Seismic True-Amplitude Imaging

Jörg Schleicher Martin Tygel Peter Hubral

SEG Geophysical Developments Series No. 12 Robert H. Stolt, volume editor Stephen J. Hill, series editor

SOCIETY OF EXPLORATION GEOPHYSICISTS The international society of applied geophysics Tulsa, Oklahoma, U.S.A.

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All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transcribed in any form or by any means, electronic or mechanical, including photocopying and recording, without prior written permission of the publisher.

Published 2007 Printed in the U.S.A.

Library of Congress Cataloging-in-Publication Data

Schleicher, Jörg. Seismic true-amplitude imaging / Jörg Schleicher, Martin Tygel, Peter Hubral. p. cm. -- (SEG geophysical developments series ; no. 12) Includes bibliographical references and index. ISBN 1-56080-143-3 (volume) -- ISBN 0-931830-41-9 (series) 1. Seismic reﬂection method. 2. Earth--Internal structure. I. Tygel, M. II. Hubral, Peter. III. Title. IV. Series.

QE538.5.S34 2007 551.110284--dc22 2007025618

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And to all who contribute to the advancement of geophysics without receiving the deserved recognition

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About the Authors ...... xii Foreword ...... xv Volume Editor’s Preface ...... xvii Authors’ Preface ...... xix Acknowledgments ...... xxv List of Symbols and Abbreviations ...... xxvii Variables and symbols ...... xxvii Latin lowercase letters ...... xxvii Latincapitalletters...... xxxi Calligraphic capital letters ...... xxxvii Greek lowercase letters ...... xxxix Greek capital letters ...... xli Other symbols ...... xliv Indices and accents ...... xliv Subscripts ...... xliv Superscripts ...... xliv Mathematical accents ...... xlv Operational symbols ...... xlvi

Chapter 1: Introduction ...... 1 True-amplitude Kirchhoff migration ...... 1 True-amplitude Kirchhoff demigration ...... 9 True-amplitude Kirchhoff imaging ...... 12 Additional remarks on true amplitude ...... 16 Overview ...... 17

Chapter 2: Description of the Problem ...... 21 Earth model ...... 21 Macrovelocity model ...... 22 Wavemode selection ...... 22 Coordinate system ...... 22 Measurement conﬁgurations ...... 23 Measurement surface ...... 23 Measurement conﬁguration ...... 24 Data-space description ...... 29 Hagedoorn’s imaging surfaces ...... 30 The diffraction-traveltime, or Huygens surface ...... 31 The isochronous surface ...... 31 Hagedoorn’s imaging conditions ...... 32

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Mapping versus imaging ...... 33 Migration and demigration: Mapping ...... 33 Generalized Hagedoorn’s imaging surfaces ...... 34 Uniﬁed approach: Mapping ...... 36 Seismic-reﬂection imaging ...... 46 Summary ...... 53

Chapter 3: Zero-Order Ray Theory ...... 55 Wave equations ...... 55 Ray ansatz ...... 57 Homogeneous medium ...... 57 Inhomogeneous medium ...... 58 Time-harmonic approximation ...... 59 Time-domain expressions ...... 60 Validity conditions ...... 61 Eikonal and transport equations ...... 63 Acoustic case ...... 63 Elastodynamic case ...... 65 Rays as characteristics of the eikonal equation ...... 69 Slowness vector ...... 69 Characteristic equations ...... 71 Rayﬁelds ...... 72 Ray coordinates ...... 72 Transformation from ray coordinates to global Cartesian coordinates . . . 72 Ray Jacobian ...... 73 Solution of the transport equation ...... 73 Solution in terms of the ray Jacobian ...... 73 Point-source solutions ...... 75 Caustics ...... 78 Computation of the point-source solution ...... 79 Homogeneous medium ...... 79 Inhomogeneous medium ...... 79 Ray-centered coordinates ...... 80 Transformation from ray-centered to global Cartesian coordinates ..... 81 Transformation from ray to ray-centered coordinates ...... 82 Ray Jacobian in ray-centered coordinates ...... 83 Ray-tracing system in ray-centered coordinates ...... 83 Paraxial and dynamic ray tracing ...... 85 Paraxial ray tracing ...... 85 Dynamic ray tracing ...... 86 Paraxial approximation ...... 88 Initial conditions for dynamic ray tracing ...... 89 Ray-centered propagator matrix Πˆ ...... 90 Rays at a surface ...... ˜ 91 Vector representations ...... 91

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Surface representation ...... 93 Transformation from local Cartesian coordinates to ray-centered coordinates ...... 93 Transformation from local to global Cartesian coordinates ...... 95 Relationship between the slowness-vector representations ...... 96 Surface-to-surface propagator matrix Tˆ ...... 99 Rays across an interface ...... ˜ 101 Boundary conditions ...... 101 Dynamic-ray-tracing matrices ...... 102 Ray Jacobian across an interface ...... 103 Primary reﬂected wave at the geophone ...... 104 Ray amplitude at the geophone ...... 104 Complete transient solution ...... 108 Summary ...... 109

Chapter 4: Surface-to-Surface Paraxial Ray Theory ...... 111 Paraxial rays ...... 111 Traveltime of a paraxial ray ...... 113 Inﬁnitesimal traveltime differences ...... 113 Surface-to-surface propagator matrix ...... 116 Paraxial traveltime ...... 120 Ray-segment decomposition ...... 124 Chain rule ...... 124 Ray-segment traveltimes ...... 128 Meaning of the propagator submatrices ...... 130 Propagation from point source to wavefront ...... 130 Propagation from wavefront to wavefront ...... 131 Fresnel zone ...... 133 Deﬁnition ...... 134 Time-domain Fresnel zone ...... 136 Projected Fresnel zone ...... 136 Time-domain projected Fresnel zone ...... 140 Determination ...... 140 Other applications of the surface-to-surface propagator matrix ...... 141 Geometric-spreading decomposition ...... 141 Extended NIP-wave theorem ...... 143 Summary ...... 147

Chapter 5: Duality ...... 149 Basic concepts ...... 149 Duality of reﬂector and reﬂection-traveltime surface ...... 151 Basic assumptions ...... 151 One-to-one correspondence ...... 152 Duality ...... 152

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Basic deﬁnitions ...... 152 Diffraction and isochronous surfaces ...... 153 Useful deﬁnitions ...... 154 Expressions in terms of paraxial-ray quantities ...... 157 Duality theorems ...... 158 First duality theorem ...... 158 Second duality theorem ...... 159 Proofs of the duality theorems ...... 160 First duality theorem ...... 160 Second duality theorem ...... 162 Fresnel geometric-spreading factor ...... 164 Curvature duality ...... 169 Beylkin determinant ...... 169 Summary ...... 170

Chapter 6: Kirchhoff-Helmholtz Theory ...... 173 The Kirchhoff-Helmholtz integral ...... 175 Kirchhoff-Helmholtz approximation ...... 177 Asymptotic evaluation of the Kirchhoff-Helmholtz integral ...... 182 Geometric-spreading decomposition ...... 183 Phase shift because of caustics ...... 186 Summary ...... 187

Chapter 7: True-Amplitude Kirchhoff Migration ...... 189 True-amplitude migration theory ...... 192 Underlying assumptions ...... 192 Diffraction stack ...... 195 Evaluation at a stationary point ...... 197 Evaluation elsewhere ...... 197 Evaluation result ...... 198 True-amplitude weight function ...... 199 Traveltime functions ...... 199 Traveltime difference and Hessian matrix ...... 200 Geometric-spreading factor ...... 201 Final weight function ...... 203 Alternative expressions for the weight function ...... 206 True-amplitude migration result ...... 207 Comparison with Bleistein’s weight function ...... 209 Free-surface, vertical displacement ...... 210 Particular configurations ...... 210 Zero-offset (ZO) configuration ...... 211 Common-offset (CO) configuration ...... 211 Common-midpoint-offset (CMPO) configuration ...... 211 Common-shot (CS) configuration ...... 212

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Common-receiver (CR) configuration ...... 212 Cross-profile (XP) configuration ...... 213 Cross-spread (XS) configuration ...... 213 True-amplitude migration procedure ...... 213 Summary ...... 215

Chapter 8: Further Aspects of Kirchhoff Migration ...... 219 Migration aperture ...... 219 Minimum aperture ...... 220 Application...... 223 Pulsedistortion...... 223 Geometric approach ...... 226 Mathematical derivation ...... 228 Geometric interpretation ...... 231 Synthetic example ...... 232 Resolution ...... 234 Mathematical derivation ...... 235 Synthetic example ...... 238 Vertical-fault example ...... 240 Multiple weights in Kirchhoff imaging ...... 245 Multiple diffraction-stack migration ...... 245 Three fundamental weights ...... 249 Synthetic example in 2D ...... 251 Summary ...... 255

Chapter 9: Seismic Imaging ...... 259 Isochron stack ...... 261 Asymptotic evaluation at the reﬂection-traveltime surface ...... 262 Isochron stack in the vicinity of the reﬂection-traveltime surface ...... 263 Isochron stack elsewhere ...... 264 True-amplitude kernel ...... 264 Diffraction-stack and isochron-stack chaining ...... 268 Chained solutions for problem 1 ...... 269 Chained solutions for problem 2 ...... 277 General remarks on image transformations ...... 282 Summary ...... 284

Appendix A: Reflection and Transmission Coefficients ...... 287 Reflection coefficients ...... 287 P-P reflection ...... 287 SV-SV reflection ...... 289 SH-SH reflection ...... 290 P-SV reflection ...... 290 SV-P reflection ...... 291

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Transmission coefﬁcients ...... 292 P-P transmission ...... 292 SV-SV transmission ...... 292 SH-SH transmission ...... 293 P-SV transmission ...... 293 SV-P transmission ...... 293

Appendix B: Waves at a Free Surface ...... 295 P-waves at a free surface ...... 295 S-waves at a free surface ...... 297 SV-waves at a free surface ...... 297 SH-waves at a free surface ...... 298 General remark on elastic waves ...... 299 Acoustic waves at a free surface ...... 299

Appendix C: Curvature Matrices ...... 301

Appendix D: Relationships to Beylkin’s Determinant ...... 305

Appendix E: The Scalar Elastic Kirchhoff-Helmholtz Integral ...... 307 The anisotropic, elastic Kirchhoff integral ...... 307 Anisotropic Kirchhoff-Helmholtz approximation ...... 309 The Kirchhoff-Helmholtz integral for an isotropic medium ...... 311

Appendix F: Derivation of the Scalar Elastic Kirchhoff Integral ...... 315 A scalar wave equation for elastic elementary waves ...... 315 Direct waves ...... 316 Transmitted waves ...... 319 Reﬂected waves ...... 320

Appendix G: Kirchhoff-Helmholtz Approximation ...... 323 Plane-wave considerations ...... 323 Local plane-wave approximation ...... 325

Appendix H: Evaluation of Chained Integrals ...... 327 Cascaded conﬁguration transform ...... 327 Cascaded remigration ...... 330 Single-stack remigration ...... 333

Appendix I: Hessian Matrices ...... 335 Conﬁguration-transform Hessian matrix ...... 335 Remigration Hessian matrix ...... 337

References ...... 341

Index ...... 351

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Jörg Schleicher is an associate professor of applied mathematics at the Institute for Mathematics, Statistics, and Scientiﬁc Computing of the State University of Campinas (IMECC/UNICAMP) in Brazil. From September 1995 through September 1996, he was a visiting scientist at IMECC/UNICAMP, with joint grants from the Brazilian Research Council CNPq and the Alexander von Humboldt Foundation. From 1990 through 1995, he was a research fellow at the Geophysical Institute of Karlsruhe University in Germany. Schleicher is a founding member of the Wave Inversion Technology Consortium in Karlsruhe. His research interests include all forward and inverse seismic methods, particularly Kirchhoff modeling and imaging, amplitude-controlled imaging methods, migration velocity analysis, ray tracing, and model- independent stacking. Schleicher received a B.Sc. in physics in 1985, an M.Sc. in physics in 1990, and a Ph.D. in geophysics in 1993, all from Karlsruhe University. In 1998, he received SEG’s J. Clarence Karcher Award. Schleicher is a member of SEG, EAGE, DGG, SBGf, and SBMAC.

Martin Tygel is full professor in the Applied Mathematics Depart- ment and head of the Laboratory of Computational Geophysics at the State University of Campinas (UNICAMP), Brazil. The laboratory, which was founded in 2001, conducts a variety of applied geophysics projects linked to academia and industry. From 1995 through 1999, Tygel was president of the Brazilian Society of Applied Mathematics (SBMAC). He was a visiting professor at the Federal University of Bahia (PPPG/UFBa), Brazil, from 1981 through 1983 and at the Geo- physical Institute of Karlsruhe University, Germany, in 1990. Tygel was an Alexander von Humboldt fellow from 1985 through 1987 at the German Geological Survey in Hannover. His research interests are in processing, imaging, and inversion of geophysical data. His emphasis is on methods and algorithms that have a sound basis in wave theory and that ﬁnd signiﬁcant practical application. He is a founding member of theWave Inversion Technology Consortium in Karlsruhe. Tygel received a B.Sc. in physics from Rio de Janeiro State University in 1969, an M.Sc. in 1976, and a Ph.D. in 1979, both in mathematics from Stanford University. In 2002, he received EAGE’s Conrad Schlumberger Award. He is a member of SEG, EAGE, SBGf, and SBMAC.

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Peter Hubral is a retired full professor of applied geophysics at Karlsruhe University in Germany, specializing in seismic waveﬁeld imaging and inversion. From 1974 through 1985, he was with the German Geological Survey in Hannover, and in 1970–1973, he was with Burmah Oil of Australia. He had many assignments overseas. Hubral directed the Wave Inversion Technology Consortium in Karlsruhe from its foundation in 1997 until his retirement in 2006. He has written numerous papers and is a coauthor of three previous books, one with Theodore Krey on velocity computation from traveltimes, one with Martin Tygel on transient waves in 1D layered media, and one with Serge Shapiro on 1D random media. Apart from geophysics, Hubral is interested in ancient cultures. Hubral received an M.Sc. in geophysics in 1967 from the Technical University of Clausthal in Germany and a Ph.D. in 1969 from Imperial College, London University. He is an honorary member of EAEG/EAGE and SEG. Hubral received EAEG’s Conrad Schlumberger Award in 1978, SEG’s Reginald Fessenden Award in 1979, and EAGE’s Erasmus Award in 2003.

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Some of the tasks of a career in research come with honor and with pleasure. Being asked to write a foreword for this book is one of those cases. This volume is an important contribution to the understanding of true-amplitude seismic imaging through the unique school of thought led by these three authors, Jörg Schleicher, Martin Tygel, and Peter Hubral. It is also an exposition stemming from years of research and knowledge gained from successful implementations of their work. There is significant overlap between the research of these authors and their colleagues and the research in some of the same areas by my colleagues and me. After all, we have in common the fact that we are trying to extract information about the structure of the earth’s interior from actively generated seismic data. Furthermore, we want to learn as much as we can from the data about changes in medium parameters through narrow regions of rapid structural variations that we simplify by thinking of them as reflectors. The parallels in our collected works are indicated clearly by the citations to one another’s work in our research papers. However, the differences are important too, and they are to be celebrated. They are the highlights of the uniqueness of our points of view and our approaches to the same problems and objectives. The differences enrich both schools of thought and all others as well. Thus, I am delighted that this book now collects the ideas of the authors in a cohesive and sequential manner. In reviewing or reading papers by the authors or in attending talks that they present, I am always anticipating an “Aha!” I have rarely if ever been disappointed. Now the sources of all those instances of “Aha!” are collected in this one work. The authors are masters of the geometric interpretation of both the structure of the data and its depiction and its relation to the structure of reflectors in the earth. This makes Seismic True-Amplitude Imaging a must-read book for a full grasp of the authors’ distinctive and important point of view. So I invite the reader to read on and learn.

1Norman Bleistein is University Emeritus Professor and research professor of geophysics at Colorado School of Mines, where he is also former director of the Center for Wave Phenomena in the Department of Geophysics and of its Consortium Project on Seismic Inverse Problems. He is a guest professor of geophysics at the Chinese University of Geosciences, Wuhan, and is associated with various journals as advisory board member or adjudicator. Bleistein is an honorary member of SEG.

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Seismic True-Amplitude Imaging is a ray-theoretical exposition of seismic imaging processes, unapologetic in that it bristles at any suggestion that ray theory is not wave theory. This attitude is justified in part by being right — ray theory does come from an asymptotic approximation to the wave equation — and by the fact that similar asymptotic assumptions tend to sneak like viruses into so-called full-wave-theoretical processes as well. On other issues, the book is more accommodating. After noting that the original Kirch- hoff diffraction integral was devised only for forward-wave propagation, the authors concede that the term Kirchhoff migration has passed into common usage, and they learn to live with it. Similarly, although the term true amplitude is a lexicological minefield, it is widely applied to processes that seek to faithfully preserve amplitude information. The authors are comfortable with that, although they apply a very specific definition of the term in their text. This book uses the term imaging in the widest possible sense, which was an eye-opener for me. I have tended to use imaging as a synonym for migration, but I can’t do that anymore. The authors note that seismic data in any stage of processing is likely to contain discrete events or images, perhaps geometrically distorted but nevertheless pictures or images of the earth’s interior. This observation is facilitated by the ray-theoretical point of view, because it is largely in asymptopia that such images form. In consequence, any operation that affects or changes the images in the data can be considered an imaging process. This brings under the imaging umbrella a variety of processes, including partial and residual migration, conversion of one experimental configuration into another, and so on. Strictly, I suppose, Kirchhoff imaging employs a ray-theoretical model for propagation and a far-field diffractive model for reflections. It is best suited for a medium composed of regions where earth properties change slowly, divided by reflecting surfaces where properties change abruptly. Where those conditions are met imperfectly, the method may tend to impose this model, because this is how the method views the world. That is not necessarily bad — geophysicists tend to view the world that way too. Kirchhoff depth migration has enjoyed a long period of preeminence.Advances in com- puter power in recent years have allowed contemplation of full-wave-equation depth- migration algorithms, but the authors of Seismic True-Amplitude Imaging show no deep concern for the future of ray-theoretical imaging. In that, they are probably justified, for several reasons. First, migration methods, once established, never die. The inverse seismic problem is so tough and complex that no single technique, however powerful, can be universally successful. New techniques may be added to the arsenal but are not likely to completely displace proven methodologies.

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Second, the asymptotic approximation is very well met under most circumstances by seismic data, and even when that is not the case, ray-theoretical techniques often are able to extract useful information. Third, ray theory is well suited to true-amplitude processing because amplitudes can be computed explicitly at every point and related back to the underlying earth properties. Fourth, Kirchhoff-based imaging is uniquely suited to composite operations (e.g., demigration using one velocity structure followed by migration using another velocity structure) because, in asymptopia, a composite operation can be condensed into a single operation. For extremely complex geology, ray theory might have difﬁculty providing a complete description of wave propagation, even where it is technically possible to do so. A full-wave- equation method, in contrast, may provide a complete description without extra effort. That might give some advantage to a full-wave method, but the blessings are mixed. In ray theory, one knows exactly which waves are where. If some portion of the complete waveform (perhaps a multiple or a converted wave) does not contribute positively to the desired image, a ray-theoretical method might eliminate it. Even where a full-wave method can produce the better image, one would likely want to run a Kirchhoff algorithm concurrently to aid in analysis. Seismic True-Amplitude Imaging provides a clear, readable, and reasonably complete presentation of Kirchhoff imaging theory. Although subjects such as beam forming and multipath imaging are not presented in detail, the tools to deal with them are present. For those of us not steeped in ray theory, the book provides a good introduction and tutorial, then digs deeply and profoundly into a theory of generalized imaging.

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In the world of seismic exploration, any method that concerns the determination of an image of subsurface reflectors from seismic reflections or diffractions is called seismic depth migration. This method requires an a priori given reference velocity model, which acts as an initial guess of the actual depth velocity model that is to be constructed. Depending on the geologic complexity of the earth, the reference velocity model, generally called the macrovelocity model, needs to be vertically and/or laterally inhomogeneous, elastic isotropic, or anisotropic. If the migration procedure consists only of transforming interpreted and picked traveltimes of selected reflections (such as the primary reflections from sought-for key horizons), the method is termed map migration. Manipulating the seismic reflections that are to be migrated by using algorithms that are based on or derived from the wave equation (assuming any given propagation medium) leads to what has become known as wave-equation migration. That seismic traces as a whole, and not only interpreted (picked) reflection events, can be used by wave-equation migration methods has substantially simplified and improved the seismic imaging and inversion processes and the subsequent interpretation of the migrated results. It also has given the amplitudes of the migrated events a certain physical significance that map migration cannot provide. In close correspondence to seismic migration, there are a variety of other seismic- imaging methods [e.g., the dip-movement (DMO), migration-to-zero-offset (MZO), or redatuming processes, etc.] that also transform one image or section, which may be in the time or depth domain, into another. In this sense, any collection of traces (e.g., a constant- offset or time-migrated section) generally is called an image. It therefore is necessary to refer to the process that has created an image (e.g., a common-offset depth migration, zero- offset time migration, etc.). In this way, we will speak of the migrated image, MZO image, etc. The whole set of imaging methods can be referred to as seismic imaging. In many of the wave-equation migration methods, the geometric simplicity of map migration (which in general involves constructing rays, wavefronts, isochrons, or maximum- convexity surfaces) is largely lost. Unfortunately, many geophysicists and seismic interpreters have become accustomed to this situation. Some are inclined to believe that good migrated images can be achieved only at the expense of losing the geometric insight. However, this is not true. A kinematic conception such as the one proposed by Hagedoorn (1954) for migration can and should be maintained in connection with all imaging processes. In fact, until not too long ago, the general belief among seismic explorationists was that although the ray method is quite valuable for forward seismic modeling (e.g., the construction of synthetic seismograms for a given earth model), traveltime inversion (e.g., the construction of an initial macrovelocity model from picked traveltimes), and reflection

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tomography (e.g., the refinement of an initial macrovelocity model with the help of picked traveltimes of some key primary reflections), it has little to offer for seismic-reflection imaging and wave-equation migration. This situation has now dramatically changed. For more than 10 years, we have used the seismic-ray method to develop imaging algorithms. To some extent, what we present here is a didactic reorganization of our previous work, including our up-to-date view of the subject. One of our principal aims is to confirm that ray theory no longer should be considered a “stepchild” in the fields of wave-equation migration and seismic imaging. In fact, it is a very useful part of these fields because it can handle the kinematic (related to reflection traveltimes) and dynamic (related to reflection amplitudes) aspects of wave-equation migration in an exact way that is geometrically and physically appealing. Subsurface images in either the time or depth domain can be constructed, as will be shown, on a ray-theoretical basis from specific (e.g., compressional-primary or shear- wave) reflections that have been recorded using various measurement configurations [e.g., zero-offset, common-offset, common-shot, common-receiver, or vertical-seismic-profiling (VSP) measurements]. All reflections that are imaged by seismic-migration methods provide migrated amplitudes in addition to the subsurface reflector positions. In this book, some emphasis is put on correctly handling the amplitudes of specific elementary reflections in a 3D prestack migration. For definiteness, we principally consider P-P primary reflections; however, we also show that the same approach also easily handles shear or converted waves. From the depth-migrated elementary reflections, a quantitative measure of angle-dependent plane-wave interface-reflection coefficients can be obtained.This is highly desirable because it provides the input to the so-called AVO (amplitude-versus-offset) techniques. We know of no other approach than the ray-theory-based one in which wavefield amplitudes can handle in reflection imaging in a similar geometrically easy way. The simple principles of ray theory are equally valid in the presence of an inhomogeneous, 3D layered earth, and for arbitrary measurement configurations. The earth and the distribution of the petrophysical parameters are geometric. Ray theory, which is also a geometric wave-equation theory, is ideally complementary. Clearly, much of the seismic world restricts the meaning of wave-equation migration to differential wave-equation migration. We find this use to be misleading because it implies that ray-based migration methods are entirely different and have nothing to do with the wave equation; however, as we pointed out, ray-theoretical approaches also are based on the wave equation, and it is proper to refer to them as such. In principle, the theory presented in this work can be seen as a generalization of Hagedoorn’s original ideas, in which time-to-depth migration was performed with the help of either maximum-convexity curves or isochrons. As is well known, Hagedoorn’s purely kinematic migration concepts found a full wave-equation-based equivalent formulation in what has become known as Kirchhoff depth migration. This is a wave-equation-migration technique that is based on the weighted summation (or stacking) of seismic trace amplitudes on seismic records along measurement-configuration-specific diffraction-time curves (surfaces). These are auxiliary surfaces that are constructed using the a priori given macrovelocity model. As shown below, much more general imaging tasks (e.g., MZO, redatuming, etc.) also can be achieved similarly. All these methods can be generally referred to as Kirchhoff-type imaging procedures. They all are based on the weighted stacking of the

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seismic data along auxiliary surfaces that are constructed within the given macrovelocity model and that are specific to the imaging purpose under consideration. Strictly speaking, the attribute Kirchhoff for this depth-migration technique is slightly misleading. In fact, the Kirchhoff (modeling) operator consists of a wavefield forward extrapolation operator. It is based on the idea of extrapolating true physical wavefields that are recorded on a surface in a direction away from the sources. This is realized as a superposition of Huygens elementary waves. Depth migration is unrelated to this forward extrapolation. In the early days of wave- equation migration (restricted to a homogeneous macrovelocity model), the migration operation was conceived as wavefield extrapolation that was designed to propagate the recorded wavefield backward in time toward fictitious exploding sources at the reflectors. Because this extrapolation was derived from a modification of the Kirchhoff integral, the new operation was called Kirchhoff migration. Reinhard Bortfeld vehemently rejected this term. His reason was—and we very much share his view—that with the original Kirch- hoff integral, one only can perform a wavefield forward extrapolation. Seismic post-stack migration [of a common-midpoint (CMP) stack or an NMO (normal moveout) /DMO/stack section, which is assumed to approximate the response at the earth’s surface of a hypothetical exploding-reflector-model wavefield] evidently is a backward extrapolation of a hypothetical wavefield. It only can be achieved with an operator that results from the Kirchhoff integral after a trick is applied. This trick involves changing of the propagation direction of the elementary Huygens waves, which formally appear in the Kirchhoff integral on the measurement surface that surrounds the real or secondary sources. The Kirchhoff-type- wavefield backward extrapolation operator that results from applying the trick, has since become known in the nongeophysical community as the Porter-Bojarski integral. Moreover, no seismic records other than the common-shot record represent the response of one wavefield because they cannot be described by a single physical experiment. As a consequence, we cannot claim that Kirchhoff migration is based on the Kirchhoff (modeling) integral; however, what we can accept is that Kirchhoff migration can be regarded as a physical inverse to Kirchhoff modeling. This is because Kirchhoff migration recovers the Huygens elementary waves that are the input to Kirchhoff modeling. We must acknowledge, though, that the term Kirchhoff migration has now become common use in the seismic community. For this reason, and because of the above physical considerations, we accept the term. Recall, however, that before the term Kirchhoff migration was introduced, there existed the term diffraction-stack migration. It also was based very much on the original ideas of Hagedoorn (1954), but involved no more than summing the amplitudes of a CMP stack section along maximum convexity curves, e.g., diffraction-time curves or surfaces. No weights were used in the diffraction-stack migration. No wavefield-extrapolation concepts were necessary, and no exploding-reflector model was required to describe the CMP stack section to justify the diffraction-stack-migration operation that was performed mostly in the time domain. It therefore is absolutely legitimate to call a Kirchhoff migration as proposed in this book a weighted diffraction-stack migration. Because we will describe here a Kirchhoff migration that also concerns amplitudes, we require weights and descriptions of the reflections in the seismic records in terms of solutions of the wave equation. Moreover, we require that shots

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and receivers be reproducible, i.e., that they have identical characteristics, when moved along the measurement surface. In fact, it was Newman (1975) who first recognized the need to modify the ordinary diffraction stack to handle migration amplitudes correctly and to give time-migrated primary reflections a quantitative, physically well-defined value, which then was referred to as a true amplitude. We understand the attribute true in the sense of faithful, rather than in accordance with verity. A true amplitude is nothing more than the amplitude of a recorded primary reflection (e.g., a zero-offset reflection) that is compensated (e.g., multiplied) by its geometric-spreading factor. Because all depth-migrated seismic reflections discussed in this book are true-amplitude reflections, we feel justified to call the proposed wave-equation Kirchhoff-migration method a true-amplitude migration, as well. Admittedly, many geophysicists think that a true amplitude is the designation for an unmanipulated amplitude, e.g., of a primary reflection as it was recorded in the field. Some also assume that if an amplitude is kept unchanged in a certain seismic process, then the process is a true-amplitude process. In our terminology and in Newman’s, this is not so. In fact, as the reader will learn in this book, the construction of a true-amplitude reflection from a reflection that was recorded in the field involves not only a scaling of the considered reflection amplitude by using the geometric-spreading factor, but also the reconstruction of the analytic source pulse multiplied by the reflection coefficient. Elementary-wave reflections (like primary reflections) might have suffered modifications because of caustics along the raypath between source and receiver. The reason for the resultant source-pulse distortion (and the need for the source-pulse reconstruction) is, as the theory shows, that the geometric-spreading factor might not be a real positive, but also could be a negative or imaginary quantity. In short, migrated true-amplitude reflections provide a good and physically well-defined measure for estimating angle- dependent plane-wave reflection coefficients that may vary laterally along a curved target reflector. In this book, we also describe other true-amplitude imaging processes, such as a true- amplitude MZO. In these more general cases, the term true amplitude, which loosely means that geometric spreadings are accounted for in the best possible way, requires a more pre- cise, problem-specific definition. This is provided in Chapter 1. For example, true-amplitude MZO means that the geometric-spreading factor of an input common-offset primary reflection is transformed into the geometric-spreading factor that pertains to the corresponding zero-offset reflection obtained after the MZO transformation; however, note that, just like in true-amplitude migration, reflection and transmission coefficients of primary reflections remain unaltered by any true-amplitude process. For up-to-date, comprehensive collections of research publications devoted to the subject of this book, see Hubral (1998) and Tygel (2001). Another important contribution to the subject is Bleistein et al. (2001). The proposed theory of true-amplitude migration very much is part of what scientists in other disciplines (e.g., nondestructive testing, radar, etc.) also may call reflection tomography. Each of these important subjects has developed its own terminology to describe very similar methods and results.

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In Goldin’s (1987a, 1987b, 1990) terminology, the theory described below also could be formulated with the method of discontinuities. Goldin also uses the concept of a true- amplitude migration. In the terminology of Beylkin (1985a, 1985b) and Bleistein (1987), the methods proposed here also may be called seismic migration/inversion. However, because we consider ourselves exploration geophysicists at heart, we opted for the simple title of this book, giving up any desire to be the most general and universal. We are fully aware that true-amplitude and migration unfortunately are terms that are very specific to our profession.As indicated above, the techniques described in this book resemble in part those developed by Beylkin, Bleistein, Bortfeld, Goldin, Newman, and many others. They summarize the research that we have performed over the last decade. Nevertheless, we hope this book will offer sufficient new and compact results of interest to readers who look for conceptually and geometrically appealing seismic full-wave equation migration methods in 3D media. The proposed true-amplitude migration and imaging methods not only provide a good understanding of the geometry that is involved in the imaging process, but also give the imaged amplitudes a lithologically significant value. Ray theory is confined to the description of seismic waves in smooth elastic or acoustic media separated by interfaces along which the medium parameters change in a discon- tinuous manner. Consequently, the migration procedures also expect the real earth to be representable by a medium of this type. Because ray theory now is well developed for more complex media (e.g., anisotropic, absorbing, slightly scattering, etc.), we are con- vinced that, in principle, the approaches offered here for an isotropic elastic medium can be extended to all such media in which ray theory offers a good description of the seismic wave-propagation phenomena. An excellent source for learning more about the ray method and its use in forward seismic modeling is Cervený (2001). As already indicated, true-amplitude migration is not the only topic dealt with in this book. There is a variety of additional seismic imaging procedures, all of which can more or less keep the amplitudes well controlled. The first of these is true-amplitude demigration. Under a given measurement configuration, demigration transforms a true- amplitude depth-migrated image into its corresponding true-amplitude seismic record. True-amplitude migration and demigration provide the building bricks for the unified theory of reflection imaging that is developed in this book. All true-amplitude imaging processes (see Chapter 9) result from the chaining or cascading of a true-amplitude migration and demigration. Popular in seismic-reflection imaging is the process of MZO that is related closely to the DMO process. The MZO and DMO processes also can be handled in a true-amplitude manner, as can the remigration (velocity continuation or residual migration) and other imaging processes. Remigration or residual migration involves improving a true-amplitude depth-migrated image by taking into account a better macrovelocity model, and using the roughly depth-migrated image as an input. Other imaging processes that can be treated analogously are shot continuation and true-amplitude redatuming. Shot continuation simulates a displaced common-shot section from a neighboring one. Redatuming requires changing the seismic traces from one measurement surface to another. Because we treat various true-amplitude imaging processes, we have entitled this book Seismic True-Amplitude Imaging.

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Finally, we stress again that the theory offered in this work relies on the validity of the ray-theoretical description of the seismic-wave propagation in the media under consideration. In particular, and in conformity with the ray assumptions, it addresses the imaging of selected elementary (e.g., essentially primary) reﬂections. Given the comprehensive scope and versatility of the ray method in the formulation of the theory of seismic-wave propagation, we feel that seismic exploration and reservoir imaging under the current approach has a promising future. Its consistent use should improve the understanding not only of seismic-ray theory, but also of the seismic-reﬂection imaging problem in general.

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In preparing this book, we have relied on the help of many colleagues and friends.We are grateful for innumerable scientiﬁc discussions and technical contributions, including those of Norman Bleistein, Ricardo Biloti, Elizabeth Davie, Robert Essenreiter, Alexander Görtz, Sonja Greve, Valeria Grosfeld, Christian Hanitzsch, Zeno Heilmann, Thomas Hertweck, Christoph Jäger, Herman Jaramillo, Makky S. Jaya, Frank Liptow, Jürgen Mann, Volker Mayer, Rowena Mills, Amélle Novais, Claudia Payne, Mikhail Popov, Rodrigo Portugal, Matthias Riede, Lúcio T. Santos, Robert H. Stolt, Anne H. Thomas, Amanda Van Beuren, Kai-Uwe Vieth, Andrea Weiss, and Yonghai Zhang.

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This list of symbols provides short definitions for most of the symbols, indices, accents, and abbreviations that are used in this book. All symbols are defined at least briefly where they first appear in the text, but most entries here also provide one or more references to a chapter, section, figure, table, equation, or appendix that contains the most detailed explanation of the symbol. Wherever possible, we used conventional symbols.

Variables and symbols Latin lowercase letters

aG constant receiver position vector that describes the conﬁguration (Chapter 2, “Measurement conﬁgurations” section)

aS constant source position vector that describes the conﬁguration (Chapter 2, “Measurement conﬁgurations” section) c acoustic velocity (Table 1 of Chapter 3)

cij kl components of the elastic tensor (Appendix E) cˆ conversion coefﬁcient vector, describes the recorded components of the particle displacement on a free surface (Chapter 7, “Free-surface, vertical displacement” section; Appendix B)

eˆk coordinate unit vectors of the ray-centered coordinate system qˆ in the qk direction (k = 1, 2, 3) (Chapter 3, “Ray-centered coordinates” section)

fm scalar model parameter (equations F-4 and F-5 of Appendix F) f [t] seismic source wavelet, source pulse, or source signal that is assumed to be reproducible if more than one experiments or shots are involved (equation 16 of Chapter 3) fˆ vectorial source term in the elastodynamic wave equation 6 of Chapter 3 g source strength, directional characteristics, radiation pattern (Chapter 3, “Point- source solutions” section)

gm scalar model parameter (equations F-4 and F-5 of Appendix F) hB Beylkin determinant (equation 76 of Chapter 5) h half-offset vector (equation 5 of Chapter 2) hˆ elastic polarization vector (Appendix E)

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G hˆ polarization vector of the receiver ray (Appendix E) ref hˆ polarization vector of the reﬂected ray (Appendix E) hˆ S polarization vector of√ the source ray (Appendix E) i imaginary unit, i = −1 ˆ ik coordinate unit vectors of the global Cartesian coordinate system rˆ in the rk direction (k = 1, 2, 3) (Chapter 5, “Fresnel geometric-spreading factor” section) jˆ xˆ k coordinate unit vectors of the local Cartesian coordinate system in the xk direction (k = 1, 2, 3) (Chapter 3, “Validity conditions”; Chapter 5, “Fresnel geometric-spreading factor” section) k bulk modulus (Table 1 of Chapter 3)

0 length scale of the inhomogeneities of the medium ψ characteristic length of medium (equation 24 of Chapter 3) mD stretch factor of migration (equation 16 of Chapter 5; Chapter 8, “Pulse distortion” section) m(r) prestretch factor (equation 3 of Chapter 9) m midpoint vector (equation 8 of Chapter 2) n number of transmitting or reﬂecting interfaces in a system of seismic layers (Chapter 3, “Ray amplitude at the geophone” section)

nI stretch factor of demigration (equation 17 of Chapter 5; equation 11 of Chap- ter 9) nˆ unit normal vector to the ray (equation 76 of Chapter 3)

nˆ M unit normal vector to a (real or hypothetical) interface at depth point M (Figure 3 of Chapter 7)

nˆ R unit normal vector to the reflector at the specular-reflection point MR (Figure 3 of Chapter 7) nˆ (x) surface normal in local Cartesian coordinates (equation 153 of Chapter 3) p ray parameter or horizontal slowness; the ray parameter is defined only in laterally homogeneous media or with respect to some reference direction (Appendices A and B only) p acoustic pressure (equation 4 of Chapter 3) p 2D vector that represents a measure for the 3D slowness vector of the paraxial ray and that is obtained through double projection of the latter; an index indicates the point where it is taken (equation 171 of Chapter 3) p 0 2D projected slowness vector of the central ray (Chapter 3, “Vector representations” section) p pˆ p 2D projection of the 3D slowness vector p of a paraxial ray into the tangent plane at P0 by a single projection (Chapter 3, “Relationship between the slowness-vector representations” section)

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p(q) 2D slowness vector in ray-centered coordinates; it is a paraxial quantity because for a central ray, p(q) = 0 always (Chapter 3, “Paraxial ray tracing” section) p (q) p(q) 0 initial value of (equations 142 of Chapter 3) pˆ 3D ray slowness vector (equation 56 of Chapter 3) pˆ 0 3D ray slowness vector of the central ray (Chapter 3, “Vector representations” section) pˆ G slowness vector of the receiver ray at the scattering point (Appendix E) pˆ p 3D slowness vector of a paraxial ray (Chapter 3, “Vector representations” section) pˆ (x) p 3D slowness vector in local Cartesian coordinates (equation 179 of Chapter 3) pˆ (q) 3D slowness vector in ray-centered coordinates (Chapter 3, “Ray-tracing system in ray-centered coordinates” section) pˆ ref slowness vector of the incident ray after specular reflection at the scattering point (Appendix E) pˆ pˆ T 3D projection of p into the tangent plane at P ; first step of the double projection (Chapter 3, “Relationship between the slowness-vector representations” section) q 2D ray-centered coordinate of a point on the paraxial ray (Chapter 3, “Paraxial ray tracing” section) q q 0 initial value of (equations 142 of Chapter 3) qˆ 3D ray-centered coordinate of a point on the paraxial ray (Chapter 3, “Ray- centered coordinates” section) r upper 2D subvector of rˆ; horizontal coordinate vector; with index, horizontal location vector of the respective point (Chapter 2, “Earth model” section and Figure 4) r 0 horizontal coordinates of a central point P0 (equation 165 of Chapter 3) r CT stationary point of configuration transform (Chapter 9, “Chained solutions for problem 1” section) r G 2D receiver coordinate vector (Chapter 2, “Measurement configurations” section) r M depth point coordinate (equation 74 of Chapter 4) r P horizontal coordinates of a generic point P (Chapter 2, “Paraxial ray tracing” section) r R horizontal coordinates of the reflection point MR (Figure 4 of Chapter 2) r S 2D source coordinate vector (Chapter 2, “Measurement configurations” section) r∗ stationary point of the Kirchhoff demigration integral in equation 1 of Chapter 9 (Chapter 9, “Asymptotic evaluation at the reflection-traveltime surface” section) rˆ 3D coordinate vector in global Cartesian coordinates, with index-location vector of the respective point (Chapter 2, “Earth model” section)

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rˆG 3D receiver coordinate vector (Chapter 3, “Ray ansatz” section) rˆS 3D source coordinate vector (Chapter 3, “Ray ansatz” section) s arc length of a ray (equations 65 and 67 of Chapter 3) t time variable tˆ unit tangent vector of the ray; one of the Frenet vectors (equation 72 of Chapter 3) uˆ real elastic particle displacement vector (equation 1 of Chapter 3) v wave-propagation velocity of the elementary wave mode under consideration, speciﬁed as α, β,orc; an index indicates the location at which it is taken (equation 54 of Chapter 3 and related discussion)

v0 wave velocity at the coincident source-receiver point of a normal ray (Chapter 7, “Particular conﬁgurations” section)

vG wave velocity at the receiver point G (equation 210 of Chapter 3) + vk medium’s velocity on the reflection ray at a reflection/transmission point at interface k (Chapter 3, “Ray amplitude at the geophone” section) − vk medium’s velocity on the incident ray at a reflection/transmission point at interface k (Chapter 3, “Ray amplitude at the geophone” section)

vM wave velocity at an arbitrary depth point M (equation 45 of Chapter 7) vR wave velocity at a reflection point MR (equation 18 of Chapter 5) vS wave velocity at the source point S (Chapter 3, “Point-source solutions” section) v(r,z) input velocity field for remigration (Chapter 2, “Unified approach: Mapping” section) v(˜ ρ,ζ) ouput velocity field for remigration (Chapter 2, “Unified approach: Mapping” section) wˆ multiple-migration weight vector (Chapter 8, “Three fundamental weights” section) x 2D local Cartesian coordinate system defined in the tangent plane to a given surface that passes through its origin, and whose index indicates the location of its origin; represents a measure for the distance of the paraxial ray from the central ray in the plane tangent to the surface (Chapter 3, “Vector representations” section) x G 2D local Cartesian coordinate system in the plane tangent to the measurement surface at G (Chapter 2, “Measurement configurations” section) x M 2D local Cartesian coordinate system in the plane at an arbitrary depth point M (equations 49 and 50 of Chapter 3) x R 2D local Cartesian coordinate system in the plane tangent to the target reflector at MR (Chapter 6, “Asymptotic evaluation of the KHI” section) x S 2D local Cartesian coordinate system in the plane tangent to the measurement surface at S (Chapter 2, “Measurement configurations” section)

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xˆ 3D local Cartesian coordinate system; xˆ = (x,x3), where the x3-axis points in the direction normal to a given surface that passes through its origin. Its index indicates the location of its origin (Chapter 3, “Vector representations” section) z vertical (depth) coordinate (Chapter 2, “Earth model” section)

zR vertical (depth) coordinate of the reﬂection point MR (Chapter 8, “Pulse distortion” section)

Latin capital letters

A aperture of seismic migration (equation 5 of Chapter 7); it generally is equal to the aperture of the seismic experiment, i.e., the surface in which all end points of the parameter vector ξ lie in the seismogram section, and thus areas over which data exist (Chapter 2, “Measurement conﬁgurations” section) A upper left 2 × 2 submatrix of a propagator matrix Tˆ of a paraxial ray in the ˜ vicinity of a known central ray; it describes the dependence˜ of the coordinates of the endpoint of the paraxial ray on those of its initial point. Without an index, A refers to the whole primary reﬂected ray, whereas with index 0, 1, or 2, it refers to˜ the corresponding ray segment with that index (equation 190 of Chapter 3) B migration input amplitude (Chapter 8, “Pulse distortion” section)

BCR configuration transform input amplitude (Appendix H) B upper right 2 × 2 submatrix of a propagator matrix Tˆ of a paraxial ray in the ˜ vicinity of a known central ray; it describes the dependence˜ of the coordinates of the endpoint of the paraxial ray on its slowness vector at its initial point. Without an index, B refers to the whole primary reflected ray, whereas with index 0, 1, or 2, it refers˜ to the corresponding ray segment with that index (equation 191 of Chapter 3) Ca reflection (a = r) or transmission (a = t) coefficient (Appendix G) C lower left 2 × 2 submatrix of a propagator matrix Tˆ of a paraxial ray in the ˜ vicinity of a known central ray; it describes the dependence˜ of the slowness vector at the endpoint of the paraxial ray on the coordinates of its initial point. Without an index, C refers to the whole primary reflected ray, whereas with index 0, 1, or 2, it˜ refers to the corresponding ray segment with that index (equation 192 of Chapter 3) CMP common midpoint: denotes a seismic experiment in which the source and receiver are dislocated so that their common midpoint remains fixed; all rays of the involved ray family are assumed to pertain to the paraxial vicinity of the normal ray that emerges at the common midpoint (Chapter 2, “Measurement configurations” section) CMPO common-midpoint offset: denotes the CMP experiment if the rays do not belong to the paraxial vicinity of the normal ray at CMP; the O reminds us that the central ray is now an arbitrary offset ray (Chapter 2, “Measurement configurations” section)

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CO common offset: denotes a seismic experiment in which source and receiver are dislocated so that their common offset remains fixed (Chapter 2, “Measurement configurations” section) CR common receiver: denotes a seismic experiment in which the sources are dislocated along the seismic line and the common receiver remains at a fixed location (Chapter 2, “Measurement configurations” section) CS common source (or common shot): denotes a seismic experiment in which the receivers are dislocated along the seismic line and the common source (or shot) remains at a fixed location (Chapter 2, “Measurement configurations” section)

DG denominator of the free-surface conversion coefﬁcients (equation B-8 of Ap- pendix B)

DR denominator of the elastic, isotropic reflection coefficients (equation A-3 of Appendix A) D lower right 2 × 2 submatrix of the propagator matrix Tˆ of a paraxial ray in the ˜ vicinity of a known central ray; it describes the dependence˜ of the slowness vector at the endpoint of the paraxial ray on that at the start point. Without an index, D refers to the whole primary reflected ray, whereas with index 0, 1, or 2, it refers˜ to the corresponding ray segment with that index (equation 193 of Chapter 3) E demigration aperture (equation 1 of Chapter 9)

EY Young’s modulus (Table 1 of Chapter 3) F [t] analytic source signal assigned to f [t] (equation 18 of Chapter 3)

Fmig[z] analytic source signal of a migrated reflection event; ideally the same as F [t] (equation 4 of Chapter 7) F surface curvature matrix (equation 152 of Chapter 3) ˜ G geophone (or receiver) position (Chapter 2, “Measurement configurations” section) G receiver position in the (paraxial) vicinity of G (Chapter 2, “Measurement configurations” section) G upper left 2 × 2 submatrix of Gˆ (Chapter 3, “Transformation from local ˜ Cartesian coordinates to ray-centered˜ coordinates” section) G(r) upper left 2 × 2 submatrix of Gˆ (r) (Chapter 3, “Transformation from local to ˜ global Cartesian coordinates” section)˜ Gˆ transformation matrix from local Cartesian coordinates xˆ to ray-centered coor- ˜ dinates qˆ (Chapter 3, “Transformation from local Cartesian coordinates to ray- centered coordinates” section) Gˆ (r) transformation matrix from local Cartesian coordinates xˆ to global Carte- ˜ sian coordinates rˆ (Chapter 3, “Transformation from local to global Cartesian coordinates” section) ˜ H CC Hessian matrix of TCC(ξ, r, N) (equation 52 of Chapter 9; Appendix I) ˜

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H D diffraction-traveltime Hessian matrix (equation 9 of Chapter 5) ˜ HF Fresnel-zone matrix (equations 87–89 of Chapter 4) ˜ H I Hessian matrix of TI (equation 12 of Chapter 5) ˜ H IS Hessian matrix of δIS (equation 10 of Chapter 9) ˜ HP projected Fresnel-zone matrix (equation 95 of Chapter 4) ˜ HR reflection-traveltime Hessian matrix (equation 8 of Chapter 5) ˜ H traveltime-difference Hessian matrix (equation 65 of Chapter 5) ˜ H Hessian matrix of T (ξ, r) (equation 10 of Chapter 5) ˜ H Hessian matrix of the migration output (equation 25 of Chapter 8) ˜ Hˆ transformation matrix from ray-centered coordinates qˆ to global Cartesian coor- ˜ dinates rˆ (Chapter 3, “Transformation from ray-centered to global Cartesian coordinates” section) HT abbreviation for the Hilbert transformation (equation 19 of Chapter 3) I 2 × 2 unit matrix (equation 2 of Chapter 2) ˜ Iˆ 3 × 3 unit matrix (equation 38 of Chapter 3) ˜ Iˆ 4 × 4 unit matrix (equation 148 of Chapter 3) ˜ J ray Jacobian (equation 75 of Chapter 3) + Jk Jacobian of the reflection ray at a reflection/transmission point at interface k (Chapter 3, “Ray amplitude at the geophone” section) − Jk Jacobian of the incident ray at a reflection/transmission point at interface k (Chapter 3, “Ray amplitude at the geophone” section)

KCC true-amplitude kernel for cascaded conﬁguration transform (equation 39 of Chapter 9)

KCR true-amplitude kernel for cascaded remigration (equation 60 of Chapter 9) KCT true-amplitude kernel for single-step conﬁguration transform (equation 56 of Chapter 9)

KDS true-amplitude weight function (or kernel) of Kirchhoff migration in integral 5 of Chapter 7 (general form in equation 46 of Chapter 7)

KIS true-amplitude kernel for Kirchhoff demigration (equation 27 of Chapter 9) KKH true-amplitude kernel for Kirchhoff-Helmholtz modeling (equation 9 of Chap- ter 6)

KRM true-amplitude kernel for single-step remigration (equation 67 of Chapter 9) K surface-curvature matrix (Appendix C) ˜ KI curvature matrix of the isochron (equation 62 of Chapter 5) ˜ KR curvature matrix of the target reﬂector (equation 62 of Chapter 5) K˜ K − K difference of curvature matrices I R (equation 67 of Chapter 5) ˜ ˜ ˜ M arbitrary point in the depth domain (Chapter 2, “Hagedoorn’s imaging surfaces” section)

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MCT depth point where the isochrons of the input and output conﬁgurations in a con- ﬁguration transform are tangent (Chapter 9, “Chained solutions for problem 1” section)

MI subsurface point on the isochron (Chapter 2, “Hagedoorn’s imaging surfaces” section)

MID point on the isochron z = ZI (r; ND) of ND (Chapter 2, “Chained solutions for problem 2” section)

Mp P-wave modulus (Table 1 of Chapter 3) MR reﬂection point, where the central ray is reﬂected according to the rules of Snell’s law

MRM the dual point in the depth domain of NRM (Chapter 9, “Chained solutions for problem 2” section)

M reflector point (Chapter 5, “Duality of reflector and reflection-traveltime surface” section) M depth point in the paraxial vicinity of M

MR paraxial reﬂection point (Chapter 4, “Projected Fresnel zone” section) M 2 × 2 submatrix of Mˆ (equation 178 of Chapter 3) ˜ ˜ Mˆ 3 × 3 traveltime Hessian matrix in ray-centered coordinates (equations 173 and ˜ 174 of Chapter 3) Mˆ (x) 3 × 3 traveltime Hessian matrix in local Cartesian coordinates (equation 173 of ˜ Chapter 3) N arbitrary point in the time-trace domain (the seismic section) (Chapter 2, “Hagedoorn’s imaging surfaces” section) ˜ ˜ ND point on the output model Huygens surface TD(ξ; M) (Chapter 9, “Chained solutions for problem 2” section)

NR dual point to MR on the reﬂection traveltime surface (Chapter 2, “Hagedoorn’s imaging surfaces” section)

NRM time point where the Huygens surfaces of input and output models in a remigration are tangent (Chapter 9, “Chained solutions for problem 2” section)

N point on the reflection-traveltime surface (Chapter 5, “Duality of reflector and reflection-traveltime surface” section) N second-derivative (or Hessian) matrices of two-point traveltimes in local Car- ˜ tesian coordinates. Meaning of indices: N A is point source at point A, with ˜ B second derivatives taken at point B; N AB is mixed second derivatives with respect to first the coordinates of A˜, then those of B (equations 33–35 of Chapter 4) NIP normal (ray) incidence point, the point where the normal ray meets the reflector (Chapter 4, “Extended NIP-wave theorem” section) O 2 × 2 zero matrix (equation 2 of Chapter 2) ˜

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P generic (ray) point (Chapter 3, “Point-source solutions” section)

P0 initial point of a ray (Chapter 3, “Point-source solutions” section) P1 point on ray 1 (Chapter 4, “Infinitesimal traveltime differences” section) P2 point on ray 2 (Chapter 4, “Infinitesimal traveltime differences” section) PG factor appearing in the elastic free-surface conversion coefficients (equations 6–8 of Appendix B)

Pw point on a wavefront on ray 2 (Chapter 4, “Inﬁnitesimal traveltime differences” section) P paraxial ray point (Chapter 3, “Vector representations” section and Figure 3) P dynamic-ray-tracing matrix (equation 134 of Chapter 3) ˜ P 1 upper left 2 × 2 submatrix of the propagator matrix in notation of Cervený.ˇ It ˜ describes the dependence of the slowness vector in ray-centered coordinates at the endpoint of the paraxial ray p(q) on the coordinates of the initial point q (equations 146 of Chapter 3)

P 2 lower left 2 × 2 submatrix of the propagator matrix in notation of Cervený.ˇ It ˜ describes the dependence of the slowness vector in ray-centered coordinates at the endpoint of the paraxial ray p(q) on that at the initial point p(q) (equations 146 of Chapter 3) PV abbreviation for the principal value of an integral (equation 19 of Chapter 3) Q generic volume containing all sources in the Kirchhoff integral (Appendix F) Q ray Jacobian matrix; dynamic-ray-tracing matrix; upper left 2 × 2 submatrix of ˜ Qˆ (equation 108 of Chapter 3) ˜ Q1 upper right 2 × 2 submatrix of the propagator matrix in notation of Cervený;ˇ it ˜ describes the dependence of the ray-centered coordinates at the endpoint of the paraxial ray q to those of the initial point q (equations 146 of Chapter 3)

Q2 lower right 2 × 2 submatrix of the propagator matrix in notation of Cervený;ˇ it ˜ describes the dependence of the ray-centered coordinates at the endpoint of the paraxial ray q on the slowness vector at the initial point p(q) (equations 146 of Chapter 3) Qˆ 3 × 3 transformation matrix from ray coordinates γˆ to ray-centered coordinates ˜ qˆ (Chapter 3, “Transformation from ray to ray-centered coordinates” section) Qˆ (r) 3 × 3 transformation matrix from ray coordinates γˆ to global Cartesian coor- ˜ dinates rˆ (Chapter 3, “Transformation from ray coordinates to global Cartesian coordinates” section) R generic volume containing all scatterers (secondary sources) in the Kirchhoff integral (Appendix F)

Rc amplitude-normalized reflection coefficient (Chapter 3, “Ray amplitude at the geophone” section; Appendix A) ¯ Rc energy-normalized or reciprocal reflection coefficient at the target reflector (equation 220 of Chapter 3)

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RPP P-P reflection coefficient (Appendix A) RPS P-S reflection coefficient (Appendix A) RSP S-P reflection coefficient (Appendix A) RSS SV-SV reflection coefficient (Appendix A) RSSH SH-SH reflection coefficient (Appendix A) R configuration rotation matrix (Chapter 2, “Measurement configurations” sec- ˜ tion) S source position (Chapter 2, “Measurement configurations” section)

SG factor appearing in the elastic free-surface conversion coefﬁcients (equations 6–8 of Appendix B) S paraxial source point (Chapter 2, “Measurement conﬁgurations” section) Sˆ ray-tracing-system matrix (equation 138 of Chapter 3) ˜ T period of a monofrequency wave traveling along the ray (equation 85 of Chap- ter 4)

Tk amplitude-normalized transmission coefﬁcient at interface k (Chapter 3, “Ray amplitude at the geophone” section) ¯ Tk energy-normalized or reciprocal transmission coefﬁcient at interface k (equation 219 of Chapter 3)

TPP P-P transmission coefficient (Appendix A) TPS P-S transmission coefficient (Appendix A) TSP S-P transmission coefficient (Appendix A) TSS SV-SV transmission coefficient (Appendix A) TSSH SH-SH transmission coefficient (Appendix A) Tˆ local Cartesian surface-to-surface propagator matrix; a 4 × 4 matrix of a paraxial ˜ ray in the notation of Bortfeld (1989), made up of four 2 × 2 matrices, A, B, C, and D. It describes the assumed linear connection between the parameters˜ ˜ of˜ the paraxial˜ ray in the beginning and the ending points of the central ray. Without an index, it refers to the whole primary reflected ray, whereas with an index 0, 1, or 2, it refers to the ray segment that is associated with that index (equation 194 of Chapter 3) U seismic trace, i.e., the recording of the scalar amplitude of the principal component of the particle displacement as an (analytic) function of time (Chapter 2, “Measurement configurations” section; equation 227 of Chapter 3; equation 1 of Chapter 7)

U0(ξ) approximate seismic-trace amplitude for negligible A (equation 3 of Chapter 7) U(˜ η,τ) desired simulated seismic record in the output space (equation 57 of Chapter 9) Uˆ analytic elastic particle-displacement vector (equation 21 of Chapter 3) V an arbitrary volume under investigation (Chapter 3, “Solution of the transport equation” section; Appendix E; Appendix F)

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V3 vertical component of the particle velocity (Appendix B) V velocity derivative matrix (equation 127 of Chapter 3) ˜ Wˆ 4 × 1or4× 2 matrix of paraxial ray quantities (equations 136 and 139 of Chap- ˜ ter 3) X 2 × 2 residual matrix (equation 184 of Chapter 3) ˜ XP cross profile: denotes a seismic experiment, in which source and receiver are dislocated perpendicular to each other, i.e., a cross profile (Chapter 2, “Mea- surement configurations” section) XS cross spread: denotes a seismic experiment, in which source and receiver are dislocated on perpendicular lines, i.e., a cross spread (Chapter 2, “Measurement configurations” section)

YR 2 × 2 auxiliary matrix (equation 42 of Chapter 5) ˜ Z spatial Hessian matrix (Appendix C) ˜ ˜ ZCR Hessian matrix of ZCR (ξ, r; M) (Appendix I) ˜ ZI isochron Hessian matrix (equation 13 of Chapter 5) ˜ ZR reﬂector Hessian matrix (equation 14 of Chapter 5) Z˜ Z − Z Hessian-matrix difference I R, i.e., Hessian matrix of difference function ˜ ˜ ˜ ZI − ZR (equation 66 of Chapter 5) Z Hessian matrix of depth function Z (equation 15 of Chapter 5) ˜ ZO zero offset: denotes a seismic experiment in which source and receiver are coincident, i.e., have zero offset, and are dislocated jointly along the seismic line (Chapter 2, “Measurement conﬁgurations” section)

Calligraphic capital letters

A amplitude factor describing the accumulated transmission losses along the ray (equation 218 of Chapter 3) B upper left 2 × 2 submatrix of Bˆ ; projection matrix from local Cartesian coordi- ˜ nates x to the global Cartesian˜ coordinates r (equation C-10 of Appendix C) Bˆ 3 × 3 rotation matrix involved in the transformation from local Cartesian ˜ coordinates xˆ to global Cartesian coordinates rˆ (equation 164 of Chapter 3) D 2 × 2 traveltime difference matrix between the Hessian traveltime matrices of ˜ the CMPO-diffraction and CMPO experiments (equation 129 of Chapter 4) F general function of six coordinates (r,z,ξ,t) that deﬁnes the Huygens and isochron surfaces (equation 3 of Chapter 5) G Green’s function, i.e., solution of the wave equation for a point source with a δ-impulse as time signal (equation 1 of Chapter 6) H Hamiltonian of the ray equations (equation 58 of Chapter 3) L point-source geometric-spreading factor (normalized); also called spherical divergence (equation 91 of Chapter 3)

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N kernel of the anisotropic, elastic Kirchhoff-Helmholtz integral (equation E-18 of Appendix E)

OD depth obliquity factor (equations 63 and 64 of Chapter 5) ODS obliquity factor of the diffraction-stack integral (equation 45 of Chapter 7) OF Fresnel obliquity factor (equation 114 of Chapter 4) OKH obliquity factor of the Kirchhoff-Helmholtz integral (equation 13 of Chapter 6) P acoustic amplitude factor (equation 23 of Chapter 3) T traveltime along a ray; frequently also called eikonal (Chapter 3, “Time- harmonic approximation” section)

T0 traveltime of the entire central ray (Chapter 4, “Paraxial traveltime” section) T01 traveltime of the descending ray segment of the central ray (Chapter 4, “Ray- segment traveltimes” section)

T02 traveltime of the ascending ray segment of the central ray (Chapter 4, “Ray- segment traveltimes” section)

T1 traveltime of the descending ray segment (source ray) (Chapter 4, “Ray-segment traveltimes” section)

T2 traveltime of the ascending ray segment (receiver ray) (Chapter 4, “Ray-segment traveltimes” section)

TCC ensemble of stacking lines for the cascaded conﬁguration transform (equation 38 of Chapter 9)

TCT stacking line for the single-step conﬁguration transform (equation 55 of Chap- ter 9)

TD diffraction-traveltime function (Chapter 2, “Hagedoorn’s imaging surfaces” section; Chapter 5, “Basic deﬁnitions” section)

TI 4D function that represents the ensemble of Huygens surfaces for all points MI on the isochron N (equation 6 of Chapter 5) TR reflection-traveltime function (Chapter 2, “Measurement configurations” section) T CO R common-offset reflection traveltime (Chapter 2, “Unified approach: Mapping” section) T ZO R zero-offset reflection traveltime (Chapter 2, “Unified approach: Mapping” section) T T − T traveltime difference D R (equation 8 of Chapter 7) T 4D function that represents the ensemble of Huygens surfaces for all points M on the target reflector R (equation 4 of Chapter 5) Tε length of the source wavelet, pulse length: f [t]=0 ∀ t/∈ (0, Tε) (equation 90 of Chapter 4) U generic amplitude factor of the principal component of the wavefield, U =|Uˆ | (equation 54 of Chapter 3)

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U(0) zero-order amplitude coefficient (equation 25 of Chapter 3) U(1) first-order amplitude coefficient (equation 25 of Chapter 3) U(P ) P-wave amplitude factor of the principal component of the wavefield (equation 43 of Chapter 3) U(S) S-wave amplitude factor of the principal component of the wavefield (equation 52 of Chapter 3) U(S) 1 S-wave component amplitude factor (equation 47 of Chapter 3) U(S) 2 S-wave component amplitude factor (equation 47 of Chapter 3) Uˆ vectorial displacement amplitude factor (discussion related to equation 12 of Chapter 3) c Uˆ displacement amplitude at a free surface (Appendix B) X amplitude factor after Kirchhoff migration (equation 10 of Chapter 8)

ZCR ensemble of stacking lines for cascaded remigration (equation 59 of Chapter 9) ZI isochron surface function (Chapter 2, “Hagedoorn’s imaging surfaces” section; Chapter 5, “Basic deﬁnitions” section)

ZM measurement surface function (Chapter 2, “Measurement conﬁgurations” section)

ZR target reflector function z = ZR(r) (Chapter 2, “Earth model” section) ZRM stacking line for single-step remigration (equation 66 of Chapter 9) Z Z − Z depth surface difference I R (equation 15 of Chapter 9) Z 4D function that represents the ensemble of isochron surfaces for all points N on the reflection-traveltime surface R (equation 7 of Chapter 5) ZR remigrated (more accurate) reflector image z˜ = ZR(ρ) (Chapter 2, “Unified approach: Mapping” section)

Greek lowercase letters

α P-wave velocity; possesses an index according to position (Table 1 and equation 41 of Chapter 3; Appendix A) β S-wave velocity; possesses an index according to position (Table 1 and equation 42 of Chapter 3; Appendix A)

βM in-plane dip angle at M (equation 27 of Chapter 9) βP in-plane surface dip angle at a point P (Chapter 3, “Transformation from local to global Cartesian coordinates” section)

βR in-plane reﬂector dip angle at MR (Chapter 5, “Basic concepts” section) γ 2D ray coordinate vector (Chapter 3, “Transformation from ray coordinates to global Cartesian coordinates” section)

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γˆ 3D ray coordinate vector (Chapter 3, “Transformation from ray coordinates to global Cartesian coordinates” section)

δCC phase function of the cascaded transformation transform (equation 49 of Chap- ter 9)

δCR phase function of the cascaded remigration (equation 69 of Chapter 9) δCT phase function of the single-step transformation transform (equation H-8 of Appendix H)

δIS phase function of the demigration integral (equation 7 of Chapter 9) δjk Kronecker delta; it equals one for j = k and otherwise is zero δRM phase function of the single-step remigration (equation H-23 of Appendix H) δ(t) Dirac delta function ζ output depth coordinate after remigration (Chapter 2, “Unified approach: Map- ping” section) η output configuration parameter after configuration transform (Chapter 2, “Unified approach: Mapping” section) + ϑk scattering angle at a reflection/transmission point at interface k (Chapter 3, “Ray amplitude at the geophone” section) − ϑk incidence angle at a reflection/transmission point at interface k (Chapter 3, “Ray amplitude at the geophone” section)

ϑ0 the angle the normal ray makes with the surface normal at the coincident source- receiver position S0 = G0 (equation 119 of Chapter 4) ϑG ray emergence angle at point G (equation 217 of Chapter 3) ϑP emergence angle at P (Chapter 3, “Transformation from local Cartesian coordinates to ray-centered coordinates” section)

ϑS ray emergence angle at point S (equation 217 of Chapter 3) ± ϑk incidence and reflection/transmission angles between the ray segment at interface k and the interface normal (Chapter 3, “Rays across an interface” section) ± ϑM incidence and reflection angles between the ray segments at M and the interface normal nˆ M ; the upper index is omitted where incidence and reflection angles = + = − are equal, i.e., where ϑM ϑM ϑM (equation 114 of Chapter 4) ± ϑR incidence and reflection angles between the ray segments at MR and the interface normal nˆ R; the upper index is omitted where incidence and reflection angles = + = − are equal, i.e., where ϑR ϑR ϑR (equation 64 of Chapter 5) κ compressibility (Table 1 of Chapter 3 only) κ KMAH index (Keller, Maslov, Arnold, and Hörmander). This index counts the number of caustics along a raypath, i.e., the number of points where the ray tube shrinks to zero. A focus point increases the KMAH index by two because the ray tube shrinks to zero in two dimensions; an index may indicate a specific ray segment (Chapter 3, “Caustics” section)

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λ Lamé parameter (Chapter 3, “Wave equations” section) μ Lamé parameter, shear modulus (Chapter 3, “Wave equations” section) ν ray variable that increases monotonically along the ray; possible variables are s, T , and σ (equation 62 of Chapter 3)

ξ configuration parameter; a 2D parameter (ξ1,ξ2) that describes the seismic measurement configuration; horizontal coordinate vector of the data space (Chapter 2, “Earth model” section) ∗ ξ stationary point of the Kirchhoff migration integral in equation 5 of Chapter 7 (Figure 2 and equation 10 of Chapter 7) ξ RM stationary point of remigration (Chapter 9, “Chained solutions for problem 2” section; Appendix H) density of the ambient medium at the current position (Chapter 3, “Wave equations” section) + k medium’s density on the reflection ray at a reflection/transmission point at interface k (Chapter 3, “Ray amplitude at the geophone” section) − k medium’s density on the incident ray at a reflection/transmission point at interface k (Chapter 3, “Ray amplitude at the geophone” section) ρ horizontal components of global Cartesian coordinates in the depth domain after remigration (Chapter 2, “Unified approach: Mapping” section) σ Poisson’s ratio (Table 1 of Chapter 3 only) σ ray variable that increases monotonically along the ray; optical length of the ray (equation 64 of Chapter 3) τ output time coordinate after configuration transform (Chapter 2, “Unified approach: Mapping” section) ϕ in-plane rotation angle between the projected slowness vector of the central ray p 0 and the x1-axis (Chapter 3, “Transformation from local Cartesian coordinates to ray-centered coordinates” section)

ϕr in-plane rotation angle between the r1-axis and the vertical plane (Chapter 3, “Transformation from local to global Cartesian coordinates” section)

ϕx in-plane rotation angle between the x1-axis and the vertical plane (Chapter 3, “Transformation from local to global Cartesian coordinates” section) ω angular frequency

Greek capital letters

M diffraction-traveltime surface (Figure 4 of Chapter 2) R reflection-traveltime surface (Figure 4 of Chapter 2) CO R common-offset reflection-traveltime surface (Chapter 2, “Unified approach: Mapping” section) ZO R zero-offset reflection-traveltime surface (Chapter 2, “Unified approach: Map- ping” section)

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G 2 × 2 conﬁguration matrix that describes the receiver position subject to the ˜ chosen seismic experiment (equation 14 of Chapter 2)

M projection matrix (equation 103 of Chapter 4) ˜ S 2 × 2 conﬁguration matrix that describes the source position subject to the ˜ chosen seismic experiment (equation 14 of Chapter 2) ˆ 3 × 3 matrix; Christoffel matrix (equation 39 of Chapter 3) ˜ ± − + R angles that the incident (R ) and reﬂected (R ) rays at MR make with the vertical axis (Figure 5 of Chapter 8) upper left 2 × 2 submatrix of ˆ ; projection matrix from ray-centered coordi- ˜ nates q to the local Cartesian coordinates˜ x (equation 159 of Chapter 3) ˆ 3 × 3 rotation matrix from ray-centered coordinates qˆ to the local Cartesian ˜ coordinates xˆ (equation 157 of Chapter 3) traveltime derivative matrix in local Cartesian coordinates x (equation 102 of ˜ Chapter 4) (r) traveltime derivative matrix in global Cartesian coordinates r (equation 11 of ˜ Chapter 5) ˆ ray-centered 4 × 4 propagator matrix of a paraxial ray in notation of Cervenýˇ ˜ (Cervenýˇ and Ravindra, 1971; Cervenýˇ et al., 1977; Cervený,ˇ 1987). It is made up of the 2 × 2 matrices Q1, Q2, P 1, and P 2. Matrices P 1, Q1 (plane-wave matrices) are obtained by˜ dynamic˜ ˜ ray tracing˜ along the˜ central˜ ray with the initial conditions of a plane wave at the starting point (P 1 = O, Q1 = I). ˜ ˜ ˜ Correspondingly, the matrix pair P 1, Q1 (point-source matrices) is obtained˜ by dynamic ray tracing along the central˜ ˜ ray with the initial conditions of a point source at the starting point (P 2 = I, Q2 = O) (equation 147 of Chapter 3) ˜ ˜ ˜ surface of the volume V (Chapter 3, “Solution˜ of the transport equation” section; Chapter 6)

1 top surface of ray tube (Chapter 3, “Solution of the transport equation” section) 2 bottom surface of ray tube (Chapter 3, “Solution of the transport equation” section)

M measurement surface (Figure 4 of Chapter 2) N isochronous surface or isochron associated with point N (Chapter 2, “Hage- doorn’s imaging surfaces” section and Figure 4; Chapter 5, “Basic deﬁnitions” section)

R reflecting interface, target reflector (Chapter 2, “Unified approach: Mapping” section and Figure 4) ˜ R more accurate target reflector image after remigration (Chapter 2, “Unified approach: Mapping” section)

ϒCC output amplitude factor of cascaded conﬁguration transform (Appendix H, “Cascaded conﬁguration transform” section)

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ϒCR output amplitude factor of cascaded remigration (Appendix H, “Cascaded remigration” section)

ϒCT output amplitude factor of single-step conﬁguration transform (equation H-9 of Appendix H)

ϒDS output amplitude factor of Kirchhoff (diffraction-stack) migration (equations 48 and 49 of Chapter 7)

ϒIS output amplitude factor of Kirchhoff (isochron-stack) demigration; (equations 5 and 6 of Chapter 9)

ϒKH output amplitude factor of Kirchhoff-Helmholtz modeling (equation 19 of Chapter 6)

ϒRM output amplitude factor of single-step remigration (equation H-24 of Appen- dix H) migration output (equation 5 of Chapter 7) b time-dependent migration output (equation 6 of Chapter 7)

TA analytic true-amplitude signal, desired result of true-amplitude migration (equation 4 of Chapter 7)

0(r) migrated trace amplitude as input for demigration (equation 3 of Chapter 9) ˆ multiple-migration output vector (equation 49 of Chapter 8) 2 × 2 in-plane rotation matrix of the ray coordinates q in direction of the ˜ Cartesian coordinates x; upper left 2 × 2 submatrix of ˆ (equation 159 of Chapter 3) ˜ ˆ 3 × 3 in-plane rotation matrix of the ray coordinates qˆ in direction of the ˜ Cartesian coordinates xˆ (equation 157 of Chapter 3) ˆ r 3 × 3 in-plane rotation matrix between the r1-axis and the vertical plane (Chap- ˜ ter 3, “Transformation from local to global Cartesian coordinates” section) ˆ x 3 × 3 in-plane rotation matrix between the x1-axis and the vertical plane (Chap- ˜ ter 3, “Transformation from local to global Cartesian coordinates” section) demigration output (equation 1 of Chapter 9) ˆ vector function (equations F-10 and F-20 of Appendix F) plane perpendicular to the ray (Chapter 3, “Transformation from local Cartesian coordinates to ray-centered coordinates” section)

0 tangent plane to the considered surface at P (Chapter 3, “Vectorrepresentations” section)

M coordinate plane at point M; its normal halves the angle between the upgoing and downgoing ray segments; thus, M is tangential to the isochron and a possible reﬂector at M (Figure 3 of Chapter 7)

R tangent plane to the target reﬂector at MR (Figure 3 of Chapter 7) T tangent plane to the considered surface at P (Chapter 3, “Vectorrepresentations” section)

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Other symbols

0 2D zero vector (Chapter 3, “Dynamic ray tracing” section) 0ˆ 3D zero vector (equations 9 and 10 of Chapter 3) ∇ 2D differential operator nabla with respect to horizontal global Cartesian coordinates; ∇ = (∂/∂r1,∂/∂r2). An index indicates that derivatives are taken with respect to other coordinates (Chapter 3, “Wave equations” section) ˆ ˆ ˆ ∇ 3D differential operator nabla; ∇ = (∂/∂r1,∂/∂r2,∂/∂r3). The symbols ∇·, ∇ˆ ×, and ∇ˆ signify the divergence, curl, and gradient operations, respectively. An index indicates that derivatives are taken with respect to other coordinates (equation 1 of Chapter 3)

Indices and accents In this sublist, we explain the indices that may vary, indicating that the variable they specify is the same but is taken at a different location or ray segment. Indices that distinguish variables from other, unrelated variables that use the same letter are not explained here.

Subscripts

i, j, k, numbering indices that take on values from 1 to 3 for 3D quantities, 1 or 2 for l,m,n 2D quantities, or 1 to n when numbering the interfaces in a system of seismic layers G speciﬁes quantities that belong to the geophone position G M speciﬁes quantities that belong to an arbitrary depth point M

R specifies quantities that belong to the reflection point MR or its dual point NR S specifies quantities that belong to the source position S 0 specifies quantities that belong the one-way normal ray and its paraxial vicinity 1 specifies quantities that belong to the descending ray segment of the central ray and its paraxial vicinity 2 specifies quantities that belong to the ascending ray segment of the central ray and its paraxial vicinity specifies a difference quantity

specifies quantities that belong to the reflection-traveltime surface R specifies quantities that belong to the reflector R

Superscripts

ref specifies quantities that belong to the reflected field s specifies quantities that belong to the scattered field

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T denotes the transpose of a vector or of a matrix −T denotes the inverse of the transpose (transpose of the inverse) of a matrix (q) specifies quantities in ray-centered coordinates (r) specifies quantities in global Cartesian coordinates (x) specifies quantities in local Cartesian coordinates + specifies ray quantities taken at an interface immediately after reflection or transmission, i.e., on the outgoing side of the interface − specifies ray quantities taken at an interface immediately before reflection or transmission, i.e., on the incidence side of the interface; nomenclature in accordance with Ursin (1990) −1 denotes the inverse of a matrix

Mathematical accents da a˙ (dot over symbol) time derivative: a˙ = dt aˇ (check over symbol) denotes quantities in the Fourier domain a˜ (tilde above symbol) marks quantities belonging to the output space of a configuration transform or a remigration a (bold symbol) 2D vector aˆ (bold symbol with hat) 3D vector aˇ (bold symbol with check) 3D vector in the Fourier domain a∗ (asterisk after vector symbol) marks the stationary points of the migration and demigration integrals a (prime after vector symbol) denotes quantities at the endpoint of a transmitted ray or at the reflection point of a reflected ray a (double prime after vector symbol) denotes quantities at the endpoint of a reflected ray A¯ (short bar over symbol) distinguishes the reciprocal (i.e., energy-normalized) ¯ ¯ reflection and transmission coefficients (Rc, T ) from the standard (amplitude- normalized) ones (Rc, T ) A (long bar over symbol) denotes points in the paraxial vicinity of a corresponding point without bar A (tilde below bold symbol) 2 × 2 matrices ˜ Aˆ (tilde below bold symbol with hat) 3 × 3 matrices ˜ Aˆ (tilde below bold symbol with double hat) 4 × 4 matrices ˜ A∗ (asterisk after matrix symbol) marks propagator matrices that pertain to the ˜ reverse ray

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Operational symbols

· symbolizes the scalar (or inner) product of two vectors (∇ · a indicates the divergence operation), i.e., the sum of the products of the corresponding components of these vectors (in matrix notation: a · b = aT b) × symbolizes the vector product of two vectors (∇ × a indicates the curl operation) or any kind of multiplication at a line break

Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/4158418/9781560801672.fm.pdf by guest on 24 September 2021 “1127fm” — 2007/6/26 — page xlvi — 16:59 — Stage II — #46