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
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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
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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
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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.
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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.
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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
This Page Intentionally Left Blank
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
1.1. GOVERNING EQUATIONS AND PLANE-WAVE PROPERTIES
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.
1.1. GOVERNING EQUATIONS AND PLANE-WAVE PROPERTIES
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)
1.1. GOVERNING EQUATIONS AND PLANE-WAVE PROPERTIES
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
CHAPTER 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION
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.
1.1. GOVERNING EQUATIONS AND PLANE-WAVE PROPERTIES
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
C H A P T E R 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION
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
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
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
C H A P T E R 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION
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
CHAPTER 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION
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
1.2. PLANE WAVES IN TRANSVERSELY ISOTROPIC MEDIA
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, I~l << 1, [7[ << 1) by expanding (most often, linearizing) seismic velocities, and group and polarization angles in e, 6, and 7. Below, the linearized approximations are used to gain valuable analytic insight into different wave-propagation problems. At the same time, it is preferable to apply exact equations in actual modeling, inversion and processing algorithms. Although originally designed just for weakly anisotropic models, Thomsen anisotropic parameters turned out to be convenient in TI (and even lower-symmetry) media with arbitrary strength of velocity anisotropy. One of the most important features of Thomsen notation, discussed below, is the reduction of the number of parameters responsible for P-wave kinematic signatures from four to three.
1.2.3
Exact and approximate phase and group velocity
Phase velocity The SH-wave phase-velocity function [equation (1.36)] can be rewritten in terms of the parameter 7 as
VsH(O) -- Vso ~/1 + 27 sin2 0.
(1.54)
Clearly, 7 is the only anisotropic coefficient that controls the kinematics of the SHwave. Predominantly positive values of 7 found from experimental data imply that the SH-wave velocity increases from the symmetry axis (0 - 0) towards the isotropy plane. Next, we use Thomsen parameters to study the behavior of the phase velocities of P- and SV-waves. Dividing both parts of equation (1.43) by the squared vertical velocity V~0 [equation (1.44)] and substituting the anisotropic coefficients e [equation (1.46)] and 5 [equation (1.47)] yields the exact P - S V phase-velocity function expressed through the Thomsen parameters (Tsvankin, 1996): V2(0)
=
l + e s i n 20
V~o
f
2 f
I
4 sin 2 0 (25 cos 2 0 - e cos 20) +
4e 2 sin4 ~9
(1.55)
where f -- 1
V~~ = 1
c55
V/~o
c33
(1.56)
22
C H A P T E R 1. E L E M E N T S O F 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
is the only term containing the S-wave vertical velocity. In the special case of e = ~, the P-wave phase velocity (corresponding to the plus sign in front of the radical) becomes
Vp(O) - Vpo ~ i + 23 sin 20. Equation (1.57) can be transformed into the SH-wave phase-velocity expression (1.54) by simply replacing Vpo with Vs0 and ~ with 7. Therefore, by analogy with the SHwave, the P-wave slowness surface (not the phase-velocity surface) has an elliptical shape if e = 5. For that reason, transversely isotropic models with equal values of e and 6 are called elliptical. The simplicity of the elliptical condition expressed in terms of Thomsen parameters is another advantage of that parameterization over the conventional notation. As in purely isotropic media, the phase velocity of the SV-wave in elliptical models is constant. Substituting e = 5 in equation (1.55) with the minus sign in front of the radical, we find Vsv(O) = Vso. (1.58) Hence, the kinematics of SV-waves is not influenced by anisotropy if e = 3. Elliptical anisotropy, however, does change the SV-wave polarization vector and radiation pattern, which do not depend solely on the phase-velocity function (see Chapter 2). The elliptical model is hardly typical for the subsurface because e > 3 for most formations. (For SH-waves, transverse isotropy always means elliptical anisotropy with the degree of the velocity variations determined by the parameter 7-) Still, the notion of elliptical anisotropy is important because some signatures of crucial importance in seismic processing depend on the deviation of a particular TI medium from elliptical. The "anellipticity" parameter r/introduced in Chapter 4 proves to be one of the most significant measures of anisotropy in P-wave processing. For general TI media with e -r 3, equation (1.55) can be simplified further by separating out under the radical a term containing e - ~: V2(0)
=
l + e s i n 20
V~o
f
2 +
f~( ~
1+
2Esin2O) 2 f
2(e - ~) sin2 20 f
(1.59)
While somewhat more compact than the original equation (1.43) parameterized by the stiffnesses, equation (1.59) still does not readily convey the influence of ~ and on the phase-velocity function. To gain analytic insight into the behavior of phase velocity, equation (1.59) can be simplified under the assumption of weak anisotropy (Icl << 1, 151 << 1). Our next goal is to derive an approximate P-wave phase-velocity function based on that assumption and show that it is practically independent of the shear-wave vertical velocity Vs0.
1.2. PLANE WAVES IN TRANSVERSELY ISOTROPIC MEDIA
23
Expanding the radical in equation (1.59) in a Taylor series and dropping terms quadratic in the anisotropies e and 5 yields for the P-wave
v (o)
= 1 + 25 sin 2 9 cos 2 0 + 2e sin 4 0.
(1.60)
V~0
Taking the square root and linearizing equation (1.60) further in c and 5 results in Thomsen's (1986) weak-anisotropy approximation: Vp(O) - Vpo (1 + 5 sin 2 0 cos 2 9 + e sin 4 0),
(1.61)
Vp(O) - Vvo [1 + 5 sin 2 9 + (e - 5) sin 4 0].
(1.62)
or, equivalently, The simplicity of equation (1.61) and the separation of the "isotropic" velocity Vpo from the anisotropic terms provide a clear insight into the behavior of the Pwave phase-velocity function. As expected from equation (1.49) for 5, this parameter controls the velocity Vp near the symmetry axis (i.e., for small 9). In contrast, the parameter e dominates phase velocity near the isotropy plane, where the ~-term goes to zero (see examples in Figure 1.8 below). The trigonometric coefficients of the eand /Lterms are equal to each other at 0 = 45 ~ Thus, a simple inspection of the values of e and 5 is sufficient for a qualitative description of P-wave phase velocity. When considering velocity variation with angle in VTI media, there is often the impression that this phenomenon can be fully described by simply contrasting the horizontal and vertical P-wave velocities (i.e., using the parameter ~). Accordingly, these two velocities can be used to derive P-wave velocity trigonometrically along any direction. This is a misconception because in non-elliptical media the horizontal and vertical velocities are insufficient for predicting P-wave velocity along any oblique direction. It should be emphasized that the approximate velocity Vp from equation (1.61) is independent of the shear-wave vertical velocity Vso. To estimate the contribution of Vso, the weak-anisotropy approximation can be refined by adding terms quadratic in e and 5. Note that, if the medium is elliptical, the weak-anisotropy formula (1.62) reduces to the exact expression (1.57). Hence, our refinement to the weak-anisotropy approximation for the squared phase velocity should be proportional to the difference e-5. Equation (1.59) is ideally suited for our purposes because it contains an explicitly separated non-elliptical term. Expanding the radical in equation (1.59) in a Taylor series and retaining the terms quadratic in e and 5 leads to
v (e)
= 1+ 25 sin 2 0 cos 2 O+ 2e sin 4 O+ ~4 ( e - 5 ) ( c sin 2 0+5 cos 2 0) sin 4 9 cos 2 9 . (1 .63)
V~0
Equation (1.63), with its linear and quadratic terms in e and 5, can be called the "moderate-anisotropy approximation" for the squared P-wave phase velocity. As
24
C H A P T E R 1. E L E M E N T S O F B A S I C T H E O R Y OF 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
expected, the quadratic term vanishes for elliptical anisotropy. However, if we continued the derivation to find a similar expansion for the phase velocity itself, it would contain a quadratic term that does not go to zero for ~ - 5; this is clear from the fact that equation (1.61) is not exact for elliptical anisotropy. Because of these quadratic terms, equation (1.63) is numerically more accurate than equation (1.60), and may be used with larger ~ and 5. However, equation (1.63) serves best to develop analytic insight into the influence of Vso on P-wave phase velocity. The contribution of Vso [or f; see equation (1.56)] is limited to the quadratic term in equation (1.63). For a practically important range of Vso corresponding to 2.5 > Vpo/Vso > 1.5, f changes from 1.19 to 1.8, which amounts to a variation of about 50% in f and in the quadratic term as a whole. This could lead to a tangible change in phase velocity provided the quadratic term itself were significant. However, the quadratic term vanishes for 0 - 0 ~ and ~ - 90 ~ and remains relatively small at intermediate angles. Of course, this conclusion applies to the range of c and 5 for which the approximation (1.63) can be used. For instance, for a model with e - 0.5, 5 = 0, equation (1.63) predicts just a 1.8% maximum variation in the P-wave phase velocity over the range in Vpo/Vso from 1.5 to 2.5. Moreover, estimates of the influence of Vso based on equation (1.63) turn out to be overstated because the contribution of the neglected higher-order terms reduces the influence of Vso even further. For the same model with c - 0.5, 5 - 0 and the same range of Vpo/Vso, the exact phase velocity varies by no more than 0.6%. Figure 1.8 shows the influence of Vso on the exact P-wave phase velocity for several combinations of e and 5. With increasing difference e - 5, the curves corresponding to the two extreme values of Vso diverge slightly from each other, but the overall contribution of the variations in the shear-wave vertical velocity remains practically negligible, even for uncommonly strong velocity anisotropy with e > 0.5. Comparison of the two media with e - 0.8 in Figure 1.8 demonstrates that the magnitude of the influence of Vso is controlled more by the difference between e and 5 than by the individual values of the coefficients: the dependence of the P-wave phase velocity on Vso is somewhat less pronounced for the model with larger 5 but smaller e - 5. It is intuitively comfortable to reach this conclusion of the negligibility of Vso in the expression for P-wave phase velocity. However, since Vso appears several times in the exact equation (1.59), this analytic and numerical verification of intuition is necessary. It should be emphasized that in the conventional notation the influence of the stiffness c55 - pV~o on P-wave velocity cannot be ignored. We have been able to reduce the number of parameters by switching to Thomsen notation because the combined influence of c55 and C13 is absorbed by the parameter 5 [equation (1.47)]. Phase velocity is a fundamentally important function in anisotropic wave propagation because it determines (along with its derivatives) the group-velocity vector [equations (1.14)-(1.16)] and, consequently, seismic traveltimes. In general, any kinematic P-wave signature in transversely isotropic media can be represented as
F,.i~ot L ( 5 ) + Q(e, 5, Vso),
(1.64)
where K~ ~ is the isotropic signature (c - O, 5 - 0), L is the term linear in the
1.2.
PLANE
WAVES
IN TRANSVERSELY
e=0.2
ISOTROPIC
MEDIA
25
8=0.1
e=0.5
6--0.1
4.5
E
40
>~3.2 . . . . . . . . . . . . . . . .
3.5-
................
,
,
~=0.8
8----0.1
5.0
3.0
t 9
o
,
,
~=0.8
6--0.6
5.0
,
, 0
, , 30 60 P h a s e Angle (degrees)
90
F i g u r e 1.8" D e p e n d e n c e of P - w a v e p h a s e v e l o c i t y on t h e s h e a r - w a v e v e r t i c a l v e l o c i t y Vso. T h e d a s h e d c u r v e c o r r e s p o n d s t o V p o / V s o Vpo - 3 k m / s is fixed.
1.5, t h e g r a y c u r v e t o V p o / V s o - 2.5;
26
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
anisotropies e and 6, and Q denotes the quadratic and other higher-order terms in e and 5, which contain Vso. The negligibly weak dependence of the P-wave phase velocity on the S-wave vertical velocity implies that the influence of Vso should be small in all kinematic problems involving P-waves. In terms of the conventional notation, the main implication of this result is that it is impossible to resolve the stiffness elements c55 and c13 from P-wave traveltime data, which constrain only a combination of c55 and c13 (Thomsen's ~). Similar to the derivation for the P-wave, linearizing equation (1.59) gives the SV-wave phase velocity in the weak-anisotropy limit as Vsv(O) = Vso (1 + a sin 2 0 cos 2 0),
(1.65)
or (7
Vsv(O) - Vso (1 + ~ sin 2 20),
(1.66)
where a is the following combination of Thomsen parameters:
/ a-
-
.
(1.67)
\Vso]
Equation (1.65) shows that the parameter a largely determines the kinematic signatures of SV-waves in TI media (Figure 1.9). However, in contrast to the result for P-wave propagation, the influence of the individual values of the Vpo/Vso ratio, c, and 5 on SV-wave propagation becomes more pronounced with increasing values of the anisotropic coefficients. For weak anisotropy, the SV-wave phase velocity is symmetric with respect to 0 = 45 ~ where it reaches either a maximum (for positive a) or a minimum (for untypical a < 0). The maximum of the exact velocity curve, however, moves from 45 ~ towards smaller angles with increasing a (Figure 1.9c,d). The accuracy of the linearized approximations for SV-waves (Figure 1.9) is generally lower than that for P-waves because the contribution of the squared vertical-velocity ratio often make8 much larger than ~ and 6. The linearized phase velocity of SV-waves can be regarded as a simplified version of the P-wave velocity function without the sin 4 0-term. Indeed, the P-wave approximation (1.61) can be transformed into equation (1.65) if Ve0 is replaced by Vso, 5 by a, and e set to zero. Therefore, there is no need for deriving kinematic signatures of SV-wave8 for weak transverse isotropy; all of them can be found directly from the corresponding P-wave signatures by making the above substitutions. G r o u p velocity Since the VTI model is azimuthally isotropic, phase velocity of all three modes is independent of the azimuthal angle r and the group-velocity component orthogonal to the vertical propagation plane [equation (1.16)] goes to zero. The group-velocity vector for each mode can be found by substituting the appropriate phase-velocity function
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
27
0---0.24 E ~
1.5
>
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
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
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.
1.2. PLANE WAVES IN TRANSVERSELY ISOTROPIC MEDIA
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
C H A P T E R 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION
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
1.2. PLANE WAVES IN TRANSVERSELY ISOTROPIC MEDIA
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
CHAPTER 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION
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 <_ 0.2, 151 < 0.2 is not substantial. The analysis of polarization in the weak-anisotropy approximation is valid for typical transversely isotropic media with cla + c5~ > 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
1.3. PLANE WAVES IN ORTHORHOMBIC MEDIA
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
C H A P T E R 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION
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
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
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
1.3. PLANE WAVES IN ORTHORHOMBIC MEDIA
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
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
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-
1.3. PLANE WAVES IN ORTHORHOMBIC MEDIA
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
30 60 90 Polar angle (deg)
0
30 60 90 Polar angle (deg)
>,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
30 60 90 Polar angle (deg)
0
30 60 90 Polar angle (deg)
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
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
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
CHAPTER 1. ELEMENTS OF BASIC THEORY OF ANISOTROPIC WAVE PROPAGATION
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.
Then the three solutions of equation (1.114) can be represented as (Korn and Korn, 1968) Yl,2,3
-
-
2
cos ~ + k
,
(1.117)
where k= 0,1,2, and cos t3 --
q 2r
9
'
0<~<~-.
-
_
The phase velocity is found from pV 2 = y - a/3.
(1.118)
lB. G R O U P - V E L O C I T Y V E C T O R AS A F U N C T I O N OF PHASE V E L O C I T Y
57
The largest root (k = 0) of equation (1.117) yields the phase velocity of the Pwave, while the other two roots (k = 1, 2) correspond to the split shear waves. For equation (1.114) to have three distinct roots, Q should be negative; if Q - 0, two of the roots are identical. For the Christoffel equation considered here, the two smaller roots may coincide with each other, which leads to shear-wave singularities in certain slowness directions. In principle, the two bigger roots may also become identical (then it no longer makes sense to talk about P and S-waves), but this is an unusual occurrence that requires a large magnitude of velocity anisotropy. Equation (1.117) is valid for an arbitrary anisotropic medium provided the appropriate Christoffel matrix is substituted into equations (1.111)-(1.113).
1B
G r o u p - v e l o c i t y v e c t o r as a f u n c t i o n of phase velocity
Here, the group-velocity vector in arbitrary anisotropic media is expressed as a function of phase velocity Y(0, r represented through the spherical polar (0) and azimuthal (r phase angles. For purposes of this derivation, it is convenient to use an auxiliary Cartesian coordinate system Ix, y, z] rotated by the angle r around the x3 axis of the original system Ix1, x2, x3] (Figure 1.2). We start with the general expression for group velocity introduced in the main text (e.g., Berryman, 1979):
v ~ = o(kv)
o(kv)
Ok~ x +
Oky y +
o(kv)
Ok~ z,
(1.119)
where V is the phase velocity and k is the wave vector. First, we obtain the two components of the group-velocity vector confined to the vertical plane y = 0 (Va~ and VGz); in the following, we will call them the "in-plane" components. From equation (1.119) it follows that
Va~
O(kV) =
ok~
Ok
-
OV
v-5~+k ~
(1.120)
To evaluate Fax, both V and k~ can be represented as functions of the phase angle 0 with the vertical (z) axis. Keeping the components ky and kz constant and substituting k - Ck~ + k~ + kz2 leads to
I
Okx k~,kz=const
k - sin0.
The second term in the right-hand side of equation (1.120) takes the form
OV
=k
k~
ky ,kz --const
OV
=cons/) ( lkz const)
(1.121)
58
C H A P T E R 1. ELEMENTS 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
Since k~
-
kz tan ~,
Okx kz = cos 2 O" O0 kz~-Const Taking into account that kz/k - cos0, we obtain =
k~
-
O0 ~=~on~t
(1.122)
cos0.
(1.123)
Substituting equations (1.121) and (1.123) into equation (1.120) yields cos0.
Vex - V sin O + -~- r
(1.124)
Similarly, the second in-plane component of the group-velocity vector is given by Vc~ = V cos 0 -
ovJ
- ~ ,=~o~t sin0.
(1.125)
Equations (1.124) and (1.125) for Vax and VGz fully determine the group-velocity vector in vertical symmetry planes of any anisotropic media. The transverse component of the group-velocity vector can be represented as
o(ky) Vvy=
Ok~
ky
OV
= V Z + k~. kx,kz-const Oky
(1.126)
The first term in the right-hand side of equation (1.126) vanishes because the transverse component of the wave vector (ky) in the xz plane is zero. Both the phase velocity V and ky in the second term are conveniently represented through the azimuthal phase angle r defined with respect to the plane y - 0: _
0k Expressing ky through the second horizontal component k~ yields ky - k~ tan r Therefore,
!
kx
0r kx,kz=const Since the derivative should be taken at r
COS 2
0,
Ok21 - kx . C~r kz,kz=const
(1.128)
lB. GROUP-VELOCITY VECTOR AS A FUNCTION OF PHASE VELOCITY
59
When computing the derivative of phase velocity with respect to r it has to be taken into account that both the polar angle 0 and the azimuthal angle r change with ky and r Therefore, we need to evaluate
ov I _
=
~
ov I
ooI
ov
0r
+-~
=
O--const O~ kx ,kz=const
From simple geometry, kx
1
kz c o s ~ '
and 00
or Substituting r
kx ,kz=const
kz cos 2 0 = k z cos 2 r sine.
0 shows that =0.
~176 L
0 r k~,kz=const
If the y-axis points in the positive C-direction (counterclockwise from the xl-axis),
or or
--1,
and equation (1.129) reduces to
or[
_ ov]
- -
~
0r k~,k~=~on~t
o
(1.130)
0r O=~on~t
Using equations (1.128) and (1.130), equation (1.127) can be rewritten as OV Oky
10V k~ 0r O=const
(1.131)
Finally, the transverse component VGy [equation (1.126)] takes the form k OV vGy = k~ 0r
O=const
1 OVI . sin 0 0r [O=const
(1.132)
The y-axis in equation (1.132) points in the direction of increasing r i.e., counterclockwise from the xl-axis of the original coordinate system.
This Page Intentionally Left Blank
Chapter 2 Influence of anisotropy on point-source radiation and AVO analysis Velocity variations with angle in anisotropic media influence not only the traveltimes of seismic waves but also their polarizations and amplitudes. As follows from the Christoffel equation (1.10), the polarization vectors of plane waves in the presence of anisotropy are not strictly parallel or perpendicular to the propagation (slowness) or ray direction. This suggests that polarization directions of body waves with realistic curved wavefronts should also be distorted by anisotropic velocity fields. Analysis of both polarization and amplitude anomalies, however, requires going beyond planewave theory and investigating 3-D wavefronts generated by such sources of seismic energy as a concentrated force, explosion and dislocation. The first section of this chapter gives a description of point-force radiation patterns and body-wave polarizations in anisotropic media. Numerical analysis based on the evaluation of Fourier-Bessel (Weyl-type) integrals is supported by a closed-form solution for the far-field displacement derived using the stationary-phase approximation. Both analytic and numerical results show that the amplitudes of P-waves and, in particular, S-waves in anisotropic media may be substantially distorted by focusing and defocusing of energy, usually associated with maxima and minima (respectively) in the angle-dependent velocity. The second section is devoted to the basics of amplitude-variation-with-offset (AVO) analysis in transversely isotropic media with a vertical symmetry axis (VTI). The dependence of reflection coefficients on the incidence angle is quite sensitive to the presence of anisotropy on either side of the interface. Furthermore, if the overburden contains anisotropic (e.g., shale) layers, AVO signatures are also influenced by the amplitude focusing phenomena above the target horizon. The results of the first section make it possible to obtain simple approximations for the anisotropy-related AVO "focusing factor" and compare its contribution with that of the anisotropic terms in the reflection coefficient. 61
62
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
2.1
Point-source radiation in anisotropic m e d i a
Most early methods of body-wave computation for anisotropic models were based on ray theory ((~erven)~, 1972). Ray-tracing algorithms are relatively efficient and possess significant flexibility in treating complex subsurface geometries and various anisotropic symmetries (e.g., Gajewski and P~en~k, 1987; Guest and Kendall, 1993). However, the ray method in anisotropic media encounters difficulties stemming, for example, from the reduced accuracy of the solutions in the vicinity of shear-wave cusps, singularities, and in areas of rapid amplitude variations (see below). Analytic ray-theory (i.e., far-field) approximations for radiation patterns of point sources in anisotropic media were given in Tsvankin and Chesnokov (1990a), BenMenahem et al. (1991) and Gajewski (1993). Gajewski (1993) also developed an efficient numerical scheme to evaluate the curvature of the slowness surface needed to calculate radiation patterns, and gave a detailed discussion of body-wave amplitudes in models containing aligned fluid-filled and dry cracks. Van der Hijden (1987) generalized the Cagniard-de Hoop method to account for the influence of anisotropy in horizontally layered media and described the design of appropriate integration contours. Another approach to modeling full wavefields in the presence of anisotropy is the so-called Alekseev-Mikhailenko method based on combining partial separation of variables with a finite difference scheme (Martynov and Mikhailenko, 1984). One of the most widely used seismic modeling tools, the reflectivity method, was extended to anisotropic media by Booth and Crampin (1983), Fryer and Frazer (1987), Taylor (1987) and others. The anisotropic reflectivity technique produces an exact three-dimensional wavefield, but the model is restricted to a stack of horizontal homogeneous layers. The main complication in the synthesis of reflectivity seismograms in azimuthally anisotropic media is the time-consuming double-contour integration with respect to the horizontal components of the slowness vector. Also, the accuracy of this integration is rather sensitive to the internal parameters of reflectivity algorithms (e.g., to the integration step over slowness and the size of the smoothing windows at the ends of the integration interval). To make the reflectivity methodology more practical, Tsvankin and Chesnokov (1990a,b) introduced an approximation for the influence of azimuthal anisotropy on the amplitude and phase terms of the integral kernel. This technique reduces the double integral for individual reflections to a single Fourier-Bessel integral over horizontal slowness, although it cannot be applied to the full reflectivity matrix for layered media (see the discussion below). 2.1.1
Green's
function
in homogeneous
anisotropic m e d i a
Exact integral solution The integral representation of the Green's function in a homogeneous anisotropic medium is derived (following Tsvankin and Chesnokov, 1990a) in Appendix 2A by
2.1.
P O I N T - S O U R C E R A D I A T I O N IN A N I S O T R O P I C M E D I A
63
applying Fourier transformations to the wave equation. The displacement from a point force located at the origin of the coordinate system is given by the following summation over plane waves:
1F
u(t, x) -- ~
4--~2 v--1
i0 /0
S(w, x) (I)(aJ)e i~t dw,
V(.) (p) e-i~[por r162
(2.1)
Po dpo de,
(2.2)
where z = x3 is the vertical coordinate of the receiver, r is the horizontal sourcereceiver distance, a is the azimuth of the source-receiver line with respect to the xl-axis, (I)(w) is the Fourier transform of the source pulse, p is the slowness vector with the components {Pl - P0 cos r P2 = P0 sin r P3}, and U (~) is the displacement vector of the elementary plane wave [equation (2.27)]. It is assumed that z is positive; if z < 0, the minus in front of (iw/4~ 2) should be changed to a plus. Equation (2.2) represents a decomposition of point-force radiation into elementary plane waves defined by the slowness vector p, so it can be called the Weyl integral for anisotropic media. The plane-wave decomposition of the wavefield from an arbitrary source (e.g., a point dislocation) has the same form as equation (2.2), but with a different term U (~) that depends on the seismic moment tensor (Tsvankin and Chesnokov, 1990b). The wavefield contains three wavefronts corresponding to the P-wave (u - 1) and two split S-waves (u - 2, 3). For each mode, the vertical slowness p~) needs to be found from the horizontal slownesses pl,2 using the Christoffel equation (1.10). If z > 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.
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
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
2.1. POINT-SOURCE RADIATION IN ANISOTROPIC MEDIA
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
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
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
2.1. POINT-SOURCE RADIATION IN ANISOTROPIC MEDIA
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.
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
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
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
(~,~~
X2
Figure 2.7: Same as in Figure 2.5 but for a smaller source-receiver distance (R - 5
km).
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
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.
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
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
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
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[ << 1, [5[ << 1) are given by equations (1.61) and (1.73), respectively. Substituting these equations into approximation (2.8), applying a Taylor-series expansion, and dropping the terms quadratic in 5 and ~ yields F~
Ap(R, O) - 47rp V~,oR
1 - 2(~ - 5) sin 2 20 + 5 sin 2 0 1 + 25 "
(2.9)
The term 1 +25 has been retained in the denominator to make the weak-anisotropy approximation exact in the symmetry direction (0=0). Indeed, equation (2.30) shows that the exact far-field P-wave amplitude in the symmetry direction is given by Fa/[47rp V~oR (1+25)], where Fa is the vertical component of the force. Equation (2.9) may, of course, produce errors at oblique incidence if e and 5 are not small. The factor V~0 (1 + 25) is equal to the squared P-wave normal-moveout (NMO) velocity from a horizontal reflector derived in Chapter 3. Hence, the influence of transverse isotropy on the amplitude for 0 - 0 and on the NMO velocity is determined by the same expression (1 + 25), which controls the curvature of the P-wave phasevelocity surface in the symmetry direction. Thus, the focusing (or defocusing) of the P-wave energy at vertical incidence depends on the single anisotropic coefficient 5, the parameter responsible for nearvertical P-wave propagation (see Chapter 1). If 5 < 0, the velocity function has a maximum at 0 - 0 ~ and the amplitude at vertical incidence increases as a result of the focusing of energy; conversely, if 5 > 0, a velocity minimum at 0 - 0 ~ leads to lower amplitudes due to the defocusing. The velocity maximum or minimum in the symmetry direction represents the only "3-D" velocity extremum in transversely isotropic media: phase velocity increases (or decreases) away from the symmetry axis at the same rate in all azimuthal directions, not just in the incidence plane. Therefore, the focusing (or defocusing) of energy in the symmetry direction is more pronounced than that for any other velocity extremum with the same value of dV2/dO2. This peculiarity of the symmetry direction is reflected in equation (2.8) by the value of the ratio sin ~ / s i n O at r - 0 - 0 [equation (1.73)]" sin r lira = 1 + 25. 0-~0 sin 0 In contrast, sin r
0-
1 at any other velocity extremum.
2.2.
RADIATION P A T T E R N S AND AVO ANALYSIS IN VTI MEDIA
85
While distortions of the amplitude in a given direction are certainly diagnostic of anisotropy, the property of radiation patterns of most importance in AVO analysis is the anisotropic correction to the angular amplitude distribution. The lowest-order anisotropic angular term in equation (2.9) depends on the difference e - ~. If ~ - ~ > 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
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
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.
RADIATION P A T T E R N S AND AVO ANALYSIS IN VTI MEDIA
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).
2.2. RADIATION PATTERNS AND AVO ANALYSIS IN VTI MEDIA
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.
2.2. RADIATION PATTERNS AND AVO ANALYSIS IN VTI MEDIA
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.
RADIATION P A T T E R N S AND AVO ANALYSIS IN VTI MEDIA
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
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
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).
2.2. RADIATION PATTERNS AND AVO ANALYSIS IN VTI MEDIA
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.
RADIATION P A T T E R N S AND AVO ANALYSIS IN VTI MEDIA
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
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
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 <~ 151, the non-elliptical term usually makes the most significant contribution to the total error. Thus, for typical values of the anisotropic coefficients, the difference e - 5 determines, to a large degree, the angular variation of the P-wave NMO velocity. If e - 5 > 0
116
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
(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
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
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
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
~=-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
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
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 <0.15-0.2, nonhyperbolic moveout does not seriously distort the P-wave moveout velocity on spreads close to the CMP-reflector distance. A detailed description of nonhyperbolic moveout is given in Chapter 4. A p p a r e n t versus t r u e dip
In the discussion above, NMO velocity was analyzed as a function of the reflector dip r However, since the true dip is difficult to obtain directly from seismic data, in seismic processing (e.g., in the constant-velocity DMO correction) NMO velocity is expressed through the ray parameter (horizontal slowness) p of the zero-offset ray: Vnmo(P ) =
Vnmo(0).
-- ,
~/1 - p2V2mo(0)
(3.25)
3.2. NMO VELOCITY FOR VERTICAL TRANSVERSE ISOTROPY
125
where p is determined from reflection slopes on the zero-offset section to(y) (to is the two-way time, y is the CMP coordinate)"
P-
sin r 1 dto = V(r -2 dy "
(3.26)
Equation (3.26) [but not equation (3.25)] is valid for both isotropic and anisotropic media. In homogeneous isotropic media, Vnmo(0) and V(r are equal to the medium velocity V, and equation (3.25) is identical to the cosine-of-dip relationship (3.6). If equation (3.25) is applied in anisotropic media [instead of the cosine-of-dip relationship (3.6)], the true dip angle r is effectively replaced with the apparent dip A
r
sin r - p Vnmo(0)
--
Ynm~ sin r V(r
;
cos r - ~/1 - p2V2mo(O) .
(3.27)
In contrast to isotropic media, for vertical transverse isotropy the zero-dip NMO velocity Vnmo(0) given by equation (3.11) (for P-waves) generally does not coincide with the phase velocity at the dip angle V(r and r ~ r This means that the error of equation (3.25) in VTI media differs from the error of the cosine-of-dip relationship (3.6) discussed above. As shown in Figures 3.10 and 3.11, the correction using the apparent dip leads to higher errors (deviations from unity) for both e - 5 > 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
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
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
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
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
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
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
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
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
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
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
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
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 << 1) the terms quadratic in 5 in equation (3.38) can be dropped, which leads to Vnmo(0 ) =
Vpo(1 + 5 cos 2 u ) =
(3.40)
Vho r .
Thus, in elliptical media the NMO velocity from a horizontal reflector remains close to the horizontal phase velocity, whether the elliptical axes are tilted or not. Equation (3.35) for an arbitrary tilt of the symmetry axis can also be simplified under the assumption of weak anisotropy (15] << 1): Vnmo ( r
=
W0
cos r
- u)] . 1 + ~ + ~ sin2(r - L,) + 2~ sinu cosSin(r162
(3.41)
Figure 3.21 shows that the tilt has a significant influence on the dip dependence of NMO velocity in elliptical media. When one of the elliptical axes is vertical (u = 0 ~ or v = 90~ deviations from the cosine-of-dip relationship (3.6) are entirely controlled by the phase-velocity variations [equation (3.37)]. Although the difference between the vertical and horizontal velocity for the model in Figure 3.21 is significant (close to 2070), the anisotropy-induced error in the cosine-of-dip relationship for both u = 0 ~ and u = 90 ~ is relatively small. In contrast, for the symmetry axis tilted at 45 ~ and -45 ~ the anisotropic signature is much more pronounced. Note that for u = 45 ~ the NMO velocity goes to infinity at dips below 90 ~ since the section of the wavefront propagating towards the reflector does not contain the full range of phase angles, as was discussed in detail above. In general (non-elliptical) TI media, with e -J: ~, equations for elliptical anisotropy are valid for SH-waves only. The above analysis for the elliptical P-wave remains entirely valid for the SH-wave if Vpo is replaced with Vso and ~ with 7- For instance, the exact NMO equation (3.35) for the SH-wave takes the form Vnmo(r
Vso v / l + 2 7 v / l + 2 7 s i n 2 ( r cosr
) [1_2
sinusin(r cosr
"
(3.42) For SV-waves, the elliptical condition (e = 5) implies that the phase and group velocities are independent of angle, and the NMO velocity is described by the isotropic cosine-of-dip formula (3.6). Below we demonstrate that the NMO equations for elliptical anisotropy take the same form as those for isotropic media when the dip r is replaced with the ray parameter (horizontal slowness) of the zero-offset ray. Weak transverse isotropy Qualitative moveout analysis for non-elliptical media is convenient to perform in the weak-anisotropy approximation. The P-wave NMO velocity, linearized in the
142
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=0.2
: :
:
: "
:
:
:
:
:
Figure 3.21: Dip dependence of P-wave moveout velocity for elliptical anisotropy (c=5=0.2). The dotted curve is the exact NMO velocity calculated from equation (3.35); the dashed curve is the weak-anisotropy approximation from equation (3.41). Both curves are normalized by Vnmo(0) / cos r where Vnmo(0) is the exact NMO velocity from a horizontal reflector.
3.3. NMO VELOCITY FOR TILTED TI MEDIA
143
parameters e and 5, is derived in Appendix 3B: Vnmo(r cos r
Vpo ! 1 + 5 + 5 sin 2 $ + 3(~ - 5) sin 2 r (2 - sin 2 r
-
+
2 sin u sin $ COS (~
[5 + 2(e - ~)sin 2 q~]
(3.43) f
where r r u. For u - 0, equation (3.43) becomes equivalent to the VTI equations (3.21) and (3.23)" V~mo(r COSr -- Vp0 [1 + 5 + 5 sin 2 r + 3(e - 5) sin 2 r (2 - sin 2 r
(3.44)
The terms on the first line of equation (3.43) have the same form as that in VTI equation (3.44) but with the dip r replaced by the difference r u. The last term of equation (3.43) is a pure contribution of the tilt of the symmetry axis; note that it goes to zero not only for u - 0, but also for a dipping reflector normal to the symmetry axis ( r 0). In the latter case, as discussed above, all anisotropic angular terms in equation (3.43) vanish, and the dip dependence of NMO velocity is described by the isotropic cosine-of-dip relationship. For a horizontal reflector (r - 0) and an arbitrary tilt angle, the P-wave NMO velocity is given by Vnmo(0) - Vpo [1 + 5 - 5 sin 2 u + (e - 5) sin 2 v (7 cos 2 v - 1)].
(3.45)
Equation (3.45) shows that the contribution of tilt to the NMO velocity from a horizontal reflector is mostly controlled by the difference e - 5. For models with fixed values of the Thomsen parameters, the tilt can change Vnmo(0) in a significant way. For example, for VTI media (u - 0), V n m o ( 0 ) - Vpo (1 + 5), whereas for HTI media (v - 90~ Vnmo(0) - Vpo (1 + 5 - e). Note that the ratio of the zero-dip NMO velocity and vertical velocity is responsible for the anisotropy-induced mis-ties in time-to-depth conversion (see section 3.4). The ratio of the cosine-of-dip corrected NMO velocity [equation (3.43)] and the zero-dip value Vnmo(0) [equation (3.45)] determines the error in the dip dependence of NMO velocity caused by the anisotropy. The presence of a separate "tilt" term in equation (3.43), as well as the form of the argument ( r v) in other terms, are indicative of a strong influence of tilt on the variation of the P-wave NMO velocity with dip. Since the "tilt" term depends on e and 5 separately, we can no longer expect that deviations from the cosine-of-dip dependence will be largely controlled by the difference between e and 5, as is the case in VTI media. This conclusion is supported by the numerical results below. The weak-anisotropy approximation for SV-waves can be obtained from P-wave equation (3.43) by making the substitutions discussed above: VPo ~ Vso , (~ ~ o-, and e --+ 0: Vnmo(r cos r
-
Vso (1 + a + a sin 2 r - 3a sin 2 r (2 - sin 2 r k
144
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
+ 2a
sin p sin r cos 2r f
(3. 46)
N u m e r i c a l r e s u l t s for g e n e r a l T T I m e d i a Figures 3.22 and 3.23 show the exact P-wave NMO velocity (3.5) and the weakanisotropy approximation (3.43) at different tilt angles for two media with small and moderate values of the anisotropic parameters c and 5. For both models the accuracy of the weak-anisotropy approximation is satisfactory at all tilt angles. As was the case for VTI media, the behavior of the normalized NMO velocity strongly depends on the sign of ~ - 5. It is interesting that for the model parameters used in Figure 3.23 the NMO velocity is identical for u = +45 ~ and also for u = 0 ~ and v = 90 ~ This is explained by the vanishing parameter e for this model: if e - 0, the P- and SV-wave phase velocities are symmetric with respect to the 45 ~ angle with the symmetry axis. In this case the phase velocity and, consequently, the NMO velocity do not change when we rotate the symmetry axis by 90 ~ Figure 3.24 illustrates the influence of tilt on the exact P-wave NMO velocity for three models with a positive e - 5 = 0.2. As predicted by the weak-anisotropy approximation, the tilt causes profound changes in the normalized NMO curves. If the symmetry axis is tilted towards the reflector (u > 0), the cosine-of-dip corrected NMO velocity remains almost flat in the range 0 < r < u and then increases for large dips. A "negative" tilt (u < 0), which corresponds to the symmetry axis tilted away from the reflector, reverses the NMO curve and makes the cosine-of-dip corrected NMO velocity decrease with dip. (For a vertical symmetry axis, this type of behavior arises for negative values of c - 5.) Also, when the symmetry axis is tilted, the dip dependence of the P-wave NMO velocity is no longer tightly controlled by the difference e - ~. Deviations of the P-wave NMO curves from those for vertical transverse isotropy become pronounced at relatively mild tilts (Figure 3.25). Only for tilt angles up to + ( 1 0 - 15) ~ is the dip dependence of NMO velocity close to that for VTI media. A tilt of +20 ~ is sufficient to cause substantial changes in the NMO curves; for u = 20 ~ the NMO velocity increases sharply at r > 50 ~ as the dip approaches the maximum phase angle ~maxcontained in the wavefront. Although the above analysis was performed for a single homogeneous transversely isotropic layer, the results can be extended to vertically heterogeneous media by using a 2-D Dix-type equation introduced in section 3.4. This equation is valid in symmetry planes of any anisotropic model that includes a dipping reflector beneath a vertically stratified overburden. Each layer, for instance, can be transversely isotropic with an in-plane symmetry axis tilted at an arbitrary angle. The normal-moveout velocity for such a model is represented by the rms average of the single-layer NMO velocities analyzed here. In Chapter 6, we continue the discussion of dip-dependent NMO velocity in TTI media by expressing it as a function of the ray parameter p of the zero-offset ray.
3.3.
NMO VELOCITY
FOR TILTED
TI MEDIA
e=0.2
145
5=0.1
E 1.4
N
-~ 0.8 . . . . . . . .
:,}'~'~
: . . . . . . . . "..~ *~. 9 .". . . .
o 0.7
..............
E t__
x. 9149
0
44
_"
Ib~
"
"
%:
0
20 40 Dip (degrees)
60
v=90 ~ . 4
.
o
.......
.N
.
.
.
.
.
9
.
.
: ........
: ........
:
:
B e
..:l.
;./ .~
9
o
9
~
.
~
~
Figure 3.22: P - w a v e N M O velocity in T T I m e d i a for c=0.2 a n d / ~ = 0 . 1 . T h e dotted curve is the exact N M O velocity calculated 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 (3.43). Both curves are normalized by Vnmo(0)/cos r where Vnmo(0) is the exact N M O velocity from a horizontal reflector.
146
CHAPTER
3.
NORMAL-MOVEOUT
VELOCITY
e=0
IN LAYERED
ANISOTROPIC
MEDIA
=0.1
v=--45 ~ 9
-
0
-
"
~
"
E -o
:
.N 0.9 . . . . . . . . -~
~. . . . ,~ ~ . . . . . . . . . .,~
.~,
:
:
:. . . . .
-
v=45 ~
v=_90 ~ 1.1 o
........
.9 . . .Q.~ I I p
:
~-..
,
........
~.,,,
9
9
I
I
.9 . . . .
: ~
i
F i g u r e 3.23: N o r m a l i z e d P - w a v e N M O velocity for c - 0 a n d ~=0.1. T h e d o t t e d curve is t h e e x a c t N M O velocity c a l c u l a t e d from e q u a t i o n (3.5); t h e d a s h e d curve is t h e 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 r o m e q u a t i o n (3.43).
3.3.
NMO V E L O C I T Y F O R TILTED TI MEDIA
147
e-5=0 2 v=_-45 ~ 1.0
.
.
.
.
.
0
9
"
0 Z
,
,
0
2'0
40
6'0
Dip (degrees)
v=90 ~ 1.60
~ 1.0
0 Z
,
,
0.8
0
20 40 Dip (degrees)
60
Figure 3.24: The exact normalized P-wave N M O velocity for models with c - 5=0.2" e = 0 1 , 5 = - 0 1 (solid black); e = 0 2 , 5 = 0 (gray); e = 0 3 , 5=0.1 ( d a s h e d )
148
CHAPTER
3.
NORMAL-MOVEOUT
VELOCITY
IN LAYERED
ANISOTROPIC
MEDIA
e-fi=0.2
1.4
1.8"
0
0
|
I
I
I
Dip (degrees)
I
Dip (degrees)
v=20 ~
o
E i..,.
t . ... ...
. . . . . . . . . . .
:i
;
i
"
~"
.......
~/:
"
........
. ........
.;..~
I
!
~ o.8
0
z ,
,
,
0.6
!
Figure 3.25" The exact normalized P-wave NMO velocity for mild tilt angles. The anisotropic parameters are the same as in Figure 3.24" e=0.1, ~=-0.1 (solid black); e=0.2, c~=O (gray); e=0.3, 5=0.1 (dashed).
3.4. NMO VELOCITY IN LAYERED MEDIA AND TIME-TO-DEPTH CONVERSION
3.4 3.4.1
149
NMO velocity in layered media and time-to-depth conversion 2-D Dix-type NMO equation for dipping reflectors
The conventional Dix (1955) equation, routinely used in seismic velocity analysis, is derived for a horizontally layered, isotropic medium. It turns out, however, that an averaging NMO equation of the same form remains valid for more complicated, anisotropic models above a dipping reflector. Let us consider a medium comprised of a stack of horizontal homogeneous layers above a dipping reflector (Figure 3.26). The incidence plane is assumed to coincide with the dip plane of the reflector and a plane of symmetry in all layers (although the symmetries themselves may be different). For example, if certain layer is orthorhombic, the incidence plane should represent one of the three mutually orthogonal symmetry planes. The normal-moveout velocity for this model (it can be called effective) is obtained in Appendix 3C: 1
gn2m~
N
-- ~0 /~1 [gn(/m)~
t~i)(p)'
(3.47)
where t~i) is the interval traveltime in layer i computed along the zero-offset ray, N t(00 is the total zero-offset time and V(~o(p) is the interval NMO velocity to -- E~=I for the ray parameter p of the zero-offset ray. Since the slowness vector of the zerooffset ray is orthogonal to the reflector (bottom of layer N), the ray parameter is expressed through the dip r as p = sin r162 (Figure 3.26). It follows from the derivation in Appendix 3C that the interval NMO velocities Vn(/m)oare evaluated for non-existent reflectors perpendicular to the slowness (phase-velocity) vector of the zero-offset ray in each layer; this is discussed in more detail below. Equation (3.47) is quite general in the sense that it does not assume any specific type of anisotropy, although it does require the incidence plane to be a plane of symmetry. For isotropic media, formula (3.47) becomes equivalent to the NMO expression of Shah (1973). Thus, as in the conventional Dix equation, the effective normal-moveout velocity over a stratified anisotropic medium is equal to the root-mean-square of the interval NMO velocities weighted by the interval zero-offset traveltimes. For anisotropic layers the interval velocity Vn(/m)oshould be found using the single-layer equation (3.5) introduced above. A more profound difference between equation (3.47) and the conventional Dix formula, however, is in the nature of the interval NMO velocities, which no longer correspond to any physical interfaces in the model. It should be emphasized that this complication is caused by the reflector dip rather than by the velocity anisotropy in the overburden. Even if the model is purely isotropic, the zero-offset ray from a dipping reflector and its slowness vector are tilted, and the interval NMO velocities
150
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
r
Figure 3.26: Schematic depth section for a stratified anisotropic model that includes a dipping reflector beneath a stack of horizontal homogeneous layers. It is assumed that the incidence plane is the dip plane of the reflector and a symmetry plane of the medium. V (N) is the phase velocity in the layer immediately above the reflector, x (~) (i = 1, 2, ...N) are the horizontal ray displacements in each layer for the one-way ray traveling from the zero-offset reflection point to the surface.
3.4.
NMO V E L O C I T Y IN LAYERED MEDIA AND T I M E - T O - D E P T H CONVERSION
151
should be computed for non-horizontal (imaginary) reflectors in the overburden. The NMO velocity in any layer i (including the one immediately above the reflector) can be formally obtained by applying the Dix equation to the NMO velocities from the top [Vnmo(i- 1)] and bottom [Vnmo(i)] of the layer: [Vn(~o]2 = V~mo(i) to (i) - Vn2mo(i- 1 ) t o ( i - 1) to(i) - to(i - 1) '
(3.48)
where t o ( i - 1) and to(i) are the two-way traveltimes to the top and bottom of the layer, respectively. Despite its familiar appearance, equation (3.48) cannot be used in the same straightforward fashion as the conventional Dix formula. Suppose we are interested in finding the interval NMO velocity in the medium immediately above the dipping reflector (i = N). While the effective NMO velocity Vnmo(N) can be determined directly from the data, Vnmo(N- 1) represents the NMO velocity from interface N - 1 tilted in such a way that is becomes orthogonal to the slowness vector of the zerooffset ray in the ( N - 1 ) - t h layer. Hence, the recovery of v~(N)o(p) is impossible without obtaining the NMO velocities in the overlying medium for non-existent reflectors with different dips. This task is not trivial for anisotropic media because conventional velocity analysis for the overburden provides us only with NMO velocities and traveltimes for horizontal reflectors. However, if the overburden contains a dipping interface (such as a fault plane) intersecting horizontal layers, it may be possible to combine the NMO velocity of the dipping event with that of the horizontal event in the same layer to find the NMO velocity for any other dip. This result for P-waves in VTI media is obtained in Chapter 6. A Dix-type velocity-analysis algorithm for vertical transverse isotropy based on equations (3.47) and (3.48) is described in Chapter 7. 3.4.2
Horizontally conversion
layered
media
and
time-to-depth
Anisotropy-induced errors in depth Complications associated with NMO velocities from non-existent intermediate interfaces do not arise if the reflector is horizontal. Since the overburden also contains only horizontal boundaries, the slowness vector of the zero-offset ray remains vertical (p - 0) everywhere along the raypath, and equation (3.47) reduces to the rms average of the interval NMO velocities from horizontal reflectors. Note, however, that unless the horizontal plane is a plane of symmetry, the zero-offset ray may deviate from the vertical direction, and to may be different from the vertical reflection time. Let us consider in more detail the Dix-type equation for horizontally layered VTI media. The horizontal (isotropy) plane in VTI media is a plane of symmetry, so the zero-offset ray, which corresponds to a vertical slowness vector, is also vertical. Hence, the effective NMO velocity is equal to the rms average of the interval values of the
152
C H A P T E R 3. N O R M A L - M O V E O U T VELOCITY IN LAYERED ANISOTROPIC MEDIA
zero-dip NMO velocity weighted by the vertical traveltimes: 1
V:mo(N) -
N
E [Vn(2o(0)]
')
(3.49)
i=1
Using equations (3.11), (3.13) and (3.14) for the interval NMO velocities, equation (3.49) can be rewritten for each mode (P, S V , S H ) as
Vnmo(N)-
0, rms (1
(3.50)
where V0, rms is the root-mean-square average of the true vertical velocities V(i) weighted by t~i), and ~ is the appropriate anisotropic coefficient r averaged over the stack of layers, 1 N = V02,rmst0 E .=
(as )
The parameter r is equal to 5 for P-waves, to a for SV-waves, and to 7 for S H waves. Equation (3.50) reduces to the rms averaging of the vertical velocities only if the medium is isotropic or in the unlikely situation that the average value of r is zero. As discussed in section 3.5, equations (3.50) and (3.51) can be used to describe reflection moveout in stratified orthorhombic media, if the vertical symmetry planes have the same azimuths in all layers. Application of the Dix formula (3.48) to the effective NMO velocity (3.49) yields the interval NMO velocity, which contains a contribution of the anisotropic parameters (5, a or 7). Therefore, ignoring the influence of anisotropy in conventional processing and treating the output of the Dix differentiation as the true vertical velocity leads to distortions in time-to-depth conversion. According to equation (3.11), the fractional (relative) error in reflector depth produced by conventional P-wave processing in VTI media is close to the value of 5. For example, if 5 has a typical average value of 0.1, the reflector depth will be overestimated by about 10%. Of course, anisotropy is just one of the reasons for time-depth mis-ties, which may be also caused by other factors such as reflector dip, lateral and vertical heterogeneity, and deviations from hyperbolic moveout. Still, in many exploration areas, the anisotropic contribution to the NMO velocity seems to be the most pervasive reason for errors in converting time to depth. E s t i m a t i o n of 5 f r o m surface a n d b o r e h o l e d a t a
Assuming that the P-wave stacking (moveout) velocity determined from reflection data coincides with the analytic NMO velocity, the interval NMO velocity can be computed from the Dix equation. Then, if the medium has VTI symmetry and the interval vertical velocity is estimated from borehole data (e.g., from check shots or well logs), the parameter 5 can be recovered using equation (3.11):
5--~
1{[,0,]2 } Vpo -
-1
.
(3.52)
3.4.
N M O V E L O C I T Y IN L A Y E R E D M E D I A A N D T I M E - T O - D E P T H
CONVERSION
153
The Dix equation (3.48) is designed for a step-wise velocity model composed of coarse homogeneous layers. In practice, we are often dealing with continuous (closely tabulated) stacking velocity functions obtained, for example, by spatial averaging of many individual stacking (moveout) velocities measured within a certain area. In the limit of small layer thickness, equation (3.48) reduces to the derivative of [Vn2moto] with respect to vertical time. Since differentiation is a relatively unstable procedure sensitive to noise in the data, the function [Vn2moto] has to be smoothed prior to recovering the interval NMO velocity (e.g., by fitting [Vn2moto] to a straight line in a window moving along the time axis). Smoothing of this kind was used to produce the field data examples displayed in Figures 3.27-3.30. Figure 3.27a shows a comparison between average velocities derived from check shots and those obtained from reflection data acquired in offshore Africa near Angola (Alkhalifah et al., 1996). The separation of the two curves continues to increase down to about 2 s two-way time, then remains constant for the remainder of the measurements; the maximum separation amounts to a 12% mis-tie in the average velocity. Recasting the two sets of measurements in Figure 3.27a in terms of interval velocity resulted in the estimated interval values of 5 [equation (3.52)] in Figure 3.27b. After reaching a relatively large value of 0.35 at 2.2 s, 5 rapidly drops almost to zero with increasing vertical time. The overall shape of this/~ curve correlates well with our understanding of the geologic section, which is composed of Tertiary shales down to about 2.3 s. Evidently, these shales are primarily responsible for the anisotropy, and become increasingly anisotropic with depth (for discussion of anisotropic processing in this area, see Chapters 7 and 8). Similar results for an exploration area offshore Zaire were reported by Ball (1995). Figures 3.28-3.30 display estimates of the depth-varying parameter/~ at different locations over a hydrocarbon field in the Gulf of Mexico. (These results were obtained by Tsvankin and Thomsen and described in an internal report of Amoco Production Research, 1992.) Systematic analysis of time-depth mis-ties over the field using a data base of stacking and check-shot velocities showed that 5-curves do not change sharply between adjacent boreholes, unless they belong to different tectonic blocks. A consistent spatial pattern of the variations of (~ is indicative of the strong correlation between this parameter and the distribution of seismic horizons; this is confirmed by simplified lithologic columns along the boreholes. The horizontal lines beneath the plots in Figures 3.28-3.30 correspond to sandy intervals documented in the boreholes, with shales occupying the intervals between the sands. An excellent agreement between the vertical variations of 5 and lithology is evident. Shaly horizons, as in the offshore Africa study above, are characterized by positive values of 5, while near-zero or small negative 5 values in most cases coincide with sands. Also, the most significant anisotropy is observed in the deeper shale layers. Another case study describing anisotropy-induced mis-ties in time-to-depth conversion in the Gulf of Mexico was presented by Meadows and Abriel (1994).
154
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
5 0.4
1.0-
b Figure 3.27: Anisotropy-induced mis-ties between borehole data and surface seismic data in offshore Africa. (a) Average P-wave vertical velocity from check-shot data (gray curve) and average velocity derived from surface reflection data (black curve), both as functions of vertical time. The black curve is obtained by applying the Dix equation to the stacking velocity and then converting the resulting interval velocity into the average velocity. (b) The corresponding parameter 5.
Figure 3.28: Depth-dependent parameter 5 obtained from stacking and check-shot velocities in the Gulf of Mexico. Sand intervals are marked by the horizontal lines on the plots; shales are alternating with the sands. The two curves correspond to different sizes of the smoothing vertical-time window: 300 ms (solid line) and 450 ms (dotted).
3.4.
NMO VELOCITY IN LAYERED MEDIA AND T I M E - T O - D E P T H CONVERSION
o
s~
10~
ls~
Figure 3.29: Same as Figure 3.28 at a different well location in the area.
Figure 3.30: Same as Figure 3.28 at a different well location in the area.
155
156
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
Significant differences between the stacking and vertical velocities in shale layers in the North Sea were documented by Banik (1984) who interpreted this discrepancy in terms of elliptically anisotropic model. The ubiquitous nature of time-depth mis-ties related to transverse isotropy is confirmed by results from Southeast Asia (Desegaulx et al., 1994) and South America (Uzcategui and Mujica, 1995). Clearly, anisotropy is the most common reason for errors in estimating vertical velocity and reflector depth from surface seismic data.
3.5 3.5.1
Elements of 3-D analysis of N M O velocity Equation of the N M O ellipse
The previous results in this chapter were restricted to the 2-D problem, with the incidence plane assumed to represent both a vertical symmetry plane of the medium and the dip plane of the reflector. Here, we remove all restrictions on the model complexity and give an exact analytic representation of azimuthally dependent normal-moveout velocity using the results of Grechka and Tsvankin (1998b). Suppose a pure (non-converted) mode (P or S) was recorded on common-midpoint (CMP) lines with different azimuthal directions over a medium with arbitrary heterogeneity and anisotropy (Figure 3.31). NMO velocity Vnmo(a) is defined in the conventional way through the initial slope of the t2(x 2) curve (t is the reflection traveltime, x is the offset) on a CMP line with the azimuth a [equation (3.98)]. The derivation of the NMO equation in Appendix 3D is based on expanding the one-way traveltime from the zero-offset reflection point to the surface in a double Taylor series in the horizontal coordinates Ix1, x2]. This leads to the following expression for the exact NMO velocity of any pure mode: 1
= Wll cos 2 a + 2W12 sin a cos a + W22 sin 2 a .
(3.53)
Vn2mo (od)
Op~ where P i - OT/Oxi ( i - 1, 2) Here W is a symmetric matrix defined as Wij _ To o~j, are the horizontal components of the slowness vector for rays between the zero-offset reflection point and the surface location [x~, x2], T(x~,x2) is the one-way traveltime from the zero-offset reflection point, and To = ~-(0, 0) is the one-way traveltime at the CMP location xl = x2 = 0 (Figure 3.31). The derivatives needed to obtain the matrix W are evaluated at the common midpoint. Therefore, the exact normal-moveout velocity of any pure-mode reflection at a given spatial location is fully controlled by just three parameters - the components of the matrix W. Plotting the value of NMO velocity in each azimuthal direction a yields a certain curve in the horizontal plane. To determine the type of this "NMO curve," equation (3.53) can be expressed through the eigenvalues )~1 and )~2 of the matrix W:
Vn mo( )
= )~ c o s 2 ( a - fi) + )~2 sin2(a - fi),
(3.54)
3.5. ELEMENTS OF 3-D ANALYSIS OF NMO VELOCITY
157
X2
CMP
lines ~ , , ~ / ~ a
zero-offset
~
/Xl
~~.~. ~
finite-offset ray
ray
reflection point
Figure 3.31: Common-midpoint gathers with different azimuthal orientation around the same CMP location. Reflection-point dispersal is ignored because it does not influence NMO velocity (see Appendix 3D).
158
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
X2
v2 X1
Vnmo
Figure 3.32: NMO ellipse with the semi-axes V1 = 1 / v / ~ and V2 = 1/v~2. where fl is the rotation angle determined by one of the eigenvectors of W (Appendix 3D). Typically, the eigenvalues A1 and A2 are positive. Indeed, if A1 < 0 or A2 < 0, Vn2moin some azimuthal directions becomes negative as well, implying that CMP reflection traveltime initially decreases with offset. For A1 > 0 and A2 > 0, equation (3.54) [which is fully equivalent to equation (3.53)] describes an ellipse with the axes rotated by the angle/~ with respect to the original coordinate frame [xl, x2] (Figure 3.32). Thus, the azimuthally-dependent NMO velocity Vnmo(a) from equation (3.53) usually has an elliptical shape in the horizontal plane. The orientation of the ellipse and the values of its semi-major and semi-minor axes are determined by the derivatives of the ray parameter with respect to the horizontal coordinates. Explicit expressions for the axes of the NMO ellipse in VTI media (for dipping reflectors) and orthorhombic media (for horizontal reflectors) are given below. It should be emphasized that the NMO-velocity equation (3.53) is valid for any sufficiently smooth reflectors and general anisotropic heterogeneous velocity fields. It was assumed, though, that the traveltime field exists at any azimuth near the CMP point (which is not true in shadow zones) and can be adequately described by a Taylor series expansion for the squared traveltime t2(x~), where x~ is the offset on the survey line with azimuth a. While this expansion is routinely used in seismic processing, it quickly degenerates in the presence of strong lateral velocity variation. Also, analytic approximations for reflection traveltimes may become invalid for shear waves in anisotropic media, especially in the vicinity of cusps (triplications) on the wavefront (see Chapter 1). For typical subsurface models, however, the Taylor series expansion does provide a good approximation for the traveltime, and the hyperbolic moveout equation is sufficiently accurate (at least, for P-waves) on conventionallength CMP spreads.
3.5. ELEMENTS OF 3-D ANALYSIS OF NMO VELOCITY
159
For normal-moveout velocity to represent an ellipse, the eigenvalues A1 and A2 of the matrix W should be positive. In some anomalous cases, however, the reflection traveltime may decrease with offset (e.g., for turning waves, as discussed in Hale et al., 1992, or when lateral velocity variation in the overburden is large), yielding a negative V:m o. Then /~1 and/or/~2 are also negative, and NMO velocity cannot be described by an elliptical curve. Equation (3.54) remains formally valid even in this case, but the hyperbolic moveout approximation may become inaccurate on relatively short spreads. If A1 or A2 is equal to zero, the reflection traveltime along one of elliptical axes is constant, and the normal-moveout velocity in this direction is infinite. For instance, NMO velocity goes to infinity on the dip line of a vertical reflector in a homogeneous isotropic, VTI and HTI medium. For TI media with a tilted symmetry axis, Vnmo becomes infinite for steep dips either below or above 90 ~ depending on the anisotropic parameters of the medium (see section 3.3). As follows from equation (3.54), the NMO ellipse for A1 = 0 or A2 = 0 degenerates into two straight lines parallel to the direction in which Vnmo = ce. Such cases, however, cannot be considered as common, and equations (3.53) and (3.54) describe an ellipse for most models of practical interest in reflection seismology. The simplest example of the elliptical azimuthal dependence of normal-moveout velocity is Levin's (1971) NMO equation for a dipping reflector beneath a homogeneous isotropic medium: V2
Vn2m~ r - 1
cos 2 c~ sin 2 r
(3.55)
where a is the azimuth of the CMP line with respect to the dip direction. To demonstrate that equation (3.55) does represent an ellipse, it can be rewritten as 1 V2mo (0~, r
=
sin 20~ V2
+
COS20L
(V/cosr
(3.56)
Clearly, the elliptical axes are parallel to the dip and strike directions, with the semi-axes given by V (strike line) and V/cosr [dip line; see equation (3.6)]. Equation (3.56) can be easily obtained as a special case of the general NMO expression (3.53). 3.5.2
NMO
ellipse
in VTI
media
Equation (3.53) can be used to give analytic description of NMO velocity from both horizontal and dipping reflectors in anisotropic media with any symmetry. Here, previous 2-D results of this chapter are extended to 3-D by obtaining azimuthally dependent NMO velocity for vertical transverse isotropy. Consider a dipping reflector beneath a horizontally layered VTI medium. Then NMO velocity on the dip line is given by equation (3.5) for a single layer and by the Dix-type averaging formula (3.47) for a vertically heterogeneous overburden. The
160
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
equation of the NMO ellipse can be used to develop an exact expression valid for NMO velocity on a CMP line with arbitrary orientation. Due to the axial symmetry of the overburden, the model under consideration has a vertical symmetry plane that coincides with the dip plane of the reflector. (In the special case of a horizontal reflector, the properties of all vertical planes are identical, and NMO velocity is azimuthally independent.) If one of the coordinate axes (Xl or x2) is aligned with the dip plane, the NMO equation (3.53) yields an ellipse without any rotation since W12 : 02T/OXlOX2in this case goes to zero. Therefore, one of the axes of the NMO ellipse for any model that contains a vertical symmetry plane is always aligned with the symmetry-plane direction. For the VTI medium treated here, the second axis of the ellipse is parallel to the reflector strike, and the NMO equation (3.53) gives a remarkably concise result:
n:o(O, l -- Vnmo(0 , r
cos 2 ol -~- Vn~n2o -~, r
) sin 2 a ,
(3.57)
where the azimuth a is measured from the dip plane and the semi-axes of the NMO ellipse are denoted as Vnmo(OL-- 0, r (dip line) and Vnmo(OL -- 2' r (strike line); as before, r is the dip. For simplicity, the following discussion is restricted to a single homogeneous VTI layer. Then the dip-line NMO velocity is given by equation (3.5) which is valid not just for vertical transverse isotropy, but also in vertical symmetry planes of any other homogeneous anisotropic models. To construct the NMO ellipse (3.57), one also needs the normal-moveout velocity on the line parallel to the strike of the reflector. The approach used to derive equation (3.5) in Appendix 3A leads to the following expression for the strike-line Vnmo (Grechka and Tsvankin, 1998b): Vnmo
~ , (~ - V(t~)
-Jr V(r tan r d--~-0=r "
In contrast to the more general dip-plane NMO formula (3.5), equation (3.58) is limited to vertical transverse isotropy. Equations (3.57), (3.5) and (3.58) are sufficient to obtain azimuthally dependent NMO velocity in homogeneous VTI media for all three pure modes (P, SV, SH). If the medium is isotropic, the derivatives of phase velocity vanish, and equation (3.58) reduces to the strike-line component of Levin's (1971) expression (3.56):
Vnmo g, r
-- V ( r
V.
(3.59)
For the special case of elliptical anisotropy, substitution of the phase-velocity function [e.g., equation (1.54)] into equation (3.58)yields Ynmo ( 2 ' ( ~ ) -
Ynmo(0, O ) -
const
(3.60)
for any dip r Therefore, the normal-moveout velocity in the strike direction is equal to the zero-dip NMO velocity, as in isotropic media, but does not coincide with the
3.5. ELEMENTS OF 3-D ANALYSIS OF NMO VELOCITY
161
true vertical velocity. As shown above, the dip-line NMO velocity in elliptical media [equation (3.18)] deviates from the isotropic cosine-of-dip dependence in accordance with the phase-velocity variations. To study the influence of the anisotropic parameters on Vnmo(~, r in general (nonelliptical) VTI media, we again use the weak-anisotropy approximation (Ic,/~, 71 <~ 1). The dip-line NMO velocity of P-waves, linearized in c and/~, was obtained in equation (3.23). The analytic and numerical results above prove that Vnmo(0, r is mostly governed by the difference between the parameters c and 5. This conclusion turns out to be even more accurate for the normal-moveout velocity measured in the strike direction. For P-waves in weakly anisotropic VTI media, Vnmo(2, (~) [equation (3.58)] contains the parameters s and 5 only as the difference s m (~;
Vnmo ~, r
- Vnmo(0, 0)[1 + (e - ~) sin 2 r (2 - sin 2 r
(3.61)
As discussed above, for typical positive e - ~ P-wave NMO velocity on the dip line grows with r more rapidly than the isotropic cosine-of-dip dependence [equation (3.23)]. Likewise, the strike-line NMO velocity [equation (3.61)] increases with dip if e - ~ > 0, rather than being constant as in isotropic (and any elliptically anisotropic) media. However, since the term that contains e - g is three times smaller in the strike-line expression, the NMO velocity in the strike direction is less sensitive to the anisotropy than that in the dip direction. To explain this result, note that for the same CMP spread in the dip and strike directions, rays on the dip-line gather span a wider range of angles with the symmetry axis. Hence, the influence of anisotropy on CMP traveltimes and, consequently, on NMO velocity is more significant in the dip direction. Similar to the dip-line equation (3.7), the strike-line NMO velocity (3.58) can be rewritten in terms of the slowness components of the zero-offset ray as Ynmo
2,P
--
p(pq,_
q) p(r
'
where p = sin g / V is the horizontal slowness, q = q(p) = cos g / V is the vertical slowness, q' - dq/dp, and q" - d2q/dp 2. 3.5.3
NMO
ellipse in orthorhombic
and
HTI
media
The concept of the NMO ellipse becomes particularly important for moveout analysis in azimuthally anisotropic media where NMO velocity varies with azimuth even for a horizontal homogeneous layer. The discussion here is based on the results of Grechka and Tsvankin (1998b) for two common azimuthally anisotropic m o d e l s - orthorhombic (with a horizontal symmetry plane) and HTI. Properties of wave propagation and Thomsen-style notation for orthorhombic anisotropy are described in Chapters 1 (sections 1.1 and 1.3) and 2 (section 2.1). The NMO ellipse for horizontal transverse isotropy is obtained as a special case of the ellipse for orthorhombic media using the results of section 1.3.
162
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
N M O v e l o c i t y in s y m m e t r y planes Consider a horizontal orthorhombic layer with the vertical symmetry planes [xl,x3] and [x2, x3]. If a CMP line is parallel to the xx- or x2-axis, the NMO velocity of any pure mode can be found from the exact 2-D NMO equation (3.8). To evaluate Vnmo as an explicit function of the medium parameters, it is convenient to use the analogy with vertical transverse isotropy. As shown in section 1.3, kinematic signatures and polarization vectors in the symmetry planes of orthorhombic media are described by the same equations as those in VTI media. Therefore, the symmetry-plane NMO velocities can be obtained from the VTI equations (3.11), (3.13) and (3.14) with the substitution of the appropriate anisotropic parameters. For a line in the x 1-direction, the recipe of section 1.3 gives Vnmo,p(OL -- 0, r
0) --
Vpo ~/1 -[- 2(~(2) ,
(3.63)
Vnmo,sv(O, O) -- VSO ~/1 + 2a(2),
(3.64)
Vnmo,su(O,O) -- Vsl V/1 + 27 (2) ,
(3.65)
where Vpo is the vertical P-wave velocity [equation (1.91)], and Vso [equation (1.92)] and Vsl are the velocities of the vertically traveling shear waves polarized in the xland x2-directions (respectively). The anisotropic coefficients 5 (2), 7 (2) and c(2) are 5(2)) [see equaintroduced in equations (1.93)-(1.95)], and a(2) __ (Vpo/Vso)2 ( s tion (2.7)]. The superscript "(2)" refers to the fact that the anisotropic parameters are determined in the Ix1, x3]-plane orthogonal to the x2-axis (not to be confused with the layer index for stratified media). Equations (3.63)-(3.65) remain entirely valid in HTI media with the symmetry axis parallel to the xl-direction. Likewise, the NMO velocities on a line parallel to the x2-axis take the form Unmo,P ( ~ , 0 )
-- U p 0 ~ l - ~ - 2 ( ~ (1) ,
Unmo,SV -~, 0 -- US1
-~-
(3.66)
,
Unmo,SH(~,O ) - Vs0 ~/1 + 27(1),
(3.68)
where a (1) - (Vp0/Vsl) 2 (e(1)_ 5(1)), and the anisotropic coefficients in the [x2,x3]plane are defined in equations (1.96)-(1.98). As discussed in section 1.3, the term "SV-wave" actually refers to different shear modes (Sx and $2) in the two vertical symmetry planes (the same is true for the SH-wave). The simplicity of equations (3.63)-(3.68), which are valid for any strength of velocity anisotropy, is a good illustration of the advantages of the Thomsen-style notation introduced in section 1.3. For HTI media with the symmetry direction Xl, [x2, x3] represents the isotropy plane (in which all velocities are independent of propagation direction), so 5 (1) - o -(1) ~- ,),(1) ___ 0. Hence, for this model the NMO velocities (3.66)-(3.68) are simply equal to the corresponding vertical velocities.
3.5.
E L E M E N T S OF 3-D ANALYSIS OF NMO V E L O C I T Y
163
To find normal-moveout velocity in the symmetry planes of horizontally layered orthorhombic models, it is sufficient to apply the conventional Dix equation to the single-layer NMO velocities. This conclusion follows from the general NMO formula (3.47) valid for any vertical symmetry plane in anisotropic media. Of course, for a stack of orthorhombic layers to have a throughgoing vertical symmetry plane, the symmetry-plane directions in all layers have to be aligned. The analogy between the symmetry planes of orthorhombic media and transverse isotropy also applies to the description of nonhyperbolic moveout (see Chapter 4). The kinematic equivalence with VTI media also helps to describe the NMO velocity for reflections from a dipping reflector, if the dip plane coincides with one of the symmetry planes of orthorhombic media. Although the exact 2-D NMO equation (3.5) cannot be expressed as a simple function of the anisotropic coefficients in the presence of dip, the weak-anisotropy approximations for VTI media [e.g., equation (3.23)] can be applied in both vertical symmetry planes with substitution of the appropriate set of the anisotropic coefficients. Moreover, equation (3.23) remains valid for P-waves outside the symmetry planes of weakly anisotropic orthorhombic media, as long as the CMP line is confined to the dip plane of the reflector and the VTI coefficients are replaced with the azimuthally dependent parameters 5(r and c(r [equations (1.108) and (1.109)]. A z i m u t h a l variation of N M O velocity While symmetry-plane kinematic signatures can be obtained by analogy with VTI media, normal moveout outside the symmetry planes is influenced by azimuthal velocity variations. As shown above for the VTI model with a dipping reflector, one of the axes of the NMO ellipse for any medium with a vertical symmetry plane is always parallel to the symmetry-plane direction. Therefore, for a horizontal orthorhombic layer, both elliptical axes lie in the vertical symmetry planes [xl, x3] and Ix2, x3], and the NMO ellipse can be written as [see equation (3.57)] V~m2o(a, 0) - Vnm2o(0, 0) COS2 a + Vnm2o ~, 0 sin 2 ~,
(3.69)
where a is the azimuth of the CMP line with respect to the xl-axis. Since the symmetry-plane velocities Vnmo(0, 0) and Vnmo (2' 0) can be easily obtained by analogy with vertical transverse isotropy [for P-waves, from equations (3.63) and (3.66)], equation (3.69) provides a concise analytic description of azimuthally-dependent normal moveout of all three pure modes. Fitting an ellipse to azimuthal measurements of moveout velocity (a minimum of three) yields the orientation of the symmetry planes and the NMO velocities within them. After reconstructing the NMO ellipse, one can obtain the normal-moveout velocity in arbitrary azimuthal direction. For the most common orthorhombic models with a single system of fractures embedded in a VTI background (Figure 1.4) or with two orthogonal fracture systems, the vertical symmetry planes are parallel and normal to the fractures. Therefore, azimuthal moveout analysis can be used to supplement or
164
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
even replace shear-wave splitting in detecting the predominant fracture orientation(s) in the subsurface (e.g., Lynn et al., 1996). The methodology of azimuthal moveout inversion for horizontally layered anisotropic media was described by Grechka and Tsvankin (1999b), who also presented a field-data application of their algorithm. In the following, the discussion is limited to P-wave moveout; shear-wave NMO ellipses in a horizontal orthorhombic layer are described in detail by Grechka, Theophanis and Tsvankin (1999). P-wave NMO velocity can be found as an explicit function of the anisotropic parameters by substituting equations (3.63) and (3.66) into equation (3.69): V.2 (a, 0) - V~0 (1 -+- 2(~(1)) (1 + 2/i (2)) nmo,P 1 + 25(2) sin 2 a + 25(1) cos 2 a
(3.70)
Although only two anisotropic coefficients ((f(1) and 5 (2)) influence P-wave NMO velocity from horizontal reflectors, the other three P-wave kinematic parameters (e(1), e(2) and 5(3); see section 1.3) do contribute to the quartic (nonhyperbolic) moveout term (Chapter 4) and, therefore, to the moveout velocity estimated from finite-length spreads. Description of P-wave NMO velocity in orthorhombic media for dipping reflectors can be found in Grechka and Tsvankin (1999a). Normal-moveout velocities for vertical and horizontal transverse isotropy can be obtained as special cases of equation (3.70). For instance, for HTI media with the symmetry axis Xl, the coefficient 5 (1) = 0, and equation (3.70) becomes 1 + 25 (2)
V-2nmo,P(OL,0) -- V20 1 + 25(2) sin 2 OL
(3.71)
Clearly, any orthorhombic medium with 5 (1) = 0 or (f(2) = 0 is fully equivalent to horizontal transverse isotropy in terms of the azimuthally-dependent P-wave NMO velocity from horizontal reflectors. In the weak-anisotropy approximation (15(1)1 << 1, 15(2)1 << 1), the terms quadratic in the anisotropic coefficients in equation (3.70) can be dropped to obtain Vnmo,p(C~,0) - Vpo(1 + 5 (1) sin 2 c~ + 5 (2) cos 2 a) - Vpo[1 + 5(r
(3.72)
where 5(r is given by equation (1.108). Equation (3.72) has the same form as the weak-anisotropy approximation (3.22) for VTI media. To explain this result, note that anisotropy-related out-of-plane phenomena in weakly anisotropic media can be ignored, while P-wave phase velocity in any vertical plane is described by the VTI equation (1.61) with azimuthally-dependent coefficients ~ and 5 [see equation (1.107)]. Reflection moveout for both transversely isotropic and orthorhombic media is generally nonhyperbolic, and the analytic NMO velocity may differ from the moveout (stacking) velocity measured on finite CMP spreads. For typical VTI media, as shown in this chapter, the hyperbolic moveout equation parameterized by the NMO velocity adequately describes P-wave reflection moveout on conventional-length spreads (i.e., spreads close to or smaller than the reflector depth).
3.5.
E L E M E N T S OF 3-D ANALYSIS OF NMO V E L O C I T Y
165
Figure 3.33: Accuracy of the hyperbolic NMO equation for P-wave traveltimes in a horizontal orthorhombic layer. The dots mark the best-fit moveout velocity obtained from the exact traveltimes for the spreadlength equal to the reflector depth; the solid curve is the NMO ellipse calculated using equations (3.63), (3.66) and (3.69). The medium parameters a r e V p o - 2.0 km/s, c(1) -- 0.110, 5 (1) -- -0.035, e(2) _ 0.225, 5(2) = 0.100, 5(3) = O. The numerical test in Figure 3.33 (Grechka and Tsvankin, 1998b) illustrates the (small) influence of nonhyperbolic moveout on azimuthally-dependent P-wave traveltimes in an orthorhombic layer. The difference between the analytic NMO ellipse and the finite-spread moveout velocity was computed for an orthorhombic medium whose vertical symmetry planes have the properties of two VTI models used in Chapter 4 below- Dog Creek shale (Figure 4.5) and Taylor sandstone (Figure 4.1). As expected from the equivalence with vertical transverse isotropy, the errors in moveout velocity caused by nonhyperbolic moveout in the symmetry planes are the same as those for the corresponding VTI models (0.8% for Dog Creek shale and 2.6% for Taylor sandstone). Between the symmetry planes, the error changes monotonically with azimuth, and the finite-spread moveout velocity remains close to the theoretical NMO ellipse for any orientation of the CMP line. Therefore, in typical orthorhombic media, the analytic expressions for NMO velocity are sufficiently accurate for P-wave moveout on CMP spreads that do not exceed the reflector depth. The same observation was made by Tsvankin (1997a) and A1Dajani and Tsvankin (1998) for P-waves in HTI media. For models with unusually pronounced deviations from hyperbolic moveout, NMO velocity can be estimated by nonhyperbolic semblance analysis using the results of Chapters 4 and 7.
166
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
Appendices for Chapter 3 3A
2-D N M O equation in an anisotropic layer
Suppose the CMP line is confined to the dip plane of a reflector overlaid by a homogeneous anisotropic layer (Figure 3.1). We assume that the incidence plane represents a plane of mirror symmetry of the medium, which makes the problem two-dimensional because neither phase- nor group-velocity vector of any reflected ray has out-of-plane components. Our goal is to find an analytic expression for the normal-moveout velocity as a function of the reflector dip r [equation (3.4)]: Vn2mo(r
lim
d(t2)
(3.73)
.
For models with a horizontally homogeneous overburden, the horizontal component of the slowness vector (ray parameter) p does not change between the reflector and the surface. In this case it is convenient to represent the NMO velocity in the following way (Hale et al., 1992):
_2 lim
Vn2m~162-
~dh
(3.74)
to h-.O d p '
where h - x / 2 is half the source-receiver offset (h > 0 in the down-dip direction) and to is the two-way traveltime along the zero-offset ("normal-incidence") ray. Equation (3.74) is a special case of the more general 3-D representation (3.99) of NMO velocity obtained in Appendix 3D. Note that the zero-offset ray (h = 0) is not necessarily perpendicular to the reflector in the presence of anisotropy; it is the phasevelocity vector associated with the zero-offset ray that has to be orthogonal to the reflector (Figure 3.1). Equation (3.74) is derived from one-way traveltimes under the assumption that reflection-point dispersal can be ignored. As shown by Hubral and Krey (1980, Appendix D), the difference between the true (specular) and zero-offset reflection points changes only the quartic and higher-order moveout terms and does not influence NMO velocity; their conclusion holds for anisotropic media as well. Therefore, the depth z0 of the reflection point for any half-offset h (Figure 3.1) can be considered equal to the depth of the zero-offset reflection point. Then h = z0 (tan r - tan r where ~0 and r are the group-velocity (ray) angles for the zerooffset and nonzero-offset rays respectively. Now equation (3.74) becomes 2z0 lim d t a n r Y:m~162
to
h-~O
dp
(3.75) "
3A. 2-D N M O E Q U A T I O N IN AN A N I S O T R O P I C L A Y E R
167
The group angle r in a vertical symmetry plane can be found from equations (1.14) and (1.15) for the in-plane components of the group-velocity vector: 1 dV
tan 0 -4 v 49 tan r - 1 tan0dV V
(3.76) "
d0
Although equation (3.76) coincides with the corresponding expression (1.71) for vertical transverse isotropy, the vertical axis in our model is not necessarily an axis of symmetry. The derivative d tan r in equation (3.75) may be written as dtan r
d tan r dO
dp
dO
dp
Using equation (3.76), we find 1 d2y
d tan r d0
1 + f~ dO2 cos 2 ~ (1 tan 0 dY)2 " V
de
Since p - sin O / V , dO
V cos ~)(1
dp
tanvOdVdo) '
SO
d tan r = dp
1 d2V~
V (1 + Vd--~) [cos t~ (1 t a n 0 d V ) ] 3 V
d0
(3.77) "
The vertical distance from the zero-offset reflection point to the surface is given by Z0
--
1 VG(r
to cos r
Taking into account that Va(r cos C0 is the vertical component of the groupvelocity vector [equation (1.15)] yields 1
(tan0dV)
Zo - -~ Vto cos 0
1
V
dO
"
The phase angle 0 for the zero-offset ray is equal to the dip r and z0 becomes 1 ( Z o - ~ V ( O ) t0cosr 1
tan r dV) Y(r d~ '
(3.78)
where the derivative dy should be calculated at the angle r Substituting equations (3.77) and (3.78) into formula (3.75) for NMO velocity, we finally obtain /
Vnmo((~)_ V(0) COSr
1
d2 V
1 + y~-r 1 - - tanOd--V-'V V(dp) d8
'
where both derivatives of phase velocity should be evaluated at the dip r
(3.79)
168
3B
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
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 - w a v e N M O velocity in T T I m e d i a
Here NMO equation (3.5) is linearized in the anisotropic coefficients for P-waves in weakly anisotropic TI media with a tilted symmetry axis. First, we expand equaand [tanCV'(r162 which go tion (3.5)in the anisotropic terms [ V " ( r 1 6 2 to zero in isotropic media. Dropping the quadratic and higher-order terms in this expansion transforms equation (3.5) into Vnmo((~) -
V((~)
cosr
1 -~
v"(r
2v(r
~- t a n
r v'(r v(r
.
(3.80)
For P-waves, the phase-velocity function fully linearized in the anisotropic coeiticients e and ~ is given by equation (1.61). If the symmetry axis is tilted at the angle from the vertical, the weak-anisotropy approximation (1.61) becomes Vp(O) - Vpo (1 + 6sin 2 0cos 2 0 + esin 4 0),
(3.81)
with 0 - 0 - u. The derivatives of Vp (3.81) to be used in the NMO equation (3.5) are V'(O) - Vpo sin 20 (6 cos 20 + 2e sin 2 0),
(3.82)
V"(O) - 2Vpo [ 5 cos 40 + 2e sin 2 0 (1 + 2 cos 20)].
(3.83)
Since the phase velocity and its derivatives [equations (3.81) - (3.83)] should be evaluated at the angle 0 - r they become functions of the variable r - r u. It is convenient to rewrite equation (3.80) as Vnmo(r
cosr
V((~)
v"($) [1 2v(~)
- v'(r
~- tan r V(r
v'(r
]
+ (tan r - tan $) V((~)
" (3.84)
The term
c ( r-
v"($) + tan r- v'(~) v(r- [ 1 + 2v(r v($)
(3.8s)
has exactly the same form as [ V n m o ( r for a vertical axis of symmetry (u - 0, - r The weak-anisotropy approximation for VTI media was found in equation (3.23)" Vnmo((~) cos (~ -
Vpo[ 1 -~- (~ -~- (~ sin 2 r + 3(e - 5) sin 2 r (2 - sin 2 r
Replacing r in equation (3.86) with r allows us to represent C(r tion (3.85) as C(r
- Vpo [1 + a + di sin 2 q~+ 3(e - 5)sin 2 (~ (2 - sin 2 r
(3.86) in equa-
(3.87)
3C. 2-D D I X - T Y P E E Q U A T I O N IN L A Y E R E D A N I S O T R O P I C MEDIA
169
Using equations (3.81) and (3.82), we find the weak-anisotropy approximation for the remaining term in equation (3.84):
r'(r (tan r - tan r
V (r
_ 2 sin u sin r [ ~ + 2(6 - ~) sin 2 r
(3.88)
COSr
Substituting equations (3.87) and (3.88)into equation (3.84) yields Vnmo(r
COS ~)
--
Ypo{ 1 + 6 + 6 sin 2 r + 3(6 - 6) sin 2 r (2 - sin 2 r +
2 sin u sin r
[~ + 2(6- ~)sin 2 r
(3.89)
COS (~
Equation (3.89) is the weak-anisotropy approximation for the P-wave NMO velocity in TTI media fully linearized in the anisotropic parameters 6 and ~.
3C
2-D Dix-type equation in layered anisotropic media
The goal of this appendix is to generalize the single-layer NMO equation (3.5) for a stratified anisotropic medium above a dipping reflector (Figure 3.26). It is assumed that the CMP line is aligned with the dip direction of the reflector, and the dip plane coincides with a plane of mirror symmetry in all layers. For such a model, the kinematics of wave propagation can be treated in 2-D because the phase- and groupvelocity vectors do not deviate from the incidence (dip) plane. The same assumption was made in the derivation of the single-layer equation (3.79) in Appendix 3A. It is convenient to express the NMO velocity using the same equation (3.74) as in the single-layer problem: Vnemo--lim d ( x 2 ) d ( t 2)
=--
2 lim
dh
(3.90)
to h~O alp'
where h = x / 2 is half the source-receiver offset (h > 0 in the down-dip direction), p is the ray parameter (horizontal slowness), and to is the two-way zero-offset traveltime. Since the medium above the reflector is laterally homogeneous, the ray parameter of any given ray remains constant between the reflector and the surface. Reflection-point dispersal has no influence on the NMO velocity, and the horizontal ray displacement h can be represented as h -
x (i) - Xo
,
where N is the number of layers above the reflector, x (/) is the horizontal displacement of the ray in layer i, and x0 is the total horizontal displacement of the zero-offset ray
170
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
between the CMP location and the reflection point (Figure 3.26). Equation (3.90) now becomes
N d(x(i)) lim/~1 dp " Vn2m~ to2 h-~O
(3.91)
"~.
To identify the interval values of NMO velocity in equation (3.91), let us draw an imaginary reflector through the intersection of the zero-offset ray with the bottom of layer i. The normal to the reflector is chosen to coincide with the slowness (phase) vector that corresponds to the segment of the zero-offset ray in this layer. Next, imagine that the intersection of the zero-offset ray with the top of layer i represents a common-midpoint location of a gather parallel to the actual CMP line. Then the segment of the zero-offset ray in layer i will coincide with the raypath of the zero-offset CMP reflection from the imaginary interface. In accordance with equation (3.90), the NMO velocity from the imaginary reflector at this CMP location would be given by [Vn(/m)o(p)]2 -
t~i~p) ~(')~(o d(x(i))dp '
(3.92)
where p corresponds to the zero-offset ray, t~i) is the two-way traveltime along the zero-offset ray in layer i, and x~i) is the horizontal displacement of the zero-offset ray in layer i. It follows that the sum in equation (3.91) contains the squared NMO velocities from the reflectors normal to the slowness vector of the zero-offset ray in all layers. Therefore, substituting equation (3.92) into equation (3.91) yields a Dix-type averaging formula: 1 y %~
-- t--o E [gn(2o(P)] 2 i=1
t~i)(P)-
(3.93)
Although this expression looks identical to the conventional Dix equation, the interval NMO velocities in equation (3.93) correspond to imaginary reflectors with different dips determined by the orientation of the slowness vector of the zero-offset ray in each layer.
3D
3-D NMO equation in heterogeneous anisotropic media
Let us consider reflection traveltimes of a pure mode (P or S) recorded in CMP geometry over an arbitrary heterogeneous, anisotropic medium. As before, in deriving the NMO velocity we ignore the difference between the true (specular) reflection point and the zero-offset reflection point R (Figure 3.34) and obtain reflection moveout using the one-way traveltime 7(xl,x2) between the zero-offset reflection point and the surface location [xl, x2] (see Appendix 3A).
3D. 3-D NMO EQUATION IN HETEROGENEOUS ANISOTROPIC MEDIA
171
X2
A+(X1 ,X2)
0 / ~ ; A-(-xl
|
/ x~ ! ! !
| %
zero-offset ray zero-offset reflection point
Figure 3.34: In the derivation of NMO velocity, the real CMP reflection raypath from A + to A- (not shown on the plot) can be replaced with a non-specular raypath going through the zero-offset reflection point. Generalizing the approach first suggested by Hale et al. (1992) for the 2-D problem, T(xl, x2) can be expanded in a double Taylor series in the vicinity of the common midpoint" --
7-(-]-Xl ' --~-X2)
=
TO + ~ x l X l
0~-
0~-
(~2T X21 } ~ X0l X 2
-~- ~ x 2 X 2 -} (~X2 2
(~:Tl(~X2
~ 0
Jr- . . . .
(3.94)
(~X2 2
Here ~-+ = 7-(+xl, +x2), ~_ = 7 - ( - x l , - x 2 ) , ~-0 is the one-way zero-offset traveltime, (-+-xl,-+-x2) are the coordinates of the source A + and receiver A-, and the derivatives are evaluated at CMP location O (Figure 3.34). It is assumed that the traveltime T(Xl,X2) is a function differentiable at least twice at the common midpoint xl - x2 = 0. Keeping only the quadratic and lower-order terms in the Taylor series expansion (3.94), we can represent the squared two-way CMP traveltime as t 2 ( x l , x 2 ) - (7-+ + T_) 2 -- t~ + 4(Wll x~ + 2 W 1 2 x l x 2 + W 2 2 x ~ ) ,
(3.95)
where to - 27-0 is the two-way zero-offset traveltime and W is a symmetric matrix a2T defined as W i j - T o oxioxj"
172
CHAPTER 3. NORMAL-MOVEOUT VELOCITY IN LAYERED ANISOTROPIC MEDIA
The coordinates xl and x2 can be expressed in terms of the azimuth a of the CMP line (Figure 3.34) and half the source-receiver offset, h: X1
---
x2 =
h cos oL
hsina.
(3.96)
The squared two-way CMP traveltime from equation (3.95) now becomes: t2(h, c~)
t~ + 4 h 2 (Wll cos 2 c~ + 2W12 sin c~cos c~ + W22 sin 2 a ) .
(3.97)
Using the definition of the normal-moveout velocity Vnmo 4h 2 t 2 ( h , oz) - t 2 -Jr- Vn2mo(OZ) ~ - . . .
(3.98)
,
we obtain from equation (3.97): -1
Vn mo( )
--
[Wll COS20~ + 2W12sin ozcos c~ + W22 sin2 0~]
=
t--o LO~Xlc~
2 lOP1
Opl
Op2
Op2
c~ + (~x2 + ~ x l ) sin c~ cos c~ + ~
sin 2 c~
, (3.99)
where pi = OT/Oxi, (i = 1, 2) are the horizontal components of the slowness vector p of rays emanating from the zero-offset reflection point; Pl and p2 are measured at the surface. To simplify equation (3.99), the coordinate axes can be aligned with the eigenvectors of the matrix W. The required rotation angle/3 is given by (assuming W12 -r 0) W22
W l l -~- v/(W22
/ ~ - tan-1 Introducing the eigenvalues
W l l ) 2 _~_
2W12
"
(3.100)
of W,
equation (3.99) after the rotation can be written as
VnmZo(O~)= A~ COS2(O~- fi) + A2 sin2(o~- fi).
(3.102)
Chapter 4 Nonhyperbolic reflection moveout Chapter 3 was devoted to the influence of anisotropy on normal-moveout velocitythe parameter that controls reflection traveltimes on spreadlengths no longer than about the distance between the CMP and the reflector. If the medium is anisotropic and/or heterogeneous, hyperbolic moveout equation parameterized by NMO velocity loses accuracy with increasing offset. It should be emphasized that angle-dependent velocity makes reflection moveout nonhyperbolic even in a single homogeneous layer (unless the anisotropy is elliptical). Anisotropy-induced deviations from the short-spread hyperbola for a typical VTI model (Taylor sandstone, Figure 4.1) are illustrated by Figures 4.2-4.4. If the layer were isotropic (e = 5 = 0), the moveout would be purely hyperbolic for any sourcereceiver offset. The residual moveout in Figure 4.2 is calculated as the difference between the exact traveltimes and the best-fit hyperbola found by the least-squares method. Since the spreadlength Xm~xis limited by the reflector depth z, the residuals for both P- and SV-waves at all offsets are small (less than 2 ms). The error of the best-fit hyperbola, however, rapidly increases with spreadlength and for the SV-wave reaches 60 ms when X m a x : 2 Z (Figure 4.3). It is important to notice that even if the time residuals are small, the finitespread (effective) moveout velocity of the best-fit hyperbola may differ from the NMO velocity (Figure 4.4). Whereas Vnmo is determined analytically in the zero-spread limit, the finite-spread moveout velocity is obtained by fitting generally nonhyperbolic data with a hyperbolic function. For instance, the low residuals in Figure 4.2 were achieved with a moveout velocity for the P-wave that is 2.6% higher than Vnmo. Therefore, if in this example we had used a hyperbola parameterized by the NMO velocity, the time residuals would have been much higher. Typically, the difference between the finite-spread and NMO velocity, as well as the residual moveout, becomes pronounced for spreadlengths exceeding the reflector depth (Figure 4.4). For that reason, offsets x > 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<~ in some of the layers. As a result, the influence of layers with negative a on deviations from hyperbolic moveout is considerably dampened by the vertical heterogeneity. The interval quartic moveout coefficient for P-waves in VTI media can be expressed through just two p a r a m e t e r s - the NMO velocity from a horizontal reflector and the anellipticity coefficient r/. It is clear from equations (4.13) and (4.15) that the interval values of these two parameters control the coefficient A4 in stratified VTI models. Explicit expressions for A4 and P-wave long-spread moveout in layered VTI media as a whole are introduced in section 4.3. Equation (4.16) can be used to derive a Dix-type differentiation formula for the interval quartic moveout coefficient A~N). Suppose the reflection events from the top and bottom of layer N were used to retrieve the effective values of the NMO velocity and the coefficient A4. Then the interval NMO velocity Vn(mNo ) can be computed from the conventional Dix equation. Next, equation (4.16) can be rearranged to get
g(N)-
t o (1N ) ~N
[(V(/m)~
4
A~i) (V(/m)~ (t~i) )2] t~i) ,
(4.17)
where
g(N) -
V4mo(N)[1 - 4 A4(N) V4mo(N)t~(N)].
(4.18)
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
182
I
12
Figure 4.7: Quartic Taylor series (4.1) with the exact moveout coefficients (tT, the dotted curve) and the more elaborate approximation (4.23) (tA, solid) in a layer of Dog Creek shale; z = 3 km. After obtaining the quantities g ( N ) and g ( N - 1) using the reflections from the top and bottom of the layer, the interval quartic coemcient can be found as A~N) _
1
4~ vnmo
[ (gn(No))4
L
g ( N ) to(N) - g ( N - 1)to(N - 1)] to(N) - t o ( N -
1)
J
(4.19)
Application of equations (4.18) and (4.19) to the inversion of long-spread P-wave moveout in VTI media is discussed in Chapter 7.
4.2
Nonhyperbolic moveout equation
Although the quartic Taylor series (4.1) provides valuable analytic insight into properties of nonhyperbolic moveout, it rapidly loses accuracy at moderate offsets (Figure 4.7). Due to the increasing quartic moveout term (which is negative for the example in Figure 4.7), the Taylor-series curve sharply deviates from the exact traveltimes for x > 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
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
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.
NONHYPERBOLIC REFLECTION MOVEOUT
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.
NONHYPERBOLIC REFLECTION MOVEOUT
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
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
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.
NONHYPERBOLIC REFLECTION MOVEOUT
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
NONHYPERBOLIC REFLECTION MOVEOUT
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)
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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
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
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
CHAPTER 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES
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
CHAPTER 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES
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.
A P P L I C A T I O N TO A VTI LAYER
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
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
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.
A P P L I C A T I O N TO A VTI LAYER
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
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
~'). 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
CHAPTER 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES
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
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
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
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
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
CHAPTER 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES
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
CHAPTER 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES
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 [~] << 1 and ]5] << 1, phase velocities of P- and S ( S V ) - w a v e s can be approximated by (see Chapter 1): Vp(O) - - V p o [1 + (~sin 2 0 + (e - ~)sin 4 0] (5.100) and Vsv(O) = Vso (1 + a sin 2 0 cos 2 0),
where 0 is the phase angle with the symmetry axis.
(5.101)
242
CHAPTER 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES
S
P
Figure 5.16: Geometry for the derivation of the weak-anisotropy approximation for P S moveout. OP and OS are the slowness (phase-velocity) directions of the downgoing P-wave leg and upgoing S-wave leg of the converted wave, and 0p and 0s are the corresponding phase (slowness) angles with the reflector normal ON. OM, the projection of the slowness vectors onto the reflector (denoted as Pint), should be identical for both P- and S-waves. Our first goal is to express the group velocity and group angle for both P- and S-waves (Figure 5.1) through the slowness Pint. Let us denote the angle between the phase-velocity (slowness) vector of the downgoing P-wave and the reflector normal (pointing upward) by 0p (Figure 5.16). Then the phase angle 0 with the symmetry axis is equal to 0p - r and equation (5.100) can be written as _
V p ( O p ) -- Vpo
[1 + 5 sin2(0p - 0) + (e - 5)sin4(0p - 0)].
(5.102)
In the linearized weak-anisotropy approximation, group velocity (at the group angle) is equal to phase velocity (Chapter 1). Also, it is possible to use the isotropic relationship between the phase angle 0p and Pint because 0p is contained only in terms already linear in the anisotropic parameters. Hence, the group velocity of the P-wave is given by VG,p(Pint) -- Vpo [1 -~- (~ sin2(0iso,p -- 0) + (~ -- 5) sin4(0iso,P -- 0)],
(5.103)
where sin Oiso,P - Pint
YPO ;
cos 0iso,P - - V / 1 - (Pint YP0) 2.
(5.104)
Next, the P-wave group angle with the vertical Cp [equation (5.49)] has to be found as a function of Pint. The phase angle 0p is expressed as _
sin
0p -
Pint Y p - Pint YPo (1 -~- O/anis,P),
(5.105)
5D. W E A K - A N I S O T R O P Y
APPROXIMATION
FOR PS-MOVEOUT
and
cos Ov - - ~ / I -
(PintVPO) 2
IN VTI M E D I A
243
(Pint Vpo)2 ] (-Pi~tY--po)2 Oianis'P
I- I
-
(5.106)
'
where OLanis,P - - ~
sin2(0iso,P - r + (e - ~)sin4(0iso,P - r
(5.107)
Using the weak-anisotropy relationship between the group and phase angles in TI media [equation (1.73)] and taking into account that the P-wave is taken to propagate upward from the conversion point yields tan Cp - - tan(0p - r
+ 25 + 4(e - 5)sin2(0iso,P - r
(5.108)
Thus, the weak-anisotropy approximation makes it possible to find explicit expressions for the group velocity and group angle in terms of the slowness projection Pint.
Substituting sin 0p and cos 9p from equations (5.105) and (5.106) into equation (5.10 and further linearizing in the anisotropic parameters, we obtain tan Cp
=
[1 + 25 + 4(c - ~) sin2 (0iso,P
- tan(0-iso,P - r
tan -~- ~ a n i s , P
]
~iso,P
sin(0iso,P - r
r
- - (~)
(5.109 ) "
From equation (5.108) it also follows that 1
cos r
= -
-
1
{1 + sin2(0iso,P - r
cos(e~ - r
+ 4(~ - 5)sin2(~iso,P - r
Combining equations (5.103) and (5.110) gives the term find the traveltime along the P-wave leg:
1 VG,pcos~p
=
1 -Vpocos(0iso,p ~ ~anis,P
(5.110)
(1/VG,pcosCp)needed
to
[ - r
1 + 2(c - 5) sin4(0iso,e - r
cosr
COS 0iso, P COS ( 0 i s o , g - - (~)
]
(5.111)
9
Similar algebraic transformations provide the corresponding expressions for the S-wave leg: tan Cs
-
tan(0-iso,S - r
-~- OLanis'S
[1 + 2a - 4a sin2(/~i~o,S - r
tan 9iso,S sin(giso,S - r 1
VG,SCOS ~2s
vs0 cos (Oiso,~ - r
(5.112) - - (~)
[[1 - 2a sin4(0iso s - r cosr
-[- OLanis,S
-
'
cos eiso,s cos (0~so,s - r
(5.113)
244
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
where OLanis,S - - (7
sin2(0iso,S - r
- - (/))
(5.114)
and sin 0-iso,S -Pint Vso;
COS0-iso,S -- ~/1 -- (Pint Vs0)2-
(5.115)
Note that the terms involving the anisotropic coefficients in equations (5.112) and (5.113) can be found from the anisotropic terms in the corresponding P-wave equations (5.109) and (5.111) by making the substitutions discussed in Chapter 1: Vpo should be replaced with Vso, 5 with a, and c set to zero. Equations (5.109), (5.111), (5.112), and (5.113) are sufficient for obtaining the traveltime and offset for the PS-wave using the general relationships (5.52)-(5.56).
5D.2
Moveout attributes
Next, symbolic software M a t h e m a t i c a is used to derive the weak-anisotropy approximations for the attributes of the P S traveltime function. The slope of the moveout curve at zero offset can be found by linearizing the exact equation (5.81) in ~ and 5. The horizontal slownesses (pp and Ps) of the zero-offset reflection have to satisfy Shell's law [equation (5.83)] and the condition q,P
pp and ters:
Ps
(5.116)
q,s .
are found by linearizing the vertical slownesses in the anisotropic parameqp-qeo
1 - 25pP [3+(e
3) pp2 1/2o] ,
(5.117)
qPo qs -
qso
2
(1 - a p s V~0),
(5.118)
where 11 qPo-V~}o-P~, qso -
(5.119)
- Ps "
Substitution of equations (5.117) and (5.118)into equations (5.83) and (5.116) leads to a system of two equations for the horizontal slownesses pp and Ps. Splitting pp and Ps into the isotropic and anisotropic parts, solving this system, and using the obtained horizontal slownesses in equation (5.81) yields sin r
dt d x x=0
:
2Vpo(l+~)
[(1-~2)+4~(a-5)]
sin 3 r
+
+
[ ~ (1 + ~) - (7 (~3 + 9~2 + 8)] 2Vp0~(l+~) sin 5 r 2 Y.0 ~ (1 + ~) ~ ( ~ + 5 ~ + 5~ + 1);
(5.121)
5D. WEAK-ANISOTROPY APPROXIMATION FOR PS-MOVEOUT IN VTI MEDIA
245
Vpo/Vso. To find the attributes of the traveltime minimum, we need to determine the corresponding horizontal slowness Pv-min_ "-Dmins -- pmin. The linearized form of pmin c a n be found from Snell's law [equation (5.9)] by substituting the approximations for the vertical slownesses [equations (5.117)and (5.118)]: -
pmin -
-
_min (1 + Piso
A p min
),
(5.122)
where Piso _min is the isotropic solution [equation (5.97)] and tan r
A p m i n __ --
2pmso n
(qpo Aqv + qso Aqs) 9
(5.123)
_min Here qpo and qso are given by equations (5.119) and (5.120) with pp - Ps - Piso, and
Aqp =
(~.min~2 Piso ) [(~ _.[_ (s __ 5)
2
_min,2 (Pi~o) V~0],
(5.124)
qPo minx2 A q s -- O" [Piso ) Vff0.
(5.125)
If necessary, the slowness projection onto the interface corresponding to the traveltime minimum can be found from equation (5.122) as pmin int-
sine ( 1 VSO
2
1 )
(5.126)
Ypo'
where only the leading dip term was retained. Equation (5.126) is fully equivalent to the isotropic result (5.97). The weak-anisotropy approximation for NMO velocity is derived from the exact equation (5.92). Expanding the linearized version of equation (5.92) in sine up to the quadratic term and substituting pmin from equation (5.122) leads to the following form of Vnmo,PS: K_ 2
nmo,PS
_
+
1
sin2r
Vpo Vso - 8 ~VpoVso~-73
[3V4 ~
-
2Vl~oVso+6V~oV~0 _ 2VpoV~0 + 3V~0]
c [-4 Vl~oV~o+ sin 2 r (3 Vp4o- 11 V~oVso 2 VpoV40(vP0 + Vso) - VpoVso + 3 VpoV~o + 6 V~o )
6 (V o "[-
2
2oVs4o(
Vvo) [-4 V2oV~o+ sin 2 r (3 V40 - 8 V~oVso o + Vso) - 6 VpoVso 2 2 - 8 VpoV~o+ 3 V~0)] .
(5.127)
It is convenient to describe the moveout attributes in terms of the ray parameter (horizontal slowness) of the pure P-wave reflection recorded at zero offset (P~o). Neglecting the cubic and higher powers of sin r we can represent PPo as
246
CHAPTER 5. REFLECTION MOVEOUT OF MODE-CONVERTED WAVES
sin r
sin r ~.
= vp(r
(5.128)
vp0
Equation (5.128) has a purely "isotropic" form because the anisotropic terms in PPo are multiplied with high powers of sin r not included in our approximation. Substituting ~ and PPo and expressing c through a yields the following expression for NMO velocity: V. -2 nmo,PS (PPo)
--
V.-2nmo,PS(0) - P2~ (3t~ 4 - 2t~ 3 -~-6m 2 - 2~ +
_
Pp02( ~ _ 1 ) [ 6 a ( ~ + 1 ) 2 - ( a - 5 ) ~ ( 3 ~ 2 ~ ( ~ + 1)
3)
2-2~+3)],(5.129)
where V.-2
1
nm~
[ 1 - 2(a+t~/))]
-- VpoVso
1 + ,~
(5.130) "
The approximate normalized offset Xmin/tmin is derived in a similar way. Lineariz-
ing the exact equation (5.18) for Xmin/tmin,substituting pmin [equation (5.122)] and retaining only the linear and cubic terms in sin r we find Xmin
__
tmin
-
sine Vs0 [ ] sin3 r Vs0 [ 2 ~;- 1 + 2 ( n ~ - a) + 16 t~2 (t~5 -~- 3t~3 - 3t~2 - 1) + 45~3(~ 2 + 1) + 4a(2~ 3 + 5~ 2 + 6~ + 1)].
(5.131)
Likewise, the leading dip term in the weak-anisotropy approximation for the spatial derivative of the minimum traveltime (5.19) is given by dtmin
dtmin
= tan r ~
dYcMp
5E
dzcMp
= PPo (1 + ~).
(5.132)
3 - D d e s c r i p t i o n o f P S m o v e o u t in l a y e r e d media
Here the results of Appendix 5A are extended to mode-converted waves recorded in arbitrary azimuthal direction, first over a homogeneous layer and then over stratified anisotropic media.
5E.1
Single anisotropic layer
Suppose the mode conversion occurs at a plane dipping interface beneath a homogeneous layer with arbitrary anisotropic symmetry. Similarly to the 2-D case treated in Appendix 5A, the traveltime along the P-wave leg can be found as tp =
Zr g~ cos r
=
Zr , g~
(5.133)
hE. 3-D D E S C R I P T I O N
OF
P S M O V E O U T IN L A Y E R E D M E D I A
247
where, as before, z~ = R D is the depth of the reflection point R, Cp is the angle between the P-wave ray and the vertical and ge - VG,p is the P-wave group velocity (Figure 5.13). The vertical component g3 of the group-velocity vector, expressed through the slowness vector, has the form (Grechka, Tsvankin and Cohen, 1999) 1 - - = q - P~ q,~ - P2 q , 2 , g3
(5.134)
where pl and p2 are the horizontal components of the slowness vector, q - P3 is the vertical component, q,1 - O q / O p l , and q,2 - O q / O p 2 . Substituting equation (5.134) into equation (5.133) yields the P-wave traveltime as a function of the slowness components: tp -
(5.135)
Zr (qe - P~v q,~e - P2v q,2e) ,
where q,~e - O q p / O p ~ p (j - 1, 2). Taking into account the contribution of the S-wave leg gives the following expression for the PS-traveltime" t - tp + t s -
Zr (qv -- PiP q,~v -- P~v q,2v + qs -- P~s q,~s -- P2s q,2s)"
(5.136)
The slowness vectors of the P and S-waves are related to each other by Snell's law at the reflector (i.e., their projections on the reflector should be identical). The coordinates of the source and receiver can be expressed through the horizontal component of the group-velocity vector and the length of the ray segment between the reflector and the surface. Assuming that the source (A) excites P-waves and denoting le = A R (Figure 5.13), the horizontal coordinates of the source with respect to the conversion point R are given by x~
1p - g,~ ~ -
Zr g~ g~cosr
g'v
=Zr~g~
(i
--
1 2) ,
,
(5 137)
where g3P should be positive because both 1p > 0 and ge > 0. For example, if the x3-axis points upward, the group-velocity vector should correspond to an upgoing wave.
As shown in Grechka, Wsvankin and Cohen (1999), g~/g3 = -q,~ (i = 1, 2), which helps to rewrite equation (5.137) as x~p - --Zr q,~p (i -- 1, 2).
(5.138)
Hence, the source-receiver vector A C (Figure 5.13) is described by X -- AC
-- {(Xls
--
Xlp),
(X2s -- X 2 p ) } -- Z r {(q, lP -- q, IS)'
(q,2P - q,2s)}.
(5.139)
To find all source-receiver pairs with a common midpoint and form a CMP gather, zr needs to be replaced with the vertical distance between the CMP and the reflector (ZcMp -- O B , Figure 5.17). First, let us determine the distance B D (see the inset
248
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
Figure 5.17: Relationship between the depth zr = D R of the conversion point R and the vertical distance ZcMe = O B between the CMP and the reflector. O N and O H are the dip and strike directions of the reflector (respectively), r is reflector dip. Note that M N _ L O N , O M = B D and ZCMe -- Zr = R M = Q N = O N tan r
5E. 3-D D E S C R I P T I O N O F
PS
M O V E O U T IN L A Y E R E D M E D I A
249
in Figure 5.13) between the CMP and the projection of the reflection point onto the surface. Since 1 BD - ~ AC- DC (5.140) and D C = - z r {q,~s, q,~s }, we can use equation (5.139) to obtain Zr B D - {BD1, BD2} - -~ {(q,~p + q,~s), (q,2P + q,2s)} "
(5.141)
As illustrated by Figure 5.17, zcMP can be represented as
zcMP = zr + B D c o s ( / N O M ) tan r
(5.142)
where r is reflector dip a n d / N O M is the difference between the azimuths of the dip plane and line B D . To ensure the correct sign of zcMP - zr, c o s ( / N O M ) has to be positive if point D is located updip from point B:
c o s ( / N O M ) - B D I ~ + BD2 ~e BD
(5.143)
where ~1, ~2 is a horizontal unit vector in the updip direction. Using equation (5.141), we find Zr B D c o s ( / N O M ) - -~ [(q,lP + q, ls) ~1 + (q,2P + q,2s) ~2] . (5.144) Equations (5.142) and (5.144) make it possible to obtain zr as a function of zcMp: zcMP Zr= I+Az' Az - tan2 r [(q, 1P
(5.145)
+ q, as) ~l + (q,2P + q,2s) ~2]
"
(5 " 146)
Substituting ZCMP from equation (5.145) into equation (5.136) yields the final expression for the traveltime of the PS-wave:
t - - I + AzZCMP(qP --P~Pq,~P --P2Pq,2P +qs --P~sq,~s --P2sq,2s), where Az is determined in equation (5.146). tot (5.139) is given by X - - 1 AzZCMP 21_ { ( q , IP - -
5E.2
Layered
q, IS)'
(5.147)
Similarly, the source-receiver vec-
(q,2P - q,~s)}"
(5.148)
media
Suppose a plane dipping reflector is now overlaid by a stratified anisotropic medium (all interfaces above the reflector are assumed to be horizontal) with arbitrary symmetry in each layer. As before, the sources and receivers are collected into either
250
CHAPTER
5.
REFLECTION
MOVEOUT
OF MODE-CONVERTED
WAVES
common-conversion-point (CCP) or CMP gathers with different orientation (Figure 5.13). To find the traveltime and offset on CCP gathers, it is sufficient to sum up the single-layer solutions from the previous section. Using equation (5.136), the traveltime of the wave converted at interface N can be written as N
t - tp + t s - y ~ z (t) (q(t) _ Pip z~, l P - P2P aa(t) + q(t) ,2P
_ Pls
O(g) a , lS
P2s
~7(~)) a,2S
(5.149)
~=1
where z (t) is the thickness of layer g; for layer N, the thickness Z (N) should be measured above the conversion point. The horizontal components of the slowness vector for both waves (PiP and P~s, i = 1, 2) remain constant between the reflector and the surface because the medium along the raypath is horizontally homogeneous. Likewise, for the source-receiver vector equation (5.139) yields x - {(x~
- x~),
(~.
- x~,)}
-
z(~) to(~) - o(~)) l,lP a, lS
z(~) (q(~) - q!~))] ,2P
~
"
~g=l
(5.150) Therefore, after expressing P~s through p~p (or vice versa) using Shell's law at the reflector, CCP gathers in layered media can be generated in a rather straightforward way from equations (5.149) and (5.150). Moveout equations for CCP gathers can be adapted for CMP geometry by replacing Z (N) with the thickness of layer N beneath the CMP (z(U)). Since all interfaces (u) can be expressed in the form of the single-layer in the overburden are horizontal, z CMP equation (5.142): Z CMP (u) -- Z (u) + B D c o s ( / N O M ) tan r ~ (5 " 151) where B D is the distance between the CMP and the projection of the reflection point onto the earth surface, a n d / N O M is the azimuth of the line BD with respect to the dip plane of the reflector (Figure 5.17). Using the relationship between BD and the source-receiver vector AC = x [equation (5.140), see the inset in Figure 5.13)] and equation (5.150) leads to
BD--~1 { I__~i z(~)(q(t)+q'([))l i~ z(~)(q'(~)~+ ~ )]}
(5.152)
The term [ B D cos(/NOM)] is the projection of vector BD onto the dip direction of the reflector [see equations (5.143) and (5.144)]" BD cos(/NOM)
1 {I__~z(~).(~)a(t)] .(t)) (~,~P + ,,~s) ~1 + i~= z(t),~(t) (~,2P + ~,2s
- -~
(5.153) as before, ~1, ~2 is a horizontal unit vector in the updip direction. Next, equation (5.153) can be substituted into equation (5.151)" z(N)
CMP
- - z ( N ) "~-
tan~ 2
(q, lP +
,Is
,2s
5154
hE. 3-D DESCRIPTION OF PS MOVEOUT IN LAYERED MEDIA
251
which can be treated as a linear equation for z (N) with the solution
z(N) __ tan(p {[ ~ N/=1 z(t) (q g) + q(e),1)]8 ~1 + [r/N_-11 Z(/)(q,(2~)p -~ q(/),2s)] ~'2} 2
z(N)__ CMP
1+ ~
[(q(N) + q(N))~I + (q(N) _+_q(N)) ~2] 2
,
,1S
\
,2P
,
(5. 55)
N-1 in the numerator vanish, and If the model consists of a single layer, the sums ~/=1 equation (5.155) reduces to equation (5.145). Using z (y) from equation (5.155) in equations (5.149) and (5.150) yields the converted-wave traveltime and source-receiver vector in CMP geometry. The location of the CMP gather is defined by the thickness z C(y) of layer N beneath the midpoint. MP
5E.3
2-D relationships
for symmetry
planes
3-D equations developed above can be simplified in the special case of a PS-wave recorded in a vertical symmetry plane of a layered anisotropic medium. The incidence plane is also assumed to coincide with the dip plane of the reflector, which reduces the problem to 2-D. P-waves propagating in vertical symmetry planes are coupled to only the in-plane polarized shear wave ( S V ) and, therefore, generate a single mode conversion. If the dip plane is aligned with the xl-axis, the projection of the slowness vector on the axis x2 vanishes, and the traveltime (5.149) simplifies to N t - - tp + t s -- E
"(/) + q~/) - P ~ s q(/)). z(~) (q(~) - P~P u,~p ,~s
(5.156)
/=1
The corresponding source-receiver offset can be found from equation (5.150) as N X - - XlS -- X l p - - E
z(/) (q,(1/)P - - a o , (/)) IS "
(5.157)
i=1
Equations (5.156) and (5.157) are sufficient for modeling CCP gathers. To obtain a CMP gather, z (N) in equations (5.156) and (5.157) should be replaced with z(N)p. Substituting {1 - 1 and {2 - 0 into 3-D expression (5.155) yields z(N) tanr ~-~N-1 Z(g) (g) q(/)) z(N) __-- CMP -- 2 /----1 (q,iP + ,is
1 + ~2
(q(~ + o(N)) , ~,IS
(5.158)
This Page Intentionally Left Blank
Chapter 6 P-wave time-domain signatures transversely isotropic media
in
The most critical step in extending seismic inversion and processing to anisotropic media is to identify the medium parameters responsible for measured reflection signatures. In Chapter 1 it was demonstrated that Thomsen notation makes it possible to reduce the number of parameters that govern P-wave kinematics in VTI media from four to three - Vpo, ~ and 5. Therefore, if amplitude preservation is not an issue, for purposes of P-wave depth imaging (e.g., prestack depth migration) one needs to reconstruct the spatial distributions of those three parameters. The subject of this chapter is time-domain P-wave processing, which includes normal-moveout (NMO) and dip-moveout (DMO) corrections, prestack and poststack time migration. As shown here, for laterally homogeneous VTI models above a horizontal or a dipping target reflector, P-wave time-domain signatures depend on the interval values of just two combinations of Vpo, e and 5. Furthermore, both "time-processing" parameters can be recovered from P-wave reflection moveout alone, without using borehole information or multicomponent (S or PS) data. According to the results of Chapter 4, long-spread P-wave moveout in a horizontal VTI layer is controlled by the vertical traveltime (tpo) and two medium parameters, one of which is the NMO velocity commonly measured in seismic processing: Vnmo(O)
-
Vp o V/ i Jr- 2 (~ .
(6.1)
The other parameter is the anisotropic coefficient r], which quantifies the anellipticity of the medium [equation (4.14)]" r / - 1 + 23"
(6.2)
V~mo(0) is responsible for the initial slope of the t2(x2)-curve, while 77 determines the deviation of the traveltime function from a hyperbola. If the medium is elliptical (e =/~, 77 - 0), moveout in a single layer is purely hyperbolic, although the moveout velocity is generally different from the vertical velocity. In layered VTI media, longspread moveout of P-waves depends on the interval values of Vnmo(0) and ~ and interval vertical traveltimes. 253
254
CHAPTER 6. P-WAVE TIME-DOMAIN SIGNATURES IN TRANSVERSELY ISOTROPIC MEDIA
We start out by proving that the same parameters, Vnmo(0) and 7, also govern P-wave NMO velocity from dipping reflectors, if it is expressed through the slopes of reflection events on the zero-offset section. Then the analytic description of longspread P-wave moveout from Chapter 4 and numerical modeling are used to show that Vnmo(0) and 7/are sufficient for describing the impulse response of P-wave poststack time migration. This leads to a general two-parameter representation of P-wave timedomain reflection signatures (first developed by Alkhalifah and Tsvankin, 1995) that has far-reaching implications for velocity analysis and time processing in VTI media. The last section of the chapter discusses the possibility of describing P-wave NMO velocity in terms of Vnmo(0) and ~ for TI models with a tilted symmetry axis.
6.1
6.1.1
P - w a v e N M O v e l o c i t y as a f u n c t i o n o f ray parameter 2-D analysis
for a VTI
layer
First, let us assume that the CMP line is confined to the dip plane of the reflector where NMO velocity is described by the 2-D equations (3.5) and (3.7):
Vnmo-- V((~) 1 + v~-r COSr
[ 0=r _-1-- t~nCd__Y_V v(r dO 0=r
ld2vI
Jqpq~ q IP(r
.
(6.3)
Here V(O) is the phase velocity as a function of the phase angle 0, r is the dip, p = sin0/V and q = cos0/V are the horizontal and vertical components of the slowness vector, q' = dq/dp, and q" - d2q/dp 2. Since the slowness (phase-velocity) vector of the zero-offset ray is orthogonal to the reflector, Vnmo is evaluated at the phase angle equal to the reflector dip, 0 = r The NMO velocity in equation (6.3) can be expressed as a function of either the reflector dip r or the ray parameter p. To make the most appropriate parameter choice for the inversion procedure, we have to take into account that reflection data carry no explicit information about the dip of the reflecting interface. In contrast, the ray parameter corresponding to the zero-offset reflection can be measured from zero-offset data as 1 dto (6.4) P -
i dy '
where to(y) is the two-way traveltime on the zero-offset (or stacked) section, and y is the common-midpoint position. If the medium above the reflector is laterally homogeneous (but possibly vertically heterogeneous), the ray parameter does not change between the reflector and the surface, and
1 dto 2 dy = p(r
sin r - V(r
(6.5)
6.1. P-WAVE NMO VELOCITY AS A FUNCTION OF RAY PARAMETER
255
Therefore, in general it is convenient to use the representation of NMO velocity directly in terms of the ray parameter [as in the second equation (6.3)]. Note that the function Vnmo(P) is of primary importance in seismic processing because it determines the operator used in dip-moveout (DMO) correction (e.g., Hale, 1984). In the numerical analysis of NMO velocity, the replacement of r by the ray parameter p does not pose any serious problems. The Christoffel equation can be expressed through the slownesses p and q, which leads to a quadratic equation for q2 (Appendix 6A). Differentiating the Christoffel equation yields the derivatives of q with respect to p needed in equation (6.3). Alternatively, p and q can be recalculated into the phase angle and phase velocity and substituted into the function Vnmo(r It is most important to note that using p as the argument may change the influence of the elastic coefficients on the NMO velocity since the form of the equation will be different (p is a function of r and the anisotropic parameters). Elliptical and weak anisotropy The dependence of NMO velocity (6.3) on the ray parameter can be elucidated by considering the special cases of elliptical and weak anisotropy. If the medium is elliptical (e = ~), Vnmo(P) is given by (Appendix 6A) Vnmo (P) -
Vnmo (0)
(6.6)
11 -p2Vn2moidi " Equation (6.6) is a good illustration of the difference between the NMO equations expressed through dip and ray parameter. As shown in Chapter 3, if the dip r is used as the argument, the velocity Vnmo(r in elliptical media deviates from the isotropic cosine-of-dip dependence, with the error proportional to the ratio of the phase velocities V(r - y/1 + 25 sin 2 r [equation (3.19)]. Therefore, the normalized velocity [Vnmo(r cos(r contains a contribution of the anisotropic parameter d. In contrast, Vnmo expressed through the ray parameter [equation (6.6)] is a function of just the zero-dip NMO velocity with no explicit anisotropic terms. In fact, equation (6.6) coincides with the corresponding expression (3.25) for isotropic media; the influence of the anisotropy in formula (6.6) is absorbed by Vnmo(0) [equation (6.1)]. This result means that for elliptical models the trade-off between Vpo and (~ in equation (6.1) cannot be resolved from the dip-dependence of P-wave NMO velocity. Moreover, reflection moveout for elliptical anisotropy is purely hyperbolic and does not provide any information other than Vnmo(0), which is identical (for a vertical symmetry axis) to the horizontal velocity. On the other hand, NMO velocity as a function of p can be easily reconstructed from Vnmo(0), which makes isotropic dipmoveout processing suitable for elliptical media. For qualitative analysis of Vnmo(P) for non-elliptical models, we employ the weakanisotropy approximation (le[ << 1, [~[ << 1). Substitution of p into the NMO equa-
256
CHAPTER 6. P-WAVE TIME-DOMAIN SIGNATURES IN TRANSVERSELY ISOTROPIC MEDIA
tion (6.3) and linearization in e and ~ yields (see Appendix 6A) Vnm~ [1 + (e - 5) f(g)], Vnmo(P) - v/1 _ g
(6.7)
where
f(g)
--
g (4g 2 - 9g + 6) , 1-g
g--
p2
%o(0
) 9
Note that for elliptical anisotropy (e = 5) the weak-anisotropy approximation (6.7) reduces to the exact NMO equation (6.6). Again, it is interesting to compare equation (6.7) with the corresponding weakanisotropy equation (3.23) derived as a function of the dip r Although the difference - 5 was emphasized in Chapter 3 as the most influential parameter in the function Vnmo(~)), equation (3.23) does have a separate term dependent on 5. In contrast, when the dip is replaced by the ray parameter, the NMO velocity explicitly contains the anisotropic coefficients only in the form of the combination e - 5. [Of course, 5 is also hidden in the velocity Vnmo(0).] Instead of the three original unknown parameters (Vp0, e and 5), the weak-anisotropy approximation for NMO velocity depends on only two combinations of them - Vnmo(0) and e - 5. In terms of a possible inversion procedure, moveout (stacking) velocities from two distinct dips should yield enough information to recover the two effective parameters and reconstruct the NMO velocity as a function of ray parameter. In the most common case, when Vnmo(0) has been found by conventional NMO analysis of horizontal events, the reflection from a single dipping interface makes it possible to recover the difference ~ - (~. Clearly, the trade-off between Vpo, ~ and 5 cannot be resolved from P-wave NMO velocity alone, no matter how many reflectors (dips) are used. Normal moveout from more than two dipping reflectors just provides redundancy in the estimation of Vnmo(0) and e - ~. Although these conclusions have been drawn for weak transverse isotropy, the numerical analysis below leads to a similar result for VTI media with arbitrary strength of the anisotropy. General VTI media Next, we study the NMO-velocity function by performing a nonlinear inversion of the exact equation (6.3) for e and/~ using the Newton-Raphson method. The input data for the inversion procedure include the P-wave NMO velocities and ray parameters for two different reflector dips; one of the reflectors can be (but is not necessarily) horizontal. Starting with the isotropic model (e = ~ = 0) as the initial guess was sufficient to achieve a fast enough convergence in the numerical tests described here. In the first example (Figure 6.1), P-wave NMO velocity is computed for a horizontal reflector and a reflector dipping at 50 ~ in a typical VTI model with e = 0.2 and 5 = 0.1. Then these two values of Vnmo(P), along with the corresponding ray parameters, are inverted for e and 5 using a certain value of the vertical velocity Vpo.
6.1.
P-WAVE NMO V E L O C I T Y AS A F U N C T I O N OF RAY P A R A M E T E R
257
0 2
0 - 0 . 1
~
~
'
Figure 6.1: Parameters e and 5 obtained from the NMO velocities corresponding to a horizontal reflector and p = 0.23 s/km (50 ~ dip). The values of PRO used in the inversion are shown on the horizontal axis. The model parameters are Vpo = 3.0 km/s, e = 0.2 and 5 = 0.1.
For the actual Vpo = 3.0 km/s, the inversion algorithm produces the correct values of both anisotropic parameters (Figure 6.1). Varying the vertical velocity from 2.6 km/s to 3.5 km/s yields a family of solutions for which the difference between e and 6 is close to the exact value ( e - 6 = 0.1). Therefore, in agreement with the weak-anisotropy approximation [equation (6.7)], the inversion of the P-wave NMO velocity from dipping reflectors provides a good estimate of the difference e - 5. An important property of the family of solutions (models) shown in Figure 6.1 is that all of them have practically the same NMO velocity (as a function of the ray parameter) for all possible dipping reflectors, not just for the two dips used in the inversion. This is illustrated by Figure 6.2, which shows that the NMO velocity for any given reflector dip (i.e., any value of p) is almost identical within the range of solutions in Figure 6.1. Therefore, the inversion procedure using the NMO velocities for any two dipping reflectors (of course, the dips should not be close to each other) will produce the same family of equivalent models as in Figure 6.1; these models can be referred to as the "equivalent solutions," or ES. Thus, normal-moveout velocities measured at two different dips provide enough information to obtain the NMO velocity for any ray-parameter value. This indicates that Vnmo(P) is controlled by just two combinations of the parameters Vpo, ~, and 5, rather than by the three parameters individually. Otherwise, measurements at two
258
CHAPTER 6. P-WAVE TIME-DOMAIN SIGNATURES IN TRANSVERSELY ISOTROPIC MEDIA
r
-"~ ....
p=C.i b
s/iota
, _.
!2 o_ .s_/,k_m._. _ [m ,
,
,
,
,
,
3.4
Figure 6.2" NMO velocity for the family of solutions in Figure 6.1 and four different values of the ray parameter (i.e., four different dips for each model). dips (i.e., for two values of p) would have been insufficient to reconstruct the function Vnmo (p)-
Another example, for a medium with stronger anisotropy, is shown in Figure 6.3. As for the previous model, all ES corresponding to different vertical velocities have the same NMO velocity for the full range of dips [including the same Vnmo(0)]. However, in contrast with the previous example, the anisotropy is too pronounced for the weakanisotropy equation (6.7) to hold, and the inversion does not provide an accurate value of ~ - 6, unless we have a good estimate of the vertical velocity. In essence, equation (6.7) and the above examples demonstrate that the exact NMO velocity expressed through p depends on just the zero-dip velocity Vnmo(0) and some combination of the anisotropy coefficients close to the difference e - 6 . Therefore, Vnmo(0)-
con'st
(6.8)
within the family of ES. Since Vnmo(0 ) -- Vpov/1 + 26, equation (6.8) provides a relationship between Vpo and 6 that accurately describes the curves 6(Vpo) in Figures 6.1 and 6.3. The combination of e and 6 that defines the family of ES deviates from the difference e - 6 with increasing anisotropy. To find this relationship between ~ and 6, we examine the behavior of group-velocity curves for equivalent solutions. Figure 6.4 shows the group velocity as a function of the group angle for three ES corresponding to Vpo = 2.8, 3.0 and 3.2 km/s from Figures 6.1 and 6.3. For both models (Fig-
6.1.
P-WAVE NMO VELOCITY AS A FUNCTION OF RAY PARAMETER
"-'~-~..
259
~: ..... ...... ,,~ . . . .
R ,
~
~.__~
Figure 6.3: Inverted values of e and 6 as functions of Vpo for a model with Vpo - 3.0 km/s, c - 0.3 and 5 - - 0 . 1 . One of the reflectors is horizontal, the other is dipping at 40 ~.
>
_
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
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
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
CHAPTER 6. P-WAVE TIME-DOMAIN SIGNATURES IN TRANSVERSELY ISOTROPIC MEDIA
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
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
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
E
.
.
.
9
.
"
"
C
m
9
,
.
.
.
.
.
.
i
.
~ ~
.
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
.
.
:
!
i
:
o
E e-
9 .
.
.
.
.
9.
.
.
.
.
.
9 .
.
.
.
.
."
.
.
.
.
.
s
.
.
.
.
o Z
1.0 pVnmo(O)
o'.4 0'.6 pVnmo(O)
v=_45 ~
v=90 ~
_
~
~
.
:
:
~
. . . . . i. . . . . . : . . . ~ <
o'.2
0
.....
: .....
"...i
....
..... i...... i ...... ~..-/,. : : : /, E 1.4
$0.4 0.2
1.0
_
9
Z
o.a
.....
9
:. . . . . .
:.
9 9
. .
....
..f_l
.
. S .
.
.
.
.
~ : j J
'. .... ::...... i ..... i ..... !-.~ 1.0
_
Figure 6.13: Normalized P-wave NMO velocity as a function of the ray parameter. The models correspond to e - 5=0.2 (same as in Figure 3.24)" e=0.1, 5 =-0.1 (solid black); ~=0.2, 5=0 (gray); ~=0.3, 5=0.1 (dashed).
6.4.
MOVEOUT
ANALYSIS
FOR
TILTED
SYMMETRY
277
AXIS
v=-45 ~
o
0.4
............................
Z
"
"
"
v=90 ~ 2.2 E 9
:
9 .
.
.
, ~ , ~
.
9 :
.....
~,,~
) 1.8 "o N
"--
F i g u r e 6.14: - 0.1, 5 -
Normalized -0.071
P-wave
NMO
.....
:. . . . . .
: ......
: ....
:
:
: 9f i
~
.....
v e l o c i t y for m o d e l s w i t h t h e s a m e r/ -
0.2:
( s o l i d b l a c k ) ; ~ - 0.2, 5 - 0 ( g r a y ) ; c - 0.3, 5 - 0.071 ( d a s h e d ) .
278
C H A P T E R 6.
.....
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 I S O T R O P I C M E D I A
'....~. ~.
. . . . .
> 2.0
: ....
9
9
. ~
~. . . . . .
" . . . . .
;
. 9
9
. . . . .
...................
=
i
0
~
-~.-
i
i
,
,
!
:,"
~lP,
0
1.2
.
.
.
.
.
.
.
.
.
.
2.5
.
o
0
E t-
......
.....
i ....
.,
.... !
.
E
i
L_ 0
z 1.0 0
0.2
0.4
0.6
pVnmo(O)
0.8
1.0
0
I
I
!
1.0
pVnmo(O)
Figure 6.15" Normalized P-wave NMO velocity for models with different q" q=0.1 (solid black); q=0.2 (gray); q=0.3 (dashed). For all models, (~ - 0 and c - q.
Note that for small 5 the parameter r/is close to the difference ~ - 5 , so the plots for constant e - ~ contain a partial answer to this question. For instance, in Figure 6.13 there is some separation between the curves with the same e - 5, but the general behavior of the NMO velocity for fixed c - 5 is similar. Figure 6.14 reproduces the result from Figure 6.13, but this time the values of 5 were adjusted to make r / = 0.2 for all three models. The curves moved much closer to each other, although a perfect coincidence of the NMO velocities for the full range of dips was achieved for only VTI media (~ = 0~ Figure 6.15 shows the normal-moveout velocity at the same tilt angles as in Figure 6.14 but for differing values of q. While for vertical and horizontal orientations of the symmetry axis r / h a s a pronounced influence on the NMO-velocity curves, at intermediate tilt angles the resolution in q is considerably lower. For ~ = +45 ~ the difference between the curves corresponding to r / = 0.2 and q = 0.3 in Figure 6.15 is quite small.
6.4.
M O V E O U T ANALYSIS F O R TILTED SYMMETRY AXIS
279
v=--45 ~
v=45
0
0
...... 9
9
9
~.
:. . . . . .
: .....
.N
~
~ ,
1:0 pVnrno(0)
pVnmo(0)
Figure 6.16: Sensitivity of P-wave NMO velocity to 7/ at intermediate tilt angles. The curves correspond to r/=0.05 (solid black); r/=0.1 (dotted); fl=0.15 (gray); r/=0.2 (dashed). For all models, 6 = 0 and e = r/.
As mentioned above, for weak anisotropy the deviation of the P-wave NMO velocity from the isotropic dependence (6.18) is proportional to the difference c - 6, which is close to ~/for small values of 6. This is illustrated by Figure 6.16, which shows that even at intermediate tilts the normalized NMO velocity still is almost linearly dependent on 7/ ~ ~ - 6 for small and moderate values of ~ and 6. However, with increasing 7/this is no longer the case, and the curves for q > 0.15 are relatively close to each other (Figures 6.15 and 6.16). Apparently, at v = 4-45 ~ the influence of the terms quadratic in the anisotropic coefficients leads to a more complicated relationship between the NMO velocity and the parameters c and 6 than that in VTI media. A detailed analysis, confirmed by Figures 6.14-6.16, shows that the NMO velocity remains somewhat sensitive to 6 as r/ reaches the 0.15-0.2 range, but does not change much with increasing ~. As a result, for ~7= 0 . 1 5 - 0.2 and higher, the DMO signature at intermediate angles depends on the individual values of e and 6 and is not fully controlled by 7/. For completeness, Figure 6.17 displays the P-wave NMO velocity at different tilt angles for (less typical) models with negative r/. [If r / = 0, the model is elliptical and Vnmo(P) is described by the isotropic dependence (6.18).] Note that at all tilts the signature for r / < 0 is reversed compared to that for positive values of fl (Figure 6.14). Nonetheless, there is also a similarity between models with positive and negative r/: in both cases, ~7 controls the dependence Vnmo(P) more tightly for near-vertical and near-horizontal orientations of the symmetry axis than at intermediate tilt angles. The difference between the dip-dependence of NMO velocity for vertical transverse isotropy and TI media with a tilted axis of symmetry makes it necessary to study the transition between the two signatures at mild tilt angles. Even a tilt of 4-20 ~ is sufficient to eliminate the sharp increase in the normalized NMO velocity typical for
280
CHAPTER 6. P-WAVE TIME-DOMAIN SIGNATURES IN TRANSVERSELY ISOTROPIC MEDIA
11=-0.1 2.2 O
E t'0.6-
E 1.4
EO.4.
0
0
z
Z
1.0
v=90 ~ =0.8-
..... ii.....
>
"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
CHAPTER 6. P-WAVE TIME-DOMAIN SIGNATURES IN TRANSVERSELY ISOTROPIC MEDIA
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 << 1, 15] << 1) VTI media was expressed in Chapter 2 through the dip r [equation (3.23)]: Vnmo(r
[1 + 5 sin 2 r + 3(e - 5) sin 2 r (2 - sin 2 r
-- Vnm~ COS (~
(6.28)
where Vnmo(0) = Vpo (1 +5). To express the phase angle r through the ray parameter, we use the linearized approximation (1.61) for the phase velocity Vp: sin r _ P
--
Vp(~))
-
Vpo
sin r [1 + 5 sin 2 r + (~ - 5) sin 4 r
(6.29)
SO
sin r - p Vpo [1 + 5 sin 2 r + (e - 5) sin 4 r
(6.30)
In the anisotropic terms of equation (6.30), the angle r can be replaced by its isotropic Vpo, which gives counterpart given by sin r sin r - p Vpo [1 + 5 p2 Vp20+ (e _ 5) p4 V/~0].
(6.31)
Substituting sin r = p Vpo into the anisotropic terms of equation (6.28), using equation (6.31) to find cos r and further linearizing Vnmo(P) in e and 5, we obtain Vnmo(P)- Ynmo(0)[1 + ( e - 5)/(g)], X/1 - g f
(6.32)
g (4g 2 - 9g + 6) p2 ; 9 = Vn o(0), 1-g where it is assumed that g < 1. In the derivation of equation (6.32), the vertical velocity Vpo was replaced in the anisotropic terms with Vnmo(0) since the difference between Vpo and Vnmo(0) will change only terms quadratic in the anisotropic parameters. -
6B. NMO VELOCITY IN TILTED ELLIPTICAL MEDIA
NMO
6B
285
v e l o c i t y in t i l t e d e l l i p t i c a l m e d i a
For elliptically anisotropic models with a vertical symmetry axis, NMO velocity depends on the ray parameter p in the same way as in isotropic media [equation (6.6) or (6.27)]: Vnmo(P) -
V~mo(0) , ~/1 - p2 Vn2mo(0)
(6.33)
with Vnmo(0 ) --
Vpo x/1 + 25.
Here, the NMO equation (3.35) in terms of the dip r is used to prove that the isotropic relationship (6.33) holds in elliptically anisotropic media with arbitrary tilt of the elliptical axes. It is easier to carry out this proof by transforming equation (6.33) into equation (3.35) rather than the other way around. Using the phase-velocity equation (3.32), the ray parameter can be written as
P-
sin r _ sin r V(r - Vp0 V/1 + 25 s i n 2 ( r
(6.34) v;'
where u is the tilt of the symmetry axis. The zero-dip NMO velocity Vnmo(r = 0) for arbitrary tilt can be found from equation (3.35):
Vpo x/1 + 25 Ynmo(0) -- ~ i - ~
~6 sin 2
.
(6.35)
Equations (6.34) and (6.35) can be substituted into equation (6.33) for Vnmo(P):
Vpo x/1 § 2~ V/1 + 2~ sin2(r Vnm~ (r
--
d
u) ,
(6.36)
where d - {1 + 25 [sin 2 y + sin2(r - u)] + 452 sin 2 ~ sin2(r - ~) - (1 + 2~) sin 2 (~}1/2 , The denominator d can be further simplified to d = cos r - 25 sin u sin(r - u), so equation (6.36) [and, therefore, equation (6.33)] becomes identical to expression (3.35) for Vnmo(r
This Page Intentionally Left Blank
Chapter 7 Velocity analysis and parameter estimation for VTI media The results of the previous chapters clearly demonstrate that anisotropy in general and transverse isotropy in particular have a substantial influence on the traveltimes of reflected waves and their normal-moveout velocities. Since reflection moveout is the main source of information for velocity analysis, conventional (i.e., those based on the assumption of isotropy) algorithms are bound to produce distorted velocity models and images in the presence of anisotropy. For example, as shown in Chapter 3, the difference between the stacking (moveout) and vertical velocity in anisotropic media leads to mis-ties in time-to-depth conversion and the wrong depth scale of seismic sections. Also, anisotropy causes serious difficulties in imaging of dipping reflectors, such as fault planes (Chapter 6; also, see Lynn et al., 1991, and examples below). Once the importance of anisotropy in seismic processing is accepted, two further impediments to taking its presence into account must be overcome: (1) processing algorithms that include anisotropy are more complex than those that ignore it, and (2) even with such processing capability at hand, estimating the anisotropy parameters required by these algorithms is a highly challenging task. The second problem has always seemed to be especially intimidating for exploration seismologists. Even for the relatively simple VTI model treated in this chapter, P-wave propagation is governed by four stiffness coefficients or, alternatively, four Thomsen (1986) parameters- the vertical velocities Vpo and Vso and the anisotropic coefficients e and ~. This would seem to imply that velocity analysis for VTI media will not only be complicated in practice (estimating the one critical parameter for isotropic m e d i a velocity- is difficult enough), it will also require additional measurements beyond those obtained in conventional seismic surveys [e.g., cross-hole, vertical seismic profile (VSP), or auxiliary shear-wave data]. As we shall see, however, velocity analysis for purposes of time-domain P-wave imaging is much easier and more straighforward than might be thought. In short, it is feasible in routine practice. The first step towards solving the parameter-estimation problem for VTI media was taken in Chapter 1 by proving that all kinematic signatures of P-waves depend on just three Thomsen parameters out of four (Vp0, e and/~; the influence of Vs0 is practically negligible). Further analysis in Chapters 4 and 6 led to several conclusions of crucial importance for seismic imaging: 287
288
C H A P T E R 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
1. If a medium contains a dipping target reflector beneath a laterally homogeneous (e.g., horizontally layered) VTI overburden, all P-wave time-domain processing steps (NMO and DMO correction, prestack and poststack time migration) are controlled by just two parameters - the NMO velocity from a horizontal reflector [Vnmo(0)] and the anisotropic parameter ~ - in each layer. 2. The two time-processing parameters govern both long-spread (nonhyperbolic) moveout from horizontal reflectors and NMO velocity of dipping events. Therefore, it should be possible to obtain them from surface P-wave data without using shear waves or borehole information. 3. For models with a laterally homogeneous VTI overburden, however, P-wave reflection traveltimes do not constrain the vertical velocity Vpo - a parameter needed for depth imaging of P-wave data. This chapter introduces practical methodologies for estimating the parameters Vnmo(0) and r/ from P-wave reflection moveout recorded in both 2-D and 3-D geometries. [One more algorithm for time-domain velocity analysis, based on the DMO method of Fowler (1984, 1988), is described in Chapter 8.] Then the results of Chapter 5 are used to combine dip-dependent reflection moveout of P-waves with that of converted PSV-waves in an inversion algorithm capable of building VTI velocity models in depth. The fact that the parameter ~ is a function of Vnmo(0) and the horizontal velocity Vhor points to an alternative way of performing velocity analysis in VTI media not yet explored in the literature. If it is possible to obtain an accurate value of Vhor (e.g., from head waves traveling along a horizontal reflector or from cross-hole tomography), then Vnmo(0) and Vhor can be used to carry out time-domain processing of P-wave data. Dipping events or nonhyperbolic moveout - the main sources of direct information about r / - then are not needed at all. Of course, head waves can help in estimating the horizontal velocity for only the shallow part of the section. A point worth mentioning here is that refraction surveys in VTI media do not yield the vertical velocity Vpo, unless the horizontal and vertical velocities coincide with one another (i.e., e = 0). Although the chapter deals just with vertical transverse isotropy, the limited equivalence between orthorhombic and VTI media (see sections 1.3 and 3.5) implies that all 2-D results in Chapters 6-8 are fully applicable to both vertical symmetry planes of orthorhombic media. As long as the acquisition line is confined to the dip plane of the structure, 2-D seismic processing steps for VTI media and the symmetry planes of orthorhombic media can be carried out using the same algorithms, although the amplitude treatment in orthorhombic models should be different. For example, to perform P-wave time processing in a symmetry plane of orthorhombic media, it is necessary to obtain the corresponding zero-dip NMO velocity and the coefficient ~. Furthermore, 2-D VTI algorithms remain valid for P-waves outside the symmetry planes of orthorhombic media under the assumption of weak anisotropy.
7.1. P-WAVEDIP-MOVEOUTINVERSIONFOR ~?
7.1 7.1.1
289
P-wave dip-moveout inversion for Inversion in the dip plane of a VTI layer
Suppose the CMP line is confined to the dip plane of a reflector overlaid by a homogeneous VTI medium. As shown in Chapter 6, the parameters Vnmo(0) and r] can be obtained from P-wave NMO velocities of two different dipping events. If one of the reflectors is horizontal (the most common case), its NMO velocity gives a direct estimate of Vnmo(0). Then the NMO velocity and the reflection slope (i.e., the ray parameter p) of the dipping event can be used to estimate the coefficient ~ by inverting the exact NMO equation (6.3). This inversion was discussed in section 6.1, with the results for two typical VTI models displayed in Figures 6.1 and 6.3. It is important to evaluate the sensitivity of this single-layer inversion procedure to errors in the measured moveout velocities. As illustrated by Figure 6.5b, the NMOvelocity curves corresponding to r/=0.1, 0.2, and 0.3 are well resolved within a wide range of dips, which indicates that the inversion procedure is reasonably stable. To quantify errors in r/, the change in the velocity Vnmo(P) can be represented as
oqVnmo(p) aWnmo(p) AT]. AWnmo(P)- (~Vnmo(0) A Vnmo(0)-~0~
(7.1)
Normalizing the velocities yields
AWnmo(P) Wnmo(0)oqVnmo(P) AVnmo(0) 1 ~176 Vnmo(P) : Wnmo(P) 0Wnmo(0) Vnmo(0) "~- Wnmo(P) C~T]
A77 .
(7.2)
Using equation (7.2), the error in r/can be expressed through the errors in the NMO velocities,
FVnmo(p) AVnmo(p) A~v- L f~ Vnmo(P) where
fl Vnmo(0)]
mVnmo(0)
f2
Wnmo(0) ,
OVnmo(P) fl-
OVnmo(0) ;
f2-
(7.3)
OVnmo(P) 0,977
If the NMO velocity from a horizontal reflector were measured exactly or with a relatively high accuracy, the error in r/would be controlled just by the first term in formula (7.3). However, assuming that the errors in Vnmo(P)and Vnmo(0)are independent and have comparable magnitude, we can characterize the sensitivity of by
E = ~/Vn~m~ + [/1Cnmo(0)]~. Ihl
(7.4)
Figure 7.1 shows E as a function of the ray parameter of the dipping event for the two VTI models used above in the inversion for r/. In both examples, as expected, the sensitivity function (i.e., the inversion error) decreases with dip, but then reaches
290
CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
Figure 7.1" Sensitivity parameter E [equation (7.4)] as a function of p corresponding to the dipping reflector for (a) the model from Figure 6.1 (Vp0=3.0 km/s, e - 0.2, 6 - 0.1), and (b) the model from Figure 6.3 (Vp0=3.0 km/s, e - 0.3, 6 - - 0 . 1 ) .
f-
J 9 - . : . . ....
~ f
J
.~0.5
Figure 7.2: Dependence of the inverted values of 71 (a) and ~Iho r (b) on errors in the measured NMO velocities for dips of 0 ~ and 40 ~ The black lines correspond to errors in the NMO velocity for the dipping reflector only; the gray lines correspond to identical errors in both Vnmo values. The model parameters are Vpo = 3.0 km/s, e = 0.2 and 6 = 0.1 (7 = 0.083).
7.1.
P - W A V E D I P - M O V E O U T I N V E R S I O N F O R 77
291
a minimum at a dip of about 46 ~ (a) and 51 ~ (b). Clearly, for mild dips the NMO velocity is not sufficiently influenced by ~/ (see Figure 6.5b). The minimum in E is caused by the higher sensitivity to errors in Vnmo(0) at steep dips; if the error in Vnmo(0) is negligibly small, then the minimum does not exist, and the inversion for r] is most stable at large dips approaching 90 ~. However, there are two factors that make steep (beyond 50-60 ~) reflectors, even if they are available, less desirable in the inversion algorithm. First, at large dips the inversion for ~ becomes more sensitive to errors in the ray parameter (see Figure 6.5b). Second, the recovery of the NMO velocity itself from reflection traveltimes becomes less stable because the magnitude of moveout decreases with dip due to a higher NMO velocity. As the dip approaches 90 ~ the traveltime becomes almost independent of offset and loses its sensitivity to the NMO velocity. Below it is demonstrated that the problems caused by large dips may be overcome in 3-D geometry by using the moveout on the strike line of the reflector. It is instructive to examine numerically the sensitivity of 77 and the horizontal velocity Vhor to errors in the measured NMO velocities (Figure 7.2). The percentage error in Vhor and the absolute error in ~ are smaller than the percentage errors in the input values of Vnmo, which indicates that the inversion is quite stable. For example, a 5% error in both NMO velocities causes less than a 3% error in Vhor. The percentage error in 77 is much larger, but this can be expected for so small an anisotropic coefficient.
7.1.2
2-D inversion in vertically heterogeneous media
Coarse-layer model Next, consider a horizontally layered VTI model with a throughgoing dipping interface, such as the fault plane in Figure 7.3, with P-wave moveout recorded in the dip plane of the reflector. The interval values of the zero-dip NMO velocity can be estimated by applying the conventional Dix equation to measured NMO velocities from horizontal reflectors. Then NMO velocities of dipping events can be inverted for the interval values of r] using the Dix-type equation (6.11). Note that determination of the interval velocities of dipping events is impossible without obtaining the NMO velocities in the overlying medium for the ray parameter p of the zero-offset ray (Figure 7.3). This task is not trivial because conventional velocity analysis for the horizontally layered overburden provides us with only NMO velocities and traveltimes corresponding to p = 0. The inversion procedure in the first (subsurface) layer, based on the single-layer equation (6.3) and the measured values of the NMO velocities Vn(lm)o(0)and Vn(im)o(p(1)), yields the parameter r/(1) in this layer (see above). Then Vn(lm)o(0)and r/(1) can be used to calculate the interval NMO velocity in the first layer for the ray parameter p(2) (reflection slope) corresponding to the dipping reflector in the second layer. Also, in accordance with equations (6.12)-(6.14), the vertical traveltime in layer 1 can be recalculated into the interval traveltime t(1)(p (2)) along the oblique zero-offset ray
292
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
Figure 7.3: Horizontally layered VTI medium with a dipping fault plane. NMO velocities from the dipping reflector can be used to find the interval values of the anisotropic parameter r/. The ray parameters p(i-1) and p(i) correspond to dipping events with the zero-offset reflection points in layers i - 1 and i. The layer-stripping procedure requires recalculating interval NMO velocity and zero-offset time in layer ( i - 1) from the value p(i-1) to p(i).
7.1. P-WAVE DIP-MOVEOUT INVERSION FOR r/
293
reflected from the dipping interface in the second layer. After having determined the NMO velocity of the dipping event reflected in the second layer and its zero-offset traveltime from the data, we can apply equation (6.11) to obtain the interval velocity Yn(~o(p(2)). In combination with the interval value Vn(2m)o(0) found from the horizontal events, the velocity Y(2m)o(p(2)) is inverted for the parameter 7](2) in the second layer. The layer-stripping procedure is continued downward, as long as both horizontal and dipping events are available.
Continuous velocity functions The layer-stripping algorithm outlined above is designed for coarse-layer subsurface models. In practice, however, one often deals with continuous moveout-velocity functions connecting velocity picks made at different vertical times. For this situation, it is convenient to apply an integral form of the Dix-type equation (6.11), with the NMO velocity expressed through the ray parameter (horizontal slowness) p and vertical slowness q according to equation (6.3): V:m~ (P' t~
-
1 fto(p) t o (p) Jo
q" p q' - q
dt
'
(7.5)
where the integration is performed over the traveltime t = t(p) measured along the generally oblique zero-offset ray; to(p) - fo ~ dt is the total zero-offset traveltime for a certain velocity pick. The oblique interval traveltime dt is related to the corresponding interval vertical traveltime dT- by equation (6.13):
dt = Vpo (q - p q') d~'.
(7.6)
Substituting equation (7.6)into equation (7.5)leads to Y:mo (P, 7.) __ -t0(p,1 T-------~ fo T Vpo(r q"(r de,
(7.7)
where the integration variable ~ now has the meaning of the vertical time. Integrating equation (7.6) yields the zero-offset time corresponding to the velocity pick Vnmo(P, T):
to(p,
- -J0
- p
(7.8)
The horizontal slowness p in equation (7.8) does not change along the ray because the model above the reflector is laterally homogeneous. Equations (7.7) and (7.8) express the NMO velocity and zero-offset traveltime of dipping events in vertically heterogeneous VTI media in a form suitable for moveout inversion. For horizontal events (p = 0), the NMO equations (6.11) and (7.5) become equivalent to the integral version of the conventional Dix (1955) formula, Vn mo (P = O, ) =
2 (0, ~) d~ f o V.nmo,int
(7.9)
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CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
where Ynmo,int(0,~) is the interval zero-dip NMO velocity [for P-waves, V.nmo,int 2 --qint/qint ,t -- YP0,int 2 (1 -~- 2(~int)]. Obtaining the interval NMO velocity from equation (7.9) essentially amounts to differentiating the measured function Vn2moT, which inevitably leads to amplification of errors made in velocity picking. To mitigate this instability, equation (7.9) can be solved, for example, by the technique described in Grechka et al. (1996). Their approach is based on approximating the velocity picks by Chebyshev polynomials and finding the interval velocity Ynmo,int in the Chebyshev domain. The desired smoothness of the solution and the degree to which errors in the effective velocities propagate into the interval values are regulated by choosing the appropriate number of the polynomials. Once the interval zero-dip NMO velocity has been obtained, the interval parameter flint can be found from the NMO velocity and zero-offset traveltime of dipping events [equations (7.7) and (7.8)]. The input data include the triplets of the NMO velocity Vnmo(P,to) and the corresponding values of the zero-offset traveltime to and the ray parameter p (reflection slope on the zero-offset sections). These triplets can be picked from zero-offset time sections generated by scanning the stacking velocity (i.e., constant-velocity stacks) or from semblance velocity panels at a number of adjacent CMP locations. A stable way to reconstruct the smooth component of the time-varying interval r/int(T)-function from the triplets {Vnmo, to,p} is based on representing T]int(T) aS the sum of Chebyshev polynomials (Han et al., 2000). For a trial solution flint (T) (specified at each iteration) and the zero-offset traveltime t0(p, 7) of a particular velocity pick, we first find the corresponding vertical time T using equation (7.8). Then the trial flint-function and time T are used to compute the NMO velocity Vnmo(P,T) from equation (7.7). Finally, ~nt(7) is updated to minimize the difference between the picked and computed NMO-velocity values for all velocity picks. A somewhat different algorithm based on the same NMO equations is described in Alkhalifah (1997a).
Synthetic example Numerical examples illustrating the accuracy and stability of the 2-D DMO-inversion algorithm in a homogeneous VTI medium were given at the beginning of this chapter. Figures 7.4 and 7.5 demonstrate an application of the inversion method based on Chebyshev polynomials to synthetic data generated for a vertically heterogeneous VTI model. The input data included the zero-dip NMO velocity Vnmo(0) sampled at 40 vertical times 7 and 40 triplets {Vnmo,t0,p} representing NMO velocities of dipping events computed from the exact equations as a function of the zero-offset time to and ray parameter (reflection slope) p. The values of p fall into the range 0.13 - 0.26 s/km, which corresponds approximately to reflector dips from 20 ~ to 40 ~. For noise-free input data, the algorithm accurately reconstructs the interval curves of both Vnmo(0) and r] (Figure 7.4). Figure 7.5 shows the inversion results for the same model, but this time the input
7.1.
P - W A V E D I P - M O V E O U T I N V E R S I O N F O R 7?
295
,4
..........................
..........................
2 Vnmo(O )
Figure 7.4: Application of the algorithm of Han et al. (2000) based on Chebyshev polynomials to the inversion of synthetic data. Effective (measured) NMO velocity from horizontal reflectors (left plot) and NMO velocities of dipping events (40 picks, not shown) were inverted for the interval Vnmo(0) (dashed line, middle plot) and r] (dashed line, right plot). The thin solid lines on the plots with the interval parameters mark their actual values.
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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
0.4
.......
o.6
. . . . . . . .
::...... ~.i ...... ::......
i
i .......
I
o.8 ...... i....... i~ '; ...... i 1
1.2 1.4 . . . . . . . . . . . . . . .
Figure 7.5: Same as in Figure 7.4, but the input NMO velocities contain Gaussian noise with a standard deviation of 2%.
7.1. P-WAVE DIP-MOVEOUT INVERSION FOR ~
297
NMO velocities are contaminated with a 2% error. As expected, this error gets somewhat amplified in the process of generating interval values, but the curves of Vnmo(0) and 7/ correctly reproduce the time-dependent trends of both parameters. The maximum error in the interval 77 does not exceed 0.06, and is less than 0.04-0.05 for most of the interval. The divergent trend of the inverted Vnmo(0) at late times ~- --+ 2 s in Figure 7.5 is explained by the small contribution of the very end of the time interval to the effective value of Vnmo(0). Since the dips used in the inversion were relatively mild (many are under 30~ such an accuracy is quite satisfactory. Without the regularization (smoothing) inherent in the Chebyshev polynomial method, errors in the effective NMO velocities would cause instability and much larger errors in the interval parameters than those here.
7.1.3
3-D inversion of azimuthally varying NMO velocity
The 2-D DMO-inversion algorithm is designed to obtain Vnmo(0) and r/from NMO velocities measured for different dips (e.g., a horizontal and a dipping reflector) in the dip plane of the reflector(s). Equation (6.15) for the NMO ellipse in VTI media can be used to extend this methodology to arbitrary azimuthal direction of the CMP line. Alternatively, Vnmo(0) and 77 can be found from the azimuthal dependence of NMO velocity for a single dipping reflector. Here we discuss these 3-D moveout-inversion methods for a homogeneous VTI medium; extension to multiple layers requires application of the 3-D generalized Dix equation of Grechka, Tsvankin and Cohen (1999). Before performing the inversion, it is important to verify whether or not it is possible to reconstruct the NMO ellipse from reflection data in the presence of anisotropyinduced nonhyperbolic moveout. Figure 7.6 displays a comparison of the analytic NMO ellipse derived in the zero-spread limit with the best-fit moveout velocities determined from the exact traveltimes. Clearly, equation (6.15) accurately describes P-wave moveout for conventional spreadlengths equal to (or even somewhat larger than) the distance between the CMP and the reflector. (Also note the high accuracy of the weak-anisotropy approximation for the model in Figure 7.6 that has moderate values of e, 5, and r/). Hence, azimuthally dependent NMO velocity can be found via hyperbolic semblance velocity analysis and used in the inversion for the parameter 77. Single C M P line First, suppose that P-wave reflection traveltimes were recorded on a single CMP line, but this time with arbitrary azimuthal orientation. As follows from equation (6.15) and the analysis of the dip- and strike-line NMO velocities in section 6.1, the P-wave NMO velocity from a dipping reflector is a function of four parameters: Vnmo( Og,P) :
f(~, p,
Vnmo(0), 77).
(7.10)
Knowledge of the angle c~ between the CMP line and the dip plane of a certain reflector is sufficient for recovering the ray parameter (horizontal slowness) p of the
298
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
30
Figure 7.6: Azimuthally varying P-wave moveout velocity in a homogeneous VTI model with V n m o ( 0 ) ~-- 0.926 km/s and q = 0.2 (Vp0 = 1.0 km/s, e = 0.1, ~ = -0.071) for the reflector dip r = 45 ~ The azimuth c~ (the numbers around the plot) is measured with respect to the dip plane of the reflector. The dots are the moveout (stacking) velocities obtained by fitting a hyperbola to the exact traveltimes computed for spreadlength equal to the CMP-reflector distance; the solid curve is the ellipse calculated using the exact NMO equations (6.15), (6.3) and (3.62); the dashed curve is the NMO ellipse in the weak-anisotropy approximation [equations (3.23) and (3.61)].
7.1.
P - W A V E D I P - M O V E O U T I N V E R S I O N F O R ~7
299
zero-offset reflection. Since half of the zero-offset reflection slope is equal to the projection of the ray parameter on the CMP line, p can be obtained as
1 P=2cosa
dto(y) dy '
(7.11)
where to is the two-way zero-offset traveltime on the line with azimuth a, and y is the CMP coordinate. If the zero-dip value Vnmo(0) has been determined using a horizontal reflection event, normal-moveout velocity from a single dipping reflector is sufficient to estimate the parameter ~, which remains the only unknown in the NMO function (7.10). The inversion procedure based on traveltime measurements in a single azimuthal direction is illustrated by the synthetic example in Figure 7.7. The exact NMO equation (6.15) was inverted for ~ using ~, p, Vnmo(O),and Vnmo(Ol,p) as the input parameters. Since the P-wave phase velocity (or slowness), needed to calculate NMO velocity, formally is a function of four parameters (here the model was parameterized by Vnmo(0), r/, Vso and 6), it is strictly necessary to specify two more p a r a m e t e r s - Vs0 and 6 - for computation purposes. However, in agreement with our earlier conclusion that NMO velocity depends only on Vnmo(0) and ~, use of intentionally incorrect values of Vs0 and 6 in the inversion did not prevent our algorithm from recovering the exact value of 77 (for error-free input parameters). The main purpose of the example in Figure 7.7 was to study the stability of r] estimation by introducing errors in the three parameters [p, Vnmo(0), and Vnmo(a, P)] that would be obtained from reflection traveltimes. In Figure 7.7a, Vnmo(a,P) was held at the correct value to examine the influence of errors in p and Vnmo(0), while in Figure 7.7b the ray parameter p was assumed to be exact. The accuracy of the inversion for ~ in the presence of realistic errors in p, Vnmo(0) and Vnmo(a,p) is quite satisfactory. For instance, if the error in V~mo(0) does not exceed -+-2.5% (a reasonable value for semblance velocity analysis) and the other parameters are exact, the maximum error in ~ is just +0.03. (Given that ~ << 1, we cannot expect small relative errors in this parameter.) The results obtained in Figure 7.7 for a line at an azimuth of 30 ~ from the dip direction are similar to the error estimates made above for the dip plane of the reflector (Figure 7.2). The stability of the inversion procedure decreases, however, as the CMP line gets closer to the strike direction because the NMO velocity becomes less sensitive to r/. T w o or m o r e C M P lines In general, when the orientation of the CMP line with respect to the dip plane is unknown, one needs data from lines with at least three different azimuthal directions to reconstruct the NMO ellipse from moveout measurements. However, the minimum number of azimuths can be reduced from three to two by using, in addition to normal moveout, the zero-offset reflection traveltimes along the two CMP lines. Indeed, it is clear from equation (7.11) that the zero-offset reflection slopes for two different
300
CHAPTER
7.
VELOCITY
ANALYSIS AND PARAMETER
............................... !....... I! . . . . . . .
9- o
.qD 9 9 9 :
..........
. . . . . . .
::
..... . . . . . .. . . . . . .i .. . .. . .. . . . .
9
."
~,,.
i ~ . . 9. ,~,.
i i
-~---'~---~'-.--'"o
:
- . . -
.... " " - : .......... i. . . . . . . : . ~ .-........ ..... : i
0.1 .......... :.......... : : : ~
....... ..........
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0
FOR VTI MEDIA
0.25 -~. ..................................
: .......... :
ESTIMATION
'
I
Vnm o (0)
Vnm o (0)
Figure 7.7: Stability of q estimation using the P-wave normal-moveout velocity measured on the CMP line that deviates by a = 30 ~ from the dip direction of the reflector. The actual parameters a r e V n m o ( 0 ) : 2.0 km/s, r / = 0.15 (Vso= 1.2 km/s, 5 = 0.0), p = 0.35 s/km, Vnmo(C~,p) = 3.24 km/s. The values Vs0 = 0.8 k m / s and 5 = 0.2 used in the inversion were intentionally incorrect. (a) Vnmo(C~,p) = 3.24 k m / s (correct value) and p = 0.33 s / k m (dotted line), 0.35 s / k m (solid, correct value), and 0.37 s / k m (dashed). (b) p = 0.35 s / k m (correct value) and Vnmo(C~,P) = 3.08 k m / s (dotted), 3.24 k m / s (solid, correct value), and 3.40 k m / s (dashed). azimuths can be used to find the ray parameter p and the dip direction of the reflector (i.e., the azimuths of the axes of the NMO ellipse). With the orientation of the NMO ellipse obtained from zero-offset reflection trayeltimes, the normal-moveout velocities on the same two lines are sufficient to uniquely determine the elliptical semi-axes. The resulting dip-line and strike-line NMO velocities for a single dipping event can be inverted for the key parameters Vnmo(0) and ~/ using equations (6.3) and (3.62). If Vnmo(0) has already been found using horizontal events, NMO velocities in the dip and strike directions give independent estimates of r/, which provides useful redundancy in the inversion. Clearly, it is preferable to include as many different azimuths as possible to enhance stability in the reconstruction of the NMO ellipse. The inversion of four azimuthal moveout measurements from a single dipping reflector in the VTI model of Dog Creek shale is shown in Figure 7.8. The azimuth of the semi-major axis of the NMO ellipse (a0 = 19~ the actual value is 20 ~ and the semi-axes themselves were found by fitting an ellipse to the moveout velocities obtained from the exact traveltimes. Then the inversion of the dip-line and strike-line NMO velocities [equations (6.3) and (3.62)] yielded the values Vnmo(0)-- 2.065 k m / s (instead of the actual 2.054 km/s) and r / = 0.100 (instead of 0.104). The small errors in both parameters are due to the influence of nonhyperbolic moveout on the input moveout velocity; deviations from hyperbolic moveout for the model of Dog Creek shale were analyzed in detail in Chapter 4.
7.1.
P - W A V E D I P - M O V E O U T I N V E R S I O N F O R 7/
301
Figure 7.8: Inversion of the P-wave NMO ellipse reconstructed from the exact reflection traveltimes in the VTI model of Dog Creek shale (see Figure 4.5). The azimuth of the dip plane of the reflector is c~0 = 20 ~ the dip r = 50 ~ The dots mark the moveout (stacking) velocity measured at four azimuths (0 ~ 45 ~ 90 ~ 135 ~ by least-squares fitting of a hyperbola to the exact traveltimes for the spreadlength equal to the CMPreflector distance. The dashed curve is the best-fit ellipse that approximates the four values of moveout velocity; the solid curve is the theoretical NMO ellipse for this model. The model parameters a r e V n m o ( 0 ) = 2.054 km/s and r / = 0.104 (Vpo -~ 1.875 km/s, e = 0.225, ~ = 0.100). The semi-axes of the best-fit ellipse are 4.280 km/s and 2.243 km/s (the theoretical values are 4.259 km/s and 2.238 km/s, respectively).
302
CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
Inversion for large dips As discussed above, the inversion in the dip plane of the reflector experiences difficulties in handling large dips (e.g., flanks of salt domes) because of the small magnitude of the reflection moveout. Although the dip-line velocity Vnmo(0,p) for large dips remains sensitive to 7, the estimation of the NMO velocity itself from the reflection traveltimes becomes less stable. The 3-D algorithm can overcome this problem by using the normal-moveout velocity in the direction of reflector strike. Indeed, for a vertical reflector in a homogeneous VTI medium the dip-line NMO velocity [equation (6.3)] is infinite, and the equation (6.15) of the NMO ellipse reduces to Unmo (oL, 2 ) - Unto~
(Tln ;f
(7.12)
According to equation (7.12), the NMO ellipse degenerates into two straight lines parallel to the dip plane of the reflector. Rays reflected from a vertical interface propagate in the horizontal plane where the velocity (Vhor) is independent of angle. As a result, the strike-line NMO velocity (3.58) from a vertical reflector is equal to the horizontal velocity:
"nmo
71-
( 13)
For P-waves, Y h o r - Ynmo(0)~l -~- 2/], and the strike-line velocity Vnmo(2, 2) makes it possible to obtain r], provided Vnmo(0) has been determined from horizontal events. In general, equation (7.12) allows one to find Vnmo(~, ~) and r] from the NMO velocity measured at any known azimuth (except for the dip line), but the accuracy of the moveout-velocity estimation decreases away from the strike direction. 7.1.4
Field-data
example
Next, we test the DMO-inversion algorithm and illustrate the advantages of accounting for anisotropy in processing using a seismic line from offshore Africa (Figure 7.9, courtesy of Chevron Overseas Petroleum). This case study was originally presented by Alkhalifah et al. (1996) and Alkhalifah (1996a). The section in Figure 7.9 was obtained using a sequence of conventional NMO, DMO and time-migration algorithms designed for isotropic media. While horizontal and mildly dipping reflectors are imaged well, steep fault planes (such as the one at a time of 1.5-2.5 s to the left of CMP 1000) are almost invisible and can be identified only by the terminations of subhorizontal interfaces. Similar difficulties in the imaging of fault planes in the Gulf of Mexico and several other areas were discussed by Lynn et al. (1991), who attributed the problem to the influence of transverse isotropy. To demonstrate that the loss of dipping energy is indeed caused by anisotropy, it is useful to examine constant-velocity CMP stacks (stacks generated at certain
7.1.
P - W A V E D I P - M O V E O U T I N V E R S I O N F O R 7/
303
Figure 7.9" Time-migrated seismic line from offshore Africa produced by isotropic processing. The gray bar to the left of CMP 1000 shows the CMP gathers examined in Figures 7.10 and 7.11.
304
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
V
2 2400
2550
o~ 1
V
2 Figure 7.10: Constant-velocity stacks for the area below the gray bar in Figure 7.9 after the conventional sequence of NMO and constant-velocity DMO corrections (without accounting for anisotropy). The values on top of each panel indicate the stacking velocity.
7.1.
P-WAVE DIP-MOVEOUT
I N V E R S I O N F O R rl
305
constant values of stacking velocity) after application of NMO correction followed by conventional constant-velocity DMO (Figure 7.10). The goal of DMO processing is to focus both horizontal and dipping events on the same velocity panel. However, while subhorizontal reflectors are imaged best at a stacking velocity of about 2200 m/s, the dipping event goes into focus at a much higher velocity (2400-2450 m/s). As a result, the conventional processing sequence produces a weak, blurry image of the dipping fault plane. Conventional DMO has failed because the stacking (moveout) velocity increases with dip much faster than implied by the isotropic equation (6.6) (see Figure 6.5). If the DMO problem had been caused by vertical velocity gradient, then the dipping event would have been imaged at a lower stacking velocity than that of the horizontal event (Hale and Artley, 1993). Therefore, it is the increase in the stacking velocity for dipping reflectors due to anisotropy that caused the DMO algorithm to break down. To apply the DMO-inversion method discussed above, the best-fit stacking velocities and the corresponding ray parameters for the subhorizontal and dipping events were picked from constant-velocity stacks in the area of the fault (under the gray bar in Figure 7.9). The inversion procedure, based on equation (6.3) (assuming a homogeneous VTI overburden), yielded a moderate value of 71 - 0.07, which was used to reprocess the data by means of a TZO (transformation to zero offset; Alkhalifah, 1996b) ray-tracing algorithm adapted for VTI media. (Anisotropic processing methods are discussed in more detail in Chapter 8.) The anisotropic TZO succeeded in focusing both the subhorizontal and dipping events on the same velocity p a n e l the one corresponding to the best-fit stacking velocity for the subhorizontal reflectors (Figure 7.11). The anisotropic processing sequence described above was implemented under the assumption that the medium above the dipping reflector is homogeneous. However, analysis of the time dependence of the zero-dip stacking (moveout) velocity [which can be approximated by Vnmo(0)] shows a pronounced velocity gradient of about 0.7 s -1. Therefore, the obtained value of 77- 0.07 can be regarded as an effective parameter resulting from the combined influence of vertical heterogeneity and anisotropy; it is reduced from the value associated with anisotropy alone (see the results for factorized VTI media in Chapter 3). While this effective ~ enabled us to correct for the anisotropy in DMO processing (because we had just a single dipping event), it can produce errors in anisotropic poststack migration or inversion for the individual values of ~ and 5. Figure 7.12 shows time-dependent parameters Vnmo(0) and r] obtained from the NMO velocity of the dipping event measured at five different points along the fault plane. The region above 2 s, characterized by positive r/values reaching 0.12, corresponds to a relatively thick shale formation, which is expected to be transversely isotropic. This example confirms that a proper correction for vertical heterogeneity in the DMO inversion increases the estimated interval r] within anisotropic formations. Application of the anisotropic TZO algorithm using the parameters from Figure 7.12 produces the zero-offset section in Figure 7.13. The fault-plane reflection
CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
306
2200
2250
2300
2.4 Figure 7.11" Constant-velocity stacks after transformation to zero offset (TZO) adapted for homogeneous VTI media; ~ - 0.07.
1-
V-
Figure 7.12" Inverted interval values of time.
Vnmo(0)
and q as functions of the vertical
7.1.
P-WAVE DIP-MOVEOUT
I N V E R S I O N F O R ~7
307
v
(D
2.5 Figure 7.13: Zero-offset section generated by the anisotropic TZO algorithm using the time-dependent parameters from Figure 7.12. The arrows point to regions of improved continuity of subhorizontal interfaces relative to that in Figure 7.14. that was missing after isotropic processing is now well focused; it suggests an average dip for the fault of about 45 ~. The section in Figure 7.13 is similar to the panels in Figure 7.11 obtained using a constant effective value 7/= 0.07. However, if we had dipping events over a range of vertical times, their preservation in the TZO process would require using the time-dependent 7/function from Figure 7.12. For comparison, Figure 7.14 shows zero-offset sections obtained by conventional DMO techniques designed for isotropic homogeneous (a) and vertically heterogeneous (b) media. The constant-velocity DMO correction used here is based on the frequencywavenumber method of Hale (1984), while the V(z) DMO is accomplished via the TZO method of Artley and Hale (1994). Clearly, in comparison with the VTI result in Figure 7.13, the fault is poorly imaged by both isotropic algorithms. The impulse response of a representative anisotropic DMO (TZO) operator used for these data is given in Figure 7.15. Even though the values of r/are quite moderate, the shape of the impulse response is far from the familiar isotropic ellipse. A detailed analysis of the VTI DMO impulse response is given in Chapter 8 (section 8.2).
308
C H A P T E R 7.
~15 v
O~
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
9
CD
2.5 Figure 7.14: Zero-offset section from Figure 7.13, but this time after NMO and (a) homogeneous isotropic DMO correction, and (b) V(z)isotropic DMO. The NMO correction is based on stacking velocities obtained from conventional semblance analysis.
7.1.
P-WAVE DIP-MOVEOUT
I N V E R S I O N F O R 7/
309
~1.4 ~15 2 !
N16 Figure 7.15: Impulse response of the transformation to zero offset (TZO) for the VTI model with the parameters from Figure 7.12. The offset is 1.5 km.
Figure 7.16: Common-midpoint gathers for CMP location 700 after (a) anisotropic TZO, (b) NMO and homogeneous isotropic DMO, and (c) NMO and V(z) isotropic DMO.
310
CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
Not only did the anisotropic (VTI) TZO algorithm do a better job of imaging the dipping fault plane, it also improved the continuity of subhorizontal reflectors (see the arrows in Figures 7.13 and 7.14) by correcting for anisotropy-induced nonhyperbolic moveout. Figure 7.16 shows reflections from subhorizontal interfaces at CMP location 700 after application of the anisotropic TZO, as well as of the two isotropic DMO methods preceded by the hyperbolic NMO correction. Both isotropic algorithms (b,c) fail to align horizontal events just above 2 s because they do not properly account for deviations from hyperbolic moveout at large offsets. In contrast, the anisotropic correction flattens the moveout for the whole offset range, thus ensuring a properly stacked event on the zero-offset section. Figure 7.17a displays a close-up of the fault imaged with an anisotropic phase-shift time-migration algorithm (see Chapter 8, section 8.3) using the inverted parameters from Figure 7.12. The zero-offset section for the migration was obtained by the anisotropic TZO code. The output of the conventional processing sequence in Figure 7.17b is much inferior in terms of imaging the fault plane, which confirms the advantages of anisotropic DMO (TZO) in preserving dipping energy (the influence of anisotropy on the migration step in this case is less significant). In Chapter 8, we discuss processing of a more complicated section, from the same area, that contains multiple fault planes.
7.1.5
Discussion
Both P-wave time-processing parameters for VTI media (Vnmo(0) and 7) can be estimated from surface reflection data by inverting the dip-dependence of P-wave moveout velocity. The 2-D algorithm for homogeneous models requires NMO velocities and the corresponding reflection slopes (ray parameters) to be measured in the dip plane of the reflector for two different dips; more than two dips provide redundancy that may increase the accuracy of the inversion. The zero-dip NMO velocity Vnmo(0) is typically obtained directly from hyperbolic semblance analysis of horizontal (or subhorizontal) events, while the NMO velocity from a dipping interface is used to estimate the anisotropic parameter 77. The moveout inversion for r/becomes more accurate with increasing dip, but for steep (near-vertical) reflectors estimation of Vnmo itself becomes less stable due to the small magnitude of reflection moveout. In vertically heterogeneous (e.g., layered) media, the parameters Vnmo(0 ) and r/ can be reconstructed as functions of vertical time using the 2-D Dix equation (6.11) generalized for symmetry planes of anisotropic media. The medium should contain a throughgoing dipping reflector, such as a fault plane, that generates P-wave reflections providing information about r/for a wide depth (vertical time) range. To reduce the instability in the Dix-type differentiation and reconstruct the smooth components of the vertical variation in Vnmo(0) and r/, it is convenient to represent both parameters through Chebyshev polynomials. The parameters Vnmo(0) and ~ can also be estimated from data acquired outside the dip plane of the reflector. The equation of the NMO ellipse, introduced in Chap-
7.1.
P-WAVE
DIP-MOVEOUT
INVERSION
F O R 7/
311
1 CD V
CD
E
im
2 Figure 7.17: Time-migrated section produced by (a) anisotropic phase-shift migration of the data from Figure 7.13 using the parameters from Figure 7.12, and (b) isotropic phase-shift migration of the data from Figure 7.14a.
312
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
ter 3, makes it possible to extend the 2-D r/-estimation algorithm to a CMP line with arbitrary orientation (the angle between the line and the dip plane has to be known). For wide-azimuth 3-D surveys, the NMO ellipse of a dipping event can be found using the NMO velocities and reflection slopes on just two differently oriented lines. Then the semi-axes of the NMO ellipse can be inverted for Vnmo(0) and r/, which means that P-wave reflection traveltimes from a single dipping interface are sufficient for timedomain velocity analysis. Also, the problems experienced by the dip-line inversion in handling near-vertical reflectors can be overcome by using CMP-line azimuths close to the strike direction. One of the ways to include the DMO-inversion technique in the processing flow is to apply a Fowler-type dip-moveout method (Fowler, 1984), which transforms CMP data into constant-velocity stacks calculated for a range of stacking velocities. Application of Fowler D MO in anisotropic parameter estimation is discussed in Chapter 8 (section 8.1).
7.2
Inversion 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
In the absence of dipping events, it may be possible to carry out P-wave velocity analysis in VTI media using nonhyperbolic (long-spread) moveout from horizontal reflectors. The key anisotropic parameter r/, needed for time processing (or, alternatively, the horizontal velocity Vhor), controls the deviation of the P-wave moveout curve from a hyperbola on long CMP spreads (Chapter 4). Here, following the results of Grechka and Tsvankin (1998a) and Alkhalifah (1997b), the moveout equations developed in Chapter 4 are used to estimate r/and Vhor via a two-dimensional nonhyperbolic semblance search. The parameter-estimation algorithm is introduced first for a single VTI layer and then extended to vertically heterogeneous media using a Dix-type differentiation procedure that involves the nonhyperbolic term of the moveout equation. Stability analysis shows that the horizontal velocity is relatively well-constrained by long-spread moveout if the spreadlength exceeds two reflector depths, with errors comparable to those in the NMO velocity. In contrast, the parameter r/is influenced by the trade-off between the NMO and horizontal velocity and, therefore, is more sensitive to errors in the reflection traveltimes. Synthetic examples are followed by application of nonhyperbolic moveout inversion to a VSP (vertical seismic profiling) survey and offshore reflection data.
7.2.1
Single
VTI
layer
Modified m o v e o u t e q u a t i o n P-wave reflection traveltime in a VTI layer can be expressed through the normalmoveout velocity from a horizontal reflector [Vnmo - Vnmo(0)] and the horizontal
7.2. INVERSION OF P-WAVE NONHYPERBOLIC MOVEOUT
313
velocity (Vhor)as [equation (4.29)1 X2 t 2 -- t 2 4- Vn2m---~ -
(Vh2or- V : m o ) x 4 V:mo (to0 V2mo -~- Vh2or X2) 9
(7.14)
Chapter 4 also introduces an equivalent equation (4.28) in terms of Vnmo and the parameter r/. The form (7.14) is more suitable for inversion purposes because the horizontal velocity proved to be better constrained than is ~ by long-spread moveout. Also, it is more convenient to implement nonhyperbolic velocity analysis if both moveout parameters have the dimension of velocity (Vnmo and Vhor). Numerical analysis of equation (7.14) in Chapter 4 shows that its deviation from the exact traveltime for the spreadlength Xma x -- 2Z (Z is the reflector depth) is less than 1% of the vertical time to. Still, the parameters that provide the best fit of equation (7.14) to the exact moveout are somewhat different from the actual model values. In the example from Figure 4.10, the best-fit ~ is close to 0.13, while the actual value is 0.16. Such an error means that P-wave moveout on CMP spreads with Xma x -- 2Z is not highly sensitive to 77; this conclusion is confirmed by further numerical tests below. Equation (7.14) was designed to ensure the correct behavior of the moveout for infinitely large offsets (x -4 cx)) (see Chapter 4). At intermediate offsets (e.g., x/z 2), however, this moveout approximation can be improved further by empirically changing the denominator of the nonhyperbolic term. In particular, as shown below, introducing the coefficient C = 1.2 in front of Vh2or X 2 in the denominator minimizes deviations from the exact traveltimes for the most practical range of offsets 1.hz < x < 2.5z (Grechka and Tsvankin, 1998a). This modification changes the moveout at large offsets, while keeping the correct values of the quadratic and quartic moveout coefficients. Thus, the following version of equation (7.14) will be used in the inversion below: X2 t 2 ( x ) -- t~ -Jr- Vn2m-'---~-
(Vh2or -- Vn2mo)x 4 Vn2mo (to2 V2m o -~- C gh2or X2) '
(7.15)
with C = 1.2. Figure 7.18 shows contours similar to the ones in Figure 4.10 but now calculated using the modified approximation (7.15) for three different spreadlength-to-depth (Xmax/Z)ratios. The correction of the traveltime equation has moved the center of the contours of the traveltime residuals Atma~ in Figure 7.18 to the correct position (Vnm o = 2.0 km/s and Vho r -- 2.3 km/s), as well as substantially reduced Atmax at the center of the contours. This means that the actual parameters substituted into equation (7.15) provide the best fit to the exact traveltimes, which was not the case with equation (7.14). Note that the contours of the time residual Atma~ in Figure 7.18 form an elongated "valley" in the [Vnmo,Vhor]-plane. For example, the contour Atmax = 4 ms in Figure 7.18b covers a significant range of both NMO and horizontal velocities for which the traveltime at all offsets is quite close to that for the actual model.
314
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 MEDIA
i
11 i i
/I'
2 /./',
2 5- I . / ~ . Y / . / . / . . J /
~
.o.j-....!///...~
2.5-~.;
,.::1
I/~
.... .r
Figure 7.18: The influence of V n m o and Vhor on the P-wave reflection traveltimes in a horizontal VTI layer for three different spreadlengths Xm~x: Xm~x/Z= 1 (a); Xmax/Z- - 2 (b); Xmax/Z~-- 3 (c). The contours display the maximum difference/ktmax (in ms) between the traveltimes for a reference model (computed using ray tracing) and those for models with the same to but different Vnmo and Vhor shown on the axes [computed from equation (7.15)]. The model parameters a r e V n m o = 2 km/s and Vhor = 2.3 km/s (r/= 0.16). Models A, B and C are used in further numerical tests below.
7.2. INVERSION OF P-WAVE NONHYPERBOLIC MOVEOUT
315
The changes in Vnm o and Vhor within a given contour Atmax = const are comparable but have opposite signs. Since r/is approximately determined by the ratio Vhor/Vnmo [equation (4.26)], the variation in r/ over the range of the kinematically equivalent models (i.e., those within the contour Atmax = 4) may be approximated by the sum of the corresponding percentage changes in the horizontal and NMO velocity. Therefore, if the errors in the NMO and horizontal velocities are close to one another (and opposite in sign), a certain relative error in Vhor translates into about twice as large absolute error in the parameter ~. As expected, Figure 7.18 indicates that both Vhor and the corresponding r/ become better constrained with increasing maximum offset Xma• For larger Xmax/Z, the contours for a given maximum time residual become tighter and tilt toward the Vnmo-aXis, thus making the variation in Vhor for a fixed Atmax less pronounced. The meaning of this result becomes clear if we recall that Vhor and r/are responsible for the nonhyperbolic moveout term, which makes a significant contribution to the traveltime only at offsets exceeding the reflector depth. Therefore, the inversion of nonhyperbolic moveout for Vhor and r] becomes more stable with increasing spreadlength. It is seldom possible, however, to use spreads Xmax > 2.5z in practice due to limitations of the acquisition design and difficulties in applying semblance analysis to reflections at near-critical or post-critical incidence angles. Figure 7.19 shows ray-traced traveltime curves for the models labeled by letters A, B (the actual model) and C in Figure 7.18b. Despite a significant variation in 77 (from 0.09 to 0.24) among the three models, the maximum values of the traveltime differences (/ktABand /ktCB) picked from Figure 7.18b are about 3 ms. Figure 7.19, generated using an exact ray-tracing algorithm, confirms this result: the moveouts tA, tB, and tc are very close to each other at all offsets, with deviations smaller than 3 ms. It is noteworthy that the maximum separation between the curves occurs at an intermediate offset close to the reflector depth rather than at the largest offset Xma• = 2z. The absolute variation in r/ from Model A to Model C (0.15) even exceeds the sum of the corresponding relative changes in Vnmo and Vhor (5-6% for both velocities). The conclusions drawn from Figures 7.18 and 7.19 have important consequences for evaluating the feasibility of nonhyperbolic moveout inversion. For spreadlength reaching two reflector depths, the horizontal velocity becomes almost as tightly constrained by reflection traveltimes as is the NMO velocity. At the same time, the absolute variation in r/over the family of kinematically equivalent models is at least twice as large as the corresponding relative change in Vhor. Hence, even small long-period traveltime errors may result in significant deviations of ~ from the correct value. Below, this conclusion is verified by performing the actual inversion of long-spread moveout in the presence of different kinds of traveltime errors. N o n h y p e r b o l i c s e m b l a n c e analysis If reflection traveltimes on a certain spreadlength are known, the parameters to, Vnmo, and Vhor can be obtained, for instance, by least-squares fitting of equation (7.15) to
316
CHAPTER
7.
VELOCITY
ANALYSIS
AND
PARAMETER
ESTIMATION
.............
S ......... S
FOR
VTI
% ............. %
MEDIA
9
........S s:"..................... :.~...... . i S
9
:
~
'
....... '-.,..................... ! , .... ~~ i ..............
............. Q
e t
' .......... " ~,~ "
Figure 7.19: (a) Moveout curves (tA, ts, and to) obtained by ray tracing for the models marked by A (dashed), B (thin solid), and C (dotted) in Figure 7.185. (b) The traveltime differences AtAB -- t A - ts (dashed) and A t c s -- t c - ts (dotted). The parameters are: Vnmo = 1.95 km/s, Vhor = 2.37 km/s, 7/ = 0.24 (Model A), Vnmo = 2.0 km/s, Vhor = 2.3 km/s, q = 0.16 (Model B, actual values), and Vnmo = 2.05 km/s, Vhor = 2.23 km/s, r] = 0.09 (Model C). the moveout curve (see a VSP-data example below). However, due to the presence of noise and the need to automate data processing, picking of reflection traveltimes on CMP data is seldom done in practice. Instead, the best-fit moveout (stacking) velocity and the corresponding hyperbolic moveout curve are usually found from semblance analysis on CMP gathers, preferably with spreadlength close to reflector depth (e.g., Taner and Koehler, 1969). Similarly, the inversion for Vhor or ~ can be accomplished by semblance analysis along nonhyperbolic moveout curves. For a given value of the vertical traveltime to, one now needs to perform a two-dimensional semblance scan over Vnmo and Vhor to find a model that yields the maximum value of semblance. The semblance coefficient is usually calculated as (Taner and Koehler, 1969)
rE.n x F(x t)]
S
(to,
Vnmo,
r])
--
Z'~tto-'to--T/2L X - - X m i n ' x-'to+T/2 ~"~ X m a x F2(x, t) M A.~t,o=to_T/2~ . . ~ X . _ X m i n
,
(7.16)
where M is the number of traces. The amplitudes F(x, t) and their squares F2(x, t) in equation (7.16) are summed along the nonhyperbolic moveout curves t (t~, Vnmo, Vhor, X) [described by equation (7.15)] originated at the vertical (zero-offset) traveltimes t~ within a smoothing window T centered at the time to. In the examples below (Grechka and Tsvankin, 1998a), reflected arrivals are generated by placing at ray-traced traveltimes a wavelet that has the same shape and amplitude for all offsets. The offsets for the traces were equally spaced with a 40-m interval; for the most typical maximum offset Xm~x = 2z = 2.0 km used below, that
7.2. INVERSION OF P-WAVE NONHYPERBOLIC MOVEOUT
317
gives M - 50. The length of the smoothing window was T = 20 ms, which corresponds to five time samples. The amplitudes F(x, t) between time samples were obtained by linear interpolation. The results of the nonhyperbolic semblance search using data from a single-layer VTI model are displayed in Figure 7.20. The coordinates of the semblance maximum are close to the actual parameters Vnmo and Vhor; therefore, the estimate of ~ is sufficiently accurate as well. [Note that application of the original nonhyperbolic equation (7.14) leads to errors in all three parameters, particularly in 7.] The semblance contours in Figures 7.20b,c and the contours of the maximum traveltime difference in Figure 7.18b look quite similar. This can be expected since the semblance values for models with close traveltime curves cannot differ much from each other. Again, Vnmoand Vhor can simultaneously vary along the diagonal "ridge" in the semblance contours without producing significant changes in the values of semblance (Figure 7.20). Confirming the results of the traveltime analysis, Figure 7.20 shows that within this family of models with close semblances, higher values of Vnmo are compensated by lower values of Vhor, and absolute variations in 7? are at least twice as large as the corresponding percentage changes in Vhor. The influence of spreadlength on the sensitivity of reflection moveout (this time measured by the value of semblance) to Vnmo and Vhor is illustrated by Figure 7.21. For a relatively short (in the context of nonhyperbolic moveout inversion) spread of 1.hz in Figure 7.21a, the contours are tilted towards the Vho~-axis, and the horizontal velocity is constrained much less tightly than is Vnmo. However, for a longer (but still feasible) spread equal to twice the reflector depth (Figure 7.21b), the range of variation in Vhor within the family of kinematically equivalent models becomes closer to that of Vnmo. For less common spreads close to or exceeding 2.5z (Figures 7.21c and 7.21d), the horizontal velocity has even more influence on the semblance values than does the NMO velocity. Asymptotically, for infinitely large offsets the effective moveout velocity approaches Vho~, and the reflection moveout is controlled by the horizontal velocity. Influence of t r a v e l t i m e e r r o r s on Vhor a n d r] To study the influence of realistic traveltime distortions on the inversion results, an error function ~-(x) was added to the reflection traveltimes t(x) from Figure 7.20 without changing the trace amplitudes. The function T(x) in Figure 7.22a was chosen to coincide with the time residual for Model A (Figures 7.18b and 7.19b). The maximum of T(x) in this case is less than one time sample (4 ms) in conventional processing, and its predominant period is between 1.5 Xma~ and 2 Xm~x. Such a class of time distortions may result from contamination of reflection times by coherent noise (errors in the statics correction, ambient noise etc.) or by the influence of small highor low-velocity lenses. In accordance with the form of the error function, the pair (Vnmo = 1.96 km/s, Vhor = 2.36 km/s) that provides the maximum semblance in Figure 7.22a practically
318
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
,.o ~llllllllllllillllllllllll ~'" illlllllllllillllllllll~lllillllllllll ~"~ Illllllllllllllllllllllllllllllll~lllll
Illll
"4 lllllIllllllllllillIlllllllllilllllIlllll~ll
~.~ ....... i..... ~:~ ~.
~::
Figure 7.20: (a) Synthetic seismogram of the P-wave reflected from the bottom of a VTI layer with the same parameters as in Figures 4.10 and 7.18: to = 1 s, Vnmo = 2 km/s and Vhor = 2.3 km/s (7/ = 0.16). The spreadlength is equal to two reflector depths; the source pulse is a Ricker wavelet with the dominant frequency 40 Hz. (b) Semblance contours in the coordinates Vnmo and Vhor computed using equation (7.15) for to = 1 s. (c) Same as (b), but expressed through Vnmo and r/.
7.2.
INVERSION OF P-WAVE N O N H Y P E R B O L I C M O V E O U T
212 . . . . . . . . .
~
~.~
i,"i
319
i ..... ....
i. . . . . .
>
2.4
6
1.8
2.1
1.9
2.0
2.1
2.2
i
.
2.2 . . . . . . . .
i .~23o
>2.4-
9
2.5-
.~ .... i....... i .... oj
>2.4-
~.s_
i
Figure 7.21" Semblance contours for different spreadlengths plotted in coordinates [Vnmo, Vhor]. (a) Xmax/Z- 1.5; (b) X m ~ x / Z - 2; (C) Xmax/Z- 2.5; (d) Xmax/Z- 3. The model parameters are the same as those in Figure 7.20.
320
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
~2 0r ~
9
....
"4
iii i
9
.
iiiiiii
,
i
=.=k....i ....... i..../..i .... bi
E o r--
O.
.
9
Figure 7.22: Results of semblance analysis for the model from Figure 7.20 after the addition of traveltime errors with small magnitude. (a) The traveltime error is equal to the residual for Model A from Figure 7.18b (see Figure 7.19b); (b) the error is equal to the residual for Model C from Figure 7.18b (see Figure 7.19b); (c) same as (a) plus random noise with a maximum of 3 ms; (d) linear error function with T(0) = 4 ms and T ( X m a x ) : - 4 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
321
coincides with the parameters of Model A [a small difference is due to the slight inaccuracy of equation (7.15)]. Note that the semblance maximum moves along the diagonal ridge in Figure 7.20b, with the increase in Vhor compensated by the smaller value of Vnmo. While Vnmo and Vhor are off by only -2% and +2.6%, respectively, the opposite sign of these errors yields the best-fit value of r / = 0.23 instead of the correct ~ = 0.16. Figure 7.22b shows the result of the semblance analysis with an added error function equal to the time residual for Model C from Figures 7.18b and 7.19b. As expected, the coordinates of the semblance maximum in this case are close to the parameters of Model C (Vnmo = 2.05 km/s, Vhor = 2.25 km/s, ~ = 0.1), with the error in ~ equal to (-0.06). Thus, a low-frequency noise function with a magnitude of just i 3 ms or less causes the variation of ~ between 0.1 and 0.23. Figure 7.22c reproduces Figure 7.22a, but with the addition of random noise on top of the traveltime residual for Model A. The values of semblance became substantially smaller, and the semblance maximum moved a little bit further from the actual solution. The best-fit r/in this case is equal to 0.25, which is 0.09 higher than the actual value. In general, the shape of the semblance objective function is not sensitive to purely random errors in the traveltimes. Similar results were obtained for other slowly varying error functions with a small magnitude. For instance, Figure 7.22d was generated for a linear error function equal to 4 ms at zero offset and (-4 ms) at the end of the spread. Such an error mitigates the increase in traveltime with offset, and the effective NMO velocity becomes higher. To provide a good fit at large offsets, the horizontal velocity has to be smaller than the actual value, and the semblance analysis yields Vnmo = 2.07 km/s and Vhor = 2.24 km/s. The corresponding r/is as low as 0.085, which is about one-half of the actual value. The semblance analysis in Figure 7.22 confirms the conclusion drawn from the study of reflection traveltimes (Figures 7.18 and 7.19) that the horizontal velocity is almost as tightly controlled by reflection moveout as Vnmo, provided the spreadlength is no smaller than two reflector depths. At the same time, long-period traveltime errors that can be considered as insignificant in the practice of data processing may cause errors in inverted value of r/reaching • These errors are entirely due to the trade-off between Vnmo and Vhor on long-spread gathers (i.e., if we knew the correct value of Vnmo, we could determine Vhor and ~ with high accuracy). 7.2.2
Nonhyperbolic
velocity
analysis
for layered
media
Analytic formulation The above results were limited to nonhyperbolic moveout inversion in the simplest single-layer model. In Chapter 4 it was shown that the nonhyperbolic equation (7.14) retains the same form in horizontally layered media, provided the NMO and horizontal velocities are replaced by effective values that absorb the influence of the vertical heterogeneity above the reflector.
322
CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
Following Appendix 4B, the traveltime of the P-wave reflected from the N-th (i - 1, 2 . . . N) interface in vertically stratified VTI media can be written as
t2(x, N) -
x2 [V2or(N)-Vn2mo(N)]x 4 . tg(N) + Vn2mo(N) - Vn2mo(N)[t~(X) V4mo(N ) + C Vh2or(N) x 2 ]'
(7.17)
the value of the coefficient C, introduced in equation (7.15), is discussed below. Since all interfaces, including the reflector, are horizontal, the normal-moveout velocity Vnmo(N) is computed through the interval NMO velocity and interval vertical traveltime using the conventional Dix equation,
1 ~.= (Vn(i)m~ t~i) " to(N)
Vn2m~
(7.18)
The effective horizontal velocity for layered media can be defined as (Appendix 4B) V h o r ( N ) -- V n m o ( N ) V / 1 + 2 r / ( N )
(7.19)
,
with the effective parameter 77(N) given by
~(N)--~
V4mo(N) to(X)
E(Vn(2o) 4
i=1
(1 + 87?(i))
- 1
.
(7.20)
In Chapter 4 it was shown that the interval values of the quartic moveout coefficient can be obtained by the Dix-type differentiation of the combination of the effective parameters denoted as g(N) [equation (4.18)]. From equation (7.20) it follows that
g(N) - V2mo(N)[1 + 877(N)] - Vn2mo(N)[4Vh2or(N) -- 3Vn2mo(X)],
(7.21)
or
9(N)
1 N i ~ 1(Vn(2o) 4 (1 + 8r/(i)) t~i) to(N)
=
1
N
to(N) ~(Vn(/m)~
[4(Vh(:~)2 -- 3(Vn(/m)~ t~i)"
(7.22)
Applying the Dix equation to g(N) from equation (7.22) makes it possible to find the interval values v(i) v hor and r/(i)"
Vh(ori) - Vn(~~ I and ~(i) _
1
4 (Vn(2o)4
g(i) to(i) - g(i - 1)to(i - 1) to(i) - to(i- 1)
[g(i)to ( i ) - g ( i - 1)t0(i- 1) to(i) - to(i- 1) 8 (P~2o) 4 1
_
t
3 4 '
(7.23)
(7.24)
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
323
This formalism provides an analytic basis for a parameter-estimation approach that can be summarized as follows: 1) Carry out semblance analysis of the reflections from the top and bottom of layer i using the nonhyperbolic moveout equation (7.17). The semblance search produces the effective values of to, Vnmo and Vhor which serve as input data for the interval parameter estimation. [Clearly, for the bottom of the shallowest layer (i = 1), the effective and interval values are equal to one another.] 2) Compute the interval normal-moveout velocity in layer i from the Dix equation, (V(/m)o)2
Vn2mo(i)to(i)
to(i)
Vn2mo(i
1)to(i
1)
(7.25)
to(i- 1)
3) Using equation (7.21), obtain the auxiliary effective parameters g(i) and g(i-1) corresponding to the reflections from the top and bottom of the layer of interest. 4) Carry out the Dix differentiation of g(i) and use the interval NMO velocity V(~o to compute the interval values of the horizontal velocity and ~ from equations (7.23) and (7.24). Interval parameter estimation First, it is necessary to show that equation (7.17) [with the effective parameters from equations (7.18)-(7.20)] gives an adequate description of P-wave moveout in stratified media. For the single-layer model, the accuracy of the moveout equation on intermediate spreads was increased by inserting C=1.2 into the denominator of the nonhyperbolic moveout term [equation (7.15)]. In the presence of vertical variations in the elastic properties, we do not always find the value C=1.2 to be optimal. Numerical tests performed for various stratified VTI models (Grechka and Tsvankin, 1998a) showed that for a vertical velocity gradient of 0.5-0.6 s -1 and relatively small effective r/values up to 0.1-0.15 it is preferable to use C = I or even C < 1 (although the accuracy provided by C=1.2 is not much inferior). As was the case for a single layer, the value C=1.2 usually gives an advantage for larger r/_> 0.2. It should be mentioned that the traveltimes calculated using C = I and C=1.2 for the spreadlength-to-depth ratio Xmax/Z-- 2 are close to each other (in most cases, the difference is less than 0.3-0.5% of to). Still, varying C from 1 to 1.2 does have some influence on the inverted value of the effective r/because, as shown above, VTI media with substantially different r/may have similar traveltimes for spreads Xmax/Z~ 2. In principle, after obtaining a model from nonhyperbolic moveout inversion (using, for instance, C=1.2), it is possible to find the best-fit C by comparison of equation (7.17) with ray tracing and use this updated value to produce the final inversion output. Table 7.1 summarizes the results of a numerical test performed for a model consisting of four homogeneous VTI layers with a typical increase in Vpo with depth (the average gradient is close to 0.6 s -1) and values of r/ ranging from zero in the near-surface layer to a maximum of 0.2 in the third layer. The effective moveout
324
CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
Layer
Vnmo ( k m / s )
actual
found
Effective values Vhor ( k m / s )
error
(%)
actual
found
error
actual
~7 found
(%)
,,
1 2 3 4
2.098 2.216 2.392 2.459
2.100 2.225 2.390 2.450
0.1 0.4 -0.1 -0.4
2.098 2.318 2.698 2.792
2.100 2.340 2.760 2.860
0.1 0.9 2.3 2.4
0.000 0.136 0.144
0.000 0.053 0.167 0.181
actual
found
0.000 0.100 0.200 0.140
0.000 0.109 0.261 0.206
o.o .7
error (abs) 0.000 0.006 0.03i o.o37
Interval values Layer
Vnmo (km/s) actual found error
Vhor ( k m / s )
actual
found
(%) -2.098 2.519 2.779 3.033
2.100 2.546 2.755 2.962
0.1 1.1 -0.9 -2.3
error
(%) 2.098 2.759 3.288 3.431
, .
2.100 2.811 3.399 3.521
0.1 1.9 3.4 2.6
error (abs) 0.000 0.009 0.06~1 0.066
Table 7.1: Accuracy of the nonhyperbolic moveout approximation for P-wave reflection traveltimes in layered VTI media. The effective parameters Vnmo(N), Vhor(N) and r/(N) ("found") were obtained from semblance velocity analysis [using equation (7.17) with C--1.2 in the denominator of the nonhyperbolic term] of synthetic seismograms computed by ray tracing (Figure 7.23); the spreadlength-to-depth ratio for all events is equal to 2. The "actual" effective values were calculated from equations (7.18)-(7.20). The lower table shows the results of the layer-stripping of the recovered effective parameters [using equations (7.23)-(7.25)] versus the actual interval values. The model consists of four homogeneous VTI layers with the following parameters: layer 1 - z (depth of the b o t t o m ) = 0.7 kin, Vpo = 2.0 km/s, e = 0.05, /~ = 0.05; layer 2 - z = 1 km, Vpo = 2.42 kin/s, e = 0.15, 5 = 0.0417; layer 3 - z = 1.5 km, Vpo = 2.6 km/s, e = 0.3, /~ = 0.0714; layer 4 - z = 1.7 km, Vpo = 2.9 km/s, = 0.2, 5 = 0.0469; the corresponding values of Vnmo, Vhor and r/ are given in the table.
7.2. INVERSION OF P-WAVE NONHYPERBOLIC MOVEOUT
o.o1 111 III!11
325
llilllll
IIIII!1111
Illiilll I Illlllllllll 11]II i111111 .1111 IlllllllllllIIIIi1111111I111 IIIIII i111111111111111IIIIIIIIIIIIIIIIIIIIl11111111111Ir~ Figure 7.23: Synthetic seismograms of P-wave reflections generated by ray tracing for the model from Table 7.1. The traces were muted to maintain the spreadlength-todepth ratio of 2 for all four events. The wavelet amplitude and dominant frequency are identical on all traces. coefficients Vnmo(N), Vhor(N) and q(N), which were obtained from ray-traced seismograms (Figure 7.23) for all four interfaces using equation (7.17), are sufficiently close to the analytic values computed from equations (7.18)-(7.20). [We did not try to optimize the value of C, using (7-1.2 in equation (7.17).] Despite the approximate character of the moveout equation (7.17) and some amplification of errors during the layer stripping, the interval values of Vnmo and Vhor have been recovered with acceptable accuracy (for all events, the spreadlength was twice as large as the reflector depth). Note, however, that the relatively small errors in Vnm o and Vhor for the two deepest layers have opposite signs (as was the case in the single-layer model), and the errors in the interval q for these layers reach 0.06. Since moveout in stratified media is well-described by the same equation that we studied in detail for the homogeneous VTI model, our analysis of the accuracy in the estimation of Vnmo, Vho~ and rt must remain valid for the effective values of all three parameters. As an example, Figure 7.24a displays semblance contours for the reflection from the bottom of the layered model discussed above (Table 1). Again, the contours make up an elongated ridge along which the effective Vnmo(N) and Vhor(N) can be varied without producing tangible differences in the value of semblance (compare with Figure 7.20). The stability of the layer-stripping algorithm is illustrated by the test in Figure 7.24b,c. Errors up to about +3% were introduced in the effective parameters
326
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 VTI M E D I A
~.~2.8 .... . . . . . . . . . . . . . .
-20 ' 2 . 3 8 2 . 4 2.42 2 . 4 4 2 . 4 6 2 . 4 8
2 . 5 2.52
0.6
oli .
-0.2
.
.
.
C
'
2.38 2.4 2.~,2 2.~,4 2.46 2.~,8 2'.s 2.~2 Effective V n m o (kin/s)
Figure 7.24: (a) Semblance contours for the reflection from the bottom of the model (interface 4) described in Table 7.1. The semblance analysis was carried out using equation (7.17) on ray-traced synthetic data (Figure 7.23) for the spreadlength-todepth ratio of 2. (b) Percentage errors in the interval velocities Vnmo (solid) and Vhor (dashed) in the bottom (fourth) layer; those interval values were calculated from the effective velocities picked along the ridge in the semblance contours marked by the dots on plot (a). The effective moveout parameters for the reflection from interface 3 were not changed. (c) Interval parameter q (the actual r/ = 0.14) found from the distorted interval values of Vnmo and Vhor shown on plot (b).
7.2.
INVERSION OF P-WAVE N O N H Y P E R B O L I C M O V E O U T
327
Vnmo(4) and Vhor(4) for the fourth interface, which simulated the movement of the semblance maximum along the ridge in the semblance contours under the influence of long-period traveltime errors with small magnitude (on the order of several ms). As expected, the layer stripping leads to a substantial magnification of errors in both velocities (the thickness of the layer is less than 15% of its depth), but the interval values of Vnmo and Vhor were estimated with similar accuracy. This and other tests for a representative set of VTI models show that the inversion of the nonhyperbolic moveout term for Vhor does not introduce much higher errors than does the conventional Dix differentiation for the NMO velocity, provided the maximum offset reaches at least twice the reflector depth. Therefore, nonhyperbolic moveout inversion can be used to reconstruct the "low-frequency" component of the horizontal velocity as a function of vertical time. As before, however, the errors in the interval values of the NMO and horizontal velocity in Figure 7.24b have opposite signs. Since the interval Vnmo and Vhor are more distorted than in the single-layer model, the error in the interval parameter r] easily becomes unacceptably l a r g e - comparable to r] itself (Figure 7.24c). 7.2.3 vsP
Field-data
examples
s u r v e y over a c a r b o n a t e r e s e r v o i r
The performance of nonhyperbolic moveout inversion is demonstrated here on a walkaway VSP (vertical seismic profiling) data set acquired in 1995 by the Reservoir Characterization Project of the Colorado School of Mines at Vacuum Field in Lea County, New Mexico (Grechka and Tsvankin, 1998a). The objective of the survey was to use both surface seismic and VSP to monitor changes in the properties of a carbonate reservoir after injection of CO2. The data used in the inversion were recorded by a downhole receiver placed above the reservoir at a depth of 1005 m. The wavefield was excited at several hundred surface source locations with offsets of 60 to 1370 m from the receiver borehole. The times of the first P-wave arrivals were picked from the seismograms and used as substitutes for the reflection traveltime from an imaginary interface at the receiver depth (Figure 7.25). The data points plotted in squared coordinates (t 2 - x 2) in Figure 7.25a cluster around a curved line, indicating the presence of substantial nonhyperbolic moveout. Equation (7.15) was used to find the set of moveout parameters that gives the best fit (in the least-squares sense) to the observed traveltimes; the corresponding moveout is marked by the solid line in Figure 7.25a. The inverted values of the NMO and horizontal velocities a r e V n m o - - 3536 m/s and Vho r : 4145 m/s, yielding r/ = 0.19. Note that the largest offset-to-depth ratio was over 2.7 (the offset should be doubled for comparison with surface-seismic reflection data), which is quite favorable for nonhyperbolic moveout analysis. If equation (7.15) is used to fit the data with a range of values of Vnmo and Vhor, the rms time residuals (Figure 7.25b) look similar to the contours of the maximum
328
CHAPTER 7. VELOCITY ANALYSIS AND PARAMETER ESTIMATION FOR VTI MEDIA
:
:
:~
.
0.8 v
{D ~
E 0.7
.
.
.
.
.
; i.~
~
:
. . . .
{D
>
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
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
.
.
.
.
.
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 . . . . . . . . . . . . . . . . . . .
q~
9
b
r
,2
..............................
=4
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0.1 ........... "..' 9 C
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.
.
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.
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3.2
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
-~
]
C Vpo 0.3 9
.:..I
0.2 ..~"
-~
1.2
0.1 .
e
1
9 .:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
f
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
.........
9
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.
oi
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9 .....
9
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9
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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
:i
.
. . .
:
. . . . .
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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.
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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.
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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
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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.
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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
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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.
8.1. FOWLER-TYPE TIME-PROCESSING METHOD
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.
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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).
8.1. FOWLER-TYPE TIME-PROCESSING METHOD
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
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FOR
9
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,~ .'" - . , , "
MEDIA
'
N
(a) q=O.O0 0
5
.................................... I..................................I . . . . . . . . . . . . . .
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...... ~ . . . .
" . . . . ~ -~-,~L~,.,.:~
,~ ~' .~.."~ , : ~ ' ~- = N _ ~ . I .' p ~ . .~ , ~ " ~ , ~ -~~ ~
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. ..:7~.~..:..:~.~:~!..!!~:~i~{. .~.!~.~.i~:(~~..7~~i.~i~`~:i~!~:~~)~~ m Yi-iTI:=:,-Y:7:; 1
Figure 8.2: O u t p u t Fowler D M O / S t o l t m i g r a t i o n stacked panels for t h e correct Vnmo(0) - 3000 m / s a n d different values of q. Panel (d) c o r r e s p o n d s to t h e correct m o d e l p a r a m e t e r s .
8.1.
FOWLER-TYPE
TIME-PROCESSING
....
9
.,~.~...~.~
METHOD
363
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2.0 ~~ (a) Vnmo(O)=2700m/s
(b) Vnmo(O)=2850m/s ~ l
(c) Vnmo(O)=3000m/s ...............................................
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10
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---.::.i7.;:. : i:-:: ~t~~~:i~<".'
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Figure 8.3: Fowler D M O / S t o l t migration stacked panels for the correct r / - 0.15 and different values of Vnmo(0). Panel (c) corresponds to the correct model parameters.
364
CHAPTER
8.
P-WAVE
1
3
IMAGING
FOR
2
VTI
MEDIA
3
=
9-:: '~,; ;!::2!:: .::!.;:i~s
(a) Vnmo(O)=2700 m/s i;:N.; . . . . . . . . .
!
I
',,. ~".~-::~.:
(b;Vnmo(O)=2850 m/s K::i~;i!l
....
: -".::,
....
"?'
' :~ .... < ~':?::.: ~::'i~'~::~;:~'i ' t : y : : '~: .,~': i' .4~
:r
' ~.
,
....
..%2'.-
'...;:,'~:.,):.
.:..~,"
:~.
..~
/-.,:"~::,.,~..&~:~,~:;.-.,:~:,:!,:*:~i~ ~,.:',:,'..
(d) Vnmo(0)=3150m/s
(c) Vnmo(O)=3000 m/s
. . . . . .
9 ]i~ ;;i~if:i"i!~:'.....~ :~ '~ ';" '~ ....... ='~ : .....
.
.
9"~ ;!",
"~.
;~b
9
~~. ~ . ; " . . ~
:~ "
.... : . . . . .
"..
I.
I
.
, ~--~.,
- :
..
,.
-.,,.,-,
" :,.
:.i:
'
[.,.,
,:!:!
.,i::~i;i i;.i:; ...,-
N r:";"
.
'*:I
~ : ~ . ~
i: -.
.-.,
_L______L :_:-tl ! ::_::__2.~i,~-~": ,
.. 9
_
..
.
.
(f) Vnmo(0)=34S0 m/s F i g u r e 8.4: F o w l e r D M O / S t o l t a l g o r i t h m ( 7 / - 0).
m i g r a t i o n s t a c k e d p a n e l s g e n e r a t e d by t h e i s o t r o p i c
8.1. FOWLER-TYPE TIME-PROCESSING METHOD
365
the depth axis after time-to-depth conversion without influencing the focusing of reflections. The cost associated with the anisotropic algorithm is almost identical to that of the isotropic method for the same total number of output panels. Although the two-parameter search in VTI media requires producing more panels, it is possible to compute four or five hundred output migration panels at an increase in computational effort of only about 20% (most of the computing time goes into the generation of the constant-velocity stacks needed for the DMO resampling step). The real limitation becomes disk space, input/output, and the resources required for displaying all output panels.
8.1.4
Field-data example
The results below were obtained by Anderson et al. (1996) using a data set acquired by Chevron Overseas Petroleum in offshore Africa. Data from the same area have been used to test parameter-estimation methods in Chapter 7. The main goal of this case study is to image reflectors associated with fault planes beneath a thick anisotropic shale sequence. The angle of the fault in Figure 8.5 is about 40 ~ while the subhorizontal reflectors have dips less than 5~ The medium has an average vertical velocity gradient of about 0.6-0.7 s -1, which cannot be honored precisely by Fowler's algorithm designed for homogeneous models. To account for the velocity gradient, the final output sections were interpolated ("painted") using panels with different (time-dependent) values of Vnmo(0) o b t a i n e d from the conventional semblance velocity analysis of subhorizontal events. Each panel was generated for a constant (independent of time) 77 value, and the best-fit r/was chosen by visual inspection of the fault-plane reflection. In principle, it is possible to account for vertical variations in r/ as well by interpolating between triplets [Vnmo(0), rl, to] specified for each subsurface location. Therefore, although the algorithm operates with a homogeneous model, the parameters of the output section [Vnmo(0 ) and 77] may vary in reflection time and space. Figure 8.5 shows six DMO-corrected stacked panels for values of ~ ranging from 0 to 0.20. As discussed above, Vnmo(0) on each panel is a time-varying function estimated from semblance analysis. A segment of the fault plane appears in its unmigrated position at vertical times of about 1.7-2 s and midpoints 850-900. The best image of this DMO-corrected fault-plane reflection is produced for an r/value between 0.08 and 0.12. The output of the complete DMO/migration sequence for the same range of ~ is displayed in Figure 8.6. The fault-plane event migrates from about midpoint 850 and time 1.95 s on the DMO section (Figure 8.5) to its final location at midpoint 950 and time 1.5 s on the migrated section (Figure 8.6). Interestingly, the best migrated fault-plane image corresponds to 0.12 < ~ < 0.16, which is larger than the bestfit 77 identified for the DMO step. Also, note that the fault-plane event tends to migrate further horizontally for larger values of r/. It should be emphasized that the
366
C H A P T E R 8.
P - W A V E I M A G I N G F O R VTI M E D I A
V
._E 1
(bl
1.2
.E_ 1.6 1.8
(d)
(c)
v
o~ 1 (D
.E_ 1
E Figure 8.5: Fowler DMO panels of the field data set from offshore Africa for different values of q. Each panel is interpolated (painted) using a time-dependent velocity function Vnmo(0).
8.1.
FOWLER-TYPE
TIME-PROCESSING
METHOD
367
800
900
v
.E_ I
(b)
v
o~ 1 (D
.E_ 1 b-
(d)
9
o~ 1 9
V
(D
.E_ 1. p9
(,
(f)
Figure 8.6' Fowler DMO/Stolt migration panels of the field data set for different values of q. Each panel is interpolated (painted) using a time-dependent velocity function Vnmo(0).
368
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
anisotropic DMO-migration sequence yields a much clearer fault-plane reflection than that obtained by the isotropic algorithm [see panel (a) for 77= 0]. The discrepancy between the r/values estimated by just DMO versus DMO plus migration is most likely caused by the influence of the velocity gradient and, possibly, also by a vertical gradient in the parameter r/. Increasing vertical velocity is known to "squeeze" the DMO operator (Hale and Artley, 1993; Lamer, 1993; Tsvankin, 1995a). The r/-scan based on homogeneous Fowler DMO adapts to the gradient by underestimating the true value of r/(see Chapter 7). In contrast, that same vertical velocity gradient tends to "broaden" the migration operator, causing the algorithm to adapt by overestimating 77during the migration step. Clearly, the interplay between the parameter r/and vertical velocity gradient helps Fowler's algorithm to construct satisfactory images of fault planes in vertically heterogeneous media. This suggests that an appropriate use of this method may be to obtain a DMO-corrected stacked section that preserves fault-plane reflections and migrate it by a more sophisticated, V(z) anisotropic algorithm. Ultimately, however, an even better job of imaging and ~ estimation could be done with an approach that accounts for both anisotropy and heterogeneity (see the field-data example at the end of this chapter).
8.1.5
Discussion
The idea of Fowler's DMO method is to combine dipping and horizontal events, stacked at different velocities, on a single section corresponding to the true zerodip NMO velocity. Fowler DMO for isotropic media is implemented by mapping (resampling) in the frequency-wavenumber domain using the cosine-of-dip dependence of stacking velocity. For vertical transverse isotropy, the same goal is accomplished by applying the analytic equation for the NMO velocity of dipping events introduced in Chapter 3. The importance of Fowler DMO as a velocity-analysis tool is much more significant in VTI media, where it offers a practical way of obtaining the key parameters Vnmo(0) and r/in the process of DMO correction. Although VTI implementation requires a two-dimensional scan over Vnmo(0) and 77, the added computational effort is relatively minor because most of the computing time is spent on generating constant-velocity stacks at the first stage of DMO processing. Also, there are two ways to speed up this parameter-estimation procedure. First, the NMO velocity for horizontal reflectors [Vnmo(0)] can be found from conventional semblance analysis (as was done in the field-data example), which leaves only 77undetermined. Second, the constant-velocity stacks, produced at the initial stage of the Fowler DMO process, can be used to perform preliminary estimation of Vnmo(0) and 7/. DMO-corrected constant-velocity stacks are well-suited for poststack migration using Stolt's F-K algorithm. To extend Stolt migration to vertical transverse isotropy, the constant medium velocity in the frequency-resampling procedure has to be replaced with the angle-dependent phase-velocity function for VTI media. The overall
8.2.
D I P M O V E O U T BY F O U R I E R T R A N S F O R M
369
structure of Stolt's isotropic algorithm remains essentially unchanged. Although each DMO-corrected and migrated section is produced for homogeneous media, it is possible to handle vertical and lateral heterogeneity in an approximate way by painting (interpolating) the final image from sections corresponding to different pairs of Vnmo(0) and 7. This approach, however, may lead to inaccuracies in parameter estimation because the best-fit Vnmo(0) and r/absorb the influence of both anisotropy and heterogeneity and take different values at the DMO and migration steps. Therefore, more work is needed to evaluate the potential of Fowler's method as a velocity-analysis tool in heterogeneous, anisotropic media. Both synthetic and field-data examples show that the isotropic DMO-migration sequence cannot handle general (non-elliptical) VTI models. While one can find an isotropic velocity that images one or two dipping events well, it is practically impossible to focus both horizontal and dipping reflectors without taking anisotropy into account. In contrast, the algorithm adapted for VTI media generates a highquality image for a wide range of dips up to 75 ~ Despite some degree of tradeoff between the two VTI processing parameters, the correct pair of Vnmo(0) and produces the best-focused overall image in the synthetic test. On the whole, the Fowler-type DMO method, combined with the velocity-analysis procedure and Stolt poststack migration, represents a complete inversion-processing sequence capable of estimating the effective anisotropic parameters and producing migrated images for the best-fit model. The algorithm can also be used in the vertical symmetry planes of orthorhombic models where the kinematic signatures are described by the same equations as those in VTI media.
8.2
Dip moveout by Fourier transform
DMO by Fourier transform, originally developed by Hale (1984), is a well-established method that has proved its robustness in conventional processing. In contrast to Fowler DMO, Hale's algorithm operates with NMO-corrected data and, if the medium is isotropic and homogeneous, does not require knowledge of velocity. For VTI media, application of Hale DMO should be preceded by a velocity-analysis procedure designed to estimate the parameters Vnmo(0) and 77 (see Chapter 7). Extension of Hale's constant-velocity DMO to anisotropic media, presented here, follows the work of Anderson and Tsvankin (1997). The DMO operator, to be applied to common-offset data corrected for normal moveout, is based on the same exact expression (3.5) for dip-dependent NMO velocity as that in Fowler's method. Concise analytic solution obtained in the weak-anisotropy approximation shows that the anisotropic correction to Hale's DMO operator is quite significant for non-elliptical models; hence, it is important to take anisotropy into account for any DMO method. Impulse responses and synthetic examples demonstrate the accuracy of this DMO technique for typical VTI media and reveal the complications caused by anisotropy in the DMO process.
370
8.2.1
CHAPTER 8. P-WAVE IMAGING FOR VTI MEDIA
Hale's D M O m e t h o d
The DMO algorithm of Hale (1984) is designed to correct common-offset data for the dip-dependent term in the hyperbolic moveout equation. For both isotropic and anisotropic media, the original event time t on conventional-length CMP spreads is close to a hyperbola parameterized by NMO velocity, 4h 2 t 2 -- t02 -~- g 2 m o ( r
,
(8.7)
where h is half the source-receiver offset. The NMO equation for horizontal reflectors, applied to the data prior to DMO processing, results in the corrected time tn given by 4h 2 -- t
--
(88)
Vn2mo (0) "
Combining equations (8.7) and (8.8) leads to the following expression for the zero-offset time in terms of the NMO-corrected time:
[ 1
to2 - t2n + 4h 2 V:mo(0)
1 I
-- Vn2mo(r
"
Equation (8.9), valid for the hyperbolic portion of the moveout curve in both isotropic and anisotropic media, is required for the coordinate transformation used in Hale DMO. In a homogeneous, isotropic medium, equations (8.9) and (8.1) yield +
4h2 sin2 r
'
(8 10)
or, introducing the ray parameter p [equations (8.1) and (8.2)], t~ -- t2n +
4h2p 2 .
(8.11)
Hale's isotropic DMO algorithm corrects for the dip-dependent moveout term in equation (8.11) (4h2p2). Therefore, the process of generating the zero-offset section is split into the NMO and DMO steps, with the DMO operator being completely independent of velocity. A convenient way to estimate the ray parameter is to transform the data into the frequency-wavenumber domain [see equation (8.2]. Hale's (1984) DMO method is based on the following Fourier-transform relationships between the time-space domain and the frequency-wavenumber domain for the zero-offset wavefield P0:
Po(to, yo, h) -
/dw/dk ~
~ Po(w, k, h) e -~(~to-k~o)
Po(w, k, h) - / dt~ f dy~ P~ (t~, y~, h) JT e i(~t~A-ky~) ,
(8.12) (8.13)
where Pn(tn, yn, h) denotes the data after the normal-moveout correction (as before, y is the CMP location), JT is the transformation's Jacobian, and A = to/t~; the
8.2.
D I P M O V E O U T BY F O U R I E R T R A N S F O R M
371
relationship between the zero-offset and common-offset NMO-corrected data is defined as P0(t0, Y0, h) - P~ (t~, yn, h). The dip dependence of NMO velocity is contained in the term A that relates the zero-offset and NMO-corrected time at a given half-offset h. To emphasize the form that is also valid for anisotropic media, equation (8.9) can be used to express A as a function of NMO velocity: A-
t0_
1+
t-:
4h2[ 1
1]
-
Vnmo(0) V:mo(P)
.
(8.14)
For isotropic media, equation (8.11) makes it possible to put A in the form used by Hale (1984), where
A-
~
1+
4h2p2 I t---~--~-
k2h2
1-~ ~ t ~ .
(8.15)
In Hale's (1984) original algorithm, it is assumed that reflection-point dispersal can be ignored, and y0 - Yn. Therefore, the Jacobian JT of the transformation (8.13) is defined as Oto tn = A_1 (8.16) -
=
to
The assumption Y0 - Yn does not influence the kinematic aspects of the DMO process but leads to errors in amplitude preservation (e.g., Bleistein, 1990). Bleistein (1990) and Black et al. (1993) developed "true-amplitude" variants of Hale DMO by replacing the identification y0 - y~ with a more accurate relationship. Both methods modify to [as defined in equation (8.7)] to honor the movement of the surface location associated with a specular reflection point during the transformation to zero offset. The phase factors in equation (8.13) used by these "true-amplitude" algorithms coincide with Hale's phase factor, but both algorithms require a different Jacobian based on the derivative [0(t0, Y0)]/[0(t~,y~)]. In essence, the more complicated Jacobian corrects a relatively small amplitude error for the steeper reflections (see Figure 8.7 below). Bleistein (1990) assumes that (1) the input data have not been corrected for spherical divergence, (2) NMO is done by resampling (mapping) without any scale factors, and (3) the spectral density of the input wavelet is preserved. The corresponding value for JT in Bleistein's algorithm then is
JT = A-~(2A2-
1).
(8.17)
Black et al. (1993) assume that (1) the input data have been corrected for spherical divergence, (2) NMO is done by resampling without any scale factors, and (3) the peak amplitude of the input wavelet is preserved. This leads to the following value for JT: JT = A-3(2A 2 - 1). (8.18)
372
C H A P T E R 8. P-WAVE IMAGING F O R VTI MEDIA
In the presence of anisotropy, the specular reflection point moves differently than in isotropic media, and the Jacobians of Bleistein (1990) and Black et al. (1993) become inaccurate. The method of Anderson and Tsvankin (1997), discussed here, is based on the original Jacobian introduced by Hale (1984) and is not designed to preserve amplitudes during DMO processing. However, as illustrated below, the amplitude distortions caused by the simplistic form of the Jacobian are relatively minor and become somewhat visible only for large dips.
8.2.2
2-D Hale D M O for anisotropic media
Hale's (1984) DMO operator can be extended to anisotropic media using the 2-D NMO equation (8.3). To satisfy the assumptions behind equation (8.3) and make the problem two-dimensional, the vertical incidence plane has to coincide with a vertical symmetry plane of the medium and the dip plane of the reflector. For example, if the medium is transversely isotropic, the incidence plane should either contain the symmetry axis or be orthogonal to it. To make Hale's DMO operator suitable for anisotropic media, the 2-D NMO equation (8.3) for Vnmo(P) has to be substituted into the general expression (8.14) for factor A. It is convenient to rewrite equation (8.14) in the form A -
where
D(p) is the
4h 2 1 + t-T D ( p ) ,
(S.19)
DMO-correction factor given by
D(p)-
1 1 Vn2mo(O) - Vn2mo(P) .
(8.20)
If the medium were isotropic and homogeneous, equation (8.19) would reduce to the conventional DMO formula (8.15). In anisotropic media, however, the velocity Vnmo(P) has a more complicated form determined by the exact equation (8.3). Other than that, the zero-offset section in anisotropic media can be produced by an NMO-DMO sequence similar to the one used in conventional processing. The time tn is a result of the conventional NMO procedure that eliminates the zerodip moveout term 4h2/Vn2mo(O).The influence of anisotropy in the normal-moveout correction is hidden in the value of Vnmo(0). The only possible complication at this stage is anisotropy-induced deviations from hyperbolic moveout, which are small on conventional-length CMP spreads. To transform the NMO-corrected data to zero offset, the anisotropic DMO operator should compensate for the dip-dependent term (4h2/t~)D(p) in equation (8.19). The main difference between D(p) in anisotropic media and its isotropic counterpart p2 [equation (8.15)] is that the former has to be obtained from equation (8.3), which depends on the parameters of the anisotropic velocity field. Therefore, DMO processing should be preceded by an inversion procedure designed to reconstruct the NMO velocity as a function of the ray parameter.
8.2.
D I P M O V E O U T BY F O U R I E R T R A N S F O R M
373
Assuming that Vnmo(P) and D(p) have been obtained, the rest of Hale's DMO algorithm remains essentially unchanged. Factor A and the Jacobian JT = A-1 are computed from equation (8.19) and used in the integral transforms (8.13) and (8.12) designed to produce the zero-offset section from the NMO-corrected, constant-offset sections. If the dependence D(p) is tabulated, the entire anisotropic NMO-DMO sequence takes no more computing time than does Hale's method for isotropic media. The only extra step to be included due to the presence of anisotropy is the parameterestimation procedure. This algorithm ignores anisotropy-induced nonhyperbolic moveout, which seems to be a shortcoming compared to isotropic DMO methods. Although isotropic constantvelocity DMO is also based on the hyperbolic moveout equation, it is supposed to work at large offsets because reflection moveout in a homogeneous isotropic model is purely hyperbolic. Such a model, however, is an idealization of realistic subsurface formations, which are, at a minimum, vertically heterogeneous. Any kind of heterogeneity leads to nonhyperbolic moveout on long spreads that cannot be properly handled by conventional NMO and DMO algorithms. Also, as discussed in the section on Fowler DMO, nonhyperbolic moveout is more of a problem in the NMO correction for horizontal reflectors than in dip-moveout removal. Large offsets, at which deviations from a hyperbola for dipping events may become significant, are usually muted out because of NMO stretch. If there is a need to preserve traces at these uncommonly large offsets, DMO processing should be performed by more accurate (but more costly) ray-tracing techniques, such as the TZO (transformation to zero offset) algorithm of Alkhalifah (1996b).
8.2.3
Application to V T I media
D M O - c o r r e c t i o n factor While the above algorithm can be used for arbitrary "2-D anisotropy" (i.e., in any vertical symmetry plane of anisotropic media), the following discussion is restricted to vertical transverse isotropy, where Vnmo(P) and D(p) are governed by the key timeprocessing parameters Vnmo(0) and 77. As shown in Chapter 7 and in the section on Fowler DMO above, Vnmo(0) and 77can be obtained from dip-dependent P-wave NMO velocity expressed through the ray parameter. Alternatively, it is possible to estimate (albeit with a lower accuracy) from nonhyperbolic P-wave moveout on long spreads. To understand the performance of Hale D MO in VTI media, one has to examine the dependence of the DMO-correction factor D(p)[equation (8.20)] on the parameters Vnmo(0) and r/. For the special case of elliptical anisotropy (r/ = 0), P-wave NMO velocity is the same function (8.1) of the ray parameter and Vnmo(0) as that in isotropic media. The difference between the isotropic and elliptically anisotropic expressions for Vnmo(P) is only in the value of Vnmo(0), which is equal to the horizontal velocity (but is different from the vertical velocity) if the medium is elliptically anisotropic. Therefore, conventional constant-velocity Hale DMO, along with any other isotropic time-processing algorithm, is perfectly suitable for elliptical anisotropy (see
374
CHAPTER 8. P-WAVE IMAGING FOR VTI MEDIA
Chapter 6). The most convenient way to get analytic insight into the behavior of D(p) for nonelliptical VTI media is to apply the weak-anisotropy approximation (1~1 << 1,151 << 1). Using the NMO equation (6.7) linearized in e and 5, D(p) from equation (8.20) can be simplified to D(p) = p2 [1 + 2(~ - 5)(492 - 9g + 6)], (8.21) where g - p2 Vn2mo(0); recall that in the linearized approximation e - 5 = r]. Although it might be dangerous to use weak-anisotropy solutions for quantitative estimates, it is clear from equation (8.21) that transverse isotropy makes a significant contribution to the DMO factor. For instance, for g = 0.25, which roughly corresponds to a dip of 30 ~ the anisotropic term 2(~ - 5)(4g 2 - 9g + 6) reduces to 8 ( c - 5), which means that the isotropic and anisotropic parts of the DMO factor are comparable even for relatively small ( c - 5). This conclusion is in good agreement with the analysis of the function Vnmo(P) in Chapter 6. It is also important that the influence of the anisotropy cannot be ignored even at mild dips (small g); in fact, the relative magnitude of the anisotropic term decreases with dip. Of course, the contribution of the dip term as a whole at mild dips is relatively small; also, the weak-anisotropy approximation becomes inaccurate for large dips. Although the anisotropic DMO operator does depend on the zero-dip NMO velocity through g -- p2 V : m o ( 0 ) , the influence of errors in Vnmo(0) on D(p) becomes pronounced only for large p, corresponding to steep events. Numerical
implementation
The implementation of Hale's DMO technique for VTI media discussed here is based on the work of Anderson and Tsvankin (1997). First, as in Fowler's algorithm, the function Vnmo(P) and the DMO-correction factor D(p) are tabulated in constant increments in p. The algorithm uses the exact equations for NMO and phase velocity in VTI media, so there are no restrictions requiring anisotropy to be weak. The input parameters to be determined beforehand (see Chapter 7) are the NMO velocity for a horizontal reflector Vnmo(0) and the coefficient r/. D(p) is then used to compute factor A needed in Hale's transformations. After the conventional NMO correction, data on common-offset gathers are Fouriertransformed over the midpoint axis, providing complex-valued traces as a function of the wavenumber. The time integral over tn is computed numerically, yielding the zero-offset section in the F-K (w-k) domain [equation (8.13)]: P0((,d,
k, h) - f Pn(tnk' h) eiwtnAd~n,
(8.22)
where p = k/2w, and A(tn, k,w) is determined from equation (8.19). The direct numerical integration over tn in equation (8.22) is the most time-consuming part of the algorithm. Finally, a two-dimensional inverse Fourier transform produces the output DMO-corrected, common-offset gather in the t-x domain.
8.2.
D I P M O V E O U T BY F O U R I E R T R A N S F O R M
8.2.4
375
Synthetic examples
VTI DMO impulse response Figures 8.7-8.12 (after Anderson and Tsvankin, 1997) demonstrate basic properties of the Hale-type DMO operator devised for vertical transverse isotropy and illustrate anisotropy-induced distortions in the DMO impulse response. If the medium is isotropic, the VTI DMO algorithm produces results identical to those of conventional Hale DMO (Figure 8.7a,b). Bleistein's (1990) "true-amplitude" impulse response in Figure 8.7c has the same shape as that of the standard Hale's response but the amplitudes on the steeper dips are slightly higher. Overall, the difference between these two responses is relatively minor. In agreement with the analytic results discussed above, the VTI operator reduces to the conventional Hale's operator not only for isotropic but also for elliptically anisotropic media (E = ~, Figure 8.8). Although velocity anisotropy for the model in Figure 8.8 is quite pronounced (e = 5 = 0.2), it has no influence on the kinematics of the DMO process. The only difference between the isotropic and elliptically anisotropic DMO operators is that the latter is computed using the values of Vnmo(P) and D(p) from table lookup. However, even relatively small deviations from elliptical anisotropy have strong influence on the DMO operator. Figure 8.9 demonstrates the broadening of the DMO ellipse that occurs for a VTI model with a typical moderate positive r]. Note that the accuracy of the weak-anisotropy approximation (8.21) in generating the DMO operator in Figure 8.9b is quite satisfactory. Still, it is preferable to use the exact VTI operator in routine processing applications because the weak-anisotropy approximation becomes less accurate for larger values of 77. The broadening of the DMO impulse response becomes more pronounced with increasing r/(Figure 8.10). It is noteworthy that for models with significant anellipticity (i.e., large It/[), the impulse response is not necessarily limited to the source-receiver interval and may have triplications. In contrast, for small negative values of 71, the DMO response becomes more narrow (Figure 8.11). With decreasing r / < 0, the DMO response curves downward with a shape reminiscent of that typical for the isotropic inverse DMO operator (Figure 8.12). Clearly, the familiar DMO ellipse cannot describe the impulse response for VTI models with either positive or negative r/.
Comparison of isotropic and VTI DMO corrections The next example shows the performance of the VTI DMO algorithm in a homogeneous VTI model with four reflectors and dips ranging from 0~ to 75 ~ (Figure 8.13). Synthetic common-offset data were computed by anisotropic ray tracing for offsets from 0 to 3 km in increments of 0.1 km. The zero-offset section that may be considered as an ideal output for the sequence of NMO and DMO corrections is shown in Figure 8.14. The vertical line at midpoint location 2.75 km marks the CMP gather selected for the comparison of isotropic and
376
C H A P T E R 8.
P-WAVE IMAGING F O R VTI MEDIA
b
C
0.20-
Figure 8.7: DMO impulse responses on a common-offset gather for a homogeneous isotropic medium. Here, and in Figures 8.8-8.12, the offset equals 400 m and the trace spacing is 25 m. (a) Conventional Hale operator; (b) the VTI operator (the VTI response is identical to the standard Hale response for isotropic models); (c) Bleistein's (1990) operator (Bleistein's "true-amplitude" DMO response has the same kinematics but somewhat higher amplitudes for larger dips).
8.2.
D I P M O V E O U T BY F O U R I E R T R A N S F O R M
377
Figure 8.8: VTI DMO impulse responses for an elliptically anisotropic medium with
c=a=0.2 (n=0). VTI DMO in Figures 8.15 and 8.16. Figure 8.14 helps to identify the key model reflections on CMP gathers with the same events on the zero-offset section. The NMO-corrected common-offset synthetic data were processed first by the conventional isotropic DMO method of Hale (Figure 8.15), and then by the VTI DMO algorithm (Figure 8.16). Clearly, VTI DMO aligns the events out to a fairly large offset, irrespective of dip, allowing CMP stack to sum reflections without distortion. In contrast, after application of isotropic DMO, all dipping events in Figure 8.15 are misaligned because their moveouts remain undercorrected. It should be emphasized that the synthetic data were generated by a ray-tracing algorithm that is not related to the analytic formula used in the anisotropic DMO code. Figure 8.16 shows that the horizontal events still have significant nonhyperbolic moveout and will require either nonhyperbolic moveout correction or muting on the longer offsets prior to the stack. This example, however, exaggerates the influence of nonhyperbolic moveout because standard processing rarely includes offsets up to 3 km at times less than 1.5 s. For example, the maximum offset for the reflection from the second horizontal interface (depth = 1.5 km) reaches twice the reflector depth. The dipping events also exhibit nonhyperbolic moveout, but, in agreement with the conclusions of Chapter 3, to a far lesser degree than that for the horizontal events. As a result, the VTI DMO operator, based on the hyperbolic moveout equation, still provides a sufficiently accurate description of moveout from dipping reflectors on CMP spreads twice as large as the reflector depth (Figure 8.16). If it is necessary to preserve even larger offsets in the stack, an NMO/DMO sequence based on anisotropic ray tracing may be necessary to correct for nonhyperbolic moveout.
378
CHAPTER
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P-WAVE
IMAGING
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VTI
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'
i i
Figure 8.9: VTI DMO impulse responses for a medium with e - 0.2 and 5 - 0.1 ( r / - 0.083). (a) The response computed from the exact NMO equation (8.20); (b) the response computed from the weak-anisotropy approximation (8.21).
8.2.
DIP
MOVEOUT
BY FOURIER
TRANSFORM
379
I I q t
b~ t
t
i
I
i
E
Figure 8.10" VTI DMO impulse responses for e -
0.4 and 5 -
0.0 (q=0.4).
0.4
I q I
i I
!
i I
p
I
j
!I
E !
i '
I
I
0.20i
'
I
l
[
i
I
9
III1~111
I
III
Figure 8.11" VTI DMO impulse responses for e - 0.0 and ~ - 0.1 ( r / - - 0 . 0 8 ) .
380
C H A P T E R 8. P-WAVE IMAGING F O R VTI MEDIA
-0.,
0
0.2
0.4
(
(
Figure 8.12: VTI DMO impulse responses for e - 0.0 and 5 - 0.2 (q - -0.14).
30
E v
c"
f~ CI
CD
3t Figure 8.13: The model used for generating synthetic data. The reflectors are embedded in a homogeneous VTI medium with the following parameters: Vpo=3 km/s, e=0.2, and 5=0.05 (V~mo(0) =3.15 km/s, q=0.14).
8.2.
DIP M O V E O U T BY F O U R I E R T R A N S F O R M
381
if} V
9
Figure 8.14" Zero-offset synthetic data generated by anisotropic ray tracing. The vertical line at 2.75 km marks the CMP used for comparing the isotropic and VTI DMO corrections in Figures 8.15 and 8.16.
382
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
Figure 8.15: CMP gather after application of NMO and isotropic DMO. "30 deg," "45 deg," etc. are the reflector dips corresponding to the recorded events.
8.2.
D I P M O V E O U T BY F O U R I E R T R A N S F O R M
1
383
2
i
Figure 8.16: CMP gather after application of NMO and VTI DMO.
384
8.2.5
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
Discussion
Possible extensions of t h e a l g o r i t h m With use of table lookup, the VTI DMO algorithm is no slower than the generic Hale's method in isotropic media. However, direct numerical integration over the NMO-corrected time makes Hale's DMO operation relatively time-consuming, even for isotropic media. The conventional way to speed up Hale's isotropic DMO algorithm is to use the log-stretch method (Notfors and Godfrey, 1987; Liner, 1990). Generalization of this method for anisotropic media, however, is not straightforward because only the kinematics of the small-offset, near-elliptical portion of the DMO impulse response can be preserved. The variations of amplitude and triplications seen along the VTI D MO response cannot be adequately approximated by straightforward stretching and squeezing in the log-stretch domain. Still, it would be worthwhile to explore the possibility of extending the log-stretch method to VTI media and estimating the magnitude of the associated errors. The algorithm can be upgraded to perform V(z) DMO by replacing the singlelayer NMO formula with the more general Dix-type equation (3.47) for vertically heterogeneous VTI media. Alternatively, since the "homogeneous" VTI DMO operator has the added degree of freedom provided by the parameter ~, in many cases it can be adapted for V(z) media by applying an effective value of r/that absorbs the influence of both the anisotropy and heterogeneity. The anisotropic D MO operator introduced here can be used in symmetry planes of more complicated media such as TI with a tilted axis of symmetry or orthorhombic. Application of Hale DMO to those models, however, has to be preceded by a velocityanalysis procedure designed to reconstruct the dip-dependent NMO velocity from seismic data. Summary The 2-D analytic expression for normal-moveout velocity developed in Chapter 3 makes extension of Hale (1984) DMO to symmetry planes of anisotropic media relatively straighforward. The NMO-DMO processing sequence (including Hale's integral transforms) remains essentially unchanged in the presence of anisotropy, but the transformation's Jacobian has been modified to account for the anisotropic behavior of the NMO velocity of dipping events. The anisotropic correction to the DMO operator for VTI media is determined by the zero-dip NMO velocity Vnmo(0) and the anisotropic coefficient r / - the timeprocessing parameters that can be estimated from P-wave reflection data (see Chapter 7). After the inversion step has been completed, the NMO velocity and Hale's Jacobian are computed as a function of the ray parameter. Then the transformation of NMO-corrected constant-offset sections to zero offset is carried out using the conventional isotropic formalism. Tests on synthetic data show that the VTI DMO operator efficiently flattens Pwave moveout on conventional-length spreads for any dip, while application of Hale's
8.3.
TIME AND DEPTH MIGRATION
385
isotropic algorithm to non-elliptical media leaves dipping events severely misaligned. Two interesting points are brought out by the examples. First, the DMO impulse response in strongly non-elliptical media is not restricted to the region between the source and the receiver and may exhibit triplications. Second, reflections from steep interfaces are less influenced by nonhyperbolic moveout and appear more flat after DMO than are those from horizontal reflectors; this conclusion is in good agreement with the results of Chapter 3. Therefore, nonhyperbolic moveout (ignored in the algorithm discussed here) is not expected to impede the performance of the anisotropic DMO correction.
8.3
Time and depth migration
Migration errors produced by conventional isotropic imaging algorithms in the presence of anisotropy are well documented in the literature (e.g., Larner and Cohen, 1993; Alkhalifah and Larner, 1994) and illustrated here by a number of examples in Chapters 6-8. The most common anisotropy-induced distortions in migration include an inaccurate depth scale of the image, mispositioning of dipping features, and (in prestack migration) poor focusing of dipping and, sometimes, even horizontal reflections. Generalization of isotropic migration algorithms for VTI media is accomplished by using either the appropriate phase-velocity equations or an anisotropic ray tracer in the migration operator. Stolt poststack time migration based on P-wave velocity equations for VTI media was introduced in section 8.1 as part of the Fowler-type time-processing sequence. Here, we use the work of Alkhalifah (1995, 1996a) to describe extensions of phase-shift (Gazdag) time migration and Gaussian beam depth migration to VTI media. Phase-shift migration techniques for vertical transverse isotropy were also developed by Kitchenside (1991) and Meadows and Abriel (1994). VTI extensions of Kirchhoff migration, not discussed here, can be found in Sena and Toks5z (1993) and Ball (1995). For other types of anisotropic imaging algorithms, see papers by Uzcategui (1995) and Uzcategui and Mujica (1995) on migration using explicit depth extrapolation and by Phadke et al. (1994) on 45 ~ wave-equation migration.
8.3.1
Phase-shift (Gazdag) migration
Fourier-domain implementations of migration are restricted to laterally homogeneous media, excluding some approximations designed to handle smooth lateral velocity variation. This inherent limitation of time imaging, however, allows migration codes in the Fourier domain to be comparatively efficient. In addition, the assumption of lateral homogeneity simplifies the velocity-analysis problem for VTI media by allowing for the two-parameter representation of P-wave time-domain signatures (see Chapter 6).
386
CHAPTER
8.
P-WAVE
IMAGING
FOR VTI MEDIA
Gazdag (1978) suggested a phase-shift method for downward extrapolation of seismic wavefield in the frequency-wavenumber (w-k) domain. The wavefield (W) transformation in Gazdag migration is described by
W(w, k, z) - W(w, k, z - O)A(p,z)e-i~fo [c~162162
,
(8.23)
where V(z) is the depth-dependent velocity, 0 is the plane-wave propagation (phase) angle at depth z, and p = k/(2w) is the ray parameter of the zero-offset ray. For isotropic media, 0 is expressed through p as sin O(p, z) = V(z) p .
A(p, z) is an amplitude factor designed to correct for the influence of vertical heterogeneity. It is often omitted in practice, in part because it goes to infinity as the depth z approaches the turning point where 8(p, z) = +90 ~ Equation (8.23) is the WKBJ solution (e.g., Aki and Richards, 1980, p. 416) of the wave equation expressed in the frequency-wavenumber domain (Helmholtz equation), d2W [ w2 ---5dz + V 2(z)
1 k 2 W - O.
For migration of the zero-offset seismic section f(t, x), W(w, k, z = 0) is assumed to represent the Fourier-transformed data F(w, k) at the surface (z = 0). Inverse Fourier transformation of equation (8.23) from wavenumber k to distance x yields
W(w,x,z) -
If
A(p,z) e-i~fo~[c~176162
f(w,k)dk.
(8.24)
Then, evaluation of the inverse Fourier transformation from frequency w to time t at t = 0 (the imaging condition) results in the subsurface image,
g(x,z)
= _
1
27r ./ e-i"~t W(w'x'z)lt=o dw 1 ]
JfdwfA(P'z)e-i~f~176176
4~2
(8.25)
Equation (8.25) is a concise mathematical representation of the phase-shift migration method. Output data after time migration are usually expressed as a function of the twoway vertical time 7-9 Substituting 7- - f~ d t - 2 f~ dA and d t - 2 dA into equaV V tions (8.24) and (8.25) produces the wavefield in the time domain:
l/
W(w, x, T) - ~
g(X,T)--~
~-
.4(p, T) e-i~ fo cos~(p,t)dt+ ikx F(w, k) dk,
1/ dw / A(p,v)e-'~f[c~
(8.26)
(8.27)
8.3. TIME AND DEPTH MIGRATION
387
where 0(~-) - O(z) and A(~-) - A(z); z - 0.5 f~ V(T)dT. The above formalism remains valid for VTI media, but the phase velocity V becomes a function not only of depth, but also of the phase angle 0: sin O(p, ~-) - V (0, 7-)p. Also,
dt=2~ Therefore, equations (8.26) and (8.27) have to be replaced by 1 W(w, x, ~-) - ~ f 2~(p, 7") e - ~ fo [cosO(v,t)Vpo(t)/v(O,t)]dt + ~k~F(w, k) dk ,
(8.28)
and 1
g(x, ~-) - -~2 / dw f f~(p, v-) e - i ~
for [cos O(p,t)Vpo(t)/V(O,t)]dt + ikx f@d, k)
dk .
(8.29)
Despite being generally slower than Stolt (1978) migration (discussed above), the method based on equations (8.28) and (8.29) can handle (in the ray-theory approximation) the kinematics in vertically heterogeneous VTI media. It should be emphasized that in Gazdag migration it is possible to account for arbitrary vertical variation in all parameters without assuming a factorized anisotropic model. Figure 8.17 shows Gazdag migration for a vertically heterogeneous VTI model with a wide range of dips. All dipping reflectors in Figure 8.17 are well-focused, whereas Stolt migration for the same model produces poor images of the two steepest reflectors due to the approximate treatment of vertical heterogeneity (Alkhalifah, 1996a). The cost of the Gazdag migration, however, is about 5 to 6 times that of the Stolt migration, with both algorithms being practically as efficient as their isotropic counterparts. For other synthetic and field-data tests of Gazdag's VTI algorithm, see Chapter 6 (Figure 6.10), Chapter 7 (Figure 7.17), and section 8.5 below (Figure 8.37). An important question is whether or not dipping events (e.g., fault-plane reflections) can be used to evaluate the performance of migration and estimate migration velocities. If the zero-dip NMO velocity in VTI media has been obtained from semblance analysis of horizontal events, 77 remains the only parameter responsible for imaging the fault at its anticipated position. As shown in the discussion of Fowler DMO above (section 8.1), it is indeed possible to pick the best-fit 7/for homogeneous VTI media using dipping reflectors on time sections obtained by Fowler DMO and Stolt poststack migration. Poststack migration alone is not as sensitive to 7] as is the DMO-migration sequence and cannot be effective in anisotropic velocity analysis for reflector dips smaller than 45-50 ~ (Alkhalifah and Larner, 1994). For larger dips, a scan over migrated images obtained for different r] values can help to estimate the optimum r] for migration (see the example in Figure 6.11). If ~ varies with depth, this approach yields an effective value of r] that absorbs the influence of both the anisotropy and heterogeneity. Of course, it is possible to estimate
388
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
q) v
E b-
F i g u r e 8.17: G a z d a g m i g r a t i o n o f a z e r o - o f f s e t s e c t i o n g e n e r a t e d for a v e r t i c a l l y h e t erogeneous VTI model with
Vpo(z)-l.5+O.6zk m / s ,
e -
T h e r e f l e c t o r s a r e d i p p i n g a t 30 ~ , 45 ~ , 60 ~ , 75 ~ a n d 90 ~ .
0.1 a n d ~ -
0 (q -
0.1).
8.3.
T I M E AND D E P T H M I G R A T I O N
389
interval ~ as a function of vertical time using the DMO-inversion method described in Chapter 7 (section 7.1). In principle, interval r/values in migration velocity analysis can also be obtained from the layer-stripping relationships for P-wave nonhyperbolic moveout of horizontal events (see section 7.2) by taking advantage of the inherent similarity between the time-migration diffraction curve and reflection moveout from horizontal interfaces (Alkhalifah, 1996a).
8.3.2
Gaussian beam migration
Time-migration algorithms described above are devised for laterally homogeneous media and have to be replaced by depth-imaging methods in areas of complex structure and/or substantial lateral velocity gradients. For VTI media, P-wave depth migration requires knowledge of the spatial distribution of the vertical velocity Vpo and Thomsen parameters e and 5. Estimation of these parameters for laterally heterogeneous VTI models is a highly challenging problem currently being investigated using tomographic-style methods (Grechka et al., 2000a,b). For media with moderate structural complexity, it may be possible to estimate a laterally varying VTI velocity field by interpolating and extrapolating 1-D inversion results obtained for different portions of the model. This approach is applied in a synthetic test of VTI processing algorithms below (see section 8.4). However, 1-D inversion methods operating with P-wave moveout yield only the two time-processing parameters (Vnmo(0) and ~), and require additional information for building a model for depth migration (see Chapter 7). Alternatively, in some cases anisotropic parameter estimation can be performed just for the laterally homogeneous part of the model. Indeed, the shallow subsurface often consists of sediments with mild lateral velocity variations, including anisotropic shale layers. The structure below the sediments may be much more complex (e.g., it may include salt domes and carbonate formations), but only weakly anisotropic. Reflections from fault planes in the shallow part of the model and from dipping structures at depth, combined with such additional information as check shots or PS data, can be used to estimate the anisotropic parameters of the overburden. Then this anisotropic velocity model can be combined with the isotropic velocities found for the bottom (complex) part of the subsurface to perform depth migration. Basics of the G B M m e t h o d in anisotropic m e d i a
One of the depth-imaging methods that has proved its efficiency and versatility for isotropic media is Gaussian beam migration (GBM). Among the attractive features of GBM are robustness with respect to ray caustics and shadows, the ability to image dips exceeding 90 ~ with turning waves, straightforward extensions for migration of prestack and 3-D data, and acceptable computational speed (Hill, 1990). Gaussian beam migration is considered here because it is particularly well-suited for anisotropic media. First, since ray tracing amounts to a small portion of the total computation in GBM, the additional cost incurred by accounting for anisotropy is
390
C H A P T E R 8. P-WAVE IMAGING F O R VTI M E D I A
typically only about 40% or less (Alkhalifah, 1995). Second, GBM is capable of processing data from arbitrary anisotropic and heterogeneous media with any reflector orientation. Third, the method properly handles caustics associated with anisotropy, such as SV-wave triplications (cusps) in strongly non-elliptical VTI media (see Chapter 1). Fourth, GBM is not limited to migration of the first P-wave arrivals. Description of Gaussian beam ray tracing and migration can be found in (~erven:~ and Pgen~{k (1984), Hanyga (1986), Hill (1990) and Hale (1992). Here, we follow the work of Alkhalifah (1995, 1996a) on 2-D anisotropic poststack Gaussian beam migration in a vertical symmetry plane of the medium. Extension of the method to 3-D is straightforward since the general anisotropic ray-tracing equations of (~erven:~ (1972) and Hanyga (1986) are not limited to two dimensions. As described by Hale (1992), Gaussian beam migration can be conveniently performed by subdividing a common-midpoint stack section into overlapping subsets of stacked traces. The stacked traces within each subset are multiplied by a centered Gaussian taper function over midpoint. These windowed data are then slant-stacked to form beams in a range of directions characterized by different ray parameters. For a single beam, the subsurface image h(x, z) at a point defined by the lateral position x and depth z is calculated from the following equation (Hale, 1992): (8.30) where p, as before, is the ray parameter, D is a local slant stack of windowed surface seismic data, and bH is its Hilbert transform. AR and and imaginary parts of the complex-valued amplitude, and 7R(X, z,p) are the real and imaginary parts of the complex-valued time, both G aussian beam ray tracing. The slant stack D is computed from
1)( 7-R, ~-I) -- - 1~
/ e_iw~.R+lWlT I B (w , p) dw ,
the GaussianA1 are the real and TI(X, z, p) calculated by
(8.31)
where B(w, p) is the local slant stack of the surface seismic data in the frequency (w) domain, given by
p)-
I
k).
271"
F(w, k) is the 2-D Fourier transform of the seismic data f ( x , t) recorded at the surface z=0. Hale (1992) and Alkhalifah (1995) use the following loop to implement Gaussian beam migration (the subscript j denotes a beam):
8.3. TIME AND DEPTH MIGRATION
391
G a u s s i a n b e a m migration:
for all points (x, z) {
}
h(x, z) = 0
for all xj = j A (all beam center locations) { compute f y(t, x) by shifting and tapering data f (t, x) for all p (all reflection slopes) { compute filtered slant stack bj(7, p) of f j(t, x) for all points (x, z) within beam { compute complex-valued 7-y(p, x, z) and Ay(p, x, z) accumulate contribution to h(x,z) from bj(T,p)
}
Here, 7-(p,x, z) and A(p,x, z) are the complex-valued time and amplitude, given by ~- =
TR + i sgn(w)~-I
A = AR+isgn(w) Ai, and A is the spacing between the beam center locations xj. Application to anisotropic media requires calculating the complex-valued time and amplitude using anisotropic ray tracing. The needed equations, based on the results of (~erven:~ (1972) and Hanyga (1986), can be found in Alkhalifah (1995, 1996a). The anisotropic GBM can be extended to three dimensions by using the 3-D version of dynamic and kinematic ray tracing, combined with 3-D application of beam expansions. An important issue in Gaussian beam modeling and migration is a proper choice of the initial beam width 1. The beams should be as narrow as possible because the method becomes inaccurate if the velocity field changes rapidly over the beam width. However, if the value of 1 is too small, the beam quickly increases in width along the raypath. Hill (1990) used the following expression to compute 1 for isotropic models: l-
V~vg
(8.32)
fmin
where fmin is the minimum frequency in the data and Vavg is the average velocity. Alkhalifah (1995) shows that equation (8.32), with Vavgcomputed as the average of the horizontal and vertical velocities over the entire grid, is applicable to transversely isotropic media as well. Velocity variations with angle, however, change the angular distribution of rays and, therefore, the beam width for a given ray (group-velocity)
392
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
direction. This "focusing" phenomenon and its implications for the amplitude variation along the wavefront in anisotropic media were discussed in detail in Chapters 1 (e.g., see Figure 1.11) and 2. The beam width is smallest in propagation directions with the highest wavefront curvature and the largest concentration of rays. In contrast, for wavefront segments with less curvature and lower density of rays the beam is broader. Hence, although the number of rays used to calculate the wavefield in GBM may be the same for both isotropic and anisotropic media, the output of the anisotropic algorithm may be influenced by the different properties of the Gaussian beams. Anisotropic ray tracing involves much more complicated and computationally expensive equations (e.g., the Christoffel equation) than does the conventional ray tracing in isotropic media. As a result, P-wave ray tracing in heterogeneous media, required to implement Gaussian beam migration, is a much slower process in the presence of anisotropy. However, isotropic ray tracing amounts to only about 2% of the total cost of GBM for heterogeneous models. Therefore, the substantially increased computing time for ray tracing in anisotropic media raises the cost of the full GBM process by a relatively modest amount - generally by 40% or less (by only 10% for TI media). Although GBM is generally more expensive than Kirchhoff migration, it has an advantage in terms of extension to anisotropic media. For Kirchhoff migration, ray tracing consumes the major portion of the processing time, so this method must bear the full additional cost of accounting for anisotropy. Also, in contrast to Kirchhoff migration, GBM can handle caustics and is not limited to migration of the first Pwave arrivals. The computer-memory requirements of anisotropic GBM are similar to those for isotropic media. The main difference is the additional number of input velocity files required to specify the parameters of a given anisotropic model.
Synthetic e x a m p l e s Figure 8.18 shows the response of GBM in a homogeneous VTI medium to impulses at three different times. The model has the anisotropy parameters of the weatheredgypsum TI medium of Thomsen (1986) including the uncommonly large ~ > 1. The wavefronts, as well as the amplitude distribution along them, are distinctly different from their isotropic counterparts (for more details, see Chapter 1, section 1.2). In particular, the SV-wavefront exhibits triplications (caustics), which are handled without difficulty by the Gaussian beam migration. Also, note the higher amplitudes on the P-wavefront near the velocity maximum at vertical incidence. The performance of anisotropic G BM was tested using the structural model shown in Figure 8.19. Figure 8.20 displays synthetic zero-offset sections for this structure embedded in an isotropic medium (a) and a factorized VTI medium (b), both with depth-dependent vertical velocity (see the discussion of factorized media in Chapter 3, section 3.2). Due to the substantial magnitude of this shift, isotropic migration
8.3.
TIME AND DEPTH MIGRATION
393
Figure 8.18: Impulse responses of Gaussian beam migration at times 0.8 s, 1.6 s and 2.3 s for P-waves (a) and SV-waves (b) in a strongly anisotropic homogeneous VTI medium. The parameters are Vpo = 2 km/s, Vso = 1 km/s, ~ = 1.1 and 5 = -0.14.
C H A P T E R 8. P-WAVE IMAGING F O R VTI M E D I A
394
E j=2-
_
Midpoint (km) Figure 8.19: Structural model used in the migration tests below. vertical reflector is at midpoint position 1.2 km.
Note that the
applied to data from VTI media is expected to leave steeply dipping features mispositioned. This expectation is confirmed by Figure 8.21a, which shows the anisotropic data from Figure 8.20b migrated using the standard isotropic GBM algorithm. The velocity chosen for the isotropic migration is the best-fit stacking velocity for horizontal reflectors obtained from hyperbolic semblance analysis. The image in Figure 8.21a is clearly undermigrated; in addition, the structure is placed at too great a depth because the stacking velocity for positive 5 exceeds the root-mean-square (rms) vertical velocity. The section produced by anisotropic GBM using the correct velocity model (Figure 8.21b) is free of those distortions. Note that the difference in computing time between the GBM algorithm devised specifically for factorized VTI media and its isotropic version is less than 10%. Since Gaussian beam migration is based on the high-frequency approximation (ray tracing), its application in layered media requires some degree of smoothing of the input velocity field. For anisotropic media, smoothing must be applied to the spatial distribution of all relevant medium parameters. Figure 8.22a shows a 2-D model composed of four homogeneous transversely isotropic layers, with the vertical velocity increasing from top to bottom. The model was smoothed by convolving the velocity field with a 2-D Gaussian-shaped, pointspread function with a width (half-amplitude point) of 0.1 km in all directions (Figure 8.22b). Although the smoothing is shown just for the vertical velocity, all needed velocity fields (corresponding to the TI elastic coefficients) are smoothed in the same way.
8.3.
TIME AND DEPTH MIGRATION
395
Figure 8.20: Synthetic zero-offset seismograms for the structure in Figure 8.19 embedded in (a) an isotropic medium, and (b) a factorized VTI medium with e - 0.134 and ~ = 0. The vertical velocity for both models is given by V p o ( Z ) - 3000 + 0.5 z m/s. The sections were generated using a Kirchhoff modeling program that can handle factorized TI media. The dot in the bottom right corner helps to estimate the anisotropy-induced shift of the reflections from the vertical interface.
396
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
f:l.
f:l.
cl
D 3
4
(a)
Figure 8.21: Migration of the anisotropic zero-offset data shown in Figure 8.20b using (a) isotropic GBM; (b) anisotropic GBM with the correct model parameters. The importance of the velocity smoothing is evident from the GBM impulse responses shown in Figure 8.23. In areas of large discontinuities in the original model (Figure 8.22a), the impulse responses in Figure 8.23a almost disappear, a striking shortcoming of the high-frequency approximation. Such an extreme degradation of the impulse response, seldom observed for isotropic media, is likely caused by the discontinuities in all parameters of TI media. This problem could be avoided be applying the necessary boundary adjustment in ray propagation using Snell's law or an anisotropic perturbation technique (Jech and Pgen~fk, 1989). Such modifications, however, reduce the efficiency of the migration process, which makes this approach less attractive, particularly considering the substantial uncertainty in the subsurface velocity model. In contrast, the impulse responses for the smoothed model in Figure 8.23b remain strong and continuous throughout the medium. Note that the tilt of the symmetry axis in the second layer leads to the rotation of the two deeper impulse responses in Figure 8.23b. Application of anisotropic GBM to layered TI media is illustrated by Figure 8.24. Figure 8.24a shows a zero-offset synthetic section for the model in Figure 8.22a, except that the symmetry axis in the second layer is vertical. Then the synthetic data were migrated by the anisotropic GBM algorithm using the the smoothing of the velocity fields discussed above (see Figure 8.22b). The resulting image in Figure 8.24b is well-focused and closely matches the correct depth model.
8.3.
TIME AND DEPTH MIGRATION
397
a
8=0:0 e=0.3 .
~=0..1-4 ~0.5.. 5=.02 a=0.2 4
0
I
0
b E
v2|
-o
9
(-.
c3
{D
D
Figure 8.22: (a) Four-layer transversely isotropic model used for tests of Gaussian beam migration in Figures 8.23 and 8.24. The symmetry axis is vertical in all layers except the second one, where it is tilted by 30~ from the vertical (see the arrow). From top layer to bottom, the vertical velocity (indicated by the shading level) is 1.0 km/s, 1.5 km/s, 2.5 km/s, and 3.0 km/s; the anisotropic coefficients are marked on the plot. (b) Smoothed version of the above model.
398
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
E v
r f:l. {D
s 3
4 Figure 8.23: GBM impulse responses at CMP location 2.5 km for (a) the original model in Figure 8.22a, and (b) the smoothed model in Figure 8.22b.
8.3.
T I M E AND D E P T H M I G R A T I O N
399
Figure 8.24: (a) Synthetic seismogram generated by anisotropic finite-differences for the model shown in Figure 8.22a, but with a vertical symmetry axis in the second layer. (b) Anisotropic GBM migration of section (a) using smoothed velocity fields.
400
CHAPTER 8. P-WAVE IMAGING FOR VTI MEDIA
8.4
S y n t h e t i c e x a m p l e for a m o d e l from the Gulf of M e x i c o
Here, VTI processing algorithms are applied to a 2-D synthetic data set generated by an anisotropic finite-difference code (this section is based on the paper by Han et al., 2000). The model was built by J. Leveille and F. Qin of Amerada Hess Corp., who considered it to be representative of several exploration areas in the Gulf of Mexico. Although the overall structural complexity of the model is moderate, it includes a salt body surrounded by sedimentary layers and a relatively steep fault plane (Figure 8.25). The synthetic data were computed by F. Qin using the parameters listed in Table 8.1. The anisotropic parameters can be considered as "best-guess" values not based on any particular case study. Judging by the magnitude of the coefficients e and 6, some of the intervals may be considered as moderately or even strongly anisotropic, with the parameter e approaching 0.3 in a thin layer at a depth of about 4.5 km (see Figure 8.25b). The average value of 6 is about 0.1, which is close to the Dix-inversion results for the Gulf of Mexico data discussed in Chapter 3 (Figures 3.28-3.30). The model in Figure 8.25, however, does not contain the strongly anisotropic shale layers at depth that produced the large 6 values in Figures 3.28-3.30. Also, the key time-processing parameter r/ is relatively small throughout the model, with a maximum value of 0.09 and average close to 0.05 (Figure 8.25d). Whereas such 7/ values are not expected to cause serious problems in the focusing of reflection events, it is still instructive to evaluate possible image distortions and the performance of isotropic algorithms for such "quasi-elliptical" VTI media, which have moderate values of 6.
8.4.1
Parameter estimation
The first step of the processing sequence is estimation of the parameter r/ using dipping events reflected from the right flank of the salt body and from the fault plane. The inversion procedure, based on the 2-D Dix-type equation (3.47), is described in Chapter 7 (section 7.1). The main steps in reconstructing the time-dependent interval function 7]int(T ) w e r e as follows: 1. Common-shot gathers were resorted into common-midpoint gathers. The inversion for r/was performed using the CMP gathers that contain reflections from the top of the salt body (CMP locations from 4.9 to 6.7 km) and from the fault plane (locations from 11.0 to 16.8 km). Thus, only about 30% of the data (Table 8.1) were actually included in estimating 7/. 2. Conventional semblance analysis was applied to find the NMO velocity Vnmo(0, r) of subhorizontal events and the triplets {Vnmo,to, p} (see section 7.1) for the reflections from the salt body and the fault plane. The ray parameters (horizontal slownesses) p for dipping events were estimated using the lateral time shifts of the corresponding semblance velocity maxima.
8.4.
S Y N T H E T I C E X A M P L E F O R A M O D E L F R O M T H E G U L F OF M E X I C O
401
3500
Ypo
Figure 8.25" Depth section showing parameters of the Amerada VTI model" the P-wave vertical velocity Vpo (a), and the anisotropic coefficients 5 (b), E (c), and r/
(d).
402
C H A P T E R 8.
Model size: Number of shots: Shot spacing: Number of receivers per shot: Receiver spacing: Dominant frequency: Recording time: Sampling interval: Aperture for modeling:
P-WAVE IMAGING FOR VTI MEDIA
21,945 m x 9,144 in 361 61m 500 12.2 m 20 Hz 8s 4 ms -3,048 m to +6,096 m
Table 8.1: Parameters used in the finite-difference computations for the model from Figure 8.25. 3. The zero-dip velocities Vnmo(0, T) and the triplets {Vnmo,to,p} were inverted for the interval functions Vnmo,int(0,T) and r/int(W) using the algorithm based on equations (7.7)-(7.9). Both Vnmo,int(O,z) and r/int(~-) were represented by Chebyshev polynomials. There are two main sources of errors in estimating the interval Vnmo and r / - incorrect model assumptions and errors in velocity picking. The inversion procedure is designed for laterally homogeneous media above each dipping reflector, whereas in our model most "subhorizontal" interfaces have dips up to 15~ Ignoring the dips in evaluating Vnmo(0) for this model leads to velocity errors (estimated from the isotropic cosineof-dip relationship) reaching 3.5%. Uncertainty in the velocity picking may give rise to errors of similar magnitude. Figure 8.26 displays a typical CMP gather located at 15.0 km and the corresponding semblance panel. The arrival reflected from the fault plane is recorded at a zero-offset time of 3.85 s. Apparently, this reflection has a much higher stacking (NMO) velocity than those of the subhorizontal events near zero-offset times of 3.67 s and 4.05 s (Figure 8.26b). Partly due to the interference with the event at 3.67 s, the fault-plane reflection produces a relatively broad semblance maximum (Figure 8.26a) covering approximately 0.15 km/s along the velocity axis. As a result, we can expect the uncertainty in velocity picking on the order of 3-4%. Naturally, this error propagates into the interval values of Vnmo and r/ with amplification, thus causing instability in the straightforward Dix-type differentiation. Application of Chebyshev polynomials, however, amounts to a smoothing operation that helps to stabilize the inversion procedure and eliminate spurious points on the interval curves. The time-dependent curves of the estimated interval zero-dip NMO velocity and interval parameter r/for the left portion of the model are shown in Figure 8.27. The inversion for r/was carried out using the NMO velocities and zero-offset traveltimes of reflections from the right flank of the salt body (Figure 8.25). Due to the regularization (smoothing) properties of the inversion algorithm, the estimated r/-curve
8.4.
SYNTHETIC
EXAMPLE
FOR A MODEL
FROM
THE GULF
OF MEXICO
403
]
E~ 1 7 6
.o.,~
4.0 0.1
9
4.1 ~
4.2
-
b
Figure 8.26" Typical semblance panel (a) and the corresponding CMP gather (b). (solid line in Figure 8.27b) represents a sufficiently accurate but smoothed version of the actual discontinuous ~-function (dashed). Figure 8.28 displays the parameter-estimation results for the right side of the model obtained using reflections from the dipping fault plane. Note that the inversion algorithm was unable to detect the low-~ layer at a vertical time of 3 s (Figure 8.28b), which was too thin to produce a noticeable change in the effective stacking (NMO) velocity (for the level of velocity errors described above). Apart from this problem, the algorithm adequately reproduced the general (low-frequency) trend of the interval rl-function.
8.4.2
Depth migration
Whereas the parameters Vnmo(0) and r/ are sufficient for time-domain processing, depth imaging requires knowledge of the vertical velocity Vpo and the parameters c and 5. To build a model for depth migration, P-wave reflection moveout has to be combined with other data, such as reflection traveltimes of shear or converted waves (see Chapter 7); this joint inversion, however, was not attempted on this synthetic data set. Nonetheless, Han et al. (2000) performed prestack depth migration using a 45 ~ finite-difference migration algorithm extended to VTI media. The main goal of migrating the data was to evaluate image distortions caused by replacing the correct anisotropic velocity field with the following models: 1. Anisotropic model defined using the inverted 77 and the "best-guess" vertical velocity chosen equal to the zero-dip NMO velocity. 2. Purely isotropic model with the velocity equal to either the zero-dip NMO velocity or the true vertical velocity.
404
C H A P T E R 8. P-WAVE IMAGING F O R VTI MEDIA
0
0.02
a
0.04
0.06
b
Figure 8.27: Interval values of (a) Vnmo and (b) q (solid lines) estimated as functions of the two-way vertical traveltime using reflections from subhorizontal interfaces and from the right flank of the salt body. The velocity analysis was performed for CMP locations from 4.9 through 6.7 km. The dashed line shows the correct q-function for location 5.4 km.
"
-i
L
_..
,.
,
,__/_'.i
b
Figure 8.28: Interval values of (a) Vnmo and (b) r/(solid lines) estimated using reflections from subhorizontal interfaces and from the fault plane. The velocity analysis was performed for CMP locations from 11.0 through 16.8 km. The dashed line shows the correct T/-function for location 13.8 km.
8.4.
SYNTHETIC EXAMPLE FOR A MODEL FROM THE GULF OF MEXICO
405
22
E v tO
Cl
Figure 8.29" Prestack depth migration for the correct anisotropic model.
Figure 8.29 shows a section obtained by prestack depth migration using the correct VTI model from Figure 8.25. Both the salt body and the fault plane are imaged reasonably well (also, see the close-up in Figure 8.31a). The left part of the bottom of the salt body, however, is almost invisible because of the limited recording aperture (see Table 8.1). Also, some of the multiple arrivals, such as those inside the salt, were only partially attenuated in the stacking of the migrated data. The migrated images in Figures 8.30 and 8.31b were computed for an anisotropic velocity model based on the parameter-estimation results described above. Since the time-dependent r/-function was determined at only two spatial locations, the r/section was built by interpolating and extrapolating the curves from Figures 8.27b and 8.28b. This smoothed and rather crudely interpolated version of the actual q field, however, did not degrade the quality of image, except for a slight deterioration in the focusing of the fault plane [compare sections (a) and (b) in Figure 8.31]. In general, fine details of the q-section do not have much influence on the migration results. Also, the magnitude of r/in the model was so small (on average, q ~ 0.05, Figure 8.25d) that high accuracy in restoring r/was unnecessary. To specify the vertical velocity for the anisotropic migration in Figures 8.30 and 8.315, it was assumed that the parameter 5 = 0 [i.e., Vpo = Vnmo(0)], which leads to the incorrect depth scale for the whole image. Specifically, the percentage depth error is expected to be close to the average value of ~ above the reflector. Indeed, the depth of subhorizontal reflectors in the lower right part of Figure 8.30 is overstated by about 7%, while the average ~ in this part of model is about 0.1 (Figure 8.25c). Except for the depth error, the image in Figure 8.30 is quite close to the benchmark section from Figure 8.29. For more structurally complicated models, however, it may be necessary to know all three relevant parameters (V0, c and 5) to ensure proper focusing of reflection events.
406
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
E v r
O
121
Figure 8.30: Migration using an anisotropic model obtained by interpolation and extrapolation of the inverted q-functions (Figures 8.27b and 8.28b). The vertical and NMO velocity are taken equal to each other (5 - 0, e - q).
Figure 8.32 can be considered as the best possible output of the conventional isotropic processing sequence. The isotropic velocity model used to generate Figure 8.32 is based on the correct NMO velocity Vnmo(0), conceivably obtained by error-free semblance velocity analysis. Comparing the lower right portions of Figures 8.32 and 8.29, we notice that the subhorizontal reflectors are mispositioned in depth, which is also the case in Figure 8.30. The overall quality of the image, however, is comparable to that in Figures 8.29 and 8.30. This is confirmed by a close-up of an area near the fault-plane in Figure 8.31 [compare section (c) with (a) and (b)]. For example, there are no crossing features (conflicting dips) in the vicinity of the fault, which would have been indicative of the presence of anisotropy. The continuity and crispness of the fault-plane reflection on the isotropic image (Figure 8.31c) are somewhat inferior to those on the VTI images, but the difference is not dramatic. The acceptable quality of this isotropic result is explained by the small values of the time-processing coefficient r/. Ignoring r/in this model does not cause substantial distortions in the stacking of dipping events, while the correct Vnmo(0) ensures that the horizontal reflectors are focused well. Another option in choosing the isotropic migration velocity is illustrated in Figures 8.33 and 8.31d. This time, the data were migrated using the correct vertical velocity Vp0, which in some cases may be obtained from check shots and well logs. As expected, all subhorizontal reflectors in Figures 8.aa and 8.31d are correctly positioned in depth. However, the image quality is considerably lower than that on the isotropic section migrated with the correct NMO velocity Vnmo(0). The difference between Vp0 and Vnmo(0) causes misstacking of subhorizontal events throughout
8.4.
SYNTHETIC EXAMPLE FOR A MODEL FROM THE GULF OF MEXICO
407
18
E v
r f:l.
El
14
16
18
14
16
18
E v
r f:l.
D
Figure 8.31" Comparison of migration results for an area near the fault plane. (a) Anisotropic migration for the correct model; (b) anisotropic migration using the estimated parameters; (c) isotropic migration using the NMO velocity Vnmo(0); (d) isotropic migration using the vertical velocity Vpo.
408
C H A P T E R 8.
P-WAVE IMAGING FOR VTI MEDIA
E v t-"
O
D
Figure 8.32" Isotropic migration with the medium velocity equal to the correct NMO velocity Vnmo (0) .
E v l--
O
a
Figure 8.33: Isotropic migration with the medium velocity equal to the correct vertical velocity Vpo. The arrows mark some of the intersecting reflectors.
8.4.
S Y N T H E T I C E X A M P L E F O R A M O D E L F R O M T H E GULF OF M E X I C O
409
the model. Since the wrong values of Vnmo(0) also distort the NMO velocity of dipping events, the reflections from the fault plane (Figure 8.31d) and from the flank of the salt body (Figure 8.33) are poorly focused and positioned. Also, notice the intersecting reflectors with different dips marked by the arrows in Figure 8.33.
8.4.3
Discussion
This synthetic data test illustrates parameter-estimation and imaging issues for VTI media with a small time-processing parameter r] (average r/ ~ 0.05) and a larger parameter /i (average 5 ~ 0.1) responsible for time-to-depth conversion. Dipping reflections from the fault plane and the salt body were available over only two limited ranges of CMP locations on the left and right sides of the model (as is often the case in practice). Therefore, 1-D q-estimation could be carried out separately for those two groups of dipping events, with subsequent interpolation and extrapolation of the ri-curves for the whole model. Although this procedure reconstructs only large-scale variations in 77, a smoothed version of the r/-field is usually sufficient for imaging purposes. Depth migration of the inverted anisotropic velocity model produces a high-quality image close to that generated for the exact velocity field. Some inaccuracies in r] estimation do not cause visible distortions in the focusing and positioning of reflection events, in part because the magnitude of r/in the model is rather small. However, since the vertical velocity cannot be determined from surface P-wave data, the image has the wrong depth scale (it was assumed that the vertical and NMO velocities are equal to each other). The same mispositioning of subhorizontal events is observed on the depth section generated for a purely isotropic model with the velocity equal to the correct zerodip NMO velocity Vnmo(0). Due to the small ri-values, the overall image quality is comparable to that of the anisotropic section, except for some degradation in the focusing and continuity of the fault-plane reflection. Since one needs to use the vertical velocity Vpo to image horizontal events at the correct depth, the data were also migrated with the isotropic velocity model based on Vpo. Although reflector depths in this case are indeed accurate, both the horizontal and dipping reflections are poorly focused because in anisotropic media the vertical velocity is inappropriate for stacking reflection events of any dip. Note that the difference between the vertical velocity and the zero-dip NMO velocity is determined by the anisotropic coefficient 5, which is greater than r] in this model. Thus, isotropic processing algorithms are inherently inadequate in dealing with anisotropic velocity fields. No single velocity is sufficient for generating a section with both good focusing and the correct spatial position of reflection events, even for layer-cake geometry. Only anisotropic depth migration based on accurate estimates of the parameters Vnmo(0), e and 5 can provide a crisp image with reflectors located at the correct depth.
410
8.5
C H A P T E R 8. P-WAVE IMAGING F O R VTI MEDIA
Field-data example with multiple fault planes
The 2-D line treated here (following the results of Alkhalifah et al., 1996) comes from the offshore Africa data set partly examined in Chapters 3 and 7 above. The anisotropy in the study area is associated mostly with massive shale formations and manifests itself in substantial mis-ties in time-to-depth conversion (Figure 3.27) and poor imaging of dipping fault planes (Figures 7.10 and 7.17). A clear picture of the faulting of the Cretaceous section is essential for understanding the extent of the oil fields in this area. As we shall see below, accounting for anisotropy helps in imaging even some subhorizontal and gently dipping features, which are not properly stacked after the conventional NMO correction due to the influence of nonhyperbolic moveout. The seismic section in Figure 8.34 exhibits a more complicated subsurface structure than the section analyzed in Chapter 7 (Figure 7.17), with reflections from a large number of dipping fault planes. The upper layers (above 3.0 s at CMP location 750 and above 2.1 s at CMP location 1050) are largely composed of high-pressure, anisotropic shales. The 3-D processing sequence used to generate the image in Figure 8.34 was based primarily on the assumption of isotropy. The only exception was use of a "stretched" DMO algorithm designed to compensate empirically for the influence of anisotropy on the dip-dependent NMO velocity and DMO impulse response. The stretched DMO, however, can provide only an approximation of the actual DMO signature in VTI media (see section 7.2 above); therefore, its application met with only partial success in this complicated area. While this processing sequence proved capable of imaging most shallow fault planes, it did not preserve some important dipping events beneath the shale formation.
8.5.1
Parameter estimation
Application of anisotropic (VTI) time-imaging algorithms should be preceded by estimation of the zero-dip NMO velocity and parameter r]. Since dipping events occupy a large time (and depth) interval, the velocity-analysis algorithm should be capable of accurately reconstructing the vertical variation in 77; clearly, a single effective value would not be sufficient. Figure 8.35 shows a stacked seismic section of the same data produced by conventional NMO and 2-D DMO algorithms without either taking anisotropy into account or applying DMO stretch. The arrows in Figure 8.35 mark the dipping events used in the inversion for the interval parameter 77. Although these dipping reflections seem to span almost 4.5 s of data, the vertical (migrated) time corresponding to the deepest measurement of stacking velocity of fault-plane reflections is only about 3.5 s. The inversion process based on the Dix-type equation (3.47) and a piecewiselinear approximation for Vnmo(0) and ~ yielded the time-dependent interval parameters shown in Figure 8.36. Smoothing was used to remove the sharp edges that arise at times where stacking velocity was measured. The predominant velocity variation in the section is vertical, which facilitates application of the 1-D q-estimation algorithm. In the area between CMP locations 400 and 800 and for the vertical time not
8.5.
FIELD-DATA E X A M P L E W I T H M U L T I P L E FAULT P L A N E S
411
1.0
E 3.0
4.0 Figure 8.34: Time migration of offshore Africa data, based on 3-D isotropic processing.
412
C H A P T E R 8.
P-WAVE IMAGING F O R VTI MEDIA
~25 E " ~- 3.0 3.5 4.0 4.5
Figure 8.35" Stacked section for the same line as in Figure 8.34 after application of NMO and 2-D isotropic homogeneous DMO. The arrows point to the dipping reflections used to estimate the parameter r/.
8.5. FIELD-DATA EXAMPLE WITH MULTIPLE FAULT PLANES
413
20OO I
m
_
_
3
_
~
4""
_
4
Figure 8.36: Interval values of Vnmo(0)and 77as functions of the vertical time estimated from reflections recorded between CMP locations 600 and 850. exceeding 3 s, the lateral variation in the stacking velocity of subhorizontal events is limited by 5%. Based on the locations of the velocity picks (Figure 8.35) and the moderate magnitude of lateral heterogeneity, the results in Figure 8.36 can be considered representative of the area between CMP locations 450 and 950. In the water layer, Vnmo(0)=l.5 km/s and, obviously, ~/was set to zero. Down to about 2.5 s, the interval values of 7/in Figure 8.36 increase with vertical time and then rapidly drop back to zero. No adequate dip-moveout information on r/was available at vertical times exceeding 3.5 s. As in conventional velocity analysis, further refinement of the spatial distribution of r/can be achieved through migration velocity analysis, which can be based, for instance, on a depth-focusing approach. Although the ~/-curve in Figure 8.36 does not correspond to the same location as the ~-curve from the same area discussed in Chapter 3 (Figure 3.27), it is interesting that the overall shape of the two curves is similar. Both anisotropic parameters are positive in the Tertiary shale section, with particularly large values near its bottom. Similar estimates of the parameter r/in this area were obtained by Toldi et al. (1999) using the inversion of nonhyperbolic moveout. Potentially, this correlation between the two anisotropic coefficients can be used in time-to-depth conversion. Indeed, while time imaging requires knowledge just of the coefficient r/, the vertical velocity (or (~) is the key parameter for generating a depth
414
C H A P T E R 8.
P-WAVE I M A G I N G F O R VTI M E D I A
image. As discussed in Chapters 6 and 7, in models with laterally homogeneous VTI overburden, P-wave reflection traveltimes alone are not sufficient to recover the true vertical velocity. In the early phase of field development, velocity information from surface-seismic measurements is often combined with the initially sparse well control to determine the depth structure away from a crestal position. This combination is typically based on spatial interpolation of the mis-ties in velocities obtained from VSP and surface seismic data. With a moderate degree of correlation between the 5 values computed from the mis-ties at the wells and nearby r] values extracted from the surface data, we have the intriguing possibility of using r] measurements away from the wells to help us predict 5, and thus refine the time-to-depth relationships at those locations.
8.5.2
Time imaging
Application of 2-D algorithms in this case study is justified by the fact that the line is close to the dip direction of the structure, and most of the recorded reflections correspond to features within or near the vertical incidence plane. Still, some events may represent out-of-plane reflections that should be processed by 3-D methods. Those out-of-plane events are expected to stack better at a lower velocity (closer to the isotropic NMO velocity), and therefore focus better on the isotropic image. An example of this kind is the reflection near CMP location 800 between 3.5 s and 4.0 s in Figure 8.35. Based on examination of parallel lines in the same area, it was identified as an out-of-plane event. Figure 8.37a shows the same section as in Figure 8.34, but this time after a twodimensional conventional isotropic processing sequence (i.e., phase-shift time migration of the section from Figure 8.35). For comparison, Figure 8.37b displays the 2-D data imaged with phase-shift VTI time migration (using the inverted parameters of Figure 8.36) applied to the stacked section obtained by the TZO (transformation to zero offset) algorithm of Alkhalifah (1996b) devised for vertically heterogeneous VTI media. This comparison gives a clear picture of the benefits of taking anisotropy into account in seismic imaging; the improvements here are numerous and significant. Probably the most dramatic example is the fault located at CMP location 870 between vertical times of 2.5 and 3 s (Figure 8.37b). An interpreter using the isotropic processing result might readily either extend the reflections across this fault (invisible on the isotropic section) or identify a minor subsidence to the left of the fault. The anisotropic image and the inverted interval values of ~, however, suggest that shales extend all the way down to 3 s under CMP location 800, thus indicating a much larger subsidence. The anisotropic section also shows a noticeable improvement in the continuity of the dominant fault plane that extends from under CMP location 900 (a time of about 1 s) all the way down to 3 s beneath CMP location 500. Another example is the region of the nearly horizontal events near CMP location 500, at 2.5 s. The improved focusing and continuity of those events is a result of the nonhyperbolic moveout
8.5.
F I E L D - D A T A E X A M P L E W I T H M U L T I P L E FAULT P L A N E S
415
cD V
{D
E
Figure 8.37: Comparison between the results of isotropic and anisotropic 2-D processing. (a) Section obtained by isotropic phase-shift migration of the section shown in Figure 8.35; (b) the result of anisotropic TZO and phase-shift migration using the parameters from Figure 8.36.
416
C H A P T E R 8. P-WAVE IMAGING F O R VTI MEDIA
correction included in the anisotropic processing sequence as part of TZO. Such gently dipping features of limited lateral extent are often important for understanding the fine structure of reservoirs. It is interesting that the dipping events on the 2-D anisotropic migrated section (Figure 8.37b) are focused better than are those on the section produced by 3-D isotropic DMO and migration (Figure 8.34). This, of course, is not to say that 2-D anisotropic processing always produces a better image of the subsurface than does 3-D isotropic processing. As evidenced by the out-of-plane events on the 2-D sections presented here, treating just anisotropy or three-dimensionality alone is necessarily incomplete. Where the subsurface has even modest three-dimensional complexity, it is best to incorporate the VTI algorithms into 3-D processing (e.g., see Ball, 1995). The fault reflections above 2 s on the 3-D isotropic image (Figure 8.34) were accurately positioned despite the presence of anisotropy. Perhaps, the stretched DMO helped to focus the fault-plane reflections, and artificially high velocities may have been used in the time migration to compensate for presence of anisotropy. Such use of higher velocities may be helpful if the locations of the dipping events are known in advance, such as where the positions of fault-plane reflections are supported by the reflections from bedding. The artificially high velocities, however, seem to have caused overmigration of reflections at depth; an example is the bottom event under CMP location 900 in Figure 8.34.
8.6
Discussion
Despite the clear improvements achieved by the anisotropic (mostly VTI) imaging methods in the examples given throughout the book, it may seem that accounting for anisotropy in processing is too costly. However, although anisotropic DMO and migration codes are somewhat more complicated algorithmically than their isotropic counterparts, they are often almost as efficient in terms of computing time and memory requirements. This is the case, for instance, for Fowler's and Hale's D MO methods discussed above. Likewise, the TZO algorithm of Alkhalifah (1996b) is based on traveltime tables calculated once for any given model and is, therefore, practically as fast as the corresponding isotropic algorithm. In migration codes that make repeated use of ray tracing, the overall cost of adding anisotropy depends on the contribution of ray tracing to the total computing time (this contribution is small, for example, for Gaussian beam migration). Hence, it is no longer justified to apply isotropic imaging methods in the presence of transverse isotropy, even if adjustments of the processing parameters can bring about local improvements in image quality. Processing generalized for VTI media is neither too complex nor too costly, nor is anisotropy in the earth's subsurface unusual. In fact, recent advances in seismic inversion and processing for azimuthally anisotropic media (e.g., Grechka and Tsvankin, 1998b, 1999, 2000) are leading to the development of practical imaging methods for far more complicated models than vertical transverse isotropy.
8.6.
DISCUSSION
417
To an extent, routine processing (devised for isotropic media) often includes a certain degree of freedom, e.g., a juggling of velocities, that allows the processor to overcome some limitations imposed by ignoring the true presence of anisotropy. With such freedom, the processed result can often be steered toward a "geologically plausible" solution, although with only limited success for the offshore Africa data set examined above. Despite all attempts to artificially adjust the parameters of the isotropic processing flow, it failed to image fault planes in a massive shale formation. It should be emphasized that the strength of the anisotropy (in time imaging for VTI media, measured by the parameter 77) in this study area was not anomalous, and similar results have been obtained in other regions, such as Trinidad (Alkhalifah, 1996a) and the Gulf of Mexico (Meadows and Abriel, 1994; Bartel et al., 1998). Also, with the advent of prestack depth migration replacing the conventional processing sequence of NMO-DMO-stack-poststack migration, the possibilities of turning knobs on isotropic algorithms become much more limited. It might be thought that the added parameter 77, here, accomplishes little more than the benefit of having an extra degree of freedom in processing. It is important to note, however, that for the offshore Africa data, the time-dependent rl-function obtained from the inversion of dip-moveout data yielded not only an improved stack of the dipping events after DMO, but also improved migrated position of the corresponding fault planes and improved nonhyperbolic moveout correction of the gently dipping features. Furthermore, the significant improvement in image quality is just one of the benefits of accounting for transverse isotropy in seismic processing. Another benefit, that may be equally important, is the possibility of obtaining a new physical parameter of the subsurface (r/) from P-wave reflection data. This coefficient has the potential of becoming an important tool in lithology discrimination from surface seismic data, even if an acceptable image could have been obtained without applying anisotropic processing. For instance, in some areas of the Gulf of Mexico the influence of transverse isotropy on DMO is counteracted by a vertical velocity gradient (Lynn et al., 1991), which enables isotropic DMO and poststack migration (with some "adjustments" of velocity) to image fault-plane reflections. However, even in this case r/may serve as a useful lithologic tool that would allow the interpreter to discriminate between shales and sands using only surface seismic data.
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Author Index Abaseev, 104 Abriel, 153, 385, 417 Aki, 1, 2, 8, 66, 79, 82, 83, 91, 103, 104, 386 Al-Chalabi, 176, 181 Al-Dajani, 165, 177, 185 Alekseev, 62 Alfaraj, 203 Alkhalifah, vi, 20, 153, 178, 189, 254, 265, 294, 302, 305, 312, 329, 358, 373, 385, 387, 389-391, 410, 414, 416, 417 Anderson, 355, 359, 360, 365, 369, 372, 374, 375 Artley, 305, 307, 368 Auld, v, vi, 20 Bakulin, 9, 10, 12, 36, 37 Ball, 153, 385, 416 Banik, 81, 91, 156 Bartel, 417 Bednar, 215, 219, 231,351 Behle, 211, 229 Ben-Menahem, 62, 98, 100 Berryman, 5, 20, 57 Black, 371,372 Bleistein, 64, 371,372, 376 Booth, 62 Byun, vi, 183 Cerven~, 62, 390, 391 Chapman, 47 Cheadle, 41 Chesnokov, 62-65, 67, 79, 85, 98, 104 Cohen, 112, 128, 204, 235, 247, 261, 262, 264, 265, 297, 385 Corrigan, 98, 101,102, 183 Crampin, v, 4, 38, 39, 46, 62, 79, 81 Dellinger, 173, 269, 273 Desegaulx, 156 Dix, vii, 149, 293 Duren, 81
Fatti, 185 Fedorov, v, vi, 8 Fowler, viii, 288, 312, 353-355, 359 Frazer, 62, 63, 98 Fryer, 62, 63, 98 Fuchs, 64 Gajewski, 41, 62, 64, 77, 98, 102 Gazdag, viii, 264, 385-387 Gidlow, 185 Godfrey, 384 Graebner, 81 Granli, vi, 199 Grechka, 38, 40, 49, 65, 156, 160, 161, 164, 165, 188, 199, 204, 225, 227, 235, 247, 262-264, 281, 294, 297, 312, 313, 316, 323, 327, 337, 338, 347, 389, 416 Guest, 62 Hake, 177, 181 Hale, viii, 159, 166, 171,255, 305, 307, 353, 354, 368-372, 375, 376, 384, 390 Han, 294, 400, 403 Hanyga, 103, 390, 391 Helbig, v, vi, 1, 4-6, 8, 10, 20, 35-38, 43, 46, 51, 56, 190, 269, 273 Hill, 389-391 Hubral, 166, 207 Isaac, 130, 272 Jech, 47, 396 Keith, 81 Kendall, 62 Kim, 82, 92, 94 Kiselev, 67, 77 Kitchenside, vi, 358, 385 Koehler, 109, 181,316 Korn, 56 Krey, 166, 207
430
Author Index
Kumazawa, 67 Larner, 128, 265, 368, 385, 387 Lawton, 130, 272 Le Stunff, 269, 347, 353 Leafy, 111 Levin, 111-113, 116, 159, 160, 190, 229, 355 Liner, 384 Lynn, H., 164 Lynn, W., vi, 287, 302, 417 MacBeth, viii Martinez, 82 Martynov, 62, 79 Meadows, 9, 153, 385, 417 Mensch, 41, 46, 47 Mikhailenko, 62, 79 Muir, 173, 269, 273 Mujica, 156, 385 Musgrave, v, vi, 1, 4-6, 8, 20, 46, 103, 190 Nolte, 350 Notfors, 384 Obolentseva, 38, 65 Payton, 1 Phadke, 385 Pratt, 47 P~en~{k, 41, 47, 62, 77, 91,390, 396 Radovich, 190 Rasolofosaon, 41, 46, 47 Ray tracing, 29 Richards, 1, 2, 8, 66, 79, 82, 83, 91, 103, 104, 386 Robertson, 98, 101, 102 Rommel, 35, 51, 54, 74 Riiger, 38, 46, 81, 82, 91, 92, 94, 100 Sams, 20 Sarkar, 215 Sayers, 11, 47 Schoenberg, 20, 35-37, 43, 51, 56 Sena, 183, 184, 385 Seriff, 207, 229 Sexton, 334
Shah, 149 Sriram, 207, 229 Stolt, viii, 354-356, 358, 387 Taner, 109, 181,316 Taylor, 62 Tessmer, 211,229 Theophanis, 40, 164, 225, 227 Thomsen, v, vi, 1, 2, 17-24, 26, 36, 40-44, 46, 49, 51, 54, 55, 81, 82, 91, 92, 102, 107, 113, 116, 118, 131, 136, 143, 153, 161, 162, 174, 177, 179, 181, 184, 185, 187, 189, 193, 199, 219, 227, 228, 232, 253, 269, 270, 287, 336, 345, 361,389, 392 ToksSz, 385 Toldi, 333, 413 Tsvankin, vi, 21, 35, 36, 38, 40, 46, 49, 51, 54, 62-65, 67, 74, 77, 79, 82, 85, 87, 91, 98, 100, 104, 110, 115, 130, 131, 153, 156, 160, 161, 164, 165, 177, 178, 181, 184, 185, 188, 199, 204, 225, 227, 228, 232, 235, 247, 254, 262-264, 281, 297, 312, 313, 316, 323, 327, 336-338, 368, 369, 372, 374, 375, 416 Uren, 140, 178, 273 Uzcategui, 156, 385 Van der Hijden, 62 Vavry~uk, 91 Vestrum, 272 Wild, 46 Williamson, 334 Winterstein, 9 Wright, 81 Yedlin, 38, 39
Subject Index tilted TI, 12, 112, 130, 159, 168, 208, 272, 285, 384, 397 triclinic, 8, 9 VTI, 11, 12, 14, 17, 21, 34, 37, 44, 51, 61, 67, 72, 81,105, 113, 132, 151, 152, 159, 163, 164, 173, 177, 181, 185, 187, 190, 195, 197, 200, 207, 208, 228, 241, 254, 264, 269, 283, 287, 353 Aperture, 265, 360, 361,402, 405 Apparent dip, 124, 274 Attenuation, 81,199 Azimuthal anisotropy, 6, 8-10, 12, 36, 62, 64, 67, 72, 74, 77, 80, 111, 115, 130, 156, 161, 177, 185, 223, 272
Acquisition, 199, 219, 270, 288, 315, 340 Aliasing, 356, 359 Amplitude, 38, 54, 61, 62, 64, 67, 77, 79-81,105, 107, 253, 269-271, 288, 316, 361,386, 390, 392 focusing caused by anisotropy, 30, 32, 61, 66, 67, 71, 80, 82, 84, 95, 100, 101,392 of converted waves, 199, 215, 219, 351 preservation in DMO, 371,375, 384 Amplitude variation with offset (AVO), 81,335 AVO gradient, 91, 94, 101 for converted waves, 215 for P-waves, 84, 91, 101,271 for S-waves, 94, 101 Anisotropic parameters for HTI media, 44, 164 for orthorhombic media, 40, 44, 46, 51, 67, 71, 72, 115, 162, 164, 165 for VTI media, s e e Thomsen parameters parameter U, 22, 119, 178, 181, 187, 197, 253, 260-264, 266, 270-272, 288, 289, 312, 334, 353, 354, 369, 387, 389, 400, 403, 405, 406, 409, 410, 414, 417 parameter ~l for tilted TI media, 274 Anisotropic symmetry, 1 general TI, 11, 14, 21, 34, 69 HTI, 12, 14, 38, 44, 110, 130, 132, 159, 161,177 monoclinic, 9, 10 orthorhombic, 9, 10, 12, 14, 35, 36, 67, 85, 95, 110, 111, 115, 149, 152, 161, 177, 178, 187, 208, 228, 232, 288, 340, 351, 369, 384 431
432
Subject Index
Cusp (triplication) on shear wavefront, 5, 34, 38, 51, 62, 79, 98, 102, 111, 135, 158, 190, 200, 231 Data processing, 18, 20, 34, 81, 110, 111,130, 271,287, 353 depth-domain, 152, 271, 273, 334, 389, 40O for converted waves, 200, 334 time-domain, 179, 264, 270, 273, 281, 289, 312, 354, 369, 385, 400, 410 Density, 2, 4, 56, 81, 91, 103, 216 Dipping layer, 12, 112, 130, 272 Dipping reflector, 49, 110, 114, 115, 118, 130, 159, 169, 200, 208, 219, 233, 239, 241, 246, 249, 254, 265, 269, 271, 272, 283, 285, 289, 337, 346, 354, 369, 387, 389, 392, 400, 410 Displacement, 2 for plane waves, 3, 37, 38, 63-65, 105 from a point source, s e e Green's function Dix equation 2-D, anisotropic, 130, 144, 149, 151, 159, 169, 170, 261, 281, 291, 293, 310, 384, 400, 402, 410 3-D, anisotropic, 264, 297 conventional, 149, 151-154, 163, 182, 185, 189, 197, 291, 293, 322, 323, 327, 336, 340, 350 for converted waves, 207, 225, 227 for quartic moveout term, 181, 182, 189, 197, 312, 322, 323, 327, 333 DMO Fowler, 353, 354, 416 Hale, 354, 369, 416 Transformation to zero offset, s e e TZO Eigenvalue, 4, 16, 63, 156, 159, 172 Eigenvector, 4, 16, 34, 63, 66, 158, 172 Elliptical anisotropy, 15, 16, 22-24, 28, 30, 39, 44, 45, 85, 89, 91, 95, 98, 100, 102, 113-115, 118, 119, 124, 125, 131, 140, 156, 161,
173, 178, 181, 182, 187, 212, 253, 255, 256, 261-263, 269, 271, 272, 279, 281, 283, 285, 335, 342, 357 Equivalence Orthorhombic/HTI, 14, 37, 162, 164, 177 Orthorhombic/VTI, 11, 14, 37, 40, 44, 47, 51, 69, 72, 81, 162, 163, 165, 177, 178, 187, 208, 228, 232,288, 351,369 Factorized medium, 125, 136, 305, 387, 392, 394, 395 Fault-plane reflection, 151, 287, 291, 302, 310, 340, 350, 365, 387, 389, 400, 410 Fermat's principle, 237 Field-data examples, 153, 302, 327, 365, 410, 417 Finite-difference modeling, 62, 399, 400 Focusing of amplitude, s e e Amplitude focusing of imaged reflections, 265, 305, 307, 354, 361, 365, 369, 385, 387, 396, 400, 405, 406, 409, 4 1 4 , 416 Fractures, 1, 10, 12, 111, 163 micro-corrugated, 9 multiple sets, 9, 10 Frequency, 3, 5, 63-65, 98, 105, 217, 321, 355, 358-360, 386, 391, 394, 396 Frequency domain, 65, 80, 386, 390 Frequency spectrum, 103 Frequency-wavenumber domain, 307, 354-356, 359, 368, 370, 386 Gaussian noise, 296, 338, 342, 347, 349 Geometrical seismics, 54, 64, 69, 77, 81, 132 Green's function, 62, 103 far-field approximation, 64, 82, 105 Group angle, 5, 21, 65, 66, 69, 80, 166, 233, 234 for orthorhombic media, 38, 46, 54, 67, 69, 71, 72, 77
Subject Index for TI media, 26, 34, 82-85, 89, 100, 102, 107, 132, 194, 195, 242, 243, 258 Group velocity, 5, 57, 65, 69, 233, 247 for orthorhombic media, 38, 45, 49, 51 for TI media, 26, 95, 132, 136, 192, 195, 210, 242, 258, 270 Group-velocity surface, 15, 38, 51 Group-velocity vector, 4, 5, 16, 24, 26, 57, 63, 74, 80, 82, 110, 111,131, 135, 166, 167, 169, 201, 203, 205, 223, 234, 235, 247, 391 Head wave, 288 Helmholtz equation, 386 Heterogeneity, 1, 3, 82, 156, 170, 173, 235, 365, 369, 387, 389, 392, 413 vertically heterogeneous media, 125, 128, 144, 159, 169, 281, 291, 305, 307, 310, 329, 333, 336, 358, 368, 384, 386, 387, 410 Hooke's law, 2, 3, 17 Horizontal slowness, s e e Ray parameter Horizontal transverse isotropy (HTI), s e e Anisotropic symmetry, HTI Intrinsic anisotropy, 1, 11, 20, 305, 333, 365, 410 Inversion field-data examples, 302, 327, 365, 410 joint of P and PS data, 334 of dip moveout, 256, 289, 387, 400, 410 of NMO ellipse, s e e NMO ellipse of NMO velocity, s e e NMO velocity of nonhyperbolic moveout, 312 stability issues, 289, 291,294, 297, 299, 300, 302, 310, 312, 315, 317, 325, 333, 335-338, 342, 346, 347, 402 using Fowler DMO, 356 Jacobian (for Hale DMO), 370-373, 384
433
Lam~'s constants, 13 Layered media, 144, 149, 159, 169, 180, 185, 189, 190, 197, 219, 228, 249, 261, 269, 281, 291, 310, 321,329, 333, 340 Longitudinal direction, 5 Mapping, s e e Resampling Migration finite-difference, 403 Gaussian beam, 385, 389, 416 Gazdag, 264, 265, 385, 415 Kirchhoff, 353, 385, 392 poststack depth, 385, 389, 416 poststack time, 264, 265, 354-356, 358, 385, 387, 411,415 prestack depth, 403 Stolt, 354-356, 358, 387 Mode conversions, s e e Converted waves Monoclinic media, s e e Anisotropic symmetry, monoclinic Moveout (finite-spread) velocity, 110, 113, 116, 118, 124, 128, 139, 173, 176, 190, 229, 300, 377 azimuthally dependent, 165, 297 Moveout attributes for converted waves, 200, 201,204, 209, 211, 213, 214, 226, 231, 338, 341,342, 350 NMO ellipse, 156, 170 application in VTI inversion, 297, 310, 335, 336, 346, 351 for converted waves, 225, 227 for HTI media, 161 for orthorhombic media, 161 for VTI media, 159, 263 NMO velocity, 109, 166, 169, 170, 173, 180, 181,207, 285, 354, 369 expressed through ray parameter, 254, 272, 283 for converted waves, 200, 204-206, 211,214, 227-230, 232, 239, 245 for HTI media, 161 for orthorhombic media, 54, 161 for tilted TI media, 130, 168, 272 for VTI media, 18, 54, 84, 95, 100, 113, 159, 178, 180, 181, 183,
434 NMO
Subject Index
185, 187-189, 192, 195, 197, 253, 264, 266, 270-272, 288, 312, 313, 315, 321, 323, 325, 327, 329, 333, 353, 387, 400, 403, 410 inversion for VTI, 288, 289, 335, 337, 354, 400, 410 Nongeometrical phenomena, 54, 77, 81, 87 Nonhyperbolic moveout, 110, 113, 173, 195 for azimuthal anisotropy, 165, 297 for converted waves, 228, 230 for P-waves, 116, 124, 139, 187, 197, 270, 288, 297, 310, 358, 361, 373, 377, 414 for SV-waves, 190 general equation, see TsvankinThomsen moveout equation inversion for VTI media, 312 nonhyperbolic semblance analysis, 315
190, 192, 241, 242, 255, 270, 273, 281, 283, 299, 353, 358, 359, 368, 374, 385, 387 Phase-velocity surface, 4, 39, 40, 80, 82, 84, 125 Phase-velocity vector, 4-7, 16, 19, 28-30, 34, 46, 63, 74, 77, 80, 110, 131, 135, 149, 166, 169, 201,233, 242, 254 Plane wave, 1-3, 8, 14, 36, 56, 61, 63-66, 71, 80, 103-105 Point-source radiation, 4, 5, 15, 28, 35, 54, 61,103-105, 131 Polarization vector (angle), 1, 3-6, 8, 10, 61, 63, 105, 201,223 for curved wavefront, 5, 61, 65-67, 72, 77, 79, 80 for orthorhombic media, 38-40, 43, 45, 46, 51, 54, 67, 72, 77, 79, 80 for TI media, 14-17, 19, 22, 34, 83, 85, 95, 98, 269, 270 Propagation phenomena, 82, 84, 92, 94, 100, 101
Orthorhombic media, see Anisotropic symmetry, orthorhombic Overmigration, 266, 361,416
Quartic moveout coefficient for converted waves, 228 for pure modes, 176, 190
Parameter estimation, see Inversion Particle motion, 34, 67 nonlinear, 67, 77 Particle motion (polarization) diagrams, 67 Phase angle, 6, 14, 29, 34, 37-39, 65, 69, 71, 80, 82, 83, 85, 102, 107, 131, 132, 135, 136, 141, 144, 167, 176, 195, 207, 234, 241-243, 254, 255, 283, 284, 356, 386, 387 Phase velocity, 3-7, 11, 56, 57, 65-67, 69, 72, 80, 104, 110-112, 167, 168, 176, 207, 254, 356 for orthorhombic media, 37-39, 41, 45-47, 51, 54, 164 for TI media, 14, 15, 17-19, 21, 26, 28, 29, 32, 83, 84, 87, 89, 94, 95, 105, 107, 113-115, 118, 119, 131, 132, 135, 136, 140, 141, 144, 160, 168, 174, 177, 179,
Radiation pattern, 62, 66, 77, 80 for orthorhombic media, 54, 67, 69, 75, 80 for TI media, 22, 34, 81, 82, 84, 94, 100, 101, 105, 271 Ray parameter (horizontal slowness), 62-64, 80, 105, 107, 119, 124, 125, 141, 144, 149, 156, 158, 161, 166, 169, 172, 203, 205-208, 210, 221, 223, 225-227, 231, 237, 241, 244, 245, 250, 254, 255, 262, 271, 272, 274, 281, 283, 285, 289, 291, 293, 294, 297, 299, 300, 305, 310, 338, 341, 346, 350, 355-357, 359, 370, 372, 373, 384, 386, 390, 400 Ray tracing, 62, 77, 81, 128, 132, 185, 188, 199, 204, 216, 223, 225, 231, 305, 314, 315, 347, 360, 375, 377
(continued)
Subject Index using Gaussian beams, 390-392 Reflection coefficient, 38, 45, 81 for P-waves, 19, 91,101,270, 271 for S-waves, 94, 100, 101 Reflection moveout, s e e Moveout velocity, NMO velocity, nonhyperbolic moveout Reflection-point dispersal, 157, 166, 169, 170, 177, 371 for converted waves, s e e Conversionpoint dispersal Reflectivity method, 62, 80 Regularization, 297, 402 Resampling, 355-357, 359, 365, 368, 371 Seismic processing, s e e Data processing Shale anisotropy, 11, 12, 20, 30, 82, 92, 94, 112, 116, 153, 179, 190, 305, 333, 365, 410 Shear-wave singularity, 4, 57, 62, 66 for orthorhombic media, 38, 39, 79 for TI media, 15, 16 Shear-wave splitting, 4, 13, 16, 38, 39, 43, 57, 63, 72, 79, 104, 164, 201 Slope of CMP moveout, 215, 235 for converted waves, 204-206, 209, 212, 215, 221, 226, 231, 239, 244, 338, 340, 341,350 in squared coordinates, 110, 156, 190, 192, 253 Slope on zero-offset section, 125, 208, 254, 289, 291, 299, 338, 341, 342, 346, 350, 357, 372, 390 Slowness surface, 4, 5, 16, 38, 62, 80, 98, 125, 136, 182 concave, 5, 34, 98, 194 ellipsoidal, 15, 22, 39 Slowness vector, 3-6, 11, 14, 30, 34, 38, 46, 51, 57, 61-63, 66, 104, 105, 110, 112, 149, 151, 156, 166, 170, 172, 204, 205, 223, 225, 226, 231, 234, 242, 247, 251,254, 261,346, 356 Snell's law, 203-206, 221, 223, 225-227, 231, 234, 239, 244, 245, 247, 250, 396
435
Stacking velocity, s e e Moveout (finite-spread) velocity Stationary-phase approximation, 61, 64, 67, 69, 72, 77, 80, 85, 87-90, 95, 98, 105, 108 Stiffness matrix, 8 for HTI media, 12 for isotropic media, 13 for monoclinic media, 9 for orthorhombic media, 10, 36 for tilted TI media, 12 for triclinic media, 8 for VTI media, 12, 14 Stiffness tensor, 2, 3, 7, 16, 17, 36, 65, 75, 103 Strain tensor, 2, 8 Stress tensor, 2, 8 Symmetry, s e e Anisotropic symmetry Symmetry axis of general TI media, 11, 14, 82, 102, 110, 178, 372 of HTI media, 12, 14, 37, 44, 162, 164 of tilted TI media, 12, 112, 130, 159, 272, 384, 394 of VTI media, 11, 15-18, 20, 21, 23, 34, 37, 81, 95, 100, 161, 190, 208, 255, 394 Symmetry planes horizontal, 9, 176, 180, 199, 207 of general TI media, 11, 110 of HTI media, 12, 14, 162, 177 of monoclinic media, 9 of orthorhombic media, 10, 14, 36, 37, 40, 44, 47, 51, 67, 72, 77, 80, 162, 177, 178, 187 of tilted TI media, 130, 272 of VTI media, 14, 16, 28, 30, 32, 81, 113, 160, 177, 208, 228, 253, 287, 353 vertical, 12, 14, 58, 64, 66, 105, 110, 149, 166, 167, 169, 176, 180, 200, 219, 227, 233, 234, 251, 384 Synthetic seismograms, 62, 67, 265, 316, 324, 360, 375, 387, 392, 396, 400 for converted waves, 215, 219
436
Subject Index
Thomsen parameters, 2, 17-21, 24, 26, 36, 82, 92, 102, 107, 113, 115, 118, 131, 136, 143, 168, 187, 189, 195, 209, 243, 244, 253, 269-273, 283, 285, 287, 334, 336, 337, 339, 341, 343, 361, 389 Tilted transverse isotropy (TTI), s e e Anisotropic symmetry, tilted TI Time-to-depth conversion, 110, 114, 151, 232, 269, 271, 287, 347, 409, 410, 413 Transmission coefficient, 64 Transmission losses, 81 Transverse isotropy (TI), s e e Anisotropic symmetry, TI Traveltime, s e e Moveout velocity, NMO velocity, nonhyperbolic moveout Triclinic media, s e e Anisotropic symmetry, triclinic Triplication, s e e Cusp Tsvankin-Thomsen moveout equation, 184, 230, 232 TZO (transformation to zero offset), 305, 307, 310, 373, 414, 416 Undermigration, 266, 361,394 Velocity, s e e Phase velocity, Group velocity, Moveout velocity or NMO velocity Velocity analysis, s e e Inversion Velocity dispersion, 5 Velocity gradient lateral, 269, 389 vertical, 125, 136, 305, 323, 356, 365, 368, 417 Vertical transverse isotropy (VTI), s e e Anisotropic symmetry, VTI Voigt recipe, 8 VSP (vertical seismic profiling), 34, 82, 183, 327, 414 Wave equation anisotropic, 2, 3, 7, 17, 63, 103, 104 isotropic, 13, 386 Wave vector, 5
Wavefront, 4, 5, 16, 61, 63, 65, 79, 80, 82, 205, 392 ellipsoidal, 15, 28, 101 for orthorhombic media, 38, 39, 67, 71 for TI media, 19, 30, 83, 92, 98, 102, 131, 141,194, 200, 231,392 Weak-anisotropy approximation, 21, 270 for converted-wave moveout, 200, 208, 232, 241 for DMO operator, 374 for group velocity, 29, 32 for NMO velocity, 115, 119, 141, 161, 163, 164, 168, 256, 261, 263, 273, 284, 297 for nonhyperbolic moveout, 178, 182, 195 for phase velocity, 23, 26 for polarization, 35 for radiation pattern, 82, 84, 95, 101, 105 for reflection coefficient, 91, 94, 100, 101 for velocity in orthorhombic media, 45, 46, 54 Weyl integral, 61, 63, 64, 80, 104, 105 WKBJ solution, 386 Zero-offset migration, s e e Migration, poststack Zero-offset migration impulse response, 264 Zero-offset migration operator, 354 Zero-offset ray, 110, 112, 119, 124, 135, 141, 144, 149, 151, 161, 166, 169, 254, 261, 272, 291, 293, 346, 386 Zero-offset reflection, 109, 130-132, 156, 166, 169, 170, 244, 254, 263, 283, 291, 294, 299, 341, 346, 402 for converted waves, 200, 209, 215, 221,226 Zero-offset section, 111,125, 208, 254, 264, 265, 305, 307, 310, 341, 355, 357, 370, 372, 374, 375, 386, 392, 396
