third smallest value is a solution that optimally partitions the first part into two, and so on. The computational cost is typically less than O(n3∕2 ), where n is the number of nodes in the graph. This shows that the Ncut method is computationally expensive. An example of applying the Ncut to seismic images is shown in Figure 7.6a in order to separate the salt diapir from the other layers. Clearly, the salt diapir was segmented much better than the other segmentation methods stated earlier in this chapter (Figure 7.6b and c). The number of partitions (segments) was equal to three, which required computing three eigenvector images of the seismic image. The last one was used to draw the line that separates the salt from the other layers.

7.4 Automatic Fault Extraction Seismic data sets typically contain a large number of faults at many different spatial scales, and knowing the location of faults and fault networks is critical to understanding a geologic system. Conventional approaches for picking faults are largely manual, and involve laborious hand-picking of discontinuities one fault at a time. This is time consuming, resulting in hundreds of man-hours of work, performed by trained geologists.

Seismic Image Segmentation

(a)

(b) 20 40 60 80 100 120 50 100 150 200 250 300 350 400 450 (d)

(c)

(e)

(f)

(g)

Figure 7.6 (a) A seismic image showing a salt diapir, (b) the computed edges, (c) the boundary of the segmented salt diapir, and (d) the segmented salt diapir. There were three Ncut eigenvector images that were used to segment the salt diapir from the other layers. These are seen in (e)–(g). (See color plate section for the color representation of this figure.)

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Seismic data cube (Amplitude)

Thinned fault property

“Semblance” property

Fault property

Flat fault property

Figure 7.7 The basic principle of image segmentation for automatic fault detection. (See color plate section for the color representation of this figure.)

Seismic Image Segmentation

Hence, automated fault tracking and extraction in 3D seismic data sets is desirable. In recent years, several methods have been proposed for automatic fault extraction by Randen et al. [67] and Gibson et al. [99]. Randen et al. [67] used fault-enhancing attributes to detect faults. They first created an attribute cube that captured the faults by highlighting local high energy along the fault surfaces and low energy everywhere else. However, the attributes that they utilized in their work produced relatively little connectivity within the surfaces. Additionally, intersecting faults and noise made surface extraction from these data difficult. They handled this challenge by conditioning the attribute cubes upon extraction of the surfaces. The conditioning allowed separation of the fault surfaces based on dip and azimuth, such that cubes with nonintersecting faults could be created. From these, the surfaces were extracted as connected components. Pedersen et al. [100] introduced the ant tracking algorithm. This uses the principles from ant colony systems to extract surfaces appearing like faults in very noisy data and write them as an ant track cube. This cube contains only what is likely to be true fault information. Jacquenmin and Mallet [101] proposed a method based on a cascade of two Hough transforms in order to extract fault surfaces. The basic idea of their algorithm is that the intersection of a fault by a series of (x, z) cross-sections is approximately a family of straight lines. Each of these straight lines is transformed into a point in a first parametric space due to the first Hough transform. For each fault, the set of points so obtained constitutes (approximately) a new straight line in the parametric space, which is then transformed into a point of a second parametric space through a new Hough transform. Reverse transformations then allow the rebuilding of each fault as a set of points. After extraction, each subset of points can be approximated by a surface patch to provide a 3D geometric representation of the associated fault. In their results, they showed that a series of 3D points associated with more or less planar patches corresponded to faults (Figure 7.7). In the case of approximately planar faults, their method was fully automatic. However, in the case of heavily curved faults, it was necessary to interactively merge some subsets of points extracted separately and corresponding to the same fault. Research continues in this field to automatically extract the shape of geologic structures such as faults and channels in which hydrocarbons are potentially trapped.

7.5 Summary Seismic image segmentation requires careful preprocessing. Most of the time, segmentation is preferably performed on the attribute images rather than the amplitude values directly. This chapter explored many basic and advanced segmentation algorithms and applied them to seismic images.

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Glossary Note. The following is a list of good references to most of the terms in this glossary: • Sheriff’s Encyclopedic Dictionary of Applied Geophysics [102]. • http://www.glossary.oilfield.slb.com/. Absolute sea level change Worldwide sea level change mostly due to deglaciation. Accommodation space Space in which sediments can be deposited. Acoustic impedance Seismic P-wave velocity multiplied by density of a rock. Additive Gaussian noise Statistical noise which is added to the signal. The probability density function is normal. Amplitude variation with offset (AVO) The variation in the amplitude of a seismic reflection with angle of incidence or source–geophone distance. Anticline A fold in which layers are convex upward. Arch and dome A concentric fold structure formed by upward stress. Asymptotically Approaching a value or curve arbitrarily closely. Barrier reef Long coral reefs usually formed at the edge between a continental shelf and slope (Figure G.1). Bright spot An increase of amplitude caused by hydrocarbon accumulation. Bubble effect In marine seismic acquisition, the gas bubble produced by an air gun oscillates and generates subsequent pulses that cause source-generated noise. Calcite Calcium carbonate mineral CaCO3 and the main constituent of limestones. Carbonate rock Sedimentary rock composed of mainly calcium and magnesium minerals. Cementation Precipitation of cement between rock grains. Check-shot survey The procedure of measuring directly the traveltime from a source on the surface to a geophone positioned in a well. Clastic rock Sedimentary rock composed of mainly fragments from previous rocks. Clay Aluminum silicate minerals and the main constituent of shales. Coherence attribute The attribute that measures the similarity of signals. Coherence volumes 3D data set that results from applying the coherence attribute to the 3D seismic data. Coherency Measure of the lateral change of the seismic waveform. Complextrace Trace (signal) that has an imaginary part as a result of a mathematical transformation. Convergent tectonic plate boundary This occurs when two plates collide. Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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Glossary

re Sho

at) ef fl

Sea le vel

(Re eef er r

Inn

Re lim ef est one

est

f cr

Ree

ter

Ou

ore f (F

)

f ree

ree

Figure G.1 Types of coral reefs. Source: http://pubs.usgs.gov/fs/2002/fs025-02/. (See color plate section for the color representation of this figure.)

Correlative conformity Extension of an unconformity where the geologic time break disappears. Crustal subsidence Down warping of the Earth’s crust due to sediment loading. Cubic spline A method of interpolation using up to cubic power. Curvature The amount by which a geometric object deviates from being flat or straight. Decimation In wavelets, is the downsampling. Density log A record of rock bulk density as a function of depth in a borehole. Diagenesis Alteration of a sedimentary rock after its deposition by physical and chemical processes. Diagenesis can enhance or destroy primary porosity. Differential geometry Discipline that uses differential calculus and linear algebra to study problems in geometry. Dim spot A decrease of amplitude caused by hydrocarbon accumulation. Dix velocity The seismic velocity of a layer determined from stacking velocities and time. Dolomite Calcium–magnesium carbonate mineral CaMg(CO3 )2 and the main constituent of dolomites. Dolomitization Transformation of calcite to dolomite through ion replacement of Ca ions by Mg ions. It can enhance porosity because dolomite’s volume is smaller than that of calcite. Fault dip angle The angle that the fault plane makes with the horizontal. Fault throw The amount of vertical separation of a layer generated by the fault. Finite-difference method A method of approximating a derivative by taking the difference of the function at two discrete points.

Glossary

Flat spot A horizontal seismic reflection attributed to an interface between two fluids, such as gas and water or gas and oil. Foot wall The side of a fault that lies below a dipping fault plane (see Hanging wall). Fresnel zone The area of a reflector from which most of the energy of a reflection is returned. Gas chimney A region of low-concentration gas escaping and migrating upward from a gas accumulation. Graben A down-dropped block bounded by normal faults. Grain solution It occurs in shallow limestones where water is rich in carbonic acids that can dissolve calcite grains. Hanging wall The side of a fault that lies above the fault plane (see Foot wall). Horizon slice A view of 3D seismic data showing points lying on the same horizon (reflector). Isotropic Having the same physical property no matter which direction it is measured. Kerogen Organic matter after initial burial in a source rock. It matures into bitumen, which produces oil or gas depending on the kerogen type. Klauder wavelet The autocorrelation of a Vibroseis sweep. Lithology The mineral composition of a rock. Magmatic intrusion Forceful emplacement of magma beneath sedimentary rocks causing them to rise and form convex-up structures. Meandering Extensive curvature of a river channel due to low ground relief. Migrated seismic data Seismic data after performing the seismic migration process. The main objective of migration is to move dipping reflectors to their correct positions to produce a more accurate image of the subsurface. Mistie Occurs when interpreted seismic data do not match results of drilling a well. Multiple (reflection) A reflection more than once from the same interface. Permeability Ability of a sedimentary rock to allow fluids to flow. Petroleum Crude hydrocarbon products, including oil and gas. Pinnacle and patch reefs Isolated, round, comparatively small coral reefs. Polarity reversals Occur when the cap rock has a slightly lower seismic velocity than the reservoir and the reflection has its sign reversed. Porosity Ability of a sedimentary rock to store fluids. Primary (reflection) The first reflection from a subsurface interface. P-wave without S-wave anomaly An anomaly in multicomponent seismic data sets that is observed only on P-wave data but not on S-wave data, indicating its fluid origin. Quartz Silicon oxide mineral SiO2 and the main constituent of sandstones. Radial and concentric normal faults These are often associated with arches and domes. Figure G.2 shows a plan view of a typical dome with prominent concentric faults. Random noise Irregular part of signal that is not coherent. Reflection Seismic wave that is reflected from an interface where there are changes in acoustic impedance. Reflectivity Reflection coefficient (reflectance). Reflectivity series A time series of reflection coefficients at normal incidence. Ricker wavelet A zero-phase wavelet that represents the second derivative of the Gaussian function.

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Glossary

3 km

N

Figure G.2 Horizon slice from a 3D seismic volume over a mud dome showing prominent concentric (T) normal faults (VSA source: Butler, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org). (See color plate section for the color representation of this figure.)

T

Rock matrix The solid part of a sedimentary rock. Seismic depth section Seismic section after converting its vertical axis from time to depth. Seismic horizontal resolution The ability to distinguish closely spaced features laterally on a seismic section. Seismic vertical resolution The ability to distinguish closely spaced reflections vertically on a seismic section. Seismic wavelength The distance between successive peaks or troughs of a seismic wave. Sonic log A record of rock P-wave velocity as a function of depth in a borehole. Speckle Granular noise that inherently exists in a signal. Stacked seismic data Seismic data after going through normal move-out correction, muting, and stacking processes. It represents an initial image of the subsurface. Stacking velocity The average velocity we get by fitting a hyperbola to the true time–distance curve of a primary seismic reflection.

Glossary

Static shift A constant vertical shift in a seismic trace due to lateral variations in shallow layers. Stratigraphy The study of rock strata and their classification. Subsampling Resampling of a signal at a coarser sampling interval. Synthetic aperture radar (SAR) Image representation form of radar data. Synthetic seismic data Seismic data generated by forward modeling on a computer. Tectonic Deformation and/or fracturing of rocks under stress. Time-average equation An equation that relates the seismic P-wave velocity V in a sedimentary rock to those of its solid matrix Vm and pore fluid Vf in terms of porosity 𝜙: 𝜙 1−𝜙 1 + . = V Vm Vf Time slice A view of 3D seismic data showing points having the same two-way time. Time-variant A physical property that changes with time. Unconformity Surface representing rock erosion or nondeposition. It represents a break in the geologic record. Vertical seismic profiling (VSP) Measurements of the response of a geophone at various depths in a borehole to sources on the surface. Welllog A record of one or more physical measurements as a function of depth in a borehole.

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k

(a)

k

k (b)

(c)

(d)

(e)

Figure 1.2 Various displays for a seismic section image: (a) wiggle display, (b) variable area display, (c) wiggle-variable area display, (d) gray-scaled variable density display, and (e) colored variable density display. Note that the vertical axis represents the two-way traveltime increasing downward, while the horizontal axis represents the distance from left to right.

Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

k

k

500 ms TWT

k 500 ms TWT

k

1 km

1 km (a)

(b)

Figure 2.7 (a) Seismic section showing normal faults; (b) interpreted section (VSA author: Butler, 2015; data courtesy of Fugro N.V.). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

k

(a)

(b)

Figure 2.8 (a) Seismic section showing a reverse fault; (b) interpreted section (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

k

2 km

2 km (a)

(b)

Figure 2.9 (a) Seismic section showing a flower structure associated with a strike-slip fault; (b) interpreted section (VSA author: Stewart, 2015). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

500 ms TWT

k

on-lapping basin fill

500 ms TWT

dip decreases up section

15 km

15 km

(a)

(b)

Figure 2.10 (a) Seismic section showing reflection termination at faults; (b) interpreted section (VSA author: Butler, 2015; data courtesy of Fugro N.V.). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

Figure 2.11 Seismic section showing reflection offset across faults (VSA author: Butler, VSA, 2015; data courtesy of Fugro N.V.). Source: Courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

500 ms TWT

k

1 km

500 ms TWT

(a)

1 km

(b)

Figure 2.12 (a) Stacked seismic section showing differential reflection dip across faults; (b) interpretation showing fault locations (VSA author: Butler, 2015; data courtesy of Fugro N.V.). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

(a)

k Agbada Formation

k

high amplitude on fold crest = Direct Hydrocarbon Indicator

detachment at top of Akata Shale

(b)

Figure 2.14 (a) Seismic section showing a fault producing a reflection; (b) interpreted section. (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

500 ms 2 km

(a)

k

artefacts in forelimb

single thrust strand

500 ms 2 km

(b)

Figure 2.15 (a) Seismic section showing seismic fault zone exhibiting amplitude loss; (b) interpreted section (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

(a) channel complex

fold located between strike-slip zones

channel complex

5 km

k

thrust anticline located by basement step?

(b)

Figure 2.16 (a) Seismic section showing compressional folds; (b) interpreted section (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

V = 2H (a)

k

k

V = 2H (b)

Figure 2.17 (a) Seismic section showing a fold generated by a salt diapir; (b) interpreted section (VSA author: Stewart, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

200 ms(TWTT)

N

S

10 km

(a)

200 ms(TWTT)

k

N

S

10 km

(b)

Figure 2.18 (a) Seismic section showing a compactional folds; (b) interpreted section (VSA author: Jackson, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

(a) ridge split by faults

frontal anticline

pitch out against flank

sand pitches out against ridge 5 km

k

k

(b)

strike-slip fault zone transpressional ridge

Figure 2.19 (a) Seismic section showing a mud diapir; (b) interpreted section (VSA author: Jackson, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

k

(a)

(b)

Figure 2.20 (a) A 3D seismic horizon slice showing channels; (b) interpreted slice (VSA author: Sylvester, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

A

Eroded anticline

Seafloor

Channel valleys

NW

SE

100 ms (twtt)

5 km

Base of channels sequence Normal faults

Figure 2.21 A 2D seismic section showing channels (VSA author: Torvela, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

Cretaceous k

k

Devonian Figure 2.22 A 2D seismic section showing channels (indicated by arrows) with false synclines below them due to low velocity of channel fill material (VSA author: Posamentier, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

Isolated Patch Reefs

1 km

100 ms Horizon “C” Horizon “B” Abundant Patch Reefs

Figure 2.23 A 2D seismic section showing patch reefs with strong reflections from reef tops due to high contrast with overlying layers (VSA author: Posamentier, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

Moat

~100 m

k

Debris aprons

1 km

5 km

Figure 2.24 A 3D seismic volume showing a patch reef (VSA author: Posamentier, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

(a)

ESE

WNW

k

k

3 km (b)

Figure 2.25 (a) A 2D seismic section of a truncation trap formed by an angular unconformity; (b) interpreted section (VSA author: Hunt, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

500 ms

5 km

(a)

500 ms

5 km

k

k (b)

Figure 2.26 (a) Seismic section showing downlaps and toplaps; (b) interpreted section (VSA source: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

S

500 ms(TWTT)

5 km

N

S

(a)

k

fan

500 ms(TWTT)

faulted topography associated with the Middle Jurassic-Early Cretaceous rift event N

(b)

5 km

S

Figure 2.27 (a) Seismic section showing onlaps; (b) interpreted section (VSA source: Jackson, 2015; data courtesy of CGG Veritas) Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

500 msTWT

k

5 km

(a)

k 500 msTWT

k

5 km

(b)

Figure 2.28 (a) Seismic section showing erosional truncation; (b) interpreted section (VSA source: Stewart, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

C H8 (~2 Ma)

B

C’

Pockmark

H7 (~5 Ma)

H6 (8.2 Ma)

H5 (~10 Ma)

H4 (16.5 Ma)

k

k

1s TWT

Figure 2.30 Seismic section showing parallel reflection configuration (VSA author: Jobe, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

20 km

Figure 2.31 Seismic section showing chaotic reflection configuration (VSA author: Butler, 2015; data courtesy of Fugro N.V.). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

500 ms

5 km

Figure 2.32 Seismic section showing prograding (sigmoidal) reflection configuration (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

k

(a)

mass transport complexes

examples of channel complexes

Bottom Simulating Reflector gas hydrates

?sand pinch-outs onto flank of anticline

high amplitude on fold crest = Direct Hydrocarbon Indicator

North

Agbada Formation

South

1 s TWT

k

slide scars

complex forelimb fold-thrust zone

5 km

detachment at top of Akata Shale

(b)

Figure 2.33 (a) Seismic section showing bright spot; (b) interpreted section (VSA author: Butler, 2015; data courtesy of CGG Veritas). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

k

k

1.8

Top Sand

OWC

2.2

Figure 2.35 Seismic section showing a flat spot due to an oil–water contact at about 2 s, as confirmed by drilling and logging results. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

k

k

k

k

1090

542 1000

500 900

450 800

400 700

350 600 2000

2248

k

2496

2744

2992

3240

3480

3704

Figure 2.37 Seismic volume showing P effect (left) but no S effect (right) due to fluid effects. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

k

k

k

s H1 0.5

1.0

H2

1.5

H3 H4 H5

2.0

H6 (a) km

0.5

k

k

1.0 1.5 2.0 2.5

(b)

Figure 2.40 (a) Synthetic seismic data generated from the velocity model in (b) using finite-difference modeling. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

k

k

s

0.5

1.0

1.5

2.0

(a) s

0.5

k

k 1.0

1.5

2.0

(b)

Figure 2.41 (a) Synthetic seismic stacked section derived from the velocity–depth model in Figure 2.40b. (b) Stacking velocity field derived from (a), where colored curves indicate picked primary reflections. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

k

k

km

0.5 1.0 1.5 2.0 2.5

(a) km

0.5 1.0

k

k

1.5 2.0 2.5

(b)

Figure 2.42 (a) Velocity–depth model derived from the time and stacking velocities in Figure 2.41b using the Dix method for time-to-depth conversion. (b) True velocity–depth model. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

k

k

km

0.5

1.0

1.5

2.0

2.5

(a) s

k

0.5

k

1.0

1.5

2.0

(b)

Figure 2.46 (a) Velocity–depth model showing lateral velocity variation in the near-surface layer; (b) synthetic stacked seismic section derived from this model. Note the false structural high under the near-surface lateral velocity increase. Source: Yilmaz, http://wiki.seg.org/wiki/Seismic_Data_Analysis. Used under CC-BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0/deed.en [1].

k

k

4000.0

5000.0

k

k

6000.0

Figure 2.51 Image for Problem 2.3.

2

3

+

TWT (sec)

4

Amplitude

5

−

Figure 2.52 Image for Problem 2.4.

k

200 ms (twtt)

k

SE

NW

Figure 2.53 Image for Problem 2.5.

k

k

Figure 2.54 Image for Problem 2.6.

k

k

1 km

1 Sec

2 Sec

3 Sec

4 Sec

4 Sec

Figure 2.55 Image for Problem 2.7.

5 km

k 500 ms

k

Figure 2.59 Image for Problem 2.11.

k

k

Figure 2.60 Image for Problem 2.12. 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

k

(b)

(a)

0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4 (c)

(d)

Figure 4.4 (a) A seismic image, (b) its instantaneous amplitude, (c) its instantaneous phase, and (d) its instantaneous frequency.

k

k

k

60 50 40 30 20 10

(a)

(b) 0.5 0.4 0.3 0.2 0.1 0 –0.1 –0.2

k

–0.3 –0.4 –0.5 (c)

(d)

Figure 4.5 (a) A seismic image, (b) its instantaneous amplitude, (c) its instantaneous phase, and (d) its instantaneous frequency.

k

k

k

(a)

(b)

k

k (c)

(d)

Figure 5.2 Edge detection example: (a) original image, (b) result of Sobel horizontal kernel, (c) result of Sobel vertical kernel, and (d) gradient magnitude.

k

k

(a)

(b)

k

k

(c)

(d)

Figure 5.3 Edge detection example: (a) original image, (b) result of Prewitt horizontal kernel, (c) result of Prewitt vertical kernel, and (d) gradient magnitude.

k

k

(a)

(b)

k

k (c)

(d)

Figure 5.4 Edge detection using the Canny edge detector: (a) original image, (b) default values, 𝝈 = 2, T1 = 0.01, T2 = 0.1, (c) 𝝈 = 1, T1 = 0.01, T1 = 0.01, and (d) 𝝈 = 0.5, T1 = 0.11, T2 = 0.2.

k

k

(a)

(b)

k

k (c)

(d)

Figure 5.5 Edge detection using the LOG edge detector: (a) original image, (b) default value (𝝈 = 0.0034), (c) 𝝈 = 0.005, and (d) 𝝈 = 0.0078.

k

k

(a)

k

k

(b)

(c)

Figure 5.7 (a) Input of time slice seismic data; (b) after applying the Laplacian operator (Equation 5.20); (c) after applying the Laplacian operator (Equation 5.21).

k

k

(a)

k

k

(b)

Figure 5.8 (a) Input time slice; (b) after applying the difference method of calculating coherence using Luo et al. algorithm. Source: [43].

k

k

(a)

k

k

(b)

Figure 5.10 (a) Input time slice. (b) Dip attribute slice computed using Equation 5.23.

k

k

(a)

k

k

(b)

Figure 5.17 (a) Input time slice. (b) Randomness attribute slice computed using Immerkær mask. Note that in image (b) the noise is more emphasized than the channels.

k

k

Pos. 0.0 Neg. (a)

High

Low (b)

k

(c)

(d)

Figure 5.18 (a) Seismic time slice. (b–d) Single-frequency time slices of (b) 30 Hz, (c) 50 Hz, and (d) 100 Hz. The channels are clearly visible on the 100 Hz time slice.

Increasing energy

Increasing wavelength 0.0001 nm 0.01 nm Gamma rays

10 nm X-rays

Ultraviolet

1000 nm

1 mm

Infrared

1 cm

1m

100 m

Radio waves Radar TV FM

AM

Visible light

400 nm

500 nm

600 nm

700 nm

Figure 6.1 Color wavelengths comprising the visible range of the electromagnetic spectrum. Source: https://9-4fordham.wikispaces.com/Electro+Magnetic+Spectrum+and+light.

k

k

k

The Color Space

Blue (B)

Cyan (C) White (W)

Magenta (M)

Black (K)

Green (G)

Red (R) Yellow (Y)

Figure 6.2 A 3D subspace normalized color cube of the color space, where red, green, blue, cyan, magenta, yellow, black, and white are at the corners of the cube. Note also that the gray scale is only a diagonal linking the black corner to the white one. By coloring seismic images and seismic image attributes, one will be able to see more details than with gray-scaled ones.

White (W)

k

k Green (G) Cyan (C)

Yellow (Y)

Intensity Saturation

Hue

Blue (B)

Red (R)

Magenta (M)

Black (K)

Figure 6.3 The HSI model defined by a double-sided cone. The hue axis rotates through various colors at various angles ranging from 0 to 360∘ so it can be used to map seismic attributes such as azimuth. The saturation axis is useful for dip–azimuth attribute images. The intensity is perpendicular to saturation.

k

k

Positive

0

180

Negative Single polarity attribute color maps

0

360

A gradational color A dual polarity A cyclical color map attribute color map map with dual polarity attribute color map

Figure 6.4 Examples of commonly used color bar maps for various seismic attribute images.

0.5 0.4 0.3

k

0.2 0.1 0 0.1 0.2 0.3 0.4 0.5

(a)

(b) 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

(c)

(d)

Figure 6.5 Variable-density displays using various color maps for (a) seismic image amplitudes with a gray-color map, (b) seismic image amplitudes in (a) but with a dual-property attribute color map, (c) the dip magnitude of (a) with a single-polarity gray-color map, and (d) the instantaneous amplitude of (a) with a single-polarity color map.

k

k

k

(a)

1.5

1.5

1

1

0.5

0.5

0

0

–0.5

–0.5

–1

–1

–1.5

–1.5

(b)

Figure 6.6 Overlay of the instantaneous phase attribute of a seismic image with (a) black positive peaks variable area and (b) black positive peaks and gray negative lobes variable area.

k

k

(a)

(b)

(c)

(d)

Figure 6.7 Variable-density displays using various color maps for (a) seismic image amplitudes, (b) its instantaneous amplitude, (c) a blend-mix of both (a) and (b) with 𝜆 = 0.65, and (d) part of (a) mixed with (b) at the same locations with 𝜆 = 0.65.

k

k

k

k

Figure 7.1 The basic principle of image segmentation.

k

k

(a)

(b)

k

k

(c)

(d)

Figure 7.3 (a) A seismic image showing a salt diapir, (b) the eigenvector of the amplitude envelope attribute, (c) the boundary detected between the salt diapir and the layers above it, and (d) the segmented salt diapir.

k

k

(a)

(b)

k

k (c)

(d)

Figure 7.5 (a) A seismic image showing a salt diapir, (b) and (c) the two selected color classes, and (d) the segmented salt diapir. Note that the POCS parameters were 𝜌 = 0.8, 𝜖 = 0.35, and 𝑤1 = 𝑤2 = 0.5.

k

k

(a)

(b) 20 40 60 80 100 120

(c)

50 100 150 200 250 300 350 400 450 (d)

k

k

(e)

(f)

(g)

Figure 7.6 (a) A seismic image showing a salt diapir, (b) the computed edges, (c) the boundary of the segmented salt diapir, and (d) the segmented salt diapir. There were three Ncut eigenvector images that were used to segment the salt diapir from the other layers. These are seen in (e)–(g).

k

k

Seismic data cube (Amplitude)

k

“Semblance” property

Fault property

Flat fault property

Thinned fault property

Figure 7.7 The basic principle of image segmentation for automatic fault detection.

k

k

k

re

Sho

f Ree ef (

flat)

Sea le vel

e er r

Inn

Re lim ef est one

est

f cr

Ree

ter

Ou

ore f (F

f) ree

ree

Figure G.1 Types of coral reefs. Source: http://pubs.usgs.gov/fs/2002/fs025-02/.

k

k

k

k

3 km

N

T

k

k

Figure G.2 Horizon slice from a 3D seismic volume over a mud dome showing prominent concentric (T) normal faults (VSA source: Butler, 2015). Source: courtesy of Virtual Seismic Atlas (VSA), 2015 (www.seismicatlas.org).

k

151

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2 3

4 5 6 7

8 9

10

11 12

13

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157

Index 2D sesimic lines 24, 100 3D seismic data 2, 24, 26, 27, 87, 98, 100, 143 3D seismic surveys 114 3D space 103, 104, 107

a acquisition and processing effects 48 additive Gaussian noise 109 additive noise 109 algae 26 amplitude 1, 3, 11, 27, 33–35, 84–88, 98, 100, 101, 108, 114, 123–125, 127, 130, 131, 135, 138, 143 analytical velocity functions 43 anisotropy 42 Arabian Gulf 19, 22, 98 asymptotically 110 AVO 35 azimuth 2, 98, 100, 101, 126, 127, 143

b bedding over bowtie 44

48

c cavities 9 center block 111 channels 2, 11, 24, 26, 27, 48, 53, 54, 56, 80, 87, 88, 91, 98, 101, 115, 133, 134, 143 chaos attribute 108 chaotic 33, 108

climate 29 coherence attribute 96 coefficients 98 cube 98 volumes 96 coherency 87 coherent noise 59 collapse features 102 compressive stresses 18 concordances 30 conformable and nonconformable dip 48 convolution 3, 6, 39, 80, 89, 112 corals 10, 26 correlated 112, 138 curvature 42, 44, 87, 98, 100–103, 105–108, 115

d deconvolution 2, 40, 101, 136 deep effects 44, 48 depositional environment 33, 34, 55 derivative 85, 87–90, 96, 98, 99, 103, 106, 107 deterministic deconvolution 40 differential compaction 19, 27, 100 sedimentation 27 differentiation 88 diffractions 11, 26 dip 10, 11, 27, 30, 35, 42, 44, 48, 98, 100, 101, 108, 115, 126, 127, 131, 143 direct hydrocarbon indicators (DHI) 27 directional derivative 106

Seismic Data Interpretation using Digital Image Processing, First Edition. Abdullatif A. Al-Shuhail, Saleh A. Al-Dossary and Wail A. Mousa. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

158

Index

disagreement between well-log and seismic data 35 discontinuity 88, 89, 98, 100, 133, 134 discrete Fourier transform (DFT) 78, 114 display variable area 3 variable density 3 wiggle 3 displaying seismic 3, 123 distal deposition 30 divergent 33 Dix method 42 double-inverse method 40 downdip 44 downlaps 30, 33 dynamite 40

fractures 11, 21, 89, 98, 101, 113 fracturing 8, 9

g gentler and wider convex-up structures 44 geologic 1–3, 7, 44, 87, 88, 91, 115, 133, 140, 143 geometric effects 44 spreading 2 geometry 103–105 geophone 2 geophysical 1, 88, 115 gradient 43, 88–91, 93, 98, 100, 101, 106, 109 Gulf of Mexico 19

h e East Texas 27 edge 26, 87–91, 93, 96, 99, 101, 109–113, 115, 116, 118, 121 edge normal 91 eigenvector 100, 135, 140 erosional truncations 30 explorationists 88, 100

f facies 27, 33, 34 false highs 44, 48 false lows 44 fault 10, 11, 19, 48, 87, 91, 98, 102, 108, 114, 115, 127, 140, 143 faults 11, 35, 44, 48, 88, 89, 96, 98, 100–102, 108, 115, 140, 143 finite difference 39 element 39 volume 39 first-order derivative 89 flat regions 108, 110 folding 10, 18, 19, 21, 26, 27 folds 11, 18, 19, 21, 51, 102 formation tops 35 Forties 21 forward modeling 35 fractal 101

highs beneath reverse faults horizons 100, 101, 103

48

i image analysis 88, 133 pattern recognition 88 processing 1, 2, 6, 10, 44, 87, 88, 96, 109, 115, 123, 125, 126 segmentation 87, 88, 133 Immerkær mask 109, 111–113 internal seismic forms 33 interpretation 1–3, 7, 10, 11, 27, 35, 41, 44, 49, 57, 87, 88, 96, 98, 100, 108, 115, 126, 127, 130 Intisar (Idris) 26 inverse modeling 40 inverse operator 40 inversion 40, 44, 136

l Laplacian filters 109, 110 layer dip 42 least-squares fitting 40 linear filter 80 localization 88, 90 lows beneath normal faults luminance 88, 123

48

m minimum-phase equivalent

40

Index

n

random noise 3, 59, 87 ray-tracing 39 rectangular 56 reefs 10, 11, 26, 27, 34, 48, 52 reflectance 88, 123 reflection coefficient 3 free 33 time 3 reflections 11, 26, 27, 29, 30, 33–35, 42, 44, 87, 98 reflectivity function 3 reflectors 42, 56, 96, 98, 100–102 reservoir 1, 2, 7, 8, 10, 11, 24, 44, 87, 88, 100, 113 reverse (thrust) faults 11 reverse faults 11

seismic arrivals 87 attributes 3, 44, 83, 84, 87, 88, 100, 102, 108, 115, 123, 124, 126, 127, 130, 133, 138 convolution model 3 data 1–3, 6, 44, 49, 77, 87–89, 96, 98, 100–103, 108, 114, 115, 127, 133, 143 event 3 exploration 59 facies analysis 33 interpretation tools 27 modeling 35, 39 noise 59, 100 quality 108 sequence stratigraphy 27, 29, 49 signals 3, 59, 99 surveys 2, 43, 91, 98, 113 trace 3 seismogram 3 semblance 100 shale gas 8 shallow effects 44 shape matrix 105, 106 sideswipes 44 signal deterioration 21 signal-to-noise 2, 87, 90 smoothing 80, 87, 88, 90, 96, 101 sonic-log 44 source rock 7, 8, 26 spectral decomposition 114, 115 spike-deconvolution 40 stratigraphic trap 10, 22 strike-slip (wrench) faults 11 strike-slip faults 11 structural features 11 trap 10, 24 subsurface 1, 3, 11, 35, 39–41, 59, 87, 88, 91, 100, 115 system tract 30, 33, 55

s

t

noise reduction 96 normal faults 11

o oblique (combined) faults 11 onlaps 30, 33 orientation 88–90, 100, 108

p P-wave 35, 43, 56, 59 petroleum migrates 7 reservoir 8, 10, 27 physical modeling 35 pitfalls 44, 48, 49 primary 2, 9, 35, 42, 48, 124, 125 problems arising from seismic data processing 35 prograding 33 proximal deposition 30 pseudorandom noise 59

r

S-wave 35 scattering 11 secondary 9, 91, 124, 125, 130 sediment supply 29, 34

tangent vector 105, 107 textures 110, 133 thinning of reflections 44 tight gas 8

159

160

Index

time slice 24, 26, 54, 91, 96, 98, 102, 115, 121 time-to-depth conversion 35, 41–44 toplaps 30, 33 trap 7, 10, 11, 18, 19, 21, 22, 26, 27, 48, 50–54, 56, 57, 143 traveltime 3 tristimulus 88 tying seismic and well data 35

u unconformities due to a multicycle wavelet 48 unconventional resources 8

v variance 108–110, 112, 121 velocity analysis 2, 41, 100, 134 increase with depth 44 information 44 Vibroseis 40

w wavelet 39, 40, 48 source 3 weighting 90, 100 well 26, 35, 39–41, 43, 44, 87, 88, 108 window 7, 108