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HANDBOOK OF GEOPHYSICAL EXPLORATION

SEISMIC EXPLORATION

VOLUME 29

SEISMIC SIGNATURES AND ANALYSIS OF REFLECTION DATA IN ANISOTROPIC MEDIA

HANDBOOK OF GEOPHYSICAL EXPLORATION

SEISMIC EXPLORATION

Editors: Klaus Helbig and Sven Treitel

Volume 1. Basic Theory in Reflection Seismology ~ 2. Seismic Instrumentation, 2nd Edition ~ 3. Seismic Field Techniques 2 4A. Seismic Inversion and Deconvolution: Classical Methods 4B. Seismic Inversion and Deconvolution: Dual-Sensor Technology 5. Seismic Migration (Theory and Practice) 6. Seismic Velocity Analysis ~ 7. Seismic Noise Attenuation 8. Structural Interpretation 2 9. Seismic Stratigraphy 10. Production Seismology 11.3-D Seismic Exploration 2 12. Seismic Resolution 13. Refraction Seismics 14. Vertical Seismic Profiling: Principles

3rd Updated and Revised Edition 15A. Seismic Shear Waves: Theory 15B. Seismic Shear Waves: Applications 16A. Seismic Coal Exploration: Surface Methods 2 16B. Seismic Coal Exploration: In-Seam Seismics 17. Mathematical Aspects of Seismology 18. Physical Properties of Rocks 19. Shallow High-Resolution Reflection Seismics 20. Pattern Recognition and Image Processing 21. Supercomputers in Seismic Exploration 22. Foundations of Anisotropy for Exploration Seismics 23. Seismic Tomography 2 24. Borehole Acoustics ~ 25. High Frequency Crosswell Seismic Profiling 2 26. Applications of Anisotropy in Vertical Seismic Profiling 1 27. Seismic Multiple Elimination Techniques 1 28. Wavelet Transforms and Their Applications to Seismic Data

Acquisition, Compression, Processing and Interpretation 1 29. Seismic Signatures and Analysis of Reflection Data

in Anisotropic Media

~In preparation. 2planned.

SEISMIC EXPLORATION

Volume 29

SEISMIC SIGNATURES AND ANALYSIS OF REFLECTION DATA IN ANISOTROPIC MEDIA

by

Ilya TSVANKIN Professor of Geophysics Center for Wave Phenomena Department of Geophysics Colorado School of Mines Golden, CO, USA

) 2001 P E R G A M O N An Imprint of Elsevier Science Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo

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First edition 2001

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Dedicated to the memory of my parents, Daniel and Maya

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Preface

This book provides background information about anisotropic wave propagation and discusses modeling, inversion and processing of seismic reflection data in anisotropic media. Seismic anisotropy is hardly a new topic in the geophysical literature, with the first contributions made by Polish scientist M.P. Rudzki in the last decade of the 19th century and the beginning of the 20th century (for a detailed historical overview, see Helbig, 1994). Also, a comprehensive theoretical treatment of wave propagation in anisotropic solids has been developed in crystal acoustics (Fedorov, 1968; Musgrave, 1970; Auld, 1973).

Still, for most of its history seismic inversion and processing has been based on the assumption that the subsurface is isotropic, despite the general acceptance of the fact that most geologic formations possess a certain degree of anisotropy. The reluctance to treat anisotropic models was quite understandable because the problem of reconstructing even isotropic velocity fields from seismic data acquired at the Earth surface (and, sometimes, in boreholes) is difficult and ill-posed without simplifying assumptions. Why then add another level of complexity that may not be constrained by the available data? Also, the mathematics needed to describe anisotropic wave phenomena seemed too involved and often counterintuitive for most geophysicists.

The change in the attitude toward anisotropy in the exploration community can be traced back to the mid-1980's, when the work of Stuart Crampin, Rusty Alford, Leon Thomsen and others made it clear that processing of shear-wave data requires accounting for S-wave splitting caused by azimuthal anisotropy (commonly related to natural fractures). In contrast, the influence of anisotropy on compressional (P) waves, which represent a majority of data being acquired in oil and gas exploration, is not nearly as dramatic. Although P-wave velocity in anisotropic media can change significantly with propagation angle, P-waves do not split into two modes and their reflection moveout on conventional-length spreads (close to reflector depth) typically is hyperbolic. Hence, it has been customary for processors and interpreters to artifi- cially adjust the parameters of the conventional (i.e., isotropic) processing flow when working with P-wave data from anisotropic media. This approach, however, has produced distorted velocity models and proved to be inadequate in compensating for the full range of anisotropic phenomena in P-wave imaging, particularly in prestack depth migration or when working with multicomponent data.

One of the most pervasive anisotropy-induced distortions in P-wave processing is the wrong depth scale of seismic models caused by the difference between the vertical and stacking (moveout) velocities in anisotropic media. Also, ignoring the angle dependence of velocity creates serious problems in imaging of dipping reflectors (such as faults) beneath or inside anisotropic formations. Massive acquisition of

vii

viii Preface

large-offset offshore data has revealed another common manifestation of anisotropy - nonhyperbolic moveout on long spreads that cannot be reproduced with isotropic models.

This mounting evidence of the need to account for anisotropy in seismic processing prompted an increased effort in anisotropic velocity analysis and imaging in the late 1980's and early 1990's (e.g., Byun et al, 1989; Kitchenside, 1991; Lynn et al., 1991). While extending migration and dip-moveout (DMO) methods to anisotropic media is largely a technical issue, practical implementation of the anisotropic processing algorithms was hampered primarily by the difficulties in parameter estimation. Inverting for the several anisotropic parameters needed to characterize even the simplest anisotropic model - transverse isotropy - seemed to be well beyond the reach of reflection seismology.

The breakthrough that happened during the past decade was in identifying the key parameters for time and depth imaging in anisotropic media and developing practical methodologies for estimating them from seismic data. For example, time-domain processing of P-wave data in transversely isotropic media with a vertical symmetry axis (VTI) was proved to be controlled by a single anisotropic coefficient (77) that can be determined from P-wave reflection traveltimes (Alkhalifah and Tsvankin, 1995). The research in anisotropic velocity analysis and parameter estimation, spearheaded by the Center for Wave Phenomena (CWP) at the Colorado School of Mines, was built on the pioneering work of Thomsen (1986), who introduced a new notation for TI media that greatly simplified analytic description of seismic signatures. In addition to improving seismic images of exploration targets, anisotropic parameters were shown to provide valuable information for lithology discrimination and characterization of fracture networks. Those results, which finally made anisotropic processing a practical endeavor with far-reaching exploration benefits, are the main focus of this book.

The most recent development that has put anisotropic models at the forefront of seismic processing is the technology of multicomponent ocean-bottom seismology (OBS). The high-quality converted-wave (PS) data acquired on the sea floor were effectively used in several exploration scenarios, most notably for imaging targets beneath gas clouds (e.g., Granli et al., 1999). Isotropic processing of PS-waves, however, often turned out to be inadequate because the influence of anisotropy on mode conversions generally is more substantial that that on P-waves. Mis-ties between PP and PS sections (such as different depths of reflectors) could not be removed without taking anisotropy into account. Hence, significant attention in the book is devoted to the kinematic properties of converted waves in anisotropic media and velocity-analysis methods operating with PP and PS data.

Although the emphasis of the book is on applications of anisotropic models in reflection seismology, some background information about anisotropic wave propagation is given in C h a p t e r 1 and the first section of Chapter 2. A more detailed discussion of the theoretical aspects of seismic anisotropy can be found in Helbig (1994) [other useful references are the books by Fedorov (1968), Musgrave (1970) and Auld (1973) mentioned above]. Chapter 1 also introduces Thomsen notation for transverse

Preface ix

isotropy and then extends it to the more complicated, but possibly quite realistic, orthorhombic model. Note that one of the main reasons for the difficulties in moving anisotropy into the mainstream of seismic processing was in the conflicting notations used in the anisotropic literature. As demonstrated throughout the book, Thomsen parameters not only simplify the description of a wide range of seismic signatures, they also provide valuable insight into the influence of anisotropy on seismic velocities and amplitudes.

C h a p t e r 2 deals with the dynamic aspects of wave propagation in anisotropic media. The Green's function for a homogeneous anisotropic medium is derived as a Weyl-type integral over plane waves, and then simplified for the far-field using the stationary-phase approximation. The analytic results and numerical modeling are used to study the influence of anisotropy on body-wave polarizations and radiation patterns from point forces, including the dramatic phenomenon of focusing and de- focusing of energy associated with angle-dependent velocity. The second section of Chapter 2 discusses the amplitude-variation-with-offset (AVO) response for P- and S-waves in VTI media. Anisotropy may cause serious (and comparable) distortions in both the reflection coefficient and the amplitude distribution along the wavefront propagating through the overburden.

Normal-moveout (NMO) velocity - a signature of critical importance in the velocity analysis of reflection d a t a - is the subject of C h a p t e r 3. A general 2-D NMO equation is used to give a concise analytic description of dip-dependent NMO velocity for homogeneous TI models with a vertical and tilted axis of symmetry. Extension of the classical Dix equation to symmetry planes of layered anisotropic media helps to relate the effective and interval NMO velocities for dipping reflectors and to express anisotropy-induced errors in time-to-depth conversion for VTI media in terms of Thomsen parameter 5. The chapter also presents a 3-D treatment of azimuthally dependent NMO velocity based on the equation of the NMO ellipse, with explicit solutions given for TI and orthorhombic media.

Discussion of reflection traveltimes in anisotropic media is continued in Chap- t e r 4, which is devoted to nonhyperbolic (long-spread) moveout. The influence of anisotropy on large-offset traveltimes in horizontally layered media is explained using the quartic (fourth-order) moveout coefficient. The most important result of this chapter is a general nonhyperbolic moveout equation (Tsvankin and Thomsen, 1994), which remains sufficiently accurate for P- and PS-waves in a wide range of anisotropic models. For P-waves in VTI media, this equation is rewritten in terms of the "anellipticity" parameter r/ which, as shown in Chapter 6, plays a key role in time-domain processing.

C h a p t e r 5 generalizes the results of Chapters 3 and 4 for reflection moveout of mode-converted waves. Instead of modifying the traveltime series t(x) to account for the asymmetry of PS-wave moveout, the traveltime-offset relationship is expressed in parametric form through the components of the slowness vector. This represen- tation, developed for both 2-D and 3-D (wide-azimuth) geometry, helps to generate common-midpoint (CMP) gathers without two-point ray tracing and leads to closed-

x P r e f a c e

form expressions for NMO velocity and other moveout attributes of PS-waves. The formalism of Chapter 5 provides a foundation for the joint traveltime inversion of P and PS data in VTI media discussed in Chapter 7.

Analysis of time-domain signatures of P-waves for transverse isotropy with a vertical and tilted axis of symmetry is presented in C h a p t e r 6. For laterally homogeneous VTI models above the target reflector, P-wave moveout is shown to depend on just two medium parameters- the NMO velocity for a horizontal reflector Vnmo(0) and the coefficient 7. These parameters are sufficient to perform all P-wave time-processing steps in VTI media including NMO and DMO corrections, prestack and poststack time migration. Chapter 6 also contains a general overview of P-wave signatures in VTI media and summarizes the advantages of Thomsen notation.

C h a p t e r 7 addresses one of the most important problems of anisotropic processing- parameter estimation in VTI media. Synthetic examples and case studies demonstrate that velocity analysis for purposes of time-domain P-wave imaging is feasible in routine practice. The time-processing parameters Vnmo(0) and 77 can be estimated from surface P-wave data alone using either dip-dependent NMO velocity or nonhyperbolic moveout for horizontal reflectors. To build VTI velocity models in depth, dip-dependent reflection moveout of P-waves is combined with that of converted PSV-waves in both 2-D and 3-D inversion algorithms.

P-wave DMO and migration methods for vertical transverse isotropy are discussed in C h a p t e r 8. Extension of Fowler DMO to VTI media results in a complete time-processing sequence that includes parameter estimation, DMO correction and poststack Stolt migration. Another DMO algorithm, designed for NMO-corrected data acquired in symmetry planes of anisotropic media, represents a generalization of Hale's isotropic DMO by Fourier transform. Basic features of TI migration and the distortions caused by applying isotropic codes to anisotropic data are described in the section devoted to phase-shift (Gazdag) time migration and Gaussian beam depth migration. Field-data examples illustrate significant improvements in P-wave imaging achieved by the anisotropic methods and the possibility of using the derived anisotropic coefficients in lithology discrimination.

While seismic signatures and processing algorithms for TI media with a vertical and (to a less extent) tilted symmetry axis are treated in sufficient detail, inversion/processing methods for lower-symmetry models have been largely left out of the book. The rapid advances in the analysis of wide-azimuth multicomponent data from azimuthally anisotropic media, which make this area of research one of the most exciting in anisotropic seismology, will be the main topic of a follow-up monograph. Also, the book contains a theoretical and numerical analysis of shear-wave splitting (Chapters 1 and 2), but does not describe processing of split S-wave data - a subject addressed in many journal publications and a monograph by MacBeth (2001).

The book is written in such a way that it should be useful for both graduate students and more experienced geophysicists working in research, exploration or development. There is no doubt that proper understanding of anisotropic processing requires working knowledge of the mathematical tools used in anisotropic wave propagation.

Preface xi

However, I have always believed that success in dealing with anisotropy requires cut- ting through the sometimes overwhelming complexity of anisotropic mathematics and identifying the mathematical details critical in solving a particular problem. There- fore, most mathematical derivations are explained at the simplest possible level and relegated from the main text into appendices; some more involved mathematical results are referenced but not reproduced in the book. To make sure that the main conclusions and their practical implications are not buried under details, they are highlighted in the discussion and summary sections.

I am deeply grateful to many people without whom this book would not have been written. Evgeny Chesnokov invited me to his laboratory in the mid-1980's and exposed me for the first time to the exciting field of seismic anisotropy. Leon Thomsen introduced me to the exploration aspects of anisotropy and provided guidance through the first steps of my career in the United States. Fruitful collaboration with Leon has been indispensable in developing the key ideas of this monograph. Many results described in the book have been obtained by the A(nisotropy)-Team at CWP, and I owe a dept of thanks to my CWP colleagues, especially to Vladimir Grechka, Ken Larner and the late Jack Cohen. Ken Larner has been particularly instrumental in developing and supporting the anisotropic program at CWP.

Significant contributions to the material in the book have been made by Tariq Alkhalifah, formerly a CWP student, John Anderson, who completed his PhD at CSM while being an employee of Mobil, and John Toldi of Chevron. The book has also benefited from the results of CWP students Andreas Riiger, Abdulfattah A1-Dajani, Baoniu Han and Tagir Galikeev. I would like to thank my colleagues Phil Anno, Andrey Bakulin, Richard Bale, Pat Berge, James Berryman, Leonid Brodov, James Brown, Bok Byun, John Castagna, Dennis Corrigan, Stuart Crampin, Joe Dellinger, Dan Ebrom, Paul Fowler, James Gaiser, Dirk Gajewski, Dave Hale, Andrzej Hanyga, Zvi Koren, Peter Leary, Yves Le Stunff, Jacques Leveille, Frank Levin, Xiang-Yang Li, Heloise Lynn, Colin MacBeth, Mark Meadows, Michael Mueller, Francis Muir, Gerhard Pratt, Ivan P~en6lk, Fuhao Qin, Patrick Rasolofosaon, Jazz Rathore, BjSrn Rommel, Colin Sayers, Michael Schoenberg, Arcangelo Sena, Serge Shapiro, Risto Siliqi, Jaime Stein, Paul Williamson, Peter Wills, Don Winterstein, and others for many stimulating discussions on various aspects of seismic anisotropy.

The idea of the book was suggested to me by the editors of this series, Klaus Helbig and Sven Treitel. Thorough reviews by Vladimir Grechka, Klaus Helbig, Ken Larner and Andreas Riiger have helped to substantially improve the text. John Stockwell and Barbara McLenon of CWP have provided invaluable assistance with setting up the LaTeX files and preparing the manuscript for publication.

My research in anisotropy at the Colorado School of Mines has been supported by the Consortium Project on Seismic Inverse Methods for Complex Structures at CWP and by the Office of Basic Energy Sciences of the United States Department of Energy.

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Contents

1 E l e m e n t s of bas ic t h e o r y of a n i s o t r o p i c wave p r o p a g a t i o n 1 1.1 Governing equations and plane-wave properties . . . . . . . . . . . . 2

1.1.1 Wave equation and Hooke's law . . . . . . . . . . . . . . . . . 2 1.1.2 Christoffel equation and properties of plane waves . . . . . . . 3 1.1.3 Group (ray) velocity . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Anisotropic symmetry systems . . . . . . . . . . . . . . . . . . 7

1.2 Plane waves in transversely isotropic media . . . . . . . . . . . . . . . 14 1.2.1 Solutions of the Christoffel equation . . . . . . . . . . . . . . . 14 1.2.2 Thomsen notation for transverse isotropy . . . . . . . . . . . . 17 1.2.3 Exact and approximate phase and group velocity . . . . . . . 21 1.2.4 Polarization vector and relationship between phase, group and

polarization directions . . . . . . . . . . . . . . . . . . . . . . 34 1.3 Plane waves in orthorhombic media . . . . . . . . . . . . . . . . . . . 36

1.3.1 Limited equivalence between TI and orthorhombic media . . . 37 1.3.2 Anisotropic parameters for orthorhombic media . . . . . . . . 40 1.3.3 Signatures in the symmetry planes . . . . . . . . . . . . . . . 44 1.3.4 P-wave velocity outside the symmetry planes . . . . . . . . . 46 1.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Appendices for Chapter 1 1A Phase velocity in arbitrary anisotropic media . . . . . . . . . . . . . . 56 1B Group-velocity vector as a function of phase velocity . . . . . . . . . 57

2 In f luence of a n i s o t r o p y on p o i n t - s o u r c e r a d i a t i o n a n d AVO ana lys i s 61 2.1 Point-source radiation in anisotropic media . . . . . . . . . . . . . . . 62

2.1.1 Green's function in homogeneous anisotropic media . . . . . . 62 2.1.2 Numerical analysis of point-source radiation . . . . . . . . . . 67 2.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.2 Radiation patterns and AVO analysis in VTI media . . . . . . . . . . 81 2.2.1 Radiation patterns for weak transverse isotropy . . . . . . . . 82 2.2.2 P-wave radiation pattern . . . . . . . . . . . . . . . . . . . . . 84 2.2.3 P-wave reflection coefficient in VTI media . . . . . . . . . . . 91 2.2.4 AVO signature of shear waves . . . . . . . . . . . . . . . . . . 94 2.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Appendices for Chapter 2 2A Derivation of the anisotropic Green's function . . . . . . . . . . . . . 103 2B Weak-anisotropy approximation for radiation patterns in TI m e d i a . . 105

xiii

xiv CONTENTS

3 N o r m a l - m o v e o u t v e l o c i t y in l a y e r e d a n i s o t r o p i c m e d i a 109 3.1 2-D NMO equation in an anisotropic layer . . . . . . . . . . . . . . . 110

3.1.1 General expression for dipping reflectors . . . . . . . . . . . . 110 3.1.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.2 NMO velocity for vertical transverse isotropy . . . . . . . . . . . . . . 113 3.2.1 Horizontal reflector . . . . . . . . . . . . . . . . . . . . . . . . 113 3.2.2 Elliptical anisotropy . . . . . . . . . . . . . . . . . . . . . . . 114 3.2.3 Weak-anisotropy approximation for general VTI media . . . . 115 3.2.4 Dip-dependent NMO velocity of P-waves . . . . . . . . . . . . 118

3.3 NMO velocity for ti l ted TI media . . . . . . . . . . . . . . . . . . . . 130 3.3.1 Absence of reflections from steep interfaces . . . . . . . . . . . 131 3.3.2 Dip-dependent P-wave NMO velocity . . . . . . . . . . . . . . 139

3.4 NMO velocity in layered media and t ime-to-depth conversion . . . . . 149 3.4.1 2-D Dix-type NMO equation for dipping reflectors . . . . . . . 149 3.4.2 Horizontally layered media and t ime-to-depth conversion . . . 151

3.5 Elements of 3-D analysis of NMO velocity . . . . . . . . . . . . . . . 156 3.5.1 Equat ion of the NMO ellipse . . . . . . . . . . . . . . . . . . . 156 3.5.2 NMO ellipse in VTI media . . . . . . . . . . . . . . . . . . . . 159 3.5.3 NMO ellipse in orthorhombic and HTI media . . . . . . . . . 161

Appendices for Chapter 3 3A 2-D NMO equation in an anisotropic layer . . . . . . . . . . . . . . . 166 3B Weak-anisotropy approximat ion for P-wave NMO velocity in T T I media l68 3C 2-D Dix-type equation in layered anisotropic media . . . . . . . . . . 169 3D 3-D NMO equation in heterogeneous anisotropic media . . . . . . . . 170

4 N o n h y p e r b o l i c r e f l e c t i o n m o v e o u t 173 4.1 Quart ic moveout coefficient . . . . . . . . . . . . . . . . . . . . . . . 176

4.1.1 General 2-D equation for a single layer . . . . . . . . . . . . . 176 4.1.2 Explicit expressions for VTI media . . . . . . . . . . . . . . . 177 4.1.3 Layered media . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.2 Nonhyperbolic moveout equation . . . . . . . . . . . . . . . . . . . . 182 4.2.1 Weak-anisotropy approximations . . . . . . . . . . . . . . . . 183 4.2.2 General long-spread moveout equation . . . . . . . . . . . . . 184

4.3 P-wave moveout in VTI media in terms of the parameter z] . . . . . . 185 4.3.1 Single layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.3.2 Layered media . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4.4 Long-spread moveout of SV-waves in VTI media . . . . . . . . . . . 190 4.4.1 Models with negative a . . . . . . . . . . . . . . . . . . . . . . 190 4.4.2 Positive a and models with cusps . . . . . . . . . . . . . . . . 190

Appendices for Chapter 4 4A Weak-anisotropy approximat ion for long-spread moveout . . . . . . . 195 4B P-wave moveout in layered VTI media . . . . . . . . . . . . . . . . . 197

C O N T E N T S xv

5 Ref lec t ion m o v e o u t of m o d e - c o n v e r t e d waves 199 5.1 Dip-dependent moveout of PS-waves in a single layer (2-D) . . . . . . 200

5.1.1 Paramet r ic representat ion of P S t ravel t ime . . . . . . . . . . 201 5.1.2 At t r ibutes of the P S moveout function . . . . . . . . . . . . . 204

5.2 Applicat ion to a VTI layer . . . . . . . . . . . . . . . . . . . . . . . . 208 5.2.1 Weak-anisotropy approximat ion for P S moveout . . . . . . . . 208 5.2.2 Recovery of PS-wave moveout curve . . . . . . . . . . . . . . 213

5.3 3-D t r ea tmen t of PS-wave moveout for layered media . . . . . . . . . 219 5.3.1 2-D expressions for vertical symmet ry planes . . . . . . . . . . 219 5.3.2 3-D description of P S moveout . . . . . . . . . . . . . . . . . 223 5.3.3 Moveout a t t r ibu tes in layered media . . . . . . . . . . . . . . 226

5.4 PS-wave moveout in horizontally layered VTI media . . . . . . . . . 228 5.4.1 Taylor series coefficients . . . . . . . . . . . . . . . . . . . . . 228 5.4.2 Nonhyperbol ic moveout equat ion . . . . . . . . . . . . . . . . 230

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Appendices for Chap te r 5 5A 2-D description of P S moveout in a single layer . . . . . . . . . . . . 233 5B 3-D expression for the slope of CMP moveout . . . . . . . . . . . . . 235 5C NMO velocity for converted-wave moveout . . . . . . . . . . . . . . . 239 5D Weak-anisotropy approximat ion for P S - m o v e o u t in VTI media . . . . 241

5D.1 Paramet r ic expressions for the t ravel t ime curve . . . . . . . . 241 5D.2 Moveout a t t r ibu tes . . . . . . . . . . . . . . . . . . . . . . . . 244

5E 3-D description of P S moveout in layered media . . . . . . . . . . . . 246 5E.1 Single anisotropic layer . . . . . . . . . . . . . . . . . . . . . . 246 5E.2 Layered media . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5E.3 2-D relat ionships for symmet ry planes . . . . . . . . . . . . . 251

6 P-wave t i m e - d o m a i n s i g n a t u r e s in transverse ly i sotropic m e d i a 253 6.1 P-wave NMO velocity as a function of ray pa rame te r . . . . . . . . . 254

6.1.1 2-D analysis for a VTI layer . . . . . . . . . . . . . . . . . . . 254 6.1.2 Dip plane of a layered medium . . . . . . . . . . . . . . . . . . 261 6.1.3 3-D analysis using the NMO ellipse . . . . . . . . . . . . . . . 263

6.2 Two-paramete r description of t ime processing . . . . . . . . . . . . . 264 6.2.1 Migrat ion impulse response . . . . . . . . . . . . . . . . . . . 264 6.2.2 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . 266

6.3 Discussion: Nota t ion and P-wave signatures in VTI media . . . . . . 269 6.3.1 Advantages of Thomsen parameters . . . . . . . . . . . . . . . 269 6.3.2 Influence of vertical transverse isotropy on P-wave s igna tu res . 270

6.4 Moveout analysis for t i l ted symmet ry axis . . . . . . . . . . . . . . . 272 6.4.1 NMO velocity as a function of ray pa ramete r . . . . . . . . . . 272 6.4.2 Pa rame te r ~ for t i l ted axis of symmet ry . . . . . . . . . . . . 274 6.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

XVI C O N T E N T S

Appendices for Chap te r 6 6A Dependence of NMO velocity in VTI media on the ray pa ramete r . . 283

6A.1 Building the function Vnmo(P) . . . . . . . . . . . . . . . . . . 283 6A.2 Elliptical anisotropy . . . . . . . . . . . . . . . . . . . . . . . 283 6A.3 Weak transverse isotropy . . . . . . . . . . . . . . . . . . . . . 284

6B NMO velocity in t i l ted elliptical media . . . . . . . . . . . . . . . . . 285

7 V e l o c i t y a n a l y s i s and parameter e s t i m a t i o n for V T I m e d i a 287 7.1 P-wave dip-moveout inversion for r/ . . . . . . . . . . . . . . . . . . . 289

7.1.1 Inversion in the dip plane of a VTI layer . . . . . . . . . . . . 289 7.1.2 2-D inversion in vertically heterogeneous media . . . . . . . . 291 7.1.3 3-D inversion of azimuthal ly varying NMO velocity . . . . . . 297 7.1.4 Fie ld-data example . . . . . . . . . . . . . . . . . . . . . . . . 302 7.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

7.2 Inversion of P-wave nonhyperbolic moveout . . . . . . . . . . . . . . 312 7.2.1 Single VTI layer . . . . . . . . . . . . . . . . . . . . . . . . . 312 7.2.2 Nonhyperbol ic velocity analysis for layered media . . . . . . . 321 7.2.3 Fie ld-data examples . . . . . . . . . . . . . . . . . . . . . . . 327 7.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

7.3 Joint inversion of P and P S da ta . . . . . . . . . . . . . . . . . . . . 334 7.3.1 S-waves in paramete r es t imat ion for VTI media . . . . . . . . 335 7.3.2 2-D inversion of horizontal and dipping events . . . . . . . . . 337 7.3.3 3-D inversion of wide-azimuth da ta . . . . . . . . . . . . . . . 346 7.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

8 P - w a v e i m a g i n g for V T I m e d i a 353 8.1 Fowler-type t ime-processing me thod . . . . . . . . . . . . . . . . . . . 354

8.1.1 Fowler DMO in isotropic media . . . . . . . . . . . . . . . . . 355 8.1.2 Extension to VTI media . . . . . . . . . . . . . . . . . . . . . 356 8.1.3 Synthetic example . . . . . . . . . . . . . . . . . . . . . . . . 360 8.1.4 Fie ld-da ta example . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

8.2 Dip moveout by Fourier t ransform . . . . . . . . . . . . . . . . . . . . 369 8.2.1 Hale's DMO method . . . . . . . . . . . . . . . . . . . . . . . 370 8.2.2 2-D Hale DMO for anisotropic media . . . . . . . . . . . . . . 372 8.2.3 Applicat ion to VTI media . . . . . . . . . . . . . . . . . . . . 373 8.2.4 Synthetic examples . . . . . . . . . . . . . . . . . . . . . . . . 375 8.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

8.3 Time and depth migrat ion . . . . . . . . . . . . . . . . . . . . . . . . 385 8.3.1 Phase-shift (Gazdag) migrat ion . . . . . . . . . . . . . . . . . 385 8.3.2 Gaussian beam migrat ion . . . . . . . . . . . . . . . . . . . . 389

8.4 Synthet ic example for a model from the Gulf of Mexico . . . . . . . . 400 8.4.1 Pa rame te r es t imat ion . . . . . . . . . . . . . . . . . . . . . . . 400 8.4.2 Depth migra t ion . . . . . . . . . . . . . . . . . . . . . . . . . 403

CONTENTS xvii

8.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 8.5 Fie ld-data example with multiple fault planes . . . . . . . . . . . . . 410

8.5.1 Pa ramete r es t imat ion . . . . . . . . . . . . . . . . . . . . . . . 410 8.5.2 Time imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

8.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

R e f e r e n c e s 419

A u t h o r I n d e x 429

S u b j e c t I n d e x 431

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Chapter 1

Elements of basic theory of anisotropic wave propagation

A medium (or a region of a continuum) is called anisotropic with respect to a certain parameter if this parameter changes with the direction of a measurement. If an elastic medium is anisotropic, seismic waves of a given type propagate in different directions with different velocities. This velocity anisotropy implies the existence of a certain structure (order) on the scale of seismic wavelength imposed by various physical phenomena. In typical subsurface formations, velocity changes with both spatial position and propagation direction, which makes the medium heterogeneous and anisotropic. The notions of heterogeneity and anisotropy are scale-dependent, and the same medium may behave as heterogeneous for small wavelengths and as anisotropic for large wavelengths (e.g., Helbig, 1994). For example, such small-scale heterogeneity as fine layering detectable by well logs may create an effectively anisotropic model in the long-wavelength limit.

Anisotropy in sedimentary sequences is caused by the following main factors (e.g., Thomsen, 1986):

�9 intrinsic anisotropy due to preferred orientation of anisotropic mineral grains or the shapes of isotropic minerals;

�9 thin bedding of isotropic layers on a scale small compared to the wavelength (the layers may be horizontal or tilted);

�9 vertical or dipping fractures or microcracks.

It is common to see anisotropy produced by a certain combination of these factors. For instance, systems of vertical fractures may develop in finely layered sediments, or the thin layers themselves may be intrinsically anisotropic. As a result, subsurface formations may possess several anisotropic symmetries, each with a different character of wave propagation (subsection 1.1.4).

This chapter is devoted to the basics of wave propagation in anisotropic media with an emphasis on velocities and polarization of plane waves. Many general theoretical developments below (especially those in the first section) have been discussed in detail by Helbig (1994) and in several other monographs (e.g., Musgrave, 1970; Aki and Richards, 1980; Payton, 1983). The main purpose of revisiting anisotropic wave

2 CHAPTER 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION

propagation here is to present several analytic results in the form most suitable for application in seismic inversion and processing, and to establish a convenient notation that simplifies analysis of seismic data. In particular, section 1.2 introduces Thomsen parameters for transverse isotropy and demonstrates their advantages in understanding the influence of anisotropy on seismic signatures. Extension of Thomsen notation to the more complicated orthorhombic model is presented in section 1.3.

1.1 Governing equations and plane-wave properties

1 . 1 . 1 W a v e e q u a t i o n a n d H o o k e ' s law

The wave equation for general anisotropic heterogeneous media follows from the second Newton's law applied to a volume AV within a continuum. Expressing the tractions (contact forces) acting across the surface of AV in terms of the stress tensor ~-ij yields (e.g., Aki and Richards, 1980)

02ui OTij P cgt 2 Oxj = fi , (1.1)

where p is the density, u = (Ul, u2, u3) is the displacement vector, f = (fl, 5 , f3) is the body {external) force per unit volume, t is the time and xi are the Cartesian coordinates. Summation over j = 1, 2, 3 {and all other repeated indices below} is implied; i = 1, 2, 3 is a free index.

For a medium with a given density and a certain spatial distribution of applied body forces f(x), equation (1.1) contains two unknowns: the displacement field u and the stress tensor Tij. Hence, the wave equation cannot be solved for displacement unless it is supplemented with the so-called "constitutive relations" between stress and strain (or stress and displacement). In the limit of small strain, which is sufficiently accurate for most applications in seismic wave propagation, the stress-strain relationship is linear and is described by the generalized Hooke's law:

Tij = CijkZ ekl . (1.2)

Here Cijkl is the fourth-order s t i f fness tensor responsible for the material properties (it is discussed in detail below), and ekl is the strain tensor defined as

ekl -- -~ ~ + ~Xk " (1.3)

Equivalently, Hooke's law can be written through the compl iance tensor 3 i j k l ,

eij = Sijkt TkZ . (1.4)

Restricting the wave-propagation theory to linearly elastic media by adopting Hooke's law (1.2) is the most crucial simplifying assumption in both isotropic and

1.1. GOVERNING EQUATIONS AND PLANE-WAVE PROPERTIES 3

anisotropic wave propagation. Allowing quadratic and higher-order terms in the stress-strain relationship leads to a n o n l i n e a r wave equation that is quite difficult to solve. For example, nonlinear terms in displacement preclude application of such powerful tools of linear theory as the principle of superposition, Fourier transforms, etc.

Substituting Hooke's law (1.2) and the definition (1.3) of the strain tensor into the general wave equation (1.1), and assuming that the stiffness coefficients are either constant or vary slowly in space (so that their spatial derivatives can be neglected), we find

02ui 02uk P-b~- - c~jk~ OxjOzt = f ~ (1.5)

Equation (1.5) is valid for linearly elastic, arbitrary anisotropic, homogeneous (or weakly heterogeneous) media. Most of the results in this book are ultimately based on solutions of the wave equation (1.5). The wave equation for isotropic media can be obtained by using the isotropic form of the stiffness tensor ciikl [see equation (1.28) below].

1.1.2 Christoffel equation and properties of plane waves

To give an analytic description of plane waves in anisotropic media, we make equation (1.5) homogeneous by dropping the body force f:

02ui 02uk P - - ~ - - Cijkl i)XjOXl = O. (1.6)

Physically, the homogeneous wave equation describes a medium without sources of elastic energy. As a trial solution of equation (1.6), we use a harmonic (steady-state) plane wave represented by

Uk - Uk e i~(n~xi/v-t) , (1.7)

where Uk are the components of the po l a r i z a t i on vector U, w is the angular frequency, V is the velocity of wave propagation (usually called p h a s e velocity), and n is the unit vector orthogonal to the plane wavefront (the wavefront satisfies n j x j - V t -

cons t ) . As demonstrated below, another quantity particularly useful in anisotropic wave theory is the s l o w n e s s vector p = n / V .

Substituting the plane wave (1.7) into the wave equation (1.6) leads to the so- called Christoffel equation for the phase velocity V and polarization vector U:

G21 G22 - p V 2 G23 U2 - 0 . (1.8) G31 G32 G33 - pV 2 U3

Here G~k is the Christoffel matrix, which depends on the medium properties (stiffnesses) and the direction of wave propagation:

Gik -- Cijkl n j n t . (1.9)


As follows from the intrinsic symmetries of the stiffness tensor [see equation (1.19)], the Christoffel matrix is symmetric (Gik = Gki). Note that the density p can be removed from the Christoffel equation by using density-normalized stiffness coefficients.

Introducing Kronecker's symbolic 5ik (bik -- 1 for i = k and 5ik - 0 for i % k), equation (1.8) can be rewritten in a more compact form,

[Gik - pV25ik] Uk = 0. (1.10)

The Christoffel equation (1.8) or (1.10) describes a standard 3 x 3 eigenvalue (pV 2) - eigenvector (U) problem for the symmetric matrix G. The Christoffel matrix is positive definite (Musgrave, 1970, Chapter 6), and its three eigenvalues are real and positive (otherwise, the velocity V can become complex). The eigenvalues are found from

det [Gik - pV25ik] = O, (1.11)

which leads to a cubic equation for pV 2. Solutions of equation (1.11) in terms of the elements Gik can be found in Appendix 1A. For any given phase (slowness) direction n in anisotropic media, the Christoffel equation yields three possible values of the phase velocity V, which correspond to the P-wave (the fastest mode) and two S- waves. Therefore, an anisotropic medium "splits" the shear wave into two modes with different velocities and polarizations (see below). In certain directions the velocities of the split S-waves coincide with each other, which leads to the so-called shear-wave singularities discussed by Crampin (1991), Helbig (1994), and others. Isotropy may be considered as a degenerate type of anisotropic media in which two S-wave velocities always coincide with each other.

Plotting the phase velocity of a given mode as the radius-vector in all propagation directions n defines the phase-velocity surface. Likewise, plotting the inverse value 1 /V in the same fashion results in the slowness surface, whose topology is directly related to the shape of wavefronts from point sources and to the presence of shear-wave singularities. As discussed in detail in Musgrave (1970) and Helbig (1994), the ray direction (i.e., the direction of the group-velocity vector, see below) is orthogonal to the slowness surface. In homogeneous isotropic media the phase-velocity and slowness surfaces, along with the corresponding wavefronts, are spherical.

After the eigenvalues have been determined, the associated eigenvectors U for each mode can be found from any two of the three equations (1.8). While the magnitude of the eigenvectors is undefined (each of them can be multiplied with any number), their orientation determines the polarization of plane waves (1.7) propagating in the direction n. The plane-wave polarization vector in isotropic media is either parallel (for P-waves) or orthogonal (for S-waves) to the slowness vector. In the presence of anisotropy, however, polarization is governed not only by the orientation of the vector n, but also by the elastic constants of the medium which determine the form of the Christoffel matrix G.

Since the matrix G is real and symmetric, the polarization vectors of the three modes (i.e., the eigenvectors) are always mutually orthogonal, but none of them is necessarily parallel or perpendicular to n. Thus, except for specific propagation


directions, there are no pure longitudinal and shear waves in anisotropic media. For that reason, in anisotropic wave theory the fast mode is often called the "quasi-P"- wave and the slow modes "quasi-S1" and "quasi-S2." 1

Still, in certain directions the fast wave can be polarized parallel to n, and the two slow waves polarized in the plane perpendicular to n (see the discussion of "longitudinal" directions in Helbig, 1994). It should be emphasized that the orthogonality of the polarization vectors does not hold for non-planar wavefronts because the three body waves recorded at any receiver location correspond to different slowness directions. Polarization of body waves excited by point sources is described in Chapter 2.

1 . 1 . 3 G r o u p ( r a y ) v e l o c i t y

The group-velocity vector determines the direction and speed of energy propagation (i.e., it defines seismic rays) and, therefore, is of primary importance in seismic traveltime modeling and inversion methods. The difference between the group- and phase-velocity vectors may be caused by velocity variations with either frequency (velocity dispersion)or angle (anisotropy).

As illustrated by the 2-D sketch in Figure 1.1, the group-velocity vector in a homogeneous medium is aligned with the source-receiver direction, while the phase-velocity (or slowness) vector is orthogonal to the wavefront. Since in the presence of anisotropy the wavefront is not spherical, the group- and phase-velocity vectors generally are different. As mentioned above, the group-velocity vector is perpendicular to the slowness surface, which helps to relate triplications (cusps) on shear wavefronts to concave parts of the slowness surface (e.g., Musgrave, 1970). Note that cusps cannot exist on P-wavefronts because the slowness surface of the fastest mode is always convex.

Unlike phase velocity, which can be obtained directly from the Christoffel equation, group velocity depends on the phase-velocity function and, in some representations, on the polarization vector. In its most general form, the group-velocity vector can be written as (e.g., Berryman, 1979)

o(kv) Va-grad(k) (kV) - - ~ i1+

a(kv) O(kV) i2 + i3, (1.12)

Ok2 Ok3

where k = (kx, k~, kz) is the wave vector, which is parallel to the phase-velocity vector and has the magnitude k = w/V (~ is the angular frequency), and il, i2, and ia are the unit coordinate vectors. Differentiation with respect to each component of the wavenumber has to be performed with the other two components held constant. Note that although equation (1.12) does involve frequency, group velocity in homogeneous non-dispersive media is frequency-independent.

The partial derivatives of kV in equation (1.12) can be evaluated using the Christoffel equation, which gives an expression for the j- th component of VG in

1For brevity, the qualifiers in "quasi-P-wave" and "quasi-S-wave" will be omitted.


/ / / / f f

f J

*~1~ ~ ~ / ~ ~ , i ' /

wave vector k

source

J

wavefront

Figure 1.1: In a homogeneous anisotropic medium, the group-velocity (ray) vector points from the source to the receiver (angle ~b). The corresponding phase-velocity (wave) vector is orthogonal to the wavefront (angle 0).

terms of the phase velocity and plane-wave polarization (Musgrave, 1970):

1 Vaj - - ~ cijkt Vi Vk nl . (1.13)

The polarization vector U in equation (1.13) is assumed to have a unit magnitude. It is possible, however, to exclude the polarization vector from the group-velocity

expressions. For example, Helbig's (1994) equation for Va contains only phase velocity and its derivatives with respect to the components of the unit vector n. A particularly convenient (especially for azimuthally anisotropic media) expression for group velocity can be obtained in the coordinate system associated with the phase (or slowness) vector.

Let us introduce an auxiliary Cartesian coordinate system [x, y, z] with the horizontal axes rotated by the angle r around the x3-axis of the original coordinate system [Xl, X2, X3] , SO that the phase-velocity vector lies in the [x, z] coordinate plane (Fig- ure 1.2). Since both group-velocity components in the [x, z]-plane (Vaz and Vaz) are calculated for ky = 0, they are independent of out-of-plane phase-velocity variations.

Treating the phase-velocity vector as a function of the polar angle 0 with the vertical axis, and of the azimuthal angle r leads to (see the derivation in Appendix 1B)

a(kv) av] - cos/), (1.14) Okx = V sin 0 + ~ r

V a z - O(kV) = V c o s 0 - OV] sin0. (1.15) Okz ~ r

The transverse component of the group-velocity vector Vay depends solely on azimuthal phase-velocity variations and is fully determined by the first derivative of


X3----Z ,,~ V G ~G in

r x

X1

Figure 1.2: The vectors of group (VG) and phase (V) velocity in anisotropic media. The vector V lies in the [x, z] plane of an auxiliary Cartesian coordinate system Ix, y, z]; r is the angle between the horizontal projection of V and the xl-axis of the original coordinate system. In general, VG deviates from the phase-velocity vector in both the vertical (Ix, z]) plane and azimuthal direction. V~ is the projection of VG onto the Ix, z] plane.

phase velocity with respect to the azimuthal phase angle r (Appendix 1B):

O ( k V ) _ 1 OV (1.16) VGy = Oky - sin8 0r e=~on~t "

Equations (1.14)-(1.16) express the group-velocity vector in arbitrary anisotropic media through 3-D variations of the phase-velocity function.

From the representations of group velocity given above it follows that the projection of the group-velocity vector onto the phase (slowness) direction is equal to phase velocity:

IVI = (VG-n) . (1.17) Hence, the magnitude of the group-velocity vector is always greater than or equal to that of the corresponding phase-velocity vector. Equation (1.17) is particularly convenient in derivations involving seismic traveltimes and reflection moveout. Below we present simplified weak-anisotropy approximations for group velocity in transversely isotropic media in terms of the anisotropic parameters.

1.1.4 Anisotropic symmetry systems

The contribution of the medium symmetry to the wave equation (1.5) and the Christof- fel equation (1.8) is controlled by the stiffness tensor cijkl, whose structure determines


the Christoffel matrix (1.9) and, consequently, the velocity and polarization of plane waves for any propagation direction.

While a general fourth-order tensor has 34 = 81 components, c~jkt possesses several symmetries that reduce the number of independent elements. First, due to the symmetry of the stress and strain tensors, it is possible to interchange the indices i and j, k and l:

Cijkl - - Cj ikl ; Ci jk l - - C i j l k . (1.18)

Also, from thermodynamic considerations (Aki and Richards, 1980; Helbig, 1994),

Cijkl = CkUj. (1.19)

As follows from equations (1.1S) and (1.19), the medium with the lowest possible symmetry is described by a total of 21 stiffness elements, and the tensor Cijkl can be represented in the form of a 6 • 6 matrix. This operation is usually accomplished by replacing each pair of indices ( i j and kl ) by a single index according to the so- called "Voigt recipe:" 11 --+ 1, 22 ~ 2, 3 3 - + 3, 23 -+ 4, 13 -+ 5, 12--+ 6. The transformation of the index pair i j into the corresponding index p can be formally described by the equation

p = iSij + (9 - i - j ) ( 1 - ~ij) . (1.20)

Since the pairs of indices can be interchanged [equation (1.19)], the resulting "stiffness matrix" is symmetric.

Each anisotropic symmetry is characterized by a specific structure of the stiffness matrix, with the number of independent elements decreasing for higher-symmetry systems. Here we describe just a few symmetries of most importance in seismological applications and refer the reader to crystallographic literature (e.g., Fedorov, 1968; Musgrave, 1970) and to Helbig (1994) for a more comprehensive analysis.

Triclinic m e d i a

The most general anisotropic model with 21 independent stiffnesses is called triclinic:

Cll c12 c~3 c~4 c15 c16 c12 c22 c23 c24 c2~ c2~

c(trc)__ C13 c23 c33 c34 c35 c36 . (1.21) C14 C24 C34 C44 C45 C46 C15 C25 C35 C45 C55 C56 C16 C26 C36 C46 C56 C66

With a special choice of the coordinate system it is possible to eliminate the elements c34, c3~ and c45 (Helbig, 1994, p. 116). Although there are reasons to believe that some subsurface formations (especially those with multiple fracture sets) possess triclinic symmetry, the large number of independent parameters so far has precluded application of this model in seismology.


Figure 1.3: Two systems of parallel vertical fractures generally form an effective monoclinic medium with a horizontal symmetry plane. In the special cases of two orthogonal (r + r - 90 ~ or identical systems the symmetry becomes orthorhombic.

Monoc l in i c media

The lowest-symmetry model identified from seismic measurements is monoclinic (Win- terstein and Meadows, 1991), which has "only" 13 independent stiffness coefficients. In contrast to triclinic models, monoclinic media have a plane of mirror symmetry with the spatial orientation defined by the underlying physical model. For instance, if a formation contains two different non-orthogonal systems of small-scale vertical fractures embedded in an azimuthally isotropic background, the effective medium becomes monoclinic with a horizontal symmetry plane (Figure 1.3).

In the special case of two identical or orthogonal vertical fracture sets the model has orthorhombic symmetry (see below). Three or more sets of vertical fractures generally make the effective medium in the long-wavelength limit monoclinic (or even triclinic, depending on the symmetry of the background). Potential importance of monoclinic media in seismic exploration is corroborated by abundant geological (in- situ) evidence of multiple vertical fracture sets. An interesting example of monoclinic media with a vertical symmetry plane is that of a single vertical system of rotationally non-invariant fractures with micro-corrugated faces in isotropic host rock (Bakulin et al., 2000c).

If the symmetry plane of a monoclinic medium is orthogonal to the x3-axis, the stiffness matrix has the following form:

c~ c12 c13 0 0 c16 c~2 c22 c23 0 0 c26

c(mnc) _ c13 c23 c33 0 0 c36 (1.22) 0 0 O C44 C45 0 " 0 0 0 C45 C55 0

C16 C26 C36 0 0 C66

10 CHAPTER I. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION

s,ooe, p,a~ I x2+ ;,,el /

Figure 1.4: Orthorhombic model caused by parallel vertical fractures embedded in a finely layered medium. One of the symmetry planes in this case is horizontal, while the other two are parallel and perpendicular to the fractures.

The number of stiffnesses in equation (1.22) can be reduced from 13 to 12 by aligning the horizontal coordinate axes with the polarization vectors of the vertically propagating shear waves, which eliminates the element c45 (Helbig, 1994).

O r t h o r h o m b i c media

Orthorhombic (or orthotropic) models are characterized by three mutually orthogonal planes of mirror symmetry (Figure 1.4). In the coordinate system associated with the symmetry planes orthorhombic media have 9 independent stiffness coefficients.

One of the most common reasons for orthorhombic anisotropy in sedimentary basins is a combination of parallel vertical fractures with vertical transverse isotropy (see below) in the background medium, as illustrated by Figure 1.4. Orthorhombic symmetry can also be caused by two or three mutually orthogonal fracture systems or two identical systems of fractures making an arbitrary angle with each other. Hence, orthorhombic anisotropy may be the simplest realistic symmetry for many geophysical problems (Bakulin et al., 2000b).

In the Cartesian coordinate system associated with the symmetry planes (i.e., each coordinate plane is a plane of symmetry), the orthorhombic stiffness matrix is written as

cll 02 c13 0 0 0 C12 C22 C23 0 0 0

C(ort)_ C13 C23 C33 O 0 0 0 0 0 c44 0 0 " (1.23) 0 0 0 0 c55 0 0 0 0 0 0 c66


X 3

Figure 1.5: VTI model has a vertical axis of rotational symmetry and may be caused by thin horizontal layering.

The Christoffel equation (1.8) in the symmetry planes of orthorhombic media turns out to have the same form as in the simpler transversely isotropic model. The equivalence between the symmetry planes of orthorhombic and TI media helps to develop a unified notation for the two models and gain important insights into wave propagation for orthorhombic anisotropy (see section 1.3).

Transversely isotropic media

The vast majority of existing studies of seismic anisotropy are performed for a transversely isotropic (TI) medium, which has a single axis of rotational symmetry. All seismic signatures in such a model, also called hexagonal, depend just on the angle between the propagation direction and the symmetry axis. Any plane that contains the symmetry axis represents a plane of mirror symmetry; one more symmetry plane (the "isotropy plane") is perpendicular to the symmetry axis. The phase velocities of all three waves in the isotropy plane are independent of propagation direction because the angle between the slowness vector and the symmetry axis remains constant (90~

The TI model resulting from aligned plate-shaped clay particles adequately describes the intrinsic anisotropy of shales (Sayers, 1994a). Shale formations comprise about 75% of the clastic fill of sedimentary basins, which makes transverse isotropy the most common anisotropic model in exploration seismology. Most shale formations are horizontally layered, yielding a transversely isotropic medium with a vertical symmetry axis (VTI). Another common reason for TI symmetry is periodic thin layering (i.e., interbedding of thin isotropic layers with different properties) on a scale small compared to the predominant wavelength (Figure 1.5).

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