Seismic Signatures and Analysis of Reflection Data in Anisotropic Media

HANDBOOK OF GEOPHYSICAL EXPLORATION SEISMIC EXPLORATION VOLUME 29 SEISMIC SIGNATURES AND ANALYSIS OF REFLECTION DATA I...

Author: I. Tsvankin

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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

~In preparation. 2planned.

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

SEISMIC EXPLORATION

V o l u m e 29

SEISMIC SIGNATURES AND ANALYSIS OF REFLECTION DATA IN ANISOTROPIC MEDIA

by Ilya T S V A N K I N Professor of Geophysics Center for Wave P h e n o m e n a Department of Geophysics Colorado School of Mines Golden, CO, USA

) 2001 PERGAMON An Imprint of Elsevier Science A m s t e r d a m - London - N e w York - Oxford - Paris - Shannon - Tokyo

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9 2001 Elsevier Science Ltd. All rights reserved.

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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 artificially 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 P P 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 P P 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 defocusing 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 C h a p 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 representation, 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

Preface

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 P S 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 cutting through the sometimes overwhelming complexity of anisotropic mathematics and identifying the mathematical details critical in solving a particular problem. Therefore, 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.


Contents 1

2

E l e m e n t s of b a s i c t h e o r y o f a n i s o t r o p i c w a v e p r o p a g a t i o n 1.1 Governing equations and plane-wave properties . . . . . . . . . . . . 1.1.1 Wave equation and Hooke's law . . . . . . . . . . . . . . . . . 1.1.2 Christoffel equation and properties of plane waves . . . . . . . 1.1.3 Group (ray) velocity . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Anisotropic symmetry systems . . . . . . . . . . . . . . . . . . 1.2 Plane waves in transversely isotropic media . . . . . . . . . . . . . . . 1.2.1 Solutions of the Christoffel equation . . . . . . . . . . . . . . . 1.2.2 Thomsen notation for transverse isotropy . . . . . . . . . . . . 1.2.3 Exact and approximate phase and group velocity . . . . . . . 1.2.4 Polarization vector and relationship between phase, group and polarization directions . . . . . . . . . . . . . . . . . . . . . . 1.3 Plane waves in orthorhombic media . . . . . . . . . . . . . . . . . . . 1.3.1 Limited equivalence between TI and orthorhombic media . . . 1.3.2 Anisotropic parameters for orthorhombic media . . . . . . . . 1.3.3 Signatures in the symmetry planes . . . . . . . . . . . . . . . 1.3.4 P-wave velocity outside the symmetry planes . . . . . . . . . 1.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 2 3 5 7 14 14 17 21 34 36 37 40 44 46 51

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

56 57

I n f l u e n c e o f a n i s o t r o p y o n p o i n t - s o u r c e r a d i a t i o n a n d A V O a n a l y s 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 . . . . . . . . . . . . . 2B Weak-anisotropy approximation for radiation patterns in TI m e d i a . . xiii

103 105

xiv

3

CONTENTS

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 N M O 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 N M O velocity of P-waves . . . . . . . . . . . . 118 3.3 NMO velocity for tilted TI media . . . . . . . . . . . . . . . . . . . . 130 3.3.1 Absence of reflections from steep interfaces . . . . . . . . . . . 131 3.3.2 Dip-dependent P-wave N M O velocity . . . . . . . . . . . . . . 139 3.4 NMO velocity in layered media and time-to-depth conversion . . . . . 149 3.4.1 2-D Dix-type N M O equation for dipping reflectors . . . . . . . 149 3.4.2 Horizontally layered media and time-to-depth conversion . . . 151 3.5 Elements of 3-D analysis of NMO velocity . . . . . . . . . . . . . . . 156 3.5.1 Equation of the NMO ellipse . . . . . . . . . . . . . . . . . . . 156 3.5.2 N M O ellipse in VTI media . . . . . . . . . . . . . . . . . . . . 159 3.5.3 N M O ellipse in orthorhombic and HTI media . . . . . . . . . 161 Appendices for C h a p t e r 3 3A 2-D N M O equation in an anisotropic layer . . . . . . . . . . . . . . . 166 3B Weak-anisotropy approximation for P-wave NMO velocity in T T I m e d i a l 6 8 3C 2-D Dix-type equation in layered anisotropic media . . . . . . . . . . 169 3D 3-D NMO equation in heterogeneous anisotropic media . . . . . . . . 170

4

Nonhyperbolic reflection moveout 4.1 Quartic moveout coefficient . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 General 2-D equation for a single layer . . . . . . . . . . . . . 4.1.2 Explicit expressions for VTI media . . . . . . . . . . . . . . . 4.1.3 Layered media . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonhyperbolic moveout equation . . . . . . . . . . . . . . . . . . . . 4.2.1 Weak-anisotropy approximations . . . . . . . . . . . . . . . . 4.2.2 General long-spread moveout equation . . . . . . . . . . . . . 4.3 P-wave moveout in V T I media in terms of the p a r a m e t e r z] . . . . . . 4.3.1 Single layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Layered media . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Long-spread moveout of SV-waves in VTI media . . . . . . . . . . . 4.4.1 Models with negative a . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Positive a and models with cusps . . . . . . . . . . . . . . . . Appendices for C h a p t e r 4 4A Weak-anisotropy approximation for long-spread moveout 4B P-wave moveout in layered VTI media . . . . . . . . . . . . . . . . .

.......

173 176 176 177 180 182 183 184 185 187 189 190 190 190 195 197

CONTENTS

5

xv

R e f l e c t i o n 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

D i p - d e p e n d e n t moveout of P S - w a v e s in a single layer (2-D) . . . . . .

200

5.2

5.1.1 P a r a m e t r i c representation of P S traveltime . . . . . . . . . . 5.1.2 A t t r i b u t e s of the P S moveout function . . . . . . . . . . . . . Application to a V T I layer . . . . . . . . . . . . . . . . . . . . . . . .

201 204 208

5.3

5.2.1 W e a k - a n i s o t r o p y a p p r o x i m a t i o n for P S m o v e o u t . . . . . . . . 5.2.2 Recovery of P S - w a v e moveout curve . . . . . . . . . . . . . . 3-D t r e a t m e n t of P S - w a v e moveout for layered m e d i a . . . . . . . . .

208 213 219

5.3.1 5.3.2

219 223

5.4

5.5

2-D expressions for vertical s y m m e t r y planes . . . . . . . . . . 3-D description of P S moveout . . . . . . . . . . . . . . . . .

5.3.3 Moveout a t t r i b u t e s in layered m e d i a . . . . . . . . . . . . . . P S - w a v e moveout in horizontally layered V T I m e d i a . . . . . . . . . 5.4.1 Taylor series coefficients . . . . . . . . . . . . . . . . . . . . .

226 228 228

5.4.2 Nonhyperbolic moveout equation . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

230 231

Appendices for C h a p t e r 5 2-D description of P S moveout in a single layer . . . . . . . . . . . . 3-D expression for the slope of C M P moveout . . . . . . . . . . . . . N M O velocity for converted-wave moveout . . . . . . . . . . . . . . . W e a k - a n i s o t r o p y a p p r o x i m a t i o n for P S - m o v e o u t in V T I m e d i a . . . . 5D.1 P a r a m e t r i c expressions for the traveltime curve . . . . . . . . 5D.2 Moveout a t t r i b u t e s . . . . . . . . . . . . . . . . . . . . . . . . 3-D description of P S moveout in layered m e d i a . . . . . . . . . . . . 5E.1 Single anisotropic layer . . . . . . . . . . . . . . . . . . . . . . 5E.2 Layered m e d i a . . . . . . . . . . . . . . . . . . . . . . . . . . . 5E.3 2-D relationships for s y m m e t r y planes . . . . . . . . . . . . .

233 235 239 241 241 244 246 246 249 251

P - w a v e t i m e - d o m a i n s i g n a t u r e s in t r a n s v e r s e l y isotropic m e d i a

253 254 254 261

5A 5B 5C 5D

5E

6

6.1

6.2

6.3

6.4

P-wave N M O velocity as a function of ray p a r a m e t e r 6.1.1 2-D analysis for a V T I layer . . . . . . . . . . . . . . . . . . . 6.1.2 Dip plane of a layered m e d i u m . . . . . . . . . . . . . . . . . .

.........

6.1.3 3-D analysis using the N M O ellipse . . . . . . . . . . . . . . . T w o - p a r a m e t e r description of time processing . . . . . . . . . . . . . 6.2.1 Migration impulse response . . . . . . . . . . . . . . . . . . . 6.2.2 Brief s u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . .

263 264 264 266

Discussion: N o t a t i o n and P-wave signatures in V T I m e d i a . . . . . . 6.3.1 A d v a n t a g e s of T h o m s e n p a r a m e t e r s . . . . . . . . . . . . . . . 6.3.2 Influence of vertical transverse isotropy on P - w a v e s i g n a t u r e s . Moveout analysis for tilted s y m m e t r y axis . . . . . . . . . . . . . . .

269 269 270 272

6.4.1 6.4.2 6.4.3

272 274 281

N M O velocity as a function of ray p a r a m e t e r . . . . . . . . . . P a r a m e t e r ~ for tilted axis of s y m m e t r y ............ Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XVI

CONTENTS

Appendices for C h a p t e r 6 6A D e p e n d e n c e of N M O velocity in V T I media on the ray p a r a m e t e r 6A.1 Building the function Vnmo(P) . . . . . . . . . . . . . . . . . . 6A.2 Elliptical anisotropy . . . . . . . . . . . . . . . . . . . . . . . 6A.3 Weak transverse isotropy . . . . . . . . . . . . . . . . . . . . . 6B N M O velocity in tilted elliptical m e d i a . . . . . . . . . . . . . . . . . 7

8

. .

283 283 283 284 285

V e l o c i t y a n a l y s i s and p a r a m e t e r e s t i m a t i o n f o r V T I m e d i a 7.1 P - w a v e dip-moveout inversion for r/ . . . . . . . . . . . . . . . . . . . 7.1.1 Inversion in the dip plane of a V T I layer . . . . . . . . . . . . 7.1.2 2-D inversion in vertically heterogeneous m e d i a . . . . . . . . 7.1.3 3-D inversion of azimuthally varying N M O velocity . . . . . . 7.1.4 F i e l d - d a t a example . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Inversion of P-wave nonhyperbolic moveout . . . . . . . . . . . . . . 7.2.1 Single V T I layer . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Nonhyperbolic velocity analysis for layered m e d i a . . . . . . . 7.2.3 F i e l d - d a t a examples . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Joint inversion of P and P S d a t a . . . . . . . . . . . . . . . . . . . . 7.3.1 S-waves in p a r a m e t e r e s t i m a t i o n for V T I m e d i a . . . . . . . . 7.3.2 2-D inversion of horizontal and dipping events . . . . . . . . . 7.3.3 3-D inversion of wide-azimuth d a t a . . . . . . . . . . . . . . . 7.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 289 289 291

P-wave imaging for VTI media 8.1 Fowler-type time-processing m e t h o d . . . . . . . . . . . . . . . . . . . 8.1.1 Fowler D M O in isotropic m e d i a . . . . . . . . . . . . . . . . . 8.1.2 Extension to V T I media . . . . . . . . . . . . . . . . . . . . . 8.1.3 Synthetic example . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 F i e l d - d a t a example . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Dip moveout by Fourier t r a n s f o r m . . . . . . . . . . . . . . . . . . . . 8.2.1 Hale's D M O m e t h o d . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 2-D Hale D M O for anisotropic media . . . . . . . . . . . . . . 8.2.3 Application to V T I media . . . . . . . . . . . . . . . . . . . . 8.2.4 Synthetic examples . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 T i m e and d e p t h m i g r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Phase-shift (Gazdag) m i g r a t i o n . . . . . . . . . . . . . . . . . 8.3.2 Gaussian b e a m migration . . . . . . . . . . . . . . . . . . . . 8.4 Synthetic example for a model from the Gulf of Mexico . . . . . . . . 8.4.1 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 D e p t h m i g r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . .

353 354 355 356 360 365 368 369 370 372 373 375 384 385 385 389 400 400 403

297 302 310 312 312 321 327 333 334 335 337 346 347

CONTENTS

8.5

8.6

xvii

8.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i e l d - d a t a example with multiple fault planes . . . . . . . . . . . . . 8.5.1 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Time imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

409 410 410 414 416

References

419

Author

Index

429

Subject

Index

431


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

C H A P T E R 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

1.1.1

Governing equations and plane-wave properties Wave

equation

and

Hooke'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 P cgt2

OTij 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: (1.2)

Tij = CijkZekl .

Here Cijkl is the fourth-order stiffness tensor responsible for the material properties (it is discussed in detail below), and ekl is the strain tensor defined as ekl -- -~

~

+ ~Xk

Equivalently, Hooke's law can be written through the compliance tensor eij = Sijkt TkZ .

(1.3)

" 3ijkl ,

(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 P--~-

02uk (1.6)

- Cijkl i)XjOXl = O.

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 p o l a r i z a t i o n 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 c o n s 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 socalled Christoffel equation for the phase velocity V and polarization vector U:

G21

G22 - p V 2

G31

G32

G23 G33 -

pV 2

U2

- 0 .

(1.8)

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)

4

C H A P T E R 1. E L E M E N T S OF BASIC THEORY OF A N I S O T R O P I C WAVE PROPAGATION

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 Swaves. 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


5

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

Group

(ray) velocity

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)

Va-grad(k)(kV) -

a(kv) o(kv) i3, - ~ i 1 + O(kV) i2 + Ok2

(1.12)

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.

6

C H A P T E R 1. E L E M E N T S OF BASIC 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

i'/ 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 (Figure 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] r

cos/),

(1.14)

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

sin0.

(1.15)

Okx = V sin 0 + ~

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

1.1.

GOVERNING EQUATIONS AND PLANE-WAVE P R O P E R T I E S

7

~ Gin

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):

VGy =

O(kV)_ 1 OV Oky - sin8 0r e=~on~t "

(1.16)

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 = ( V G - 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 Christoffel equation (1.8) is controlled by the stiffness tensor cijkl, whose structure determines

8

CHAPTER 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION

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 - - Cjikl ; Cijkl - - C i j l k . (1.18) Also, from thermodynamic considerations (Aki and Richards, 1980; Helbig, 1994), (1.19)

Cijkl = CkUj.

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 (ij and kl) by a single index according to the socalled "Voigt recipe:" 11 --+ 1, 22 ~ 2, 3 3 - + 3, 2 3 - + 4, 1 3 - + 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 -

(1.20)

~ij) .

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:

c(trc)__

Cll c12 c~3 c~4 c15 c16 c12 c22 c23 c24 c2~ c2~ C13 c23 c33 c34 c35 c36 C14

C24

C34

C44

C45

.

(1.21)

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.


9

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. Monoclinic media The lowest-symmetry model identified from seismic measurements is monoclinic (Winterstein 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 (insitu) 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(mnc) _

c~ c12 c13 c~2 c22 c23 c13 c23 c33 0

0

0 0 0

0 0 0

c16 c26 c36

O

C44

C45

0

0

0

0

C45

C55

0

C16

C26

C36

0

0

C66

(1.22) "

s,ooe,

p,a~ I

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). Orthorhombic 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 C(ort)_

C12 C22 C23 C13 C23 C33

0 0 0

0 0 0

0 0 0

0 O

0 0

0 0

c44 0 0 0 c55 0 0 0 c66

"

(1.23)


11

X3 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).

12


The stiffness matrix of VTI media is given by 0

0

Cll -- 2C66

Cll

Cll

C13

0

0

0

C13

C13

C33

0

0

0

c(vti) __

Cll -- 2C66 C13

0

0

0

0

c55

0

0

0

0

0

0

c55

0

0

0

0

0

0

C66

"

(1.24)

The matrix c(vti) has the same nonzero elements as that for orthorhombic media [equation (1.23)] but the relationships between the c{j's reduce the number of independent stiffnesses in VTI media from nine to five. Seismic signatures and processing of reflection data in VTI media are the main focus of this book. In some cases, transversely isotropic layers may be dipping, which leads to a tilt of the symmetry axis with respect to the earth surface (TTI medium). For example, uptilted shale layers near salt domes are expected to produce an effective TTI model with a large inclination of the symmetry axis. Tilted transverse isotropy should also be rather typical for overthrust areas, such as the Canadian Foothills, where shale layers are often bent by tectonic processes and may have dips exceeding 45~. Yet another physical reason for TTI media is a system of parallel dipping fractures in an isotropic background. To obtain the elastic parameters of tilted TI media, the stiffness tensor for vertical transverse isotropy corresponding to the matrix (1.24) has to be rotated in accordance with the orientation of the symmetry axis. Tilting the symmetry axis all the way to horizontal leads to a mode] called "horizontal transverse isotropy," or HTI (Figure 1.6). In most cases, HTI media are caused by a system of parallel vertical circular ("penny-shaped") cracks embedded in an isotropic background. Hence, horizontal transverse isotropy is the simplest possible model of a formation with vertical fractures. Description of seismic fracturecharacterization methods for HTI media can be found in Baku]in et a]. (2000a). Deviations from the circular crack shape, misalignment of the crack planes, the addition of a second crack system, or the presence of anisotropy in the background inevitably lower the symmetry of the effective medium to orthorhombic or less. The HTI mode] has two mutually orthogona] vertical planes of symmetry- the symmetry-axis plane and the isotropy plane (Figure 1.6). If the symmetry direction coincides with the xl-axis, the stiffness tensor can be obtained from that for the VTI model by interchanging the indices 1 and 3. The corresponding stiffness matrix has the form

c(hti) _

Cll

C13

C13

0

0

0

C13

C33

C33 -- 2C44

0

0

0

c13 C33 -- 2C44

c33

0

0

0

0

0

0

C44

0

0

0 0

0 0

0 0

0 0

(1.25) "

c55 0 0 c55

Both HTI and VTI media can be treated as special cases of the more complex orthorhombic model.


13

e,r,-ax,s

c//"

/

/

/

/

~ I" [.... I"" !'" I' t ~~~

j

~

]_~ymmetry ,..//~.

Figure 1.6: HTI model due to a system of parallel vertical cracks. Vertically traveling shear waves in HTI media split into two modes polarized parallel and perpendicular to the crack faces (i.e., in the vertical symmetry planes).

Isotropic media If the medium is isotropic (i.e., all directions of wave propagation are equivalent), c~jkl becomes a fourth-order isotropic tensor given by

(iso) Cijkl -- )k~ij~kl -t- ~(~ik~jl "31-~il~jk) ,

(1.26)

or, in the two-index notation,

c(iso)_

A+2# A

A A+2#

A

A

0 0 0

0 0 0

A A

A+2# 0 0 0

0 0 0 0 0 0

0 0 0

# 0 0 0 tt 0 0 0 #

'

(1.27)

where A and # are the Lam~'s constants. Substituting the tensor (1.26) into the general wave equation (1.5), we arrive at the familiar equation of motion for homogeneous isotropic media:

02Ui [)--~-

02Uj OXj

-- (,~ "4-~l)OXi

-- It

02Ui OXj OXj

=

f'"

(1.28)

14


1.2

Plane

1.2.1

w a v e s in t r a n s v e r s e l y

isotropic

media

S o l u t i o n s of t h e C h r i s t o f f e l e q u a t i o n

Phase velocity and polarization of body waves propagating through transversely isotropic media can be obtained from the Christoffel equation (1.8) with the stiffnesses specified in equation (1.24). Although the stiffness matrix (1.24) corresponds to vertical transverse isotropy (i.e., the symmetry direction is x3), the orientation of the symmetry axis with respect to the coordinate system may be arbitrary, as long as the model is homogeneous. All body-wave properties in an unbounded homogeneous TI medium, as illustrated by the results of this section, depend just on the angle between the slowness vector and the axis of symmetry. The Christoffel matrix (1.9) for VTI media becomes all

-

Cll n 2 ~- c66 n22 -~- c55 rt32 ,

a22

-

c66 n 2 -t- Cll n 2 -~- c55 n 2 ,

(1.29) (1.30) (1.3~) (1.32) (1.33) (1.34)

a ~ - c~ ( ~ + n~) + ~ h i , G12 = (cll - c66) n l n 2 , G13 = (c13

+ c~5) n~n3,

G23 = (c13 + c ~ ) n2n3 .

Since in TI media all planes containing the symmetry axis are equivalent, it is sufficient to study wave propagation in a single vertical plane. Choosing the Ix1, x3]-plane (n2 = 0) and substituting equations (1.29)-(1.34) into the Christoffel equation (1.8) yields

Ecl,n c55

0

,c,3 c 5 n, n3

0

c66n~ + cb~n] - p V 2

0

(c~ + ~ ) nln~

o

c~1 + ~,~ - py ~

= 0.

]

U2

v~

(1.35)

It is important to mention that equation (1.35) has the same form as that of the Christoffel equation in the symmetry-axis plane of HTI media and, moreover, in all three symmetry planes of orthorhombic media. Hence, kinematic signatures and polarizations in VTI media and in symmetry planes of orthorhombic media are given by identical equations. As discussed in section 1.3 below, this equivalence helps to extend analytic relationships developed for transverse isotropy to orthorhombic media. Since in the [xl,x3]-plane G12 = G23 = 0, system (1.35) splits into independent equations for pure transverse motion (/-/1 = U3 = 0) and in-plane motion (U2 = 0). Phase velocity can be found by setting the determinant of the matrix [Gik -- pV25ik] to zero. Expressing the unit vector n in equation (1.35) in terms of the phase angle

1.2. PLANE WAVES IN TRANSVERSELY ISOTROPIC MEDIA

15

0 with the symmetry axis (nl = sin0; n3 = cos0) gives the phase velocity of the transversely polarized mode (U2 -~ 0, U1 = U3 = 0) as

vsH(o)

|

sin e +

cos e

(1 36)

9

P

Equation (1.36) describes the so-called SH-wave with polarization vector confined to the horizontal plane. For vertical propagation (0 = 0), the SH-velocity is equal to ~/csi~/p, while in the horizontal direction VSH(90~ -- ~/c66/P. Therefore, the magnitude of velocity anisotropy of the SH-wave depends on the fractional difference between two stiffnesses- c66 and c55. According to equation (1.36), if the slowness 1/VsH is plotted as the radius-vector in the direction 0, it traces out an ellipse with the axes in the vertical and horizontal directions. (Note that the SH-wave phase velocity itself is not elliptical.) As shown below, an elliptical shape of the slowness surface leads to an ellipsoidal wavefront from a point source (group-velocity surface). For that reason, the SH-wave anisotropy in transversely isotropic media is called elliptical; the anisotropy of P- and SV-waves becomes elliptical only for a subset of TI models. The in-plane polarized modes (P-SV) are described by the first and third equations (1.35):

[Cllsin20--kCa5COS20-pV 2 (C13 + C55 ) sin 0 c o s 0

(Cx3+C55)sinOcosO c55 sin 2 O d- C33 COS2 0 -- pV 2

] [gl IU3

0

(1.37)

"

Clearly, phase velocity and polarization of P- and SV-waves depend on four stiffness coefficients - c11, c33, c55 and c13; the fifth stiffness, c66, influences only SH-wave propagation. If the waves travel along the symmetry axis (0 = 0), equation (1.37) further simplifies to

[c55-pV 2 0

0 C33 --

pV 2

] lUll_

U3

0

"

(138)

One of the solutions of equation (1.38) corresponds to a pure P-wave polarized in the direction of propagation:

Vp(O - O) - ~

;

U~ - 0, U3 - 1.

(1.39)

The other solution is a shear (SV) wave with a horizontal in-plane polarization:

Vsv(O - O) - ~ p

;

U~ - 1, U3 - 0.

(1.40)

Comparison of equations (1.40) and (1.36) shows that the SV- and SH-waves have the same vertical velocity, which creates a shear-wave singularity in the x3 (symmetry)

16


direction. Since at the symmetry axis the slowness surfaces of the SV- and SH-waves touch each other tangentially, this is the so-called "kiss" singularity. The SV and SH slowness surfaces may also cross each other at oblique propagation angles leading to a circular "intersection" singularity. Whenever two eigenvalues of the Christoffel matrix are equal, as is the case for 0 = 0 in VTI media (and for any direction in isotropic media), the corresponding eigenvectors, or polarizations, are not uniquely defined. Although we found the SVand SH-waves at vertical incidence to be polarized in the xl- and x2-directions (respectively), for 0 = 0 the very separation of the Christoffel equation (1.35) into P - S V and SH-waves is invalid. It is easy to verify that for a vertical velocity of ~/c55/p any combination of U1 and U2 can form the corresponding eigenvector. Also, in VTI media the coordinate system can be rotated by any angle around the vertical axis without changing the stiffness tensor. This means that the polarization vectors of the vertically traveling SV- and SHwaves in VTI media can lie anywhere in the horizontal plane. For wavefronts generated by point sources, the polarization of the S-wave at vertical incidence depends on the properties of the source. In the isotropy plane, equation (1.37) reduces to

Vp(O - 90 ~ - V/clip

u , = 1,

Vsy(O - 90 ~ - Vsv(O = 0 ~ - ~/Y-~,

= 0;

U1 = 0, U3 - 1.

(1.41)

(1.42)

Equations (1.39)-(1.42) show that the vertical and horizontal velocities are generally different for the P-wave (unless c33 : c11) and equal to each other for the SV-wave. The velocity of the SV-wave, however, does vary at oblique incidence angles 0 ~ < 0 < 90~ the only TI model with a constant Vsy is elliptical (see below). Clearly, both P- and SV-waves have a pure (longitudinal and shear) polarization along the x 1-axis. Another distinctive feature of wave propagation both parallel and perpendicular to the symmetry axis is that the phase- and group-velocity vectors coincide with each other. This result follows directly from the group-velocity equations (1.14) and (1.15), if we take into account that the phase-velocity function in the Ix1, x3]-plane is symmetric with respect to 0 = 0 ~ and 0 = 90 ~ and the derivative dV/dO vanishes in the vertical and horizontal directions. Due to the rotational invariance with respect to the symmetry axis, equations (1.41) and (1.42) are valid for any slowness direction in the horizontal (isotropy) plane. It is important to notice that while the velocity of each mode in the isotropy plane is independent of direction, the SV- and SH-waves travel with different velocities, which produces shear-wave splitting. Unfortunately, the relative simplicity of the solutions of the Christoffel equation for P and SV-waves cannot be maintained for oblique propagation angles. Setting the determinant of the matrix [Gik - pV25ik] in equation (1.37) to zero gives the

1.2.

P L A N E WAVES IN T R A N S V E R S E L Y I S O T R O P I C M E D I A

17

following equation for the phase velocity:

2pV2(9) = (Cll + c55)sin 2 0 + (c33 + c55)cos 2 0 +~/[(Cll - c~5)sin 2 0

(c33- c5~) cos 2 0]2 + 4(c13 + c55)2 sin 2 0 cos 2 0, (1.43)

where the plus sign in front of the radical corresponds to the P-wave, while the minus sign corresponds to the SV-wave. Substitution of the phase velocity of each mode into either of the two equations (1.37) can be used to find the corresponding polarization vector U. Although equation (1.43) is not excessively complicated and can be efficiently used for numerical computations, it provides little insight into the dependence of phase velocity on the elastic properties of the medium. A convenient way to simplify the phase-velocity function and other seismic signatures in TI media is to replace the standard notation by Thomsen (1986) parameters. 1.2.2

Thomsen

notation

for transverse

isotropy

The matter of notation may seem trivial, but it is of utmost importance in such multi-parameter problems as analysis of seismic signatures in anisotropic media. Historically, wave propagation was described using the elastic (stiffness) coefficients cij. Since both Hooke's law (1.2) and the wave equation (1.5) are expressed through the stiffnesses, these coefficients are convenient to use in all types of forward-modeling algorithms. The problems arise when it is necessary to go beyond specific examples and find the effective parameters that govern seismic wavefields in anisotropic media. As demonstrated by equation (1.43) and the discussion of various seismic signatures below, the conventional notation is not well-suited for this purpose. Without understanding of the relationships between the medium parameters and seismic signatures it is hardly possible to make qualitative estimates of the influence of anisotropy on seismic wavefields and, even more important, to develop inversion and processing algorithms for anisotropic media. The main disadvantages of the conventional notation in VTI media can be summarized as follows: 1. The strength of the anisotropy is hidden in the elastic coefficients. The medium is isotropic if C l l : C33 , C55 ~-- C66 and c13 = Cll - 2c66 [compare equations (1.24) and (1.27)]. Clearly, it is cumbersome to estimate the degree of velocity anisotropy just from inspection of the elastic constants. 2. Since most reflection data are acquired at small offsets, it would be useful to have a parameter responsible for P-wave velocity near the (vertical) symmetry axis. However, no such parameter exists in the conventional notation. 3. Propagation of P- and SV-waves is described by four stiffness coefficients: c11, c33, c55, and c13. It turns out that Thomsen notation makes it possible to reduce the number of independent parameters responsible for P-wave kinematic signatures to

18


three. Also, the inversion of P-wave traveltime data for the stiffness coefficients is ambiguous because of the trade-off between c55 and c13 (see below). 4. The expressions for normal-moveout (NMO) velocities in the conventional notation are complicated (see Chapter 3). Since surface seismic processing operates with reflection moveout, it is important to have easily tractable NMO equations for VTI media.

Definition and meaning of Thomsen parameters An improvement over the conventional notation can be achieved by targeting the combinations of elastic constants most suitable for the description of seismic wavefields. An alternative parameterization based on this principle was suggested by Thomsen (1986). The idea of Thomsen notation is to separate the influence of the anisotropy from the "isotropic" quantities chosen as the P and S velocities along the symmetry axis. Five elastic coefficients of VTI media can be replaced by the vertical velocities Vpo and Vso of P- and S-waves (respectively) and three dimensionless anisotropic parameters denoted as e, (~ and 7: Ypo - ~~-~,

(1.44)

Vs0 = ~ ~ ,

(1.45)

-

Cll - - C33 2C33

(1.46) '

5 -- (03 + c55) 2 - (c33 - css) 2

(1.47)

2C33 (C33 - - C55)

7 -

C66

--

C55

9 (1.48) 2c5~ In Thomsen notation, P- and SV-wave signatures depend on the parameters Vpo, Vso, c, and ~, while the SH-wave is fully described by the shear-wave vertical velocity Vso and parameter 7Some of the advantages of Thomsen notation are immediately obvious. The dimensionless anisotropies e, 5, and 7 go to zero for isotropic media and, therefore, conveniently characterize the strength of the anisotropy. The parameter e, close to the fractional difference between the horizontal and vertical P-wave velocities [see equations (1.39) and (1.41)], defines what is often simplistically called the "P-wave anisotropy." Likewise, 7 represents the same measure for SH-waves. Although the definition of 5 is less transparent than those of e and 7, this parameter also has a clear m e a n i n g - it determines the second derivative of the P-wave phase-velocity function at vertical incidence. Differentiating equation (1.43), with a plus sign in front of the radical, and using expression (1.47) for 5 yields

d2Vp dO2

= 2Vpo ~. 0=0

(1.49)

1.2.

P L A N E WAVES IN T RANSVERSELY I S O T R O P I C MEDIA

19

Vp0

Figure 1.7: P-wave rays (black) and the wavefront (solid white) in a VTI medium with a positive e ~ 0.1 and a negative 5 ~ -0.1. The wavefront in a reference isotropic model with the velocity Vpo is marked by the dashed white line. Since the first derivative of phase velocity at 0 = 0 is equal to zero, 5 is responsible for the angular dependence of Vp in the vicinity of the vertical (symmetry) direction. According to equation (1.49), P-wave phase velocity increases away from vertical if 5 is positive and decreases if 5 < 0. Figure 1.7 shows that a negative 5 makes the P-wavefront at small angles 0 lag behind the wavefront in a reference isotropic model with the velocity Vpo. On the other hand, due to a positive value of c, the VTI wavefront in Figure 1.7 is more advanced in near-horizontal directions than its isotropic counterpart. As discussed below, for vertical transverse isotropy 5 plays a more important role than does c in processing of P-wave reflection data because it controls the normal-moveout velocity from horizontal reflectors and the small-angle reflection coefficient. There is one subtle point in the relationship between the conventional and Thornsen parameterization. From equations (1.44) - (1.48) it is clear that Vpo, Vso, e, 5, and ~/ are uniquely defined by the stiffness coefficients. The inverse transition from Thomsen parameters to the stiffnesses, however, is unique for only four coefficients (c11, c33, c55, and c6~). The fifth coefficient, c13, can be determined from equation (1.47) only if the sign of the sum c13 + c55 is specified. In principle, it is possible for c13, as well as for the sum c~3 + c55, to be negative. While phase velocities are not dependent on the sign of 03 + c55, this is not the case for P-wave polarization, which becomes anomalous in media with c13 + c~5 < 0 (e.g., the polarization vector may even become perpendicular to the phase-velocity vector;

20


see Helbig and Schoenberg, 1987). However, since the stability condition (Musgrave, 1970) requires the coefficient c5~ to be always positive, the sum c13+c5~ can be negative only for uncommon large negative values of c13. Therefore, for practical purposes of seismic modeling and processing it can be assumed that c13 + c55 > 0. Note that if 5 is small (a common case), it can be approximated by ~ ~ (c13 + 2c55 - c33)/c33, and the problem with the unique determination of c~3 does not arise. Values of the a n i s o t r o p i c coefficients Existing laboratory and field data indicate that the velocity of P- and SH-waves in the isotropy plane usually are larger than those in the symmetry-axis direction, so the parameters e and 7 are predominantly positive (Thomsen, 1986; Sams et al., 1993). The values of c in sedimentary sequences range from 0.1-0.3 for moderately anisotropic rocks to 0.3-0.5 or even higher for compacted shale formations (Thomsen, 1986; Alkhalifah et al., 1996). However, as discussed below, e has little direct influence on seismic processing. In transversely isotropic media due to thin interbedding of isotropic layers, e is always greater than 5 (Berryman, 1979); another strict constraint for this model is 7 > 0. Measurements made for TI formations at seismic frequencies indicate that typically ~ > /~ even in cases where the intrinsic anisotropy of shales dominates the contribution of fine layering. Comparison of P-wave moveout and vertical velocities over shale formations usually yields moderate positive /~ values (typically on the order of 0.1-0.2), while interbedding of thin isotropic layers typically produces small negative ~ (Berryman et al., 1999). From the definition (1.47) of 5 it is clear that the minimum possible value of this parameter corresponds to c13 - -c55"

(~min : -- C332c33-C55 ----

21 ( 1 -

V~~ .

(1.50)

The upper bound on ~ can be found from the following constraint on the stiffness coefficients of TI media (Auld, 1973)" C123 < C33 (Cll -- C66 ) .

(1.51)

Equation (1.51) shows that the maximum value of c13 for fixed Cll and c33 corresponds t o c66 ---} O:

(1.52)

C13,max --- ~/C33 C l l .

Assuming for simplicity that in equation (1.52) cll - c33 and substituting C13,m~x-- C33 into equation (1.47) gives an estimate of the maximum possible ~"

( )1

(C33 + C55)2 _ (C33 -- C5S)2

(~max :

2C33 (C33 -- C55)

V~~ - 1

= 2

~0

(1.53) "

Equations (1.50) and (1.53) put the lower and upper bounds on the value of g for a TI medium with a given Vpo/Vso ratio. For the smallest possible shear-wave


21

velocity in the symmetry direction (Vso --+ 0), 5 can become only as small as -0.5. If the ratio Vpo/Vso takes a common value of 2, then (~min : --0.38. The upper limit of ~ decreases with the Vpo/Vso ratio. While 5 may reach an unusually large value of 2 for the smallest Vpo/Vso - v/2, for typical Vpo/Vso - 2 the maximum 5 is equal to 0.67. It is likely that these estimates of 5max are overstated because they are derived for c66 = 0 in equation (1.51); on the other hand, replacing Cll with c33 in equation (1.52) somewhat reduces c13,max and 5m~x. The dimensionless anisotropic coefficients are extremely convenient in developing the weak-anisotropy approximation ([e I

1.4

...........

E ~

,,

...........

:

~

1.5

>

: 0=06

Figure 1.9: Exact phase velocity of the S V - w a v e (solid curve) and its weak-anisotropy a p p r o x i m a t i o n (1.65) (dashed) for TI models with different values of a. a) a = 0 (c = 5 = 0.1); b) a = 0.24 (~ = 0.15, 5 = 0.07); c) a = 0.48 (~ = 0.25, 5 = 0.09); d) a = 0.6 (e = 0.3, ~ = 0.1). For all models, Vso = 1.4 k m / s and Vpo = 2.42 k m / s ( W o / V ~ o = 3).

28


2 Figure 1.10: SH-wave propagation from a point source (marked by the triangle) in a vertical plane of a homogeneous VTI medium with 7 = 0.12. The rays (in black) are computed with a constant increment in the phase angle; the wavefront (white) has an elliptical shape. into the remaining two (in-plane) components of V a [equations (1.14) and (1.15)]. In the symmetry-axis direction (0 = 0~ and isotropy plane (0 = 90 ~ the derivatives of phase velocity vanish, and the group- and phase-velocity vectors coincide. Group velocity has the simplest form for SH-waves, described by the phasevelocity equation (1.54):

V~x

V~z - 1

Vd0(1 + 27) + ~

(1.68) "

The denominator of the first term [Vd0 (1 + 27)] represents the squared horizontal velocity of the SH-wave. Thus, the group velocity (1.68) plotted in the [Vax,Vaz]plane (i.e., a vertical cross-section of the wavefront) is an ellipse, with the semi-axes equal to the horizontal (isotropy-plane) and vertical (symmetry-direction) velocities (Figure 1.10). Typically, the coefficient 7 is positive, and the semi-major axis of the ellipse is horizontal. Using equation (1.68), the group velocity Va = IV~I of the SH-wave can be expressed as a function of the group angle r (Vax = Va sin r Vaz = VGcos ~b), +

Va = V/1 + 27 cos~ ~J '

(1.69)

In 3-D, the wavefront of the SH-wave has an ellipsoidal shape with a circular horizontal cross-section. This result is also valid for P-waves in the special case of elliptical anisotropy (e = 5), for which the phase velocity is given by equation (1.57). The SV-wave velocity in elliptical media does not change with direction, producing a spherical wavefront.


29

For P- and SV-waves in non-elliptical TI media the magnitude of the groupvelocity vector and the group angle r can be obtained from equations (1.14) and (1.15)"

Va-V

tanr

1+

tanO-~ pdo _ t a n O 1 tan0dV -y d0

-~-

1+

,

uao ( sin0cos0 1

(1.70)

tan0dV)

y d0

"

(1.71)

Equation (1.71) shows that at angles 0 where phase velocity increases with angle (dV/dO > 0), the group angle r is greater than 0. On the other hand, if phase velocity decreases with angle so that dV/dO < 0, then r < 0. Thus, the group-velocity vector deviates from the phase direction towards higher velocity; this conclusion remains valid for lower-symmetry models as well. At any extremum of the phase-velocity function the derivative dV/dO vanishes, and the group- and phase-velocity vectors are identical. In TI media this is always the case in the symmetry-axis direction and in the isotropy plane; in addition, P- and SV-waves (but not the SH-wave) may have velocity extrema at oblique propagation angles. Due to the complexity of the exact phase-velocity function V(0) [equations (1.43) or (1.59)], analysis of group velocity is conveniently performed in the linearized weakanisotropy approximation. Differentiating the approximate P-wave velocity (1.61), we find

dVp(O) dO

= Vp0 sin 20 (5 cos 20 + 2e sin 2 0).

(1.72)

Since the term containing the first derivative of phase velocity in equation (1.70) is squared, it has only quadratic and higher-order terms in e and d. Therefore, for weak anisotropy the group velocity as a function of the phase angle coincides with the phase velocity. However, VG should be evaluated not at the phase angle, but instead at the group angle r (see Figure 1.1). Substituting equations (1.61) and (1.72) into equation (1.71) for ~b and dropping the terms quadratic in e and ~ yields tan Cp - tan 0 [1 + 2~ + 4(e - 5) sin 2 0],

(1.73)

Cp = 0 + [5 + 2(c -/~) sin 2 tg] sin 2~9.

(1.74)

or

The presence of terms linear in the anisotropies in equation (1.73) means that the difference between the group and phase angles is more pronounced than that between the group and phase velocities [equation (1.70)]. The conclusion about the weak influence of the S-wave vertical velocity, drawn for the P-wave phase-velocity function, holds for the group velocity as well. Figure 1.11 displays the results of anisotropic P-wave ray tracing for a typical VTI model with moderate values of e and 5. The difference between the phase angle t9

30


Figure 1.11: P-wave propagation from a point source (triangle) in a vertical plane of a homogeneous VTI medium with e = 0.15 and 5 = -0.1. The rays (in black) are computed with a constant increment in the phase angle; the wavefront is shown in white. In general, the unit slowness vector n (normal to the wavefront) deviates from the ray direction. and group angle ~)p manifests itself through the deviation of the wavefront normal n (determined by 0) from the ray (determined by Cp) at a given point on the wavefront. Only in the vertical, horizontal directions and at other minima or maxima of the phase-velocity function do the two angles coincide. The group velocity is higher in the isotropy (horizontal) plane than in the vertical direction because c is positive. On the other hand, a negative value of (~ makes the velocity decrease away from the vertical, which is reflected in the shape of the wavefront at small propagation angles. Figure 1.11 not only illustrates the geometry of rays and wavefronts in VTI media, it also contains information about the dynamic properties of the wavefield. The rays in Figure 1.11 are calculated with a constant increment in the phase angle to show their uneven distribution due to the influence of anisotropy. Since the group-velocity vector always deviates from the phase vector towards larger velocity, the concentration of rays is higher at velocity maxima (i.e., near ~ = 0 ~ and t? = 90 ~ and lower at velocity minima. This phenomenon helps to explain the character of energy variation along the wavefront discussed in Chapter 2. Note that for the SH-wave (Figure 1.10) the angular distribution of rays is much more even, which indicates that for elliptical anisotropy the amplitude variation with angle is close to that in isotropic media. Examples in Figure 1.12 show P-wave rays and wavefronts in four transversely isotropic models with different anisotropic coefficients ~ and ~. As the value of e increases from 0.065 for Mesa shale to 0.19 for Green River shale and Mesa Clay shale, the wavefront gets visibly extended in the horizontal direction. A relatively

1.2.

PLA N E WAVES IN TRANSVERSELY I S O T R O P I C MEDIA

a =0.195

g =-.22

~:=0.189 5=0.204

a=0.110

5=-.035

e =0.065 5=0.059

31

Figure 1.12: Rays and wavefronts of the P-wave in four homogeneous VTI models with the parameters shown under the plots. The rays are computed with a constant increment in the phase angle.

32

C H A P T E R 1. E L E M E N T S OF B A S I C T H E O R Y O F A N I S O T R O P I C WAVE P R O P A G A T I O N

2 Figure 1.13: SV-wave propagation in a VTI medium with Vpo = 2.6 km/s, Vso = 2 km/s, e = 0.15 and /~ = -0.1 (or = 0.42). The rays (black) are computed with a constant increment in the phase angle; the wavefront is shown in white. large negative 5 for Green River shale makes the wavefront almost diamond-shaped because the velocity initially drops away from the vertical before increasing towards the isotropy plane. The distribution of rays for Green River shale is extremely uneven due to the prominent velocity maxima in the vertical and horizontal directions. As discussed above, the similarity between the linearized phase-velocity equations makes it possible to obtain kinematic weak-anisotropy approximations for SV-waves from those for P-waves by making the following substitutions: Vpo --+ Vso, 5 ~ ~, e --+ 0. Applying this recipe to equation (1.74) leads to r

O"

- 0 + cr sin 20 cos 20 -- 0 + ~ sin 40.

(1.75)

The sign of the difference (r in equation (1.75) depends on a and cos 20. For most common positive or, the group angle is greater than the phase angle for 0 < 45 ~ while for 0 > 45 ~ the opposite is true. Hence, the group-velocity vector of the S V wave deviates towards the velocity maximum at 45 ~ Indeed, a higher density of rays near the 45~ angle is clearly seen for the model with a = 0.42 in Figure 1.13. Figures 1.13 and 1.11, computed for the same model, demonstrate a typically stronger influence of the anisotropy on SV-waves as compared to P-waves. Although the vertical and horizontal velocities of the SV-wave are identical, a relatively large value of a substantially distorts (extends in Figure 1.13) the wavefront at oblique angles. Also, the concentration of rays near the velocity maximum causes a significant increase (focusing)in amplitude (see Chapter 2). Figure 1.14 displays rays and wavefronts of the SV-wave for the four models used in Figure 1.12. For Mesa shale and Mesa Clay shale the medium is close to elliptical (e ~ 5, cr ~ 0), so the SV-wavefronts are practically circular (spherical in 3-D). The


33

. . . .

.~

,,-

........

:,~

-~-

.

/

l

~ =0.195 ~ = - . 2 2 (3"-1.47

a =0.189 5 =0.204 0"=-.05

a=O.110

a =0.065

5=-.035

5=0.059

Figure 1.14" Rays and wavefronts of the SV-wave for the same models as in Figure 1.12. The rays are computed with a constant increment in the phase angle.

34


velocity maximum near 45 ~ becomes more pronounced with increasing (7, and the wavefront for Taylor sandstone looks almost rectangular. An even larger a makes the slowness surface near the velocity maximum concave, which leads to a cusp (triplication) on the wavefront for Green River shale in the vicinity of 45 ~ Cusps develop for relatively large values of a in the range 0.8-1 and more. It is interesting that most rays concentrate near the edges of the cusp, with a relatively scarce illumination in the central triplication area. While being diagnostic of anisotropy, cusps complicate analytic description of shear-wave moveout and amplitude. The kinematic signature of most interest in reflection seismology is reflection moveout (traveltime), which depends on both group velocity and group angle. Normalmoveout velocity and nonhyperbolic reflection moveout are examined in Chapters 3 and 4. 1.2.4

Polarization group

and

vector

and

polarization

relationship

between

phase,

directions

The polarization vector, or the vector of particle motion, plays an important role in the processing and interpretation of multicomponent VSP and cross-hole surveys. Also, deviation of the polarization vector from its "isotropic" direction may cause distortions of radiation patterns and reflection coefficients in anisotropic media (see Chapter 2). For a given phase (slowness) direction, the polarization vectors of P-, S V - and SH-waves are mutually orthogonal because they represent the eigenvectors of the Christoffel equation (1.35). The SH-wave polarization in VTI media is always horizontal, perpendicular to the vertical plane that contains the phase vector and the polarization vectors of P- and SV-waves. Therefore, in terms of polarization the SH-wave in TI media can be called a "pure" shear mode. In contrast to its behavior in isotropic media, however, the SH-wave polarization vector is strictly defined by the TI symmetry and cannot be arbitrarily rotated within the plane orthogonal to the propagation direction. To determine the angle v between the polarization vector of the P- or SV-wave and the vertical, we substitute the phase velocity into either of equations (1.37):

U1 _ tan

v -

U3

-

s i n ~ c o s ~ (513 -~- 555) pV 2 -

cll

(1.76)

sin 2 0 - css cos 2 0 "

As shown above [equations (1.39) and (1.41)], for 0 = 0~ (symmetry axis) and 9 = 90 ~ (isotropy plane) the P-wave is polarized in the phase direction (vp = 9) and the SV-wave polarization vector is perpendicular to the phase vector. For oblique propagation angles the polarization can be conveniently studied in the weakanisotropy limit. Using the approximate P-wave phase velocity (1.61) and carrying out further linearization of equation (1.76) in ~ and ~ gives the following expression for the P-

1.2.

P L A N E W A V E S IN T R A N S V E R S E L Y

ISOTROPIC

MEDIA

35

wave polarization angle (Rommel, 1994; Tsvankin, 1996)" tan Vp - tan 0 {1 + 2B [~ + 2(e - 5)sin 2 0]}, B

m

1 2f

(1.77)

1 2(1-

V o/V o)

Equivalently, up - 0 + B [5 + 2(e - ~ ) s i n 2 01 sin 2 0 .

(1.78)

The SV-wave polarization angle Vsv is obtained by adding +90 ~ to Vp. If the wave is generated by a point source, the polarization at any receiver location can be found by applying equation (1.78) at the phase angle corresponding to the group (source-receiver) direction; this is explained in more detail in Chapter 2. For P-waves, the direction of the polarization vector is closer to that of the groupvelocity vector (ray) than to that of the phase-velocity vector. Indeed, the weakanisotropy expressions for the P-wave group (1.74) and polarization (1.78) angles are quite similar, with the difference between the two controlled by the quantity B: .p - o - B (r

- 0).

(1.79)

Since for plausible values of the Vpo/Vso ratio B satisfies 0.5 < B < 1, the Pwave polarization vector (angle vp) points between the phase-velocity (angle 0) and group-velocity (angle Cp) vectors, being closer to the group vector. This analytic result is confirmed by Figure 1.15 and the numerical analysis of P-wave polarizations from a point source in Chapter 2. Even in more complicated orthorhombic models, the P-wave polarization and group-velocity vectors usually stay close to each other. Due to the orthogonality of the P- and SV-wave polarizations for a fixed phase direction, the polarization vector of the SV-wave is not far from perpendicular to the P-wave ray. The phase direction of the SV-wave at a given spatial location, however, may be substantially different from that of the P-wave, so the angle between the SVwave polarization and group vectors may noticeably deviate from 90 ~ (see Chapter 2). Unlike the weak-anisotropy approximations for the P-wave phase and group velocity, equation (1.78) does depend on the shear-wave vertical velocity through the parameter B. This is not surprising because equation (1.78) determines the SV-wave polarization angle as well, and l]p is not a function of just P-wave phase velocity. Variation in Vso within a realistic range can cut the difference between the P-wave group and polarization angles, which we will call the polarization anomaly, in half (Figure 1.15). However, since the anomaly as a whole is small, the influence of Vs0 on the P-wave polarization angle for moderate anisotropies lel 0. Helbig and Schoenberg (1987) show that for "abnormal" media that have negative Cla + c55, the P-wave polarization vector may even become perpendicular to the phase-velocity vector.

36

CHAPTER

0

O3 ll) (D

~o

1.

ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION

.........................................................................

e-0.3

5-0.1

:

~

I~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

:

4

Q..

I

s 2

Group Angle (degrees) Figure 1.15: The difference betweeen P-wave group and polarization angles (2/)p- L/p; exact computation) for a model with e - 0.3, 5 - 0.1. The Vpo/Vso ratio is equal to 1.5 (dashed) and 2.5 (solid).

1.3

Plane

waves

in orthorhombic

media

The stiffness tensor and general physical reasons for orthorhombic symmetry were described earlier in this chapter [see Figure 1.4 and equation (1.23)]. Although orthorhombic anisotropy is believed to be quite common for azimuthally anisotropic formations, such as fractured reservoirs (Schoenberg and Helbig, 1997; Bakulin et al., 2000b), its application in seismic inversion and processing is hampered by the large number of independent stiffness elements. For a known orientation of the symmetry planes, wave propagation in orthorhombic media is described by nine stiffnesses cij [equation (1.23)]. If the symmetry planes have to be identified, as is often the case in surface seismic surveys, the number of independent parameters to be found from seismic data increases to 12. Here, we follow Tsvankin (1997c) in introducing a Thomsen-style notation that simplifies considerably the description of a wide range of seismic signatures for orthorhombic media. Taking advantage of the fact that the Christoffel equation has the same form in the symmetry planes of orthorhombic and transversely isotropic (TI) media, the stiffnesses cij can be replaced by two vertical (P and S) velocities and seven dimensionless parameters that represent an extension of the TI coefficients ~, 5 and 7 to orthorhombic models. By design, this notation provides a uniform description of anisotropic media with orthorhombic and TI symmetry, and preserves all the attractive features of Thomsen parameters in treating wave propagation in the sym-

1.3. PLANE WAVES IN ORTHORHOMBIC MEDIA

37

metry planes. Also, we develop a weak-anisotropy approximation for P-wave phase velocity that helps to extend the analogy with TI media to 2-D kinematic signatures of P-waves outside the symmetry planes and to identify the parameters responsible for P-wave kinematics. The scope of this section is restricted to plane-wave propagation. Properties of point-source radiation in homogeneous orthorhombic media are discussed in Chapter 2. 1.3.1

Limited

equivalence

between

TI and

orthorhombic

media It is convenient to use the three mutually orthogonal symmetry planes of orthorhombic media as the coordinate planes of a Cartesian coordinate system. Then the stiffness matrix c~j [equation (1.23)] has the same zero components as that for transversely isotropic media with the symmetry axis aligned with one of the coordinate directions [e.g., see equation (1.24)]. In contrast to the TI model, however, all nine elements cij in equation (1.23) are generally independent. For certain subsets of orthorhombic media (such as models with vertical fractures in a VTI background, see Figure 1.4), the number of independent stiffnesses is reduced (Schoenberg and Helbig, 1997; Bakulin et al., 2000b), but the discussion here is not restricted to any specific type of orthorhombic anisotropy. The phase velocity V and the displacement (polarization) vector U of plane waves in orthorhombic media can be found from the Christoffel equation (1.8). Using equation (1.23), the Christoffel matrix (1.9) for orthorhombic symmetry can be found in the following form: + + (1.80) ~ 2 2 -- C66 ?22 -~- C22 n 2 -4- C44 n32 ,

(1.81)

~ 3 3 -- C55 ?,/,2 .~_ C44 1~2 + C33 n32 '

(1.82)

G12 - (c12 + c66)nln2,

(1.83)

~13 -

(c13 -~- c55) n l T t 3 ,

(1.84)

G23 -

(c23 + c44) n2n3.

(1.85)

As before, n is the unit vector in the phase (slowness) direction. First, consider a plane wave propagating in one of the symmetry planes of the medium. If vector n is confined to the [Xl, x3]-plane, the terms of the Christoffel matrix G/k involving n2 vanish [see equations (1.80)-(1.85)], and equation (1.8) becomes

[clln c55n

0

,c13c55, n13

0

c66n21 + c44n2 -- flu 2

0

(c13 -~- c55)TtlTt3

0

C55122 -~- c33r~ -- p V 2

=0.

1

U2 U3

(1.86)

38

C H A P T E R 1. E L E M E N T S OF B A S I C 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

If c44 ~ - C55, equation (1.86) reduces to the corresponding Christoffel equation (1.35) for the transversely isotropic model with the symmetry axis in the x3-direction (VTI medium). In the orthorhombic model, however, the stiffness components C44 and c55 are not equal to each other, which leads to shear-wave splitting at vertical incidence. Likewise, the Christoffel equation (1.86) reduces to that for an HTI medium with the symmetry axis pointing in the Xl-direction, if c55 = c66; for more details, see Riiger(1997, 2001)and Tsvankin (1997a). Exactly as in VTI media (see section 1.2), equation (1.86) splits into two independent equations for the in-plane ("P-SV") motion (U2 = 0) and the pure transverse ("SH") motion [U1 --- U3 - 0; see equation (1.88)]. Expressing the slowness components in equation (1.86) in terms of the phase angle ~ with the vertical (x3) axis (nl = sin ~; n3 = cos ~) yields for the in-plane polarized waves

[Cllsin20+cb cos20 (c13 + c55) sin 0 cos 0

,c13+c55, sin0cos0

c~5 sin 2 0 + c33 cos 2 0 - pV 2

U3

0

"

,187,

Equation (1.87) is identical to the Christoffel equation (1.37) for P- and SV-waves in VTI media that was thoroughly discussed in the previous section. Hence, the phase velocities of the P-wave and in-plane polarized shear (SV) wave in the Ix1, x3]-plane of orthorhombic media represent the same functions of the stiffness coefficients cij and the phase angle ~ as do the P - S V phase velocities in VTI media. For example, the Pwave phase (and group) velocities in the vertical (x3) and horizontal (xl) directions are given by the VTI expressions ~/c33/p and ~/cii/p, respectively, while both the vertical and horizontal SV-wave velocities are equal to V/C55/p. The phase-velocity function of each wave in the Ix1, x3] symmetry plane is sufficient to obtain the corresponding group (ray) velocity and group angle and, consequently, all other kinematic signatures, such as normal-moveout (NMO) velocity (see Chapter 3). This means that the kinematics of P- and SV-waves in the [xl, x3]-plane of orthorhombic media is fully described by the known VTI equations. The only exception to this analogy is cuspoidal S-wave group-velocity surfaces (wavefronts) formed in the symmetry planes of orthorhombic media near shear-wave point singularities. [For a general discussion of S-wave singularities, see Crampin and Yedlin (1981), Crampin (1991) and Helbig (1994).] Some in-plane branches of the cusps may be due to slowness vectors that lie outside the symmetry planes (Grechka and Obolentseva, 1993); cuspoidal features of this nature cannot exist in TI media. Since the displacement (i.e., polarization) vectors U of the in-plane polarized plane waves are determined from the same equation (1.87), they are also given by the corresponding expressions for VTI media. Furthermore, analysis of the boundary conditions at interfaces shows that the equivalence with vertical transverse isotropy holds for plane-wave reflection coefficients as well, provided the symmetry planes have the same azimuths above and below the boundary (Rfiger, 1998). However, body-wave amplitudes in general do not comply with this "equivalence" principle because they depend on the 3-D shape of the slowness surface, not just in the symmetry plane, but also in its vicinity. Out-of-plane velocity variations lead


39

to focusing and defocusing of energy and may have a significant influence on the distribution of energy along the wavefront within the symmetry planes (see Chapter 2, section 2.1). Equation (1.86) also has a solution corresponding to the shear wave polarized orthogonally to the Ix1, x3] plane (U1 = U3 = 0, U2 r 0): / c66 sin 2 0 + c44 cos 2 0

Vstt(O)

(1 88)

9

P

Equation (1.88) describes a "pure" shear mode with an elliptical slowness curve and an elliptical wavefront in the [x~, Xa]-plane and has the same form as equation (1.36) for SH-waves in VTI media, but with c44 in place of css. Since in orthorhombic media c44 is not equal to c~5, the vertical velocity of the SH-wave, equal to differs from the vertical velocity of the SV-wave. In fact, the two S-waves in the Ix1, Xa]-plane of an orthorhombic medium are fully decoupled because they depend on different sets of elastic constants [compare equations (1.87) and (1.88)]. Similar conclusions can be drawn for wave propagation in the [x2, x3] symmetry plane. Substituting n~=0 into the Christoffel matrix [equations (1.80)-(1.85)] and introducing the in-plane phase angle with the vertical (n2 = sin 0; n3 = cos 0) in the Christoffel equation (1.8) yields for P - SV-waves:

~/c44/P,

[c22sin20+c44cos20-pV 2 (C23 -4- C44 )

sin 0

cos 0

c44

(c23+c44) sinOcosO ][U2]_0 C33 COS2 0 -- pV 2 U3

sin 2 0 +

(1.89)

"

Equation (1.89) becomes identical to the corresponding Christoffel equation (1.87) for the [Xl, x3]-plane if we replace U2 with U~ and interchange the indices 1 and 2 in the appropriate components of the stiffness tensor %k~, i.e., C22 ~

Cll;

C44 --~ C55; C23 ~

C13.

(1.90)

Therefore, to obtain the phase velocity of both in-plane polarized modes in the [x2, x3]-plane as a function of the phase angle with the vertical, it is sufficient to make the above substitutions of the elastic constants in the known phase-velocity equations for VTI media (or for the [x~, x3]-plane of orthorhombic media). It can be demonstrated exactly in the same fashion that the Christoffel equation in the third ([x~,x2]) symmetry plane can be reduced to the VTI equation by the appropriate substitution of elastic constants. Note that the term "SV-wave" actually denotes two different modes in the vertical symmetry planes of orthorhombic media (the same holds for the "SH-wave"). To understand polarization properties of S-waves for orthorhombic anisotropy, it is helpful to review a cartoon of typical phase-velocity sheets (Figure 1.16). The outer (P-wave) phase-velocity surface is usually separated from the two sheets corresponding to the split shear waves. The shear-wave phase velocities coincide in the direction marked by point A in Figure 1.16; this is the so-called point (or conical) singularity mentioned above (e.g., Crampin and Yedlin, 1981).

40

C H A P T E R 1. E L E M E N T S OF BASIC THEORY OF A N I S O T R O P I C WAVE P R O P A G A T I O N

X3 !~33 l

SV SH

$1 0

C144 X2 "~1 Figure 1.16: Sketch of phase-velocity surfaces in orthorhombic media, aij are the ~/Cij/P. normalized stiffness coefficients defined as aij -

-

The polarization of shear waves with respect to the vertical incidence plane varies with azimuth. Suppose that c44 > c55, and the fast vertically traveling shear wave 5'1 is polarized in the x2-direction. Then $1 represents a pure transverse (SH) wave for any phase direction in the Ix1, x3]-plane. As we move along the phase-velocity surface of the Sl-wave around the x3-axis to the Ix2, x3]-plane (Figure 1.16), its polarization changes from transverse (cross-plane) to in-plane (in other words, from S H in the [xl, x3]-plane to S V in the Ix2, x3]-plane). Thus, according to the analogy with vertical transverse isotropy, the Sl-wave propagating in the [xl, x3]-plane is equivalent to the SH-wave in VTI media, while in the [x2, x3]-plane it is equivalent to the SV-wave. Likewise, the polarization of the S2-wave changes from S V in the Ix1, x~]-plane to S H in the Ix2, x3]-plane. Implications of these S-wave polarization properties in traveltime analysis of multicomponent reflection data are discussed by Grechka, Theophanis and Tsvankin (1999).

1.3.2

Anisotropic parameters for orthorhombic media

To take full advantage of the analogy with VTI media, the stiffness coefficients cij can be replaced with a set of anisotropic parameters (combinations of stiffnesses) that concisely characterize a wide range of seismic signatures for orthorhombic anisotropy. Indeed, since seismic signatures for transverse isotropy are especially convenient to describe in terms of Thomsen parameters (see sections 1.2 and 6.3), it is natural to

1.3. P L A N E WAVES IN O R T H O R H O M B I C MEDIA

41

extend that notation to the symmetry planes of orthorhombic media. As shown below, those parameters are helpful in studying wave propagation outside the symmetry planes as well. Another, similar version of Thomsen-style notation for arbitrary anisotropic media was proposed by Mensch and Rasolofosaon (1997) and P~en~{k and Gajewski (1998). Their approach, however, is based on the weak-anisotropy approximation for phase velocity and leads to a different (linearized in cij) definition of the 5 coefficients. One of the main points substantiated here and in Chapter 3 (section 3.5) is that the notation introduced here is advantageous for media with any strength of the anisotropy. Thomsen-style parameters for orthorhombic media have also been used in the physical-modeling study by Cheadle et al. (1991). Definitions of the p a r a m e t e r s First, following the recipe outlined above for VTI media, we define two vertical velocities (for P- and S-waves) of the reference isotropic model. For orthorhombic media, we can choose either of the two shear-wave velocities at vertical incidence. Here, the preference is given to the S-wave polarized in the xl-direction to make the new notation for the "P-SV"-waves in the [Xl, x3] plane identical to Thomsen notation in the VTI case. F-"---

liFO- ~ / p 3

(1.91)

Vso- ? p

(1.92)

Since the Christoffel equation (1.87) for the waves polarized in the [xl, x3] plane is identical to the corresponding equation for vertical transverse isotropy, it is possible to introduce the dimensionless coefficients c(2) and ~(2) [the superscript (2) refers to the x2-axis, which defines the normal direction to the Ix1, x3] symmetry plane] through the VTI equations (1.46) and (1.47): s

_

-_ C l l C33 2C33 '

+

-

(1.93) -

2c33 (c33 - c55)

"

(1.94)

In the definition of 3 for VTI media, the coefficient C44 rather than its equal (c55) has often been used. Since these two parameters differ for orthorhombic media, we should always use c55 in equation (1.94). As in VTI media, the coefficient 5 (2) from equation (1.94) provides the exact second derivative of P-wave phase velocity at vertical incidence in the [xl, x3]-plane: 02V I - - 2 V p 0 5(2) Since this derivative is needed to obtain P-wave small-angle 002 0=0 amplitudes (see Chapter 2) and NMO velocity (Chapter 3), this definition of 5 (2) is well-suited for describing reflection seismic signatures.

42


Next, the equivalence between equation (1.88) and the corresponding expression (1.36) for the SH-wave in VTI media can be used to introduce the parameter ,.)/(2) __ C66 -- C44 2C44 9 --

(1.95)

7 (2) is identical to Thomsen's coefficient 7 [equation (1.48)] for VTI media: it is responsible here for the velocity variations of the SH-wave in the [xl, x3]-plane. In principle, it would be convenient to specify the parameters e, ~ and 7 in the same fashion for the two other symmetry planes as well. However, if we did so, some of the anisotropy coefficients would not be independent, and the notation would suffer from redundancy. Anticipating application of orthorhombic models with a horizontal symmetry plane in reflection seismology, it is most important to simplify seismic signatures in the two vertical planes of symmetry, treating the horizontal plane differently. P - SV-waves in the Ix2, x3] symmetry plane can be described by the parameters e(1) and 5 (1) (the superscript refers to the fact that this plane is normal to the x~axis) defined analogously to c(2) and 5 (2). Using equations (1.93)-(1.95) and the substitution recipe in equation (1.90) gives r

~

C22 -- C33

2c33 =

+

-

'

(1.96) -

2C33 (C33 _ C44)

c44)

,

(1.97)

(1.98) C55. 2c55 The two vertical velocities and six anisotropy parameters introduced above can be used instead of eight original stiffness coefficients: c11, c22, c33, c44, c55, c66, c23 and c13. The only remaining stiffness c12 can be replaced with a dimensionless anisotropic parameter analogous to the coefficients ~(1) and 5 (2). ,),(1) =

=

+

C66-

-

-

2Cll (Cll -- c66)

"

(1.99)

The parameter ~(3) plays the role of Thomsen's ~ in TI equations written for the [xl,x2] symmetry plane, with the xl-direction substituted for the symmetry axis. Note that the quantities e(3) and 7 (3) would be redundant. The complete list of the Thomsen-style parameters, along with their brief descriptions, is as follows: 9

Vpo -

9

Vso -

the vertical velocity of the P-wave; the velocity of the vertically traveling S-wave polarized in the xl-direction;

. e(2) _ the VTI parameter e in the symmetry plane Ix1, x3] (close to the fractional difference between the P-wave velocities in the xl- and x3-directions);

1.3.

PLANE

W A V E S IN O R T H O R H O M B I C

MEDIA

43

9 5 (2) - the VTI parameter 5 in the [Xl,X3]-plane (responsible for near-vertical in-plane P-wave velocity variations, also influences SV-wave velocity); 9 7 (2) - the VTI parameter 7 in the [Xl, x3]-plane (close to the fractional difference between the SH-wave velocities in the Xl- and x3-directions); 9 c (1) - the VTI parameter e in the [x2, x3]-plane; 9 5 (1) - t h e VTI parameter 5 in the [x2, x3]-plane; 9 7 (1) --

the VTI parameter 7 in the Ix2, x3]-plane;

9 5(3) _ the VTI parameter 5 in the [xl,x2]-plane (xl is used as the symmetry axis). While the new parameters are uniquely determined by the nine independent stiffness coefficients of orthorhombic media, the inverse transition is unique only for the coefficients associated with the velocities along the coordinate axes (c11, C2e, c33, c44, c55, and c66). To obtain the other three coefficients (c12, c13, c23) from the corresponding 5 values, it is necessary to specify the sign of the sums (c13+c55) [equation (1.94)], (c23 + c44)[equation (1.97)], and (c12 + c60)[equation (1.99)]. As discussed above, exactly the same problem arises with the transition from Thomsen parameters to the stiffness c13 in transversely isotropic media. Since the sums under consideration can be negative only for large negative values of c13, c23 or c12, which are seldom encountered in practice, it can be assumed that (c13 + c55), (c23 + c44), and (c12 + c60) are positive. That would correspond to one of the conditions of so-called "mild anisotropy," as specified by Schoenberg and Helbig (1997), and ensures the absence of anomalous body-wave polarizations in the symmetry planes (Helbig and Schoenberg, 1987). Note that the term "mild anisotropy" does n o t mean that the magnitude of velocity variations and anisotropic parameters is small. Although the nine parameters introduced above are sufficient to characterize general orthorhombic media, one may need to use different combinations of these coefficients in specific applications. For instance, shear-wave splitting at vertical incidence is conventionally described by the fractional difference between the parameters c44 and c55 (assuming C44 > C 5 5 ) : 7(s)

r

__

C44 -- C55 = 7 (1) -- 7 (2) 2c55 1 + 27(2)

~

(1.100)

V s l - Vso Vso

' ]

_

where V s l is the vertical velocity of the (fast) Sl-wave, and Vso - ~ / c 5 5 / p (introduced above) is the vertical velocity of the S2-wave. The parameter 7 (s) represents a direct measure of the time delay between two split shear waves at vertical incidence and is identical to the generic Thomsen's coefficient 7 for transversely isotropic media with a h o r i z o n t a l symmetry axis that coincides with the Xl direction.

44


Special cases" V T I a n d H T I m e d i a Both vertical and horizontal transverse isotropy can be considered as degenerate special cases of orthorhombic media with a horizontal symmetry plane. An orthorhombic medium reduces to the VTI model if the properties in all vertical planes are identical, and the velocity of each mode in the [Xl, x2] plane (the isotropy plane) is constant (although the velocities of the two S-waves generally differ from one another). Hence, the VTI constraints, which reduce the number of independent parameters from nine to five, can be written as .

e(1) =

e(2) =

9 5 (1) - - 5 (2) : 9 7 (1) =

e, 5,

7 (2) - - 7 ,

9 ~(3) = O,

where e, 5, and 7 are Thomsen VTI coefficients. Another special case is transverse isotropy with a horizontal axis of symmetry. If the symmetry axis is parallel to the Xl-direction, then the axes x2 and x3 form the isotropy plane, and

9

9 7 (I) = O.

Wave propagation in such HTI media is fully described by the anisotropic parameters in the [xl,x3]-plane (e(2)5(2) and 7 (2)) and the vertical velocities Vpo and Vso. (The last anisotropic parameter, 5(3) in this model is not independent.) 1.3.3

Signatures

in the

symmetry

planes

By design, the notation introduced above provides a simple way of describing seismic signatures in the symmetry planes of orthorhombic media using the corresponding equations for VTI media expressed through Thomsen parameters (see section 1.2). The coefficients e(i), 5(i), and 7 0) conveniently quantify the magnitude of velocity anisotropy, both within and outside the symmetry planes. The parameters e(2) and e(1) are close to the fractional difference between vertical and horizontal P-wave velocities in the planes [xl,x3] and [x2, x3] (respectively) and, therefore, yield an overall measure of the "P-wave anisotropy" in these planes. Similarly, 7 (2) and 7 (1) govern the magnitude of the velocity variation of the elliptical SH-waves in the vertical symmetry planes. One of the most important advantages of Thomsen notation for VTI media is the reduction in the number of parameters responsible for P-wave kinematic signatures


45

from four to three (Vp0, e, a). As shown in section 1.2, the stiffness coefficient that determines the shear-wave vertical velocity (C44 --" 555 in VTI media) does make a contribution to the P-wave velocity equations, but only through the parameter 5. The analogy with VTI media implies that P-wave kinematic signatures in the vertical symmetry planes depend on the vertical velocity Vpo (the scaling factor) and a pair of the anisotropic coefficients introduced above: e(2) and ($(2) ([Xl,Xa]-plane) or e(1) and 5 (1) ([x2,xa]-plane). Similar to VTI media, the coefficients e(i) and ($(i) are responsible for phase and group velocity in different ranges of phase angles, which is extremely convenient for purposes of seismic processing and inversion. Specifically, the coefficients 5(2) ([Xl,Xa]-plane) and 6(1) ([x2, xa]-plane) determine near-vertical P-wave velocity variations and, as shown in section 3.5 below, the anisotropic term in the expression for normal-moveout velocity from horizontal reflectors. Also, the new notation simplifies the elliptical condition in both vertical symmetry planes: for example, the P-wave anisotropy in the [Xl, X3] plane is elliptical if e(2) = 5 (2) (the SV-wave in-plane velocity in this case is constant). To obtain any kinematic signature (e.g., phase and group velocity, moveout from horizontal reflectors, etc.), polarization vector, and reflection coefficients of P- and SV-waves in the [xl, Xa]-plane of orthorhombic media, it is sufficient to substitute Vpo, Vso, e (2) and ($(2) into VTI equations expressed through Vpo , VS0, s and 5, respectively. For instance, adapting the exact VTI phase-velocity equation (1.59) yields the phase velocities of the P - SV-waves in the [Xl, Xa]-plane: V2(0)

v~0

=

l + e (2) sin 20

f

2 :t: ~ f ~ ( 1 + 2e(2)2sin2f 0)

- 2(e(2) - a(2)) 2 0sin2f

f -- 1 - V~o/V~,o .

,

(1.101)

(1.102)

The plus sign in front of the radical corresponds to the P-wave, and the minus sign to the SV-wave. In the weak-anisotropy approximation, the P-wave phase velocity in the [Xl, x3]-plane can be found from the VTI expression (1.61): Vp(e) = Veo (1 + a(2)sin 2 Ocos 2 e + e(2) sin 4 0).

(1.103)

Phase-velocity equations (1.101) and (1.103) demonstrate how the new anisotropic parameters help to take advantage of the analogy between the symmetry planes of orthorhombic media and vertical transverse isotropy. The same analogy with VTI media holds for P - SV-waves in the Ix2, a3]-plane, if the VTI equations are used with the parameters Vp0, Vsl, e(1) and 5(1). The exact SH-wave phase velocity in the vertical symmetry planes can be obtained from

46


equation (1.54):

VsH{[Xl,X3]--plane} --

V s I V/-I--[ -

2~/,(2) s i n 2 0 - Vso

~ 1 + 27(1) 1 + 27(2) V/1 + 27(2) sin20 ; (1.104)

VSH{[X2,x3]--plane} -- Vso ~/1 + 27(1) sin 2 0.

(1.105)

Recall that the term "SH-wave" refers to two different shear modes in the vertical symmetry planes. Expressions for the group-velocity and polarization vectors in the symmetry planes, based on the VTI equations given in section 1.2, can be found in Tsvankin (1997c). The reflection coefficients in the symmetry planes of orthorhombic media are described by Riiger (1998). 1.3.4

P-wave

velocity

outside

the symmetry

planes

Since Thomsen notation does not provide any tangible simplification of the exact phase-velocity equations for transversely isotropic media (see section 1.2), the parameters introduced above can hardly be expected to accomplish this task for orthorhombic models. Indeed, the exact phase velocity in orthorhombic media still has to be computed numerically using the results of Appendix 1A. However, as demonstrated by the symmetry-plane analysis, the Thomsen-style notation can be helpful in describing seismic signatures by analogy with VTI media, developing concise weakanisotropy approximations, and reducing the number of P-wave kinematic parameters for any strength of the anisotropy. The focus here will be on P-wave phase velocity and other kinematic signatures outside the symmetry planes of orthorhombic media. Properties of shear waves in orthorhombic models are discussed, for example, in Musgrave (1970), Crampin (1991), Wild and Crampin (1991), Helbig (1994) and Mensch and Rasolofosaon (1997). W e a k - a n i s o t r o p y a p p r o x i m a t i o n for phase velocity The dimensionless anisotropic parameters are particularly suitable for simplifying the P-wave phase-velocity function in the limit of weak anisotropy. For the planes of symmetry, as shown above, it is sufficient just to adapt the known expressions for weak transverse isotropy. Approximate P-wave phase velocity outside the symmetry planes is obtained in Tsvankin (1997c) by linearizing the exact equations of Appendix 1A in the anisotropic coefficients: V~ - V~0 [1 + 2n 4 e(2) + 2n 4 e(1) + 2n2n 2 (5(2) + 2n~n~ ~(1)+ 2n~n 2 (2e(2)+5(3))]. (1.106) It is convenient to replace the directional cosines nj of the slowness (or phase-velocity) vector by the polar (0) and azimuthal (r phase angles, nl = sin 0 cos r

n2 = sin 0 sin r

n3 = cos 0.

1.3.

P L A N E W A V E S IN O R T H O R H O M B I C

MEDIA

47

Taking the square-root of equation (1.106) yields the phase velocity exactly in the same form as in VTI media [equation (1.61)], but with azimuthally-dependent coefficients e and 5:

Vp(9, r = Vpo [1 + 5(r sin 2 9 cos 2 9 + e(r sin 4 9],

(1.107)

where (~(r

__

(~(1)sin2 r + 5(2) COS2 r

e(r _ e(1) sin 4 r + e(2) cos 4 r + (2e(2) + ~(3))sin 2 r cos 2 r

(1.108) (1.109)

In the [Xl, x3]-plane, 5 (r = 0 ~ = (g2), and e (r = 0 ~ = e(2); in the [x2, x3]-plane, (r = 90 ~ = 5 (1), and e (r = 90 ~ = e(1). Therefore, the analogy between orthorhombic and TI media in P-wave kinematics goes beyond the symmetry planes, although for out-of-plane propagation it is limited to weak anisotropy. The structure of equation (1.107) is similar to the expansion of P-wave phase velocity in a series of spherical harmonics developed by Sayers (1994b). Here, however, instead of describing the medium by perturbations of the stiffness coefficients (as was done by Sayers), a concise approximation is obtained in terms of the dimensionless anisotropic parameters s and (~(i). Since Sayers (also see Jech and P~en~{k, 1989; Mensch and Rasolofosaon, 1997) neglects terms of order (Acij/cij) 2, whereas we neglect terms of order 52 (etc.), this linearization is slightly different. Our definitions of the parameters 5(i) using nonlinear combinations of elastic modulae are particularly convenient for applications in reflection seismology. As mentioned above, the 5 coefficients introduced here yield the exact second derivative of P-wave phase velocity at vertical incidence that is needed to describe a range of reflection seismic signatures (see Chapters 2 and 3). P a r a m e t e r s r e s p o n s i b l e for P - w a v e k i n e m a t i c s Equation (1.107) does not contain any of the three parameters (Vs0, 7 (1), 3'(2)) that describe the shear-wave velocities in the directions of the coordinate axes. Clearly, kinematic signatures of P-waves in weakly anisotropic orthorhombic media depend on just five anisotropic coefficients (c (1), 5(1) e(2) 5(2), and (~(3)) and the vertical velocity. Similar conclusions for weakly anisotropic models of lower symmetries were drawn by Chapman and Pratt (1992), Sayers (19945) and Mensch and Rasolofosaon (1997). The equivalence between the symmetry planes of orthorhombic media and transverse isotropy, discussed in detail above, implies that these six parameters are sufficient to describe P-wave phase velocity in all three symmetry planes, even for strong velocity anisotropy. An important question, to be addressed next, is whether or not P-wave phase velocity depends only on the same six parameters outside the symmetry planes and in the presence of significant velocity variations. In the numerical examples below, the influence of the parameters Vso and 7 0) on the exact phase velocity of P-wave is examined for orthorhombic models with moderate and strong velocity anisotropy. It is sufficient to consider the velocity

48

CHAPTER

1.

ELEMENTS

OF BASIC THEORY OF ANISOTROPIC

oo, iliiiili

3.2-~ ffl

~ l- 3 . 0 a.

r"

2.8

i

0

WAVE PROPAGATION

3.o

I

90

~3.6

>,3.8 o

m

3.2-

iii

3.4ca.

0

30 60 90 Polar Angle (deg)

3.0

I

0

I

30 60 90 Polar Angle (deg)

Figure 1.17: Influence of the shear-wave vertical velocity Vso on the exact P-wave phase velocity. The velocity is calculated as a function of the phase angle with the vertical at the four azimuthal phase angles shown on top of the plots. The solid curve corresponds t o Vpo/Vso= 1.5 (Vpo= 3 km/s, Vso = 2 km/s), and the dotted curve to Vpo/Vso= 2.5 (Vpo= 3 km/s, Vso = 1.2 km/s). The other model parameters are: e(1) = 0.25, e(2) = 0.15, 5 (1) = 0.05,/~(2) = -0.1, ~(3) = 0.15, 7 (1) = 0.28, ~,(2) = 0.15.

function in a single octant, for instance, the one corresponding to 0 < 0 _< 90 ~ 0 < r < 90 ~ Figure 1.17 shows the dependence of P-wave phase-velocity variations in four vertical planes on the shear-wave vertical velocity Vso (or the Vpo/Vsoratio), with all other model parameters being fixed. Clearly, the influence of Vso for a wide range of the Vpo/Vsoratios is negligible not only in the [Xl, Xa]-plane (r - 0), as was expected from the equivalence with VTI media, but outside the symmetry planes as well. This conclusion holds for two models with more pronounced phase-velocity variations shown in Figure 1.18; note that even for a medium with uncommonly strong velocity anisotropy (Figure 1.18b) the difference between the phase-velocity curves corresponding to the two extreme values of the Vpo/Vsoratio remains barely visible. The contribution of Vso to the P-wave phase velocity becomes somewhat more no-


49

3.2ca.

2.8

0

0

30 60 90 Polar angle (deg)

Figure 1.18: P-wave phase velocity for Vpo/Vso= 1.5 (solid) and Vpo/Vso= 2.5 (dotted) in two models with stronger anisotropy than that in Figure 1.17; the azimuthal phase angle is 45 ~ The other model parameters are: a) c(1) - 0.25, e(2) _ 0.45, 5 (1) = -0.1, 5 (2) = 0.2, 5 (3) = -0.15, 7 (1) = 0.28, 7 (2) = 0.15; b) e(1) = 0.45, e(2) = 0.6, 5 (1) = -0.15, 5 (2) = -0.1, 5 (3) = 0.2, 7 (~) = 0.7, 7 (2) = 0.3.

ticeable only for uncommon models with close values of the P- and S-wave velocities in one of the coordinate directions. It should be emphasized that if the conventional notation is used, the influence of the stiffness coefficients Caa, c~5, and c66 on P-wave velocity cannot be ignored. For instance, c55 does make a contribution to the value of 5(2) [equation (1.94)]. As in VTI media, Thomsen-style notation makes it possible to reduce the number of parameters that govern P-wave velocity by combining the stiffness coefficients c55 and c13 in the single parameter 5(2); the same holds for the stiffnesses Caa and c66 as well. The P-wave phase velocity for two extreme values of the parameter 7 (1) is displayed in Figure 1.19. Since none of the elastic constants governing P-wave propagation in the Ix1, x3] plane depends o n ,)/(1) (with Vso being fixed), the plot starts at an azimuth of 20 ~ Although the influence of 7 (1) slightly increases in the vicinity of the Ix2, x3] plane (r - 90~ the contribution of this parameter can be safely ignored at all azimuthal angles. Similarly, P-wave velocity is independent of the coefficient 7 (2) . Thus, P-wave phase-velocity variations in orthorhombic media are governed by just the orientation of the symmetry planes and five anisotropic parameters (e(1), 5(t), ~(2), 5(2), and 5(3)), with the vertical P-wave phase velocity serving as a scaling coefficient (in homogeneous media). The 3-D phase-velocity (or slowness) function fully determines the group-velocity (ray) vector and, therefore, all other kinematic signatures (e.g., reflection traveltime). This means that the parameters listed above are sufficient to describe all kinematic properties of P-waves in orthorhombic media. Normal-moveout velocity in orthorhombic media with a horizontal symmetry plane is discussed in Chapter 3 (section 3.5). Analytic 3-D description of P-wave moveout from dipping reflectors can be found in Grechka and Tsvankin (1999a).

50

C H A P T E R 1. E L E M E N T S O F BASIC 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

0

13..

, 30 60 90 Polar angle (deg)

I

o. 3.0 0

~ 30

60

90

!"

Polar angle (deg)

Figure 1.19: Influence of the p a r a m e t e r 7 (1) on the P - w a v e phase velocity. T h e solid curve corresponds to 7 (1) = - 0 . 1 , the d o t t e d to 7 (1) - 0.65. T h e other model p a r a m e t e r s are: Vpo- 3 k m / s , Vso- 1.5 k m / s , e (1) - 0 . 2 5 , e ( 2 ) _ 0.45, 6 (1) - - 0 . 1 , 5 (2) - 0.2, 5 (3) - - 0 . 1 5 , 7 (2) - 0.15.

1.3.

P L A N E WAVES IN O R T H O R H O M B I C MEDIA

51

Accuracy of the weak-anisotropy approximation After establishing that the weak-anisotropy approximation (1.107) contains all parameters responsible for P-wave velocity in orthorhombic media, we can study the dependence of the error of equation (1.107) on these anisotropic coefficients. As illustrated by Figure 1.20, equation (1.107) provides a good approximation (especially near the vertical) for the exact phase velocity in media with moderate velocity anisotropy, both within and outside the symmetry planes. The model in Figure 1.20, taken from from Schoenberg and Helbig (1997), corresponds to an effective orthorhombic medium formed by parallel vertical fractures embedded in a VTI background matrix. Although the accuracy of the weak-anisotropy approximation becomes lower with increasing anisotropic coefficients, the error remains relatively small for polar phase angles 0 up to about 70 ~ Higher errors of equation (1.107) near the horizontal plane are not surprising since the Thomsen-style notation is designed to provide a better approximation for near-vertical velocity variations. Still, even in the model with uncommonly strong velocity anisotropy (~(2) _ 0.6) shown in Figure 1.21, equation (1.107) deviates from the exact solution by no more than 10%. Also, note that for this example the weak-anisotropy approximation does not deteriorate outside the symmetry planes; in fact, it is in the symmetry planes where we observe the maximum error for some models. Equation (1.107) can be used to build analytic weak-anisotropy solutions for such signatures as group velocity, polarization angle, point-source radiation pattern etc. (e.g., Rommel and Tsvankin, 2000). However, since those solutions require multiple additional linearizations and involve the derivatives of phase velocity, their accuracy may be much lower than that of the phase-velocity expression itself. 1.3.5

Discussion

Analytic description of wave propagation in orthorhombic media can be significantly simplified by replacing the stiffness coefficients with two "isotropic" (vertical) velocities and a set of anisotropic parameters similar to the coefficients e, 5, and 7 suggested by Thomsen (1986) for vertical transverse isotropy. The parameter definitions introduced here are based on the analogous form of the Christoffel equation in the symmetry planes of orthorhombic and TI media. This analogy implies that all kinematic signatures of body waves, plane-wave polarizations and reflection coefficients in the symmetry planes of orthorhombic media are given by the same equations (with the appropriate substitutions of cij's) as those for vertical transverse isotropy. (The only exception is cuspoidal branches of shear-wave group-velocity surfaces formed in symmetry planes of orthorhombic media by out-of-plane slowness vectors.) Since Thomsen notation provides particularly concise expressions for seismic signatures in VTI media, the Thomsen-style parameters can be conveniently used to describe phase and group velocity, plane-wave polarization angle, etc., in the symmetry planes of orthorhombic media by adapting the VTI results. The equivalence between the symmetry planes of VTI and orthorhombic media,

52

C H A P T E R 1. E L E M E N T S O F BASIC T H E O R Y O F A N I S O T R O P I C WAVE P R O P A G A T I O N

03.0> 2.8-

2.8-

u)

~2.6(-.

c-

~- 2.4

a.

0

2.4 0

:,,3.2 .. o

2.8~2.6-

ca.

2.4 0


0


>,3.2 .

m

o

o3.0(D

9 2.8~2.6-

c-

~- 2.4

Figure 1.20: Comparison between the exact P-wave phase velocity (solid curve) and the weak-anisotropy approximation from equation (1.107) (dashed) for the "standard" orthorhombic model taken from Schoenberg and Helbig (1997). The velocities are calculated at the six azimuthal phase angles shown on top of the plots. The parameters are Vpo = 2.437 km/s, c(1) = 0.329, e(2) = 0.258, 5(1) = 0.083, 5(2) _ -0.078, 5(3) _ -0.106 (the other coefficients that do not influence P-wave velocity are Vso - 1.265 km/s, 7(1) = 0.182, 7 (2) = 0.0455).

1.3.

P L A N E W A V E S IN O R T H O R H O M B I C

>

MEDIA

53

4.2-

e3.6r t'-

g_ 3.0angle (deg)

o 4 . 2 . . . . . . . . . . !. . . . . . . . . :..;..'. .... >

0

-~ >

~3.6 3.0 0

i

i

30

60

3.6-

4 . 2 . . . . . . . . . . ". . . . . . . . . ". . . . . . . . . i i ." :

:s

90 Polar angle (deg)

9

"(D ~.t~ . . . . . . . . . >

~

~

9

" . . . . . . . . : .....s ~,.! ~," ~ 3.4-

0


0


Figure 1.21" C o m p a r i s o n between the exact P-wave phase velocity (solid curve) and the weak-anisotropy a p p r o x i m a t i o n from equation (1.107) (dashed). The model parameters are Vpo - 3 k m / s , e(1) _ 0.2, e(2) _ 0.6, 5(1) _ 0.15, 5 (2) - - 0 . 1 5 , 5 (3) = - 0 . 2 .

54


however, does not apply to body-wave amplitudes in general. In particular, pointsource radiation patterns in the symmetry planes do depend on azimuthal velocity variations and, therefore, should be studied using analytic and numerical methods developed for azimuthally anisotropic media (see Chapter 2). The influence of outof-plane velocity variations on body-wave amplitudes also means that near-field polarizations in the symmetry planes of orthorhombic media may differ from those in VTI media. Indeed, as discussed in Chapter 2, polarization in the "nongeometrical" region close to the source is generally nonlinear and depends on the relative amplitudes of at least the first two terms of the ray-series expansion. In contrast, farfield polarizations are adequately described by the geometrical-seismics (plane-wave) approximation, which can be obtained by analogy with vertical transverse isotropy. The dimensionless anisotropic coefficients conveniently characterize the magnitude of anisotropy and represent a natural tool for developing linearized weak-anisotropy approximations outside the symmetry planes of orthorhombic media. Such an approximation for P-wave phase velocity has the same form as that for vertical transverse isotropy, but with azimuthally-dependent coefficients e and ~. Hence, in the limit of weak anisotropy, all 2-D kinematic results for P-waves in VTI media can be applied to any vertical plane of orthorhombic media. For instance, the weak-anisotropy approximation for P-wave NMO velocity in the dip plane of the reflector has the same form in VTI and orthorhombic media (section 3.2). This conclusion also has important implications for 2-D processing of P-wave reflection data (see the introduction to Chapter 7). The approximation for P-wave phase velocity provides a basis for developing analytic solutions for other signatures. For example, Rommel and Tsvankin (2000) derived the weak-anisotropy approximations for the P-wave group-velocity and polarization vectors and showed that they stay close to each other even outside the symmetry planes (this result was obtained for VTI media in section 1.2). As indicated by the weak-anisotropy approximation, the exact P-wave phase velocity is governed by only the orientation of the symmetry planes and six medium parameters (Vp0, e(1), 5(1) e(2)/~(2) and /~(3)), rather than by nine stiffness coefficients cij in the conventional notation. Clearly, the Thomsen-style parameters capture the combinations of the stiffnesses responsible for P-wave kinematic signatures in orthorhombic media. Hence, P-wave inversion algorithms for orthorhombic media should target these anisotropic parameters instead of the stiffness matrix. It is important to emphasize that the Thomsen-style notation is convenient to use in orthorhombic media with any strength of the anisotropy. The most attractive features of this notation, such as the reduction in the number of parameters responsible for P-wave velocity and the simple expressions for seismic signatures in the symmetry planes, remain valid even in strongly anisotropic models. Also, as shown in Chapter 3 (section 3.5), the exact P-wave normal-moveout velocity outside the symmetry planes of a horizontal orthorhombic layer is a simple function of the coefficients 5(1) and 5(2). Hence, concise weak-anisotropy approximations can be regarded as just another (not the primary) advantage of this notation, as is the case with Thomsen notation in VTI

1.3.

P L A N E WAVES IN O R T H O R H O M B I C M E D I A

55

media (see section 6.3). The Thomsen-style parameters make up a unified framework for an analytic description of seismic signatures in orthorhombic, VTI and HTI models. Both vertical and horizontal transverse isotropy can be characterized by specific subsets of the dimensionless anisotropic parameters defined here for the more complicated orthorhombic model. This uniformity of notation simplifies comparative analysis of seismic signatures and transition between different anisotropic models in seismic inversion. The parameters of orthorhombic media are usually defined with respect to the coordinate system associated with the symmetry planes (as it was done here). In seismic experiments the orientation of the symmetry planes may be unknown and therefore should generally be determined from the data. The number of required parameters in this case increases to 12. In many typical fractured reservoirs, however, one of the symmetry planes is horizontal and the only additional (10th) parameter is the azimuth of one of the vertical symmetry planes.

56


Appendices for Chapter 1 1A

Phase velocity in arbitrary anisotropic media

In this appendix, we present explicit expressions for phase velocity V in arbitrary anisotropic homogeneous media. Equation (1.11) of the main text, which follows directly from the Christoffel equation, can be transformed into x 3 + a x 2 + bx + c = 0,

(1.110)

where x = p V 2 (p is the density) and a -

-(Gll

+ G22 + G 3 3 ) ,

(1.111)

b - G l l G 2 2 + G l l G 3 3 + G22G33 - G~2 - G~3 - G~3, C-

Gila23

2 - G l l G22G33 -- 2G12G13G23 9 -k- G22G213 -+- G 33G12

(1.112) (1.113)

Through a change of variables (x = y - a / 3 ) , it is possible to eliminate the quadratic term in the cubic equation (1.110) and reduce it to the following form (e.g., Korn and Korn, 1968): y3 + dy + q = 0, (1.114) with the coefficients

a2

d= q-2

3

+b,

(1.115)

ab + c . ( 3 ) 3 - -~-

(1.116)

Using the fact that the matrix G~k is symmetric, it can be shown that the coefficient d is negative (e.g., Schoenberg and Helbig, 1997). In this case, the roots of equation (1.114) are real if the following combination of d and q is non-positive:

Q-

+

_ 0, it is necessary to pick the vertical slownesses of upgoing waves which have a positive vertical component of the group-velocity vector; for z < 0, only downgoing waves need to be taken into account. Note that it is the group-velocity vector that defines the direction of wave propagation in anisotropic media. For models without a horizontal symmetry plane, it may happen, for example, that the group-velocity vector points up (in the positive z-direction), while the corresponding phase-velocity vector points down (see examples in section 3.3). The plane-wave displacement vector U (~) depends on the the elastic coefficients and the direction of the force, and largely controls the source directivity function discussed in detail below. For layered media the term U (~)e-,~p3 . (v)z in equation (2.2) should be replaced by the so-called matrix propagator expressed through the product of the layer propagator matrices (Fryer and Frazer, 1987). S(w, x) is the Fourier transform of the displacement field, which should be computed for a set of frequencies and subsequently inverse Fourier-transformed via equation (2.1) to obtain the synthetic seismogram. Clearly, evaluating the double integral (2.2) for each frequency is a computer-intensive operation, particularly because the vertical slownesses and polarization vectors of the elementary plane waves are the eigenvalues and eigenvectors of an asymmetric 6 x 6 matrix (derived from the Christoffel equation) and must be found numerically. Only if the medium is azimuthally isotropic, it is possible to carry

64

C H A P T E R 2. I N F L U E N C E OF A N I S O T R O P Y ON P O I N T - S O U R C E RADIATION AND AVO ANALYSIS

out integration over the azimuthal angle r analytically and reduce (2.2) to a single integral over the horizontal slowness p0. To avoid the numerical integration over r the azimuthal dependence of the displacement and vertical slowness of elementary plane waves can be approximated by several terms of the Fourier series expansion (Tsvankin and Chesnokov, 1990a)" Ui(p0, r - ~ {ai,n(p0)cos [n(r - c~)] +

bi,n(po)sin [n(r - c~)]}

(2.3)

rt---0

P3(Po, r - c(po) + d(po)cos(r - c~) + e(po)sin(r - a).

(2.4)

The representations (2.3) and (2.4) have to be developed for each mode (u - 1, 2, 3) separately. For azimuthally isotropic media the vertical slowness P3 is independent of r and the expansion (2.3) for U~ (in the case of a point force) is limited by t~ < 2. If the medium is azimuthally anisotropic, equations (2.3) and (2.4) should be considered as local expansions, with the coefficients a, b, c, d, e determined separately for each receiver azimuth a. As shown by Tsvankin and Chesnokov (1990a), expanding the azimuthal dependence of Ui and p3 in Fourier series reduces the integral over the azimuthal angle to the Bessel functions Jk (0 _< k < t~). This method is used in this section to generate the numerical examples for azimuthally anisotropic models. The technique based on the Fourier series expansions in the azimuthal angle can be also applied to the description of any individual wave in layered anisotropic media. In this case, the plane-wave displacement vector U includes the product of the appropriate reflection/transmission coefficients and p3z should be replaced by the sum of the vertical phase shifts in the individual layers (Tsvankin and Chesnokov, 1990a). Approximations similar to (2.3) and (2.4), however, are difficult to use in the computation of the full wavefield because the matrix propagator is frequency-dependent and may vary rapidly with azimuth. S t a t i o n a r y - p h a s e solution for t h e far-field While the integral representation of the Green's function is sufficient for numerical analysis of the wavefield, it does not reveal the relationship between the anisotropic velocity function and the properties (polarization, amplitude) of point-source radiation. A convenient tool for simplifying Weyl-type integrals is the stationary-phase method described, for example, by Fuchs (1971) and Bleistein (1984). The zero-order approximation of this method, which operates with just the plane-wave phase function of the integrand, provides the geometrical-seismics solution valid in the far-field (i.e., for source-receiver distances large compared with the predominant wavelength). Although such a solution can be obtained for arbitrary anisotropic media (G a jewski, 1993), for the sake of simplicity the treatment here is restricted to wave propagation in vertical symmetry planes of homogeneous models. In this case the stationaryphase point, which corresponds to the plane wave that makes the most prominent contribution to the wavefield, is confined to the vertical incidence plane that contains the source and receiver. [The only exception is possible in-plane triplications on shear

2.1. POINT-SOURCE RADIATION IN ANISOTROPIC MEDIA

65

wavefronts caused by out-of-plane phase-velocity variations; see Grechka and Obolentseva (1993).] Therefore, azimuthal velocity variations near the symmetry plane influence just dynamic properties of the geometrical-seismics approximation. As shown in Tsvankin and Chesnokov (1990a), the zero-order stationary-phase approximation of the integral (2.2) for each mode u is given by 1

Si(W, r, Z) -- Ui

O(1/v) ) 09 sinO e_i~(rsing+zcosg) V/'(r- V sin-0 + d z) o2~ O02

sinO

27TV

02002~= (r sin 0 + z cos 0)

L

1 V

cos0

+

02(l/V)+ 002

V

zsi")']

sin 0 + z cos

(2.5)

(2.6)

where 9 - - p o r - p3z is the frequency-independent part of the phase function of the integrand [equation (2.2)], V is the phase velocity in the incidence plane as a function of the phase angle 9 measured from the vertical (z) axis (V also depends on the azimuthal phase angle r and d is the coefficient in expansion (2.4) of the vertical slowness in the angle r as in equation (2.2), z is positive. The plane-wave displacement vector U is given by equation (2.27). The frequency-domain displacement Si should be evaluated at the phase angle 9 corresponding to a given ray (group velocity) angle r determined by the receiver position (tan r = r/z). Physically, the angle 9 determines the direction of the wavefront normal for the group angle r in other words, the wavefront at the receiver location can be approximated by a plane segment orthogonal to the vector n -- (sin 9, cos 9). The stationary-phase approximation (2.5) relates the displacement field to the angle-dependent phase-velocity function V(9) in the incidence plane and, therefore, provides an important insight into the influence of anisotropy on point-source radiation. In equation (2.5) anisotropy manifests itself both explicitly, through the derivatives of the velocity function, and indirectly, through the difference between the phase and group velocities and distortions of the plane-wave displacement vector U. In particular: 1. The displacement S(w, x) is evaluated in the phase (plane-wave) direction rather than in the group direction that corresponds to the source-receiver ray. Hence, in contrast to isotropic media, the displacement U has to be computed for a plane wave that does not propagate in the source-receiver direction. This causes distortions in both polarization and amplitude of body waves. 2. The polarization in the stationary-phase approximation is linear (i.e., the direction of the displacement vector is independent of time) and is determined by the plane-wave displacement U. Since the orientation of U is fully controlled by the stiffness tensor (via the Christoffel equation), the direction of the farfield displacement vector of each mode depends just on the medium properties, 1For brevity, henceforth the superscript u is omitted

66

C H A P T E R 2.

I N F L U E N C E O F 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 AND AVO ANALYSIS

rather than on the characteristics of the source (except for S-wave singularities). In contrast, shear-wave polarization for isotropic models is governed by the projection of the force onto the plane orthogonal to the source-receiver line (Aki and Richards, 1980). Even if plane waves were polarized as in isotropic media, anisotropy would lead to the deviation of the polarization vector from its "isotropic" orientation because U corresponds to the phase rather than the group direction. However, as discussed in Chapter 1, anisotropy also rotates the plane-wave polarization vector away from the phase direction (for P-waves) or from the plane orthogonal to this direction (for S-waves). Those two factors can either reinforce or compensate each other, depending on the wave type and the anisotropic parameters. 9

It follows from equation (2.5) that the polarization vectors of the three waves (P, $1, $2) recorded at a fixed receiver location are not necessarily mutually orthogonal. As shown in Chapter 1, the plane-wave polarization vectors of the three waves propagating in the same phase (slowness) direction are perpendicular to one another because they coincide with the eigenvectors of the symmetric Christoffel matrix. This property, however, does not hold for non-planar wavefronts because the three body waves recorded at a fixed point in space have different wavefront normals and, therefore, correspond to different phase directions [i.e., the vector U(~)(p (~)) (v = 1, 2, 3) is evaluated for different directions n (~) = p ( ~ ) V ] .

.

Phase velocity V(8) and the plane-wave amplitude IUI deviate from the isotropic values. For example, polarization anomalies in anisotropic media change the angle between U and the force direction F, thus causing distortions of IUI (see below). It should be mentioned that the vector U depends on all six vertical slownesses corresponding to three downgoing and three upgoing waves. Therefore, even if the phase velocity of a certain wave is almost constant, its radiation pattern can still be distorted due to the influence of anisotropy on the other waves.

6 9 The derivatives o(1/v) and ~ 082 show a direct relationship between velocity 08 and amplitude variations with angle. In particular, due to the influence of the second derivative of 1/V in equation (2.6), the amplitude often increases at velocity maxima and decreases at velocity minima. These "amplitude focusing" phenomena, which are often pronounced even for weak anisotropy, are addressed in more detail below. 7. Azimuthal velocity variations may seriously distort amplitudes even in vertical symmetry planes. Whereas the traveltime is determined by the argument of the exponential in equation (2.5) [(r sin 8 + z cosS)/V] and is fully controlled by the in-plane phase-velocity function, the amplitude also depends on the factor d in

2.1.

P O I N T - S O U R C E RADIATION IN A N I S O T R O P I C MEDIA

67

the denominator. According to equation (2.4), this factor is responsible for the azimuthal variation of the vertical slowness, which is caused solely by azimuthal anisotropy. Thus, equation (2.5) reveals 3-D amplitude focusing/defocusing phenomena associated with both in-plane and azimuthal velocity variations. 2.1.2

Numerical

analysis

of point-source

radiation

Here the technique based on evaluating Fourier-Bessel integrals (Tsvankin and Chesnokov, 1990a) is used to generate synthetic seismograms and particle motion from a point force in a homogeneous anisotropic medium. To study the influence of both polar (in-plane) and azimuthal anisotropy, it is convenient to use a model with orthorhombic symmetry described in Chapter 1 (section 1.3). The main goal of these numerical tests is to verify the conclusions drawn from the stationary-phase approximation and estimate the magnitude of polarization and amplitude distortions caused by anisotropy. A detailed analysis of body-wave amplitudes in transversely isotropic media is given in section 2.2. Polarization and amplitude anomalies Figure 2.1 shows the far-field displacement from a vertical force in the Ix1, x3] symmetry plane of orthorhombic olivine (the elastic constants are taken from Kumazawa and Anderson, 1969). The wavefield contains only P and SV arrivals because SH-waves in symmetry planes are not excited by an in-plane force. For P-waves, the traveltime decreases by almost 15% towards the horizontal plane due to an increase in the phase velocity (Figure 2.2) associated with a positive coefficient e(2). The P-wave amplitude, as in isotropic media, reaches its maximum in the force direction. The imprint of anisotropy is much more pronounced in the SV-wave propagation. First, the S V traveltime in Figure 2.1 is noticeably smaller at oblique incidence angles r with the vertical, indicating the presence of a velocity maximum near an angle of 45 ~ (Figure 2.2). According to the analogy with VTI media (see section 1.3), SVwave velocity variations in the [Xl, x3]-plane are mostly governed by the parameter a (2) defined as

(vP~

(2.7)

\ Vso] For the model in Figure 2.1, a (2) = 0.56 is positive and relatively large, which explains the shape of the SV-wave velocity curve in Figure 2.2. Second, there is a sharp increase in the shear-wave amplitude away from the vertical towards the maximum at the group angle r ~ 42 ~ (see also Figure 2.3). Third, the SV-wave particle motion becomes nonlinear near the minimum of the radiation pattern (r --+ 0~ as well as in the area of largest amplitudes (r = 40~ The growing ellipticity of the SV-wave polarization near the amplitude maximum is caused by higher-order terms of the ray series expansion, which are enhanced by rapid amplitude variations along the wavefront (Kiselev and Tsvankin, 1989). Nonlinearity of polarization (particle motion) indicates that the stationary-phase approximation

68

C H A P T E R 2. INFLUENCE OF ANISOTROPY ON POINT-SOURCE RADIATION AND AVO ANALYSIS

~ I

1_

I

' -3]'I

_

!

i

I

Figure 2.1: Displacement field from a vertical force in the s y m m e t r y plane [xl,x3] of orthorhombic olivine. The receiver moves in the halfspace z < 0 along a circle with the center at the source; the radius R = v/z 2 + r 2 = 10 km. r = t a n - l ( r / I z l ) is the angle of the source-receiver line with the vertical (i.e., the group angle). Each trace is normalized by its m a x i m u m amplitude. The "radial" component is the horizontal displacement in the incidence plane (i.e., ul). Polarization diagrams of shear waves (at right) are computed in the incidence and horizontal planes; the horizontal axis on the diagrams corresponds to the radial component ul. The central frequency of the source pulse is f0 - 10 Hz, the time increment used in the Fourier transform is 5 ms, and the time axis is in units At -- 50 ms. The source-receiver distance normalized by the average wavelength of the P-wave ( ) ~ p ) and SV-wave ( A s v ) i n the incidence plane is R/)~p -- 10.8 and R/)~sy - 20.2. The medium parameters, as defined in section 1.3, a r e Vpo : 8.69 k m / s , Vso = 4.95 k m / s , c (1) = - 0 . 1 0 , e (2) = 0.15, 5(1) = -0.14, (~(2) = - 0 . 0 3 , 5 (3) = - 0 . 2 6 , 7 (1) = 0.09, 7 (2) = - 0 . 0 1 (cl~ = 32.4, c12 = 5.9, c13 = 7.9, c22 = 19.8, c23 - 7.8, c33 - 24.9, c44 - 6 . 6 7 , has the dimension of k m / s ) .

cs5 - 8.1, c66 - 7.93, p -

0.33; q c i j / P

2.1.


69

derived above (and geometrical seismics in general) becomes inadequate in describing the wavefield. A more detailed discussion of deviations from geometrical seismics can be found later in this section. In areas of linear polarization, the P-wave is polarized close to the source-receiver direction, while the SV-wave displacement vector for a wide range of angles is far from being perpendicular to that of the P-wave (Figure 2.2). As follows from the stationary-phase approximation (2.5), polarization anomalies are caused by both the difference between phase and group velocities and the influence of anisotropy on plane-wave polarization. The results of Chapter 1 show that for P-waves in TI media these two factors tend to compensate each other, and the displacement vector does not deviate much from the group-velocity (source-receiver) vector. This conclusion and the corresponding analytic relationships remain entirely valid in the symmetry planes of orthorhombic media (see section 1.3). To explain the large magnitude of the SV-wave polarization anomalies, note that the difference between the phase and group angles is determined by the derivative of the phase-velocity function with respect to the phase angle 0 [equation (1.71)]. If this derivative is similar for both P- and SV-waves, then the angles 0 for the two modes corresponding to a given group (source-receiver) angle r are also close. In this case, since the polarization vectors of the plane P- and SV-waves propagating at a given angle 0 are perpendicular to each other, the polarization vectors of the two waves at a fixed receiver location (i.e., for a fixed r will be perpendicular as well. This means that the SV-wave polarization vector will be oriented almost "isotropically" because it is orthogonal to the P-wave polarization vector which, as established above, is close to the source-receiver direction. Therefore, SV-wave polarization anomalies exist if the derivative OV/O0 in a certain range of angles is sufficiently different (e.g., has opposite signs) for the P- and SV-waves. Of course, those qualitative considerations can by applied only to areas of linear polarization where the stationary-phase approximation is valid. For the Ix1, x3]-plane of the orthorhombic model considered in Figures 2.1 and 2.2, the polarization anomalies of the SV-wave are particularly pronounced for group angles 45 ~ < r < 75 ~ where the derivatives OV/O0 for the P- and SV-waves are relatively large and have opposite signs. In this area the P- and the SV-wave displacement vectors are far from being perpendicular, coming as close to each other as of 68.5 ~ Somewhat milder, but still noticeable deviations from the isotropic SV-wave polarization of up to 15~ are observed between r = 15~ and 40 ~ The distortions of the radiation pattern are also more pronounced for SV-waves than for P-waves (Figure 2.3). The far-field radiation patterns in isotropic media, shown by dashed lines in Figure 2.3, are circular with the center either in the direction of the force (for P-waves) or in the perpendicular direction (for SV-waves). For example, the S-wave amplitude in the absence of anisotropy is governed simply by the spherical divergence factor and by the sine of the angle between the force and the source-receiver direction. Not only do both SV-wave radiation patterns in the two symmetry planes deviate from the isotropic circle, they also differ substantially from one another due to the influence of azimuthal anisotropy. The SV-wave velocity

70

C H A P T E R 2. I N F L U E N C E O F 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 ANALYSIS

: ~" 0

:

P

0

9

,

.

#

tD o~ tD

e

..

,

L

-20 0

5.4 ~. 5.2 co

E

5

v

4.6

o,J,

9

4.40

Figure 2.2: Polarization anomalies and phase velocities of body waves in the symmetry plane [xl,x3] of orthorhombic olivine (see Figure 2.1). The polarization angles are found from the maximum amplitudes on the vertical (Uz) and radial (ur) displacement components as 7P, sv - t a n - l ( l U r / U z l ) 9 The polarization anomaly is the deviation of the polarization vector from its "isotropic" orientation" / k ~ p -- ~ r ) - ~ p , A ~ S V = r + 7sv90 ~ Areas of nonlinear particle motion are marked on the top plot by the


a=0~ .

71

,•

10

a=90 ~

_s

Figure 2.3: Radiation patterns from a vertical force in the symmetry planes [Xl,X3] (c~ = 0 ~ and [x2, x3] (c~ = 90 ~ of orthorhombic olivine. The patterns are obtained by picking the total magnitude (trough plus peak) of the displacement vector on synthetic seismograms for each group (source-receiver) direction. The dashed lines show the radiation patterns in a reference isotropic medium.

curves in the planes [ X l , X 3 ] (OL = 0 ~ (Figure 2.2) and [x2, x3] (c~ -- 90 ~ are similar, but the parameter a (2) - 0.56 [equation (2.7)] is much larger than its counterpart a (1) = 0.13 in the Ix2, x3]-plane. As a result, the magnitude of the SV-wave velocity variations for c~ = 0 ~ is higher, which leads to a larger amplitude anomaly. The most striking feature of the radiation patterns in Figure 2.3 is a sharp resonance-type maximum of the SV-wave amplitude in the [Xl,X3]-plane (c~ - 0 ~ associated with the velocity maximum near ~ ~ 42 ~ (Figure 2.2). At smaller group angles the amplitude decreases so rapidly that for r - 35 ~ it is already four times less than the maximum value. The main physical reason for this amplitude anomaly is the focusing of energy near the velocity maximum. Due to the rotation of the group-velocity vectors of elementary plane waves towards the velocity maximum, the segment of the wavefront adjacent to the maximum contains a higher density of rays (see Figure 1.13) and accumulates extra energy. Conversely, velocity minima usually lead to a lower density of rays and defocusing of energy. In the asymptotic expression (2.5) the focusing of energy in the incidence plane is associated with an increase in the derivative [02(1/V)/O~) 2] evaluated at the phase

72


angle [see equation (2.6)]. The sharp character of the SV-wave amplitude maximum in Figure 2.3 is explained by the rapid deviation of the phase angle from the group angle when the receiver leaves the area adjacent to the velocity (and amplitude) maximum. As a result, even for receiver positions close to the velocity maximum, the corresponding value of [02(1/V)/092] may be relatively small or even negative. The velocity maximum of the SV-wave in the Ix1, x3]-plane has a 3-D nature, with the velocity decreasing away from the maximum in all spatial directions. This implies that the concentration of SV-rays and focusing of energy near the velocity maximum is a 3-D phenomenon not limited to the incidence plane. Hence, in agreement with the stationary-phase approximation, azimuthal velocity variations may substantially change body-wave amplitudes even in vertical symmetry planes. This 3-D character of symmetry-plane amplitude signatures makes the equivalence between the symmetry planes of orthorhombic media and transverse isotropy (see section 1.3) invalid for body-wave amplitudes. It should be emphasized that the sharp polarization and amplitude anomalies of the SV-wave are associated with phase-velocity variations not exceeding 12% (including azimuthal anisotropy). In spite of the much more pronounced P-wave velocity anisotropy with a magnitude of almost 30%, P-wave amplitude distortions are not as severe as those of the SV-wave, although they are noticeable in some parts of the radiation pattern (Figure 2.3). The main reason for the relatively mild P-wave amplitude anomalies is the absence of pronounced velocity extrema at near-vertical and oblique propagation angles where P-wave amplitudes are relatively large. The distortions of the P-wave radiation pattern would be more significant if the force were horizontal, and the high-amplitude area were in the vicinity of the velocity maximum (see Figure 2.2). A more detailed analysis of the influence of anisotropy on P-wave amplitudes (for TI media) is given in section 2.2. Shear-wave

splitting

Figure 2.4 shows a typical picture of S-wave splitting in the symmetry plane [x2, x3] (a = 90 ~ of orthorhombic olivine. The fast shear wave $1 (SH) is separated on the transverse component, while the slow wave S'2 (SV) is polarized in the incidence plane. In the other vertical symmetry plane (a = 0~ the relationship between the velocities of the SV- and SH-waves is reversed: the slow mode has a transverse polarization (see section 1.3). For vertical propagation, the time delay between the S-waves is governed by the splitting coefficient 7 (S) [equation (1.100)] that is close to 9%. The degree of overlapping of the $1- and the S2-waves increases at intermediate group angles and, if neither of the horizontal coordinate axes lied in the incidence plane, it would be difficult to separate the shear waves on the synthetic seismograms. The orthogonality of the shear-wave polarizations, however, can be identified on polarization diagrams even near r = 50 ~ where only one lobe of the signal is free from interference (Figure 2.4). All the previous results in this section have been obtained for the vertical symmetry planes of orthorhombic media. If the incidence plane does not coincide with


7oo _ ~__ 6oo_~ .... ~oo_~:~ ?

73

____~ ?, ~~__

-~

7--

7

F

./"

x~

X1

Figure 2.4: Displacement field in the symmetry plane Ix2, x3] of orthorhombic olivine from a force with an "SH" component. The force lies in the [xt,x3]-plane at an angle of 45 ~ with the vertical; the source-receiver distance R = 5 km. The transverse component is the displacement in the direction orthogonal to the incidence plane. As before, r is the angle of the source-receiver line with the vertical. Each trace is normalized by its maximum amplitude.

74

C H A P T E R 2. IN FL UENCE OF A N I S O T R O P Y ON P O I N T - S O U R C E RADIATION AND AVO ANALYSIS

Figure 2.5: Displacement due to a vertical force within a vertical plane that bisects the symmetry planes Ix1, x3] and Ix2, x3] of orthorhombic olivine. The source-receiver distance R - 10 km. a plane of symmetry, the phase-velocity (wavefront) vector at the receiver location is tilted away from the incidence plane in the azimuthal direction, which leads to dependence of the traveltime on azimuthal anisotropy. When the incidence plane bisects the angle between the vertical symmetry planes of orthorhombic olivine, deviations of the traveltimes from the values predicted by the in-plane velocities are more significant for the P-wave (Figure 2.5). Note that the P-wave displacement in Figure 2.5 has a negligibly small transverse component and is practically confined to the incidence plane (see Rommel and Tsvankin, 2000). Moreover, analysis of the P-wave polarization shows that it remains close to the source-receiver direction (as was the case in the symmetry planes) because the difference between the group- and phase-velocity vectors is compensated by the deviation of the plane-wave polarization vector from the phase direction. Outside symmetry planes shear waves do not strictly belong to either SH or SV type and may have both in-plane and out-of-plane (transverse) components (Figures 2.5 and 2.6). Generation of out-of-plane particle motion by an in-plane source is one of the most distinctive features of wave propagation in azimuthally anisotropic media. Figure 2.6 shows that only in the horizontal plane (r - 90 ~ is the slow shear wave (5'2) of the SV type (the Sl-wave for r = 90 ~ is close to being an SH-wave).

2.1.


75

Figure 2.6: Shear-wave amplitudes picked from the seismograms in Figure 2.5. The dashed line is the SV component of the fast wave $1; dashed with circles - the SH component of the Sl-wave; solid - the SV component of the S2-wave; solid with circles- the SH component of the S2-wave. The SV component is the projection of the displacement vector onto the incidence plane (~/u 2 + U2z);. SH is the transverse displacement component. For r < 45 ~ however, both shear waves have almost equal in-plane and transverse components. The displacement vectors of the three body waves in the plane c~ = 45 ~ deviate by more than 10 ~ from being perpendicular to each other, but the polarization anomalies are not as severe as those in the Ix1, x3]-plane. Whereas the far-field body-wave polarization is controlled by the stiffness tensor, the absolute and relative amplitudes of all three modes also depend on the type and characteristics of the source (Figure 2.6). In the vicinity of the horizontal plane, a vertical force emphasizes the slow shear wave $2 whose displacement vector is close to the vertical. For r < 45 ~ the source amplitude factors for the two shear waves are close to each other, and their relative values are determined mostly by a velocity minimum of the Sl-wave that leads to energy defocusing. It is interesting that the radiation of the fast shear wave in the plane c~ = 45 ~ is concentrated mostly within the interval 40 ~ < r < 80 ~ (Figure 2.6). For a smaller source-receiver distance (Figure 2.7), the waves $1 and 5'2 interfere and cannot be readily separated on the seismograms. Still, shear-wave splitting can be recognized on polarization diagrams in the horizontal plane, even in the area of elliptical particle motion. Nonetheless, it should be mentioned that even strictly perpendicular displacement vectors cannot always be separated on polarization diagrams in just two orthogonal planes.

76


(~,~~

X2

Figure 2.7: Same as in Figure 2.5 but for a smaller source-receiver distance (R - 5

km).

2.1.


77

Nongeometrical phenomena The stationary-phase solution (2.5) is equivalent to the geometrical-seismics approximation described by the leading ("zero-order") term of the ray-series expansion. Since the vast majority of ray-tracing algorithms used in seismic modeling are based on geometrical seismics (e.g., Gajewski and P~en(:[k, 1987), it is important to investigate the significance of so-called "nongeometrical" wave-propagation phenomena which require a more elaborate treatment. In a homogeneous medium the most significant deviations from geometrical seismics should be expected near minima of radiation patterns, where the leading term of the ray series is relatively small. For radiation from a point force in isotropic media, two major nongeometrical areas are near the direction of the force (for S-waves) and near the plane orthogonal to the force (for P-waves). The results of Kiselev and Tsvankin (1989) indicate that the ray method may become sufficiently accurate near amplitude minima if one additional (first-order) term of the ray series is taken into account. In general, the first-order ray-series term depends on the spatial derivatives of the leading term, which makes geometrical seismics also inadequate in areas of strong amplitude variation even away from amplitude minima. Deviations from geometrical seismics are more pronounced in the near-field because additional terms of the ray series decay with distance more rapidly than does the leading term. If the first-order term has a non-negligible magnitude, differences in polarization and waveform between the two leading ray-series terms make body-wave particle motion nonlinear (quasi-elliptical). Ellipticity of particle motion is the simplest and most direct indication of the influence of higher-order terms of the ray-series expansion. The presence of anisotropy may either reinforce or weaken nongeometrical phenomena, depending on the character of the velocity function. For instance, if the phase-velocity vector deviates from the group (source-receiver) direction towards the amplitude minimum, the magnitude of the geometrical-seismics term becomes smaller and the first-order term contributes to the wavefield over a wider range of propagation angles. Whereas for isotropic models the influence of the first-order term usually is more pronounced for S-waves than for P-waves (Kiselev, 1983), this is not necessarily the case in anisotropic media. For example, polarization analysis of the seismograms in the [x2, x3] symmetry plane of orthorhombic olivine (Figure 2.4) shows that the nonlinearity of the P-wave polarization near the horizontal plane (r - 90 ~ is more pronounced than that of the SV-wave near the vertical direction (r = 0~ Azimuthal anisotropy introduces additional complications into nongeometrical phenomena, both within and outside the symmetry planes. Nongeometrical distortions of the SV-wave excited by a tilted in-plane force in the [x2, x3]-plane of orthorhombic olivine are shown in Figure 2.8. Near the direction of the force (r = 60 ~ the SV-wave is strongly influenced by the first-order term of the ray series, which is polarized almost like a P-wave. Since the time dependence of the first-order term is described by the integral of the leading term, the SV-wave particle motion near the force direction becomes increasingly nonlinear.

78

C H A P T E R 2. INFLUENCE OF A N I S O T R O P Y ON P O I N T - S O U R C E RADIATION AND AVO ANALYSIS

3oo~

-,x~ k-___

/

/

/

/

57 m

m

b

/

-

Figure 2.8: Displacement in the symmetry plane [x2, x3] of orthorhombic olivine from an in-plane force tilted by 60 ~ away from the vertical; R - 3 km.

2.1.


79

The uncommon character of the waveform and particle motion of the SV-wave for 50 ~ < r < 70 ~ reflects the contribution of 3-D velocity variations to the leading and additional terms of the ray series. Evidently, equation (2.5) cannot adequately describe the properties of the SV-wave for receiver locations in the vicinity of the force direction. Note that the example in Figure 2.8 corresponds to the "intermediate-field" because the source-receiver distance exceeds five average SV-wavelengths. Anisotropic velocity and amplitude variations may cause deviations from geometrical seismics that do not exist in isotropic media. For instance, additional nongeometrical phenomena are associated with cusps (triplications) on shear wavefronts discussed in Chapter 1 (Martynov and Mikhailenko, 1984); the wavefront shapes become particularly complicated in the vicinity of point S-wave singularities (e.g., Crampin, 1991). Another example is the pronounced ellipticity of the SV-wave polarization near the velocity maximum in the [Xl,X3]-plane (Figure 2.1). This is a purely anisotropic phenomenon caused by rapid amplitude variations along the corresponding segment of the wavefront. It should be emphasized that the wavefield in Figure 2.1 is computed in the far-field where the higher-order terms of the ray-series expansion are significantly dampened by their large geometrical-spreading factors. The region of nonlinear polarization of the SV-wave near the amplitude maximum is much broader at smaller source-receiver distances. As mentioned above, S-wave propagation has a purely nongeometrical character in the direction parallel to the force. If the medium is isotropic, the shear-wave geometrical-seismics term vanishes in this direction, but the wavefield still contains an arrival traveling with the shear-wave velocity and polarized like a P-wave; this anomalous event is associated with the near-field term of the Green's function (Aki and Richards, 1980; Tsvankin, 1995c). For a force aligned with one of the coordinate axes (i.e., the intersections of the symmetry planes) of orthorhombic media, there are two waves with longitudinal polarization propagating in the force direction with the velocities of the split S-waves (Tsvankin and Chesnokov, 1990a). It is interesting that a similar phenomenon, which may be called the "triple splitting" of P-waves, is caused by nonlinear elasticity (Tsvankin and Chesnokov, 1987). In this case, however, shear waves with anomalous polarization are no more than weak additional terms to the regular S-waves. Also, deviations from the conventional shear-wave polarization in nonlinear-elastic media cannot be regarded as nongeometrical because they arise even in plane-wave propagation. It is important to compare the influence of nongeometrical phenomena and shearwave splitting on the polarization diagrams. Synthetic examples above demonstrate the distinctive character of shear-wave splitting even for a relatively small time difference between the two S-waves (Figure 2.7). "Nongeometrical" polarization diagrams with quasi-elliptical particle motion of a single S-wave (Figure 2.8) do not exhibit a 90~ in the polarization direction and should not lead to ambiguous interpretation. On the other hand, if one or both split S-waves have nonlinear polarization, which is rather common in seismic surveys, detection of shear-wave splitting becomes much more difficult and may require elaborate numerical modeling.

80

C H A P T E R 2. INFLUENCE OF A N I S O T R O P Y ON POINT-SOURCE RADIATION AND AVO ANALYSIS

2.1.3

Discussion

Properties of point-source radiation in anisotropic media are conveniently studied using a Weyl-type decomposition of the wavefield into elementary plane waves. The solution in the frequency domain is then obtained in the form of a Fourier-Bessel integral over the horizontal slowness components that can be evaluated by reflectivitytype algorithms. A far-field approximation derived by the method of stationary phase helps to give a qualitative description of the polarization and amplitude anomalies associated with anisotropy. The relationship between point-source and plane-wave theory in anisotropic media is complicated by the difference between the ray (group-velocity) and phase directions. Depending on the character of angular velocity variations, the phase (slowness) vector of the plane wave that locally approximates the wavefront may substantially diverge from the source-receiver direction. In general, P, $1, and $2 modes generated by the same source and recorded at the same location have different phase directions (wavefront normals). This leads to non-orthogonality of P- and S-wave polarizations and contributes to distortions of radiation patterns. Analytic and numerical results for an azimuthally anisotropic medium of orthorhombic symmetry shows that shear waves are especially sensitive to the presence of anisotropy. Sharp polarization and amplitude anomalies of the Sl-wave excited, for example, by a point force in orthorhombic olivine are associated with moderate polar and azimuthal velocity variations limited by 12%. While the P-wave polarization (displacement) vector stays relatively close to the source-receiver direction, the polarization vectors of S-waves may deviate far from their "isotropic" positions. The most pronounced polarization anomalies of SV-waves in the symmetry planes of orthorhombic media (and in TI media as well) correspond to areas with the largest difference between the derivatives of the P- and SV-wave phase velocities with respect to the phase angle (OV/O0). The P- and SV-wave polarization vectors in the symmetry planes of orthorhombic olivine make an angle that becomes as small as 68.5 ~. Radiation patterns in anisotropic media, both within and outside symmetry planes, are distorted by focusing and defocusing of energy associated with angle-dependent velocity. If group-velocity vectors (rays) are computed with a constant increment in the phase angle, higher concentration of rays in certain segments of the wavefront indicates an increase (focusing) of energy. Since the group-velocity vector always deviates from the phase vector towards larger velocity, the amplitude typically increases in the vicinity of velocity maxima and decreases near velocity minima. These amplitude focusing phenomena are extremely sensitive to the shape of the phase-velocity (slowness) surface and cannot be ignored even for weak anisotropy. In a dramatic example of anisotropy-induced energy focusing, the radiation pattern of the SV-wave in the [xl,x3]-plane of orthorhombic olivine has a resonance-type maximum at about 40 ~ to the direction of the force. It should be emphasized that amplitude anomalies in symmetry planes have a 3-D character because they are influenced by both in-plane and azimuthal velocity variations. This prevents the analogy

2.2. RADIATION PATTERNS AND AVO ANALYSIS I N VTI MEDIA

81

between the symmetry planes of orthorhombicmedia and transverseisotropy from being extendedt o body-waveamplitudes. Whereasfar-field body-wave polarizationsin anisotropic media are determined by the medium properties(the stiffness tensor),the amplitudesdependon both the stiffnessesand the characteristicsof the source.The combinedinfluenceof the velocity variations and the sourcefactor makesshear-waveamplitude behavior particularly complicatedoutside the symmetryplaneswhere neither S-wavegenerally belongsto the "SV" or "SH" type. One of the most distinctive featuresof wave propagation in azimuthally anisotropic media is generationof a strong transversemotion by a vertical force. Numerical modeling also reveals a significant deterioration of the geometricalseismicsapproximation(usedin ray tracing) even in homogeneousanisotropicmedia. Anisotropy not only modifies isotropic "nongeometrical"phenomena,it also causes additionalpolarizationand amplitudedistortionsin areasof rapid velocity and amplitude variations. Deviationsfrom the geometricalseismics,which can be identified by nonlinearpolarizationon particle-motiondiagrams,sometimesaresubstantialevenin the far-field. In principle, it is possibleto improvethe performanceof the ray method by using a two-term ray series,but such an extensionis difficult to implement for modelswith realistic complexity.

2.2

Radiation patternsand AVO analysisin VTI media

Amplitude-variation-with-offset (AVO) analysisis one of the few explorationmethods that is widely used for the direct detection of hydrocarbons. Conventional AVO algorithmsarebasedon analyticexpressionsfor the planeP-wavereflection coefficient in isotropic media. The presenceof anisotropyon either side of the reflector may significantly distort the angular dependenceof reflection coefficients (e.g.,Keith and Crampin,1977;Wright, 1987;Graebner,1992). Banik (1987), Thomsen (1993) and Riiger (1997,2001) developedanalytic approximationsfor the reflection coefficient a t a boundarybetweentwo transverselyisotropic media with a vertical symmetryaxis. Their results, valid in the limit of weak anisotropyand small velocity and density contrastsacrossthe reflector, are reviewedbelow. While the reflection coefficientprovidesa major contribution,yet anothersubstantial distortion of the AVO signaturein anisotropicmedia is associatedwith energy focusingphenomenain anisotropiclayersabovethe reflector. The real goal of AVO is to perform analysisof the variation in reflection coefficientwith incidenceanglerather than directly investigatethe amplitude variation with offset. Hence, correction for the angularamplitudevariation causedby the wave phenomenain the overburdenis an essentialcomponentof AVO processing.Propagationfactors, usually describedin the geometrical-seismics approximation,include sourcedirectivity, energydivergence, and transmissionand attenuationlossesalong the raypath (Duren, 1992).

82

C H A P T E R 2. INFLUENCE OF A N I S O T R O P Y ON P O I N T - S O U R C E RADIATION AND AVO ANALYSIS

Although correction for propagation phenomena in realistic heterogeneous subsurface models encounters many practical difficulties (Martinez, 1993), it is well understood if the medium is isotropic. The results of the previous section show, however, that amplitude variations along the wavefront in anisotropic media can make the isotropic correction inadequate. The presence of anisotropic layers above the target horizon is quite typical for sand/shale sequences commonly encountered in AVO analysis (Kim et al., 1993). While reservoir sands can be expected to exhibit relatively weak anisotropy, shale formations are often characterized by strong transverse isotropy, i.e., pronounced velocity variations in the vertical plane (see Chapter 1). This section, mostly based on the results of Tsvankin (1995b) and Riiger (1997, 2001), describes the radiation patterns and reflection coefficients in VTI media. The asymptotic far-field solution (2.30) for point-source radiation [obtained from equation (2.5)] is used to derive a concise weak-anisotropy approximation for transversely isotropic media that relates the amplitude to the Thomsen parameters e, 5 (for P- and SV-waves), and ~ (for SH-waves). As follows from analytic and numerical results, the character of the P-wave radiation pattern in the range of angles most important for AVO analysis (0-40 ~ is primarily dependent on the difference between the parameters e and 5. Comparison of the influence of transverse isotropy on the propagation factor and on the reflection coefficients shows that the two phenomena may be of comparable importance for AVO analysis.

2.2.1

Radiation

patterns

for weak

transverse

isotropy

Far-field point-source radiation in isotropic, homogeneous, non-attenuative media is determined just by the source directivity factor and spherical divergence of amplitude (e.g., Aki and Richards, 1980). The far-field displacement (2.5) in anisotropic media is a much more complicated function that depends on the shape of the phase-velocity surface. The most significant distortion of radiation patterns in anisotropic media is caused by the phenomena defined in the previous section as focusing and defocusing of energy. Energy increases (focuses) in parts of the wavefront with a high concentration of group-velocity vectors (rays) computed for a constant increment in the phase angle, whereas defocusing corresponds to areas with a low concentration of rays. It should be emphasized that the anisotropy-induced distortions of wavefront amplitudes are quite general because they take place not only in the source layer but in any other anisotropic layer encountered by the ray. Note that the wavefront focusing phenomena are of prime interest in cross-hole and reverse VSP surveys, which employ buried sources. Let us consider the simple model of a horizontal reflector below a homogeneous VTI layer (Figure 2.9). The influence of the free surface on radiation patterns is not taken into account, nor is that of source and receiver arrays; the emphasis here is on the propagation phenomena caused by velocity anisotropy. It is important to mention that the radiation patterns below are expressed as a function of the phase or group angle with the symmetry axis, which makes them suitable for arbitrary symmetry-axis

2.2. RADIATION PATTERNS AND AVO ANALYSIS IN VTI MEDIA

83

Figure 2.9: Reflection from the bottom of a transversely isotropic layer. Anisotropy distorts the angular amplitude distribution along the wavefront of the incident wave. orientation. It is convenient to analyze the relationship between the anisotropic parameters and the energy distribution along the wavefront in the limit of weak anisotropy. The derivation in Appendix 2B leads to the following weak-anisotropy approximation for the far-field amplitude [equation (2.30)] of P-, SV- or SH-waves excited by a point force F: Fu 1 (2.8) A ( n . o) -

I s ( R . 0)1 -

v

(o)n

.

Vsin0

'

-~- ~ d----~"]

where 0 is the phase angle measured from the symmetry axis, V is the phase velocity, and R - v/z 2 + r 2 is the source-receiver distance (z is the receiver depth, r is the horizontal source-receiver offset). The source term Fu is the projection of the force F onto the displacement (polarization) vector. Similar to the more general expression (2.30), the weak-anisotropy approximation (2.8) should be evaluated at the phase angle corresponding to a given ray (group) angle r = tan -1 (r/z) of the incident wave. (As shown in Chapter 1, at velocity minima and maxima the phase- and group-velocity vectors become identical.) Equation (2.8) clearly demonstrates how point-source radiation in TI media is distorted by velocity anisotropy. Although the term F~/(47rpV2R) formally coincides with the well-known expression for the far-field point-force radiation in isotropic media (Aki and Richards, 1980), the phase velocity V(0) in equation (2.8) is angledependent, and the source term Fu is influenced by the difference between the polarization and group angles. Also, the expression as a whole should be evaluated in the phase direction (angle 0), which generally deviates from the source-receiver line. The term under the radical represents a pure contribution of the anisotropy to the radiation pattern. As shown in the previous section, the second derivative of phase velocity is largely responsible for focusing/defocusing of energy along the wavefront. It is important to distinguish the source term Fu from the rest of equation (2.8). While F~ is itself distorted by the anisotropy, the existence of the remaining anisotro-

84


pic terms means that the redistribution of energy along the wavefront happens not only in the source layer, but also in any other anisotropic layer along the raypath. Further linearization of equation (2.8) in the parameters e, 5 and 7 leads to concise expressions for P-, SV- and SH-waves that elucidate the character of anisotropyinduced amplitude anomalies in TI media. 2.2.2

P-wave

radiation pattern

Weak-anisotropy approximation The linearized P-wave phase-velocity function V(O) and group angle ~ for weak transverse isotropy ([c[ 0 (the most common case), transverse isotropy causes the P-wave amplitude to decrease away from the vertical (see the change in the density of rays in Figure 1.12). For elliptical anisotropy (e = (~), the term 2 ( c - 6)sin 2 20 vanishes, and the anisotropic angular correction reduces to 5 sin 2 0. Thus, for elliptical anisotropy P-wave angular amplitude and velocity [see equation (1.57)] variations are correlated with each other because both depend on ~ in a similar fashion. The anisotropy-induced angular correction in elliptical media (given by 5 sin 2 0) between 0 ~ and 40 ~ is quite moderate, unless 5 is unusually large. Finally, if c - 5 < 0, we can expect an increase in the P-wave amplitude with angle due to transverse isotropy. Angular distortions of the radiation pattern may also be caused by the source term Fu, which depends on the polarization vector. As follows from the results of Chapter 1 [equation (1.79)], the P-wave polarization vector does not diverge much from the group (ray) direction for moderate velocity anisotropy. This means that the source term Fu for the P-wave is almost "isotropic," i.e., it is close to the absolute value of the cosine of the angle between the force and the group-velocity vector. Numerical modeling of point-source radiation carried out in the previous section leads to the same conclusion for orthorhombic media. If the anisotropic parameters are within the range I~1 -< 0.2, 151 < 0.2, the distortions of the point-force directivity factor Fu in the angular range 0-40 ~ are limited to a few percent. For other sources, such as dislocations or explosions, the dependence of the source term on anisotropy is more complicated (Tsvankin and Chesnokov, 1990b). Radiation pattern can also be expressed as a function of the group angle r that determines the source-receiver direction [r = tan-l(r/z)]. Formally, for weak anisotropy the difference between the phase and group angles for the terms 2 ( e - 5 ) sin 2 20 and 5sin20 in equation (2.9) can be simply ignored. Nevertheless, when the absolute values of the anisotropic coefficients approach 15-2070, the accuracy of the weakanisotropy approximation is increased by evaluating expression (2.9) at the phase angle corresponding to a given source-receiver direction. N u m e r i c a l analysis To verify the accuracy of the approximations developed above it is useful to consider several numerical examples. Figure 2.10 shows P-wave amplitudes excited by a point vertical force in a weakly anisotropic model with e - 6 = 0.051. The dotted curve in Figure 2.10 is the exact result obtained by evaluating the Fourier-Bessel integrals (Tsvankin and Chesnokov, 1990a), as described in the previous section; the solid curve is the stationary-phase solution (2.30) valid in the far-field for arbitrary strength of the anisotropy; and the dashed curve is the far-field weak-anisotropy approximation (2.9) (it can hardly be distinguished from the solid curve). To demon-

86


Figure 2.10: P-wave amplitude from a point vertical force in transversely isotropic olivine (Vp0=8.328 km/s, Vso = 4.606 km/s, e = -0.008, ~ = -0.059). The dotted curve is the exact result obtained by evaluating the Fourier-Bessel integrals; the solid curve is the stationary-phase solution (2.30); and the dashed curve is the weakanisotropy approximation (2.9). All curves are normalized by the radiation pattern in the corresponding isotropic medium (e=0, 5=0). The phase-velocity curve is shown at the bottom. The receiver is located at the constant distance R from the source in the far-field (R/Ap ~ 12, where Ap is the average P-wavelength in the isotropy plane).

2.2.

RADIATION P A T T E R N S AND AVO ANALYSIS IN VTI MEDIA

87

strate the influence of anisotropy, all three curves in Figure 2.10 are normalized by the radiation pattern in the corresponding isotropic medium (c=0, 5=0), given by V~oR) (for a unit force). In addition to revealing the angular distortions, cosr this correction makes it possible to see the difference in the absolute amplitude caused by the anisotropy. For the model in Figure 2.10 (TI olivine), transverse isotropy leads to a decrease in amplitude away from the vertical down to a minimum near 45-50 ~ At a group angle of 45 ~, the exact normalized amplitude (dotted curve) is 14% lower than that at vertical incidence. Note that the P-wave phase velocity has a maximum at 0 = 0 ~ and a minimum near ~ = 49 ~ The decrease in the P-wave amplitude is caused by the focusing of energy at vertical incidence and defocusing near 45-50 ~ Although a distortion of 14% over a 45 ~ interval does not seem to be significant, it occurs in a medium with less than 2% maximum variation in the P-wave phase velocity! Due to the nongeometrical effects described above, the stationary-phase result (2.30) diverges from the exact amplitude in the vicinity of the minimum of the radiation pattern (located at r = 90~ not shown in Figure 2.10). The values of c and 5 for transversely isotropic olivine are small and, predictably, the accuracy of the weakanisotropy approximation is sufficiently high. Before proceeding with amplitude analysis for a range of TI models, it is necessary to find out what parameters have an impact on P-wave radiation. The influence of transverse isotropy on P-wave amplitudes is mostly determined by the anisotropy parameters c and 5, but the P-wave radiation pattern also depends on the vertical velocities Vpo and Vso. Equation (2.30) shows that the P-wave velocity Vpo is just a scaling coefficient that does not change the shape of the radiation pattern (for constant c, 5 and Vpo/Vso).Although the shear-wave velocity Vs0 is contained in both the exact and weak-anisotropy expressions for P-wave radiation pattern, its influence is insignificant. For Vpo/Vsoratio varying from 1.73 to 2.2, the exact far-field P-wave amplitude in the 0-40 ~ range typically changes by less than 1.5% (Tsvankin, 1995b). The olivine model in Figure 2.10 represents a weakly anisotropic medium with ~ 0 and ~ = -0.059. For larger positive E - 5 (Figure 2.11), the second derivative of the P-wave velocity function increases more rapidly with angle. Consequently, the defocusing of energy away from the vertical becomes more pronounced and spreads over a wider range of angles [equation (2.8)]. Such a decrease in amplitude with angle may be typical for P-wave propagation through transversely isotropic subsurface formations (e.g., shales), which have predominantly positive e - (~. The maximum of energy defocusing for the model with c =0.1 and ~ = -0.1 is shifted from the velocity minimum towards larger angles (Figure 2.11) because dy~ d82 continues to increase even beyond the velocity minimum. In equation (2.9), this defocusing trend manifests itself through the behavior of the two anisotropic angular terms. Note that for the medium with c =0.1 and 5 = -0.1, which is still formally considered weakly anisotropic, the drop in the normalized amplitude (using the solid curve) from 0 ~ to 40 ~ reaches 35% (Figure 2.11). The second model in Figure 2.11, despite a

88

C H A P T E R 2. I N F L U E N C E O F 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 AND AVO ANALYSIS

e-5=0.2 e-0.1

e=0.25

5=-0.1

5=0.05

1.2 1.0

~

% :

~

9

.

: .%

0.8

%

...

s

0.6 ............................. : I

Group Angle (Degrees)

~ :3.2 2.8

:3.0

Figure 2.11: Normalized P-wave amplitude from a vertical force for two models with the same e - 5 - 0.2 (note the different vertical scales). The solid curve is the stationary-phase solution (2.30); the dashed curve is the weak-anisotropy approximation (2.9). The amplitude curves are normalized by the radiation pattern in the corresponding isotropic model (e = 0, di = 0). The plots at the bottom show the exact phase velocity (solid) and its weak-anisotropy approximation (1.61) (dashed).

2.2.


89

much larger value of e and more significant phase-velocity variations, produces more moderate amplitude distortions (about 21%). As illustrated by this example, the terms "weak anisotropy" or "strong anisotropy" are meaningless without reference to a particular problem. Whereas the model with c =0.1 and 5 = -0.1 is weakly anisotropic in terms of velocity variations, it should be qualified as strongly anisotropic in terms of P-wave amplitude distortions. Also, there is no direct correspondence between the P-wave amplitude anomalies in the 0-40 ~ range and the shape of the phase-velocity function. As illustrated by this example, different signatures (e.g., group velocity, normalmoveout velocity, dip moveout, amplitude etc.) depend on different combinations Of anisotropic coefficients. These effective parameters are not easy to infer from the exact solutions. The power of the weak-anisotropy approximation is in providing a convenient tool for developing analytic insight into the influence of anisotropy on wave propagation. While the weak-anisotropy approximation gives a good qualitative description of the amplitude anomalies, it deviates from the exact solution with increasing lel and 151, as well as with increasing angle. For the model with c=0.25, 5=0.05, the exact amplitude at a group angle of 40 ~ is about 10% higher than the weak-anisotropy result. Although this error cannot be considered as large given the value of e, it may still look surprising because the weak-anisotropy phase-velocity curve in the 0-60 ~ range is quite close to the exact one (Figure 2.11). However, we should keep in mind that while deriving the weak-anisotropy expression (2.9), we linearized in e and ~ not only the phase velocity itself, but also its two derivatives, the group angle, the expression for U that contains the components of the Christoffel matrix (see Appendix 2B), and, finally, the fraction in equation (2.8). This multiple linearization may lead to much higher errors in the weak-anisotropy approximation for amplitude than those in the linearized phase-velocity function. For elliptically anisotropic media (e=5), the stationary-phase solution (2.30) reduces to a simple function of the group angle r without application of the weakanisotropy approximation: Ap (elliptical) -

Fu

47cpV~oR r

1 + 25) (1 + 25 cos 2 r

.

(2.10)

The influence of anisotropy on P-wave radiation patterns in elliptically-anisotropic models is well-correlated with the character of the velocity variations (Figure 2.12). If the value of e = 5 is positive, both the velocity and the normalized amplitude increase away from vertical but the amplitude variations in the angular range 0-40 ~ remain mild, even for models with significant velocity anisotropy. For the medium with e =/~ = 0.25, the anisotropy-induced angular variation in the normalized amplitude from 0 to 40 ~ is limited to just 1370. Elliptical anisotropy, however, may lead to a more significant change in the absolute amplitude: for the same model with e = ~ = 0.25, for example, the amplitude near the vertical is more than 30% smaller than that in the isotropic medium with the same Vpo and p.

90

CHAPTER

2.

INFLUENCE

OF ANISOTROPY

ON POINT-SOURCE

RADIATION

AND AVO ANALYSIS

e-5=0 a=5=0.1

e=5=0.25

:

~ 3.2 3.0

:

:

;

"

..... j

3.O

Figure 2.12: Normalized P-wave amplitude from a vertical force for two elliptically anisotropic models (c = 5). The solid curve is the stationary-phase solution (2.30); the dashed curve is the weak-anisotropy approximation (2.9). On the bottom, the solid line is the exact phase velocity, and the dashed line is the weak-anisotropy approximation (1.61).


91

The accuracy of the weak-anisotropy approximation (2.9) turns out to be higher for elliptical anisotropy than for media with c - 5 > 0. Even for e and 5 in the 20-25% range, the weak-anisotropy curve in Figure 2.12 does not noticeably deviate from the exact amplitude for incidence angles up to 45 ~. For negative (less typical) c - 5 , in agreement with the weak-anisotropy result (2.9) the anisotropy leads to an increase in the P-wave amplitude with angle, with the signature more tightly controlled by the difference c - ~ than is that for models with - 5 > 0 (Tsvankin, 1995b). 2.2.3

P-wave

reflection

coefficient

in VTI

media

A central issue for AVO analysis is the influence of transverse isotropy on reflection coefficients and its comparison with the anisotropy-induced distortions of the radiation patterns discussed above. Derivation of plane-wave reflection coefficients in anisotropic media is outside of the scope of this book. For a detailed analysis of reflection coefficients in transversely isotropic and lower-symmetry media we refer the reader to a recent comprehensive monograph of Riiger (2001), as well as to the papers of Banik (1987), Thomsen (1993), Riiger (1997, 1998) and Vavry~uk and P~en~k (1998). Exact reflection coefficients have a rather complicated form even in isotropic media, and become much more cumbersome in the presence of anisotropy. A substantial simplification can be achieved in the limit of weak transverse isotropy and small velocity and density contrasts at the reflector. For such weak-anisotropy, weak-contrast VTI models, the plane-wave reflection coefficient has the following form (Thomsen, 1993, with a correction of the sin 2 0 tan 2 0-term by Riiger, 1997): /~(0) :

(2.11)

/~isot(0)+ Ranis(0),

where Risot,P(0) is the reflection coefficient in the reference isotropic medium (c = 0, = 0) given, for example, in Aki and Richards (1980), and Ranis is the contribution of the anisotropy. For P-waves, Riiger (1997) gives 1 1 /~anis,P(0) --"~ ((~2 -- ~1) sin 2 0 + ~ (e2 - s

sin 2 0 tan 20.

(2.12)

Subscripts 1 and 2 refer to the media above and below the reflector, respectively. One of the convenient features of equations (2.11) and (2.12) is the separation of the "isotropic" and "anisotropic" parts of the reflection coefficient. It is noteworthy that unlike the radiation pattern, the reflection coefficient at normal incidence is not distorted by transverse isotropy, if the density and P-wave vertical velocity are the same in the isotropic and VTI models. The term multiplied with sin20 determines the initial variation of the reflection coefficient away from the vertical and is conventionally called the "AVO gradient." Since it represents the most reliable part of the angle-varying reflection response, the AVO gradient plays a key role in most applications of AVO analysis. According to equation (2.12), the anisotropic part of the AVO gradient depends just on the change

92

CHAPTER 2. INFLUENCE OF ANISOTROPY ON POINT-SOURCE RADIATION AND AVO ANALYSIS

in 5 across the reflector (no dependence on c), while the lowest-order angular term in the radiation pattern (2.9) contains the difference between e and 5. The higher-angle term in equation (2.12) is fully determined by the parameter ~. Although the anisotropic parameters typically are small compared to unity, so are the isotropic terms comprising Risot(0), unless the jump in the velocities and density across the interface is uncommonly large (Thomsen, 1993). Therefore, the difference ( ~ 2 - 51 in equation (2.12) may substantially change the AVO gradient obtained from the isotropic reflection coefficient Risot (0). Relevant numerical estimates can be found in Kim et al. (1993) and Riiger (1997, 2001). To compare the influence of transverse isotropy on the radiation pattern and on the reflection coefficient, we assume that the medium below the reflector is isotropic (e.g., the shale/sand AVO model discussed by Kim et al., 1993). Then 1 1 Ranis,P(~) -- - - ~ (~1 sin 2 0 - ~ el sin 2 0 tan 2 0.

(2.13)

For the two models shown in Figure 2.13, the importance of the propagation phenomena is quite different. If e =0.1 and 5 = -0.1 (Figure 2.13a), the anisotropy causes a 35% drop in the radiation pattern from 0 ~ to 40 ~ while the anisotropyinduced angular variations in the reflection coefficient do not exceed 0.01. Hence, for a typical value of Risot of about 0.1, transverse isotropy would make less than a 10% change in the total reflection coefficient. Thus, in this case the redistribution of energy along the wavefront is likely to be a significantly more important factor in AVO than is the influence of the anisotropy on the reflection coefficient. In contrast, for the model in Figure 2.13b, the distortions of the radiation pattern in the 0-40 ~ range are limited to 7%, as compared with the absolute change in the anisotropic part of the reflection coefficient (Ranis) of about 0.05. For typical (small) values of Risot, we can expect the distortions of the reflection coefficient to be the dominant effect of transverse isotropy on AVO for this model. Obviously, it is difficult to make a general comparison between the influence of anisotropy on wave propagation and reflection coefficients. The angular variation in the reflection coefficient depends on the difference in the anisotropic parameters across the reflector, while the radiation pattern is entirely determined by the properties of the incidence medium. Also, the influence of anisotropy on the reflection coefficient depends on the impedance contrast, i.e., it is more pronounced for weak reflectors. However, it is clear that the two phenomena often are of the same order of magnitude. More examples of P-wave reflection coefficients in transversely isotropic media are provided by Kim et al. (1993), who studied a shale/sand boundary under the assumption that only the shale (the medium above the reflector) is anisotropic. For moderate values of c and 5 in the shale, the difference between the isotropic and anisotropic reflection coefficients at an incidence angle of 40 ~ is usually limited by +0.05. In one of the typical cases considered by Kim et al. (1993) (Class 2 gas sands), this difference still amounts to a 30-35% error in the reflection coefficient, which is comparable to the distortions of the radiation pattern discussed above.


a-0.1

~=-0.1

93

a=0.2

~-0.1

0.80 -

0.005 -

-0.06

Figure 2.13: Comparison of the influence of transverse isotropy on the P-wave radiation pattern (from a vertical force) and on the reflection coefficient. (Top) The far-field amplitude of the incident wave from equation (2.30) normalized by the amplitude in the corresponding isotropic medium with e = 5 = 0. (Bottom) The angular variation in the reflection coefficient caused solely by the anisotropy [equation (2.13)]. It is assumed that the TI models with the e and 5 shown on the plot overlie an isotropic medium.

94

C H A P T E R 2.

I N F L U E N C E O F 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 A N A L Y S I S

In isotropic AVO analysis, the presence of gas is often identified by an increase in the absolute value of the P-wave reflection coefficient with angle. Kim et al. (1993) concluded that transverse isotropy above the reflector usually enhances this behavior by further increasing the absolute value of the reflection coefficient away from the vertical. The above results show that the propagation phenomena in typical TI media (with e - 5 > 0) above the reflector may lead to a decrease in the amplitude with angle and, therefore, reduce or even reverse the influence of the anisotropy on the reflection coefficient. For example, it is possible that for "bright spots" with large n0rmal-incidence reflection coefficients and a relatively slow increase in the absolute value of the reflection coefficient with angle (Class 3 gas sands), anisotropy-induced amplitude distortions above the reflector may reverse the sign of the AVO gradient.

2.2.4

AVO s i g n a t u r e of shear waves

S V-wave The reflection coefficient of SV-waves at a small-contrast interface between two weakly anisotropic VTI media can be expressed as the sum [equation (2.11)] of the corresponding coefficient in isotropic media Risot(0, (~ -- (~ -- 0) and the anisotropic term (Riiger, 2001), 1 I22o

Ranis,sv(O) -- -~ V~o

[(e2

-

(~2) -

(s

--

1 (~1)] sin 2 0 ~ ~ (a2 -- al) sin 2 O,

(2.14)

where Vpo and lFso are the average of the velocities above and below the reflector, and a - ( V p o / V s o ) 2 (s - (~). Note that in the weak-anisotropy, weak-contrast approximation transverse isotropy does not distort the higher-angle (sin2 0 tan 2 0) term in the SV reflection coefficient. Similar to the parameter ~ for P-waves, a is responsible for SV-wave velocity variation near the vertical. Therefore, it is not surprising that a determines the anisotropic contribution to the AVO-gradient term in equation (2.14). As discussed by Riiger (2001), for typical shale/sand models the anisotropic gradient term for SV-waves is often comparable in magnitude to the isotropic AVO gradient. If a shale layer characterized by a > 0 overlies isotropic sands, the anisotropic addition to the AVO gradient has a negative sign. Next, we describe radiation patterns of SV-waves and their influence on AVO analysis. In the limit of weak anisotropy, kinematic signatures of SV-waves in TI media can be obtained from the corresponding P-wave signatures by replacing Vpo with Vso, 5 with a, and setting ~ - 0 (see Chapter 1). The terms "weak" or "strong" velocity anisotropy for SV-waves mostly refer to the magnitude of the parameter a rather than to the individual values of ~ and 5, although ~, 5, and the Vpo/Vso ratio do have some separate influence on the SV-wave velocity. Since the radiation pattern in the weak-anisotropy approximation [equation (2.8)] is a function of phase velocity (except for the source term Fu), the SV-wave radiation

2.2.


95

pattern can easily be derived from the P-wave equation (2.9) by making the same substitutions (Vpo --+ Vso, ~ ~ or, e = O) Fu 1 + 2a sin 2 20 + a sin 2 0 A s v ( R , 0) - 47rp V~0R 1 + 2or "

(2.15)

Equation (2.15) is exact in the symmetry direction where the anisotropic correction factor is given by 1/(1 + 2a). As shown in Chapter 3 below, Vs0 v/1 + 2a is the SV-wave NMO velocity from a horizontal reflector. Therefore, as for the P-wave, the SV-wave amplitude in the symmetry direction (r = 0 = 0) and the NMO velocity are controlled by the same expression (1 + 2a for SV-waves). In addition to maxima or minima at 0 = 0 ~ and 0 = 90 ~ the SV-wave phase velocity function Vsy has an extremum near 45 ~ (unless ~ - /~, in which case the velocity is constant). For the most common positive a (~ > 5), Vsy has a minimum at vertical incidence followed by a maximum near 45 ~ (exactly the same as for P-wave velocity for e = 0, 5 > 0). As follows from equation (2.15), this leads to an increase in the incident-wave amplitude with angle. An important difference, however, is that a is often much larger than either 5 or e - 5 due to the contribution of the squared verticalvelocity ratio. It is also noteworthy that the polarization direction and, consequently, the term Fu is more distorted by transverse isotropy for the SV-wave than for the P-wave (see section 2.1). A sharp increase of SV-amplitudes at oblique incidence angles caused by 3-D focusing of rays near the velocity maximum (Figure 2.3) was described above for symmetry planes of orthorhombic media. For VTI media, this focusing phenomenon has the same nature, but the concentration of rays near 45 ~ happens only in the vertical incidence plane and depends solely on the in-plane velocity variation mostly controlled by the parameter a (see Figure 1.14). It should be mentioned, however, that for models with relatively large a (like the ones in Figure 2.16), the S V radiation pattern becomes more dependent on the individual values of e, 5, and Vpo/Vso. Figure 2.14 shows the SV-wave amplitude for the same weakly anisotropic olivine model as in Figure 2.10, which has positive values of c - ~ and a. The SV-wave velocity anisotropy (i.e., the maximum variation in velocity) for this model is only about 4.3%. Nevertheless, the focusing of energy near the velocity maximum causes a pronounced increase in the normalized SV-wave amplitude near the 45 ~ angle. The stationary-phase solution (2.30) (solid curve) remains close to the exact amplitude (dotted curve) within the whole angular range shown in Figure 2.14; the area near the symmetry (vertical) axis, not shown on the plot, corresponds to a minimum of the radiation pattern where the amplitude goes to zero. Although the coefficient a is not small (a -- 0.168), the error of the weak anisotropy approximation (2.15) for this model does not exceed 6%. Figures 2.15 and 2.16 display the normalized SV-wave radiation patterns from a horizontal force for a suite of models with a varying from zero (elliptical anisotropy) to 0.45. The elliptically anisotropic model (Figure 2.15a) seems to be equivalent to isotropy for the SV-wave because its phase and group velocities do not change with

96

C H A P T E R 2. IN FLUENCE OF A N I S O T R O P Y ON P O I N T - S O U R C E RADIATION AND AVO ANALYSIS

1.2

I

Figure 2.14: SV-wave amplitude from a vertical force in transversely isotropic olivine (model from Figure 2.10, a - 0.168). The dotted curve is the exact result obtained by evaluating Fourier-Bessel integrals; the solid curve is the stationary-phase solution (2.30); the dashed curve is the weak-anisotropy approximation (2.15). All three curves are normalized by the radiation pattern in the corresponding isotropic medium (~ = 0, ~ = 0).

2.2.

RADIATION

PATTERNS

A N D A V O A N A L Y S I S IN V T I M E D I A

(e=5=0.2)

97

~=0.15

(e-0.1, 5=0.05)

SV-Wave Amplitude :

1

.

0

:

-

~

1.1

:

......

Group Angle (Degrees)

"~ 1.40 1.35

a

b

Figure 2.15: Normalized SV-wave amplitude from a horizontal force. The solid curve is the stationary-phase solution (2.30); the dashed curve is the weak-anisotropy approximation (2.15). The amplitude curves are normalized by the radiation pattern in the corresponding isotropic model (e - 0, 5 = 0). The plots at the bottom show the exact phase velocity (solid) and its weak-anisotropy approximation (1.65) (dashed).

98


angle. However, even in this case the SV radiation pattern deviates from the isotropic one due to the influence of the source term Fu, i.e., as a result of the polarization anomalies. In the example with e = 5 = 0.2 shown in Figure 2.15a, the anisotropy tilts the SV-wave polarization vector towards the vertical, thus reducing the amplitude generated by a horizontal force. It is interesting that for the SV-wave from a vertical force in the same model, the influence of the anisotropy leads to higher amplitudes. This explains the distortions of the SV-wave radiation pattern found by BenMenahem et al. (1991) and Gajewski (1993) for the elliptically anisotropic model of "Wills Point shale," which has e = 5 = 0.37. It should be mentioned that the original elastic constants of Wills Point shale, experimentally determined by Robertson and Corrigan (1983), did not correspond to elliptical anisotropy; Ben-Menahem et al. (1991) adjusted the parameters reported by Robertson and Corrigan (1983) to make the model elliptical. On the whole, for elliptical anisotropy the SV-wave amplitude distortions in the 0-40 ~ range are mild, unless the value of e = 5 is uncommonly large. Although the medium with e - 5 = 0.05 (Figure 2.15b) seems to be close to elliptical and the maximum velocity variation is just about 3.5%, the SV-wave amplitude signature is distinctly different from that for elliptical anisotropy. The value of a = 0.15 is sufficient to make the normalized amplitude increase by 35% as 0 changes from 0 ~ to 40 ~ Comparison of Figure 2.15b with Figure 2.14 shows that for comparable values of a the amplitude anomaly from a vertical force is more pronounced than that from a horizontal force. This is explained by the tilt of the polarization vector of the SV-wave towards the horizontal at mild incidence angles, which amplifies the minimum of the normalized amplitude from a vertical force in Figure 2.14. For a - 0 . 3 - 0.45 (Figure 2.16), the velocity minimum at the vertical and the maximum near 45 ~ become sharper, making the SV-wave amplitude anomaly much more pronounced. In the model with a = 0.3 (Figure 2.16a), the increase in the normalized amplitude between 0 ~ and 40 ~ reaches 85% (with a breakdown of the quality of the weak-anisotropy assumption), while the SV-wave velocity variation is only about 6.3%. With increasing a, the Gaussian curvature of the slowness surface at the velocity maximum decreases, eventually leading to a parabolic point, where the curvature goes to zero (see a discussion in Gajewski, 1993). At parabolic points, the stationary-phase solution (2.30) is invalid (its denominator goes to zero, and the amplitude becomes infinite). This deficiency is common for all high-frequency ray-theory solutions, including those presented by Ben-Menahem et al. (1991) and Gajewski (1993). More accurate numerical methods (e.g., Fryer and Frazer, 1987; Tsvankin and Chesnokov, 1990a), however, remain valid even at parabolic points. For higher values of a, the SV-wave slowness surface near 45 ~ becomes concave, and the wavefront exhibits a triplication (cusp) centered at the velocity maximum (see Chapter 1). The distribution of energy within the cusps is more complicated than that in the models considered above, and may not be adequately described by the stationary-phase expression (2.30).


0=0.3

(a=O.15,~=0.05)

99

0"=0.45 (e=0.2, 5=0.05)

:-':~

Figure 2.16: Normalized SV-wave amplitude from a horizontal force for larger values of ~,

100

C H A P T E R 2. INFLUENCE OF ANISOTROPY ON POINT-SOURCE RADIATION AND AVO ANALYSIS

The error of the weak-anisotropy approximation (2.15) in the angular range 0-40 ~ does not exceed 10% if a < 0 . 2 - 0.25. In the more limited range 0-30 ~ the weakanisotropy result remains close to the exact amplitude for much higher values of a (e.g., the model with a = 0.45 in Figure 2.16b). However, for a = 0.45 and group angles over 40 ~ the weak-anisotropy approximation breaks down completely, mostly because of the inaccurate linearized expression for the group angle. Large-magnitude distortions of SV amplitudes, even in media with moderate velocity anisotropy, mean that AVO analysis for SV-waves will be inadequate without an elaborate correction for propagation phenomena. Such correction, however, requires accurate estimates of the anisotropic coefficients, especially the parameter a. SH-wave For SH-waves, any transversely isotropic model is elliptical, with the velocity variation governed by the parameter 7 (Chapter 1). The anisotropic contribution to the reflection coefficient of SH-waves in the limit of weak anisotropy and weak-contrast interface is given by (Riiger, 2001) 1 Ranis,SH(0) -- ~ (72 --

71)

tan2 0.

(2.16)

Equation (2.16) is similar to the SV-wave expression (2.14) discussed above, but the relevant anisotropic parameter for SH-waves is 7- For a typical shale/sand interface, 71 > 0 and 72 ~ 0, which makes the anisotropic reflectivity term negative. The far-field radiation pattern (2.30) in elliptical media reduces to a concise expression without using the weak-anisotropy approximation. Evaluation of equation (2.30) for SH-waves leads to (Tsvankin, 19955) F~

1

.

(2.17)

AsH(R, r - 47rpV~oR V/(1 + 27) (1 + 27 cos 2 r In any plane containing the symmetry axis, the source term Fu for SH-waves is constant: Fu - F2, where F2 is the force component perpendicular to the incidence plane. Equation (2.17) coincides with an expression obtained by Ben-Menahem (1990) in a different way. Again, as for the P- and SV-waves, the contributions of the anisotropy to the SH amplitude in the symmetry direction and to the NMO velocity (see Chapter 3) are determined by the same expression; in this case, it is 1 + 27. The SH-wave radiation pattern [equation (2.17)] is practically identical to the Pwave pattern for elliptical anisotropy [~ = 5, equation (2.10)], with Vso and 7 replaced by Vpo and 5, respectively. The only difference between the two expressions is that the source term F2 for the SH-wave is constant. For typical positive values of 7, the defocusing of energy at vertical incidence leads to an increase in the SH-wave amplitude with angle, but the influence of anisotropy is relatively mild. Even for a large 7 of 0.3 (yielding velocity anisotropy of about 26%), the amplitude increase from 0 ~ to 45 ~ is limited to just 11%. The change in the

2.2.


101

absolute amplitude caused by the anisotropy is much more pronounced, but absolute amplitudes are of less importance in AVO analysis. So far, we discussed the angular distribution of amplitude for a fixed sourcereceiver distance (R=const). It is also interesting to examine the SH-wave amplitude along the (elliptical) wavefront at a fixed time t. Then equation (2.17) becomes

F2 Usu(t, r - 47rpyj~ (Vsot) (1 + 27)"

(2.18)

Equation (2.18) shows that the SH-wave amplitude along the wavefront is constant (as in isotropic media), although the wavefront is elliptical rather than spherical; the influence of anisotropy reduces to the scaling factor 1 + 27. Note that this result does not apply to P-waves in elliptically anisotropic media because the source term Fu in equation (2.10) depends on the source-receiver direction, and, consequently, the P-wave amplitude varies along the wavefront even in this case. The conclusion that the influence of transverse isotropy on radiation patterns is much more significant for SV-waves than for SH-waves is in good agreement with experimental results by Robertson and Corrigan (1983), who measured shear-wave amplitudes in anisotropic shales with positive a. They found a strong focusing of energy of SV-waves near the 45 ~ incidence angle, while the S H radiation pattern did not deviate much from the isotropic one, despite pronounced velocity anisotropy.

2.2.5

Discussion

The low-angle (gradient) term of the plane-wave reflection coefficient in VTI media contains a contribution of the anisotropic parameter responsible for the velocity variation at small incidence angles. For P-waves, the AVO gradient depends on the difference between the parameters 5 below and above the interface; likewise, the shear-wave AVO gradients depend on changes in the parameters a (SV-wave) and 3' (SH-wave). Although these anisotropic parameters (with the possible exception of cr) in most cases are much smaller than unity, for typical weak-contrast boundaries they may be comparable to the isotropic terms in the reflection coefficient. In the commonly encountered AVO model of TI shales overlying isotropic sands, the anisotropic AVO-gradient term is expected to be predominantly negative, in particular for both shear waves. In addition, amplitude-focusing phenomena above the reflector may be of equal or greater importance to AVO analysis than the influence of anisotropy on the reflection coefficient, especially for strong reflectors or small differences in the anisotropic coefficients across the reflector. The energy focusing happens in any anisotropic formation crossed by the ray of a reflected wave, not just in the source layer. If not properly corrected for, the anisotropic directivity factor may lead to a change or even a sign reversal in the AVO gradient that is conventionally used for hydrocarbon detection. Therefore, interpretation of AVO anomalies in the presence of anisotropy requires an integrated approach that takes into account not only the reflection coefficient but also the wave propagation in the overburden.

102


The analytic solution for radiation patterns in weakly anisotropic VTI media given here relates the anisotropic directivity factor for all three modes (P, SV, SH) to Thomsen parameters. The expressions for radiation patterns are derived as a function of the phase or group angles with the symmetry axis and, therefore, can be applied to transverse isotropy with any symmetry-axis orientation. For vertical transverse isotropy, the character of the P-wave angular amplitude variations in the range of angles commonly used in AVO analysis (0-40 ~ is mostly controlled by the difference between e and 5, and is practically independent of the shear-wave vertical velocity Vso. For typical models with e - 5 > 0, transverse isotropy may cause the P-wave amplitude to decrease by 30% or more over the first 40 ~ from the vertical. Application of the elliptical-anisotropy approximation (e = 5) to P-wave amplitudes may lead to unacceptable errors even if the medium is relatively close to elliptical. If the difference e - 5 is fixed at a positive value, the influence of the anisotropy becomes stronger with decreasing e and 5. Thus, there is no correlation between the strength of velocity anisotropy and the amplitude anomalies. The distortions of the radiation pattern of SV-waves are much more significant than those of P-waves. For typical moderately positive values of the parameter a, there is a sharp SV-wave amplitude maximum near an angle of 45 ~ As a increases, the wavefront of the SV-wave develops a triplication (cusp) at oblique incidence angles, with energy largely concentrated at the edges of the cusp (see Chapter 1). The anisotropy of the SH-wave is elliptical and, in contrast to the situation for SV-waves, the distortions in the angular amplitude dependence are relatively weak, even for substantial velocity variations. Moreover, the SH-wave amplitude along the wavefront (rather than at a constant source-receiver distance) in any plane containing the symmetry axis does not change at all. These results for shear waves agree with those in the case study by Robertson and Corrigan (1983). An approximate correction for the anisotropic source directivity factor can be based on asymptotic expressions for radiation patterns in homogeneous media like those discussed here or developed by Gajewski (1993). However, the redistribution of energy along the wavefront may occur in any anisotropic layer between the reflector and the surface. In this case, it is necessary to apply numerical methods capable of allowing for anisotropy in wave propagation through layered media. If the overburden is laterally homogeneous, it may be possible to correct for the anisotropic propagation factor in AVO analysis by doing amplitude calibration at well locations. This calibration, however, is sufficient for recovering only lateral variations in the reflection coefficient, unless it is possible to obtain the reflection coefficient itself from, for example, rock-properties measurements. The greatest challenge in correcting AVO signatures for anisotropy remains to determine the anisotropic parameters with sufficient accuracy. The parameter-estimation problem for vertical transverse isotropy is discussed in Chapters 6 and 7.

2A. DERIVATION OF THE ANISOTROPIC GREEN'S FUNCTION

Appendices

2A

for Chapter

103

2

Derivation of the anisotropic Green's function

Consider a point source embedded in a linearly elastic homogeneous arbitrary anisotropic medium without initial stresses. The general form of the equation of motion (wave equation) for arbitrary distribution of body forces f(x) was given in Chapter 1 [equation (1.5)]" 02ui 02uk (2.19) p~ - cijkz OxjOzl = fi, Cijkl is the stiffness tensor and p is the density; summation over repeated indices is implied. Assuming that the source is a single point force F at the origin of the Cartesian coordinate system, equation (2.19) can be rewritten as

where

02ui P -~

02uk -- Cijkl OXjOXl

= F~ 5(x) h(t).

(2.20)

Here 5(x) is the spatial (3-D) (f-function and h(t) describes the source pulse. If the force is parallel to the coordinate direction xn, the displacement ui satisfying equation (2.20) represents, by definition, the component (in) of the Green's function (e.g., Aki and Richards, 1980). The solution of equation (2.20) can be sought in the form of a triple Fourier transform, or as a superposition of harmonic plane waves (Musgrave, 1970; Hanyga, 1984)" U(t, X)

-

-

~ dw

(271.)4

/2~ P(w, k) ei(~t-kx) dk,

(2.21)

where dk - dkldk2dk3, k has the meaning of the wave vector and P ( w , k ) is the integral kernel to be found from the wave equation. Note that the three components of the wave vector are independent of each other, so the elementary plane waves in equation (2.21) do not necessarily satisfy the homogeneous wave equation [i.e., equation (2.19) with f - 0]. For the spatial 5-function we have 1 3 (2~)

5(x)-

/7~ e_ikx dk.

(2.22)

Substituting equations (2.21)and (2.22)into equation (2.20)and solving for P(w, k) allows us to express the frequency spectrum S(w, x) of the displacement vector u as

_

1

S(w, x) - (27r)3

~ P(w, k) e -ikx dk - S(w, x) (I)(w),

(2.23)

where (I)(w) is the spectrum of the source pulse,

(~(w) -

S

oo

h(t) e -~t dt,

(2.24)

104

C H A P T E R 2.

I N F L U E N C E 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 ANALYSIS

and 1

/7

H-IF

ek.

(2.25)

The matrix H is closely related to the Christoffel matrix G [equation (1.9)]: H~k = Gik/V 2 - ,o(~ik = r Pj P l - p(Sik; V is the phase velocity and p = k/co is the slowness vector. Equation (2.25) can be reduced to a superposition of plane-wave solutions of the homogeneous wave equation by performing integration over one of the wavenumbers ki. Since the result is often used in solving reflection-transmission problems at horizontal boundaries, it is convenient to integrate over the vertical wavenumber ka. As described by Chesnokov and Abaseev (1986) and Tsvankin and Chesnokov (1990a) (for the isotropic problem, see Aki and Pdchards, 1980), this integration can be accomplished by extending the integral to the complex ka-plane and evaluating the residues at the poles of the integrand. If the vertical coordinate of the receiver xa > 0, the original integration path along the real ka-axis should be supplemented with a semicircle in the lower half-plane Im(ka) < 0, so that e -ikaxa decays for ka ~ -ie~. If xa < 0, the semicircle should lie in the upper half-plane. The poles of the integrand correspond to the slowness vectors of plane waves that satisfy the Christoffel equation (1.10) and, therefore, the homogeneous wave equation. As discussed in Chapter 1, the Christoffel equation has three solutions for each direction of the slowness or wave vector. For a given pair of the horizontal wavenumbers [kl, k2], however, there will be six possible vertical wavenumbers that correspond to three upgoing waves (i.e., waves propagating in the positive xa-direction) and three downgoing waves. If xa > 0, the integration contour contains three poles at the wavenumbers ka of the upgoing P- and two split S-waves; for xa < 0, the poles will produce the contributions of the three downgoing waves (Chesnokov and Abaseev, 1986). After evaluating the residues (denoted by "Res" below) at the poles, equation (2.23) takes the form S(w x ) -

iw

47r2

u--1

~ c~

U(~')(P~P2)e -i'(pa~I+p2~2+p~)~3) dpldp2 '

(2.26)

with 1 Ha dF] p(3~) . U (~) (Pl, P2) - Res [ D(p3i

(2.27)

Here H ad is the adjoint matrix of H, D(p3) - det[Hik], and p~) are the poles of the integral kernel in equation (2.25), which also are the roots of D(p3). The minus sign in front of the expression for S(w, x) holds for x3 > 0; if x3 is negative, the minus should be replaced with a plus. Equation (2.26) may be called the Weyl integral for anisotropic media since it represents a decomposition of point-source radiation (Green's function) into planewave solutions of the wave equation. [See Tsvankin (1995c) for a detailed discussion

2B. WEAK-ANISOTROPY APPROXIMATION FOR RADIATION PATTERNS IN TI MEDIA

105

of the isotropic Weyl integral.] Indeed, it can be shown that U (~) is parallel to the polarization vector of a plane wave with the slowness vector p {Pl,P2, p~') }. Therefore, U (~) has the meaning of the displacement (amplitude) vector of the elementary plane wave, while the exponential under the integral (2.26) describes the plane-wave phase function. Plane waves comprising point-source radiation propagate in all directions because the integration is carried out over all possible horizontal slownesses pl and p2. Note that in addition to conventional homogeneous waves, the Weyl integral (2.26) contains inhomogeneous (evanescent) plane waves that correspond to relatively large values of the horizontal slowness P0 - ~/P~ + P~ and decay in the vertical direction. It is convenient to transform equation (2.26) further by introducing polar coordinates (Pl = p0 cosr P2 = p0 sine, Xl = r cosa, x2 = r sina, x3 = z): ,.

, o cos 0 v--1

The wavefield in the time domain can be found from equations (2.21), (2.23) and (2.28) as

1/2

u ( t , X) -- ~

2B

S(CO, X) (I)(cO) e iwt d w .

(2.29)

Weak-anisotropy approximation for radiation patterns in TI media

The stationary-phase approximation (2.5) is valid for the far-field displacement in any symmetry plane of an anisotropic medium (e.g., the model can be orthorhombic or monoclinic). In the special case of transversely isotropic media with a vertical symmetry axis, phase velocity and vertical slowness depend on just the phase angle 0 with vertical, and the coefficient d in the denominator of equation (2.5) vanishes. The magnitude of the frequency-domain displacement vector [equations (2.5) and (2.6)] then becomes IS]-

[U[ sinO 2~V

d2~dO 2 = (r sin O + z cos O)

1 V

cos0

+

e(1/v)

sinO

)

(2.30)

V 0 a2"---~dO2 Crsin

d2(1/V)+ dO2

2(rcosOzsinOI21 .

V

(2.31)

r sin O + z cos O

Here the exact far-field solution (2.30) for TI media is transformed into a much simpler weak-anisotropy approximation. The derivation is given for only P-waves; the radiation patterns of S-waves are obtained in the same way. The plane-wave displacement IUI determined by equation (2.27) can be represented as an explicit function of the P-wave slowness vector p. Assuming that the

106 CHAPTER 2. INFLUENCE OF ANISOTROPY ON POINT-SOURCE RADIATION AND AVO ANALYSIS

receiver is located in the [xl,x3]-plane, from equation (2.27) are

the non-zero components of the matrix H

Hll - c l i p 2 -f- c55p~ - fl,

(2.32)

H22 - c66p~ + c55p~ - p ,

(2.33)

H33

(2.34)

-

-

c55P 2 -Jr- c33P 2 -- f l ,

(2.35)

H13 =/-/31 = (C13 -[-c55)Po P3.

As discussed in Chapter 1, P- and SV-waves are independent of the component G22 of the Christoffel matrix (and, therefore, of H22) and their displacement vectors U are confined to the Ix1, x3]-plane. For a fixed po, the squared vertical slownesses of the P-waves (p2,p) and SV-waves (P3,sv)2 are the roots of the polynomial HllH33-H213 which is quadratic in p]. Therefore, the P-wave plane-wave displacement vector can be determined from equation (2.27) as U1 = K (/71 H33 - F3 H13),

(2.36)

U3 --- K (F3 H l l - F1 H13),

(2.37)

with K

-

P3 -

P3,P

[

.

(2.38)

H11H33 - H~3 p3=p3,P

Expressing the polynomial H11H33 -- H~3 through its roots, H11H33 - H23 - c33 c55 (p2 _ p],p)

(2.39)

2(p3- P3,sv2 ) ,

equation (2.38) can be rewritten in the form K-

1 2c33 c55 P3,P (P],p - P~,sv) "

(2.40)

It is convenient to relate the vertical slownesses of the S V - and P-waves by substituting equations (2.32), (2.34) and (2.35) into equation (2.39) and setting p3 = 0: P],sv -

(cllp~

-

p)(c5~p~2

(2.41)

- p) 9

C33 C55 P 3 , P

Taking into account that for P3 - p3,P the determinant H l l H 3 3 follows from equations (2.36) and (2.37)that

IUI-- g

(Hll + H33),

where F~ is the projection of the force F onto the displacement U.

H~3 -

O, it

(2.42)

2B. W E A K - A N I S O T R O P Y

A P P R O X I M A T I O N F O R R A D I A T I O N P A T T E R N S IN TI M E D I A

107

Substituting K [equations (2.40) and (2.41)], Hll [equation (2.32)] and/-/33 [equation (2.34)] into equation (2.42)yields

Jr- P3,P (C33 -4- C55) 2 P3,P 4

C33 C55 - - (-p~

c;1

--

-- 2p

p) (p2 c55 - - p) "

(2.43)

Equation (2.43) is valid for general (not just weak) transverse isotropy. Further simplification of the amplitude IUI, however, requires applying the weak-anisotropy approximation. P-wave slownesses can be expressed through the phase angle 0 and phase velocity V (p0 = sin0/V, P3,P = cos O/V), with V obtained from Thomsen's (1986) approximate equation (1.61):

Vp(O) - liFO (1 + ~sin 2 0cos 2 0 + esin 4 0).

(2.44)

Equation (2.44), along with the expressions (1.46) and (1.47) for the parameters e and ~ in terms of the stiffnesses, can be used to find the weak-anisotropy approximation for IuI. Linearization of equation (2.43) in e and 5, after straightforward but tedious algebra, gives

IUI - 2Vp(O) p c o s

0 [1-4- 2 sin 2 0 (~ cos 20 -4- 2e sin 2 0)].

(2.45)

d2V [equation (2.31)] To obtain the weak-anisotropy approximation for the derivative e-gr r and z can be represented through the group angle r and the source-receiver distance R (r - R sin r z - R cos r Then

and

r sin 0 + z cos 0 - R cos(r - 0),

(2.46)

r cos 0 - z sin 0 tan r - tan 0 = r sin 0 + z cos 0 1 + tan r tan 0 '

(2.47)

where the group and phase angles for weak transverse isotropy are related by equation (1.73)" tan r - tan 0 [1 + 2~ + 4(e - ~) sin 2 0]. (2.48) Substituting equations (2.46)-(2.48) into equation (2.31) and dropping the terms quadratic in e and 6 leads to

d2~ [l dO2 = R V

d2(1/V)I R ( l d2V) dO2 - ~ I + V dO2 "

(2.49)

The numerator of equation (2.30) can be linearized in the anisotropic coefficients using the approximate phase-velocity function (2.44)" cos 0 V

d(1/V)do sin 0 - cOSOv[1 - 2sin 2 0 (bcos 20 + 2esin 2 0)].

(2.50)

108 CHAPTER 2. INFLUENCE OF ANISOTROPY ON POINT-SOURCE RADIATION AND AVO ANALYSIS

Finally, substitution of equations (2.45), (2.49), and (2.50) into the stationaryphase approximation (2.30) yields

Fu ]Sl - 47rp V2R /~in~

1

(2.51) 1 d 2 V ~ '"

V s i n 0 (1 -~- ~ - - ~ - ]

Equation (2.51) can be fully linearized in the anisotropic parameters e and /~. However, it is a useful intermediate result because analogous derivations lead to the same equation for both shear waves. Linearized expressions for each wave type (P, SV, SH) are discussed in the main text.

Chapter 3 Normal-moveout velocity in layered anisotropic media Reflection traveltimes (moveout) provide the most reliable information for building velocity models using surface seismic data, in both isotropic and anisotropic media. If the medium is anisotropic, an attempt to fit the traveltime-offset relationship using a purely isotropic velocity field may lead to mis-stacking of reflection events and distortions in seismic images (see examples in Chapters 6-8). Hence, understanding of the influence of anisotropy on the kinematics of reflected waves is of primary importance in seismic velocity analysis and processing. Moveout of pure (non-converted) modes on common-midpoint (CMP) gathers is conventionally approximated by the Taylor series expansion near the vertical (e.g., Taner and Koehler, 1969):

t 2 = Ao + A2x 2 + A4 x4 + . . .

(3.~)

,

where x is the source-receiver offset, and the coefficients are given by

Ao - t~,

d(t2) ~=o' A2 -- d(x2)

A 4 - 2d(x2) [d(x2)

z=o ;

(3.2)

to is the two-way zero-offset traveltime. Equation (3.1) does not include odd powers of x because CMP moveout of pure modes is symmetric with respect to zero offset (i.e., it remains the same when one interchanges the source and receiver). Later on we will replace the Taylor series (3.1) with a more accurate approximation that still includes the moveout coefficients A0, A2 and A4. The moveout parameter of most practical significance in exploration is the normalmoveout (NMO) velocity Vnmo, responsible for the hyperbolic moveout on conventionallength spreads that do not exceed the distance between the CMP and the reflector: x2

t~y, = to~ + v--~'

(3.3)

nmo

1

d(x2) l

V:mo- ~ = d(t~) ~ : 0

(3.4)

110

C H A P T E R 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA

If the traveltime is plotted in the t 2 - x 2 coordinates, the factor 1/Vn2modetermines the initial slope of the moveout curve. With increasing offset, the t2(x 2) curve deviates from a straight line due to the influence of the quartic (A4 x 4) and higher-order moveout terms. Hence, stacking velocity estimated using finite-spread moveout in conventional processing (we will also call it "moveout velocity") may differ from the analytic NMO velocity defined by equation (3.4). Anisotropy causes two major distortions of reflection moveout. First, in contrast to isotropic media, NMO velocity generally differs from the vertical velocity (in a single layer) or from the root-mean-square (rms) of the interval vertical velocities (in layered media). This is the main reason for mis-ties in time-to-depth conversion and the erroneous depth scale of seismic images often produced by conventional migration algorithms. Second, anisotropy may substantially increase deviations from hyperbolic moveout since the t(x) curve becomes nonhyperbolic even in a single homogeneous anisotropic layer. In Chapters 3 and 4, we analyze the influence of anisotropy on both NMO velocity and long-spread moveout and present a concise analytic description of reflection traveltimes in anisotropic media.

3.1 3.1.1

2-D N M O equation in an anisotropic layer General

expression

for dipping

reflectors

Let us consider a CMP gather over a homogeneous anisotropic layer; the line is perpendicular to the reflector strike (Figure 3.1). The only assumption made about anisotropy at this stage is that the phase- and group-velocity vectors do not deviate from the incidence plane, i.e., the incidence plane is both the dip plane of the reflector and a plane of symmetry. For instance, the present treatment is valid for any plane containing the symmetry axis in transversely isotropic media (plus the isotropy plane in HTI media), as well as for symmetry planes in orthorhombic media. The derivation in Appendix 3A, based on the work of Tsvankin (1995a), leads to the following expression for the NMO velocity of pure-mode reflections as a function of the phase velocity V(O) and reflector dip r [equation (3.79)]: Vnmo(r

-

V(r COSr

1 + v(r 1 - - tan r V(r

e= .

(3.5)

d._~VI dO lO=r

This equation is valid for both P- and S-waves in vertical symmetry planes of anisotropic media. Due to the fact that the phase-velocity (and slowness) vector of the zero-offset ray is orthogonal to the reflector, the normal-moveout velocity is governed by the local behavior of the phase-velocity function at the dip angle of the reflector. Equation (3.5) is relatively simple to use because it involves just phase velocity rather than the components of the zero-offset ray (note that the angle between the zero-offset ray and vertical is generally different from r Difficulties in application

3.1. 2-D NMO EQUATION IN AN ANISOTROPIC LAYER

111

Figure 3.1: Depth section showing raypaths in a common-midpoint gather over a homogeneous anisotropic layer with a dipping lower boundary. V and V~ are the phase- and group-velocity vectors, respectively. Note that the zero-offset ("normalincidence") ray is not necessarily perpendicular to the reflector. of formula (3.5) can be expected only in anomalous areas where the Taylor series expansion for reflection moveout breaks down. For example, reflection traveltime is multi-valued and exhibits rather complicated behavior near shear-wave cusps (triplications) discussed in Chapter 1. Out-of-plane phenomena cannot be neglected if the incidence plane lies outside symmetry planes in azimuthally anisotropic media; still, equation (3.5) remains a good approximation if azimuthal anisotropy is weak. Azimuthally anisotropic models with orthorhombic symmetry caused by a combination of thin horizontal layering and vertical fracture systems with a low fracture density are believed to be typical for sedimentary basins (Leafy et al., 1990; see Figure 1.4). For such media, characterized by relatively weak azimuthal anisotropy and often pronounced velocity variations in vertical planes, equation (3.5) may be acceptable even outside symmetry planes. A more general, 3-D treatment of normal-moveout velocity based on the equation of the NMO ellipse is introduced in section 3.5. If the medium is isotropic, the derivatives of phase velocity vanish, and equation (3.5) reduces to the well-known cosine-of-dip relationship obtained by Levin (1971): Vnmo((/))-

V

cosr

= Vnmo(0)

(3.6)

cosr

For homogeneous isotropic media, moveout is purely hyperbolic, and equation (3.6) gives an accurate description of reflection traveltime for any spreadlength. Levin's expression is widely used in conventional dip-moveout (DMO) correction, which represents an essential processing step in building zero-offset seismic sections suitable for

112

CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA

poststack migration. Equivalently, equation (3.5) can be represented as a function of the horizontal (p = sin O/V) and vertical (q = q(p) = cos O/Y) components of the slowness vector (Cohen, 1998): Vnmo(P) --

/ q" ]

VPqt-- q

p(r

(3.7)

'

where q' = dq/dp and q" = d2q/dp 2. Equation (3.7) is evaluated for the slowness vector of the zero-offset ray (p = sin r q = cos C/V). Although, in contrast to equation (3.5), the form (3.7) does not contain an explicit contribution of the anisotropy, it has proved convenient for application in velocity analysis, inversion and processing (see Chapters 6-8).

3.1.2

Special cases

In the special case of a horizontal reflector (r - 0), the anisotropic term in the denominator of equation (3.5) vanishes, and the NMO velocity takes the form 1 d2V 1 ~ V(O) dO2

Vnmo(0)- V(O)

(3.8) 0=0

Equation (3.8) contains only the second derivative of phase velocity, even though the first derivative in the vertical direction does not necessarily go to zero, as is the case for tilted TI media. In general, velocity variation with angle makes Levin's (1971) equation (3.6) inaccurate in the presence of anisotropy. There is, however, one special type of anisotropic media for which the cosine-of-dip relationship remains entirely valid irrespective of the strength of the anisotropy. Let us assume that the medium is transversely isotropic with a symmetry axis orthogonal to the reflector (i.e., the symmetry axis is rotated along with the reflector as its dip changes). This model is typical, for instance, for dipping shale formations in overthrust areas such as the Canadian Foothills. Then the derivative -~ dy at the dip angle goes to zero, and equation (3.5) reduces to !

_ v(r Vnmo ((~)

COS

[

1 d2V V(r dO2 o=r

(3.9)

d2V at any dip correspond to the For this particular model , the values of V and -g~ symmetry direction and, therefore, are independent of r

v(r

1

v(r

d2V dO o=r

1 - V(O)

d2V

1-t V(O) dO2

- - Vnmo(O ) .

(3.10)

0=0

Hence, equation (3.9) coincides with the isotropic equation (3.6) expressed through the zero-dip NMO velocity, as was verified numerically by Levin (1990).

3.2.

NMO V E L O C I T Y F O R V E R T I C A L T R A N S V E R S E I S O T R O P Y

113

Note that this result is derived for normal (zero-spread) moveout velocities rather than for moveout velocities measured on finite spreads. In the presence of anisotropy moveout is generally nonhyperbolic, and the NMO velocity deviates from the effective moveout (stacking) velocity with increasing spreadlength. Although the spreads used by Levin (1990) are not long (equal to the normal distance to the reflector), his results for the models with the symmetry axis perpendicular to the reflector show small errors in equation (3.6) indicative of the influence of nonhyperbolic moveout. Also, while the dip dependence of NMO velocity is not distorted by the anisotropy when the symmetry axis is perpendicular to the reflector, the zero-dip value Vnmo(0) does depend on the anisotropic parameters.

3.2 3.2.1

N M O velocity for vertical transverse isotropy Horizontal

reflector

NMO velocities of all three modes (P, SV, SH) in a horizontal VTI layer take a particularly simple form in terms of Thomsen (1986) anisotropic parameters introduced in Chapter 1. Substituting expression (1.49) for 5 into equation (3.8), we find for the P-wave V n m o , p ( O ) - - Vpo ~/1 + 2~. (3.11) Therefore, the influence of transverse isotropy on the P-wave NMO velocity is absorbed by a single p a r a m e t e r - ~. [When the symmetry axis is perpendicular to a dipping reflector, the P-wave NMO velocity is given just by equation (3.11) and the cosine-of-dip factor]. For the special case of elliptical anisotropy (e = fi), NMO velocity (3.11) is equal to the horizontal velocity. Evaluating the second derivative of the SV-wave phase velocity at vertical incidence yields d2Vsv (3.12) = 2Vsoa, dO2 0--0 where a is the anisotropic coefficient defined in equation (1.67). Thus, Vnmo,sv(O) -- VSo~/X Jr- 20".

Differentiating the phase velocity of the SH-wave [equation (1.54)] twice and substituting the result into equation (3.8) leads to an expression that has the same form as the NMO velocities of P- and SV-waves: Vnmo,SH(O) -- VSO ~/1 Jr- 2' T 9

(3.14)

It should be emphasized that equations (3.11), (3.13) and (3.14) are valid for VTI media with arbitrary strength of the anisotropy. Clearly, unless the anisotropic coefficients 5, a and ~ go to zero, NMO velocities in VTI media differ from the corresponding vertical velocities. Therefore, if seismically

114

CHAPTER 3. NORMAL-MOVEOUTVELOCITY IN LAYEREDANISOTROPIC MEDIA

derived moveout (stacking) velocities and vertical traveltimes are used to determine reflector depth, the result contains a percentage error approximately equal to the corresponding anisotropic parameter (e.g., 5 for P-waves). Anisotropy-induced errors in time-to-depth conversion are discussed in more detail in section 3.4. 3.2.2

Elliptical

anisotropy

Before analyzing dip-dependent NMO velocity in general VTI media, let us consider the special case of elliptical anisotropy. It is assumed that the vertical incidence plane coincides with the dip plane of the reflector, so that the NMO velocity is described by the 2-D equation (3.5). Phase velocity in elliptical media [e.g., equation (1.54)] can be represented through the vertical (V0) and horizontal (V90) velocities as

V(O) - V/Vo2 cos2 0 + V~ sin 2 0 .

(3.15)

The NMO velocity (3.5) then becomes Ynmo(r -

Y90 ~ cos r c~

V2 sin 2 r r + V-j0 2

(3.16)

In VTI media, however, equations for elliptical anisotropy are strictly valid only for the SH-wave. If the SH-wave phase velocity is parameterized by 7 as in equation (1.54), equation (3.16)yields

VnmoS H ( r '

--

Ks~

+ 27 ~/1 + 27sin 2 r . COS ~)

(3.17)

Using equation (3.14) for the zero-dip NMO velocity of the SH-wave leads to Vnmo SH(r -- Vnm~ ' cosr

VSH(r Vs0 '

(3.18)

where VSH(r is the SH-wave phase velocity at the dip r Therefore, for elliptical anisotropy the error of the isotropic cosine-of-dip relationship is directly determined by the magnitude of phase-velocity variations, i.e., by the ratio of the phase velocity at the dip angle to the vertical velocity. If the VTI medium is elliptical (e = 5), equation (3.18) can also be used for P-waves: Vnmo,P(r Vnmo(0) Vp((/)).'-" (3.19) cosr Vp0 The SV-wave velocity in elliptical media is constant, and the NMO velocity is described by the isotropic equation (3.6).

3.2. NMO VELOCITY FOR VERTICAL TRANSVERSE ISOTROPY

3.2.3

115

Weak-anisotropy approximation for general VTI media

A convenient way to understand the influence of anisotropy on the normal-moveout velocities of P- and SV-waves for general (non-elliptical) VTI media is to use the weak-anisotropy approximation. Although this approximation may deviate from the exact equations with increasing magnitude of anisotropic coefficients, it provides simple analytic relations elucidating the dependence of the NMO velocity on the parameters e and 5 (Tsvankin, 1995a). The equations developed below also hold in weakly anisotropic orthorhombic media with a horizontal symmetry plane, if e and 5 are replaced by the azimuthally dependent coefficients corresponding to the incidence plane [see equations (1.108)and (1.109), section 1.3]. The linearized P-wave phase velocity and its first derivative are given by equations (1.61) and (1.72). Differentiating equation (1.72) gives the second derivative of

Yp: dV~,(9) = 2Vpo [5cos49 + 2csin 2 9(1 + 2 cos 29)] d92

(3.20)

After substituting the weak-anisotropy expressions into equation (3.5) and further linearizing in c and 5, the P-wave NMO velocity takes the form Vnmo(r -- Vp(r

COS r

[1 + 5 + 2(e - 5)sin 2 r

+ 2 cos 2 r

(3.21)

For a horizontal reflector (r = 0), Vnmo(0) --

gpo (1 +

5),

(3.22)

which represents the weak-anisotropy approximation of the exact equation (3.11). Using equation (3.22), equation (3.21) can be rewritten as Ynmo(r cos r

= 1 + 5 sin 2 r + 3(e - 5) sin 2 r (2 - sin 2 r

(3.23)

Vnmo (0)

The anisotropic terms on the right-hand side of equation (3.23) quantify the error when the isotropic cosine-of-dip dependence is assumed for the NMO velocity [the "dip-moveout (DMO) error"]. The structure of equation (3.23) suggests that the Pwave DMO error in VTI media has two major components, which may be called the "elliptical error" and "non-elliptical error." Indeed, for elliptical anisotropy (c = 5) the error is determined just by the term 5 sin2r that describes angular variations in the P-wave phase velocity for that model. The second, non-elliptical, component of the DMO error is the term containing the difference e - 5. The magnitude of the trigonometric coefficients in equation (3.23) shows that, unless ] e - 51 0

116


(the most typical case), NMO velocity increases with dip faster than in isotropic media (unless 5 is negative and large). This conclusion is supported by numerical computations below. According to equation (3.23), the influence of anisotropy on the dip dependence of P-wave NMO velocity is not necessarily correlated with the magnitude of the coefficients c and 5 and the overall strength of velocity anisotropy. As an example, consider two VTI models from Thomsen's (1986) paper used in the numerical study by Levin (1990) - the shale-limestone and Cotton Valley shale. Figure 3.2 shows the moveout velocities for the two models calculated using least squares directly from traveltimes ( t 2 - x 2 curves) over a spread equal to the CMP-to-reflector distance, along with the exact NMO velocity (3.5) and the weak-anisotropy approximation (3.21). All three velocities are multiplied with the cosine of the dip r Judging by the values of 5, Cotton Valley shale may be considered as a more "anisotropic" material with respect to P-wave propagation than the shale-limestone. For Cotton Valley shale, however, the large positive ~ and small negative difference e - (~ almost cancel each other's contributions in equation (3.23), and the accuracy of Levin's (1971) isotropic expression (3.6) is quite satisfactory. All three curves for Cotton Valley shale in Figure 3.2 display only small variations (3-4%) in the corrected moveout velocity with angle, confirming the conclusion about the acceptability of the cosine-of-dip relationship for this particular model. Despite the substantial value of 5, the weak-anisotropy result for Cotton Valley shale is close to the exact NMO velocity (the difference is less than 2%). In contrast, a more typical positive difference e - 5 for the shale-limestone leads to a much faster increase in the NMO velocity with dip than in isotropic media. The accuracy of the weak-anisotropy approximation for the shale-limestone is sufficiently high as well. A more systematic comparison between the weak-anisotropy approximation and the exact NMO velocity is presented below. There is a small but noticeable difference between the finite-spread moveout velocity calculated directly from the traveltimes ( t 2 - x 2 curves) and the NMO velocity from equation (3.5) in Figure 3.2. Since the moveout velocity was determined from a least-squares fit to t 2 - x 2 curves on a finite spreadlength, it could have been distorted by nonhyperbolic moveout, while the analytic NMO velocity describes purely hyperbolic moveout on short spreads. To verify this possibility, Figure 3.2 is reproduced in Figure 3.3, but with the moveout velocity calculated on a much smaller spread (1000 m instead of 3000 m), reduced to just 1//3 of the distance from the CMP to the reflector. Now the moveout velocity recovered from the traveltimes (solid curve) practically coincides with the analytic solution for the NMO velocity (dotted curve). Therefore, the analytic and numerical results are in good agreement with each other. Equation (3.23) can be transformed into the weak-anisotropy approximation for the SV-wave NMO velocity using the substitutions introduced in Chapter 1 (Vpo Vso, ~ -+ r and ~ -~ 0): Vnmo(r cos r Vnmo (0)

[SV-wave] - 1 - 5a sin 2 r + 3a sin 4 r

(3.24)

3.2.

NMO VELOCITY

FOR

VERTICAL

TRANSVERSE

ISOTROPY

~

>

117

40-

~ 3.5 >

.

O

0

15

30 45 Dip (deg)

60

~3.0

9

i

9

i

9

n

Figure 3.2: Cosine-of-dip corrected P-wave moveout velocity for Cotton Valley shale and the shale-limestone. The solid curve is the moveout velocity calculated from the traveltimes for a spread of 3000 m (the CMP-to-reflector distance is also 3000 m); the dotted curve is the exact NMO velocity computed from equation (3.5); the dashed curve is the weak-anisotropy approximation (3.21). For Cotton Valley shale, Vpo=4.721 km/s, e=0.135, 5=0.205; for the shale-limestone, Vp0=3.306 km/s, e=0.134, ~=0.

5.0 E ~>5, . 8 -

4.5- i 4.09

0

0 > 0 I

I

3.5-

I

Figure 3.3: Same as Figure 3.2, but the spreadlength used to calculate the moveout velocity from t 2 - x 2 curves (solid curve) is 1000 m instead of 3000 m. The analytic curves of the exact NMO velocity (dotted) and the weak-anisotropy approximation (dashed) are unchanged.

118


v

o

E r 4> 3

0

I

I

Dip(deg) Figure 3.4: Influence of Vso on the cosine-of-dip corrected P-wave NMO velocity calculated from formula (3.5). The black curve corresponds to Vpo/Vso - 1.5, the gray curve to Vpo/Vso - 2.5. Vpo=3 km/s, e=0.3, and 5=0.1 are the same for both curves. In the limit of weak anisotropy, the dip dependence of the SV-wave NMO velocity is fully determined by just one anisotropic p a r a m e t e r - a. If a = 0, the medium is elliptical, and the anisotropy does not alter the normal moveout of SV-waves. For typical positive values of a, the leading anisotropic term in equation (3.24) is negative, which implies that the NMO velocity increases with dip slower than in isotropic or elliptical media.

3.2.4

Dip-dependent NMO velocity of P-waves

The dip dependence of P-wave NMO velocity (called the "DMO signature" below) in VTI media is a function of two anisotropic parameters, ~ and 5. Indeed, the vertical velocity Vpo is simply a scaling coefficient for the P-wave phase velocity, if Vpo/Vso, c, and 5 are kept constant. Therefore, the normalized NMO velocity V n m o ( r is independent of Vpo. Another parameter that can be eliminated from the P-wave dipmoveout problem is the shear-wave vertical velocity Vso (or the ratio Vpo/Vso). As demonstrated in Chapter 1, although the P-wave phase velocity formally depends on four Thomsen parameters (Vpo , Vso , s and 5), the contribution of Vso is practically negligible. Figure 3.4 shows that for a wide range of Vso, the corresponding variations of Vnmo for P-waves are insignificant. The weak-anisotropy approximation suggests that the dip-dependent NMO velocity of P-waves should be examined as a function of the difference c - 5 (Figures 3.53.8). Each plot contains the same three types of curves as in Figure 3.2: the moveout velocity calculated from t 2 - x 2 curves for the 3000-m long spread (solid), the exact analytic NMO velocity computed from equation (3.5) (dotted), and the weak-anisotropy approximation for Vnmo given by equation (3.21) (dashed). Comparison between the

3.2.

NMO V E L O C I T Y F O R VERTICAL T R A N S V E R S E I S O T R O P Y

119

first two curves helps to estimate the influence of nonhyperbolic moveout for a typical CMP spread; the difference between the second and third curves shows the error of the weak-anisotropy approximation. The whole suite of plots in Figures 3.5-3.8 suggests that the P-wave DMO signature is controlled, to a significant degree (although not entirely), by the difference c - 5. In spite of certain variations from one pair of (c, 5) to another, the general behavior and range of variation of the moveout velocity are similar for all curves with fixed e - 5 , especially for moderate anisotropies lel < 0.2, ]51 < 0.2. The dominant role of e - 5 is particularly pronounced for the most typical case c - 5 > 0 (Figures 3.7 and 3.8). As demonstrated in Chapter 6, if P-wave NMO velocity is expressed through the ray parameter p of the zero-offset ray (rather than reflector dip), it is dependent on just Vnmo(0) and a specific combination of e and 5 (denoted by 7/) close to the difference e - 5. When c - 5 - - 0 . 1 (Figure 3.5), the cosine-of-dip corrected moveout velocity decreases with dip (for mild dips), in accordance with the weak-anisotropy approximation. For a fixed negative e - 5 , the cosine-of-dip correction actually becomes more accurate (i.e., the curves are closer to unity) with increasing anisotropies e and 5. On the whole, the DMO error, determined by the magnitude of the angular variations in the cosine-of-dip-corrected moveout velocity, is relatively small (the "Cotton Valley shale" case). For elliptically anisotropic models ( e - 5 - 0, Figure 3.6), the anisotropy-induced distortions of the cosine-of-dip dependence are entirely determined by the phasevelocity variations with angle [equation (3.19)]. The DMO error for elliptical anisotropy is moderate: the difference between the corrected moveout velocity and the zero-dip Vnmo for r < 60 ~ lel < 0.2, and 151 < 0.2 is less than 15%. The finite-spread moveout velocity (solid curve) in elliptical media is equal to the analytic NMO velocity (dotted) because reflection moveout is purely hyperbolic. If c - 5 > 0 (the most common case, Figures 3.7 and 3.8), the anisotropy causes a pronounced increase in the cosine-of-dip corrected moveout velocity with dip angle. Even for relatively small c - 5=0.1, the dip-moveout error reaches 25% at 45 ~ dip and 30-35% at a dip of 60 ~ ("the shale-limestone" case). For e - 5=0.2 (Figure 3.8), the corrected moveout velocity at 60 ~ dip is consistently about 60% higher than the zerodip moveout velocity. Thus, for typical VTI media with positive c - 5, the isotropic cosine-of-dip correction severely understates moveout velocities at dips exceeding 20 ~ to 30 ~ even when the anisotropy is weak. The range of dips on the plots above was limited to 60 ~. For typical models with - 5 > 0, curves of the cosine-of-dip corrected moveout velocity tend to flatten out for r > 60 ~ which means that the DMO error at steep dips is practically constant (Figure 3.9). The weak-anisotropy approximation for the NMO velocity is close to the exact solution in the important range of small and moderate values of the parameters c and 5. For lel < 0.2, 151 _ 0.2, to be specific, the error of the weak-anisotropy result, as compared with the exact NMO velocity from equation (3.5), seldom exceeds 5% (the

120

C H A P T E R 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

e-5=-0.1 e=0.1

e--0 ~----0.1

5=0.2

1.2

.

.

o

9

.

o

~ 1.0 . "~ 0.8

e=0.2

e=0.3

5=0.3

5=0.4

1.2 9

.

o

9

o

o

"5 im

m =

~

-

9

_ _ I

.

.

.

.

.

.

.

.

.

.

.

.

.-

9

.

.

.

.

.

.

.

9

~

....

-

.

0.9

.. . . . . . . . : . . . . . . . :. . . . .:-~

~ 1.0 . . . . . . . ; . . . . . . . :

9

.

.......

o

.

.

.

.

.

.

.

.

9 .

o

.

.

.

.

.

.

......

i .......

:: .......

::.......

0.8 Dip (clog)

Figure 3.5: Cosine-of-dip corrected P - w a v e moveout velocity (in k m / s ) for VTI models with e - 5 = - 0 . 1 . The solid curve is the moveout velocity calculated from t 2 - x e curves for a spreadlength of 3000 m (equal to the distance between the C M P and reflector); the d o t t e d curve is the exact N M O velocity from equation (3.5); the dashed curve is the weak-anisotropy a p p r o x i m a t i o n from equation (3.21). On each plot, the vertical P-wave velocity Vpo is adjusted so t h a t the exact analytic Vnmo(0) (dotted curve) is equal to 1 k m / s .

3.2.

NMO VELOCITY

FOR VERTICAL TRANSVERSE

e=0

ISOTROPY

121

e=0.1 ~-0.1

5=0

1.3

1.3

>.,

59 9

,

o

9

o

~

*-.1.1 0 9

9

1.1-

g, 1.o- ~

> 0 1.0

. 9

0.9

,

I

0

60

e=0.2

5=0.2

e=o.3 1.3

.

.

.

5=0.3 .

.

-

.-

......

" . . . . ,b . . . . . . . . .

Figure -

3.6:

5 =0

Cosine-of-dip

(elliptical

corrected

anisotropy).

P-wave

moveout

velocity

for models

:

with

122


e-5=0.1 e=0

~=-0.1 9

e=0.1 .6

o

>,

~=0

-

-

9

o

,,

.r

o

9

o

9

9

9

o

9

o

9

9

o

.

,,

9

o

~ 1.2-

1.2

,

!

l Dip

e=0.2

5=0.1

1.4 9

"

"

(deg)

a=0.3

1.4 "

--

......

,

5=0.2

.: . . . . . . .

i .......

:

:

:

.

:...

~ r. ..

s .

~ 1.2

Figure 3.7: Cosine-of-dip corrected P-wave moveout velocity for models with e - - 5 =0.I.

"

3.2.

NMO

VELOCITY

FOR

VERTICAL

TRANSVERSE

ISOTROPY

123

e-5=0.2 e=0 5=-0.2

e=0.1

6=-0.1

.

......

1.4- ......

1.40

9

.." . . . . . . .

. .......

- .......

:...

9

:

....

~..: .......

0

>

~ -

. i

,

I

e=0.2 5=0 1.8

9

9

:

0

.

I

e=0.3 5=0.1 .

......

~ 1.4

o

-s

: :

.9 . . . . . . .

:. . . . . . . . :

:. . . ~ , :s

s. . ~", "

~5 1 . 4 -

:

0

> 9

I

0

Figure

15

30

3.8" C o s i n e - o f - d i p

c - ~=0.2.

45

corrected

I

I

60

P-wave moveout

velocity for models

with

.

124


e-8=0.2 ~=0.1

~=--0.1

~=0.2

1.8

~=-0

1.8

"6 1.40 0 0>

1.2

1.4-

o,lo--.....

0 0 0>

1.2

Figure 3.9: Cosine-of-dip corrected P-wave moveout velocity for steep reflectors. only exception is the model with e = 0, 5=-0.2). The above suite of plots also displays the moveout-velocity distortions at various dips caused by nonhyperbolic moveout. The influence of nonhyperbolic moveout manifests itself through the difference between the moveout velocity on a finite spread (solid) and the exact analytic NMO velocity (dotted). Only for elliptical anisotropy (e=5) is the moveout purely hyperbolic, and the dotted and solid curves fully coincide with each other. Although for the maximum offset-to-depth ratio of 1, used in the calculations, the contribution of nonhyperbolic moveout to the moveout velocity is not significant, it is clearly visible on some of the plots. It is interesting that the difference between the finite-spread moveout velocity and the analytic NMO velocity changes sign with increasing dip (i.e., the solid and dotted lines cross); moreover, the influence of nonhyperbolic moveout for steep reflectors is typically smaller than that for zero dip (e.g., Figure 3.9). On the whole, for moderate le-51 0 and e -/~ < 0, especially at large dips (r > 45~ Note that after the correction for the cosine of the apparent dip, the moveout velocities for models with the same e - 5 remain close to each other if e - 5 > 0 and become even closer if e - 5 < 0. The only class of models for which the introduction of the apparent dip has a benign influence on the accuracy of the isotropic correction is elliptical anisotropy (c - 5). For elliptical models the correction of moveout velocity with cos r instead of cos r eliminates the DMO error completely (Figure 3.10). Thus, the conventional expression (3.25) is exact for elliptically anisotropic media, of which isotropic models are a special case. This conclusion, verified analytically in Appendix 6A, follows from a more general result of Chapter 6" if NMO velocity is represented as a function of the ray parameter p, it is fully controlled by Vnmo(O ) and the parameter ~ - (e-5)/(1+2~). Influence of v e r t i c a l velocity g r a d i e n t The analysis above was carried out for a homogeneous VTI layer above the reflector. Here we consider the simplest type of vertically heterogeneous VTI m e d i a - the socalled "factorized" model with a constant gradient in vertical velocity. In factorized media, the shapes of the slowness and phase-velocity surfaces are independent of spatial location but their size can vary with location. This means that the velocity Vpo depends on the coordinates, while the Vpo/Vso ratio and the anisotropic coefficients e, 5 and 7 remain constant. Figure 3.12 illustrates the accuracy of the cosine-of-dip correction for a factorized TI medium with a vertical velocity gradient of 0.6 s -1. The moveout velocity is

126


e-5=-0.1

5--~.1

e=0 1.0

e----0.1 ~--0.2

1.0 ...... - ....

o~. . . . . . . 9. . . . . . .

0 >

0 >

0.6

0.6

0

0 Dip (deg)

e--0.1 . 3

-

5=0.1 -

e-5=0

e=0.2

5=0.2

-

"5 . . . . . . . . . . . . . . . :. . . . . ; ;-~::. 9

Figure 3.10- Accuracy of the cosine-of-dip corrections using the true and apparent dip angles for models with e - 5 < 0. P-wave normal-moveout velocity (in km/s) is calculated from equation (3.5) and corrected with the cosine of the true dip angle r (dotted curves, same as in Figures 3.5-3.8) and of the apparent angle r from equation (3.27) (solid curves). In isotropic media, the corrected NMO velocity for all dips is equal to Vnmo(0) that is set to 1 k m / s on all plots.

3.2.

NMO VELOCITY FOR VERTICAL TRANSVERSE ISOTROPY

e=0

8=-0.1

e-8--0.1

1.6

. 6

127

e=0.2 -

8=0.1 -

-

. n

o

1.2-

ii

~ i 0 9> 0

0

1.2

1.0

e=0.1

1.4

~=-0.1

e-8--0.2

1.4

e=0.2

"

8=0

"

""""

..

Figure 3.11: Accuracy of the cosine-of-dip corrections using the true (dotted curves) and apparent (solid curves) dip angles for e - 5 > 0.

128

C H A P T E R 3. N O R M A L - M O V E O U T V E L O C I T Y IN LAYERED A N I S O T R O P I C MEDIA

e-5--o.2

e-5=0.1 1.2

. . . . . . .

o >E1.1 ID

~ o.9 Z

0.8

Figure 3.12: P-wave moveout velocity corrected for the cosine of the true dip angle for VTI models with a velocity gradient of 0.6 s -1. The curves are normalized by the moveout velocity for a horizontal reflector. Each curve corresponds to a different pair of e, 5. On the left plot, ~ = 0, ~ = -0.1 (black); e = 0.1, 5 = 0 (gray); e = 0.2, = 0.1 (dashed). On the right plot, e = 0.1, 5 = -0.1 (black); e = 0.2, 5 = 0 (gray); e = 0.3, 5 = 0.1 (dashed). The distance from the CMP to the reflector and the spreadlength are 3000 m; the rms vertical velocity between the surface and a depth of 3000 m is 3500 m/s. calculated from t 2 - x 2 curves using anisotropic ray tracing. Comparison of Figure 3.12 with Figures 3.7 and 3.8 shows that for typical positive values of e - 5 , deviations from the cosine-of-dip relationship are substantially suppressed by the velocity gradient. It is noteworthy that Larner and Cohen (1993) found a similar "compensation effect" in their study of migration error in factorized transversely isotropic media. When velocity increases with depth, small-offset reflections from dipping interfaces travel more close to the vertical than they do in a homogeneous medium. This makes the "effective dip" of the reflector smaller and reduces the increase in the moveout velocity with dip angle, both in isotropic and anisotropic media. For e - (~=0.1, the influence of vertical velocity variations even leads to "overcorrection," making the cosine-of-dip corrected moveout velocity decrease with dip. Figure 3.12 is reproduced in Figure 3.13 with the correction that honors heterogeneity but ignores anisotropy [V(z) DMO, Larner, 1993]. Although in this case the D MO error caused by the anisotropy is somewhat smaller than in homogeneous media with the same e and 5 (Figures 3.7 and 3.8), it is much larger than the error of the simplest cosine-of-dip correction (Figure 3.12). Therefore, for typical factorized transversely isotropic models the correction that ignores both anisotropy and heterogeneity is often more accurate than the one that honors heterogeneity but ignores anisotropy.

3.2.

NMO VELOCITY FOR VERTICAL TRANSVERSE ISOTROPY

129

e-5=0.2

~-~=0.1 1.6 0

E > "0 1.4

.__. E 1.2 0 Z

Z

10 9

- - -

-

-

~

'

~

_

1.0

,

Figure 3.13: P-wave moveout velocity after are the same as in Figure 3.12.

~--

V(z) D M O

9

9

-

correction. All p a r a m e t e r s

e-5=0.2

e-~=0.1 1.5 O

o1.1

Z

Y

:i

i

Figure^3.14" P-wave moveout velocity corrected with the cosine of the apparent dip angle r All p a r a m e t e r s are the same as in Figure 3.12.

130


Another important conclusion from Figures 3.12 and 3.13 is that in factorized vertically heterogeneous VTI media the P-wave moveout velocity is still primarily controlled by the difference between ~ and ~, rather than by the individual values of these parameters. However, in V(z) media, dip dependence of the moveout velocity is also a function of the velocity gradient, the root-mean-square (rms) vertical velocity, and the depth of the reflector. So far the DMO error in factorized media has been estimated using the true dip angle r As discussed above, conventional processing operates with the apparent dip r given by equation (3.27):

sin r - sin r Ynm~176 0)

,

(3.28)

where the velocities Vnmo and V in FTI media depend on the depths of the zero-offset reflection points z0 (for the horizontal reflector) and Zl (for the dipping reflector). For models with e - 5 > 0, the apparent dip turns out to be smaller than the true one and, consequently, the cosine-of-dip corrected moveout velocity becomes larger (compare Figure 3.14 with Figure 3.12). If e - 5 = 0.2, the introduction of the apparent dip may lead to much higher errors and a noticeable separation of curves corresponding to different pairs of c, 5. For e - 5 = 0.1, the difference between the apparent and true dip angles is somewhat smaller. In analyzing these results, one should keep in mind that the computation of the apparent dip in FTI media using equation (3.28) is strongly dependent on the relative spatial positions of the horizontal and dipping reflectors. The results in Figure 3.14 are obtained for the reflectors located at the same distance from the CMP point. If, instead, we compare the moveout velocities of reflectors at the same zero-offset time, the apparent dip usually becomes much closer to the true one. In section 3.4 we introduce a Dix-type equation for vertically heterogeneous anisotropic media that can be used to unravel the contributions of anisotropy and verticalvelocity variation to the NMO velocity.

3.3

N M O velocity for tilted TI m e d i a

The tilt of the symmetry axis away from the vertical makes the medium azimuthally anisotropic and has a strong influence on normal-moveout velocity. Tilted TI (TTI) models are typical for dipping sediments in fold-and-thrust belts (e.g., in the Canadian Foothills) and near the flanks of salt domes and volcanic intrusions (Isaac and Lawton, 1999; Tsvankin, 1997b). In TTI media, the 2-D NMO equation (3.5) is valid in the vertical plane that contains the symmetry axis and, if the symmetry axis is horizontal, in the (vertical) isotropy plane (see Figure 1.6); the incidence plane is also assumed to coincide with the dip plane of the reflector. For an arbitrary orientation of the symmetry axis with respect to the incidence plane, equation (3.5) can be used only under the assumption of weak azimuthal anisotropy.

3.3.

NMO VELOCITY FOR TILTED TI MEDIA

131

The scope of this section, largely based on the paper by Tsvankin (1997b), is restricted to 2-D analysis of NMO velocity in the dip plane that contains the symmetry axis tilted at an arbitrary angle. To cover all possible mutual orientations of the reflector normal and the symmetry axis, the tilt angle v spans the range - 9 0 ~ < v < 90 ~ while the dip r is restricted to positive angles 0 ~ < r < 90 ~ Positive values of v mean that the axis is tilted towards the reflector, while v < 0 corresponds to the axis tilted away from the reflector (Figure 3.15). The model will again be described by the Thomsen parameters, this time defined in the rotated coordinate system associated with the symmetry axis. As before, Vpo and Vso denote the P- and S-wave velocities along the symmetry axis, which now is no longer vertical. The parameters e and 7 still quantify the difference between the velocities of P- and SH-waves in the (tilted) symmetry direction and the isotropy plane, while 5 determines the second derivative of P-wave phase velocity in the symmetry direction, as in equation (1.49). P-wave moveout in TTI media is controlled by the parameters Vpo , s 5 and the tilt angle ~. The phase angle 0 is measured with respect to the vertical, with the same sign convention as that for the tilt ~. The section starts with a discussion of the unusual phenomenon of possible disappearance of reflections from steep interfaces in homogeneous tilted TI media. Next, for the special case of tilted elliptical media, NMO velocity is obtained as an explicit function of the anisotropic parameter e = 5. Then, the NMO equation for general (non-elliptical) transverse isotropy is simplified by means of the weak-anisotropy approximation. Comparison of the weak-anisotropy solution and exact numerical results elucidates the behavior of dip-dependent NMO velocity for a representative range of homogeneous TTI models.

3.3.1

Absence

of reflections

from

steep

interfaces

Before studying normal moveout, it is necessary to find out whether or not dipping events can be recorded for the full range of dips in TTI media. The wavefront excited by a point source in homogeneous, isotropic media is spherical, with rays orthogonal to the wavefront. This means that for any dip from 0 ~ to 90 ~ there exists a portion of the wavefront parallel to the reflector. This portion generates the normal-incidence (zero-offset) reflection that will be recorded at the source location; likewise, there always exists a specular reflection for non-zero source-receiver offsets. For a vertical reflector that extends all the way to the surface, the raypaths of the reflected waves are horizontal, and the traveltimes in CMP geometry are independent of offset, which implies that the normal-moveout velocity becomes infinite. For any other dip in the range from 0 ~ to 90 ~ NMO velocity has a finite value in accordance with the cosine-of-dip dependence (3.6). Angular velocity variations in anisotropic media distort the shape of the wavefront and the angular distribution of the wavefront normals. For the special cases of vertical (VTI) and horizontal (HTI) in-plane orientations of the symmetry axis, still there are no "gaps" in the dip coverage of reflection data. Since the phase- and group-

132

C H A P T E R 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

velocity vectors coincide with each other in the vertical and horizontal direction, the wavefront in each quadrant contains the full range of phase angles despite all wavefront distortions at oblique angles of incidence (possible cusps on SV-wave wavefronts are not discussed here). Also, as for isotropic media, NMO velocity in VTI and HTI models becomes infinite for a vertical reflector. Indeed, the denominator in expression (3.5) for normalmoveout velocity can be represented as V'(r D - cos r [ 1 - tan r V'(r V(r ] = cos r - sin r V(r

'

(3.29)

where V' - dV/dO. Since for both vertical and horizontal symmetry axes the derivative of phase velocity at r = 90 ~ goes to zero, the denominator D vanishes for a vertical reflector. The situation becomes much more complicated if the symmetry axis is tilted (within the incidence plane) at an arbitrary angle. It is still true that the wavefront from a point source in a homogeneous anisotropic halfspace contains the full 90 ~ range of group (ray) angles in any quadrant, but not necessarily the full range of phase angles that determine the direction of the wavefront normal (Figure 3.15). As a result, for some anisotropic models there are no wavefront normals perpendicular to reflectors having a certain range of dips and, therefore, no corresponding zero-offset and small-offset reflections. Of course, this argument is based on the geometrical-seismics approximation; even in the absence of a specular reflection, the seismogram at the source location will contain some reflected energy that does not travel along the geometrical raypath. However, unless the reflector is close to the source, we cannot expect this non-specular reflection energy to be significant (Tsvankin, 1995c). Figure 3.15 shows a typical P-wavefront for a symmetry axis tilted towards the reflector. Due to the increase in the phase and group velocity away from the symmetry axis, the maximum angle between the wavefront normal and vertical in the lower right quadrant is limited to 0 m a x -- the value corresponding to the horizontal ray. There are no wavefront normals in the angular range 0m~x < 0 < 90 ~ and, therefore, no specular zero-offset reflections for dips larger than Cm~x - 0m~x. This schematic picture is substantiated in Figure 3.16 by the results of anisotropic ray tracing for a TTI medium above a steep reflector. Since the reflector dip in Figure 3.16 exceeds Cma~, all the rays excited by the source propagate downward after the reflection and do not return to the surface. As a result, specular reflections do not exist even for non-zero offsets, and a CMP gather at the surface will not record any specular energy. Note that the tilt of the symmetry axis in Figure 3.16 is relatively mild (20 ~ The model used in Figures 3.15 and 3.16 may be typical for uptilted sediments (such as shales) near the flanks of salt domes because the normal to the bedding (and, hence, the expected symmetry axis of the effective TI medium) is tilted towards the salt body. In this case, one can expect serious problems in imaging of steep segments of the flanks, even if the processing algorithms can handle transverse isotropy.

3.3.

NMO V E L O C I T Y F O R T I L T E D TI MEDIA

~o

133

m

SYMMETRY

2

Figure 3.15: P-wavefront for a transversely isotropic medium with the symmetry axis tilted towards the reflector (v > 0). The increase in phase and group velocity away from the symmetry axis in this model reduces the angular range of wavefront normals in the segment of the wavefront propagating towards the reflector. The maximum phase (wavefront) angle in the lower right quadrant is 0max < 90 ~

134


4 !

Figure 3.16: Rays (white) and wavefronts (black) of the P-wave excited by a point source at the surface and reflected from an interface dipping at 80 ~. The model p a r a m e t e r s are e - 0.25, 5 = 0.05 and ~=20 ~

3.3.

N M O V E L O C I T Y F O R T I L T E D TI MEDIA

135

The maximum dip that would generate a specular reflection can be found from equation (1.15) for the vertical component of the group-velocity vector: Vaz = V cos 0 - V' (0) sin 0.

(3.30)

The ray direction is horizontal if VGz = 0: V'(e) cos 0 - sin 0 ~ = 0.

v(o)

(3.31)

In the absence of cusps, equation (3.31) has two solutions (differing by +180 ~) for the phase angle of the horizontal ray. For typical wavefronts with a monotonic change in the phase angle from the horizontal to the vertical ray, equation (3.31) determines the maximum phase angles for the two lower quadrants. Thus, we can use equation (3.31) to find the largest dip r for which we can record a specular reflection in a homogeneous medium with the phase-velocity function V(O) (assuming that the reflector extends to the surface). In terms of NMO equation (3.5), the absence of phase angles corresponding to a certain range of dips results in zero and negative values of the denominator and, consequently, in infinite or non-existent NMO velocity. To understand the physical meaning of the denominator D [equation (3.29)] of the NMO equation, note that it becomes identical to the left-hand side of equation (3.31) if we substitute 0 = r This implies that D vanishes if the zero-offset ray, which corresponds to the phase-velocity vector normal to the reflector, is horizontal (as for the dip r = 0m~x in Figure 3.15). In other words, NMO velocity becomes infinite if the zero-offset reflected ray (along with nonzero-offset rays) travels along the horizontal (recording) line. For larger dips, as explained above, specular reflections do not exist at all. The magnitude of this unusual phenomenon is illustrated in Figure 3.17 for a typical TI model with an increase in phase velocity away from the symmetry axis. Even a relatively mild tilt of 25 ~ is sufficient to make it impossible to record zero-offset reflections for dips exceeding 76 ~ Note that the dependence of Cm~x on the tilt angle u is asymmetric with respect to u - 45 ~ The dip r reaches its smallest value at tilt angles of 20-30 ~ when the horizontal ray (responsible for Cm~x) corresponds to the most distorted section of the wavefront, characterized by a rapid increase in phase velocity with angle. In contrast, at tilt angles u > 50 ~ the near-horizontal section of the wavefront has a shape much closer to spherical (since the parameter 5, responsible for P-wave phase velocity near the symmetry direction, is small), and the range of "missing" dips is more narrow. Of course, a tilt of the symmetry axis towards the reflector does not cause the disappearance of steep dips for all TTI models. In some TI media, phase velocity as a function of phase angle with the symmetry axis decreases over certain ranges of angles. If, due to the tilt of the axis, the horizontal ray direction corresponds to such a range, the wavefront moving towards the reflector contains all phase angles up to (and even beyond) 90 ~ For instance, this situation occurs for (less typical) models with large 5 (5 >> e) and mild positive tilt angles.

136


8s ~5

8o

70

Figure 3.17: The dip Cmax [calculated using equation (3.31)] of the steepest reflector that generates a zero-offset P-wave ray for TI media with different tilt angles ~. The model is shown in Figure 3.15; e -- 0.25,/~ -- 0.05.

An example of a wavefront that contains the full range of phase angles in the quadrant of interest is shown in Figure 3.18. Although the wavefront in Figure 3.18 has the same shape as that in Figure 3.15, the symmetry axis is now tilted away from the reflector; we would have had the same situation in Figure 3.15 had the reflector been located to the left of the source. For the case displayed in Figure 3.18, the phase-velocity vector of the horizontal ray [Smax is a solution of equation (3.31)] points upwards, and it is possible to record specular reflections not only from any dip in the 0-90 ~ range, but also for dips between 90 ~ and (/)max - - 0max, if the aperture is sufficient. We conclude that anisotropy can make it possible to record reflections from overhang structures even in the absence of a velocity gradient. In the above discussion it was assumed that the medium above the reflector is homogeneous. Some of our conclusions can be extended in a straightforward way to factorized anisotropic media (see section 3.2) with a vertical velocity gradient. In factorized models the ratios of the elastic constants (and Thomsen anisotropic parameters) are independent of spatial position and, consequently, the shape of the slowness surface and the relationship between the phase and group velocity remains the same throughout the medium. Hence, for factorized TI media with the same shape of the wavefront as that for the model in Figure 3.15, the downgoing wavefield in the lower right quadrant still cannot contain phase (wavefront) angles exceeding 0max (Figure 3.19). However, the upgoing wavefield formed due to the ray bending does include the missing phase angles in the range 0max < 8 < 90 ~ (as well as phase angles beyond 90 ~ Therefore, in such a medium zero-offset reflections for dips (/)max < r < 90 ~ represent turning rays that exist only for appropriate spatial positions of the reflector with respect to the source.

3.3.

NMO V E L O C I T Y F O R T I L T E D TI MEDIA

:

137

~

I

0 SYMM

A, Figure 3.18: The wavefront for the same TI medium as in Figure 3.15 but with the symmetry axis tilted away from the reflector. The direction of the wavefront normal in the lower right quadrant not only spans the full 0-90 ~ range but also includes angles beyond 90 ~ The angle ~ between the symmetry axis and vertical is taken to be negative if the symmetry axis is tilted away from the reflector.

138


g

..y"

Figure 3.19: Reflections from steep interfaces in a factorized TI medium with constant and ~ and an increase in the vertical velocity with depth. The shape of the wavefront at any depth is the same as in Figure 3.15. In this case, specular zero-offset reflections from steep dips with r > Cm~x represent turning rays.

3.3. NMO VELOCITY FOR TILTED TI MEDIA

139

v=60 ~

V=30 ~ 9

~

~

9

~

,

O

E

El.0 Z

60

0.9

9

~

9

9

~

9

~

9

~

I

I

~

i

Dip angle (degrees)

Figure 3.20: The influence of nonhyperbolic moveout on P-wave moveout velocity for a model with e = 0.2 and 5 = 0.03 and two tilt angles (v = 30 ~ and v = 60~ The solid curve is the effective moveout velocity calculated from the exact traveltimes (obtained by ray tracing) for the spreadlength equal to the distance between the CMP and the reflector; the dotted curve is the exact NMO velocity computed from equation (3.5). Both curves are normalized by V n m o ( 0 ) / c o s ~ ) , where Vnmo(0) is the exact NMO velocity from a horizontal reflector.

3.3.2

Dip-dependent P-wave NMO velocity

A c c u r a c y of the hyperbolic e q u a t i o n In non-elliptical TI media, reflection moveout is nonhyperbolic even for a homogeneous medium above the reflector. Therefore, although NMO equation (3.5) can be applied for any tilt angle and any strength of the anisotropy, stacking (moveout) velocity determined on finite spreads may deviate from the "zero-spread" NMO velocity. However, as was shown above for vertical transverse isotropy, the magnitude of anisotropy-induced P-wave nonhyperbolic moveout on conventional spreadlengths (close to the CMP-to-reflector distance) is small and, moreover, decreases with reflector dip. Figure 3.20 demonstrates that this conclusion remains valid in TI media with a tilted symmetry axis as well. The finite-spread moveout (stacking) velocity in Figure 3.20 was calculated by fitting a straight line to the exact traveltimes (t 2 x 2 curves) over the spread equal to the distance between the CMP and reflector. Evidently, despite the pronounced anellipticity of the TI model used in Figure 3.20 ( e - 5 - 0.17), the finite-spread moveout velocity is close to the analytic NMO value [equation (3.5)] for the whole range of dips.

140


Elliptical anisotropy Due to the relative simplicity of NMO equations for elliptical anisotropy (c - 5), it is instructive to examine this special case separately. If the symmetry axis lies in the incidence plane and makes the angle v with the vertical, the P-wave phase-velocity function is given by Vp(O) -- VPo r

+ 25 sin 2 0,

(3.32)

0 - 0 - v. The derivatives of the phase velocity needed to evaluate the NMO velocity (3.5)are obtained from equation (3.32)as dVp(O) Vpo 5 sin 20 dO = V~,(O) - (1 + 25 sin 2 ~)1/2 '

(3.33)

cos 20 - 26 sin4 V~(O) = 2Vpo6 (1 + 25 sin 2 ~})3/2 "

(3.34)

Substituting equations (3.32)-(3.34)into equation (3.5) yields

cos r

cos r

.

(3.35)

Equation (3.35) coincides with the NMO equation by Uren et al. (1990b), obtained using a different approach and presented in a different notation. For a vertical symmetry axis ( v - 0) equation (3.35) reduces to the VTI expression (3.19)" Vnmo(r )

__

Vpo v/1 + 25 r

+ 25 sin 2 r

(3.36)

COS (~ or

Vp((~) . (3.37) Vnmo ( 0 ) VPO As discussed above, if the elliptical axes are not tilted, the error in the cosine-of-dip dependence of Vnmo is determined directly by the phase-velocity variations. For a horizontal reflector (r = 0), equation (3.35) becomes Vnmo ( r

Vnmo(0) --

c o s (~

=

v/1 + 25 = Vp0 VPO V/1 + 25 sin2v

~/ 1 +

25 cos 2 v . 1 + 25 sin2v

(3.38)

Let us compare the NMO velocity from equation (3.38) with the horizontal phase velocity Vho r - - Vpo 2d cos2 v. (3.39) Equations (3.38) and (3.39) confirm the well-known a vertical (or horizontal) elliptical axis (u = 0 ~ or v = horizontal reflector is equal to the horizontal velocity. strictly true for elliptical models with tilted axes, the

fact that for a model with 90~ NMO velocity from a Although this is no longer difference between the two

3.3. NMO VELOCITY FOR TILTED TI MEDIA

141

velocities is small since they coincide in the weak-anisotropy approximation. Indeed, for weak anisotropy (151 z are often muted out in conventional processing. This chapter gives an analytic description of long-spread, nonhyperbolic moveout in horizontally layered anisotropic models, with an emphasis on P-waves in VTI media. A natural way to account for deviations from hyperbolic moveout, adopted here, is to add higher-order terms to the quadratic Taylor series for the squared traveltime and, ultimately, to modify the series itself. An alternative approach, based on equivalent-medium theory, was developed by Dellinger and Muir (1993). 173

174

C H A P T E R 4.

E ,,

NONHYPERBOLIC REFLECTION MOVEOUT

2

o

>3.5

Figure 4.1" Phase velocities of P- and SV-waves for Taylor sandstone (Thomsen, 1986). The model parameters are V p o - 3.368 km/s, V s o - 1.829 km/s, e - 0.110, 5 - -0.035 (a - 0.492).

2-

I I v

E 0

0

ffl rr

--2-

x/z Figure 4.2" Difference between the exact traveltimes and best-fit hyperbola (residual moveout) in a horizontal layer of Taylor sandstone (Figure 4.1). The spreadlength is equal to the reflector depth, Xma~ - z - 3 km.

175

80-

60-

E

v

E E

o~

2o-

X

-20

Figure 4.3: Maximum residual moveout in a layer of Taylor sandstone as a function of the spreadlength-to-depth ratio Xma~/Z;Z -- 3 km.

% % %

I

"

I

Figure 4.4: Finite-spread (effective) moveout velocity of the best-fit hyperbola normalized by the NMO velocity; the model is Taylor sandstone.

176

4.1

C H A P T E R 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

Quartic m o v e o u t coefficient

Including the quartic (fourth-order) term in the traveltime series (3.1) yields a threeterm representation of the moveout function: x2 t ~ - - t02 -+-

" -4- n 4 x 4

Vnmo

(4.1)

Clearly, the quartic moveout coefficient A4 is the key parameter responsible for the magnitude of nonhyperbolic moveout. Analytic expressions for A4 given in this section help to evaluate the applicability of the series (4.1) and to introduce a more accurate moveout approximation that does not become divergent even at infinitely large offsets. Application of the least-squares method to the quartic expansion (4.1) shows that the finite-spread moveout velocity should be close to a quadratic function of the spreadlength Xmax. This dependence, which explains the shape of the curves in Figure 4.4, was first described by A1-Chalabi (1974). 4.1.1

General

2-D equation

for a single layer

Suppose a pure-mode reflection is recorded on a CMP line confined to a vertical symmetry plane of a horizontal anisotropic layer. It is also assumed that the horizontal plane is a plane of mirror symmetry. Using the definition of the quartic coefficient

A4, A 4 - 5 d(x 2) d(x 2) z=0' and the approach outlined in the derivation of the 2-D NMO equation (Appendix 3A) yields the following exact expression: A4 =

-

4D2 + 3D~ + D 4 12 t~ V4 (1 + D2) 4"

(4.3)

The coefficients D2 and D4 are expressed as

D2 :

and

D4 -

1 d2V Vo dO2

(4.4) 0=0

1 d4V Vo d0 4

(4.5) 0=0

where V(0), as before, is the phase velocity as a function of the phase angle 0 with the vertical; V0 - V(0). Derivation of equation (4.3) can be found in A1-Dajani and Wsvankin (1998), who also obtained the azimuthally dependent quartic moveout coefficient for an HTI layer. Less general expressions for A4 restricted to VTI media were given by Hake et al. (1984) and Wsvankin and Thomsen (1994).

177

4.1. QUARTIC MOVEOUT COEFFICIENT

If the horizontal plane is not a plane of symmetry, the specular reflection point would change with offset, and the reflection-point dispersal would influence the value of A4. It should be emphasized that whereas NMO velocity is independent of reflectionpoint dispersal, this is not the case for nonhyperbolic moveout. Therefore, equation (4.3) is valid in the vertical symmetry planes of VTI, HTI and orthorhombic media (provided the orthorhombic model has a horizontal symmetry plane). Equation (4.3) shows that the contribution of anisotropy to the quartic moveout coefficient is absorbed by just two velocity terms (D2 and D4) - the second and fourth derivatives of phase velocity at vertical incidence. For a purely isotropic layer phase velocity is constant (D2 = D4 = 0), and the quartic moveout coefficient vanishes. Nonhyperbolic reflection moveout in isotropic media is caused entirely by vertical and lateral heterogeneity.

4.1.2

Explicit expressions for V T I media

To find the quartic coefficient for P- or S-waves in a given model, it is necessary to determine the phase velocity and its derivatives in the vertical direction. For P-waves in VTI media, d2V/dO2 is given by equation (1.49), while D4 can be obtained from equation (1.59): 02 = 25; (4.6) 04 = 24 (~ - 5)(1 + 25/f) - 45 (2 + 35), f - 1-

(4.7)

V o/V2,o.

Substituting equations (4.6) and (4.7) into the general expression (4.3) leads to an explicit form of the P-wave quartic moveout coefficient: 2 ( e - ~)(1 + 28/f) t2poV4o (1+ 26) 4 "

A4,p = -

(4.8)

For SV-waves, using the phase-velocity equation (1.59) with a minus sign in front of the radical yields D 2 = 2(7,

and D4 = - 2 4 a (1 + 26/f) - 4a (2 + 3a). The quartic coefficient (4.3) then becomes 2a (1 + 23/f) -

(4.9)

(1 +

For the elliptical SH-wave with phase velocity described by equation (1.54), the quartic coefficient goes to zero: A4,SH = 0 . (4.10) It can be shown that reflection moveout in elliptical media with any orientation of the symmetry axis is purely hyperbolic (Uren et al., 1990b). Not surprisingly, A4 for

178

CHAPTER 4. NONHYPERBOLIC REFLECTION MOVEOUT

both P- and SV-waves is proportional to the difference e - 5, i.e., to deviations from the elliptical model [equations (4.8) and (4.9)]. The equivalence between the symmetry planes of orthorhombic media and transverse isotropy (see section 1.3 of Chapter 1 and section 3.5 of Chapter 3) can be used to adapt the VTI equations (4.8)-(4.10) for the vertical symmetry planes of a horizontal orthorhombic layer. For instance, if a CMP line is confined to the [xl, x3] symmetry plane, the quartic term A4 for P-waves has the form of equation (4.8), but with the e and 5 coefficients defined in equations (1.93) and (1.94). Equations (4.8) and (4.9) can be simplified using the weak-anisotropy approximation linearized in the anisotropic coefficients: AW _ _ 2(e - 5) 4,P

Aw

(4.11)

t ~ 0 g/40 ,

2a

-

4,SV -- t2so V ~ 0 9

(4.12)

In the equations for the quartic moveout term, however, such a linearization may be highly inaccurate because the neglected terms in the denominator [(1 + 25) 4 and (1 + 2a) 4] become relatively large for 5 and a as small as 0.03-0.05. Since often a >> 5, the weak-anisotropy approximation is less suitable for the SV-wave than for the P-wave. In fact, even e - 5 = 0.02 is "strong" anisotropy for the SV-wave since this may mean a = 0.08 and (1 + 2a) 4 = 1.81! An alternative way of simplifying the expression for A4 for P-waves without reducing the accuracy is to neglect the contribution of the shear-wave vertical velocity by setting Vso in equation (4.8) to zero. As shown in Chapter 1, all kinematic signatures of P-waves are practically independent of Vso. In equation (4.8), as in any other P-wave kinematic signature, Vso influences only terms quadratic in c and 5. Numerical testing (see Figure 4.10 below) proves that the shear-wave vertical velocity can be dropped from the expression for A4 without degrading the quality of the nonhyperbolic moveout equation. For Vso = 0 (f = 1), equation (4.8) reduces to 2(~ - 5) 2q A4,p = - t2po V~0(1 + 25)3 = - t~ ~ P2mo,

(4.13)

where Vnmo ~ Vnmo(0) = Vpo v/1 + 25 is the NMO velocity from a horizontal reflector and r/is the "anellipticity" coefficient defined by Alkhalifah and Tsvankin (1995) as ~-

~-5 1+25

.

(4.14)

The significance of the parameterization used in equation (4.13) and, in particular, of the coefficient r/goes far beyond the quartic moveout term and the nonhyperbolic moveout equation. As proved in Chapter 6, Vnmo and r/ fully control all P-wave timeprocessing steps (NMO and DMO correction, prestack and poststack time migration) in vertically heterogeneous VTI media.

4.1. QUARTIC MOVEOUT C O E F F I C I E N T

179

2.3

O {D t~ t'--

1.9

/

0

Figure 4.5" Phase velocities of P- and SV-waves for Dog Creek shale (Thomsen, 1986). The model parameters are Vpo - 1.875 kin/s, Vso - 0.826 kin/s, c - 0.225, 5 - 0.1 (a - 0.644). Hence, for P-waves the magnitude of A4 is governed by the parameter r], which means that there is no direct correlation between the strength of velocity anisotropy (usually quantified by c) and nonhyperbolic moveout. Furthermore, if c or e - 5 is fixed, the value of A4 becomes larger if ~ d e c r e a s e s within a conventional range. Therefore, the individual values of c and/~ by themselves cannot serve as a guide to evaluating deviations from hyperbolic moveout for a particular model. Anisotropy is a multidimensional problem, manifesting itself in different ways in different contexts, and a casual inspection of anisotropic coefficients can be dangerously misleading. Figure 4.6 illustrates the magnitude of nonhyperbolic P-wave moveout for two VTI models - Taylor sandstone (Figure 4.1) and Dog Creek shale (Figure 4.5). The residual moveout in Figure 4.6 is computed as the difference between the exact traveltimes and the best-fit hyperbola. Since the absolute values of e and 5 for Dog Creek shale are much higher than those for Taylor sandstone, one would be inclined to describe Dog Creek shale as the far more "anisotropic" material with respect to P-wave propagation. While this assessment is valid for phase velocity, anisotropy-induced deviations from hyperbolic moveout (Figure 4.6) are much more pronounced for the more "anelliptical" model of Taylor sandstone that has a larger value of r / = 0.16 (for Dog Creek shale, ~ = 0.1). The character of the nonhyperbolic moveout for SV-waves strongly depends on the sign of the parameter a - ( V p o / V s o ) 2 ( E - - 5 ) . If a > 0, the quartic coefficient (4.9) reaches a maximum near a = 0.17 and then decreases at larger a. For a > 0 the contribution of A4 to the traveltime series is often smaller for the SV-wave than for the P-wave. SV-wave moveout for positive a starts to deviate substantially from a hyperbola only for offsets x > 1.5z (Figure 4.3). In contrast, if a < 0, the influence of the term (1 + 20) 4 in the denominator of equation (4.9) leads to large absolute values of A4 and strongly nonhyperbolic

180

CHAPTER 4. NONHYPERBOLIC REFLECTION MOVEOUT

0.4

"~

-0.2 -

rr" -0.4-

-0.6

Js

'

'

I

2

x/z Figure 4.6: Residual P-wave moveout (normalized by the vertical time tpo) for the models of Taylor sandstone and Dog Creek shale. The spreadlength is twice as large as the reflector depth, Xm~x -- 2Z. moveout. For a - -0.5, the quartic coefficient becomes infinite, the NMO velocity goes to zero, and the Taylor series expansion for the S V traveltime breaks down altogether (see a numerical example below in Figure 4.11).

4.1.3

Layered media

In stratified media, the quartic moveout coefficient reflects the combined influence of layering and anisotropy. The analytic expressions below are given for 2-D wave propagation under the same assumptions as those for homogeneous media: the CMP line is confined to a throughgoing vertical symmetry plane of a horizontally layered anisotropic medium, and each layer is taken to have a horizontal symmetry plane. For this 2-D model comprised of N layers, the exact coefficient A4 for pure-mode reflections is given by

A4 = [EiNl(V(i)m~ 4 [r~

g -- tO Y~i=l(Vn(2o) 4 (Vn(~o)2 t~i)] 4

t~i) + to ~iN1 A~i) (v(i)mo)S(t~i)) a ,

(4.15)

[Y~iN-_l(V(/m)o)2 t~i)] 4

where t~i) is the interval vertical traveltime, and Vn(~o and A~i) are the NMO velocity and quartic moveout coefficient in layer i. Originally, equation (4.15) was derived by Hake et al. (1984) and Tsvankin and Thomsen (1994) for vertical transverse isotropy, but it remains valid for the more general 2-D model treated here. N (Vn(/m)o)2 t~i) is equal to the product of the squared effecNote that the sum 52i=1 tive NMO velocity Vn2moand the vertical traveltime to [equation (3.49)]. Therefore,

4.1.

QUARTIC MOVEOUT C O E F F I C I E N T

181

equation (4.15) can be rewritten as A4

=

V4moto -- ~iNl(V(/m)o)4 t~i) 4 V:mo toa +

~g A~i) )8 )3 i=1 (Vn(2~ (t~i) VnSmot]

(4.16)

The first term on the right-hand side of equations (4.15) and (4.16) has the same form as the corresponding expression for layered isotropic media (Taner and Koehler, 1969; A1-Chalabi, 1974). In contrast to isotropy, however, it contains the interval normal-moveout velocity which is different from the true vertical velocity and, in general, from the phase velocity measured in any other direction. The magnitude of this term, which vanishes in homogeneous media, depends on the contrasts in NMO velocity across layer boundaries (i.e., on the vertical heterogeneity). The second term goes to zero in isotropic or elliptically anisotropic media where A~i) - 0 and, therefore, represents a purely anisotropic contribution to the quartic moveout coefficient. According to the Cauchy-Schwartz inequality, the first term is always non-positive (Hake et al., 1984). The second term may be either positive or negative depending on the signs of the interval quartic coefficients A~i). For typical VTI media with e - 6 > 0 (a > 0; r / > 0), the second term is negative for P-waves [equation (4.8)] and positive for SV-waves [equation (4.9)]. This means that vertical transverse isotropy with positive anellipticity reinforces the contribution of layering to nonhyperbolic moveout for P-waves and mitigates or even overrides it for SV-waves. For the elliptical SHwave in VTI media, all interval A~i) - 0, and the effective A4 is fully determined by the first ("layering") term. Due to the presence of (Vn(~o)s in the second term, the effective quartic coefficient for the SV-wave in VTI media does not become infinite even if 1 + 2a (i) = 0 and A~~) - c 1.5z. This is not surprising because the series (4.1) is based on the shape of the slowness surface near the vertical and cannot account for changes in velocity at oblique incidence angles. Extending the series beyond the quartic term is hardly practical and, moreover, would provide an improvement only for a limited range of offsets. A more accurate moveout approximation may be obtained by modifying the expansion (4.1) so that it

4.2. NONHYPERBOLIC MOVEOUT EQUATION

183

converges towards the exact traveltime not just for x --+ 0, but also at large offsets approaching infinity. This approximation, marked by the solid curve in Figure 4.7, is developed below. 4.2.1

Weak-anisotropy

approximations

Even in the simplest model of a single horizontal anisotropic layer, closed-form expressions for reflection traveltime in terms of the anisotropic parameters cannot be derived without imposing certain restrictions, such as small offset-to-depth ratios, weak anisotropy or elliptical anisotropy. Here, before discussing a general moveout equation, we introduce and analyze the weak-anisotropy approximation for long-spread traveltime. As shown in Appendix 4A, P-wave moveout in a weakly anisotropic VTI layer can be represented in the following form: t~ - t~0 + A w x 2 2,P

AW x4

4,P

(4.20)

-'[- 1 + x2/(Vpo tpo) 2 '

where A w w [equation (4.11)] are the linearized 2,P = 1/Vn2mo -- ( 1 - 25)/V~o and A 4,P quadratic and quartic coefficients of the Taylor series expansion of t2(x2). The same functional form, but with the vertical traveltime tso, velocity Vso and the appropriate coefficients A w and A w, is valid for SY-waves. While equation (4.20) becomes equivalent to the quartic Taylor series at small x, it has the important advantage of being asymptotically correct at infinitely large offsets. Indeed, for x --+ c~ equation (4.20) reduces to a hyperbola parameterized by the horizontal velocity [equation (4.40)]. Formula (4.20) is not the only possible weak-anisotropy expression for reflection moveout in VTI media. Byun et al. (1989) and Byun and Corrigan (1990) suggested the so-called "skewed" hyperbolic moveout equation for long-spread P-wave traveltimes in VSP (vertical seismic profiling) geometry:

t ~ -- t20 -[-

z

~-

-~

x

2Yhor

Z 2 "[-(X/2) 2

.

(4.21)

The parameters V~ and Vhor, which have the meaning of the NMO and horizontal velocities, are supposed to be found by least-squares or semblance search for the known reflector depth z. Sena (1991) derived analytic expressions for V~ and Vhor under the assumption of weak anisotropy (defined slightly differently than here). The quartic Taylor series coefficient for P-waves corresponding to approximation (4.21) was given by Sena (1991) in the present notation as A4 P (Sena) - - 2(__~e- ~)

'

1

t~0 V40 (1 + 2fi)(1 + 2e)"

(4.22)

The linear term in Sena's result (4.22) coincides with the weak-anisotropy approximation (4.11); the additional term [(1 + 25)(1 + 2e)] -1 appears due to his definition of weak anisotropy. The quartic coefficient (4.22) is substantially different from the

184


exact expression (4.8) because it does not contain the correct terms (1 + 2(~)4 and (1 + 2~/f). Thus, despite the obvious advantages of the approximations (4.20) and (4.21) over the quartic Taylor series, the inaccurate expressions for the quartic moveout term hamper their application. These equations can still be used for long-spread moveout correction (but not for parameter estimation) if the moveout parameters are treated as fitted coefficients to be recovered from the measured traveltimes or from semblance analysis of seismograms.

4.2.2

General long-spread moveout equation

A better analytic approximation for reflection traveltime may be obtained by combining the functional form of t 2 found for weak anisotropy [equation (4.20)] with the exact Taylor series coefficients and horizontal velocity. This idea leads to the long-spread moveout equation suggested by Tsvankin and Thomsen (1994): x2 t2 - t~ + ~

A4 X4 + 1 + Ax 2'

(4.23)

where A

-

A4

.

-~ m Vnmo

'

(4.24)

Vhor = V90 is the horizontal velocity. At small offsets, the moveout approximation (4.23) reduces to the exact quartic Taylor series. Traveltimes on short spreads limited by reflector depth are primarily determined by the NMO-velocity (hyperbolic) term. The addition of the denominator to the quartic moveout term does not change the fact that the coefficient A4 remains largely responsible for the magnitude of nonhyperbolic moveout. The term Ax 2 makes equation (4.23) more accurate than the quartic Taylor series at intermediate and large offsets. Expression (4.24) for the coefficient A follows from the expansion of t 2 at large x2: ~+--~

+

t02-~Z

+...

(4.25)

To ensure the correct asymptotic behavior of t 2, the moveout velocity at infinitely large offsets should approach the horizontal velocity Vhor. Hence, the mutliplier of the x2-term in equation (4.25) should be related to the horizontal velocity as 1/Vh2or, which yields equation (4.24). This definition of the parameter A makes the traveltime converge at infinitely large offsets. Therefore, by design, equation (4.23) is accurate at both small and large offsets. In principle, the Tsvankin-Thomsen expression is still an approximation that can deviate from the exact traveltime curve in the intermediate offset range. However, as illustrated by Figure 4.7, equation (4.23) provides an excellent fit to the P-wave

4.3.

P - W A V E M O V E O U T IN VTI M E D I A IN T E R M S O F T H E P A R A M E T E R r/

185

moveout in a VTI layer, even for uncommonly long offsets and substantial anisotropyinduced nonhyperbolic moveout. In Chapter 5 it is shown that equation (4.23) gives comparable accuracy in describing reflection moveout of mode-converted waves. The importance of the moveout equation (4.23) is by no means limited to a single VTI layer. In fact, the form (4.23) is so general that it can be applied to virtually any isotropic or anisotropic model, including layered azimuthally anisotropic media (A1-Dajani and Tsvankin, 1998; A1-Dajani et al., 1998). The main issue in implementing equation (4.23) for a particular model is to obtain the three needed moveout parameters- the NMO velocity Vnmo, the quartic moveout coefficient A4 and the horizontal velocity Vho~. Of course, this is not an easy task for stratified, low-symmetry anisotropic models. Figures 4.8 and 4.9 demonstrate that the nonhyperbolic moveout equation (4.23) accurately describes P-wave traveltimes in layered VTI media. In spite of the pronounced anisotropy in the two bottom layers (they have the parameters of Dog Creek shale and Taylor sandstone) and strongly nonhyperbolic moveout of the corresponding reflections, approximation (4.23) remains close to the exact traveltimes at offsets three times as large as the reflector depth. The effective NMO velocity for the theoretical curves in Figure 4.9 was calculated from the Dix equation by rms averaging of the interval NMO velocities [equation (3.49)]. Likewise, rms averaging of the interval horizontal velocities was used to find the effective horizontal velocity Vhor needed to determine the parameter A [equation (4.24)]. The quartic moveout coefficent was obtained from equation (4.16) with the interval values of A4 computed using the exact equation (4.8). In the next section, we show how the nonhyperbolic term in equation (4.23) for P-waves in VTI media may be represented as a function of the interval values of Vnmo and the anisotropic parameter r/. On the whole, moveout equation (4.23) offers a fast and efficient way to model long-spread traveltimes in anisotropic media and to evaluate deviations from hyperbolic moveout without doing ray tracing. Also, it can be applied in moveout correction on long-spread gathers, refining the more conventional approach based on the quartic Taylor series (Gidlow and Fatti, 1990). In Chapter 7, a modification of equation (4.23) is used in the inversion of large-offset P-wave traveltimes for the anisotropic parameters of VTI media.

4.3

P - w a v e m o v e o u t in V T I m e d i a in t e r m s of the parameter

The higher accuracy of equation (4.23) compared to the quartic Taylor series is achieved at the expense of adding a third moveout p a r a m e t e r - A. Obviously, a three-parameter (Vnmo, A4, A) search at each vertical time is a much more timeconsuming procedure than is the conventional semblance analysis operating with just moveout (stacking) velocity. For P-waves in VTI media, however, it is possible to

186

CHAPTER4. NONHYPERBOLICREFLECTIONMOVEOUT

0.5 km

Vpo = 2.9

5= e

Figure 4.8: The layered VTI model used to test the analytic approximation (4.23) for P-wave traveltime. The two bottom layers have the anisotropic parameters of Dog Creek shale and Taylor sandstone.

_

*" 2 - J

,

Figure 4.9: Comparison between the nonhyperbolic moveout equation (4.23) (solid curves) and the exact P-wave traveltimes for the primary reflections from each boundary of the model in Figure 4.8. The shallowest layer is isotropic, so the corresponding reflection has purely hyperbolic moveout.

4.3. P-WAVE MOVEOUT IN VTI MEDIA IN TERMS OF THE PARAMETER 77

187

reduce the number of moveout coefficients to two by expressing the traveltime (4.23) through the zero-dip NMO velocity and the anellipticity parameter r/.

4.3.1

Single layer

Using the definition (4.14) of r/, the P-wave horizontal velocity can be rewritten in the following way (the subscript "P" will be omitted)" Vhor-

Vp 0 4 1 -~- 2(~ - Vnm o 41 + 27/.

{4.26)

Substituting approximation (4.13) for the quartic moveout term and equation {4.26) into equation (4.24) for A gives A = 2_______~. 1+ t ~ vnmo 2

(4.27)

Then the nonhyperbolic moveout equation (4.23) for P-waves in a single VTI layer takes the form x2 27/x 4 t2 -- t2 -~- Vn2m----~-

Vn2mo It 2 Vn2mo -4- (1 -+- 2?7) x2] "

(4.28)

Thus, P-wave long-spread moveout can be described by just the vertical traveltime, Vnmo and r/, with no separate dependence on the Thomsen parameters Vpo, ~ and 5. For given tpo and Vnmo, r/determines the magnitude of deviation from hyperbolic moveout; if r/ - 0, the medium is elliptical and the moveout is purely hyperbolic. Equation (4.28) remains valid for CMP lines in the vertical symmetry planes of an orthorhombic layer (see sections 1.3 and 3.5) with substitution of the appropriate NMO velocity and coefficient r/. Equivalently, introducing the P-wave horizontal velocity Vhor [equation (4.26)] instead of r/, equation (4.28) can be rewritten as t 2 - t~ + ~

X2

-

(P2or-

Vn2mo)x 4

Vn~mo Vn~mo(to~ V2mo+ V~orX~)"

(4.29)

Equation (4.29) is somewhat easier to use in moveout analysis because both parameters (Vnmo and Vhor) have the dimension of velocity. Also, as demonstrated in Chapter 7, Vhor is better constrained by long-spread P-wave traveltimes than is r/. Although the general equation (4.23) is approximate, and we have made one more small approximation by setting Vso to zero in the expression for the quartic moveout coefficient, the accuracy of equations (4.28) and (4.29) is rather high. Figure 4.10 displays the maximum difference between equation (4.29) and the exact (ray-traced) P-wave moveout in a typical VTI model computed for a range of the NMO and horizontal velocities; the spreadlength is equal to twice the reflector depth. For the actual model parameters ( V n m o - 2.0 km/s, Vhor -- 2.3 km/s) which correspond to the center of the plot, the maximum deviation of approximation (4.29) from the exact traveltime is less than 6 ms (just 0.6% of to).

188

C H A P T E R 4.


1.9 I

I

I

2.5Figure 4.10: The influence of V n m o and Vho r o n the P-wave reflection traveltime in a horizontal VTI layer. The contours display the maximum difference Atmax (in ms) between the traveltimes for a reference model (computed using ray tracing) and models with the same to but different Vnmo and Vhor shown on the axes [computed from equation (4.29)]; the spreadlength X m a x = 2 km (Xmax/Z - - 2). The model parameters are Vnmo = 2.0 k m / s and Vhor = 2.3 km/s; the corresponding r / = 0.16. Thomsen parameters are Vpo = 2.0 km/s, e = 0.16, 6 - 0; the reflector depth z - 1 km, to = 1.0 s. If equation (4.29) were exact, Atma~ for the actual model parameters (at the center of the plot) would go to zero.

4.3.

P - W A V E M O V E O U T IN V T I M E D I A IN T E R M S O F T H E P A R A M E T E R 77

189

Note, however, that the center of the contours is somewhat shifted from the correct position due to the approximate character of equation (4.29), and the best-fit ~, which corresponds to the model with the smallest time residual, is equal to 0.13 instead of the actual 0.16. Also, for the same reason the traveltime difference Atma~ for this best-fit model is not equal to zero. The error increases in VTI media with more pronounced nonhyperbolic moveout (larger r/), and for a medium with r / = 0.3 the best-fit value is as low as 77 = 0.24 (Grechka and Tsvankin, 1998a). Translation of small deviations of equation (4.29) from the exact traveltimes into sizable errors in ~ is an indication of the relatively low sensitivity of moveout on spreads Xma~ ~ 2z to this parameter. This result, which has serious implications for moveout inversion, is discussed in more detail in Chapter 7. An extensive analysis of equation (4.29) and its application to parameter estimation in VTI media can also be found in the work by Alkhalifah (1997b). 4.3.2

Layered

media

The example in Figure 4.9 shows that the original version of the nonhyperbolic equation (4.23) with the exact quartic moveout coefficient A4 accurately describes longspread P-wave moveout in a stack of horizontal VTI layers. It is more attractive, however, to explore the possibility of extending the simplified two-parameter moveout equation to layered VTI models. According to the analysis in Appendix 4B, P-wave reflection traveltime from the bottom of layer N can be found in the form of equation (4.28): t2(N) -

t2(N) +

x2 2r/(N) x 4 Vn2mo(N) - Vn2mo(N){t~(N) Vn2mo(N) + [1 + 2~(N)] x2} '

(4.30)

where the NMO velocity Vnmo(N) is obtained from the Dix equation and r/(N) is an effective parameter expressed through the interval values Vn(/m)o, r](i) and t~i) as

n(N)-

g

V~mo(N) to (N)

(V~(2o)4 (1 + St/(~)) t~') - 1 .

(4.31)

i=1

The parameter r/(N) in equation (4.31) does not depend on the shear-wave velocity Vso, which is set to zero in all layers. For a given vertical time to(N), equation (4.3o) contains just two moveout parameters, Vnmo(N) and ~(N), which makes nonhyperbolic moveout analysis feasible in practice. Equation (4.3o) was derived under the additional assumption (see Appendix 4B) that the effective horizontal velocity can be obtained from the single-layer equation, Vhor(N) - Vnmo(N)V/1 + 2r/(N) .

(4.32)

Therefore, moveout equation (4.29) in terms of horizontal velocity holds for layered media as well, provided Vnmo and Vhor are replaced with the effective quantities defined here.

190

C H A P T E R 4.


It should be emphasized that the effective moveout parameters and the equation as a whole are determined by the interval zero-dip NMO velocity and r], with no dependence on the individual values of the Thomsen parameters Vpo, ~ and 5. Thus, Vnmoand ~ fully control P-wave reflection traveltimes in horizontally layered VTI models. Implications of this result for velocity analysis and seismic processing in VTI media are discussed in Chapter 6, and a numerical analysis of the accuracy of equation (4.30) is given in Chapter 7.

4.4

4.4.1

L o n g - s p r e a d m o v e o u t of S V - w a v e s in V T I media Models with negative a

As discussed above, deviation of SV-wave reflection moveout from a hyperbola is much more pronounced for (less typical) models with negative values of a than for those with positive a. If a < 0, the influence of the term (1 + 2a) 4 in the denominator of the quartic moveout coefficient (4.9) makes SV-wave moveout nonhyperbolic even at relatively small offsets. This phenomenon was found numerically by Levin (1989) in his study of the relationship between the moveout velocities of pure and converted waves in VTI media. Figure 4.11 shows the SV-wave traveltime curve for Mesaverde mudshale the material that displayed the most anomalous behavior in Levin's calculations. For this model, the denominator of A4 is close to zero (1 + 2a = 0.006), and the Taylor series expansion practically breaks down because derivatives of the function t2(x 2) become almost infinite. The exact traveltime curve in Figure 4.11 starts out with an extremely low moveout velocity (Vnmo ~ 0) and then rapidly changes its slope due to the influence of the quartic moveout term. The moveout anomaly in Figure 4.11 is not caused by particularly strong velocity anisotropy: the difference e - 5 = -0.177. The problem is that a is negative, implying that the SV-wave phase velocity has a maximum at the symmetry axis. If this maximum becomes even stronger, the S V traveltime curve develops a cusp (triplication) near the vertical (Helbig, 1966; Musgrave, 1970). For stratified media, however, the effective SV-wave quartic moveout term (4.16) does not become infinite even if the denominator of the interval A4 is equal to zero in some of the layers. Therefore, in layered VTI models one is not likely to encounter SV-wave moveout nearly as anomalous as that in Figure 4.11.

4.4.2

Positive a and models with cusps

For more typical a > 0 reflection traveltimes of the SV-wave in a VTI layer are relatively close to a hyperbola up to a spreadlength of about Xmax = 1.5z. At larger offsets the SV-wave moveout rapidly becomes increasingly nonhyperbolic and cannot

4.4.

L O N G - S P R E A D M O V E O U T O F S V - W A V E S IN V T I M E D I A

191

2.5

Figure 4.11: SV-wave moveout in a horizontal layer of Mesaverde mudshale (Levin, 1989), which has a negative a. The moveout velocity of the best-fit hyperbola rapidly increases with spreadlength. The parameters are Vpo = 4.529 km/s, Vso = 2.703 km/s, ~ = 0.034, 5 = 0.211 (c = -0.497, 1 + 2a = 0.006), z = 3 km.

0 ~ 0o

80-

60-

Figure 4.12: Long-spread SV-wave moveout for the model of Dog Creek shale (Figure 4.5), with z - 3 km. The function tw is the quartic Taylor series (4.1) with the exact moveout coefficients; tA is the approximation (4.23).

192


be adequately approximated by either the quadratic (Figures 4.3 and 4.4) or quartic Taylor series (Figure 4.12). The high residual moveout for offsets approaching twice the reflector depth (Figure 4.3) is caused by the influence of the SV-wave velocity maximum (for a > 0) at incidence angles of 40-45 ~ (Figure 4.1). Despite the increase in phase velocity with angle, the "instantaneous" moveout velocity (Radovich and Levin, 1982) determined by the inverse slope of the traveltime curve [d(x2)/d(t2)] sharply decreases in the vicinity of the velocity maximum (Figure 4.12). As a result, the SV traveltime for x > 1.5z increases with offset much faster than predicted by the quartic Taylor se ri es (4.1). Although this result seems contradictory, it does reflect the shape of the SV-wave phase-velocity function between the vertical and its maximum near 40-45 ~ If a > 0, the NMO velocity is greater than the vertical velocity due to a sharp increase in the phase velocity with angle away from the vertical [equation (3.13)]. This explains the relatively small initial slope of the traveltime curve (large instantaneous moveout velocity) in Figure 4.12. In contrast, phase velocity changes rather slowly near its maximum, and the instantaneous moveout velocity is close to the phase velocity itself (as in isotropic media). Since the maximum phase velocity is much smaller than the NMO velocity, the moveout curve rapidly bends towards the t2-axis at incidence angles approaching 40 ~ This change in slope cannot be described by the quartic Taylor series, which depends solely on the near-vertical velocity variations. The more elaborate approximation (4.23) fails to describe SV-traveltime at intermediate offsets x > 2z as well (Figure 4.12). For P-waves, the improvement over the quartic Taylor series was achieved by introducing the horizontal velocity into the nonhyperbolic term and making equation (4.23) converge at infinitely large offsets. The P-wave traveltime curve on long spreads, however, is usually much smoother than that of the SV-wave because the phase and group velocities for the P-wave rarely have pronounced extrema between 0 ~ and 90 ~ (Figures 4.1 and 4.5). Therefore, the design of the moveout approximation (4.23) is well-suited for P-waves in typical VTI models. In contrast, for SV-waves phase velocity always has either a maximum (if a > 0) or a minimum (a < 0) near an incidence angle of 45 ~ and this extremum may be quite pronounced since lal is often large. It is the influence of this velocity extremum, rather than that of the horizontal velocity, that causes the sharp bending of the SV-traveltime curve at oblique incidence angles. Due to the small value of A4 for > 0, the contribution of the horizontal velocity (and that of the nonhyperbolic term as a whole) to equation (4.23) becomes substantial only at large horizontal offsets, far exceeding those corresponding to the velocity maximum. For that reason, both equation (4.23) and the quartic Taylor series do not deviate much from the analytic hyperbola for the whole range of offsets in Figure 4.12. Refraction of rays in layered media can make the SV-wave moveout anomaly near 45 ~ less pronounced and extend it over a wide range of incidence angles. Still,

4.4.

L O N G - S P R E A D M O V E O U T O F S V - W A V E S IN VTI M E D I A

193

Figure 4.13: Accuracy of the nonyperbolic moveout equation for the SV-wave. AtA is the residual moveout after application of the analytic approximation (4.23); /ktF is the residual moveout for the same approximation, but with fitted coefficients. The model is Dog Creek shale (Figure 4.5), with z - 3 km.

8

Figure 4.14: SV-wave moveout curve with a cusp for a model with a large positive a. The parameters are (from Thomsen, 1986)" V p o - 3.048 km/s, V s o - 1.490 km/s, - 0 . 2 5 5 , 5 - -0.05 ( a - 1.276), z - 3 km.

194

C H A P T E R 4.


to give an adequate description of long-spread SV-wave moveout, equation (4.23) has to reflect the shape of the phase-velocity function in the vicinity of the velocity maximum. The accuracy of equation (4.23) for the SV-wave may be significantly increased if the analytically derived parameters Vnmo, A4 and A are replaced with fitted coefficients determined by the least-squares method (Figure 4.13). For models with positive a < 0.5, application of fitted coefficients practically eliminates SV-wave residual moveout for spreadlengths up to at least Xmax = 2z. Of course, the fit to the exact traveltimes at intermediate offsets improves at the expense of distorting the estimates of the horizontal velocity and the parameter A. Ultimately, the high magnitude of the SV-wave velocity maximum may lead to a concave shape of the slowness surface and cusps on the SV-wavefront and moveout curve (see the discussion of cusps in Chapter 1). In Figure 4.14, the cusp occupies a wide range of offsets starting at x = 1.48z (group angle of 36.5~ The presence of a cusp, though diagnostic of anisotropy, may seriously impede the analysis of reflection moveout. Difficulties in the detection of cusps are caused not just by the multi-valued traveltimes and associated interference patterns, but also by the uneven energy distribution along the cuspoidal wavefront (see the density of rays for the model of Green River shale in Figure 1.14). Approximations based on the Taylor series expansion will properly describe only the first branch of a cusp up to the turning point.

4A. W E A K - A N I S O T R O P Y APPROXIMATION F O R L O N G - S P R E A D M O V E O U T

Appendices 4A

for Chapter

Weak-anisotropy long-spread

195

4

approximation

for

moveout

Here we obtain an analytic expression for long-spread reflection moveout in a horizontal, homogeneous, weakly anisotropic VTI layer. Denoting the angle between the group-velocity vector and vertical by r (Figure 4.15), the P-wave traveltime can be written as Vpo tpo (4.33) t~(~) - cos r V~(r ' where Va(r is the group velocity at the angle r and tpo is the two-way vertical traveltime. For weak anisotropy, the group velocity is equal to the phase velocity at the corresponding phase angle ~1 [equation (1.70)], and

Va(r

- V(O) - Vpo (1 + 6sin 2 Ocos 2 t) + esin 4 O).

(4.34)

In the anisotropic terms of equation (4.34), 0 can be replaced with the group angle r because the difference r is linear in the anisotropic coefficients [equation (1.74)]:

Ira(C) = Vpo (1 + ~ sin 2 r cos 2 r + e sin4 r

(4.35)

Squaring equation (4.33), substituting equation (4.35) and dropping terms quadratic in e and 6 in the denominator yields

t~,o t~,(r -- cos2 r (1 + 26sin 2 r cos2 r + 2esin 4 r

(4.36)

Expressing the group angle r through the offset x [tanr = x/(Vpotpo)] and linearizing in c and (f, we find

t~ -t~o + &,w~: + 1 +

A w X4 4,P

[x/(Vpo tp0)] 2'

(4.37)

where AW 2,P and AW 4,P are the linearized quadratic and quartic coefficients of the Taylor series expansion of t 2 in x2: Aw -

1

2,P-V2

nmo

AW 4,P "-- - -

-

-

1 - 25

V~0 '

2 ( c - 5) t20V40

(4.38)

(4.39) "

196

C H A P T E R 4.

S [-h


R _1

x

Figure 4.15: Source and receiver over a VTI layer with the parameters Vpo, Vso, and 5. r is the group (ray) angle, 0 is the corresponding phase angle. Equation (4.39) was obtained in the main text [equation (4.11)] by linearizing the exact quartic moveout coefficient for the P-wave. For large offsets (x --+ c~), the squared traveltime (4.33) becomes X2

(4.40)

t~ -- t~,0 (1 + 2 e - 2 5 ) + V

or,-

where V,hor,P 2 - V~0 (1 + 2c) is the squared horizontal P-wave velocity. Thus, t~(x 2) exhibits the correct asymptotic behavior as the offset goes to infinity. Equation (4.37), as any other P-wave kinematic relationship for weak transverse isotropy, can be converted into the corresponding expression for SV-waves by making the following substitutions (see Chapter 1): Vpo -+ Vso, 5 --+ a and e --+ 0. Therefore, the Taylor series coefficients for the SV-wave are given by Aw

2,SV

AW

-

-

1 - 2a

(4.41)

V~0

2a v40

(4.42) "

In the limit of large offsets, the SV-wave traveltime can be found from equation (4.40) as X2

t2sv - t2so (1 - 2a) + V~0"

(4.43)

Note that Vso represents both the vertical and horizontal velocity of the SV-wave.

4B. P-WAVE MOVEOUT IN LAYERED VTI MEDIA

4B

197

P-wave m o v e o u t in layered V T I media

P-wave reflection traveltime from the bottom of layer N in horizontally stratified VTI media can be approximated by the general equation (4.23): x2

t2(X, N) - t2(N)~t_

V2mo(N)

A4 (N) x 4 _~_ 1 -~- A(N)x 2'

(4.44)

where the parameter A(N) is related to the quartic moveout coefficient A4(N), the NMO velocity Vn2mo(N) and the horizontal velocity Vhor(N) as [equation (4.24)] d4(g)

(4.45)

A ( N ) - Vho~(N) _ Vnm2o(N).

Equations (4.44) and (4.45) are identical in form to the corresponding expressions for a homogeneous medium, but the parameters A2(N), A4(N), and A(N) should be calculated for the stack of layers above the N-th interface. Normal-moveout velocity Vnmo(N) in a layered VTI medium is given by the Dix equation, while A4(N) can be obtained from the exact averaging formula (4.16). It is convenient to express A4 through the parameter r/and NMO velocity in the same way as in a single VTI layer: A4(N) - -

2~(N)

(4.46)

t2(N ) V4mo(N ) ,

where ~(N) now represents an effective parameter that absorbs the influence of both anisotropy and layering. An explicit expression for ~(N) can be found from equation (4.16) (assuming Vso = 0 in all layers): 1

rl(N)--8

1

V4mo(N)to(N)

E(Vn(/m)o) 4 (1 + 8r/(i))

- 1

,

i--1

where V(/m)o, r/(i), and t~i) are the interval values in layer i. Since the meaning of the horizontal velocity in stratified media is not strictly defined, we simply use equation (4.26) for homogeneous media: Vhor(N) -- Vnmo(N)~1 + 2r/(N) ,

(4.48)

where Vnmo(N) and ~(N) are the effective quantities defined above. The validity of this approximation is confirmed by numerical examples in the main text. Hence, A4(N) and A(N) are now related to Vnmo(N) and ~(N) by the same expressions as in a single VTI layer, and equation (4.44) takes the form of equation (4.28):

t2(x, N) - t2(N) +

x2 2~(N) X4 Vn2mo(N) - Vn2mo(N){ t~(N) Vn2mo(N) -k- [1 + 2~(g)] x 2 }" (4.49)

Rewriting this expression in terms of the effective horizontal velocity Vhor(N) makes it analogous to the single-layer equation (4.29): X2

[gh2or(N) -

gn2mo(N)] x 4

t2(x, N) - t2o(N) + Vn2mo(N) - Vn2mo(N)[t2(N)V4mo(N ) + Vh2or(N)x2 ] 9

(4.50)


Chapter 5 R e f l e c t i o n m o v e o u t of mode-converted waves With recent advances in the acquisition of multicomponent data, including the technology of ocean bottom surveys, converted waves find an increasing number of applications in seismic exploration. For example, PS-waves help in imaging hydrocarbon reservoirs beneath gas clouds, where conventional P-wave methods suffer due to the high attenuation of compressional energy (Granli et al., 1999; Thomsen, 1999). Also, converted waves provide information about shear-wave velocities and other medium parameters that cannot be constrained using P-wave data alone. This advantage of mode conversions becomes especially important in anisotropic media due to the large number of unknown parameters and the ambiguity in estimating reflector depth from surface P-wave data. It should be emphasized that the influence of anisotropy on PS-wave moveout and amplitude is usually more significant than that on P-wave signatures, and isotropic processing methods often fail to produce accurate convertedwave images. A key difference between converted and pure reflections in common-midpoint (CMP) geometry is that mode conversion can make the moveout curve asymmetric with respect to zero offset (i.e., traveltime is no longer an even function of offset). Only in the special case of horizontal reflectors and a medium with a horizontal symmetry plane, does the converted-wave (e.g., PS-wave) reflection traveltime remain the same if we interchange the source and receiver. The asymmetry of the converted-wave moveout can be further enhanced by angular velocity variations in anisotropic media. Therefore, in general the moveout of PS-waves cannot be adequately described by the conventional traveltime series t2(x 2) (extensively used above), which contains only even powers of the offset x (see Figure 5.2 below). The goal of this chapter is to give an analytic description of the azimuth- and dip-dependent traveltimes of PS-waves in anisotropic media. The first two sections are devoted to dip-line converted-wave moveout in a vertical symmetry plane of a homogeneous anisotropic layer (this part is based on the work of Tsvankin and Grechka, 2000). The CMP traveltime-offset function, derived in parametric form and represented through the components of the slowness vector of the P and S-waves, makes it possible to generate CMP gathers without two-point ray tracing. Similar equations are obtained for common-conversion-point (CCP) gathers, routinely used in the 199

200

C H A P T E R 5. R E F L E C T I O N M O V E O U T OF M O D E - C O N V E R T E D WAVES

processing of PS data to minimize conversion-point dispersal. This formalism also leads to closed-form expressions for moveout "attributes," such as the NMO velocity near the traveltime minimum. For VTI media, the weak-anisotropy approximation is used to obtain the dip-dependent PS traveltime and offset explicitly and to simplify relationships among the moveout attributes and medium parameters. Then the parametric moveout equations are extended to multi-azimuth (3-D) PS reflection data recorded over layered, arbitrary anisotropic models. Finally, the methodology of Chapter 4 is generalized for converted waves to study the long-spread moveout of PS events in horizontally layered VTI media. The results developed here provide a foundation for the joint inversion of P and PS reflection traveltimes in VTI media discussed in Chapter 7. Note that the parametric representation of PS moveout can be useful for modeling and processing of converted waves in isotropic media as well.

5.1

Dip-dependent moveout of PS-waves in a single layer (2-D)

Consider a PS-wave recorded over a homogeneous anisotropic medium in the dip direction of a plane reflector (Figure 5.1). To distinguish between the two branches of the moveout curve, it is convenient to introduce the notion of "positive" and "negative" offsets x. Assuming that the source excites P-waves that get converted into S(SV)-waves at the reflector, an offset will be considered positive if the source is located downdip with respect to the the receiver. Correspondingly, at negative offsets the P-wave source is moved updip with respect to the receiver. Figure 5.2 shows typical traveltime curves of the PS-wave computed for a dip-line CMP gather over a homogeneous VTI layer. For only a horizontal reflector (r = 0 ~ is the PS-traveltime in Figure 5.2 an even function of x. The moveout curve becomes increasingly asymmetric with dip, with the traveltime minimum recorded at positive offsets. For dips beyond 40 ~, the minimum moves to large offsets exceeding twice the CMP-reflector distance and then disappears altogether. The general character of the converted-wave moveout in Figure 5.2 (including the asymmetry) is similar to that in isotropic media. The influence of anisotropy, however, may cause a shift of the minimum traveltime towards negative offsets (Figure 5.3, r = 10~ For the model in Figure 5.3, the traveltime for the P-wave source located updip from the common midpoint may be smaller than the zero-offset value (provided the dip is mild). This unusual phenomenon, caused by an increase in the SV-wave velocity with angle for positive values of a, is explained below [see equation (5.20)]. For large values of a reaching 0.8-1, the wavefront of the SV-wave develops a cusp centered near an angle of 45 ~ with the vertical (Figure 4.14). Depending on the ratio of the P- and S-wave velocities and the range of reflection angles recorded on a CMP gather, the PS-traveltime in this case may also contain a cusp and become multi-valued. Cuspoidal moveout requires a special treatment not discussed here.

5.1. D I P - D E P E N D E N T MOVEOUT OF PS-WAVES IN A SINGLE LAYER (2-D)

201

Xl Zr

VG.s

.I(r Figure 5.1: PS-wave in a homogeneous anisotropic layer with a dipping lower boundary. The incidence plane represents the dip plane of the reflector and a symmetry plane of the medium. The source S excites a P-wave that gets converted into a shear wave at the reflector and is recorded by the receiver R. Cp and Cs are the angles between the group-velocity vectors (rays) of the P- and S-waves and the vertical, ZCMp is the reflector depth beneath the CMP, and Zr is the depth of the conversion point on the reflector. In the parametric moveout equations it is assumed that both group-velocity vectors point towards the surface. The pronounced changes in the PS moveout curve with reflector dip suggest using different sets of moveout "attributes" (parameters) for mild and steep dips. Below we give concise analytic expressions for these attributes and for PS reflection traveltime as a whole.

5.1.1

Parametric representation of P S traveltime

Suppose the PS-wave is formed by mode conversion at a plane dipping interface beneath an anisotropic (not necessarily VTI) layer, as shown in Figure 5.1. To make the problem two-dimensional, the incidence plane is assumed to coincide with both the dip plane of the reflector and a symmetry plane of the medium. The same assumption was made in the derivation of the 2-D NMO equation (3.5) in Chapter 3. In the adopted 2-D reflection model, the phase-velocity vectors and rays of reflected waves are confined to the incidence plane. Also, the polarization vector of one of the split shear modes is perpendicular to the dip (incidence) plane, and that (SH) wave is completely decoupled from the P- and SV-arrivals. Therefore, a Por SV-wave incident upon the interface generates a single converted mode (polarized in-plane) denoted below simply as PS or SP. In Appendix 5A it is shown that the traveltime and source-receiver offset of a

202

C H A P T E R 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES

-0 2

~

15 ~ 2

30 ~ 2

Figure 5.2: Dip-line reflection moveout of the PS-wave in a homogeneous VTI layer with the parameters Vpo = 2.0 km/s, Vso = 1.0 km/s, c = 0.2, ~ = 0.1 (a = 0.4); the distance between the CMP and the reflector is 1 km. Reflector dip is shown on top of each plot. Positive offsets correspond to the P-wave source located downdip from the CMP.

5.1. D I P - D E P E N D E N T M O V E O U T OF PS-WAVES IN A SINGLE LAYER (2-D)

(D-O ~ .

-0.5

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0 Offset (km)

.

203

10 ~ .

.

.

.

.

0.5

.

.

-0.5

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0 Offset (kin)

.

.

.

.

.

0.5

-0.5

0 Offset (km)

0.5

Figure 5.3: Dip-line moveout of the PS-wave in a VTI layer with the parameters Vpo - 2.0 km/s, Vso - 1.2 km/s, e - 0.3, 5 - 0.05 (a = 0.69); the distance between the CMP and the reflector is 1 km. converted wave on a CMP gather (Figure 5.1) can be represented parametrically as qP - Pv q, v + qs - Ps q,

t CMP--ZcMP l + ~ t a n r

(5.1)

and x cMv

q,P - q , s zcMv 1 + ~1 tan r (q,v + q s, ) '

(5.2)

where zc~ e is the reflector depth beneath the common midpoint, pp and Ps are the horizontal components of the slowness vector for the P- and S-waves (respectively), qv and qs are the vertical slownesses, and q,e - d q e / d p p , q,s - d q s / d p s . The slowness vectors of the P- and S-waves are related to each other by Snell's law at the reflector [see equation (5.3) below]. Note that the xl-axis is directed updip, and the group-velocity vectors of both waves are assumed to point towards the surface (Figure 5.1). According to the convention introduced above, positive CMP offsets correspond to the P-wave source located d o w n d i p from the CMP. If the medium is isotropic, equations (5.1) and (5.2) become equivalent to the expressions developed by Alfaraj (1993), and both t and x can be obtained explicitly as functions of the slowness projection Pint onto the reflector (see Appendix hA). To determine the slowness vectors for a given value of the projection Pint, it is necessary to solve the Christoffel equation in the Cartesian coordinate system associated with the reflector. It is also possible to express the traveltime curve through the horizontal slowness (ray parameter) of either P- or S-waves and find the ray parameter of the other wave from Snell's law [equation (5.83)]: Pint

=

--(Pp COS (/) -+-qp sin r

= Ps

COSr + qs sin r

(5.3)

204

CHAPTER 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES

r is the reflector dip. As discussed below, in the computation of moveout attributes it is more efficient to operate directly with pp and Ps without involving Pint. The derivative d q / d p for both waves can be found by implicit differentiation of the Christoffel equation (e.g., Grechka, Tsvankin and Cohen, 1999). Since the incidence plane is assumed to be a plane of symmetry, the Christoffel equation written in the form q(p) - 0 generally is quartic in q. For models with a horizontal symmetry plane (e.g., VTI), the polynomial q(p) becomes quadratic for q2. Although the generation of a CMP gather using equations (5.1) and (5.2) involves solving the Christoffel equation for each reflection raypath, it does not require timeconsuming two-point ray tracing. For any pair of the slowness vectors of P- and S-waves related by Snell's law at the reflector, equations (5.1) and (5.2) produce the time and offset on the CMP gather with a given zcMP. This analytic representation of P S moveout can be conveniently used to obtain NMO velocity and other attributes of the CMP moveout curve. To avoid reflection-point (i.e., conversion-point) dispersal, P S data are often resorted into common-conversion-point (CCP) gathers, which include source-receiver pairs corresponding to the same conversion point at the reflector. In contrast to CMP gathers, the locations of sources and receivers in a CCP gather depend on subsurface geometry and velocity field. For the single-layer 2-D model considered here, the depth of the conversion point Zr is related to zcMP as (Appendix hA) Zr -

(5.4)

ZCMP

1 + g1 tan r (qp

+ q,s)

Therefore, the parametric moveout expressions (5.1) and (5.2) take a particularly simple form for CCP gathers: tee P - Zr (qp - pp q p + qs - Ps q,s), Xoop

-

-

(5.5) (5.6)

More general parametric relationships for azimuthally varying P S moveout over layered anisotropic media are presented in section 5.3.

5.1.2

A t t r i b u t e s of t h e P S m o v e o u t f u n c t i o n

Moveout attributes conventionally used in the traveltime inversion of pure-mode reflections include normal-moveout (NMO) velocity (Chapter 3) and, sometimes, higher-order moveout terms responsible for nonhyperbolic moveout (Chapter 4). Due to the generally asymmetric shape of the common-midpoint P S moveout curve with respect to zero offset, the attribute largely responsible for small-offset reflection traveltime is the slope (first derivative) of the moveout curve at x = 0. If reflector dip is mild and the P S moveout has a minimum at moderate offsets, suitable attributes also are the minimum traveltime tmin, the corresponding source-receiver offset Xmin = X ( t m i n ) , and the normal-moveout velocity Vnmo defined at Xmin-

5.1. DIP-DEPENDENT MOVEOUT OF PS-WAVES IN A SINGLE LAYER (2-D)

205

S l o p e of t h e m o v e o u t curve and p o s i t i o n of the t r a v e l t i m e m i n i m u m In Appendix 5B it is shown that the apparent slowness, or slope, of any moveout curve recorded in CMP geometry (dt/dx) is determined by the difference between the projections onto the CMP line of the slowness vectors measured at the source and receiver locations (i.e., the slowness vectors of the incident and reflected rays). This representation of moveout slope is not limited to 2-D or a single homogeneous layer and remains valid in heterogeneous anisotropic media. The derivation of dt/dx in Appendix 5B can be easily modified to obtain the known expressions for data on common-shot or common-receiver gathers. The slope of reflection moveout on a shot gather, for instance, is simply equal to the ray parameter of the reflected ray at the receiver location. The result for shot gathers (but not for CMP geometry) follows from ray theory because for wavefronts excited by a fixed point source, the gradient of the traveltime at any point should be equal to the slowness vector. From the general expression (5.78) in Appendix 5B, the slope of the t(x) CMP curve of PS-waves can be written as (using the sign conventions from Figure 5.1)

dt 1 dx = -2 (p~ - p'~) '

(5.7)

with the horizontal slownesses pp and Ps evaluated at the source and receiver locations; recall that both pp and Ps correspond to group-velocity vectors pointing toward the surface. Equation (5.7) not only provides a simple expression for the slope itself, it also helps to obtain concise solutions for NMO velocity and other attributes of the traveltime minimum. If the medium above the reflector is horizontally homogeneous (as is the case with the single-layer model considered here), both pp and Ps remain constant between the reflector and the surface. To find the moveout slope at zero offset, we determine pp and Ps from Snell's law [equation (5.3)] and the condition q p = q.s, which ensures that the group-velocity vectors of the P- and S-wave are parallel to each other [see equation (5.2)]. Equation (5.7) can also be used to find the slownesses pp-minand ps-mincorresponding to the minimum of the moveout curve. Since the derivative dt/dx vanishes at the traveltime minimum, minimum, traveltime p min P

min __ pmin

--

(5.8)

PS

Note that in the special case of a horizontal reflector (r = 0), equation (5.8) is satisfied if the slowness vectors of the incident and reflected waves are vertical (PP = Ps = Pint z 0). Therefore, the minimum of the converted-wave traveltime from a horizontal reflector always corresponds to the vertical slowness vector, but the incident and reflected rays are not necessarily vertical, unless the medium has a horizontal symmetry plane. This means that in general the traveltime minimum of the converted wave from a horizontal reflector is located at a non-zero offset Xmin ~ 0, although the slowness vectors of the corresponding P and S-waves are vertical.

206


Equation (5.8) also confirms the well-known fact that for a pure-mode reflection and arbitrary reflector dip, the traveltime curve reaches its minimum at zero offset (where Pint -" 0). Indeed, if Pint -" 0, the slowness vectors of the incident and reflected waves are orthogonal to the interface (i.e, parallel to each other), so in the absence of mode conversion, Pv = Ps. As a result, for pure modes equation (5.8) is always satisfied at a vanishing Pint, and the minimum traveltime on CMP gathers is recorded at zero offset. Using Snell's law [equation (5.3)] and equation (5.8), we obtain the following relationship between the slowness components corresponding to the traveltime minimum: 2Pmin - -(qp + as) tan r

(5.9)

The vertical slownesses qv and qs can be found as functions of pp (pv=ps) from the Christoffel equation. Therefore, equation (5.9) can be solved in a straightforward way for the horizontal slowness pmin needed to evaluate the NMO velocity and other attributes associated with the traveltime minimum. Note that Pv and Ps corresponding to both zero offset and the traveltime minimum are obtained without using the slowness projection on the interface (Pint). In Appendix 5C we give an explicit solution of equation (5.9) for isotropic media and demonstrate that the traveltime minimum exists only if 2n tan r < n2 _ 1 '

(5.10)

where n - Vp/Vs, and Vp and Vs are the velocities of the P- and S-waves, respectively. For a typical n = 2, the moveout curve of the PS-wave has a minimum for reflector dips up to 53 ~ Equation (5.10) is not exact if the medium is anisotropic, but it still provides a good first-order approximation for small and moderate values of the anisotropic coefficients. NMO

velocity

Although the CMP traveltime of converted waves from a dipping reflector is not an even function of source-receiver offset, the moveout curve near the traveltime minimum (if the latter does exist) can still be described by normal-moveout velocity defined in the same way as that for pure modes, but using the function t2(x):

{1 d2(t2)] } -1 V:mo --

~

~~'x2

Xmin

(5.11) 9

Expressing both the traveltime and source-receiver offset through the slowness components and using equation (5.7) for the moveout slope yields NMO velocity in the following form (Appendix 5C):

V2 nmo,PS --

4 (q,v " As2 + q,s , Ap) 2 ] (AN + As) 2 i-~in (q,-p -~ q,;i ~ (qP -F qs)] pmin

(5.12)

5.1. DIP-DEPENDENT MOVEOUT OF PS-WAVES IN A SINGLE LAYER (2-D)

207

I! - d2qp/dp2p, q" - d 2 where q,p ,s qs/dP2s, and

Ap - 1 + q,p tan r

As -

1 + q,s tan r

(5.13)

The parameter pmin corresponding to the traveltime minimum is determined from equation (5.9). For pure (non-converted) modes, at the moveout minimum qp - qs, q,p - q,s - q', q,p" - q,s" - q", A p = A s , and equation (5.12) reduces to the 2-D NMO equations (3.5) and (3.7)" 1

gn2mo,pure __

qtt pmin q, _ q pmi,=p(r

__ V(r

COSr

d2V ]0=r

1 + v(r 1 -- tan ~bd_y_V]0=r V(r

,

(5.14)

dO

where V ( O ) is the phase velocity as a function of the phase angle with the vertical, and p(r is the horizontal component of the slowness vector of the zero-offset ray (see Figure 3.1). It should be mentioned that conversion-point dispersal on a CMP gather, properly treated in the derivation of equation (5.12), was not accounted for in equation (3.5). In agreement with Hubral and Krey (1980), the identical result of the two derivations indicates that conversion-point dispersal has no influence on NMO velocity of puremode reflections. For a horizontal reflector (r = 0), the slowness pmin vanishes, and the parameters A p = A s = 1 [equation (5.13)]. Hence, equation (5.12) becomes simply

qi', + "] nmo,PS(r -- 0) -- --

q,s

qP + qs

.

(5.15)

]pmi-=0

The squared pure-mode NMO velocity [equation (5.14)] from a horizontal reflector is equal to ( - q " / q ) ] p = o , while the traveltime along each of the legs of the PS-wave minimum-traveltime ray (which in general are not vertical) can be written as tmin,P -ZCMp qp and tmin,S -- ZCMP qs [equation (5.64)]. Therefore, equation (5.15)can be expressed through the NMO velocities of the P- and S-waves as (tmin,P -]- tmin,S ) V.nmo,PS 2 2 ~--- tmin,P V.nmo,P -]- tmin,S V.2 nmo,S"

(5.16)

Note that 2tmin,g and 2tmin,S are the zero-offset reflection traveltimes of the pure P- and S-waves, respectively. The Dix-like equation (5.16) is derived for the 2-D problem, but it does not require up-down symmetry (i.e., the model need not have a horizontal symmetry plane). If the horizontal plane is a plane of symmetry, then the P- and S rays corresponding to the vertical slowness vector are vertical, and equation (5.16) reduces to the expression for VTI media (Seriff and Sriram, 1991): t p s o V.nmo,PS 2 2 2 = t p o V.nmo,P + t s o V.nmo,SV

(5.17)

where tpo and tso are the vertical traveltimes of the P and S-waves, and t p s o tpo + tso. Implications of equation (5.17) for obtaining the SV-wave NMO velocity from P and P S data in VTI media are discussed in section 5.4 below and in Chapter 7.

208


O t h e r a t t r i b u t e s of t h e t r a v e l t i m e m i n i m u m

Another important attribute of PS moveout is the offset Xmi n of the traveltime minimum (for pure modes, Xmi n : 0 ) . Since Xmi n contains the generally unknown reflector depth ZcMp, it is convenient to normalize it by the minimum traveltime tmin. Using equations (5.1) and (5.2) and taking into account that at the traveltime minimum PP = Ps = pmin [equation (5.8)], we find

q,P - q,s

Xmin tmin

qp + qs

-

]

pmin (q,p + q,s) pmin

By recording reflection moveout of a converted mode for a range of CMP locations in the dip plane of the reflector, we can also obtain the derivative of tmin with respect In the pure-mode case, the spatial to the CMP coordinate YcMP = zcMp/tanr derivative of the minimum (zero-offset) traveltime determines the slope of reflections on the zero-offset (stacked) section and is equal to twice the ray parameter (horizontal slowness) of the zero-offset ray. For converted waves, dtmin/dYcMe does not have such a simple interpretation, but it can still contain useful information about the medium parameters. From equations (5.1) and (5.2)it follows that dtmin

-- t a n r

dYcMp

qP + qsl _

-}-q,s) ] 1 + 5 tan r (q,p + q,s) pmin pmin (q,p

(5.19)

The spatial derivative dxmin/dYcMP is not given here because it can be expressed as a combination of Xmin/tmin [equation (5.18)] and dtmin/dYcMP [equation (5.19)].

5.2

A p p l i c a t i o n to a V T I layer

Analytic developments in the previous section are general and can be used in a symmetry plane of any anisotropic medium with arbitrary strength of velocity anisotropy. Next, these results are applied to a transversely isotropic layer with a vertical symmetry axis. Since in VTI models each vertical plane is a plane of mirror symmetry, the 2-D formalism is valid for any azimuthal orientation of the reflector (this is no longer the case if the symmetry axis is tilted). As with other kinematic VTI relationships, PS-wave moveout equations remain valid in the vertical symmetry planes of orthorhombic media (see sections 1.3 and 3.5). 5.2.1

Weak-anisotropy

approximation

for

PS moveout

For weakly anisotropic models with small absolute values of ~ and 5, CMP traveltime and source-receiver offset of the PS-wave can be derived as explicit functions of the projection of the slowness vector onto the reflector, Pint (see Appendix 5D). The results of Appendix 5D make it possible to generate reflectiOn moveout of converted

5.2.

A P P L I C A T I O N TO A VTI LAYER

209

waves for weakly anisotropic VTI media without doing ray tracing or even solving the Christoffel equation. Despite the explicit form of these weak-anisotropy approximations, they are rather lengthy and do not provide an easy insight into the influence of anisotropy on the reflection moveout of converted waves. Below we give simplified expressions for the moveout attributes discussed above by linearizing the exact equations in the anisotropic parameters. S l o p e of t h e m o v e o u t c u r v e

The weak-anisotropy approximation for the slope of the CMP traveltime curve at zero offset is obtained in Appendix 5D as [equation (5.121)]

dt = dx z=o +

+

sin r

[(1 - ~2) + 4~ (a - (f)]

2 Vpo (1 + ~) sin3 r [~5 (1 + ~) - a (~3 + 9~2 + 8)] 2Vp0a(l+a) sin 5 r 2Vpo a2 (1 + a) a (a4 + 593 + 5a + 1),

(5.20)

where a - Vpo/Vso. It is interesting that the higher-order terms in sin r (sin 3 r and sin 5 r appear only due to the influence of anisotropy. For isotropic media, the exact value of the moveout slope is simply

dt[ dx, =0

(~-a-0)

-

sinr 2

"

(5.21)

Equation (5.21) shows that in isotropic models the PS traveltime from a dipping reflector always decreases with offset at x = 0 (a > 1), and the moveout minimum should be recorded at positive x, corresponding to the P-wave leg located downdip from the reflection point. In the presence of anisotropy, however, it may happen that (dt/dx)lx=o > O, and the traveltime minimum moves into the negative offset range (where the P-wave leg is located updip from the reflection point). Indeed, if a > 0 and is relatively large so that a - 5 is close to 0.5, which is quite feasible for such VTI formations as shales, and a < 2.4, the leading (sin r term in equation (5.20) has a positive sign. If the dip is mild, and the influence of the higher-order terms in sin r is small, the zero-offset moveout slope as a whole is greater than zero. For larger dips, the sin a C-term becomes increasingly dominant and eventually reverses the sign of (dt/dx)I~=o. This conclusion is confirmed by the numerical results in Figure 5.3 (for r = 10 ~ and Figure 5.45, showing that, in VTI media, (dt/dx)l~=o may be positive at mild dips, and the PS traveltime reaches a minimum at small negative x. To explain this unusual phenomenon, recall that for positive a the SV-wave velocity increases away from the vertical up to about 45 ~. In this case, although the

210

C H A P T E R 5.

R E F L E C T I O N M O V E O U T O F M O D E - C O N V E R T E D WAVES

E x

"~ - 0 . 2

PpoVpo

PPO Vpo

Figure 5.4: Exact slope of the PS-wave moveout curve at zero offset [equation (5.7); solid line] and its weak-anisotropy approximation [equation (5.20); dotted line] in two VTI models: (a) e = 0.15, 5 = 0.05 (a = 0.28); (b) c = 0.3, 5 = 0.05 (a = 0.69). For both media, Vpo = 2.0 km/s and Vso = 1.2 km/s. Reflector dip changes from 0~ to 50 ~ in accordance with the product PPo Vpo, approximated here by sin r [equation (5.22)]. Moveout curves for model b are shown in Figure 5.3. shear-wave leg for x < 0 is longer than that at zero offset, the S-wave traveltime does not increase with Ix I nearly as fast as in isotropic media because the corresponding group velocity becomes higher. As a result, the overall traveltime of the PS-wave may decrease over a certain range of negative offsets away from x = 0. The deviation of equation (5.20) from the exact solution is insignificant for moderately anisotropic media (Figure 5.4a) and becomes noticeable only when e reaches 0.25-0.3 (Figure 5.4b). Note that the model from Figure 5.4b has a large a ~ 0.7, and the zero-offset moveout slope is positive for dips ranging from 0 ~ to about 17 ~ The weak-anisotropy approximation correctly reproduces this trend of the exact function in Figure 5.4b, but overstates the initial increase in (dt/dx)l==o with dip. For purposes of velocity analysis and processing, it is convenient to replace the reflector dip r with the ray parameter (horizontal slowness) of the pure P-wave reflection recorded at zero offset (Ppo = [PP, pure (X ---- 0)[). Unlike reflector dip, P~o can be found from surface data by measuring reflection slopes on zero-offset (stacked) P-wave sections. Neglecting the cubic and higher-order terms in sin r the P-wave ray parameter can be written as sin r p~o = vp(r

sin r Vpo

.

(5.22)

If we retain only the leading term in sin r in equation (5.20), the P-wave ray parameter can be substituted in its approximate form [equation (5.22)], yielding

dt [

dx =:0

_

Pro

2 (1 + ~) [(1 - ~2) + 4t~ (a - (f)] + . . . .

(5.23)

5.2.

APPLICATION TO A VTI LAYER

211

For a typical value ~ = 2, the coefficient multiplied with the anisotropic term a - ~ is almost three times greater than the isotropic term 1 - ~2. Therefore, we can expect that (dt/dx)[,=o, obtained as a function of Pro, can provide information about the anisotropic parameters a and ~ (mostly about the difference a - 5). A t t r i b u t e s of t h e t r a v e l t i m e m i n i m u m If the reflector is horizontal, CMP traveltime of converted waves in VTI media is reciprocal with respect to the source and receiver positions and has a minimum at zero offset. To find the normal-moveout velocity of horizontal P S events, equations (3.11) and (3.13) for the NMO velocities of P- and S-waves should be substituted into equation (5.17). After separating the anisotropic term, we find K2 nmo,PS(0)

VpoVSO [ 1 + 2(a1 ++ ~~ ) ]

"

(5.24)

In isotropic media (a - d - 0) , V.nmo,PS(0) 2 reduces to the product of the P- and S-wave velocities (Tessmer and Behle, 1988). Equation (5.24) shows that transverse isotropy may significantly distort the NMO velocity of PS-waves because for typical a = 0.5 and ~ = 2 the anisotropic term in the brackets can reach 0.3 or more. As discussed above, traveltime curves of the PS-wave have a minimum only for small and moderate reflector dips. Hence, it is convenient to find the attributes of the traveltime minimum using simplified "mild-dip" approximations, in which all terms containing the cubic and higher powers of sin r are dropped. Such a weak-anisotropy, mild-dip approximation for converted-wave NMO velocity is derived in Appendix 5D: K-2 nmo,PS

2

(PPo)

--

K-2 nmo,PS(0)-- Pro ~ (3~ 4 -- 2 ~ 3 + 6m2 - 2m + 3)

_

p2 o ( e ; - 1 ) [ 6 a ( n + l ) 2 - ( a - 3 )

2n(~+ 1)

n(3n 2 - 2 e ; + 3 ) ]

(5.25)

where Vnmo,Ps(O) is the NMO velocity from a horizontal reflector [equation (5.24)]. To estimate the contribution of the anisotropic parameters to the dip dependence of NMO velocity, we rewrite equations (5.24) and (5.25) for n = 2: V.n-2 mo,PS

1 (PPo, No -- 2 ) - - WpoWso

[1 - 1.3 (5 + 0.5a)] - 3.4p2vo - p2po(2.7a + 1.85). (5.26)

Equation (5.26) shows that, for typical positive values of a, anisotropy amplifies the increase in the NMO velocity with dip (usually cr > 5). The anisotropic dip-dependent term [P~o(2.7a + 1.85)] provides an equation for a and ~ with comparable weights for both parameters. However, even for substantial values of a ~ 0.5, the magnitude of this term can reach only 35-40% of the isotropic term (3.4P~o), and the dip-dependence of NMO velocity as a whole is not highly sensitive to the anisotropic parameters. Numerical tests show that the contribution of the anisotropic parameters to the exact NMO velocity is even somewhat smaller than that predicted by equation (5.25).

212

C H A P T E R 5. R E F L E C T I O N M O V E O U T O F M O D E - C O N V E R T E D WAVES

2.4 : . . . . . . . . . . . . . . . . . . . . . . . . .

oo

2

:

:

:

:

: ....

:o e.'*

9

PPoVPo

o: o''':

.......

" :

b

Peo%

Figure 5.5: Exact NMO velocity of the PS-wave [equation (5.12), solid line] and its weak-anisotropy approximation [equation (5.25), dotted line] in two VTI models: (a) = 5 = 0.15 (elliptical anisotropy); (b) e = 0.15, 5 = 0.05. For both media, Vpo = 2.0 km/s and Vs0 = 1.2 km/s. Reflector dip changes from 0 ~ to 30 ~ The accuracy of the weak-anisotropy approximation (5.25) is illustrated by Figure 5.5. For the whole range of dips, including a horizontal reflector, the weakanisotropy solution overstates the NMO velocity, which leads to a vertical shift between the curves. Since just the leading term in r and PPo was retained, the approximation becomes less accurate with increasing dip. Nevertheless, equation (5.25) correctly reproduces the trend of the NMO-velocity curve in the most important regime of moderate dips (r < 35-40~ For larger dips, the traveltime minimum either does not exist at all, or corresponds to unusually large source-receiver offsets seldom acquired in practice. Comparison of Figures 5.5a and 5.5b also shows that the error is higher for more "anelliptical" models with larger values of a. The main value of equation (5.25) and other approximations, however, is in providing analytic insight into the behavior of various moveout attributes. The leading dip term in the weak-anisotropy approximation for the normalized offset Xmin/tmin[equation (5.131)] has the following form: Xmin ~min

PPo Y2o 2~

[ ( a - 1) + 2 ( ~ 5 - a)].

(5.27)

For ~ = 2, the isotropic term ~ - 1 in equation (5.27) is two times smaller than the multiplier of the anisotropic term, and we can expect Xmin/tmin to be quite sensitive to the parameters 5 and a. Equation (5.131) is sufficiently accurate at mild dips, even for models with large values of a (Figure 5.6). Both the exact Xmi n and the moveout slope at zero offset

5.2. APPLICATION TO A VTI LAYER

213

0.3

0.2

0.1

,

"~.L ':

. . . . . . :. . . . ~ " -

~

........

ow.q

E

..=

E -0.1

PPOVpo Figure 5.6: Exact Xmin/tmin[equation (5.18), solid line] and its weak-anisotropy approximation [equation (5.131), dotted]. The dashed line is the exact slope of the traveltime curve at zero offset [equation (5.7)]. The model parameters are the same as in Figure 5.4b: Vpo = 2.0 km/s, Vso = 1.2 km/s, e = 0.3, 5 = 0.05 (a = 0.69). Reflector dip changes from 0 ~ to 30 ~

(dt/dx)]=:o (also shown in Figure 5.6) vanish at a dip of about 17 ~ (peoVpo ~ 0.3), where the moveout minimum is located at zero offset. For smaller dips, as discussed above, the traveltime minimum moves into the negative offset range (Xmi n < 0). The approximate derivative of the minimum traveltime with respect to midpoint (dtmin/dYcMe) is found in Appendix 5D as dtmin

dYcMP

= Pro (1 + a).

(5.28)

Clearly, equation (5.28) is purely isotropic and gives only redundant information about the ratio of the vertical velocities, which can be determined in a conventional way using the zero-offset traveltimes. On the whole, the analytic approximations presented above indicate that reflection moveout of the converted PS-wave is strongly influenced by transverse isotropy and, therefore, can be efficiently used in anisotropic parameter estimation (see Chapter 7).

5.2.2

Recovery of PS-wave moveout c u r v e

Shifted hyperbola An important practical issue is how to determine the attributes associated with the PS-wave traveltime minimum (tmin, Xmin, and Vnmo,PS) from CMP data. Since the

214


1.44

............................................

Figure 5.7: Exact CMP moveout of the PS-wave in a VTI layer (solid) and its approximation with the best-fit shifted hyperbola [equation (5.29), dashed] found by least-squares minimization. The model parameters are Vpo = 2.0 km/s, Vso = 1 km/s, c = 0.2, 5 = 0.1, r = 30 ~ first derivative of the PS-wave traveltime curve (dt/dx) goes to zero at Xmin, P S moveout can be approximated with a hyperbola centered at the traveltime minimum: 2 t2(x)

2 ( X - Xmi n -- tmi n + V.2 nmo,PS

(5.29)

A typical example illustrating the application of a shifted hyperbola to the recovery of the moveout attributes is shown in Figure 5.7. Note that the exact PS-wave traveltime t(x) (solid) is generally asymmetric with respect to Xmin due to the presence of a term cubic in ( x - Xmin), which is not included in equation (5.29). This, however, does not prevent the hyperbola (5.29) (dashed) from giving the correct position of the moveout a p e x (Xmin, tmin) and an accurate estimate of the NMO velocity. The errors in the estimates of Vnmo and Xmin/tmin [compared to the exact values given by equations (5.12) and (5.18)] are only 1.1% and 0.05%, respectively. The high accuracy achieved for the model in Figure 5.7 was ensured by having an approximately equal range of offsets on each side of the apex of the moveout curve, which mitigates the influence of the cubic moveout term. If the traveltime minimum is substantially shifted with respect to zero offset, it may be necessary either to mute out a certain range of offsets (making the estimation interval more symmetric with

5.2.


215

respect t o Xmin) or add the cubic term in x - X m i n to the moveout equation. In most field-data applications, the moveout curve of the PS-wave has to be found by means of semblance velocity analysis based on equation (5.29). For each trial value of the time tmin, the parameters Vnmo,PSand Xmin can be estimated using a 2-D semblance scan (rather than a 1-D scan over Vnmo for pure modes). Since the results of semblance analysis may be influenced by the offset-dependent amplitude and waveform of the reflection event, as well as by the offset coverage, it is useful to examine synthetic seismograms of the PS-wave (Figure 5.8). For dips up to 40-45 ~ in Figure 5.8, the amplitude of the converted wave goes to zero at relatively small offsets close to the traveltime minimum (e.g., at zero offset for a horizontal reflector), which may cause complications in the semblance analysis and in the reconstruction of the moveout curve. Although the low amplitudes are observed over a narrow range of offsets, the polarity change in the wavelet may cause errors in calculating the semblance along shifted hyperbolas (5.29). An appropriate correction can be made by computing the rms amplitude for each seismogram within the time window used in the semblance search and identifying the minimum-amplitude trace. Then, prior to calculating the semblance, one has to reverse the polarity of all traces at offsets larger than that of the moveout minimum. Alternatively, it is possible to apply to PS data modified semblance operators which take into account amplitude variation with offset (e.g., Sarkar et al., 1999). Note that if the dip exceeds 30 ~ the area of the polarity reversal moves towards longer offsets and does not interfere with estimation of small-offset traveltimes (and of the moveout slope at zero offset). The shift of the polarity reversal away from zero offset is caused entirely by the influence of anisotropy. Only if the reflector coincides with a symmetry plane of the medium does the amplitude of the PS-wave vanish on the zero-offset trace.

Local coherency method Clearly, a shifted hyperbola is not suitable for monotonic moveout curves of steep PS events (Figure 5.2, r >_ 45~ In such cases, the slope of the moveout curve at x = 0 can be obtained by approximating the traveltimes at small source-receiver offsets with a straight line or a quadratic, depending on the moveout curvature. A more general approach to reconstructing asymmetric moveout, applicable to mild dips as well, is based on measures of local coherency (Bednar, 1997). This method does not depend on the functional form of the moveout curve and, as an added benefit, yields moveout slope that can be used in the inversion for the medium parameters (see Chapter 7). An estimate of the local slope p of the traveltime curve t(x) can be obtained from seismograms u(x, t) as (Bednar, 1997)

(5.30)

216


30 ~

E

o . ,

.

75 ~

Offset (kin)

Figure 5.8: Synthetic seismograms of the PS-wave reflected from the bottom of a VTI layer. The wavefield was generated by anisotropic dynamic ray tracing. The dip r of the reflector is shown on top of each plot. The parameters of the upper layer a r e Vpo : 2.0 km/s, Vso = 1 km/s, e = 0.2, ~ = 0.1, the density p = 2 g/cm 3. The lower layer is isotropic with Vpo = 2.2 km/s, Vso = 1.1 km/s and p = 2.2 g/cm 3. The distance between the CMP and the reflector is 1 km; only positive offsets (with the P-wave source located downdip from the CMP) are displayed.

5.2.


217

a

4.2 Offset (km)

~_.~+mm--+._

_ .......... - - m I mP~r R " I H l ~ l l

i

b

+

.

.

.

.

.

,.n | I || ~, I~m m m m . ~ i - _ - : - .

~ .... mm

I ~ m| i m | . mm. mm= |m ~ ~ m..........

Figure 5.9: (a) Ray-traced seismograms of the PSV-wave on a CMP gather computed for the layered VTI model from Figure 5.12c. The data are contaminated by Gaussian noise with the variance equal to 0.5% of the maximum amplitude. (b) The estimates of local slopes (tickmarks) and the exact traveltime curve (solid line). The spatial averaging window [...] is 0.2 km (twice the offset increment); the temporal window (...} is 8 ms (twice the time increment). The windows may be increased for a different level and frequency content of noise, etc.

218

C H A P T E R 5. R E F L E C T I O N M O V E O U T O F M O D E - C O N V E R T E D

,2

-

,2

9

9

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

-~ i

~

:-

: '

-_ 4.2

.

WAVES

--~ " l

~: ~

"~"

: ~

--

:

:

i

T

J

: __.

-

i

: i

-

: ~

-

:

.

:,_T

--

i

--

-

J

Offset (km) F i g u r e 5.10: S a m e as F i g u r e 5.9, b u t t h e v a r i a n c e of t h e noise is 10% of t h e m a x i m u m amplitude.

5.3.

3-D T R E A T M E N T OF PS-WAVE MOVEOUT F O R LAYERED MEDIA

219

where (...) and [...] denote averaging over t and x, respectively; the derivatives are computed through finite differences. The performance of equation (5.30) was tested on the synthetic PS-wave data shown in Figures 5.9a and 5.10a. The tickmarks in Figures 5.9b and 5.10b indicate the estimated slopes p(x, t) for which the normalized coherency

C=

~/

/~

\ o~ a t ~ a__~

(5.31)

is larger than 0.8. Clearly, the reconstructed slopes closely follow the correct moveout curve (solid line in Figures 5.9b and 5.10b) in areas of a relatively strong signal. Even the high level of broadband noise in Figure 5.10 does not prevent the algorithm based on equation (5.30) from reconstructing the moveout function for a wide range of offsets. The tickmarks outside the P S event are associated with the noise, which sometimes produces isolated coherency values exceeding the threshold C = 0.8. Although this example was generated for a 2-D model, Bednar's (1997) method can be applied in a similar way to 3-D traveltime surfaces of PS-waves discussed in the next section and Chapter 7.

5.3

3 - D t r e a t m e n t of P S - w a v e m o v e o u t for layered m e d i a

In this section, the methodology developed above for 2-D (dip-plane) wave propagation in a single layer is extended to arbitrarily oriented source-receiver lines and vertically heterogeneous media. The model is composed of a stack of horizontal, arbitrary anisotropic layers above a plane dipping interface. The P-to-S (or S-toP) conversion is assumed to take place only at the bottom of the model (reflector); Thomsen (1999) suggested to call PS-modes of this type' "C-waves." First, we treat the 2-D problem of wave propagation in vertical symmetry planes of layered media and then proceed with an analytic description of azimuthally dependent P S moveout. Exact 2-D and 3-D parametric expressions for traveltime and offset are presented for both CCP (common-conversion-point) and CMP gathers. 5.3.1

2-D expressions

for vertical

symmetry

planes

Suppose the acquisition line is confined to the dip plane of the reflector overlaid by an arbitrary number of horizontal layers (Figure 5.11). The anisotropic symmetry does not need to be specified at this stage, but the vertical incidence plane is assumed to be a plane of mirror symmetry in all layers to make the kinematics (but not necessarily the dynamics, see Chapter 2) of wave propagation two-dimensional.

220


~'). o~

Figure 5.11" PS-conversion in a vertical symmetry plane of layered anisotropic media. Since the reflector is dipping, layer N has different thickness above the conversion point (z (N)) and below the common midpoint (z(N)p). xp (xlp) and x s ( X l s ) are the horizontal displacements of the P- and S-rays with respect to the conversion point.

5.3.

3-D T R E A T M E N T OF PS-WAVE M O V E O U T F O R LAYERED MEDIA

221

Numerical example Figure 5.12 illustrates the behavior of converted-wave moveout in a layered VTI medium for both common-midpoint (CMP) and common-conversion-point (CCP) gathers on the dip line of the reflector (Figure 5.11). Similar to the single-layer model, the offset of the traveltime minimum on CMP gathers increases with dip, which makes the moveout curve increasingly asymmetric with respect to x = 0. If the dip does not exceed 30-40 ~ the traveltime minimum can often be recorded on a sufficiently long CMP gather (Figure 5.12a). For larger dips, the traveltime monotonically decreases with offset (Figure 5.12c,e), and the short-offset moveout is largely controlled by the slope of the moveout curve at zero offset. The moveout in Figure 5.12e is truncated because the reflection point at x ~ 1 km reaches the intersection of the reflector with the bottom of the second layer (i.e., layer 3 pinches out). The dependence of converted-wave moveout on dip is more complicated in the CCP domain, as compared to the more familiar CMP domain. For the model in Figure 5.12, the traveltime minimum first moves towards negative offsets with increasing dip (Figure 5.12b,d) and then returns back almost to the zero-offset position (Figure 5.12f).

Parametric dip-plane moveout e q u a t i o n s The traveltime and offset of the PS-wave on both CCP and CMP gathers is found in Appendix 5E as a special case of the general 3-D equations by aligning the axis Xl with the dip plane of the reflector. The traveltime of the P S arrival with the conversion point at interface N (Figure 5.11) is given by N t -

+

-

E

-

,IP

+

-

*,IS

'

(5.32)

~=i

where z (t) (t~ = 1, 2, ... N - 1) are the thicknesses of the horizontal layers in the overburden, z (N) is the thickness of layer N above the conversion point, q(pt) and q(J) are the interval vertical slownesses of the P- and S-waves, p~p and P~s are the projections of the slowness vectors of the P- and S-waves on the axis xl (ray parameters) ' q(t) _ ,1P dq(t)/dp~p and a=*,IS (~) - dq(t)/dp~s Since the medium above the reflector is laterally homogeneous, the ray parameters p~p and Pls remain constant between the reflector and the surface; they are related to each other through Snell's law at the reflector. The corresponding source-receiver offset is obtained in Appendix 5E as N

x - xls - xlp - ~

z(t) (U, lp"(t) _ ~,~s ~ .

(5.33)

l=l

With the xl-axis pointing updip (Figure 5.11), the offset x in equation (5.33) is positive if the P-wave leg is located downdip with respect to the S-wave leg (the same sign convention as for the single-layer model).

222

CHAPTER

5.7

.

.

.

.

.

.

.

.

.

.

5.

REFLECTION

MOVEOUT

OF

MODE-CONVERTED

WAVES

~'r/ .......... ~"................. b_ _ ....../~ J '

.

.

~.~

r

5.4

.................

i .................

~

i

,=40 o 4.8 ~.

4.6 4.4 .

.

.

.

~=

=60 ~

Offset (km)

Offset (km)

Figure 5.12: Dip-line moveout of the PS-wave converted at a dipping interface beneath three VTI layers (the top two layers are horizontal). The left column (a,c,e) shows common-midpoint (CMP) gathers, and the right column (b,d,f) shows commonconversion-point (CCP) gathers. Each row corresponds to a different reflector dip r r = 20 ~ (a,b), r = 40 ~ (c,d) and r = 60 ~ (e,f). For positive offsets the P-wave leg is located downdip from the S-wave leg. The top layer (layer 1) has the following parameters: Vpo = 2.0 km/s, Vso = 0.8 km/s, e = 0.1, 5 = 0.05; for layer 2, Vpo = 2.3 km/s, Vso = 1.0 km/s, e = 0.2, = 0.1; for layer 3, Vpo = 2.9 km/s, Vso = 1.2 km/s, e = 0.15, 5 = 0.1. The layer thicknesses (from top to bottom) are z (1) = 0.5 km, z (2) = 1.5 km and z (3) - 2 km. For CMP gathers, z (3) is the distance between the reflector and the projection of the CMP onto the top of layer 3; for CCP gathers, z (3) is the thickness of layer 3 above the common conversion point.

5.3. 3-D T R E A T M E N T OF P S - W A V E M O V E O U T F O R L A Y E R E D MEDIA

223

Similar to the single-layer model, we need to specify one of the ray parameters (Pip or Pls) and find the other from Snell's law at the reflector. The vertical slownesses q~) and q(l) in each layer, along with the derivatives qg) and q(l) can be determined ,1P ,1S ~ from the Christoffel equation. Therefore, scanning over one of the ray parameters produces the CCP gather for a conversion point located at depth ~N l = l Z(~) To generate a CMP gather, it is necessary to relate the thickness z (N) of the N-th layer above the CCP in equations (5.32) and (5.33) to the corresponding thickness z CMP (N) beneath the common midpoint (Figure 5.11) Adaptation of the corresponding 3-D expression in Appendix 5E leads to z(N) =

z(N)

CMP-

tan~b ~--~N-1 Z(s

(l)

a(l)

2 l=1 (q,,p + ~,~s ) . 1+ ~ (o(N) + o(N)) 2 Xa, lP ~,lS ]

(5.34)

Substitution of z (N) from equation (5.34) into equations (5.32) and (5.33) yields the traveltime and offset of the PS-wave recorded on the CMP gather specified by z CMP (N) . In the special case of a single layer, the sum over the N-1 layers in the numerator of equation (5.34) is dropped, and z (N) reduces to the depth of the conversion point Zr from equation (5.4). Equations (5.32)-(5.34) make it possible to compute a CMP gather of converted waves in symmetry planes of layered anisotropic media without two-point ray tracing. It is still necessary to satisfy Snell's law at the reflector and solve the Christoffel equation in each layer for both P- and S-waves, but the whole computation of t and x for a given source-receiver pair has to be performed only once (i.e., for just one ray). Also, as mentioned above, in media with a horizontal symmetry plane (e.g., VTI) the Christoffel equation q(p) - 0 has an analytic solution because it reduces to a quadratic polynomial in q2. 5.3.2

3-D description

of PS moveout

Next, consider a more general 3-D problem of a PS-wave generated at a plane dipping interface and recorded on a line with arbitrary orientation. To form an areal (3-D) CMP gather, the sources and receivers are placed on lines with different azimuths but the same CMP location (Figure 5.13). In general, an incident P-wave excites two reflected shear modes (PS1 and PS2) propagating towards the surface with different velocities and polarizations. The formalism introduced below is valid for either PSwave with the substitution of the appropriate slowness vector. Suppose the model includes a single homogeneous anisotropic layer of arbitrary symmetry (Figure 5.13). Using the 3-D relationship between the slowness and groupvelocity vectors, the traveltime of the wave reflected (converted) at the depth Zr can be written in the form (Appendix bE)

t-tp

+ t s - z ~ ( q p - p ~ p q , ~ p - p 2 p q , 2 P +qs-P~sq,~s -P2sq,2s),

(5.35)

where the subscript "2" refers to the projections on the x2-axis, and for each wave

224

C H A P T E R 5. R E F L E C T I O N MOVEOUT OF MODE-CONVERTED WAVES

C B

D

Figure 5.13: Converted PS-wave (ray ARC) recorded on an arbitrary oriented line over a homogeneous anisotropic layer with a dipping lower boundary, zr is the depth of the conversion point, Cp and Cs are the angles between the P- and S-rays and the vertical. The inset shows the source-receiver vector (AC), the common midpoint (B) and the projection of the conversion point onto the earth's surface (D).

5.3.

3-D T R E A T M E N T OF P S - W A V E M O V E O U T F O R LAYERED MEDIA

225

O q / O P 2 . As before, the slowness vectors of the P and S-waves are related to each other by Snell's law at the reflector. The source-receiver vector x - A C (Figure 5.13) is obtained in Appendix 5E as

q,2 -

x - {(xls - xlp), (x2s - x2p)} - zr ((q,~p - q,~s), (q,2P - q , 2 s ) ) .

(5.36)

Equation (5.36) yields the following expressions for the source-receiver offset x and the azimuth c~ of the source-receiver line with respect to the xl-axis: x -Ixl-

-

+

- q.):,

a - - t a n -1 (q,2P ~ , : : _- ~ q,:s) ,:: .

(5.37)

(5.38)

Equations (5.35), (5.37) and (5.38) are sufficient for generating common-conversionpoint (CCP) gathers of the PS-wave. As in the 2-D problem, moveout in CMP geometry can be modeled by replacing Zr with the reflector depth beneath the CMP location [equations (5.145) and (5.146)]: Zr =

ZcMP

1 + t~nr [(q,lP + q,ls)~1 + (q,2P + q,2s)~2]'

(5.39)

where ~1 and ~2 are the components of a horizontal unit vector that points in the updip direction of the reflector. Equations (5.35) and (5.36), with zr defined in equation (5.39), produce the traveltime and the source-receiver vector of a PS-wave on a CMP gather described by zcMe. The corresponding expressions for horizontally l a y e r e d arbitrary anisotropic media above a plane dipping reflector are developed in Appendix 5E and will not be reproduced here. If the reflector is horizontal and all layers have a horizontal symmetry plane, PS-moveout is symmetric with respect to zero offset and, for offset-to-depth ratios limited by unity, can be described by the NMO ellipse introduced in Chapter 3 (section 3.5). This result was obtained by Grechka, Theophanis and Tsvankin (1999), who also showed that the NMO ellipse of PS-waves can be related to the NMO ellipses of the pure modes by a simple Dix-like equation. The parametric 3-D equations for traveltime and source-receiver vector are particularly convenient for generating the entire areal (3-D) CMP gather, with sources and receivers occupying a wide range of azimuths and offsets. This can be done by scanning over the two horizontal slowness components of one of the waves (e.g., p~p and P2P), obtaining the corresponding slowness vector of the other wave from Snell's law and computing the traveltime and offset using equations (5.35), (5.36) and (5.39) or the more general expressions for layered media. Such an algorithm is orders of magnitude faster than two-point ray tracing for each source-receiver pair in the 3-D CMP gather. The parametric approach, however, is less efficient in modeling a single CMP line with a given orientation. In this case, it is necessary to search for the slowness vectors

226

C H A P T E R 5. R E F L E C T I O N MOVEOUT OF MODE-CONVERTED WAVES

of the P- and S-waves which do not only comply with Snell's law at the reflector, but also satisfy equation (5.38) for the azimuth of the CMP line. Therefore, it is preferable to compute the whole 3-D gather on a grid of two horizontal slownesses (e.g., those of the P-wave) and then build needed CMP gathers by interpolation. An example of the PS-wave traveltime surface computed for a 3-D CMP gather is given in Chapter 7 (Figure 7.33).

5.3.3

M o v e o u t attributes in layered media

M o v e o u t s l o p e at zero offset

Asymmetric short-spread P S moveout on CMP gathers is largely controlled by the slope of the moveout curve at zero offset (rather than by the NMO velocity, as is the case for pure modes). The zero-offset moveout slope of the PS-wave measured on the dip line over a VTI layer proved to be quite sensitive to the anisotropic parameters (see section 5.2). A general expression for the reflection slope on a CMP line described by sourcereceiver vector x is given in Appendix 5B:

dt dx

1 - ~ [p~(h, ~ ) - p i ( - h , ~ ) ] ,

(5.4O)

where h - I x l / 2 is half the source-receiver offset, and p,(h, c~) and pR(h, o~) are the projections onto the CMP-line azimuth c~ of the slowness vectors of the incident and reflected rays. The positive direction of the CMP line is taken from the source to the receiver, and both slowness vectors are computed for upgoing waves. In general, p,(h, c~) and pR(h, o~) should be evaluated at the source and receiver locations (respectively), but in our model each ray parameter remains constant between the reflector and the surface. To obtain the moveout slope at x = 0 in the 2-D problem, one needs to compute the values of PiP = P~ and Pls = PR for the zero-offset P S ray. The source-receiver offset in CMP geometry can be found from equation (5.33) with z (N) defined in equation (5.34). Setting the offset x in equation (5.33) to zero yields an equation that can be solved for one of the ray parameters (e.g., p~p); the other ray parameter is determined from Snell's law at the reflector. A similar procedure can be devised for the 3-D problem. The slownesses of the Pand S-wave legs of the zero-offset ray can be obtained by solving the equation x = 0 [x is defined by equation (5.37) for a single layer and (5.150) for layered media] for a given zcMP. Then the slowness vectors are projected onto the CMP line, and the slope of the moveout curve at zero offset is computed from equation (5.40). N o r m a l i z e d offset of t h e t r a v e l t i m e m i n i m u m For relatively mild dips up to 30-40 ~ the common-midpoint traveltime curve of the PS-wave usually has a minimum (tmin) at a certain offset Xmin (Figure 5.12). In the

5.3. 3-D T R E A T M E N T OF P S - W A V E M O V E O U T F O R L A Y E R E D MEDIA

227

single-layer VTI model, the ratio Xmin/tmin is strongly dependent on the anisotropic parameters. For the stratified 2-D model, the values of Xmin and tmin can be found from equations (5.32), (5.33) and (5.34), where the ray parameters PiP = PI and Pls -- PR should correspond to the traveltime minimum. Since at x = Xmin the slope of the moveout curve goes to zero, the ray parameters should be equal to each other: p~p = P~s = pmin [see equation (5.40)]; pmin is determined from Shell's law at the reflector. Then the ratio Xmin/tmin is given by Xmi.

__

tmin

-- ~-~N Z(t) ~L p ~_ -q(~ ~ ~--~1 S

/-~t--1

]

kt/,1p - - t/1S ] pmin~o~) _~_ 0(~))] ~'-I,1P *,IS

.

(5.41)

pmin

where z (N) should be expressed through z(N)p using equation (5.34). Note that, in contrast to the single-layer model, the normalized offset Xmin/tmin from equation (5.41), is a function of not only the elastic parameters and reflector dip, but also the layer thicknesses. Likewise, in the 3-D model the traveltime minimum on a CMP line with a given azimuth a corresponds to equal values of the slowness projections PR (h, a) = p~( - h , a) = pmin which can be found from Shell's law. The values of Xmin and train can then be found from equations (5.35), (5.36) and (5.39) for a single layer or from the corresponding expressions in Appendix 5E for layered media. N M O velocity In the presence of vertical heterogeneity, PS-wave NMO velocity from dipping reflectots, defined for the traveltime minimum, is difficult to obtain in closed form similar to the single-layer expression (5.12). Moveout equations, however, become much simpler for media with a horizontal reflector and a horizontal symmetry plane, where P S traveltime is an even function of CMP offset. NMO velocity in vertical symmetry planes of such models can be determined from equation (5.17) introduced above for homogeneous media: tpso V.n2m o , P S -_- tpo V2 n m o , P -~- tso

2 gnmo,SV

(5.42)

where the NMO velocities of P- and S-waves should be computed from the Dix equation for the stack of layers above the reflector, tpo and tso are the one-way vertical traveltimes, and tpso = tpo + tso. Derivation of equation (5.42) is given in Tsvankin and Thomsen (1994) for layered VTI media; their formalism remains fully valid for other models with a horizontal symmetry plane. A 3-D extension of equation (5.42) to azimuthally dependent NMO velocity of PS-waves (i.e., to the NMO ellipse) can be found in Grechka, Theophanis and Tsvankin (1999), who also discuss application of NMO ellipses of pure and converted waves to traveltime inversion in orthorhombic media.

228

5.4


PS-wave moveout in horizontally layered VTI media

In general, the asymmetry of the converted-wave CMP moveout curve with respect to zero offset makes the conventional hyperbolic moveout equation and the whole traveltime series t2(x 2) inadequate even on short spreads. The scope of this section, however, is limited to horizontally layered VTI media, where reflection traveltime of PS-waves is reciprocal with respect to the source and receiver positions, and the moveout minimum is located at zero offset. Therefore, PS-wave moveout for this model can be described by the series t2(x 2) discussed in detail for pure modes in Chapters 3 and 4. The results below are also valid in the vertical symmetry planes of orthorhombic media (see sections 1.3 and 3.5). To avoid confusion with SH-waves, recall that everywhere in this chapter "S-wave" implies only the SV-mode polarized in the vertical plane.

5.4.1

Taylor series coefficients

Short-spread PS moveout in laterally homogeneous VTI media is close to a hyperbola parameterized by the NMO velocity expressed through Vnmo of P- and S-waves in equations (5.17) or (5.42). For a single layer, NMO velocity of PS-waves becomes a simple function of the vertical velocities and anisotropic parameters 5 and a [equation (5.24)]. Similar to pure modes, deviation of the PS moveout curve from the analytic hyperbola is largely governed by the quartic coefficient A4 of the Taylor series expansion for t 2, 1 d [d(t2) ] [ (5.43) A 4 - ~ d(x2 ) [d(x2) z=0 Tsvankin and Thomsen (1994) obtained the following equation for the quartic coefficient of PS-waves in horizontally layered VTI media: A4,PS

__

nmo,P tso nmo,S tpo V-8 V-8 tpo -{- ts0 V-nmo,PS 8 + ~z~4'S tpo Jr- tso V-8 nmo,PS

fit4,P

V2 2 2 tPO tso ( nmo,P Vnmo,S) (tpo + ts0) 2 4 V.nmo,PS s '

(5.44)

where the quantities A4 are the quartic coefficients for P- and S-waves (A4,p and A4,s) normalized using the two-way vertical traveltimes tpo and tso: "A4,P -- A4,p t~,o ,

(5.45)

t~o,

(5.46)

A4,S -- A4,s -

1

A4,ps - A4,PS ~ (tpo + t s 0 ) 2

(5.47)

5.4. PS-WAVE M O V E O U T IN HORIZONTALLY LAYERED VTI MEDIA

229

Expressions for A4,p and A4,s in layered VTI media are given in Chapter 4. In contrast to pure modes, the quartic moveout coefficient for converted waves does not vanish even in a single isotropic layer. While in this case A4,p --= A4,s -- O, the last term on the right-hand side of equation (5.44) may have a substantial magnitude that increases with the Vp/Vs ratio. Due to the mode conversion, a single layer for PSwaves becomes effectively equivalent to a two-layer model for pure-mode reflections. It is also noteworthy that A4,PS does not become infinite in models with negative a = -0.5 (1 + 2a = 0), where the S-wave coefficient A4,s -+ oo. Indeed, A4,s in equation (5.44) is multiplied with V.rlmo,S s ~ which cancels (for a single layer) the term (1+2a) 4 in the denominator [see equation (4.9)]. Thus, it follows from equation (5.44) that the PS-wave does not exhibit the same anomalous nonhyperbolic moveout for negative a that is typical for the S-wave (see Figure 4.11). As discussed in Chapter 4, similar cancellation of the term (1 + 2(7)4 is caused by the influence of layering on A4,s [equation (4.16)]. We conclude that the quartic moveout coefficient in VTI media goes to infinity only for S(SV)-waves in a single layer. Equations (5.17) and (5.44) express both the NMO velocity and quartic moveout coefficient for the converted wave through those parameters for the P- and S-waves. Therefore, if the Taylor series coefficients are known for any two of the three waves, it is possible to find the coefficients for the remaining one. In particular, if the source excites only P-waves, the moveout of P- and PS-waves can be used to recover the NMO velocity and quartic coefficient of the pure shear reflections. As an example, for the simplest model of a single isotropic layer the NMO-velocity equation (5.17) reduces to

Vnmo,S __

Vs_

y. 2 nmo,PS , Vnmo,g

(5.48)

Tessmer and Behle (1988) presented application of equation (5.48) to the estimation of shear-wave velocity from P and P S moveout data. Predictably, this isotropic relationship loses accuracy in VTI media [compare with equation (5.24)], as was demonstrated by the numerical results of Levin (1989) who computed all three moveout velocities from the exact traveltimes on conventional-length spreads. Seriff and Sriram (1991) showed that if the S-wave moveout velocity is derived from P and PS data using the correct VTI equation (5.17), it is much closer to the value obtained by Levin (1989) from the exact traveltimes of the S-wave. Noticeable discrepancies remained only for models with negative a, in which the effective moveout velocity measured from the S-wave moveout was substantially different from the analytic NMO velocity (see Figure 4.11). Apparently, for negative a the shear-wave NMO velocity can be extracted more accurately from the quasi-hyperbolic moveouts of P- and PS-waves [using equation (5.17)] than from the S-wave moveout itself! In Chapter 7, equation (5.17) is applied to parameter estimation in VTI media.

230


65

. . . . . . . . . . . . . . . . . . . . . . . 9

0

9

~

z

i

,oi 35

30

Figure 5.14: Long-spread moveout of the P S V - w a v e in a horizontal layer of Dog Creek shale (Figure 4.5), with the thickness z = 3 km. The rhombuses mark the exact traveltimes; the dashed line is the quartic Taylor series (4.1) with the exact moveout coefficients; the solid line is approximation (4.23).

5.4.2

Nonhyperbolic moveout equation

Combined with the NMO velocity (5.17), the quartic moveout coefficient (5.44) can be used to set up the quartic Taylor series (4.1) for converted waves. A better fit to the exact traveltimes of the PS-wave, however, can be obtained by inserting the expressions for Vnmo and A4 into the general nonhyperbolic moveout equation (4.23) (Figure 5.14). The horizontal velocity of the PS-wave, also needed in equation (4.23), was taken equal to the horizontal P-wave velocity because at large offsets the P-wave leg becomes almost parallel to the layer boundaries (for a single layer). Comparison of Figure 5.14 with the corresponding plot for the pure shear wave (Figure 4.12) shows that equation (4.23) is much more accurate for mode conversions. To explain this result, note that while P S traveltime strongly depends on the nearvertical velocity variations of the S-wave through the NMO velocity and the coefficient A4, the take-off angles of the S-wave leg are limited by the critical angle that seldom exceeds 35 ~ This means that P S traveltimes are not much influenced by the shearwave velocity maximum (for a > 0) near 45 ~ that causes pronounced deviations of shear-wave moveout from both a hyperbola and the analytic approximation (4.23). In fact, the accuracy of equation (4.23) for the PS-wave is only slightly lower than that for the P-wave (Figure 4.7).

5.5.

DISCUSSION

5.5

231

Discussion

In contrast to common-midpoint moveout of pure P- or S-wave reflections, PS traveltime in CMP geometry is generally asymmetric with respect to zero offset, and cannot be described by the familiar t2(x 2) series. In isotropic media, the moveout asymmetry of mode conversions is caused by reflector dip or lateral heterogeneity in the overburden. In the presence of anisotropy, PS moveout becomes asymmetric even in horizontally layered media if at least one of the layers does not have a horizontal symmetry plane. Resorting of PS data into common-conversion-point (CCP) gathers, used in processing to eliminate conversion-point dispersal, changes the shape of the moveout curve but does not remove its asymmetry. To describe such asymmetric moveout functions analytically and understand their properties, it is convenient to represent them in parametric form. The model considered in this chapter is composed of horizontal, arbitrary anisotropic layers above a plane dipping reflector. For multi-azimuth (3-D) PS-wave data in both CMP and CCP geometry, the traveltime and the source-receiver vector are concisely expressed through the slowness components of the P- and S-wave legs of the reflected ray. After relating the P and S slowness vectors via Snell's law at the reflector, the CMP traveltime curve can be computed without time-consuming two-point ray tracing. Another general result of this chapter, which proved extremely helpful in the analysis of PS traveltimes, is that the slope of any CMP moveout curve in heterogeneous, arbitrary anisotropic media is determined by the difference between the projections onto the CMP line of the slowness vectors at the source and receiver locations. Dip-line (2-D) moveout of PS(PSY)-waves over a homogeneous VTI layer was considered in most detail to elucidate the properties of converted-wave traveltime in anisotropic media. The position of the traveltime minimum on CMP gathers of PS-waves strongly depends on reflector dip and anisotropic parameters. As in isotropic media, the minimum is usually recorded at "positive" offsets corresponding to the P-wave leg located downdip from the S-wave leg. In VTI media, however, PS traveltime can reach its minimum at negative offsets, if the parameter a is relatively large (a = 0 . 5 - 0.8), and reflector dip is mild. Further increase in a leads to the development of cusps on the wavefront of the PS-wave, and the moveout function in a certain range of offsets becomes multi-valued. For relatively mild dips up to 30-40 ~ the moveout minimum is observed at moderate offsets and can be recorded on a conventional-length CMP gather. In this case, PS traveltimes can be used to recover moveout attributes associated with the traveltime minimum tmin(Xmin),such as the normal-moveout velocity (defined by analogy with pure modes) and the ratio Xmin/tmin- These attributes can be obtained from reflection data by approximating PS-moveout with a shifted hyperbola centered at the offset Xmi n. Alternatively, asymmetric moveout can be reconstructed by evaluating local coherency measures along the moveout curve (Bednar, 1997). If reflector dip exceeds 40 - 50 ~ the traveltime minimum either does not exist at all or cannot be captured on conventional spreads (limited by 1.5-2 distances between

232


the CMP and the reflector). For those traveltime functions, monotonically decreasing with offset, the attribute that largely controls short-spread moveout is the slope of the t(x) curve at zero offset. The weak-anisotropy approximation helps to find explicit 2-D expressions for the traveltime and offset of the PS-wave in VTI media and study the dependence of the moveout attributes on the anisotropic parameters. The attributes that proved to be most sensitive to the anisotropy are the normalized offset Xmin/tminand the slope of the traveltime curve at zero offset. In Chapter 7, the attributes of PS moveout from a dipping interface are used in the inversion for the VTI parameters. For the special case of a horizontal reflector overlaid by a stack of horizontal VTI layers, PS traveltime is symmetric with respect to zero offset and can be adequately described by the nonhyperbolic moveout equation of Tsvankin and Thomsen (1994) introduced in Chapter 4. Both the NMO velocity and the quartic moveout coefficient of PS-waves can be expressed through the moveout coefficients of P- and S-waves and the vertical traveltimes. It is important to note that for typical VTI models with moderate values of 5 and larger positive values of a, the NMO velocity of PS-waves is more distorted by the anisotropy than that of P-waves. Therefore, if moveout (stacking) velocity is used for time-to-depth conversion, the depth errors on PS-wave images are often higher than those on P images (for more details, see the discussion in section 7.3). Also, it is interesting that the moveout of horizontal PS events in VTI media with negative values of a does not exhibit the same uncommonly high magnitude of nonhyperbolic moveout as does the moveout of S-waves. Therefore, the most stable way to estimate the S(SV)-wave NMO velocity for a < 0 is to use the quasi-hyperbolic moveouts of P- and PS-waves rather the S-wave moveout itself. Due to the kinematic equivalence between the symmetry planes of orthorhombic and TI media (see Chapter 1, section 1.3), the 2-D results for vertical transverse isotropy remain valid for reflections in both vertical symmetry planes of models with orthorhombic symmetry. The only change required in the VTI equations is the replacement of e, 5 and the shear-wave vertical velocity Vso with the appropriate set of orthorhombic parameters introduced in section 1.3.

5A. 2-D DESCRIPTION OF PS MOVEOUT IN A SINGLE LAYER

233

Appendices for Chapter 5 5A

of PS moveout

2-D description layer

in a s i n g l e

The objective of this appendix is to derive a parametric representation of reflection moveout of a P S ( P S V ) - w a v e recorded in the dip direction of a plane reflector. It is assumed that the incidence plane also represents a symmetry plane of the medium, so both rays and the corresponding phase-velocity vectors of the reflected waves are confined to the incidence plane (Figure 5.1). Without losing generality, the P-wave leg is assumed to be tilted downdip from the S-wave leg. The reflection traveltime can be written as z~ t - tp + t s -

Zr

(5.49)

VG, P c o s ~r)p AV VG,S c o s ~2s '

where Gp and Gs are the group angles with the vertical for the P and S segments of the reflected ray (Figure 5.1), VG,p and Va,s are the corresponding group velocities, and z~ is the depth of the conversion point at the reflector. The source-receiver offset in terms of the group angles is given by x = xp + x s = Zr (tan Gp + tan Gs).

(5.50)

To ensure that x represents the source-receiver distance, the S-wave angle Gs is considered positive if the ray is tilted updip from the vertical (for the P-wave, the opposite is true). As a result, the offset is always positive for the P-wave leg located downdip from the S-wave leg. Equations (5.49) and (5.50) can be used to generate the common-conversion-point (CCP) gather of PS-waves. Introducing the reflector depth beneath the common midpoint (ZcMp) instead of zr yields zcMp - Zr

[ 1 + ~1 tan r (tan ~p -

tan Cs)

1,

(5.51)

where r is reflector dip. Substituting ZCMI~[equation (5.51)] for z~ in equations (5.49) and (5.50), we find N

to~ - zo~ ~

(5.52)

and XCMp-

where N

1 VG,p cos r

N~

(5.53)

ZCMP D

1 + VG,s cos r

'

1 D - 1 + ~ tan r (tan Gp - tan Gs),

~o.o~j

(5.55)

234

C H A P T E R 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES

(5.56)

Nx = tan C p + tan Cs"

To satisfy Snell's law, the P and S-waves should have the same projection of the slowness vector onto the interface at the reflection (conversion) point. Hence, this projection (Pint; the subscript stands for the interface) can be used to build a parametric representation of the CMP traveltime for the converted wave. Hereafter, we assume that Pint is the "updip" projection, which corresponds to the upgoing S-wave and downgoing P-wave. If the medium is isotropic, the group angles Cp and Cs are equal to the corresponding phase angles and can be easily expressed through Pint (taken to be non-negative), reflector dip r and the velocities Vp and Vs of the P- and S-waves: sin r

-

cos r

- r

sin Cs

--

Pint

Ypc o s

r + r

- p2nt Y 2 c o s (~ - Pint

Pint Vs cos r

cos Cs -- r

2 -- Pint

r

--

sin r

(5.57)

Ypsin r

(5.5s)

Ys2 sin r

(5.59)

Vssin r

(5.60)

- p2nt Y 2

2 -- Pint

VS2 COS (~ +

Pint

Substitution of equations (5.57)-(5.60)into equations (5.54)-(5.56) and then (5.52) and (5.53) leads to explicit expressions for the reflection traveltime and source-receiver offset in CMP geometry in terms of the slowness Pint. In anisotropic media, the transition from the slowness Pint to the group angles of the reflected waves involves solving the Christoffel equation for the slowness component orthogonal to the reflector. For P - SV-waves in a symmetry plane of an anisotropic medium, the Christoffel equation for the unknown slowness component is quartic (it becomes sextic outside the symmetry planes). Since the subsurface may contain several reflectors with different dips in the same medium, it is convenient to express the traveltime curve through the slowness components in the unrotated coordinate system associated with the earth surface. Also, note that the group angles in equations (5.49)-(5.53) are defined with respect to the vertical rather than to the reflector normal. Introducing the projections of the group-velocity vectors of P- and S-waves onto the vertical (x3, subscript "3") and horizontal (x~, subscript "1") axes gives the following equivalent form of equations (5.54)-(5.56): N = 1 D-l-~tanr Nx=

1

1

+ ~ ,

(5.61)

VG3,P VG3,S (VGI,p

VG1,s)

VGa P + VGa s VGI'P

t VG1,S 9

VG3,P VG3,S

(5.62)

'

(5.63)

5B. 3-D EXPRESSION F O R T H E SLOPE OF CMP M O V E O U T

235

Here the x3-axis is directed upward, the x~-axis is positive in the updip direction, and both group-velocity vectors are assumed to point from the reflector towards the surface. The negative sign of the P-wave term in Nx, which stems from our convention for the group angles, ensures that the offset is positive if the P-wave leg is located downdip from the S-wave leg. The group-velocity components can be related to the slowness vector in the same (unrotated) coordinate system in the following way (Cohen, 1998; Grechka, Tsvankin and Cohen, 1999)" 1 Vc3 = (5.64) q - pq~ and VG1 -- - V c 3 q',

(5.65)

where q and p are the vertical and horizontal slowness components, respectively, and q ' - dq/dp. Equations (5.64) and (5.65)can be used to rewrite equations (5.61)-(5.63) as

N - qp - pp q,p + qs - Ps q,s, 1

(5.66)

D = 1 + ~ tan r (q,p + q,s),

(5.67)

- q, -

(5.68)

Here pp and Ps are the horizontal components of the slowness vector for the P- and S-waves, and q,p - dqp/dpp, q,s - dqs/dps. Note that if the P-wave leg is tilted updip with respect to the S-wave leg, under our convention the value of Nx and the source-receiver offset are negative. Substitution of equations (5.66)-(5.68) into equations (5.52)-(5.53) gives the final parametric expressions for CMP moveout of PS-waves discussed in the main text.

5B

3-D

expression

for the slope

of CMP

moveout

Here it is shown that the apparent slowness (slope) of the CMP moveout curve for any converted or pure reflection is determined by the difference between the in-line projections of the slowness vectors at the source and receiver locations. In contrast to Appendix 5A, the medium above the reflector can be arbitrary heterogeneous and anisotropic, and the reflecting interface is not necessarily planar. Consider a pure or mode-converted reflected wave recorded on a CMP line that makes angle a with the xl-axis (Figure 5.15). If the CMP location and the orientation (azimuth) of the line are fixed, the traveltime depends on only the source-receiver offset x. The slope of the moveout curve at x0 = S R is given by dt dx

_- d(tldx+ t2) I xo

xo

,

(5.69)

236


X2

0 01

Figure 5.15: To the derivation of the slope of reflection moveout at offset Xo - S R . S O R and S I O 1 R 1 (bold lines) are the raypaths of the specular CMP reflections at offsets x0 and x - S 1 R 1 , while S I O R 1 (thin line) represents a non-specular reflection. The horizontal coordinates of the reflection points are denoted as {~0)~0)} (point O) a n d {~}sp), ~sp)} (point 01).

5B. 3-D E X P R E S S I O N F O R THE SLOPE OF CMP M O V E O U T

237

where ti (x) is the traveltime along ray segment SiOi from the source to the reflector and t2(x) corresponds to segment 01Ri from the reflector to the receiver (x = SIRi; see Figure 5.15). To relate the moveout slope to the ray parameters of the incident and reflected waves, it is convenient to add and subtract from dt/dx the slope of the non-specular traveltime curve tns computed for raypath S~ORi (i.e., the raypath going through the reflection point corresponding to x0):

dt I - d(t~s + t~s) I + d ( t l + t 2 -dx t~s-t~s) -~x ~o dx ~o

I ~o"

(5.70)

t~s in equation (5.70) corresponds to ray segment $10, and t~S to OR1 (Figure 5.15). The term that involves just the non-specular traveltime can be directly expressed through the slowness vectors of the incident and reflected rays. Indeed, the nonspecular raypaths to the source and receiver ($10 and ORi) originate at the fixed reflection point O, which implies that the gradient of the traveltime surface for each ray segment is equal to the corresponding slowness vector. Hence, d(t s +

s)

dx

d(t~S + t~s) xo

d(2h)

1 - ~ [PR(h0, a ) - p,(-h0, a)],

(5.71)

ho

where h - x/2 is the half offset, ho =- Xo/2, and p,(h0, a) and PR(h0, a) are the projections of the slowness vectors of the incident ray SO and reflected ray OR (respectively) onto the CMP-line azimuth a; the slowness vectors should be evaluated at the source and receiver locations. To ensure consistency in the signs of the slowness projections, the positive direction of the x-axis used to define p is taken from S to R, and both slownesses correspond to the group-velocity vectors pointing toward the surface. Next, let us show that the term that contains the difference between the specular (t = ti + t2) and non-specular (t ns = t~s + t~s) traveltimes in equation (5.70) goes to zero. If the reflecting interface is described by ~ = f(~1,~2) (~ is the vertical coordinate, SOl and ~2 are the horizontal coordinates), the non-specular traveltime tns for fixed source and receiver positions can be treated as a function of the coordinates ~i and ~2 of the reflection point. According to Fermat's principle, the specular reflected ray for a given offset (e.g., ray Si01R1 for offset x in Figure 5.15) gives the minimum traveltime (or a saddle of the traveltime surface) for all possible reflected arrivals excited at the same source location and recorded by the same receiver. As a result, the double Taylor series expansion of tns(~l, ~2) at the specular reflection point {~}sp)~sp)} does not contain terms linear in (~l - ~sp)) and (~2 - ~sp)). Therefore, the non-specular traveltime tnS(x, sc~~ ~0)) along SiOR1 (~[0) and sc~~ correspond to point O) can be expressed in the following way through the traveltime t(x) - t(x, ~}sp) ~sp)) for the specular ray SlOiRi (Figure 5.15)" t ns (x,

_

t(x)

1 02tas ]

2

(sp)

sp))2

238

CHAPTER 5. REFLECTIONMOVEOUTOF MODE-CONVERTEDWAVES

+ +

02t ns

(:~o) _ : ~ ) ) ( : ? ) _

2102tns 10~22 r

:?~))

(~01- ~sp))2 _[_ ....

(5.72)

Note that the position of the specular reflection point and, therefore, the derivatives of tns in equation (5.72) are functions of CMP offset x. Using equation (5.72) to find the difference between tns and t and taking the derivative with respect to x leads to d [tnS(X, ~C~~ ~0)) _ t(x)] = dDll (x) (~c~0)_ ~c~sp))2 dx dx - 2 D : : ( x ) ( ~ o ) _ ,~p)) a~ ~p) + aD!2(x) (~o) _ ~p))(~o) _ ~p)) dx dx

-D12(x)

+

(~.~o)_~.~sp))dx

-t-

((~o)_(~sp))dx

dD22(x) (~o) _ ~p)12 _ 2 D22(x)(~o) _ ~;s.))

e:? ~)

d-------Z--

dx

'

(5.73)

where 1 02t n~

Dll(X)-2

0~ 2

(5.74)

~(sp)

[

02tns D12(x) -- 0~'1 0~2 ~(sp) '

D~(x)-

(5.75)

1 i)2tns 20~ ~(~p)

(5.76)

Since all terms on the right-hand side of equation (5.73) vanish at x - x0 (~{sp) __

d [tnS(X, ~0)~0)) _ t(x)] I

~x

- 0.

(5.77)

xo

Thus, the moveout slope can be obtained from the non-specular traveltimes and is equal to one-half of the difference between the projections onto the CMP line of the slowness vectors at the source and receiver locations [equation (5.71)]:

dtl ~XX X0

- ~1 [p~(ho ~ ) - p~(-ho ~)]. ~

(5.78)

5C. NMO VELOCITY FOR CONVERTED-WAVE MOVEOUT

5C

239

N M O v e l o c i t y for c o n v e r t e d - w a v e m o v e o u t

If the reflection moveout of a converted wave on a CMP gather has a minimum tmin : t(Xmin), the traveltime n e a r tmin can be described by the normal-moveout velocity Vnmo introduced by analogy with pure modes. To find an analytic expression for Vnmo, we expand the squared CMP traveltime t2(x) in a Taylor series near the traveltime minimum: t2(x)

2 - - tmi n -t-

1 d2(t 2)

d(t2) I (X-Xmin) + 2 dx Xmin

dx 2

(x-

+

Xmin)2

"'"

(5.79)

Xmin

The first derivative d(t2)/dx at x = Xmin is equal to zero, while the second derivative yields the NMO velocity that governs the traveltime for small ( x - Xmin):

{

V:m~

I

1 d2(t 2) 2

d~

Xmin

{ ()1 }-1 d

dt

--t-d-xx-~Xxmin

(5.80)

9

Using the results of Appendix 5B, the moveout slope d t / d x can be expressed through the difference between the horizontal slownesses of the P- and S-waves: dt 1 d--x = -2 (Ps - PP) .

(5.81)

Considering both d t / d x and x as functions of the projection of the slowness vector on the interface (Pint), we can rewrite equation (5.80) as y. 2

_

nmo,PS

--

{

t d(p s - Pp ) /dpint dx / dpint

-2

I }-1 -min

(5.s2)

Pint

where Pint _min corresponds to the traveltime minimum. To evaluate the derivatives in equation (5.82), Pint can be represented through the slownesses of the P- and S-waves using Snell's law: Pint ~- --(Pp COS (~ -~- qp sin r

= Ps cos r + qs sin r

(5.s3)

The P-wave in equations (5.61)-(5.63) is assumed to travel upward from the reflector, which explains the minus sign in front of the P-wave term in equation (5.83). Differentiating equation (5.83), we obtain dPp

= - (cos r + q,p sin r - 1,

dpint dps

dpint

= (cos r + q,s sin r

where, as in equations (5.66)-(5.68), q p =_ dqp/dpp and q,s - dqs/dPs. Hence,

(5.s4) (5.s5)

240


d(ps-pp)_ dpint

--

1 [ 1 1 ] cos r 1 + q,e tan r + 1 + q,s tan r "

(5.86)

The derivative d x / d p i n t in equation (5.82) can be represented through D and N, [equations (5.67)and (5.68)] as

dx

d(Nx/D)

dpint = ZCMP

dpint

(dNx/dpint) D - (dD/dpint) Nx :

ZCMP

02

9

(5.87)

Using equations (5.84) and (5.85), the derivatives with respect to Pint can be expressed through those with respect to pp or Ps; for instance,

dqp

= -q,p (cos r + q,p sin r

(5.88)

dpint

Equation (5.88) and an analogous expression for dqs/dpint allow us to obtain the numerator in equation (5.87) in the following form:

dNx

dD

dpin.__~tD -

dpintg

qi~ (1 + q,s tan r cos r + q,p sin r

x - _

_

q" (1 + q tan r ,s P sin r cos r + q,s

.

(5.89)

Note the symmetry in equation (5.89) with respect to the subscripts "P" and "S"" the second term can be obtained by interchanging these subscripts in the first term. The expression for D, needed in equation (5.87), was derived previously [equation (5.67)]" D-

1

1 + ~ tanr

+ q,s)"

(5.90)

Equations (5.89) and (5.90) are sufficient for obtaining the derivative dx/dpint from equation (5.87). NMO velocity, as given by equation (5.82), also depends on the minimum traveltime, which can be found from equations (5.52), (5.66), and (5.67): minx tmin -- t(Pint )

--

qP - PPq, P + qs - Psq,s 1 + ~1 tanr (q,p +q,s)

ZCM P

I

.

(5.91)

.rain Pmt

Substituting equations (5.86), (5.87) and (5.91) into equation (5.82) and taking into account that at the traveltime minimum pp - Ps - pmin, yields the final expression for NMO velocity as a function of the horizontal slownesses of the P- and S-waves" V"2 4 (qi'P A2s + qi's A2P) I ' (5.92) nmo,gS =

(Ap -Jv As) 2

[pmin

(q,p _j..q,si L (qp -Jv qs)]

pmin

where

Ap - 1 + q p tan r

As - 1 + q,s tan r

(5.93)

5D. W E A K - A N I S O T R O P Y A P P R O X I M A T I O N F O R P S - M O V E O U T IN V T I M E D I A

241

The ray parameter pmin can be obtained from equation (5.9) of the main text: 2Prain -- --(qP + qs ) tan r

(5.94)

where the vertical slownesses qv and qs are related to pp = Ps through the Christoffel equation. For isotropic media, qv and qs are given simply by qp -

-pp

(5.95)

and qs -

- Ps 9

In this case, it is possible to derive an explicit expression for pmin by solving equation (5.94)with qp and qs from equations (5.95) and (15.96): pmin iso

where ~ -

~

sin r V/1 + n2 + S

(5.97)

2 - tan 2 r (n 2 - 1)2;

(5.98)

2vp

V p / V s and

S -~ r

the subscript "iso" stands for "isotropic." This solution, however, exists only if the expression under the radical in equation (5.98) is non-negative, or 2t~

tan r _< ~2 _ 1 "

(5.99)

For larger dips r the PS-wave moveout curve has no minimum.

5D 5D.1

Weak-anisotropy approximation for P S - m o v e o u t in V T I media P a r a m e t r i c e x p r e s s i o n s for t h e t r a v e l t i m e curve

Traveltime and offset for PS-reflections in VTI media can found in explicit form by carrying out linearization in Thomsen's parameters ~ and 5. For small [~]

_

2.8 2.6 ~ .

.

.

.

.

~

,

.

.

.

.

.

Figure 6.4: Group velocity Va as a function of the group (ray) angle for equivalent solutions obtained for the models in (a) Figure 6.1 (Vp0=3.0 km/s, e = 0.2, 6 = 0.1), and (b) Figure 6.3 (Vp0-3.0 km/s, e - 0.3, 5 = - 0 . 1 ) . The curves correspond to models with Vp0=2.8 k m / s and the corresponding values of e and 5 (black); Vp0=3.0 k m / s (actual values, gray); and Vp0=3.2 k m / s (dashed).

260

C H A P T E R 6. P-WAVE TIME-DOMAIN SIGNATURES IN T R A N S V E R S E L Y I S O T R O P I C MEDIA

15

. . . . .

"

"

:

:

:

:

9.

.

.

.

.

.

" . . . ~ . . . "

. . . .

E h_

:

o

z

1.0

s

i

10

Figure 6.5: P-wave NMO velocity calculated from the exact equation (6.3) and normalized by the expression for isotropic media (6.6). The dip ranges from 0 ~ to 70 ~ (a) Widely different models with the same ~7 = 0.2: e = 0.1, 6 = -0.071 (solid); = 0.2, 6 = 0 (gray); e = 0.3, 6 = 0.071 (dashed) - the curves practically coincide with each other. (b) Models with different r/: q = 0.1 (solid); r / = 0.2 (gray); ~7= 0.3 (dashed). ures 6.4a,b), all three ES yield practically the same horizontal velocity (corresponding to an angle of 90~ This result, which holds for the whole range of plausible VTI models, implies that for all ES Vhor : c o n s t (6.9) Note that the horizontal velocity Vhor = Vpo v~ + 2e can be expressed just through Vnmo(0) [equation (6.1)] and r/[equation (6.2)] as Vhor-

Vnmo(O)V/1

+ 2r/.

(6.10)

Therefore, all equivalent solutions have the same value of q, and the family of ES can be described by two parameters: V~mo(0) and r/. Only these parameters can be resolved by inverting dip-dependent P-wave NMO velocity expressed as a function of the ray parameter. In principle, two distinct dips are sufficient to recover the values of V~mo(0) and 7. Additional dipping reflectors just provide redundancy in the inversion procedure, so that, for example, a least-square approach can be used to solve the overdetermined problem. The dominant role of the parameter rl is confirmed by the results in Figure 6.5a, which shows that the P-wave NMO velocity does not depend on the individual values of the anisotropy parameters e and 6, if q is fixed. According to both plots in Figure 6.5, for the most typical case of positive q (e > 6), the conventional expression (6.6) for dip dependence of V~mo in isotropic media severely understates NMO velocities for dipping reflectors. Also, the spread of the curves in Figure 6.5b suggests that the NMO velocities corresponding to 7=0.1, 0.2, and 0.3 are well resolved over a wide range of moderate and steep dips.

6.1. P-WAVE NMO VELOCITY AS A FUNCTION OF RAY P A R A M E T E R

261

Therefore, numerical analysis of the exact NMO equation has helped to generalize the weak-anisotropy approximation (6.7) to VTI media with arbitrary strength of the anisotropy. While the weak-anisotropy P-wave NMO equation (expressed through the ray parameter) is a function of Vnmo(0) and e - 5, the P-wave NMO velocity for general transverse isotropy is fully characterized by Vnmo(0) and r/. Clearly, in the limit of weak anisotropy, r] reduces to the difference e - 5. The family of ES from our first example in Figure 6.1 can be represented by Vnmo = 3.29 km/s and r/ = 0.0833; for the example from Figure 6.3, Vnmo = 2.68 km/s and r / = 0.5. Further analytic verification for the two-parameter representation of Vnmo(P) in VTI media is provided by Cohen (1997) who derives weak-anisotropy approximations quadratic in the anisotropic coefficients. While r] is determined by the values of the zero-dip NMO velocity and the horizontal velocity, the choice of form for this parameter is not unique. We could have combined Vnmo(0) and Vhor in a different fashion to obtain, for instance, 1 + 2e instead of 1 + 25 in the denominator of r/. However, whatever its form, any such anisotropic parameter would represent a measure of "anellipticity," i.e., of deviation from the elliptically anisotropic model. For elliptical anisotropy ~ = 0, and the dependence of P-wave NMO velocity on the ray parameter is purely isotropic [equation (6.6)]. 6.1.2

Dip

plane

of a layered

medium

So far, the dip-dependent velocity Vnmo(P) was studied for only a homogeneous VTI medium. If the model consists of a stack of horizontal VTI layers above a dipping reflector, NMO velocity of any pure-mode reflection on the dip line is given by the Dix-type equation (3.47): Vn2m~

- ~0 ~ [Vn(/E)~ i--1

where t~i) is the interval reflection traveltime in layer i computed along the zero-offset ray, to - ~ =N1 t~i) is the total zero-offset time and V(~o(p ) is the interval NMO velocity evaluated for the ray parameter p of the zero-offset ray; Vn(~o(p) should be determined from equation (6.3). If the reflector is horizontal, equation (6.11) reduces to the root-mean-square average of the zero-dip NMO velocities Vn(/m)o(0). The main difference between the NMO equation (6.11) and the conventional Dix formula is that for a dipping reflector the interval terms {[V(/m)o(p)]2 t~i) (p)} should be evaluated at the ray-parameter value corresponding to the reflector dip r [iv = sin r (r As discussed in Chapter 3, the interval NMO velocities in equation (6.11) correspond to non-existent reflectors orthogonal to the slowness vector of the zero-offset ray in each layer. The above results for a homogeneous VTI medium show that Vn(i)mo(P)in each layer depends just on the interval parameters Vn(~o(0) and q(i). Equation (6.11) also contains the interval traveltimes t~i) (p) along the (oblique) zero-offset ray, which are influenced by the anisotropic parameters.

262

CHAPTER 6. P-WAVE TIME-DOMAIN SIGNATURES IN TRANSVERSELY ISOTROPIC MEDIA

p=U.Zb S/KITI

p=O 22 s/kin 1.4

p-0 ,~:.~,~,,

1.2 1

:~,~,.~,,~

..i,.... ~.~z,.,

~,..~,~,~.~,,

~,~,.~

~ - ~ . , ~

~

~

p=0 i 0 s/kin 2.8

3

3.2

~,,,-.:~.~,~

....

3.4

Figure 6.6: Curves of the normalized traveltime f - to(p)/to(O) in equation (6.13) for a range of equivalent solutions and four different values of the ray parameter p. The model parameters are Vnmo(0)=3.29 km/s and r / = 0.0833. In the special case of elliptical anisotropy (e.g., the medium can be isotropic), t~i) (p) can be expressed through the corresponding zero-offset (vertical) time t~i) (p = 0) and the NMO velocity for a horizontal reflector. Using equations from Appendix 6A, the traveltime along an oblique ray that crosses a horizontal elliptically anisotropic layer can be found as

to(p) - t0(0) ~/1 + p2 Vn2mo(P) .

(6.12)

Since the NMO velocity (6.6) for elliptical media depends on only p and Vnmo(0), the traveltime to(p) is a function of the same parameters and the vertical time t0(0). If the horizontal layers are not elliptically anisotropic, the traveltime along an oblique ray can be expressed through the vertical (q) and horizontal (p) slowness components in the following way (Grechka, Tsvankin and Cohen, 1999):

to(p) = to(O) Vpo (q - pq') ;

(6.13)

q' - dq/dp. As illustrated by Figure 6.6, the term multiplied with t0(0) in equation (6.13) is a function of just Vnmo(0) and q: to(p) : t0(0 ) f ( p , Vnmo(0), T]),

(6.14)

where f (p, Vnmo(0), T]) ~ Vpo (q - p at) turns out to be independent of the individual values of the anisotropic coefficients ~ and 6. Therefore, the interval traveltime t~i)(p) in equation (6.11) is controlled by the vertical time t~i) (0) and, like the interval NMO velocity, by the interval values Vn(~o(0)

6.1. P-WAVE NMO VELOCITY AS A FUNCTION OF RAY PARAMETER

263

and ~(i). This means that the effective NMO velocity (6.11) as a whole is also a function of t~i) (0), Vn(~o(0) and r](i). Recall that the same conclusion holds for P-wave long-spread moveout from horizontal reflectors in stratified VTI media (Chapter 4).

6.1.3

3-D analysis using the NMO ellipse

In Chapter 3 it was shown that the azimuthally dependent NMO velocity of any pure-mode reflection is generally described by an ellipse in the horizontal plane (Figure 3.32). For a homogeneous VTI layer, the axes of the NMO ellipse are parallel to the dip (azimuth a - 0 ~ and strike (a - 90 ~ directions of the reflector [equation (3.57)]: Vnmo(a, p) - Vnmo (0, P) cos 2 a -~- Vn~n2o 2 , p

sin 2 a ,

(6.15)

where p, as before, is the ray parameter of the zero-offset reflection. According to the results of the previous sections, the dip-line NMO velocity Vnmo(a - 0, p) is controlled by the zero-dip value Vnmo(0) - Vnmo(a, 0) and the anellipticity parameter r]. Note that the weak-anisotropy approximation (3.61) for the strike-line NMO velocity in a VTI layer contains the anisotropic coefficients only in the form of the difference c - 5 ~ r], which is a strong indication that Vnmo(Tr/2,p) is a function of Vnmo(0) and r] as well. This conjecture is confirmed by the numerical analysis of the exact equation (3.62) performed by Grechka and Tsvankin (1998b) for a representative set of homogeneous VTI models, with a typical example displayed in Figure 6.7. In elliptical media (r/=0), the NMO velocity on the strike line is independent of dip and equal to Vnmo(0) [equation (3.60)]. The NMO ellipse (6.15) is fully governed by the normal-moveout velocities in the strike and dip directions, which means that the zero-dip NMO velocity and the parameter r] are sufficient to determine the P-wave NMO velocity for any orientation of the CMP line relative to the reflector strike. Furthermore, Figures 6.5 and 6.7 show that, in agreement with the weak-anisotropy approximation, the exact Vnmo(P) expressed in terms of the ray parameter has the following functional form (for a fixed azimuth a): Vnmo(P)-- Ynmo(0)F(p Vnmo(0), ~).

(6.16)

Hence, if Vnmo(P) is normalized by the zero-dip NMO velocity, it depends on just the product p Vnmo(0) rather than on these two parameters individually. This result is formally obtained below for TI media with arbitrary tilt of the symmetry axis [equation (6.21)]. While the coefficient r] is responsible for anisotropy-induced distortions of the NMO ellipse in all azimuthal directions, the influence of 77 decreases away from the dip plane. Indeed, comparison of Figures 6.5 and 6.7 indicates that the sensitivity of NMO velocity to r] is considerably higher on the dip line than on the strike line. The P-wave NMO ellipse for horizontally layered VTI media above a dipping reflector depends on the interval values of the parameters Vnmo(0) and U and interval

264

CHAPTER

6.

P-WAVE TIME-DOMAIN

SIGNATURES

IN T R A N S V E R S E L Y

ISOTROPIC

MEDIA

1.2 b

"" .9

4~

t I .I

~I1~ 119

. . . . . . . . . . . . . . . . .

9 " " "i

. . . . . . . . . . . . . . . . . . . .

N

",,~

9

1.05

...........

o Z

9

g g

:"S .

.

:

.

.

.

.

.

.

*s

pVnmo(0) Figure 6.7: The strike-line P-wave normal-moveout velocity Vnmo(~, P) in a VTI layer computed from the exact equation (3.62) and normalized by the zero-dip NMO velocity. The dips range from 0 ~ to 90 ~ (a) Widely different models with the same r / = 0.2: c = 0.1, 5 = -0.071 (solid); c = 0.2, ~ = 0.0 (dashed); ~ = 0.3, /~ = 0.071 (dotted). (b) Models with different r/: q = 0.1 (solid); r/ = 0.2 (dashed); 77 = 0.3 (dotted). vertical traveltimes. This result follows from the generalized 3-D Dix equation of Grechka, Tsvankin and Cohen (1999), which is outside of the scope of this book.

6.2 6.2.1

T w o - p a r a m e t e r description of t i m e processing Migration impulse response

Long-spread P-wave moveout of reflections from a horizontal interface, controlled by Vnmo(0) and r/, can also be regarded as the diffraction curve (accurate to a certain dip) on the zero-offset section. Since time migration is based on collapsing such diffraction curves to their apexes, the values of Vnmo(0) and q should be sufficient to generate a poststack time-migration impulse response that is accurate up to that certain dip. In contrast, poststack depth migration may produce depth errors if the value of Vpo is inaccurate; this is discussed further in this section and Chapters 7 and 8. N u m e r i c a l results for poststack time migration Anisotropic migration here is performed using Gazdag's (1978) phase-shift algorithm modified for VTI media (see Chapter 8, section 8.3). Figure 6.8 shows the exact zero-offset time-migration impulse responses (right half only) for different equivalent solutions (ES) from Figures 6.1 (a) and 6.3 (b). The curves for all three ES practically coincide with each other, implying that Vnmo(0) and r/ are sufficient to generate an accurate zero-offset time-migration impulse response for all dips; this conclusion holds

6.2. TWO-PARAMETER DESCRIPTION OF TIME PROCESSING

265

a (D

v

1

t ~ j

/

6

Figure 6.8: Anisotropic zero-offset time-migration impulse response for equivalent solutions from (a) Figure 6.1, and (b) Figure 6.3. The three curves on each plot correspond to the solutions with Vpo = 2.8 km/s (solid black line), 3 km/s (solid gray), and 3.2 km/s (dash-dotted black). for vertically heterogeneous (e.g., stratified) VTI media as well. Actual imaging results are displayed in Figure 6.10, which shows anisotropic poststack time migrations of synthetic zero-offset data generated for the model in Figure 6.9 with Vp0=3.0 km/s, e=0.2 and 6=0.1. The migrated sections were obtained using (a) the actual model parameters, and (b) an equivalent solution with Vp0=2.6 km/s, e=0.433, and 6=0.3. Although the second model is substantially different from the actual one, it has the correct values of Vnmo(0) and ~ and, consequently, produces an accurate image (the artifact below the steep reflector is caused by the limited aperture). Therefore, with accurate values of Vnmo(0) and q, all lateral position errors in poststack time migration (Larner and Cohen, 1993; Alkhalifah and Larner, 1994) will be eliminated. Depth migration, however, will produce erroneous results if the wrong value of Vpo (implying that 6 is also wrong) were used. Such depth errors AD can be described by AD -

(vest) ~vPO po-1

D,

(6.17)

where l?'eStvpo is the estimated vertical velocity, and D is the true depth. Hence, the difference between the poststack depth migration impulse responses of models with the same values of Vnmo(O) and r] is limited to a simple depth stretch. S e n s i t i v i t y of m i g r a t i o n to As illustrated by Figure 6.11, errors in q lead to noticeable distortions in migrated images. Predictably, the distortions are more pronounced for the model with a larger error in q" not only do the reflectors cross, but also the reflector edges are not imaged

266

C H A P T E R 6. P-WAVE TIME-DOMAIN SIGNATURES IN TRANSVERSELY ISOTROPIC MEDIA

o

i E li

t

v

r f:l.

D

,

|

!

Midpoint (km) Figure 6.9: Homogeneous VTI model with reflectors dipping at 20 ~ 40 ~ and 60 ~ well (Figure 6.11a). The dipping events could be imaged better by increasing the value of Vnmo(0), but then horizontal reflectors would go out of focus. In many cases one can estimate the accuracy of the velocity field used in the migration (or of the migration algorithm itself) by inspecting the quality of the migrated image. For example, parabolic shapes, resulting from diffracting edges, imply overmigration, whereas hyperbolic shapes are a sign of undermigration. The sensitivity of the migration results in VTI media to the value of r/indicates that this approach can be used in parameter estimation for vertical transverse isotropy. In isotropic media, undermigration is usually corrected by increasing the migration velocity. According to equation (6.10), the corresponding correction in VTI media can be achieved by increasing r/ and, therefore, the horizontal velocity (the case in Figure 6.11, where too small values of ~ cause undermigration). If the image indicates overmigration, ~ should be reduced. However, if there is more confidence in the measured value of the NMO velocity for the dipping reflector, a proper choice would be to change both Vnmo(0) and q. In fact, given that error likely exists in both, the data processor has two parameters to be adjusted. Further discussion of the sensitivity of DMO/migration results to the VTI parameters can be found in Chapter 8. 6.2.2

Brief summary

All homogeneous VTI models with fixed values of the zero-dip NMO velocity Vnmo(O) and anellipticity parameter r/have the same NMO velocity Vnmo(P)in the prestack domain and the same time-migration impulse response in the poststack domain. There-

6.2. T W O - P A R A M E T E R D E S C R I P T I O N OF TIME PROCESSING

!

.

.

.

267

.

1

2

Figure 6.10: Anisotropic poststack time migrations of synthetic data generated for the model in Figure 6.9 using (a) the actual model values of Vpo = 3.0 km/s, ~ = 0.2 and 6 = 0.1; and (b) an equivalent solution Vpo = 2.6 km/s, ~ = 0.433 and 6 = 0.3 that has the same Vnmo(0)=3.29 km/s and r/=0.0833.

C H A P T E R 6. P-WAVE T I M E - D O M A I N SIGNATURES IN TRANSVERSELY I S O T R O P I C MEDIA

268

r 0

2

.

.

.

.

!

.

.

.

.

.

.

.

.

.

.

-"!

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

i

_

Figure 6.11" Anisotropic poststack time migrations of synthetic data for the model in Figure 6.9 using distorted values of 7: (a) 7=0.01, (b) 7=0.06; the actual r] - 0.0833. In both cases Vnmo(0)=3.29 km/s, the correct value.

6.3.

DISCUSSION: NOTATION AND P-WAVE SIGNATURES IN VTI MEDIA

269

fore, those models should also have the same time-migration impulse response in the prestack domain. We conclude that knowledge of Vnmo(0) and r] is sufficient to perform all conventional time-processing steps including normal-moveout (NMO) and dip-moveout (DMO) correction, and prestack and poststack time migration. In contrast, time-to-depth conversion and depth imaging in general require an accurate value of the vertical velocity, which cannot be obtained from P-wave reflection traveltimes alone. Likewise, P-wave time processing for horizontally layered VTI media above a dipping reflector is controlled by the interval values of the parameters Vnmo(0) and r]. It should be emphasized, however, that this two-parameter description breaks down if the overburden (i.e., the medium above the reflector) contains dipping interfaces, lateral velocity gradient or any other kind of lateral heterogeneity. In this case, Pwave NMO velocity and other time-domain kinematic signatures become dependent on the individual values of the vertical velocity and the parameters e and 5 (Le Stunff et al., 1999). The entire isotropic time-imaging sequence (NMO-DMO-time migration) remains valid for elliptically anisotropic media, where ~ = 0 (see also Helbig, 1983; Dellinger and Muir, 1988). The difference between isotropy and elliptical anisotropy becomes important only in time-to-depth conversion or depth migration. Since for elliptical media the velocity Vnmo(0), conventionally used in converting time to depth, does not coincide with the true vertical velocity, isotropic algorithms will result in the wrong reflector depth.

6.3

Discussion" N o t a t i o n and P - w a v e signatures in V T I m e d i a

The previous section concludes our analysis of kinematic signatures and body-wave amplitudes in VTI media. Since the results for vertical transverse isotropy were included in all preceding chapters, it is appropriate to summarize and review them here. The main emphasis of the discussion below is on P-wave signatures because P-waves constitute a majority of data being acquired in oil and gas exploration. 6.3.1

Advantages

of Thomsen

parameters

A proper choice of parameterization is extremely important in understanding the behavior of seismic signatures in anisotropic media. Throughout the book, velocities, polarizations and amplitudes of body waves in VTI media are expressed using Thomsen's (1986) notation. Some advantages of Thomsen parameters, explained in his original paper, were discussed in Chapter 1. A major misconception about Thomsen notation is that it is useful for only weak anisotropy. The results listed below show that application of Thomsen parameters is helpful in solving a wide range of practically important seismological problems in TI media with any strength of velocity anisotropy.

270


1. It is possible to reduce the number of independent parameters needed for Pwave traveltime analysis because the shear-wave vertical velocity Vso has a weak (usually negligible) influence on P-wave kinematic signatures, even in media with strong velocity variations. Although the P-wave reflection coefficient does depend on the jump in Vso across an interface, the contribution of transverse isotropy to the reflection coefficient is practically independent of Vso (see Chapter 2). Therefore, the influence of vertical transverse isotropy on P-wave propagation is largely controlled by just the P-wave vertical velocity Vpo and two anisotropic coefficients, e and 6, with Vpo being no more than a scaling coefficient in homogeneous media. This facilitates our understanding of anisotropic wave propagation and implementation of inversion and processing algorithms. 2. Traveltime inversion of P-wave data using the conventional notation is always ambiguous because the trade-off between the parameters c13 and c~5 = c 4 4 can never be resolved, unless some independent information about one of these coefficients is available. This ambiguity is explained by the fact that c13 and c55 contribute to P-wave phase and group velocity only through the parameter 6. 3. Basic P-wave signatures, such as phase velocity (a fundamentally important function), NMO velocity from a horizontal reflector and nonhyperbolic (longoffset) reflection traveltime, can be concisely expressed through the anisotropic parameters e and 6. 4. A simple combination of Thomsen parameters, denoted as r/, controls the influence of vertical transverse isotropy on all P-wave time-domain processing steps, including NMO and DMO correction, prestack and poststack time migration. 5. The coefficients e, 6, and 7 are well-suited for developing the weak-anisotropy approximation for a wide range of seismic signatures, including body-wave amplitudes. Systematic application of the weak-anisotropy approximation provides valuable analytic insight into the influence of transverse isotropy on seismic wavefields. This book is devoted mostly to the "generic" seismic signatures (such as phase velocity and polarization vector), as well as signatures important in reflection seismology and vertical seismic profiling. Analysis and inversion of seismic data may benefit from introducing different parameterization for specific acquisition geometries. For instance, P-wave traveltimes measured in cross-hole surveys usually correspond to near-horizontal rays and, therefore, are weakly dependent on near-vertical velocity variations. In this case, it may be more useful to describe the medium by the horizontal velocity and a combination of the anisotropic parameters responsible for the phase-velocity variation near the horizontal.

6.3.2

Influence of vertical transverse isotropy on P-wave signatures

The description of P-wave signatures above included phase, group, and normalmoveout velocities, nonhyperbolic moveout, time-migration impulse response, polar-

6.3.

DISCUSSION: NOTATION AND P-WAVE SIGNATURES IN VTI MEDIA

271

ization vector, radiation pattern and reflection coefficient. The analysis was based on a series of analytic solutions valid for VTI media with arbitrary strength of the anisotropy (the only exception is the reflection coefficient, which was studied in only the weak-anisotropy approximation). Although these solutions are convenient to implement numerically, some of them are not simple enough to elucidate the relationships among seismic phenomena and the parameters of VTI media. To gain insight into the influence of anisotropy on seismic signatures, we systematically applied the weakanisotropy approximation and checked its accuracy by comparing it with the exact equations. The contribution of vertical transverse isotropy to P-wave signatures is controlled by two dimensionless anisotropic parameters - c and 5, with the P-wave vertical velocity Vpo being a scaling coefficient (in homogeneous media). P-wave signatures of most interest in reflection seismology can be divided into two main groups. For the first group, which includes the NMO velocity for horizontal reflectors, small-angle reflection coefficient, and point-force radiation in the vertical (symmetry) direction, the influence of transverse isotropy is determined entirely by the parameter 5. The second group is comprised of the dip-dependent NMO velocity, nonhyperbolic reflection moveout, time-migration impulse response, reflection coefficient and radiation pattern in the angular range 0-40 ~ The influence of VTI on these signatures is determined by both anisotropic parameters (~ and 5) and (with the exception of the reflection coefficient) is primarily governed by the difference ~ - ~, i.e., by deviations from the elliptically anisotropic model. Due to the high sensitivity of P-wave signatures to e - ~, application of the elliptical-anisotropy approximation in P-wave processing may lead to significant distortions, even if the medium is relatively close to elliptical. For laterally homogeneous VTI models above a dipping target reflector, the influence of the anisotropy on time-domain processing is determined by the effective parameter r / - ( c - 5)/(1 + 25). In combination with the zero-dip normal-moveout velocity V~mo(0), U controls the P-wave NMO velocity expressed as a function of the ray parameter, nonhyperbolic moveout from a horizontal reflector, and poststack and prestack time-migration impulse responses. Thus, P-wave reflection traveltimes depend just on the vertical time, Vnmo(0) and 7/and, therefore, are insufficient to obtain the true vertical velocity required for timeto-depth conversion. In some cases, the vertical velocity can be determined directly if check shots or well logs are available. Other sources of information about the vertical velocity are reflection traveltimes of SV- or PSV-waves (see Chapter 7). The sets of VTI parameters required for time and depth P-wave imaging, as well as for AVO analysis, are given in Table 6.1. Clearly, there is no simple correlation between the strength of P-wave velocity anisotropy and the influence of vertical transverse isotropy on reflection moveout and amplitude. The overall magnitude of P-wave velocity variations is usually determined by the value of e (unless lel is much smaller than 151), but the parameters of more importance in seismic processing are 5 and ~. Hence, the terms "weak anisotropy" or "strong anisotropy" are meaningless with-

272


Full set

Velocities, depth imaging

Time imaging

AVO (intercept, gradient)

VPO

YPo

Ynmo (0)

VPO

or r/ 5

e or 77 5

r/ -

5

Vso

Vso

Table 6.1: P-wave parameters for vertical transverse isotropy. The column "Time imaging" refers to laterally homogeneous VTI models above the reflector. out a reference to a particular problem. For instance, while the model with e=0.1, 5--0.1 is weakly anisotropic in terms of velocity variations, it can be characterized as strongly anisotropic regarding the distortions of the P-wave radiation pattern (see section 2.2).

6.4

Moveout

a n a l y s i s for t i l t e d s y m m e t r y

axis

As discussed in Chapter 3, tilted transverse isotropy (TTI) has been recognized as a common feature of dipping clastic sequences in overthrust areas, such as the Canadian Foothills, and near salt bodies in the Gulf of Mexico and the North Sea (e.g., Isaac and Lawton, 1999; Vestrum et al., 1999). The results of Chapter 3 show that NMO velocity in TI media is quite sensitive to the tilt of the symmetry axis away from the vertical. Therefore, an important question is the extent to which the two-parameter representation of P-wave NMO velocity and other time-domain signatures holds (at least, for a certain range of mild tilt angles) for TTI media.

6.4.1

N M O velocity as a function of ray parameter

The analysis here is restricted to the vertical symmetry plane of a single TTI layer (i.e., the plane that contains the symmetry axis). As in section 3.3, NMO velocity is obtained from the exact 2-D NMO equation (6.3), but this time as a function of the ray parameter of the zero-offset ray. First, we treat normal moveout for elliptical anisotropy and then proceed with a numerical study for general TTI media. Elliptical

anisotropy

The tilt of the symmetry axis in elliptically anisotropic models has a significant influence on the P-wave NMO velocity expressed through the dip r [see equation (3.35)]. However, if the dip is replaced with the ray parameter p, the NMO velocity takes exactly the same form (6.6) as in isotropic media or elliptical media with a vertical symmetry axis (Appendix 6B): Vnmo (p) -

Vnm~ (0)

V/1 _ p2 Vn2mo(0)

.

(6.18)

6.4. MOVEOUT ANALYSIS FOR TILTED SYMMETRY AXIS

273

Hence, the dependence of NMO velocity on the ray parameter remains "isotropic" for any tilt of the elliptical axes despite the complexity of the function Vnmo(r in elliptical models [equation (3.35)]. The contribution of anisotropy (including the tilt) in equation (6.18) is hidden in the values of the zero-dip NMO velocity Vnmo(0) [equation (6.35)] and the ray parameter p. Since reflection moveout for elliptical anisotropy with any orientation of the axes is purely hyperbolic (Uren et al., 1990b), all isotropic time-related processing methods (NMO, DMO, time migration) are entirely valid in elliptical media (see also Helbig, 1983; Dellinger and Muir, 1988). Isotropic depth processing, however, will image reflecting interfaces at the wrong depths because the zero-dip NMO velocity for elliptical anisotropy differs from the true vertical velocity. The above conclusions, including equation (6.18), are also fully applicable to SH-waves in general (non-elliptical) TI models. Equation (6.18) implies that deviations from the isotropic Vnmo(P) dependence for any tilt should be dependent primarily on the difference c - 5 that quantifies the degree of anellipticity of TI media. It is clear, for instance, that the anisotropic term in the weak-anisotropy approximation for Vnmo(P) should contain ~ and 6 only in the combination e - 5. Otherwise, this term would not vanish for elliptical anisotropy, which corresponds to e = 5. General T T I media Before analyzing numerical results for non-elliptical TTI models, it is important to identify the parameters that control the dependence of the P-wave NMO velocity on p. P-wave phase velocity in the vertical symmetry plane of TTI media can be represented as the product of the velocity in the symmetry direction (Vpo) and an anisotropic term dependent on the tilt v and the anisotropic parameters ~ and defined in the coordinate system associated with the symmetry axis (the contribution of Vso can be ignored). Then, according to equation (6.5), the dip r can be written as (6.19)

r -- fl (pVpo, e, 5, v), which yields the following list of parameters for the exact velocity tion (6.3)' Vnmo- Vpo f2(r e, 6, v ) - Vpo f3(pVpo, e, 5, v).

Vnmo

from equa(6.20)

According to equation (6.20), the NMO velocity normalized by the zero-dip value can be represented by Wnmo(P) Vnmo (0)

-- f4(pWpo, e, (5, z / ) - fs(pVnmo(O), e., ~, p).

(6.21)

The functions fl-f5 can be found by substituting the phase-velocity function into equation (6.3); however, implicit definitions are sufficient in this discussion.

274


Equation (6.21) shows that the contributions of the ray parameter and the zero-dip NMO velocity are absorbed by the term pVnmo(0). This result was discussed above for VTI media, where the NMO velocity depends just on pVnmo(0) and 77 [equation (6.16)]. Essentially, changes in Vnmo(0) in TTI media (for fixed anisotropic coefficients and tilt) just stretch or squeeze the NMO velocity curve expressed through the ray parameter. Therefore, it is convenient to use pVnmo(0) as the argument and study the influence of the parameters e, 5 and ~ on the NMO velocity. In the examples below, the Pwave velocity Vnmo(P) ("the DMO signature") is calculated from equation (6.3) as a function of pVnmo(0) (for dips ranging from 0 ~ to 70 ~ and is normalized by the isotropic expression (6.18) to isolate the distortions caused by the anisotropy. Figure 6.12 shows that the influence of the shear-wave velocity Vso on P-wave normal moveout, ignored in equation (6.21), is indeed small. Some difference between the NMO velocities for the two models that span a wide range of Vpo/Vsoratios is noticeable only for a horizontal symmetry axis (v = 90~ However, this separation between the curves is caused mostly by the (weak) influence of Vso on the zerodip NMO velocity Vnmo(0). The difference in Vnmo(0) leads to a small horizontal shift between the curves which gets amplified when the NMO velocity is divided by the isotropic equation (6.18) (the normalization mitigates the increase in the NMO velocity at large dips). The introduction of the ray parameter and normalization by equation (6.18) [equivalent to the replacement of the true dip with an "apparent" dip, see equation (3.27)] led to substantial changes in the character of the NMO-velocity curves compared to those given in Chapter 3. Note, for example, that the normalized NMO velocity becomes almost identical for the vertical (- = 0) and horizontal (, = 90 ~ orientations of the symmetry axis (Figure 6.13). Comparison of Figure 6.13 with Figure 3.24 shows that the changes are especially pronounced for positive ~, which correspond to the symmetry axis tilted towards the reflector. For instance, for a tilt of 45 ~ and a dip of 45 ~ the value of pVnmo(O),which represents the sine of the apparent dip angle, is close to 0.9, which is far different from the actual sin r ~ 0.71. With a further increase in dip, pVnmo(0) approaches unity (for both ~ = 45 ~ and = -45~ and the isotropic expression (6.18) goes to infinity at dips well below 90 ~ (and below the dip Cm~x for which the exact NMO velocity becomes infinite), leading to a sharp decrease in the normalized NMO velocity at ~ = +45 ~ (Figure 6.13).

6.4.2

P a r a m e t e r r] for t i l t e d axis of s y m m e t r y

For vertical transverse isotropy, the dependence of P-wave NMO velocity on the ray parameter is fully controlled by the product pVnmo(0) and parameter r] defined in equation (6.2). If the symmetry axis is tilted, the function Vnmo(P) depends on pVnmo(0), ~, ~, and the tilt ~ [equation (6.21)]. The question to be addressed next is whether or not the influence of c and 5 on the NMO velocity is still absorbed by the parameter r/.

6.4. MOVEOUT ANALYSIS FOR TILTED SYMMETRY AXIS

e=0.3

275

~-0.1

v=_-45 ~

v=_O~

1.2

2.2

0

.

O

.

.

9

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9

,

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.

9

~ 1.4

~

:

.

iiiii

0

.

Z

0:4

pVnmo(O)

pVnmo(O)

1.2 0

1.0 0

10

pVnmo(O) Figure 6.12: The influence of Vso (the shear-wave velocity in the symmetry direction) on the P-wave NMO velocity computed from equation (6.3) and normalized by the isotropic expression (6.18). The curves correspond to models with Vpo/Vso=l.5 (black) and Vpo/Vso=2.5 (gray); for both models, c=0.3, 5=0.1. The dips here and on the subsequent plots range from 0 ~ to 70~ the tilt v of the symmetry axis is marked on the plots.

276

CHAPTER

6.

P-WAVE

TIME-DOMAIN

SIGNATURES

IN TRANSVERSELY

ISOTROPIC

MEDIA

e-5=0.2 v=_O ~

1.0

2.2

o

.

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.

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o Z

1.0 pVnmo(O)

o'.4 0'.6 pVnmo(O)

v=_45 ~

v=90 ~

_

~

~

.

:

:

~

. . . . . i. . . . . . : . . . ~

"O

"

:

:

i

# EO.40

z

0.2

F i g u r e 6.17:

Normalized

P-wave

NMO

0

v e l o c i t y for m o d e l s w i t h q -

-0.1:

6 - 0 . 1 2 5 (solid b l a c k ) ; c - 0.1, 6 - 0.25 ( g r a y ) ; ~ - 0.2, ~ - 0 . 3 7 5 ( d a s h e d ) .

c -

0,

6.4. M O V E O U T ANALYSIS F O R TILTED SYMMETRY AXIS

281

VTI media and make the DMO signature almost isotropic for a wide range of reflector dips (Figure 6.18). For the character of the normalized P-wave NMO velocity to be close to that in VTI media, the tilt should not exceed 10-15 ~ For small tilt angles within this range, the NMO velocity grows with p much faster than in isotropic media (if r] > 0), and this increase is governed largely by r]. However, it should be mentioned that even for L, = i 1 0 ~ the anisotropic signature is less pronounced than that for VTI media and the r/control over the DMO signature is less tight. At tilt angles of about +30 ~ the signature gets reversed, and the normalized NMO velocity decreases with p.

6.4.3

Discussion

In summary, only if the medium is elliptically anisotropic (e = 5), is the dependence of the NMO velocity on the ray parameter (the "DMO signature") for tilted transverse isotropy described by the same equation as for VTI (and isotropic) media. Since reflection moveout in elliptical media is purely hyperbolic, all isotropic time-related processing methods (NMO, DMO, time migration) are entirely valid for elliptical anisotropy with any orientation of the elliptical axes. Also, the NMO velocity from horizontal reflectors in elliptical models remains close to the horizontal phase velocity. In non-elliptical TTI media, the function Vnmo(P) has the same character as for vertical transverse isotropy for only a narrow range of tilt angles corresponding to near-vertical and near-horizontal orientations of the symmetry axis. A tilt of +20 ~ away from the vertical is sufficient to eliminate the anisotropic D MO signature almost entirely and make the NMO velocity much less controlled by 77. At larger tilts of • 55) ~ the signature gets reversed, with Vnmo(P) increasing slower than in isotropic media (for positive ~) and being fully governed by r/ for just small and moderate values of this parameter (1771-< 0.15 - 0.2). Although the above analysis was performed for a single TTI layer, the results can be extended to symmetry planes of vertically heterogeneous media by using the 2-D Dix-type equation (3.47) introduced in Chapter 3. A more general, 3-D treatment of dip-dependent P-wave NMO velocity in TTI media can be found in Grechka and Tsvankin (2000). They show, for example, that the NMO-velocity inversion for mild tilt angles produces a family of models suitable for time-domain processing of Pwave data. In contrast to VTI media, however, these models are not described by a constant value of ~.

282

CHAPTER

6.

P-WAVE

TIME-DOMAIN

SIGNATURES

IN TRANSVERSELY

ISOTROPIC

MEDIA

11=0.2 v=_-20 ~

v=_-lO ~

1.6 0 ~::

2.2 9

.

0

,'-

E r

._N

._.

E 1.4 O

z

1.0

v=10 ~ 2.2 1.8 E 1.4~,= 0 Z

pVnmo(0)

>~ 1.4

~N 1.2 ~1.0

~0 0 4

z

F i g u r e 6.18: N o r m a l i z e d P - w a v e N M O v e l o c i t y for m i l d tilt a n g l e s a n d r/=0.2: ~ = 0.1, 5 - - 0 . 0 7 1 (solid black); e - 0.2, 5 - 0 (gray); e - 0.3, 5 - 0.071 ( d a s h e d ) .

6A. DEPENDENCE OF NMO VELOCITY IN VTI MEDIA ON THE RAY PARAMETER

283

Appendices for Chapter 6 6A 6A.1

D e p e n d e n c e of N M O velocity in V T I media on the ray parameter Building the function Vnmo(P)

Normal-moveout velocity in an anisotropic layer can be recast as a function of the ray parameter p (horizontal slowness) corresponding to the zero-offset reflection [equation (6.3)]. For VTI media, phase velocity of P- and SV-waves in terms of the phase angle 0 with the symmetry axis is described by equation (1.43). Substituting the vertical (q = cos O/V) and horizontal (p = sin0/V) slownesses into equation (1.43) yields 2p -- (Cll + C55) p2 _~_ (C33 ~_ C55) q2

i ~[(cll - c55)p2 _ (c33 c55)q212 + 4(c13 + c55)2 p2 q2,

(6.22)

where the plus in front of the radical corresponds to the P-wave. An equivalent form in Thomsen notation can be obtained from equation (1.59). For a known value of the ray parameter p, equation (6.22) becomes quadratic in qe. After q has been found, the derivatives dq/dp and d2q/dp 2 can be computed by implicit differentiation of equation (6.22). The dependence Vnmo(P) can also be built parametrically by calculating Vnmo and p as functions of the phase angle 0 = r

6A.2

Elliptical anisotropy

Next, NMO velocity is expressed through the ray parameter for the special cases of elliptical and weak anisotropy. The normal-moveout velocity in elliptically anisotropic media (e = 5) can be represented as [equation (3.19)] Ynmo((~)- Ynmo(0) V p ( r : cos r Vpo

Ynmo(0) tan r p Vpo

(6.23)

The P-wave phase velocity for elliptical anisotropy, expressed through ~ = 5, is given by equation (1.57),

Vp(O) = Vpo V/1 + 25 sin20 ,

(6.24)

and sin r _ sin r P(r = Vp(r - Vpo ~/1 + 25 sin 2 r

(6.25)

284


Solving equation (6.25) for the dip r yields s i n e = ~/1

p Vpo

(6.26)

25p2V~ 0"

Calculating tan r from equation (6.26), substituting it into equation (6.23) and taking into account that Vnmo(0) - Vpo x/1 + 25 leads to Unmo (P) -

Vnmo (0)

.

(6.27)

V/1 - p2V~mo(0) Therefore, for elliptical anisotropy NMO velocity is the same function of the ray parameter and zero-dip moveout velocity as in isotropic media. 6A.3

Weak

transverse

isotropy

The P-wave NMO velocity in weakly anisotropic (lel

0.3 0

Squared offset (km 2 )

4.7~.1.2S.

Figure 7.25: Nonhyperbolic moveout inversion of the traveltimes recorded by a downhole receiver at Vacuum Field. (a) Traveltimes for different source positions (dots) and the best-fit nonhyperbolic moveout curve from equation (7.15). (b) Root-meansquare time residuals calculated for different pairs (Vnmo, Vhor) using equation (7.15). The offsets and traveltimes were doubled to simulate a reflection experiment. The residual for the best-fit model at the center of the contours is equal to 6.7 ms.

7.2.

I N V E R S I O N OF P - W A V E N O N H Y P E R B O L I C M O V E O U T

329

traveltime difference in Figures 4.10 and 7.18. The center of the contours corresponds to an rms residual of 6.7 ms; if Vnmo and Vhor lie on the nearest contour line, the rms residual is equal to 8 ms, a difference of only 1.3 ms. The family of kinematically equivalent models associated with this contour line is in agreement with the modeling results above, whereas the large magnitude of the time residuals is caused by a sizeable scatter in the traveltime measurements. The error bars on r/ can be estimated by picking the extreme values of this parameter corresponding to the 8 ms contour line (8 ms is close to the minimum rms residual plus one time sample); this yields the range 0.09 < 77 < 0.32. Since the receiver depth in this case was known, it was possible to calculate the average vertical velocity Vpo = 3457 m/s from the zero-offset traveltime and, using Vnmo, determine 5 = 0.02; the corresponding value of e is 0.22. Note that e can be determined directly from the vertical and horizontal velocities, without involving an estimate of r]. Therefore, the error in e depends just on the uncertainty in Vho~ and is not influenced by the trade-off between Vnmo and Vhor. Unfortunately, since only one receiver was deployed, the data did not constrain the vertical variation in the medium parameters, and the inverted anisotropic coefficients should be regarded as effective values that could be distorted by the vertical (and maybe lateral) heterogeneity above the receiver. Still, the results of nonhyperbolic moveout inversion are indicative of non-negligible anisotropy in the predominantly carbonate section above the reservoir at Vacuum Field. This conclusion is supported by reflection data acquired in the vicinity of the borehole. Offshore reflection d a t a The next example shows the inversion of long-spread P-wave reflection data for the vertically varying parameter r/and the horizontal velocity. The results below were obtained by Alkhalifah (1997b) using an offshore data set acquired in West Africa by Chevron Overseas Petroleum, Inc. The section in Figure 7.26 is well-suited for nonhyperbolic moveout inversion because it is dominated by strong horizontal (or subhorizontal) events recorded over a wide range of vertical times. The maximum offset reaches 4.3 km, which ensures satisfactory resolution of the nonhyperbolic semblance analysis for relatively shallow reflectors. The measurements of the horizontal velocity and r/ at later times, however, suffer from the insufficient magnitude of nonhyperbolic moveout due to smaller spreadlength-to-depth ratios. Figure 7.27 shows four typical nonhyperbolic semblance panels computed for different vertical times using equation (7.17) with C = 1. At the earlier times (1.24 s and 1.86 s), the semblance maximum (dark) is sufficiently narrow and yields good estimates of both the NMO and horizontal velocity. Proper stacking of shallow events in conventional processing (Figure 7.26) was achieved by muting out offsets larger than reflector depth. These offsets, however, contain crucial information for the nonhyperbolic velocity analysis and estimation of the Vhor and 7- In contrast, for reflection times exceeding 2 s the semblance maximum extends in the direction almost perpen-

330

C H A P T E R 7. V E L O C I T Y ANALYSIS AND P A R A M E T E R E S T I M A T I O N F O R V T I M E D I A

o9

v

E K-

Figure 7.26" Time-migrated seismic line from offshore Africa (the same area as in Figure 7.9). dicular to the Vnmo-aXis, which indicates that the semblance function is not sensitive enough to the horizontal velocity (compare with the synthetic result in Figure 7.21a). The effective velocities Vnmo and Vhor were picked at a number of prominent semblance maxima and used in the integral version (for continuous time dependence of all parameters) of the inversion method described above. The inverted interval parameters Vnmoand r/are shown as the black curves in Figure 7.28. The gray curves mark the upper and lower limits of the possible parameter values corresponding to the uncertainties in picking Vnmo and Vhor on the semblance panels. For example, the horizontal velocity picked at the semblance maximum for the vertical time to = 1.86 s in Figure 7.27 is equal to 2.05 km/s. If, instead, the picks are made at the edges of the black area, Vhor will deviate by about +0.04 km/s (+2%) from the best-fit value. Then Vhor = 2.01 and 2.05 km/s and the corresponding NMO velocities, along with values obtained in the same way at other vertical times, are used in the inversion algorithm to generate the gray curves in Figure 7.28. Clearly, the accuracy of the nonhyperbolic moveout inversion decreases with depth due to the lower resolution in Vhor and 7; for smaller spreadlength-to-depth ratios. Therefore, while we can trust the general increase in q up to about 1.8 s, the shape of the q-curve at later times to > 2s is unreliable. The interval of relatively high 7/

7.2. INVERSION OF P-WAVE NONHYPERBOLIC MOVEOUT

to=l .24 s.

.

.

.

331

~""

.

1.7 ~"

!i

_

i'~ 84

~

ii,

i,i ~

"

i"

_.

ii ~

.

~

- , .~ ,,i

Figure 7.27: Nonhyperbolic semblance velocity analysis at CMP location 300 in Figure 7.26 for different vertical times to. The shade becomes darker with increasing semblance. Contours of constant q values are shown in gray.

332

C H A P T E R 7. V E L O C I T Y ANALYSIS A N D P A R A M E T E R E S T I M A T I O N F O R V T I M E D I A

Figure 7.28" Interval values of Vnmoand q (black curves) as a function of the vertical time at CMP location 300. The two gray curves on each plot delineate the range of possible parameter values associated with picking uncertainties.

7.2. INVERSION OF P-WAVE NONHYPERBOLIC MOVEOUT

333

values between 1 s and 2 s in Figure 7.28 is associated with a shale formation known to exhibit substantial anisotropy (transverse isotropy) throughout the region. It should be mentioned that lateral velocity variation near the seismic line used in the inversion is rather insignificant and is not expected to distort the parameterestimation results obtained for a 1-D VTI model. A more detailed case study based on the inversion of nonhyperbolic moveout in the same area can be found in Toldi et

(1999) 7.2.4

Discussion

Nonhyperbolic semblance analysis based on the moveout equations introduced in Chapter 4 can be used to estimate the horizontal velocity and parameter r/ from long-spread P-wave traveltimes. The high accuracy of the nonhyperbolic moveout equation, however, does not guarantee stable inversion results, even in the simplest model of a single VTI layer. The horizontal velocity in VTI media becomes relatively well-constrained by reflection traveltimes if the spreadlength exceeds twice the reflector depth. There is, however, a certain degree of tradeoff between Vhor and Vnmo caused by the interplay between the quadratic and quartic term of the moveout series. Since the errors in Vhor and Vnmo have opposite signs, the absolute error in the parameter r/(which depends on the ratio Vhor/Vnmo)t u r n s out to be at least twice as large as the percentage error in Vhor. Therefore, the inverted value of r/is sensitive to small correlated errors in reflection traveltimes, with moveout distortions on the order of 3-4 ms leading to errors in r/up to +0.1, even in the simplest model of a single VTI layer. Nonhyperbolic moveout in vertically heterogeneous VTI media is caused not only by anisotropy, but also by the vertical variation in the elastic coefficients. The interval values of Vhor and r/ may be obtained via a Dix-type differentiation procedure that involves the nonhyperbolic moveout term. In essence, the above conclusions about the accuracy in the estimation of Vhor and rl remain valid for stratified media as well. The interval horizontal velocity for spreadlengths close to twice the reflector depth can be recovered with an accuracy comparable to that for NMO velocity (the error, of course, becomes high for thin layers). The trade-off between Vnmo and Vhor gets amplified by the layer-stripping process, causing large errors in the interval r/values. An important practical issue is whether or not it is possible to use the parameters obtained from nonhyperbolic moveout inversion for seismic processing. Since all kinematically equivalent models provide a good approximation for long-spread moveout (despite having substantially different values of r/), they are suitable for poststack time migration. However, the need for imaging arises in the presence of structure (i.e., dipping interfaces), and it is preferable to use the imaging targets themselves for anisotropic parameter estimation by means of the more stable DMO-inversion method introduced above. 2-D DMO inversion on dip lines may experience difficulties in using very steep events (e.g., dip up to 90 ~ and beyond, as the flanks of salt domes), with their small

334

C H A P T E R 7. V E L O C I T Y A N A L Y S I S A N D P A R A M E T E R E S T I M A T I O N F O R V T I M E D I A

magnitude of reflection moveout. Since the horizontal velocity, primarily needed to image such steep dips, is relatively well-constrained by long-spread moveout, the results of nonhyperbolic moveout inversion in this case may be used to build a starting model for anisotropic migration. Another alternative, discussed above, is to obtain the horizontal velocity (or r/) from the NMO velocity on a line parallel to the strike of a steep reflector. Nonhyperbolic moveout inversion may be more difficult to use in lithology discrimination. In the absence of pronounced vertical heterogeneity, the value of r/obtained from nonhyperbolic moveout can be considered as a crude measure of anisotropy (i.e., anellipticity) above the reflector, and, therefore, as a lithology indicator (see the case studies in Chapter 8). However, the potentially large errors in interval r/values may complicate a more detailed lithology analysis based on nonhyperbolic moveout inversion. Unfortunately, the horizontal velocity is less suitable for lithology discrimination compared to r/. The conditions needed for a meaningful inversion of nonhyperbolic moveout for r/include large offsets exceeding 2-2.5 reflector depths, uniformly high data quality, negligible lateral heterogeneity, and a significant contrast in r/between layers of interest (e.g., such as that between shales and sands). A potentially interesting application of nonhyperbolic moveout inversion is estimation of the parameter e from reflection data when the vertical velocity is known (e.g., from check shots or well logs). The large errors in the coefficient r/stem from the interplay between the horizontal and NMO velocity, whereas the vertical velocity can be obtained independently. Therefore, the vertical velocity can be combined with the horizontal velocity recovered from long-spread moveout to provide an estimate of that will not be influenced by the error in Vnmo.

7.3

Joint i n v e r s i o n of P a n d P S d a t a

As discussed in Chapter 6 and earlier in this chapter, P-wave reflection moveout for laterally homogeneous VTI media above the reflector provides enough information to build velocity models in the time domain and to carry out P-wave time imaging. The two time-processing p a r a m e t e r s - the zero-dip P-wave NMO velocity (denoted as V~mo,p(O)below)and the anisotropic coefficient r / - can be estimated from dip or nonhyperbolic moveout and used to perform NMO and DMO corrections, and prestack and poststack time migration. Additional data, however, are needed to constrain reflector depth and the anisotropic parameters e and ~ required for depth imaging. For example, the P-wave tomographic, model-based inversion algorithm for VTI media developed by Sexton and Williamson (1998) uses borehole information about vertical velocity. Here, the analytic results of Chapter 5 are used to show that the interval values of ~, (~, and the vertical velocities of the P- and S-waves can be found from surface data alone by combining P-wave moveout with the traveltimes of mode-converted PS(PSV)-waves. Using converted modes, rather than pure S-waves, avoids the need for expensive shear-wave excitation on land and makes the method suitable for off-

7.3.

JOINT INVERSION OF P AND

PS DATA

335

shore exploration using the modern technology of ocean-bottom surveys. If the data are acquired on only the dip line (i.e., in 2-D), stable parameter estimation requires including the moveout of P- and PS-waves for two different dips (e.g., from a horizontal and a dipping reflector). For 3-D surveys with a sufficiently wide range of source-receiver azimuths, it is possible to estimate all four relevant parameters (Vpo, Vso, ~ and 6) using reflections from a single mildly dipping interface. In this case, the P-wave NMO ellipse determined by 3-D (azimuthal) velocity analysis is combined with azimuthally dependent traveltimes of the PS-wave.

7.3.1

S-waves

in parameter

estimation

for VTI

media

In the presence of anisotropy, a certain subset of the medium parameters usually influences both P- and S-wave propagation. For vertical transverse isotropy, the velocities of P- and SV-waves depend on the parameters Vpo, ~ and 6; additionally, SV-wave kinematics is a function of the vertical velocity Vso. Therefore, it is natural to consider combining P and S V modes in the anisotropic velocity analysis and inversion. In the processing of multicomponent reflection data from horizontally layered VTI media, normal-moveout velocity of P-waves may be supplemented, for example, with NMO velocities of SV- and SH-waves. Still, it is clear from the exact NMO equations (3.11), (3.13) and (3.14) that horizontal events on conventional-length spreads do not provide enough information to resolve the medium parameters. Even if all three NMO velocities (P, SV, SH) in a horizontal VTI layer were obtained (plus the vertical-velocity ratio V p o / V s o = tso/tpo, independent of the anisotropy), these four measurements would be insufficient to determine the five medium parameters (Vpo, Vso, 6, E (or a), ?). In particular, unless reflector depth is known, neither vertical velocity may be determined. The combination of the NMO velocities (3.11), (3.13) and (3.14) of horizontal events and the vertical arrival times can be used to solve the inverse problem only with an artificial assumption, e.g., elliptical anisotropy or no anisotropy. Even detecting the presence of transverse isotropy in conventional-spread CMP gathers requires using at least two different reflection modes. For example, if both SV- and SH-waves are recorded, anisotropy manifests itself through the difference between their NMO velocities. If the data include reflection traveltimes of P- and SV-waves, the only property of hyperbolic moveout diagnostic of anisotropy is the difference between the ratio of the NMO velocities [Vnmo,p(O)/Vnmo,sy(O)] and the ratio of the vertical velocities (Vpo/Vso = tso/tpo). The inadequacy of the NMO velocities of horizontal events for parameter estimation represents a fundamental problem in velocity analysis for anisotropic media. While in isotropic models dipping events or nonhyperbolic (long-spread) moveout may be helpful for various applications in processing (e.g., AVO analysis, suppression of multiples, imaging), the basic P-wave velocity model can often be built using moveout (stacking) velocities of horizontal P-wave events alone. In the presence of anisotropy,

336

C H A P T E R 7. V E L O C I T Y ANALYSIS AND P A R A M E T E R ESTIMATION F O R VTI MEDIA

however, velocity analysis requires using a wider range of propagation angles by including nonhyperbolic moveout or, preferably, energy from dipping reflectors.

Inversion using only horizontal S or P S events Tsvankin and Thomsen (1995) suggested combining nonhyperbolic moveout of P- and SV-waves from horizontal reflectors to obtain all four VTI parameters (Vpo, Vso, ~ and 5), but their approach encounters practical problems stemming from the difficulties in acquiring and processing of long-spread shear data. Alternatively, input data may include dip-dependent P-wave moveout (e.g., NMO velocities from a horizontal and a dipping reflector in 2-D or the NMO ellipse from a dipping reflector in 3-D), yielding the parameters Vnmo,p(0) and r/, and the NMO velocity of SV-waves from a horizontal reflector [equation (3.13)]: Vnmo,sY(O) -- VSO v/1 + 2 a , ~

(7.26)

_

If pure shear waves are not excited, the SV-wave NMO velocity (7.26) can be estimated from the NMO velocities of the P- and converted PSV-waves observed in a conventional seismic survey. Indeed, the moveout curve of the PSV-wave from a horizontal reflector is symmetric with respect to zero offset because the VTI model has a horizontal symmetry plane. Furthermore, typically P S V traveltimes on CMP spreads no longer than reflector depth can be adequately described by the NMO velocity expressed through the pure-mode NMO velocities in equation (5.17): Vn mo. , .(0) -- t . 0 Vn mo, (0) +

V.:n m o , S V ( 0 )

,

(7.27)

where tpo and tso are the vertical traveltimes of the P and S-waves and tpso = tpo +tso. Note that if the medium is azimuthally isotropic (VTI) and waves propagate in the dip plane of a reflector, P- and SV-waves are fully decoupled from the SHwave. Also, the ratio of the vertical velocities can be found from the vertical traveltimes of P- and S-waves (or P- and PS-waves)" Vpo

Vso

=

t so . tpo

(7.28)

In principle, the parameters Vnmo,p(0), r/, Vnmo,sy(O)and Vpo/Vso are sufficient to recover all four unknowns (Vpo, Vso, ~ and 5). In vertically heterogeneous media, the interval NMO velocities of P- and S-waves can be found from the conventional Dix equation and combined with the interval r/values to perform parameter estimation. Despite being relatively straightforward, this inversion procedure turns out to be unstable, with realistic small errors in the input data propagating with considerable amplification into the inverted vertical velocities, e and ~. This instability is caused by

7.3. JOINT INVERSION OF P AND PS DATA

337

the form of the dependence of SV-wave NMO velocity on the anisotropic parameters [equation (7.26)]. After obtaining r / ~ e - 6 from P-wave data and Vpo/Vso from the vertical traveltimes, equation (7.26) can be used to find the S-wave vertical velocity. However, the multiplier (Vpo/Vso)2 translates small errors in c - 6 into substantially larger errors in a and Vso. As an example, for a typical Vpo/Vso = 2, a relatively insignificant error of 0.03 in e - 6 will cause a distortion of 0.12 in a and an error of up to 12% in Vso and, consequently, in liFO. This conclusion is verified by a numerical test below.

7.3.2

2-D inversion of horizontal and dipping events

A more stable 2-D parameter-estimation procedure, introduced by Tsvankin and Grechka (2000), uses the moveout of P and PS-waves I from a horizontal and a planar dipping reflector, with the dipping events recorded on the dip line. The inversion algorithm is based on the parametric representation of PS moveout from Chapter 5 and analytic expressions for the dip-dependent P-wave NMO velocity from Chapter 3 [e.g., see equation (6.3)]. We start with parameter estimation for the simplest single-layer model and proceed with a description of a two-stage moveout-inversion algorithm for stratified VTI media. Single

layer

P-wave reflection moveout in a VTI layer is fully governed by the parameters Vnmo,P(0) and r/and, therefore, yields two equations for the inversion. Combining horizontal P and PS events allows one to determine the zero-dip NMO velocity Vnmo,sy(O)using equation (7.27). Also, the vertical-velocity ratio ~ - Vpo/Vso can be deduced from the vertical traveltimes of P- and PS-waves. On the whole, P and PS reflections from horizontal interfaces provide three constraints (Vnmo,p(0), Vnmo,sv(O)and ~) on the four unknown medium parameters. Therefore, for a certain value of one of the parameters (e.g., 6), we can determine the other three from the horizontal events: Vpo = Vnm~

(7.29)

v/1 + 2 6 '

Vso =

Vpo

,

nm~

(7.30) -- 1

]

'

O"

c - ~-2 + 6.

(7.31)

(7.32)

Next, the moveout of dipping events on the dip line of the reflector has to be inverted for the coefficient 6. In principle, it seems to be sufficient to obtain the 1For brevity, hereafter SV- and PSV-waves are denoted simply by S and PS.

338

C H A P T E R 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA

parameter 7/using the P-wave NMO velocity from a dipping reflector. In this case, however, the estimation of 5, e and the vertical velocities is too unstable to be used in practice (see the discussion above). A significant improvement can be achieved by including moveout attributes of the PS-wave reflected from the same dipping interface. The attributes used in the single-layer problem are the slope of the moveout curve at zero offset, and, if the PSwave traveltime has a minimum t m i n ( X m i n ) o n the CMP gather, the normalized offset Xmin/tmin and the NMO velocity defined at x = Xmin. Methods of recovering asymmetric PS moveout and its attributes from reflection data are discussed in Chapter 5 (section 5.2). After obtaining the "trial" VTI parameters for a given value of 6 using equations (7.29)-(7.32), the corresponding reflector dip can be estimated from the ray parameter (reflection slope) PPo of the P-wave reflection from the dipping interface. Also, the zero-offset time of the dipping P-event makes it possible to compute the depth of the reflector for the trial model. Knowledge of the reflector position is necessary to compute the trial set of the moveout attributes of the PS-wave using the results of Chapter 5. Then the parameter 6 is found by minimizing the following objective function:

_

Vnmo,P(Ppo)-

l/meas,, nmo,P

(PPo)

+

Vmm~p(Ppo)

Vnmo,PS(Ppo) -I'-

Vmeas "nmo,PS

v"mnmo,PS eas

(PPo)

2

(PPo)

(dt/dxlz:o)meas

2 XminJtmin/ [ /XminJminme]2 -~-

(Xmin/tmin)mea s

,(7.33)

where the superscript "meas" denotes the values measured from the data, while the quantities without the subscript are computed from the exact equations (6.3), (5.7), (5.12) and (5.18). The objective function (7.33) represents an overdetermined system of four nonlinear equations for the single unknown parameter 6. For PS traveltime curves without a minimum, the objective function contains only one PS moveout attribute - the slope of the moveout curve at x = 0 [equation (5.7)]. A numerical example of the joint inversion of P and PS data based on equations (7.29)-(7.33)is displayed in Figure 7.29 (Wsvanki~ and Grechka, 2000). All input parameters were computed from the exact equations and contaminated by Gaussian noise with standard deviations simulating realistic errors in data measurements. The inversion results (dots) were obtained for 200 realizations of the input data set distorted by noise. In generating the top pair of plots (Figures 7.29a,b), the terms involving the moveout attributes of the PS-wave were excluded from the objective function (7.33). The parameter r/can be accurately estimated from the P-wave NMO velocity of the dipping event, and the e- and 6-points in Figure 7.29a are close to the line corresponding to the correct value of r/. However, as predicted by the analysis above, the inversion results for Vpo, Vso, e, and 6 exhibit significant scatter indicative of the high sensitivity of the medium parameters to errors in the input data. The standard deviations

7.3.

J O I N T I N V E R S I O N OF P A N D

PS DATA

339

1.2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1.1

0.9

b

:" 0.8

1.6

1.8

2

2.2

2.4

Veo

Vpo 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1o1

.............

j

"

1

................

.............

". . . . . . . . . . . . . . . .

rJ

.............

0.9

0.1

e 5

Vpo

Figure 7.29: Parameters ~, c, Vpo and Vs0 (dots) determined by inverting P and P S moveout data from a horizontal and a dipping reflector. The input data were distorted by random noise with the following standard deviations: 0.5% for ,~ - Vpo/Vso, 1.5% for the zero-dip NMO velocities and 2% for the moveout attributes of the dipping events. (a) and (b) Inversion without the dip-moveout attributes of the PS-wave; the dip r - 30 ~ (c) and (d) Inversion including the dip-moveout attributes of the traveltime minimum of the PS-wave and the slope of the PS-moveout curve [equation (7.33)]; the dip r = 30 ~ (e) and (f) Inversion including only the slope of the PS-moveout curve; the dip r = 50 ~ The actual values are: Vpo = 2.0 km/s, Vso = 1 km/s, e = 0.2, ~ = 0.1. The solid line on plot (a) corresponds to the correct value of ~ - 0.083.

340

C H A P T E R 7. V E L O C I T Y A N A L Y S I S A N D P A R A M E T E R E S T I M A T I O N F O R V T I M E D I A

in all four parameters are too large for this algorithm to be practical (8.6% for Vpo and Vso, 0.13 for e and 0.09 for 5). Minimization using the full objective function (7.33) (including the dip-moveout attributes of the PS-wave) leads to a dramatic reduction in the scatter for all medium parameters, with standard deviations of only 2.5% for Vpo and Vso, 0.03 for e and 0.02 for ~ (Figures 7.29c,d). Clearly, the dip-moveout attributes of the PS-wave helped to overcome the problem of error amplification in the transition from r/to the vertical velocities and anisotropic coefficients. Figures 7.29a-d correspond to a relatively mild reflector dip of 30 ~ As the dip reaches 50 ~ the CMP traveltime of the PS-wave no longer has a minimum, and only the slope of the PS moveout curve at zero offset can be included in the objective function (Figures 7.29e,f). Since the moveout attributes of both P and PS-waves become more sensitive to the anisotropic parameters with increasing dip, the inversion results for the r = 50 ~ are even better than those for r = 30 ~ (the standard deviations are 1.2% for Vpo and Vso, and 0.01 for c and 5). To mitigate possible distortions caused by conversion-point dispersal, the inversion results can be refined by resorting the data into common-conversion-point (CCP) gathers (for more details, see the discussion of inversion in layered media). Layered media The joint inversion of P and PS data can be extended to stratified VTI media composed of horizontal layers intersected by dipping interfaces (e.g., by a fault plane, see Figure 7.3). The 2-D parametric expressions (5.32), (5.33) and (5.34) for reflection moveout of PS-waves are valid if the dip plane of the reflector coincides with a vertical symmetry plane of all layers in the overburden. For example, if the model is orthorhombic with two mutually orthogonal symmetry planes, this formalism can be used for the azimuth of the reflector coinciding with either of the two symmetry planes. (For any other reflector orientation, the 2-D equations are applicable only under the assumption of weak azimuthal anisotropy; see sections 1.3 and 3.5.) For azimuthally isotropic VTI media, however, there are no restrictions on reflector azimuth because every vertical plane is a plane of symmetry. The layer-stripping procedure introduced here operates with P and PS data originally collected into CMP gathers (model refinement using CCP gathers is described below). Consider an acquisition line in the dip direction of a plane boundary intersecting horizontal VTI layers (Figure 7.3). Suppose the goal is to estimate the parameters of layer i using the known parameters of the overburden. Horizontal P and PS events are used first to determine the interval zero-dip NMO velocities of P and S-waves and the P-to-S vertical-velocity ratio. Combining the P- and PS-wave NMO velocities for the reflections from the bottom of layer i yields the corresponding S-wave NMO velocity [see equation (7.27)]. Then the interval Pand S-wave NMO velocities in layer i are found using the NMO velocities from the top of this layer via the conventional Dix differentiation. The ratio of the interval vertical velocities ~(i)_ V(p~/V(~)can be computed in a straightforward way from the

7.3. JOINT INVERSION OF P AND

PS DATA

341

zero-offset traveltimes of the P- and P S-waves. Therefore, the horizontal events provide the three quantities required by the single(i) (0~ layer inversion algorithm (V(ni)mo,p(O),V.nmo,SV ~, ) and g(i)) Equations (7.29)-(7.32) can then be used to express the interval parameters of layer i through trial values of the anisotropic coefficient 5 (i) in the same way as in the single-layer problem. The dip of the reflector in layer i and the thickness z CMP (i) - quantities needed to estimate the moveout attributes of the PS-wave below - are calculated for the trial model using the traveltime and the ray parameter •(i) ~P0 of the zero-offset P-wave reflection from the dipping interface (p(~ can be determined from the reflection slope on the zero-offset section). Hence, the horizontal P and PS reflections and the zero-offset time of the dipping P event are sufficient for completely defining the trial model that corresponds to the chosen value of 5 (i). Next, dipping P and PS events generated in layer i and recorded on the dip line can be inverted for 5(i) and, therefore, for the full set of the interval parameters. The layer-stripping algorithm described in section 7.1 makes it possible to estimate the innmo,P \lJpo(~(i ),) terval NMO velocity v(i)mo,p(p(ip~) of the dipping P-wave event. To obtain ,(i)~ it is necessary to know two interval parameters of each layer in the overburden - the zero-dip NMO velocity Vnmo,p(0 ) and the coefficient r/. As discussed above, the moveout of horizontal events and the P-wave NMO velocity nmo,P (i) (PPo) have to be supplemented with attributes of the converted-wave moveout from a dipping interface for estimating the medium parameters with sufficient accuracy. The inversion algorithm operates with the slope of the moveout curve at zero offset (dt/dxl~=o) and, for mild dips, also with the normalized offset of the traveltime minimum [equations (5.40), (5.41) and (5.34)]. NMO velocity of PS-waves is not included because it is difficult to find it in closed form for vertically heterogeneous media above a dipping reflector. The parameter 5(i) (and, consequently, the full parameter set of layer i) is obtained by minimizing the objective function

[Unmo,p(ppo) -~~-___Vnme,aSp(pP~ ) 2 ~- [(dt/dxjx_O) - x--O) meas(dt/dxlx-0)meas]2 Vnmo,p(Ppo) (dt/dxl [ (Xmin/tmin)-- (Xmin/tmin)measJ2

~cMp -~-

(Xmin/tmin)mea s

.

(7.34)

If the moveout curve of the PS-wave does not have a minimum on the CMP gather, the function 9vCMP includes only Vnmo,P(Ppo) and dt/dxlx=o. Hence, the dip moveout of the PS-wave is represented by either two or just one attribute. To increase the accuracy of the parameter estimation, it is possible to include the whole conventional-spread CMP moveout of the PS-wave into the objective function. In this case, the CMP gather of the PS-wave is generated from equations (5.32), (5.33) and (5.34) for a given value of 5 (i), and the objective function contains the rms difference between the modeled and measured PS traveltimes. This inversion procedure is applied from top to bottom in a layer-stripping fashion starting with the second layer. (The parameters of the first layer are obtained using

342

C H A P T E R 7. V E L O C I T Y ANALYSIS A N D P A R A M E T E R E S T I M A T I O N F O R V T I M E D I A

the single-layer algorithm described above.) It should be mentioned that the operations with the horizontal events are entirely based on the Dix equation and, therefore, do not involve any explicit information about the parameters of the overlying layers. In contrast, the inversion of the P and PS traveltimes from a dipping reflector cannot be carried out without estimating the parameters of the overburden. Only if a medium is known to be isotropic or elliptically anisotropic, can its parameters (Vpo, Vso, and c = ~) be extracted just from the moveout of horizontal P and PS events. In general VTI media, however, the layer-stripping procedure cannot be performed using horizontal events alone. The 2-D inversion algorithm designed for CMP gathers was tested on a horizontally layered VTI model with throughgoing dipping interfaces. It was assumed that both horizontal and dipping P and PS events were recorded for each layer, so that the input data included the NMO velocities and vertical traveltimes of the two modes from the horizontal reflectors, the NMO velocities and reflection slopes of the dipping P events, and the DMO attributes of the PS-waves. All these parameters were computed using the exact equations discussed above and distorted by Gaussian noise to simulate errors in the data. Then the layer-stripping parameter-estimation algorithm based on the objective function (7.34) was applied to 200 realizations of the noise-contaminated vector of the input parameters. For the model in Figure 7.30, the dips do not exceed 35 ~, and the objective function for the first two layers includes the normalized offset of the traveltime minimum. The scatter in the estimated parameters of the subsurface layer (Figures 7.30a,b) is the same as that in the single-layer inversion results (Figure 7.29c,d). All four parameters (Vpo, Vso, e, (5) are recovered in a reasonably stable fashion, with the quasi-linear trends close to the lines corresponding to the correct values of 77 (Figure 7.30a) and Vpo/Vso (Figure 7.305). As discussed above, the vertical-velocity ratio Vpo/Vso and r/ are the parameter combinations most tightly constrained by the data, with Vpo/Vso determined directly from the vertical traveltimes and 7/ responsible for the P-wave NMO velocity from dipping reflectors. The results for the second layer (Figures 7.30c,d) are comparable to those for layer 1 because the subsurface layer is relatively thin, and the layer-stripping does not cause a sizeable error amplification. For the bottom layer (Figures 7.30e,f), however, the scatter in all four parameters becomes noticeably higher due to the distortions in the interval quantities produced by the layer-stripping procedure and error accumulation with depth. The model in Figure 7.31 contains a steeper dipping interface, with dips in the 45-55 ~ range. Comparison of Figures 7.30 and 7.31 shows that the clouds of points become less elongated with increasing dip, which means that the inversion in all three layers becomes more stable. This result is explained by the higher sensitivity of the P-wave NMO velocity and PS-wave moveout attributes to the anisotropic parameters for larger dips. A similar influence of dip was observed for the single-layer model in Figure 7.29. If the reflector has a non-negligible curvature or irregular shape on the scale of the

7.3.

J O I N T I N V E R S I O N O F P A N D P S DATA

343

_v

0.8 . . . . . . . . . . . . . . . . . . .

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9

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5 Figure 7.30: Interval parameters 5, ~, Vpo and Vso determined by inverting P and P S moveout data from horizontal and dipping reflectors in a three-layer VTI medium. The layer parameters are the same as in Figure 5.12; the dips of the interfaces in each layer are (from top to bottom) 30 ~ 35 ~ and 30 ~ (a,b) - the results for the top layer (layer 1); (c,d) - l a y e r 2; (e,f) - l a y e r 3; the actual model parameters are marked by the crosses. The input data were distorted by random noise with a standard deviation of 0.5% for the vertical traveltimes, 1.5% for the zero-dip NMO velocities and 2~ for the moveout attributes of the dipping events.

344

CHAPTER

7.

VELOCITY

ANALYSIS AND PARAMETER

ESTIMATION

FOR VTI MEDIA

Vpo 1.2

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Figure 7.31" Inversion results for a three-layer VTI medium with steeper dipping interfaces than those in Figure 7.30 (the dips are 45 ~ 50 ~ and 55 ~ from top to bottom). The parameters 5, c, Vpo and Vso in each layer are the same as those in Figures 5.12 and 7.30. The layer thicknesses are z (1) - 0.5 km, z (2) - 1.0 km and z (3) - 2 km. (a,b) the results for layer 1; (c,d) - l a y e r 2; (e,f) - l a y e r 3.

7.3.

J O I N T INVERSION O F P AND PS DATA

345

CMP gather, the moveout of converted waves in CMP geometry may be distorted by conversion-point dispersal. Since all analytic expressions above are derived for plane interfaces, the inversion procedure for models with curved interfaces may produce inaccurate parameter estimates. To mitigate conversion-point dispersal, it is preferable to carry out both velocity analysis and processing of PS-waves on common-conversion-point (CCP) gathers. Resorting P S data into CCP gathers, however, is a costly procedure that has to be based on a known velocity model. While the lateral position of the conversion point in isotropic media is controlled by the Vp/Vs ratio alone, in VTI media it is also sensitive to the anisotropic coefficients (Thomsen, 1999). Therefore, velocity analysis on CCP gathers can be preceded by the moveout inversion in CMP geometry described above. Assuming that the inversion on CMP gathers yields a good approximation for the medium parameters, it is sufficient to search for the best-fit interval parameters in a close vicinity of the obtained values. Suppose the parameters of the overburden have been determined and can be used in refining the inversion results for layer i. First, the horizontal P S events from the top and bottom of layer i are resorted into CCP gathers using the model determined at the first stage of the inversion. Semblance analysis on these CCP gathers gives updated values of the PS-wave NMO velocities from horizontal reflectors and, therefore, a more accurate estimate of the SV-wave velocity ~I(i) ,nmo,Sy(0). With the updated v(i) ,nmo,Sy(0) and the previously found values of V.nmo,p(0) (i) and g(i) the computation of the parameters of layer i is repeated for a restricted range of 5 (0 . For each trial set of the medium parameters, we reconstruct the dip and depth of the dipping reflector and resort the traces containing the P S reflection from the dipping interface into CCP gathers. The common conversion point for the dipping P S event is chosen to coincide with the zero-offset reflection point of the dipping P event - the reflection used to estimate the dip and depth of the interface. Then the P S traveltimes on the CCP gather are generated using equations (5.32) and (5.33), and the best-fit 5(0 is determined from the following objective function:

[nmo

] 2 [ t Sl 2 ITmeas (PPo) v nmo,P

-~-

§

('PS

.]

'

(7.35)

where a t p s is the rms difference between the modeled and measured traveltimes of the dipping P S event on the CCP gather, and §'~PS is the maximum measured traveltime. The general flow of the layer-stripping algorithm remains the same as that for CMP gathers. In principle, it is possible to skip the inversion on CMP gathers altogether and carry out the parameter-estimation for each trial value of 5 directly on CCP gathers, but this algorithm is much more time consuming because it involves repeated resorting of P S data.

346

7.3.3

CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA

3-D inversion of w i d e - a z i m u t h data

The 2-D inversion algorithm requires identifying and inverting P- and PS-waves from two (horizontal and dipping) interfaces for each depth interval, which may be difficult to accomplish in practice. Here it is shown that it may be possible to obtain all model parameters using a single dipping reflector, if both P and PS data are recorded for a wide range of source-receiver azimuths (in the so-called "wide-azimuth" surveys). For simplicity, the discussion is restricted to a homogeneous VTI layer with a dipping lower boundary. As demonstrated in Chapter 3 (section 3.5), azimuthally-dependent NMO velocity of pure modes in media with any symmetry is described by an ellipse. Since the VTI model is azimuthally isotropic, the axes of the NMO ellipse for any reflection event are parallel to the dip and strike directions of the reflector. For P-waves, both axes and, therefore, NMO velocity in any direction are fully controlled by the parameters Vnmo,p(0) and r/(see section 3.5). Methods of inverting the P-wave NMO ellipse for Vnmo,p(0) and r/are discussed in section 7.1 above. The next step of the inversion procedure is to specify a trial value of one of the parameters responsible for P-wave kinematics (e.g., 6) and find the other two (Vpo and e) using the parameters Vnmo,P(0 ) and ~7 determined from the P-wave NMO ellipse. Then, as described in section 7.1, the horizontal slowness of the zero-offset P-reflection can be estimated from the reflection slopes measured in at least two different azimuthal directions on the zero-offset (stacked) section [see equation (7.11)]. For given parameters of the trial model, the horizontal slowness component can be used to compute the vertical slowness of the zero-offset ray. The slowness vector of the zero-offset ray is orthogonal to the reflector, so it provides both the dip and azimuth of the reflecting interface for the trial model. Reflector azimuth can also be determined from the orientation of the NMO ellipse, if at least three sufficiently different source-receiver azimuths are available. Next, the zero-offset traveltime of the P-wave can be recomputed into the distance between the CMP and the reflector along the zero-offset ray, thus yielding the reflector depth. Therefore, given one of the VTI parameters (e.g., (~), wide-azimuth P-wave data allow us to obtain two other parameters (Vpo and ~) and the spatial position (dip, azimuth and depth) of the reflector. The last step in the inversion scheme is to estimate 6 and the vertical shear-wave velocity Vso (a parameter not constrained by P-wave moveout) using converted-wave data by matching the PS traveltimes on a 3-D (areal) CMP gather. As in the 2-D problem, it is possible to invert the DMO attributes of the PS-wave, such as the moveout slope at zero offset [see equation (5.40)]. However, to make the inversion more stable, it is better to include the whole CMP traveltime surface and define the objective function as the rms difference between the measured traveltimes and those computed for a trial model using the parametric relationships (5.35), (5.36) and (5.39). Since the forward-modeling operation does not involve multi-azimuth and multi-offset two-point ray tracing, the algorithm allows for a fast examination of a relatively wide range of both unknown parameters.

7.3. J O I N T INVERSION OF P AND P S DATA

347

This 3-D inversion procedure was tested on ray-traced reflection traveltimes of Pand PS-waves generated for a VTI layer with the parameters listed in the caption of Figure 7.32. To determine the input PS traveltimes on a regular [x, y] grid of source locations, the traveltime surface was approximated by a 2-D quartic polynomial in the horizontal source coordinates. The times at the grid points were then distorted by Gaussian noise with a standard deviation of 1% (Figure 7.33). The same polynomial approximation was used for the PS traveltime surfaces computed for each trial model. The inversion algorithm searched for the model providing the smallest rms time residual at the grid points (rather than at the original source locations). The results of the inversion are quite close to the actual values of the VTI parameters: Vpo = 2.02 km/s (error=0.02 km/s), Vso = 1.01 km/s (error=0.01 km/s), e = 0.29 (error=-0.01), 5 = 0.09 (error=-0.01). The dip, azimuth and depth of the reflector were also reconstructed with high accuracy. It should be emphasized that the reflector dip in this test was quite mild (15~ which may hamper estimation of the parameter r/using P-wave data alone (see section 7.1). However, the high sensitivity of PS traveltimes to the anisotropic parameters even at small dips makes the inversion procedure as a whole sufficiently stable. In principle, the model can be refined by resorting the PS data into areal commonconversion-point gathers and repeating the inversion. This methodology is similar to the 2-D algorithm operating with CCP gathers, but resorting PS data in 3-D is much more costly.

7.3.4 Discussion Most applications of converted waves are focused on improved imaging of targets poorly illuminated by P-wave data. In transversely isotropic media, PS(PSV)-waves can also play an important role in parameter estimation because they are governed by the same anisotropic coefficients (e and 5) as are P-waves. While P-wave reflection data contain enough information to carry out time-domain processing (NMO, DMO and time migration), they cannot be used to constrain the depth scale of horizontally layered VTI models above a target reflector. [The work of Le Stunff et al. (1999) and Grechka et al. (2000a,b) indicates that for some piecewise homogeneous VTI media with dipping or irregular interfaces in the overburden P-wave moveout can be inverted for the vertical velocity.] In conventional processing designed for isotropic media, time-to-depth conversion is usually based on moveout (stacking) velocities estimated from reflection data. If the subsurface is anisotropic, the NMO velocities of P- and PS-waves differ from the corresponding root-mean-square vertical velocities, and reflectors on isotropic P and PS images are mispositioned in depth. Furthermore, the depth error is not the same for P- and PS-waves because their NMO velocities are governed by different anisotropic coefficients. For horizontally layered VTI media, the scale of the isotropic depth image is typically more distorted for PS-waves due to the influence of the relatively large anisotropic parameter a on their NMO velocity. The resulting mismatch

348

C H A P T E R 7. V E L O C I T Y ANALYSIS AND P A R A M E T E R ESTIMATION F O R VTI MEDIA

.........

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Figure 7.32: The sources (dots) and receivers (crosses) of the areal (3-D) C M P gather of PS-waves used in the test of the inversion algorithm. The model includes a homogeneous VTI medium above a plane dipping reflector. The VTI parameters are Vpo = 2 km/s, Vso = 1 km/s, e = 0.3, 5 = 0.1. The azimuth of the dip plane coincides with the x-axis, the depth under the C M P is 1 km, and the dip is 15 ~

7.3.

>

JOINT

INVERSION

OF P AND PS DATA

349

9

"T 9 ..

0

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.

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:

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......:

0.2

0.4

X offset (km)

Figure 7.33: Traveltimes of the reflected PS-wave (plotted as a function of the source coordinate) on the areal CMP gather shown in Figure 7.32 after the approximation by a 2-D quartic polynomial and the addition of Gaussian noise.

350

C H A P T E R 7. V E L O C I T Y ANALYSIS AND P A R A M E T E R E S T I M A T I O N F O R VTI M E D I A

in reflector depth on P and P S images helps to detect anisotropy-induced depth errors away from well locations (e.g., Nolte et al., 2000). Horizontal P and PS events, however, are insufficient for unambiguous estimation of the vertical velocities and the parameters c and 5. An efficient framework for developing an inversion technique that uses dipping P S events is provided by the analytic results of Chapter 5. The traveltime and source-receiver vector of PS-waves in both common-midpoint (CMP) and commonconversion-point (CCP) geometry can be concisely expressed through the slowness components of the P- and S-wave legs of the reflected ray. This parametric representation is valid for 3-D (multi-azimuth) PS data in horizontally layered, arbitrary anisotropic media above a plane dipping reflector; it takes a particularly simple form in vertical symmetry planes of the model. One of the most attractive features of the formalism of Chapter 5 is the possibility to compute CMP traveltime curves without two-point ray tracing, which helps to quickly generate multiple models for inversion. The parametric traveltime-offset relationships also lead to concise expressions for such moveout attributes of the PSwave, needed in the inversion procedure, as the slope of the moveout curve at zero offset and the normalized offset of the traveltime minimum. The inversion algorithm introduced in this chapter estimates the VTI parameters (the P- and S-wave vertical velocities Vpo and Vso and the anisotropic coefficients and ~) by combining P-wave moveout with the traveltimes of a PS-wave reflected from a dipping interface. The 2-D inversion, which operates with dip-line reflection data originally collected into CMP gathers, is devised for VTI media composed of horizontal layers intersected by dipping reflectors, such as fault planes. (The same model was used earlier in section 7.1 in the 2-D inversion of P-wave data.) The first step in the 2-D parameter estimation is to obtain the NMO velocities and vertical traveltimes of horizontal P and P S events and apply the Dix equation to estimate the interval NMO velocities. The combination of the interval vertical traveltimes and zero-dip NMO velocities yields three equations for the four unknown interval parameters (Vpo, Vso, e and 5). Additional information needed for unambiguous inversion is provided by P- and PS-waves reflected from a dipping boundary in the same interval. For a specified interval value of 5 and the corresponding interval parameters Vpo, Vso and e, it is possible to reconstruct the trial model in the depth domain using the zero-offset traveltime and reflection slope of the dipping P-wave event. Then the best-fit set of the interval parameters is determined by matching the P-wave NMO velocity and PS-wave moveout attributes for the dipping reflector. The accuracy in all inverted parameters increases with dip due to the higher sensitivity of the input data to the anisotropy. After ~ and the other interval parameters have been found, the parameter-estimation procedure continues downward in a layer-stripping fashion. Converted-wave data in CMP geometry, however, may be corrupted by conversionpoint dispersal on non-planar interfaces. Therefore, the inversion results can be refined by resorting P S data into common-conversion-point (CCP) gathers and repeat-

7.3.

J O I N T I N V E R S I O N O F P A N D P S DATA

351

ing the parameter-estimation procedure. Since the generation of the CCP gathers requires knowledge of the velocity model, it is preceded by the inversion of CMP moveout described above. This 2-D algorithm remains valid without any modification in the vertical symmetry planes of layered orthorhombic media. For wide-azimuth multicomponent 3-D surveys, input data for the inversion include azimuthally varying P and PS traveltimes. In this case, P and P S reflections from a single mildly dipping interface are sufficient to estimate all four VTI parameters and build an accurate depth model. The P-wave NMO ellipse from a dipping reflector in VTI media can be inverted for the zero-dip NMO velocity and the anisotropic parameter ~. The two remaining VTI parameters (e.g., (f and Vso) are then determined from the traveltime surface of the PS-wave, for the same reflector, recorded on an areal (3-D) CMP or CCP gather. Also, note that the remaining anisotropic coefficient ~/can be found from P S H conversions, which exist for all azimuthal directions outside the dip plane. The joint inversion of P and PS data produces the depth distribution of the four parameters responsible for all signatures of P- and PS(PSV)-waves in VTI media. Therefore, the results can be used in prestack and poststack depth migration of Pwaves and processing of PS-waves for models with a stratified VTI overburden. The main practical difficulty in implementing the above algorithms is the identification of pure and converted reflections from the same interface. Also, the recovery of the asymmetric P S traveltime curves (in 2-D) and surfaces (in 3-D) is a complicated operation that may be hampered by strong spatial amplitude variations and phase reversals of the converted wave. To obtain PS moveout from reflection data, semblance analysis can be replaced with a method based on local coherency measures (Bednar, 1997) that also yields the moveout slope for each source-receiver offset.


Chapter 8 P-wave imaging for VTI media If one does not look for the existence of anisotropy in P-wave data, it can often go unnoticed. Still, where the subsurface is anisotropic (e.g., in the presence of massive shale formations), conventional P-wave processing based on the assumption of isotropy yields errors in seismic images and interpretations. Such anisotropy-induced distortions as mispositioning of both horizontal and dipping reflectors and misstacking of dipping events were discussed in Chapters 3, 6 and 7, and will be further analyzed below. The most critical step in correcting for anisotropy in seismic processing is estimation of the parameters of anisotropic velocity fields. As shown in Chapters 6 and 7, P-wave time processing for VTI media with a laterally homogeneous overburden requires knowledge of two parameters (V~mo(0) and r]), which can be obtained from surface P-wave data. To estimate the vertical velocity Vpo and build VTI models suitable for depth imaging, P-wave reflection moveout has to be combined with borehole data (e.g., check shots) or traveltimes of mode-converted or pure shear waves (see Chapter 7). Note that if the overburden contains dipping interfaces or other kinds of lateral heterogeneity, P-wave reflection traveltimes become dependent on the individual values of Vpo, s and (~ and, for a certain class of models, can be used to reconstruct the velocity field in depth (Le Stunff et al., 1999). Once the needed medium parameters have been estimated, conventional isotropic P-wave processing algorithms can be extended to vertical transverse isotropy by using phase-velocity equations introduced in Chapter 1 and equations for reflection moveout from Chapters 3 and 4. Here, we describe several efficient dip-moveout (DMO) and migration techniques for VTI media based on well-established isotropic imaging algorithms. This discussion of VTI processing is by no means exhaustive, as it does not include, for example, Kirchhoff migration. The main goal of this chapter is to show how analytic results developed in the previous chapters can be applied in seismic processing and to provide practical recipes for devising anisotropic imaging algorithms. The first method introduced below, based on Fowler's (1984, 1988) isotropic DMO, is a complete time-processing sequence designed to estimate the parameters Vnmo(0) and r/from constant-velocity stacks, perform DMO correction, and then migrate the stacked data using Stolt's F-K algorithm. The 2-D NMO equation from Chapter 3 helps to generalize another well-known DMO algorithm, suggested by Hale (1984), 353

354

C H A P T E R 8.

P-WAVE IMAGING F O R VTI MEDIA

for symmetry planes of anisotropic media. DMO impulse responses computed by Hale's method are used to analyze the influence of vertical transverse isotropy on the DMO operator. Next, discussion of anisotropic phase-shift and Gaussian beam migration elucidates key issues in extending both time-migration and depth-migration algorithms to VTI media. Substantial improvements in imaging achieved by proper treatment of anisotropy are illustrated by synthetic and field-data examples.

8.1

Fowler-type time-processing method

Since dip-moveout removal is an essential step in the conventional processing flow (NMO-DMO-stack-poststack migration), any errors in DMO propagate into the final seismic image. The main difficulty in devising efficient DMO algorithms for anisotropic media had been the absence of closed-form expressions for dip-dependent NMO velocity needed to replace the isotropic cosine-of-dip relationship. This gap was filled by NMO equation (3.5), which can be used in symmetry planes of any anisotropic medium. Although this equation was derived in the zero-spread limit, it provides an accurate description of P-wave moveout on conventional-length spreads close to the reflector depth (see Chapters 3 and 4). Below, equation (3.5) is used to extend Fowler's (1984, 1988) and Hale's (1984) DMO methods to anisotropic media. The main advantage of Fowler DMO is the ability to perform velocity analysis along with the DMO correction, which makes it particularly attractive for anisotropic media. The conventional (isotropic) version of Fowler's algorithm generates a large number of stacked sections covering a wide range of stacking velocities. These sections are used for velocity analysis and can be migrated using Stolt's (1978) zero-offset frequency-wavenumber (F-K) migration operator. The combination of Fowler DMO with Stolt poststack migration is often called "Fowler prestack time migration." If, however, the velocity analysis had previously been performed, other methods are more efficient in imaging the data. Fowler prestack time migration is often applied at a preliminary stage of seismic processing allowing an interpreter to produce numerous constant-velocity prestacktime-migration displays to examine possible structures that could be imaged later with more sophisticated techniques. The multiple constant-migration-velocity panels bring pieces of complicated structure into focus, allowing an early rough interpretation, which is especially valuable in new exploration areas. Also, since out-of-plane events typically image on 2-D migrated sections at velocities different from the true medium velocities, they can sometimes be identified through the interplay of image quality and migration velocity. On the whole, Fowler D MO is designed more for parameter estimation and preliminary interpretation than for final imaging. More elaborate depth-migration methods usually are required to obtain accurate depth images of complicated structures. The constant-migration-velocity panels produced by the Fowler algorithm can be used (in the same manner as conventional constant-velocity-stack panels) to estimate the NMO velocity for horizontal reflectors. A definite advantage of Fowler's approach

8.1. FOWLER-TYPE TIME-PROCESSING METHOD

355

over simple constant-velocity stacks is that the velocities have been dip-corrected; also, the spatial positions of reflectors often are less distorted than are those on constant-velocity-stack panels. Here, following the results of Anderson et al. (1996), it is shown that the advantages of Fowler's processing method can be fully exploited in VTI media by combining it with the analytic solution for NMO velocity given in Chapter 3. Extension of Fowler DMO to VTI media involves a search for two parameters (Vnmo(O) and 77) that replace a single parameter (velocity) in isotropic media. We discuss two ways to make this two-parameter search comparable in efficiency to the velocity scan in the isotropic Fowler method and demonstrate the performance of the new algorithm on synthetic examples and field data. To produce a migrated image, Fowler DMO is followed by Stolt (1978) poststack migration adapted for transverse isotropy.

8.1.1

Fowler D M O in isotropic media

The first step of Fowler's DMO process is the generation of common-midpoint stacked panels for a range of stacking velocities. Horizontal and dipping events stack coherently on different velocity panels, thus "splitting" subsurface structures into pieces emphasizing different dips. The goal of Fowler DMO is to collect all horizontal and dipping events on the panel corresponding to the correct value of the zero-dip NMO velocity Vnmo(0). The original DMO algorithm described by Fowler (1984, 1988) is based on the dip dependence of NMO velocity valid for homogeneous, isotropic media [Levin, 1971; equations (3.6)and (3.25)]: V n m o ( ( ~ ) - Vnm~

cos r

--

Vnm~ , ~/1 -- p2V2mo(O )

(8.1)

where p - sin r is the ray parameter of the zero-offset ray, which can be represented in the frequency-wavenumber domain as

1 dto k = P - 2 dy 2w

(8.2)

Here to(y) is the two-way time on the zero-offset (or stacked) section, y is the midpoint location, k is the horizontal wavenumber corresponding to y, and ~v is the angular frequency corresponding to to. The output panel for a given Vnmo(0) is built by obtaining the dip-dependent velocity Vnmo as a function of the ray parameter p [using equation (8.1)] and transferring the events from the panel corresponding to Vnmo(P) to the panel for Vnmo(0). This operation is carried out in the frequency-wavenumber (w-k) domain, where the ray parameter is given by equation (8.2). The resampling procedure is repeated for a range of zero-dip NMO (stacking) velocities to find the value of Vnmo(0) that coherently stacks both horizontal and dipping reflections.

356

C H A P T E R 8. P-WAVE IMAGING F O R VTI MEDIA

Once the panel corresponding to the best-fit velocity Vnmo(0)has been identified, the stacked data are corrected for dip moveout and ready for poststack frequencywavenumber migration. While this approach is not cost-efficient if the medium parameters are well-known or if only one constant-velocity prestack-time-migration panel is desired, it can be conveniently used in velocity analysis. A somewhat crude final variable-velocity migration can be "painted" or interpolated from the many constantvelocity-migration panels, if the velocity increment between the output panels is fine enough to avoid aliasing (see the field-data example below). In practical terms, traditional isotropic Fowler's processing sequence requires on the order of 100 constant-velocity stacks of the input data, usually computed in constant increments of the squared slowness. Then these constant-velocity stacks are Fourier-transformed to the frequency-wavenumber domain and, for each frequencywavenumber component, the data are mapped (resampled) from stacking velocity to a set of reference DMO (true medium) velocities. Finally, the data are migrated by resampling from DMO frequency to migration frequency via Stolt's (1978) F-K algorithm.

8.1.2

E x t e n s i o n to V T I m e d i a

Fowler D M O and velocity analysis

Here, Fowler's processing sequence is generalized for homogeneous VTI media. A practical way of handling vertical velocity gradient is described in the section on field-data application. To make Fowler DMO suitable for vertical transverse isotropy, the isotropic NMO equation (8.1) should be replaced by the corresponding VTI expression (3.5) [see also equation (3.7)]:

Vnmo_ _ V(r cos r

ld2v I

1 + v~-r e=r _ 1 - - tan ~bd___VV[ Y(r

dO I0=r

p q, _ q

I

,

(8.3)

P(r

where V(O) is the phase velocity as a function of the phase angle 0, r is the dip, p = sin O/V and q = cos O/V are the horizontal and vertical components of the slowness vector, q' - dq/dp, and q" - d2q/dp 2. As discussed in Chapter 3, equation (8.3) is valid for pure modes in symmetry planes of any anisotropic homogeneous medium (the incidence plane is also supposed to coincide with the dip plane of the reflector). Here, however, it will be used only for P-waves in VTI media. To be applied in anisotropic DMO algorithms, the NMO velocity should be expressed through the ray parameter p, as in the second form of equation (8.3). Alternatively, the dependence of NMO velocity on p can be built parametrically using the function V~mo(r and the expression for the ray parameter in terms of the phase angle [p = sin r162 The main complication in implementing a Fowler-type DMO process based on equation (8.3) is the difficulty in dealing with two relevant VTI parameters [Vnmo(0)

8.1.

FOWLER-TYPE

TIME-PROCESSING METHOD

357

and r/] instead of just one scalar velocity in isotropic models. Note that the anellipticity coefficient r/vanishes not only for pure isotropy (e=6-O), but also in any elliptically anisotropic medium (e=6). According to the results of Chapter 6, elliptical anisotropy is equivalent to isotropy in Fowler DMO (as well as in time processing in general) because the dependence of the NMO velocity on the ray parameter is exactly the same as in isotropic media. Although the isotropic Fowler method remains valid for elliptically anisotropic media, the reference velocity (i.e., the NMO velocity of horizontal events) is no longer equal to the true vertical velocity. For elliptical anisotropy, the resampling procedure makes it possible to recover the zero-dip NMO velocity Vnmo(0), which is equal to the horizontal velocity [see the discussion of equation (3.11)], and not the vertical velocity appropriate for depth conversion. For non-elliptical VTI media, knowledge of Vnmo(0) and ~ is sufficient not only to build the P-wave NMO velocity as a function of the ray parameter, but also to perform all essential time-related processing steps including poststack and prestack time migration (Chapter 6). As shown in Chapters 6 and 7, Vnmo(0) and r/can be estimated from the P-wave NMO velocities and the corresponding ray parameters (reflection slopes) for two distinctly different dips. Fowler-type velocity analysis offers a practical way of including the parameter-estimation step into the processing sequence. Indeed, the idea of the isotropic Fowler algorithm is to use equation (8.1) in searching for the zero-dip NMO velocity that allows proper imaging of all horizontal and dipping events. Likewise, we can employ the analytic NMO equation (8.3) for anisotropic media to find the pair of values [Vnmo(0), 77] that would produce the best zero-offset (stacked) section after resampling in the w-k domain. This implies replacing a one-dimensional search for the true medium velocity in the isotropic Fowler algorithm by a two-dimensional search for VTI media. However, since most of the computing time in Fowler DMO is spent on generating constant-velocity stacks, a more complicated (2-D) resampling procedure barely increases the overall computational cost. Moreover, there are two ways to facilitate this search and make the process of choosing the best-fit anisotropic parameters more efficient. First, it is possible to use the constant-velocity stacks produced at the initial stage of the Fowler DMO process to pick the stacking velocities and ray parameters for two different dips (e.g., for a horizontal and dipping reflector). These data are sufficient to invert for Vnmo(0) and r/ using the numerical techniques described in Chapter 7. This preliminary estimation reduces the range of both parameters in the 2-D search. Second, the NMO velocity for horizontal reflectors Vnmo(0) can be obtained by the conventional velocity analysis prior to the DMO processing. Then the parameter estimation in Fowler DMO becomes a 1-D procedure, as in isotropic media. For transverse isotropy, however, the parameter to be determined is the coefficient r/ rather than the zero-dip NMO velocity in the isotropic algorithm. As mentioned above, the processing sequence comprising velocity analysis and Fowler DMO is not considered efficient in isotropic media because there is no need

358

C H A P T E R 8. P-WAVE IMAGING FOR VTI MEDIA

to produce multiple velocity panels after the velocity has already been estimated. The situation in transversely isotropic media is entirely different due to the presence of an additional independent parameter. Even if the zero-dip NMO velocity has been obtained from velocity analysis of horizontal events, Fowler DMO represents a practical way to recover the second parameter (77) and, at the same time, perform the DMO correction. Stolt p o s t s t a c k m i g r a t i o n

The DMO-corrected stacked panels produced by Fowler's method can be imaged using Stolt's (1978) F-K migration algorithm extended to VTI media. The only substantial change required in the isotropic version of Stolt's method is the replacement of the medium velocity by the dip-dependent phase-velocity function for VTI media (Kitchenside, 1991). For defining the time-migration depth step, the reference migration velocity is set equal to the vertical velocity Vpo. Then, Vpo k

= tan r

(8.4)

2 ~dm

and -

V(r wm Vpo

+ V~o k2 _- ~z,~ V(r 2 4 ~V~n COSr Vpo

(S.5)

where ~v is the zero-offset angular frequency after DMO but before migration, ~vm is the angular frequency after migration, and r is the dip of the reflector of interest. The rest of Stolt's isotropic algorithm remains essentially unchanged. Note that equation (8.5) reduces to the isotropic cosine-of-dip relationship if V(r = VFo. The above algorithm is designed for a homogeneous medium; a semi-empirical extension to vertically heterogeneous media was developed by Alkhalifah (1996a). In the simplest implementation of Fowler's algorithm, the processing is performed for a set of reference values of Vnmo(0) and r/. (Two ways of making this 2-D scan efficient are described above.) For each pair [ V n m o ( 0 ) , ~]], the method produces DMOcorrected constant-velocity stacks and time-migrated sections. The displays of those sections (panels) are visually inspected to determine the pair [Vnmo(0), ~] that best images events over a wide range of dips. This extension of Fowler D MO is based on the hyperbolic moveout equation and, therefore, cannot correct for nonhyperbolic moveout on long spreads. However, as shown in Chapter 3 [see the discussion of Figures 3.5-3.8], nonhyperbolic moveout in VTI media is less significant (for a fixed spreadlength-to-depth ratio) for large dips than for horizontal events. Therefore, typically NMO correction is more hampered by deviations from hyperbolic moveout than is DMO removal. In fact, since the moveout for large dips is so close to hyperbolic, and the anisotropic D MO uses the correct dipdependent NMO velocity, the algorithm described here aligns dipping reflections even at fairly large offsets.


359

Numerical implementation The numerical implementation, developed by Anderson et al. (1996), closely follows the analytic description. The constant-velocity stacks are computed in constant increments A~ of slowness-squared from zero (corresponding to infinite velocity) to a pre-defined maximum value of 1/V2min. The initial mute time on the far offsets x needs to obey tmute _~ Fmax x 2 / ~ q

(8.6)

to avoid aliasing during the resampling over stacking velocity; Fmax is the maximum frequency in the data. The DMO step requires mapping (resampling) from the dip-dependent stacking velocity to the zero-dip NMO velocity Vnmo(0) based on equation (8.3). Since Vnmo(0) does not depend on dip, it can be called the "DMO velocity" (Fowler, 1988). To aid efficiency in this step, NMO velocity is tabulated as a function of the ray parameter for each desired output reference velocity Vnmo(0) and the parameter r/. Fowler DMO is accomplished by looking up the appropriate value of the stacking velocity Vnmo(P) in the table for a given p = k/(2w) and by mapping the data from the stacking velocity to the DMO velocity Vnmo(0) in the frequency-wavenumber domain. For purposes of Stolt time migration, the data are mapped from DMO frequency to migration frequency. Once again, the algorithm can be made more efficient by tabulating the needed parameters. First, it is necessary to assume a certain value for the vertical velocity Vpo. For each desired output Vpo and ~, it is convenient to build a table of values of [tan r ] versus [Vp(~))/(Vpo cos r ] for the full range of dips up to 90 ~ Then, this table is interpolated in such a way that the abscissa is in even Stolt migration is carried out by looking up in the table the increments of tan r appropriate frequency-scaling factor for each abscissa of (Vpok)/(2c~m)and mapping to migration frequency. Note that knowledge of Vpois critical for depth conversion or depth migration, but an inaccurate value of Vpo in DMO or time migration does not influence the image quality, as long as the truly relevant p a r a m e t e r s - Vnmo(0) and ~ - are correct. While building phase-velocity tables as a function of the ray parameter, a typical choice for Vpo would be Vpo = Vnmo(0) (assuming 5=0). Also, the algorithm operates with a constant value of Vpo/Vso= 2 since the dependence of P-wave NMO velocity on Vso is so inconsequential that it can be ignored. The volume of data generated during the computation of constant-velocity stacks can be very large. The algorithm of Anderson et al. (1996) uses a set of blockmatrix transpose routines, which allow the disk files to span multiple disks and require minimal RAM memory. The block-matrix transpose code helps to access the data in constant slices of time, velocity or midpoint, as needed at various stages of the algorithm. The mapping operations required for both DMO and migration are done with 8-point sinc function interpolators tabled to be accurate to within 1/512 of a sample point.

360

C H A P T E R 8.

P-WAVE IMAGING FOR VTI MEDIA

Figure 8.1: Synthetic data for this VTI model, generated by anisotropic ray tracing, were used to test the Fowler-type algorithm. The dips are 30 ~, 45 ~, 60 ~ and 75 ~. The medium parameters are Vp0=2860 m/s, e=0.215 and 5=0.05 (Vnmo(0)=3000 m/s, q=0.15).

8.1.3

Synthetic

example

The Fowler-type processing sequence described above was tested on the homogeneous VTI model shown in Figure 8.1 (Anderson et al., 1996). The wide range of dips in the model (from 0 ~ to 75 ~ helped to thoroughly examine the performance of the algorithm. A Ricker wavelet with a peak frequency of 20 Hz was convolved with broadband synthetic data computed by anisotropic ray tracing. A total of 601 midpoint locations were recorded with a trace spacing of 25 m and a listen time of 8 s; the displays show only a 1.75 s window at the location of the target reflectors. The horizontal and temporal apertures were barely sufficient to image the 75 ~ reflector with some smearing. The full Fowler prestack migration (Fowler DMO and Stolt poststack time migration) was applied to 61-fold prestack synthetic data for offsets ranging from 0 to 3000 m in 50-m increments. Six reference velocities Vnmo(0) and six reference values of r/were used for parameter-estimation purposes, yielding a total of 36 output constant-velocity, prestack-time-migration panels. The velocity Vnmo(0) was changed from 2700 m/s to 3450 m/s in steps of 150 m/s (or from 10% too low to 15% too high in 5% increments of the correct value). The parameter r/was varied over a representative range from 0 to 0.25 in steps of 0.05. In general, fewer values of r/need to be tested because the output image is less sensitive to rl than it is to Vnmo(0).


361

To compare the performance of the DMO-migration sequence for different pairs of Vnmo(0) and r/, a subset of the output panels is displayed in Figures 8.2-8.4. Each set of panels was plotted with the same gain to allow comparisons of amplitudes generated for different parameter combinations. Figure 8.2 shows multiple r/panels for the correct value of Vnmo(0). The dipping events appear undermigrated on the lower-r/panels and overmigrated on the higher-r/ panels; also, the dipping events for the wrong values of r/are poorly focused. Since for a fixed Vnmo(0) an increase in r/leads to a higher horizontal velocity, errors in r/ can be related to undermigration and overmigration similarly to errors in isotropic migration velocity in conventional processing (see Chapter 6). Clearly, the best image by far was obtained with the correct value 77=0.15 (Figure 8.2d). Some smearing on the 75 ~ event is due to the limited spatial and temporal aperture of the data. The results for multiple velocities Vnmo(0) and the correct value of r/are displayed in Figure 8.3. As expected, the best image is obtained for the correct velocity of 3000 m/s in Figure 8.3c. It is interesting that during a different set of tests in which the largest offsets were not muted out (not shown here), the two earliest horizontal reflections were imaged best on the 200 m/s-faster panel, and the third horizontal event appeared to focus best on a panel with a 100 m/s faster velocity. This was due to the increase in the stacking velocity of horizontal events caused by nonhyperbolic moveout (see Chapter 4). A mute more typical of that used on real data cured this problem. The moveouts of dipping events are more close to hyperbolic than are those of horizontal events and focus best on the correct panel, even when all offsets are included. Figure 8.4 illustrates application of the isotropic (or elliptically anisotropic) algorithm (r/=0) to the anisotropic data. It is possible to get a fairly good image of most dipping events by using a velocity that is faster than the correct one (see Figure 8.4f), but at this faster velocity the horizontal events go out of focus. Therefore, conventional Fowler DMO cannot handle non-elliptical VTI models. On the whole, the image obtained for the correct parameters Vnmo(0) and r/clearly stands out in Figures 8.2 and 8.3. This is encouraging, as there needs to be a significant difference in image quality for an interpreter to be able to pick accurate parameter values. One can obtain acceptable images for an erroneous pair [Vnmo(0), r/], but for only a limited range of dips. For instance, if a smaller Vnmo(0) is used, a given dipping event can be brought into focus with a higher value of r/. Then, however, horizontal events cannot be imaged well because the stacking velocity for them is too low. This synthetic example also verifies the conclusion of Chapter 6 that only Vnmo(0) and r/ are required for time-domain processing. Internally, the algorithm sets the vertical velocity Vpo equal t o Vnmo(0), {i--r/, and 5=0. Those are not the true Thomsen parameters Vpo, ~ and 5 for the model in Figure 8.1. However, the correct values of Vnmo(0) and r/ used in the DMO/time-migration test still produced an accurate image. Selecting 5=0 insures that the anisotropic image temporally ties the image generated by isotropic codes. Another choice of 5 (and the vertical velocity) simply rescales

362

CHAPTER

'

8.

,

P-WAVE

IMAGING

.~>. ,:' .,..~ 'J'.. ~.~..-~".~' "~,, ~i;', ~,~.:,#, :~ ~. ': :. ~'

FOR

9

VTI

,~ .'" - . , , "

MEDIA

'

N

(a) q=O.O0 0

5

.................................... I..................................I . . . . . . . . . . . . . .

~,~i:~

...... ~ . . . .

" . . . . ~ -~-,~L~,.,.:~

,~ ~' .~.."~ , : ~ ' ~- = N _ ~ . I .' p ~ . .~ , ~ " ~ , ~ -~~ ~

" -:~.-"%~ h;~

., ';.~- ,~

~:', . o ~ "

Seismic Signatures and Analysis of Reflection Data in Anisotropic Media

Seismic Signatures and Analysis of Reflection Data in Anisotropic Media (Handbook of Geophysical Exploration: Seismic Exploration)

Seismic Signatures and Analysis of Reflection Data in Anisotropic Media, Volume 29

Seismic Analysis of Structures

Seismic waves and rays in elastic media

A Lab Manual of Seismic Reflection Processing

Photochemistry of Organic Molecules in Isotropic and Anisotropic Media

Interpreting Seismic Data

Seismic Data Acquisition

Wave Propagation and Radiation in Gyrotropic and Anisotropic Media

A Petroleum Geologist's Guide to Seismic Reflection

Seismic wave propagation in stratified media

Seismic wavefields in layered isotropic media

Radiowaves and Polaritons in Anisotropic Media: Uniaxial Semiconductors

Seismic Amplitude Inversion in Reflection Tomography (Handbook of Geophysical Exploration: Seismic Exploration) (Vol 33)

Signatures of the Dead

Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic and Porous Media

Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic Porous and Electromagnetic Media

Advances in Data Analysis

Elastic Waves in Random Media: Fundamentals of Seismic Stratigraphic Filtering

Wave propagation in layered anisotropic media: with applications to composites

Wave Fields in Real Media, Volume 38, Second Edition: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media (Handbook of Geophysical ... Exploration: Seismic Exploration)

Data collection and analysis

Analysis of panel data

Analysis of Survey Data

Analysis of Economic Data

Analysis of Panel Data

Analysis of Financial Data

Analysis of ordinal data

Analysis of messy data

Seismic Signatures and Analysis of Reflection Data in Anisotropic Media