Science, Faculty of
Earth, Ocean and Atmospheric Sciences, Department of
DSpace
UBCV
Cabrera Gomez, Jose Julian
2011-01-12T23:52:27Z
1990
Doctor of Philosophy - PhD
University of British Columbia
Plane-wave decomposition and slant stack transformation have recently gained much interest as viable routes to perform a variety of prestack processing tasks, such as velocity estimation, migration, filtering, deconvolution, and velocity inversion. To further complement the current advances, the problem of earth model parameter estimation and prestack structural imaging are addressed in this work. Unlike existing methods, the algorithms presented here make a novel and systematic use of the plane-wave domain to determine migration and interval velocities, interface dip angles and common-shot gather reflector images.
To start, a method is developed to estimate migration velocities and interface dip angles in earth models composed of planar, dipping reflecting interfaces separating homogeneous layers, and where straight-ray travelpaths to the reflecting interfaces can be assumed. The method consists of transforming a common-shot gather into the plane-wave domain, where a semblance analysis search along cosinusoid trajectories is performed. Since the cosinusoid trajectories are functions of the migration velocity and interface dip angle, selection of the maximum semblance values yields the best estimates to the desired earth model parameters.
To remove the straight-ray assumption of the velocity-dip analysis method, a recursive technique is developed to estimate
interval velocities and interface dip angles via a ray tracing algorithm. This technique essentially generates plane-wave domain traveltimes for a range of interval velocities and interface dip angles, and computes the error between the generated and observed plane-wave traveltimes. The minimum error determines the best estimates of the earth model parameters.
With the information attained in the velocity-dip analysis algorithm, a plane-wave based imaging method is developed to produce prestack common-shot gather images of the reflecting interfaces. The method consists of transforming a common-shot gather into the plane-wave domain, where a velocity-dip semblance analysis is performed. Then, the plane-wave components are downward extrapolated and recombined via a dip-incorporated inverse slant-stack transformation to produce the spherical-wave field that would have been recorded by receivers placed on the reflecting interfaces. The dip incorporation consists of redefining the angle of emergence of the plane waves. Finally, a simple mapping algorithm converts the offset and time coordinates of the reconstructed wave field to the true horizontal location and two-way vertical time of the reflection points. This results in the desired prestack migrated images of the reflecting interfaces.
In this thesis, a novel algorithm to perform plane-wave decomposition via Fourier transforms is also proposed. This algorithm consists of the application of the double fast Fourier
transform to the input data, followed by complex vector multiplications with essentially the Fourier representation of the Bessel function J0 . A numerical singularity is avoided by applying an analytical expression that approximately accounts for the singular point contribution. An inverse fast Fourier transform from frequency to time gives the desired plane-wave seismogram.
The techniques proposed in this work have yielded encouraging results on synthetic and field data examples. The examples demonstrate, for the first time, the systematic use of the plane-wave domain in processing seismic reflection data from common-shot gather data to the plane-wave domain, to velocity and dip angle analysis and to prestack structural imaging. It is believed that the results from this work will help researchers as well as practising geophysicists to become better acquainted with plane-wave domain processing.
https://circle.library.ubc.ca/rest/handle/2429/30604?expand=metadata
VELOCITY-DIP ANALYSIS IN THE PLANE-WAVE DOMAIN by JOSE JULIAN CABRERA GOMEZ B.Sc. Engineering Geophysics, Instituto Politecnico Nacional (Mexico), 1981 M.Sc. Geophysics, The University of British Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Geophysics and Astronomy) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1990 (S) Julian Cabrera, 1990 (A In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date MftrrL Z7{ \<iiO DE-6 (2/88) i i ABSTRACT Plane-wave decomposition and slant stack transformation have recently gained much in t e r e s t as v i a b l e routes to perform a var i e t y of prestack processing tasks, such as v e l o c i t y estimation, migration, f i l t e r i n g , deconvolution, and v e l o c i t y inversion. To further complement the current advances, the problem of earth model parameter estimation and prestack s t r u c t u r a l imaging are addressed i n t h i s work. Unlike e x i s t i n g methods, the algorithms presented here make a novel and systematic use of the plane-wave domain to determine migration and i n t e r v a l v e l o c i t i e s , interface dip angles and common-shot gather r e f l e c t o r images. To s t a r t , a method i s developed to estimate migration v e l o c i t i e s and i n t e r f a c e dip angles i n earth models composed of planar, dipping r e f l e c t i n g interfaces separating homogeneous layers, and where straight-ray travelpaths to the r e f l e c t i n g interfaces can be assumed. The method consists of transforming a common-shot gather into the plane-wave domain, where a semblance analysis search along cosinusoid t r a j e c t o r i e s i s performed. Since the cosinusoid t r a j e c t o r i e s are functions of the migration v e l o c i t y and interface dip angle, s e l e c t i o n of the maximum semblance values y i e l d s the best estimates to the desired earth model parameters. To remove the straight-ray assumption of the v e l o c i t y - d i p analysis method, a recursive technique i s developed to estimate i i i i n t e r v a l v e l o c i t i e s and interface dip angles v i a a ray tracing algorithm. This technique e s s e n t i a l l y generates plane-wave domain traveltimes f o r a range of i n t e r v a l v e l o c i t i e s and i n t e r f a c e dip angles, and computes the error between the generated and observed plane-wave traveltimes. The minimum error determines the best estimates of the earth model parameters. With the information attained i n the v e l o c i t y - d i p analysis algorithm, a plane-wave based imaging method i s developed to produce prestack common-shot gather images of the r e f l e c t i n g interfaces. The method consists of transforming a common-shot gather into the plane-wave domain, where a v e l o c i t y - d i p semblance analysis i s performed. Then, the plane-wave components are downward extrapolated and recombined v i a a dip-incorporated inverse s l a n t -stack transformation to produce the spherical-wave f i e l d that would have been recorded by receivers placed on the r e f l e c t i n g interfaces. The dip incorporation consists of redefining the angle of emergence of the plane waves. F i n a l l y , a simple mapping algorithm converts the o f f s e t and time coordinates of the reconstructed wave f i e l d to the true horizontal l o c a t i o n and two-way v e r t i c a l time of the r e f l e c t i o n points. This r e s u l t s i n the desired prestack migrated images of the r e f l e c t i n g interfaces. In t h i s thesis, a novel algorithm to perform plane-wave decomposition v i a Fourier transforms i s also proposed. This algorithm consists of the a p p l i c a t i o n of the double f a s t Fourier i v transform to the input data, followed by complex vector m u l t i p l i c a t i o n s with e s s e n t i a l l y the Fourier representation of the Bessel function J 0 . A numerical s i n g u l a r i t y i s avoided by applying an a n a l y t i c a l expression that approximately accounts f o r the singular point contribution. An inverse f a s t Fourier transform from frequency to time gives the desired plane-wave seismogram. The techniques proposed i n t h i s work have yielded encouraging r e s u l t s on synthetic and f i e l d data examples. The examples demonstrate, for the f i r s t time, the systematic use of the plane-wave domain i n processing seismic r e f l e c t i o n data from common-shot gather data to the plane-wave domain, to v e l o c i t y and dip angle analysis and to prestack s t r u c t u r a l imaging. I t i s believed that the r e s u l t s from t h i s work w i l l help researchers as well as p r a c t i s i n g geophysicists to become better acquainted with plane-wave domain processing. V TABLE OF CONTENTS Abstract i i L i s t of Tables v i i i L i s t of Figures x Acknowledgements xiv 1 Introduction 1 2 Plane-Wave Decomposition v i a Fast-Fourier Transforms .. 6 Fourier Transform Evaluation of the Plane-Wave Decomposition Integral 7 Treatment of the Numerical S i n g u l a r i t y 11 Examples of the Computation of Plane-Wave Seismograms v i a Fourier Transforms 18 Synthetic data 19 F i e l d data 20 3 Vel o c i t y - D i p Analysis i n the Plane-Wave Domain 34 Preliminaries 35 Plane-wave decomposition with p o s i t i v e and negative angles of emergence 35 Image-ray a r r i v a l 36 The Plane-Wave Decomposition (PWD) transformation 37 The slant-stack (r-p) transformation 38 v i Method 43 The case of a h o r i z o n t a l l y layered earth model 43 The case of an earth model composed of a planar dipping interface 45 The case of an earth model composed of several planar dipping interfaces 49 Recursive Estimation Interval V e l o c i t i e s 51 4 Ray-Tracing Algorithm f o r Velocity-Dip Estimation 64 Earth Model with two Planar Dipping Interfaces .... 64 Earth Model with N Planar Dipping Interfaces 68 Comments on the Estimation of Intercept-Time T r a j e c t o r i e s from the Observed t-x Seismograms .... 70 5 Examples of Velocity-Dip Estimation 75 The Case of Small V e l o c i t y Variations 76 The Case of Large V e l o c i t y Variations 79 F i e l d Data 84 6 Plane-Wave Domain Interface Imaging 118 Outline of the Method 119 Method 120 Step (a) : Plane-wave domain 12 0 Step (b) : V e l o c i t y - d i p analysis 121 Step (c): Plane-wave downward extrapolation . 122 Step (d): Redefinition of the angles of emergence 124 Step (e): Spherical-wave mapping 126 v i i Comments on the Imaging of Several Planar Dipping Interfaces 129 7 Examples of Plane-Wave Domain Interface Imaging 141 Imaging of Single Common-Shot Gathers 141 Imaging of a Synthetic Seismic Line 146 8 Conclusions 158 References 163 Appendix A. B r i e f Derivation of the Plane-Wave Decomposition Integral 168 Appendix B. Computation of the Angles s± , a± and bj f o r the Case of two Interfaces 173 Appendix C. Determination of the Term /V2 175 Appendix D. Intercept Time Equation f o r the Case of N Interfaces 177 Appendix E. Determination of the Angles aj and faj 179 Appendix F. Determination of the Terms Z 8 / j /Vj 182 v i i i LIST OF TABLES 1 Intercept times, angles of emergence of the normal rays and migration v e l o c i t i e s obtained f o r the case of small v e l o c i t y v a r i a t i o n s and conforming in t e r f a c e dips 77 2 Interval v e l o c i t y and dip angle values f o r the case of small v e l o c i t y v a r i a t i o n s and conforming int e r f a c e dips 77 3 Intercept times, angles of emergence of the normal rays and migration v e l o c i t i e s obtained f o r the case of small v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g interface dips 78 4 Interval v e l o c i t y and dip angle values f o r the case of small v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips 78 5 Intercept times, angles of emergence of the normal rays and migration v e l o c i t i e s obtained f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips 80 6 Interval v e l o c i t y and dip angle values f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips 80 7 Interval v e l o c i t y and dip angle values f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips (ray-tracing inversion algorithm) 81 ix 8 Intercept times, angles of emergence of the normal rays and migration v e l o c i t i e s obtained f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. Noisy Data 82 9 Interval v e l o c i t y and dip angle values f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. Noisy data 83 10 Interval v e l o c i t y and dip angle values f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips (ray-tracing inversion algorithm). Noisy data .... 83 11 Normal-ray traveltimes, migration v e l o c i t i e s and inte r f a c e dip angles f o r the two-shot-gather locations. F i e l d data 86 12 Normal-ray traveltimes, stacking v e l o c i t i e s and estimated dip angles obtained from the conventionally processed stack section 88 13 Interface depths. Imaging example 1 143 14 Interface depths. Imaging example 2 144 15 Interface depths. Imaging example 3 145 16 Migration v e l o c i t i e s , interface dip angles and normal ray traveltimes f o r the a n t i c l i n e earth model 149 17 Determination of the upward ray angles bj 180 18 Determintaion of the downward ray angles a< 181 X LIST OF FIGURES 1 Steps to compute plane-wave seismograms v i a Fourier transforms 22 2 Steps to compute plane-wave seismograms v i a Fourier transforms 23 3 Synthetic t-x seismograms 24 4 Earth Model 25 5 Plane-wave seismograms obtained v i a Method 1 26 6 Plane-wave seismograms obtained v i a Method 1 27 7 Plane-wave seismograms obtained v i a Mathod 2 28 8 Plane-wave seismograms obtained v i a Method 3 29 9 F i e l d data t-x seismograms 30 10 Plane-wave seismograms obtained v i a Method 1 31 11 Plane-wave seismograms obtained v i a Method 2 32 12 Plane-wave seismograms obtained v i a Method 3 33 13 Single layer earth model 54 14 t-x seismograms 55 15 Spherical and plane wavefronts 56 16 P o s i t i v e and negative angles of emegence for plane waves 57 17 Steps to obtain plane-wave seismograms f o r p o s i t i v e and negative angles of emergence 58 18 Total traveltime 59 19 Ray geometry of a plane wave 60 20 The normal-ray plane-wave component 61 x i 21 Processing steps i n the v e l o c i t y - d i p analysis algorithm 62 22 Approximate normal-ray t r a v e l paths i n a multi-layer earth model .... 63 23 Ray-path of a plane wave i n a two-layer earth model ... 74 24 Three-layer earth model with small v e l o c i t y v a r i a t i o n s and conforming dip angles 89 25 t-x seismograms for the earth model with small v e l o c i t y v a r i a t i o n s and conforming dip angles 90 2 6 Plane-wave seismograms for an earth model with small v e l o c i t y v a r i a t i o n s and conforming dip angles 91 27 Dip angle panel 92 28 V e l o c i t y panels for d i f f e r e n t dip angles 93 29 Three-layer earth model with small v e l o c i t y v a r i a t i o n s and opposing dip angles 94 30 t-x seismograms for the earth model with small v e l o c i t y v a r i a t i o n s and opposing dip angles 95 31 Plane-wave seismograms f o r an earth model with small v e l o c i t y v a r i a t i o n s and opposing dip angles 96 32 Three-layer earth model with large v e l o c i t y v a r i a t i o n s and opposing dip angles 97 33 t-x seismograms f o r the earth model with large v e l o c i t y v a r i a t i o n s and opposing dip angles 98 34 Plane-wave seismograms for an earth model with large v e l o c i t y v a r i a t i o n s and opposing dip angles 99 x i i 35 F i t t i n g values f o r the ray-tracing v e l o c i t y inversion example 100 36 Noisy synthetic seismograms. Large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g interface dip angles 101 37 Noisy plane-wave seismograms 102 38 Dip angle panel 103 39 V e l o c i t y panel for the dip angle of 0° 104 40 F i e l d data: common-shot gather 1 105 41 F i e l d data: common-shot gather 2 106 42 F i e l d data: plane-wave seismograms f o r shot 1 107 43 F i e l d data: plane-wave seismograms for shot 2 108 44 F i e l d data: dip angle panel for shot 1 109 45 F i e l d data: dip angle panel f o r shot 2 I l l 46 F i e l d data: v e l o c i t y panel f o r the dip angle of -17° .. 113 47 F i e l d data: v e l o c i t y panel f o r the dip angle of 17° .. 115 48 Portions of the stack section 117 49 Single dipping layer earth model 132 50 t-x and T-7 seismograms 133 51 Source and image source 134 52 Plane-wave downward extrapolation to a horizontal plane 135 53 Extrapolated T-7 and t-x seismograms 136 54 T i l t e d earth's surface 137 55 t-x seismograms on the r e f l e c t i n g interface 138 56 Shot-domain interface image 139 x i i i 57 Synthetic, two-way v e r t i c a l traveltime section 140 58 Shot-domain interface image. Three-layer earth model with conforming dip angles 150 59 Synthetic, two-way v e r t i c a l traveltime section 151 60 Shot-domain interface image. Three-layer earth model with c o n f l i c t i n g dip angles 152 61 Synthetic, two-way v e r t i c a l traveltime section 153 62 Shot-domain interface image. Three-layer earth model with c o n f l i c t i n g dip angles. Noisy data 154 63 Synthetic, two-way v e t i c a l traveltime section 155 64 A n t i c l i n e earth model 156 65 Interface image of the a n t i c l i n e earth model 157 66 Ray path of a plane wave i n a two-layer earth model ... 185 67 Travel path of the normal ray i n a two-layer earth model 186 68 Ray path of a plane wave i n a multi-layer earth model.. 187 69 Normal-ray t r a v e l paths i n a multi-layer earth model .. 188 x i v ACKNOWLEDGMENTS My most s i n c e r e thanks t o Shlomo Levy f o r h i s time spent with me d i s c u s s i n g many of the r e s e a r c h ideas p r e s e n t e d i n t h i s work. A l s o , my acknowledgments t o Dr. Matthew Y e d l i n and Dr. George Bluman f o r t h e i r a d v i c e on the development presented i n Chapter 2. My thanks t o Murray Roth f o r h i s w i l l i n g n e s s t o p r o o f - r e a d the f i n a l manuscript. Many o t h e r people helped me i n one way or another d u r i n g my t h e s i s work. My s i n c e r e g r a t i t u d e t o a l l of them. T h i s work was i n p a r t sponsored by a graduate s c h o l a r s h i p from the N a t i o n a l C o u n c i l f o r Science and Technology (CONACYT) of Mexico. 1 CHAPTER 1 INTRODUCTION One of the aims i n conventional seismic data processing i s to obtain a migrated stacked section v i a a series of processing steps, among which one finds the following basic ones: data sorting from common-shot gathers ( i . e . , f i e l d gathers) to common midpoint gathers; stacking-velocity determination v i a a semblance analysis along hyperbolic moveout t r a j e c t o r i e s ; normal moveout correction; dip moveout correction; stacking; and migration. The f i n a l section i s interpreted to represent the true horizontal s p a t i a l locations of, and the two-way v e r t i c a l traveltimes to, the r e f l e c t i n g i n t e r f a c e s . Considerable research and progress have been made to improve the preceding basic processing sequence i n order to a t t a i n better r e f l e c t o r images (for example, see Cohen and B l e i s t e i n (1979), Berkhout (1984) , erase (1989), among others) . While these works have developed very sophisticated and t h e o r e t i c a l l y rigorous imaging methods, the present work explores yet another much simpler (but i n s i g h t f u l ) route to a t t a i n images of the r e f l e c t i n g interfaces. In the plane-wave domain, a basic processing sequence i s developed to go from common-shot gathers to f i n a l migrated signatures of the r e f l e c t i n g i nterfaces. 2 A large number of authors have presented plane-wave domain methods to process seismic r e f l e c t i o n data. The scope of these methods ranges from the computation of plane-wave seismograms to multidimensional f i l t e r i n g , v e l o c i t y estimation and migration (for example, see Schultz and Claerbout, 1978; Clayton and McMechan, 1981; Schultz, 1982; Stoffa et a l . , 1981; Tatham et a l . , 1982; T r e i t e l et a l . , 1982; Cabrera and Levy, 1984; O t t o l i n i and Claerbout, 1984; Brysk and McCowan, 1986; Reshef and Kosloff, 1986; Benoliel et a l . , 1987). Although these works have demonstrated some of the t h e o r e t i c a l and p r a c t i c a l advantages gained i n the plane-wave domain when processing seismic r e f l e c t i o n data, to the author's knowledge none has addressed the use of the plane-wave domain to go a l l the way from common-shot gathers to migrated r e f l e c t i o n signatures. In t h i s work, common-shot gathers w i l l be transformed to the plane-wave domain where a semblance analysis along cosinusoid t r a j e c t o r i e s w i l l be performed. This analysis w i l l y i e l d estimates of the migration v e l o c i t i e s and interface dip angles of the r e f l e c t i n g i nterfaces. With t h i s information, plane-wave downward extrapolation, and rotation w i l l be c a r r i e d out to reconstruct the wave f i e l d that would have been recorded had the seismic receivers been placed on the r e f l e c t i n g interfaces. The resultant r e f l e c t i o n signatures w i l l be mapped to the true horizontal l o c a t i o n of, and to the two-way v e r t i c a l traveltimes to, the r e f l e c t i o n points. The f i n a l r e s u l t i s a time migrated common-shot gather (CSG). For a complete seismic survey, the plane-wave domain processed CSG's are 3 sorted into common-receiver gathers and stacked to obtain a f i n a l migrated seismic section of the earth's subsurface. This work i s divided as follows: chapter 2 reviews the subject of plane-wave decomposition and presents two fast - F o u r i e r transform based algorithms to perform plane-wave decomposition. The performance of these algorithms, demonstrated on synthetic and f i e l d data sets, i s reasonably good to process seismic data i n an industry environment. Since these algorithms were developed by the author as an extension of h i s M.Sc. research (Cabrera, 1983), they are presented here only as complementary work to t h i s t h e s i s . The reader who i s f a m i l i a r with the subject of plane-wave decomposition or who i s interested only i n the v e l o c i t y - d i p analysis and imaging methods can s t a r t d i r e c t l y i n chapter 3. Assuming earth models consisting of l o c a l l y planar dipping interfaces, chapters 3 and 4 present novel plane-wave domain methods to estimate migration and i n t e r v a l v e l o c i t i e s , and in t e r f a c e dip angles. The proposed approaches can be considered as an extension of the layer s t r i p p i n g method of Schultz (1982) with some i n t u i t i v e understanding gained from the works of T r e i t e l et a l . (1982) and S t o f f a et a l . (1981). The following steps are required i n the estimation procedures: * Decomposition of a s p l i t spread common-shot gather into i t s plane-wave components. 4 * V e l o c i t y - d i p semblance analysis to i d e n t i f y migration v e l o c i t i e s and approximate interface dip angles. * For the case of small v e l o c i t y v a r i a t i o n s , a p p l i c a t i o n of a Di x - l i k e recursive r e l a t i o n (Dix, 1955) to calculate i n t e r v a l v e l o c i t i e s . * For the case of large v e l o c i t y v a r i a t i o n s , implementation of a ray t r a c i n g algorithm to compute i n t e r v a l v e l o c i t i e s and interface dip angles. The performance of the proposed earth-model parameter estimation algorithms w i l l be shown i n several data examples presented i n chapter 5. From the understanding gained and assumptions made i n the v e l o c i t y - d i p algorithm method presented i n chapter 4, a prestack plane-wave domain imaging method i s developed i n chapter 6. This method produces common-shot gather s t r u c t u r a l images of the interf a c e r e f l e c t i o n points v i a a systematic use of the plane-wave domain. The approach consists of decomposing the recorded common-shot gather into i t s plane-wave components. In the plane-wave domain, migration v e l o c i t i e s and interface dip angles are estimated v i a the v e l o c i t y - d i p analysis algorithm of chapter 3. With t h i s information, the plane-wave components are downward extrapolated, rotated and recombined to obtain the spherical-wave f i e l d that would have been recorded by receivers placed on the r e f l e c t i n g interfaces. 5 F i n a l l y , a simple mapping algorithm i s applied to the reconstructed seismograms to obtain the desired interface image of the r e f l e c t i o n points. Since the en t i r e procedure i s based on the kinematic c h a r a c t e r i s t i c s of plane waves, the resultant subsurface images only y i e l d r e l i a b l e locations of the model interfaces (no true amplitude information i s attained) . For the case of many common-shot gathers, the resultant migrated gathers are sorted into common-receiver gathers and stacked to obtain a migrated seismic section. Chapter 7 i l l u s t r a t e s the performance of the plane-wave domain imaging approach on synthetic data. The encouraging r e s u l t s obtained here demonstrate the p o t e n t i a l use of the new technique to image seismic r e f l e c t i o n data. The l a s t chapter, number 8, presents the conclusions and suggestions f o r future work. B a s i c a l l y , i t i s concluded that assuming straight-ray travelpaths f o r plane waves and earth models composed of l o c a l l y planar dipping interfaces separating homogeneous layers, the new plane-wave domain techniques represent v i a b l e and a l t e r n a t i v e routes (to conventional seismic data processing) to go from common-shot gathers, to the plane-wave domain, to v e l o c i t y and interface dip angle estimation, to prestack s t r u c t u r a l r e f l e c t o r imaging, and to migrated common-receiver stacked sections. CHAPTER 2 6 PLANE-WAVE DECOMPOSITION VIA FAST FOURIER TRANSFORMS Plane-wave decomposition of point-source data i s a process to obtain the corresponding earth plane-wave r e f l e c t i o n response for an a r b i t r a r y angle of emergence 7 and plane-wave traveltime T. This response i s known as a plane-wave seismogram and i s denoted by U(T,7) . The idea of representing a spherical-wave front generated by a point source as a superposition of plane-wave components goes back to Whittaker (1902), Sommerfeld (1909) and Weyl (1919) (modern reviews of the pertinent theory are given i n Brekhovskikh (1960), Bath (1968), Goodman (1968), Devaney and Sherman (1973), Born and Wolf (1980) and Aki and Richards (1980)) . However, the idea of computing and using plane-wave seismograms i n seismic data processing dates back only to 1971 with the work of Muller (1971) . Since then, many authors have addressed the problem of computing plane-wave seismograms from the acquired seismic r e f l e c t i o n seismograms (for example, St o f f a et a l . (1981), T r e i t e l et a l . (1982), Cabrera and Levy (1984) , and Harding (1985)) . Although many of these works have presented methods to perform plane-wave decomposition, i n t h i s chapter two additional methods w i l l be developed. Since these methods e x p l o i t the computational e f f i c i e n c y of the f a s t Fourier 7 transform, they w i l l be useful to obtain plane-wave seismograms i n a computationally intensive processing environment. Fourier Transform Evaluation of the Plane-Wave Decomposition Integral For a given frequency w the Fourier transform of a plane-wave seismogram U ( T , 7 ) i s calculated v i a ( T r e i t e l et a l . , 1982, appendix A ) U ( w , 7 ) = rS(«,r)J 0((r«sin7)/V) dr (2.1) In t h i s equation, S i s the Fourier transform of the recorded spherical-wave f i e l d s p e c i f i e d at the r a d i a l o f f s e t r, J 0 i s the Bessel function of the f i r s t kind and zero order, V i s the surface compressional-wave v e l o c i t y , 7 i s the angle of emergence measured with respect to the normal to the earth's surface, and T i s the plane-wave traveltime (also known as intercept time or v e r t i c a l delay traveltime). The inverse Fourier transform of U ( « , 7 ) gives the plane-wave seismogram ( i . e . , the plane-wave response) U ( T , 7 ) corresponding to 8 the plane-wave component that emerged at the earth • s surface with an angle 7 and had an intercept time T. Since the above equation i s s t r i c t l y v a l i d f o r a r a d i a l l y symmetric, i s o t r o p i c earth (Muller, 1971), i n what follows the r a d i a l o f f s e t coordinate r w i l l be replaced by the Cartesian o f f s e t coordinate x and no d i s t i n c t i o n w i l l be made i n the d i r e c t i o n of t h i s coordinate. The r a d i a l wavenumber k r = «p = wsin(7)/V, where p i s the ray parameter corresponding to the Snell's ray path with angle of emergence 7, i s introduced i n equation (2.1) to give: with k r = wsin(7)/V, and x equal to the source-receiver o f f s e t . The plane-wave decomposition algorithms to be developed i n t h i s chapter are based on the solu t i o n of equation (2.2) v i a Fourier transforms. To t h i s end, the following Fourier transform p a i r i s used: 00 U(w,7) = xS(w,x)J 0(k rx) dx , (2.2) 0 — xS(«,x)exp{-ik xx) dx 2* J - 00 and 00 n XS (w,X) S x(w,k x)exp{ik xx} dk x , (2.3) - 00 9 where u i s the temporal frequency and k x i s the Fourier horizontal wavenumber va r i a b l e . Substitution of equation (2.3) into (2.2) y i e l d s U(w,7) - d k x S x ( w , k x ) J 0 ( k r x ) e x p { i k x x ) dx , (2.4) with k r = usin(7)/V. The inner-most i n t e g r a l , I ( k x ; k r ) = J 0(k rx)exp{ik xx} dx , has the following a n a l y t i c a l solution (Gradshteyn and Ryzhik ,1980, equations 6.671.7 and 6.671.8) I ( k x ; k r ) = 2 2 k r - k x 2 2 k x - k r 2 2 i f k x < k r , 2 2 i f k x > k r , 2 2 i f k x = k r. 10 Application of these r e s u l t s to equation (2.4) produces, CO n U(«,7) = u - CO •J k r 2 - k x 2 s g n ( k r 2 - k x 2 ) dk x , (2.5) where -\J k r 1 -k x* = i -J k x *-k r z' f o r k x 2>k r 2, and sgn(k r 2 - k x 2 ) = * 1 when k r 2 - k x 2 £ 0, Equation (2.5) forms the basis of the f i r s t plane-wave decomposition algorithm proposed i n t h i s chapter. Its implementation requires the following sequence of steps (see Figure 1): a. ) M u l t i p l i c a t i o n of the input seismograms, S(t,x), by t h e i r respective o f f s e t values, x; that i s S x(t,x) = xS(t,x). b. ) A two-dimensional forward (fast) Fourier transform (FFT) to obtain S x ( u , k x ) ; that i s S x(u,k x) = F F T 2 D ( S x ( t , x ) } . c. ) Numerical evaluation of the i n t e g r a l (2.5). For computational speed, t h i s i n t e g r a l may be computed as a simple summation over the v a r i a b l e k x. However, since the denominator approaches zero as k x approaches k r, a small p o s i t i v e quantity should be added to the denominator to avoid the numerical d i f f i c u l t y whenever k x 2 = k r 2 . This step must be repeated f o r a l l the desired range of frequency and r a d i a l wavenumber values; t h i s step produces U ( w , 7 ) . d. ) Inverse Fourier transform with respect to frequency to obtain the f i n a l plane-wave seismograms U ( T , 7 ) ; that i s 11 U ( T,7) = F F T - 1 { U ( W , 7 ) }. I t must be emphasized that the evaluation of (2.5) requires the numerical s t a b i l i z a t i o n of the integrand whenever k x 2 = k r 2 . Although i n p r a c t i c e the addition of a small p o s i t i v e quantity, as stated i n step (c) above, normally s u f f i c e s to achieve numerical s t a b i l i z a t i o n , f o r the sake of t h e o r e t i c a l completeness the behaviour of the integrand i n equation (2.5) w i l l be analysed around the point k x=k r. Treatment of the Numerical S i n g u l a r i t y To s t a r t , equation (2.5) i s s p l i t as follows: U(u,7) where U!(«,7) + U 2(«,7) + U 3 ( u,7) + U<( w,7) -k, U x ( u,7) = -S x(«,k x) d* x , u •00 1 k 2-k 2 r *x 0 n U 2 ( u,7) = S x ( w / k x ) dk x , (2.6) 12 s x ( « , k x ) u 3 («,-y) = dk x , and J -J k r 2 - k x 2 0 n S x(«,k x) U 4(w,7) = - dk x with k r = wsin(7)/V. Computer implementation of the above in t e g r a l s v i a a simple but computationally fa s t d i s c r e t e summation w i l l encounter numerical problems for k x values near or equal to k r, since i n these cases the denominator approaches zero. Below, an approximate a n a l y t i c a l s o l u t i o n to the above integrations for k x z k r w i l l be presented. For the sake of brevity, only the t h i r d i n t e g r a l of (2.6) w i l l be analysed. The r e s u l t w i l l be applied to the other i n t e g r a l s . The t h i r d i n t e g r a l of (2.6) reads as k r n s x ( w , k x ) U 3 (u,7) dk x . 0 with k r = usin(7)/V. 13 This expression may be rewritten as k r-c U 3 (w,7) = S x ( u , k x ) •I k r 2 - J dk x + s x ( w , k x ) dk X ' (2.7) k r-e k 2 *x where e i s a small p o s i t i v e quantity. Assuming that f o r a given value of c the f i r s t i n t e g r a l of (2.7) i s numerically well behaved, the goal i s to obtain an a n a l y t i c a l approximation to the second i n t e g r a l . The f a c t o r i z a t i o n -s] k r z - k x z = -J k r+k x -vj k r-k x and the change of var i a b l e k x=k r-<r applied to the second i n t e g r a l of (2.7) (hereafter denoted by U s 3 ) gives U s 3 ( u , 7 ) = S x( w,k_-<r ) d<r where S x( w,k r - o - ) = S x ( u , k _ - o - ) •J 2kr-< (2.8) Replacing S x (w ,k r -<r ) by i t s Taylor s e r i e s expansion around the point k r gives € ( 1 <r - \ U s 3 ( w , T ) = S x(«,k r) S^( M,k r) do- + 14 (-i) n! S x ( « , k r ) do- , (2.9) n = 2 where S x ' and S x n are the f i r s t and nth deri v a t i v e of S x ( « , k r ) with respect to k x , respectively. Performing the term-by-term integrations of (2.9) gives U 8 3 (« , i r ) = S x ( w , k r ) [2<r 1 / 2]o - S^(u,k r ) [(2/3 )<r 3 / 2]o ) S x(«,k r) ^—' n! rti+ 1/2 "I n+1/2 n = 2 = 2 S x ( u , k r ) 6 1 / 2 S » ( u , k r ) € 3 /2 ("D -„ + y — s x ( w , k r ) ^—' n! .n + l / 2 n+1/2 n = 2 (2.10) For the evaluation of t h i s expression and fo r computational speed, the following two approximations are made: f i r s t , only the f i r s t two terms of equation (2.10) w i l l be used (for a small value of c, the error introduced by t h i s approximation w i l l be of order 0( e 2) ) ; and second, the numerical computation of the f i r s t d e r i v a t i v e of S x (u,k r) with respect to k x w i l l be made v i a a f i r s t - o r d e r f i n i t e -d i fference backward derivative formula. With these approximations, 15 equation (2.10) i s rewritten as: US3(W,-Y) S x(«,k r-e) - S x ( u , k r ) + 2 € 1 / 2 s x ( W / k r ) . Some algebra and subs t i t u t i o n of S x by S x (see equation 2.8) gives U s 3( W,'V) = - -fe 3 S x(«,k r-e) S x(«,k r) + 2 . -J 2k r-€ 2kt with k r = «sin('y)/V. Reca l l i n g that U s 3 represents the second i n t e g r a l of the r i g h t hand side of equation (2.7), t h i s l a t t e r equation now reads as k r- e U 3(u,7) z S x ( w , k x ) •vj k r 2 - k x 2 dk x + (2.11a) 3 S x ( w , k r - 0 S x(«,k r) . + 2 -vj 2 k r - i -J 2k r with k r = «sin(-y)/V. Similar r e s u l t s are obtained f o r the other i n t e g r a l s of equation (2.6): - k r - c U i ( W / 7 ) = i u -oo S x(«,k x) •J k x 2-k r' dk x + (2.11b) 16 3 S x(w,-k r - e ) S x ( w , - k r ) + 2 - J 2kr+< • J 2 k r 0 n U2(w,7) ~ S x(«,k x) -k r+£ dk x + k ' r A x (2.11C) 3 S x ( W / £ - k r ) S x ( w , - k r ) + 2 2 k r - e • J 2 k r U4(w,7) r i S x(«,k x) dk x + kr+e k 2 - k ' ( 2 . l i d ) 1 — ]^ € 3 s x ( w , k r + e ) s x ( « , k r ) + 2 . - J 2 k r + i 2k, with k_ = «sin(7)/V. Equations (2.11a-d) form the basis of the second plane-wave decomposition algorithm proposed i n t h i s chapter. The computational steps (represented i n Figure 2) needed e s s e n t i a l l y are the same as those of the plane-wave decomposition method developed i n the previous section (see p.1 0 ) . In the present case, however, evaluation of Vlt U 2, U 3 and U 4 v i a equations (2.11a-d) also accounts for the integrand contribution from around the numerical 17 s i n g u l a r i t y at k x=k r. The summation of the computed U l f U 2, U 3 andU 4 y i e l d s an approximation to U(w,7) (inverse Fourier transform of U(u,7) gives the desired plane-wave seismogram U(T,7) ). The incurred approximation i s caused by the use of the f i r s t - o r d e r f i n i t e d ifference d e r i v a t i v e of S (u,k r) and by having neglected the second and higher order derivatives of S (u,k r) from the computations. The numerical evaluation of equations (2.11a-d) requires a sui t a b l e value for the quantity epsilon. This quantity needs to be large enough to make a simple numerical evaluation of the integrals i n (2.11a-d) stable, but small enough to reasonably approximate the f i r s t d e r i v a t i v e of S (u,k r) by i t s f i r s t - o r d e r f i n i t e difference d e r i v a t i v e . In theory, the choice of epsilon i s not straightforward, since the involved integrations depend on the behaviour of the Fourier representation of the input data, and on the frequency content, the angular component range and the surface v e l o c i t y V used i n the computations (epsilon has the same units as k r ) . In p r a c t i c e , epsilon can be made equal to 6 - «sin(6)/V , (2.12) so that a ser i e s of angular values B can be t r i e d . In the t e s t runs performed i n t h i s work, a value of 0.5° f o r beta gave good r e s u l t s . This value i s one h a l f of the angular sampling i n t e r v a l of the computed plane-wave seismograms. In a seismic data processing industry environment, where plane-wave decomposition needs to be applied to many data gathers (e.g., 18 200 or more), i t i s recommended to se l e c t a representative data gather, perform plane-wave decompositon v i a the algorithm presented here f o r d i f f e r e n t values of B (equation 2.12), and compare the r e s u l t s to those obtained v i a the o r i g i n a l plane-wave decomposition i n t e g r a l , equation (2.2) . Once a reasonable match i s attained, the chosen B value can be used for a l l the remaining transformations of the data gathers. Examples of the Computation of Plane-Wave Seismograms v i a Fourier Transforms. By computing plane-wave seismograms on synthetic and f i e l d data sets, the performance of the plane-wave decomposition (PWD) methods developed i n the previous sections w i l l be i l l u s t r a t e d and compared to the r e s u l t s obtained v i a the o r i g i n a l PWD i n t e g r a l . In what follows, the algorithms based on equations (2.5),(2.11) and (2.2) are referred to as Methods 1, 2 and 3 respectively. That i s , plane-wave decomposition i s performed by Method 1 v i a a complex summation along the wavenumber variable k x without any special treatment of the numerical s i n g u l a r i t y ; by Method 2 v i a the a n a l y t i c a l approximation to the numerical s i n g u l a r i t y ; and by Method 3 v i a the o r i g i n a l plane-wave decomposition i n t e g r a l . 19 Synthetic data.- Figure 3 shows a one-sided common-source gather featuring 60 equally spaced traces, a frequency band from 2 to 40 Hz, a sampling i n t e r v a l of 0.004 s, and an o f f s e t range from 25 to 1500 m. The seismograms were computed v i a a kinematic forward modelling program ( B e r r y h i l l , 1979) applied to the earth model of Figure 4. Figures 5, 7 and 8 display the plane-wave seismograms obtained v i a Methods 1, 2 and 3, respectively. The o v e r a l l energy d i s t r i b u t i o n as well as event consistency and lo c a t i o n are s i m i l a r among the three sets of r e s u l t s . The following remarks are drawn from these r e s u l t s : a. ) The seismograms obtained v i a Methods 1 and 2 show coherent, near v e r t i c a l , l i n e a r events that are not present i n the r e s u l t s of Method 3. These undesirable e f f e c t s are att r i b u t e d to greater s e n s i t i v i t y of the algorithms to FFT wrapping-around e f f e c t s (see Figure 6, where the input data used was not padded with zeroes) . I t should be noted that the l i n e a r events with a shape of "<" o r i g i n a t i n g at 1.5s, 2.2s and 2. 5s i n the plane-wave seismogram 7=0° are caused by the end of the hyperbolic r e f l e c t i o n t r a j e c t o r i e s at o f f s e t 1500m i n the input data (Figure 3) . These end e f f e c t s can be attenuated by tapering the amplitudes of the input r e f l e c t i o n t r a j e c t o r i e s . b. ) The seismograms obtained v i a Methods 1 and 2 manifest a decrease ( as compared to the seismograms of Method 3) i n energy content at small angles of emergence (see f o r example the t r a j e c t o r y at 2.4 s) . This i s att r i b u t e d to the numerical 20 behaviour of the integrands of equations (2.5) and (2.11) at k r values near to zero. In t h i s case, there are numerical d i f f i c u l t i e s because the small quantity epsilon used i n the computations behaves as an adverse weight to the integrands. For Method 1 or 2, the input data should be padded with zeroes i n both the s p a t i a l and temporal axes i n order to avoid wrapping-around e f f e c t s . The amount of zero padding should be between two and four times the axis lengths. F i e l d data.- Figure 9 displays a preprocessed shot gather characterized by a s p a t i a l sampling i n t e r v a l of 20 m, a frequency band from 10 to 60 Hz, and a time sampling i n t e r v a l of 0.002 s. The preprocessing consisted of the a p p l i c a t i o n of elevation corrections, r e f r a c t i o n s t a t i c s , data muting and amplitude balancing. Plane-wave decomposition v i a Methods 1, 2 and 3 produced the r e s u l t s shown i n Figures 10, 11 and 12, respectively. In these panels, the following i s observed: a. ) The behaviour of the refracted wave (event around T=0.1 S and 7=80°) i s s i m i l a r i n the three cases. b. ) The r e f l e c t i o n events (for example, those s t a r t i n g at (T,7) = (0.9s,10°) , (1.0s,8°) and (1.3s,2°) ) show s i m i l a r amplitude and phase behaviour, although some differences 21 are observed at small angles of emergence. The r e f l e c t i o n event around (T,7) = (0.2s,4°) d i f f e r s the most among the three sets of r e s u l t s . The observed discrepancies are a t t r i b u t e d to the numerical problems i n evaluating the inverse square-root operator for small r a d i a l wavenumbers ( i . e . , f o r small angles of emergence and temporal frequencies) . In these cases, the quantity epsilon used to s t a b i l i z e the algorithms becomes numerically large as compared to the desired wavenumber values, c.) The coherent noise (e.g., the l i n e a r events seen between (T,7) = (0.6s,2°) and large intercept times and angles of emergence), a l i a s noise (e.g., the h o r i z o n t a l events appearing at a l l intercept times and angles of emergence larger than 75°) and random background noise are s i m i l a r among the three sets of r e s u l t s . From the above, i t i s concluded that i n a production-oriented seismic processing environment the proposed methods w i l l produce plane-wave seismograms comparable to those produced by the o r i g i n a l PWD i n t e g r a l (equation 2.2). However, the new methods w i l l e f f i c i e n t l y run i n array processors and supercomputers, where f a s t -Fourier transform and vector operations have been highly optimized. 22 Figure 1. Steps required to perform plane-wave decomposition v i a Fourier transforms: (a) the input t-x record i s Fourier transformed to the frequency-wavenumber (FK) domain (b). Frequency s l i c e s from t h i s domain are m u l t i p l i e d by the inverse-square root operator (schematically shown i n (c)) to y i e l d a complex number that i s stored i n the frequency-angle of emergence domain (d). A small p o s i t i v e quantity e i s used to avoid the numerical s i n g u l a r i t y at k r. A one-dimensional inverse Fourier transform converts the domain (d) into the f i n a l plane-wave seismograms (e). 23 T-X RECORD F-K DOMAIN 0 DOMAIN F-y DOMAIN e d Figure 2. Steps required to perform plane-wave decomposition v i a Fourier transforms: (a) the input t-x record i s Fourier transformed to the frequency-wavenumber (FK) domain (b) . Frequency s l i c e s (c) from t h i s domain are m u l t i p l i e d by the inverse-square root operator. An a n a l y t i c a l expression i s used to compute the i n t e g r a l contribution from the numerical s i n g u l a r i t y at k r (see equation 2.11) . The r e s u l t from t h i s step i s stored i n the frequency-angle of emergence domain (d) . A one-dimensional inverse Fourier transform converts the domain (d) into the f i n a l plane-wave seismograms. OFSET 24 Figure 3. T-X seismograms generated by a Kirchhoff wavefield extrapolation algorithm. Offsets range from 25 m to 1500 m; receiver spacing i s 25 ra. The source function i s a Ricker wavelet with center frequency of 16 Hz. 25 -6,00 , -2,20 , 200 , 600 , 1 l 1^ 00 , 16,00 , lil_a n n • o • • Vx = 1500 m/s V2 = 1800 m/s V3 = 2200 m/s Figure 4. Earth model composed of three horizontal layers. ANGLE OF EMERGENCE 10 20 30 40 50 60 70 80 90 3.0- 1 Figure 5. Plane-wave seismograms obtained from the decomposition of the t-x seismograms of Figure 3 v i a Method 1. The seismograms were computed for angles of emergence between 1° and 90° at 1° i n t e r v a l ; the surface v e l o c i t y used i n the decomposition was 1500 m/s. The marked, near v e r t i c a l , l i n e a r event i s a wrapping around e f f e c t . The l i n e a r events shaped "<" and emanating from 7=1° at 1.5, 2.2 and 2.5 s originate from the f a r - o f f s e t ends of the r e f l e c t i o n t r a j e c t o r i e s of Figure 3. S i m i l a r l y , the p o s i t i v e curvature of the plane-wave signatures for large angles (e.g., on the f i r s t one, beyond 7=45°) i s caused by the same data ends. Figure 6. Same as i n Figure 5, except that the input seismograms were not padded with zeroes. Note the strong l i n e a r events, which are FFT wrapping around e f f e c t s . ANGLE OF EMERGENCE Figure 7. Plane-wave seismograms obtained from the decomposition of the t seismograms of Figure 3 v i a Method 2. ANGLE OF EMERGENCE 29 Figure 8. Plane-wave seismograms obtained from the decomposition of the t-x seismograms of Figure 3 v i a Method 3 ( o r i g i n a l plane-wave decomposition i n t e g r a l ) . 30 Figure 9. Shot gather a f t e r the a p p l i c a t i o n of elevation corrections, r e f r a c t i o n s t a t i c s , data muting and amplitude balancing. 31 ANGLE OF EMERGENCE Figure 10. Plane-wave seismograms obtained from the decomposition of the t-x seismograms of Figure 9 v i a Method 1. Angular sampling i s 1°. The strong energy seen around T=0.080 S, 7=80" corresponds to the f i r s t t-x r e f r a c t i o n a r r i v a l . ANGLE OF EMERGENCE Figure 11. Plane-wave seismograms obtained from the decomposition of the t seismograms of Figure 9 v i a Method 2. ANGLE OF EMERGENCE 10 20 30 40 50 60 70 80 90 Figure 12. Plane-wave seismograms obtained from the decomposition of the t -seismograms of Figure 9 v i a Method 3. Compare to Figures 10 and 11 34 CHAPTER 3 VELOCITY-DIP ANALYSIS IN THE PLANE-WAVE DOMAIN Assuming straight-ray t r a v e l paths i n an earth model composed of planar dipping interfaces separating homogeneous layers, a method to estimate migration v e l o c i t i e s , i n t e r v a l v e l o c i t i e s and interface dip angles w i l l be developed. Although the proposed method i s s t r i c t l y v a l i d f o r constant-velocity earth models, assuming straight-ray travelpaths to the r e f l e c t i n g interfaces, the proposed method can be applied to non-constant v e l o c i t y earth models. The f i r s t section of t h i s chapter presents some basic concepts of the plane-wave domain and of the plane-wave decomposition (PWD) and slant-stack transformations. The second and t h i r d sections introduce the new earth-model parameter estimation technique. A ser i e s of examples w i l l be deferred u n t i l chapter 5 i n order to complement the foregoing theory with a ray-tracing based i n t e r v a l -v e l o c i t y estimation procedure (chapter 4 ) . Preliminaries Plane-wave decomposition with p o s i t i v e and negative angles of emergence. - Since the v e l o c i t y - d i p analysis method to be developed i s based on the analysis of ray paths corresponding to plane-wave components emerging at the earth's surface with p o s i t i v e and negative angles of emergence, i t i s important to e s t a b l i s h the need to consider plane-wave decomposition for both of these d i r e c t i o n s . Consider a two-dimensional earth model composed of a planar, dipping interface separating two homogeneous, i s o t r o p i c , acoustic media characterized by P-wave v e l o c i t i e s and V 2 (see Figure 13) . A point source situated on the earth's surface generates a spherical wavefront that i s r e f l e c t e d by the interface RR' . Receivers l a i d out on the earth' s surface record the r e f l e c t e d energy to produce a set of t-x seismograms. These are schematically drawn i n Figure 14. To a f i r s t approximation, a spherical wavefront maybe thought of as the envelope of plane waves that propagate with the same speed V : along d i r e c t i o n s s p e c i f i e d by rays perpendicular to the wavefront (Figure 15) . For the case of a spl i t - s p r e a d receiver geometry, these rays determine d i s t i n c t plane-wave responses at each side of the source l o c a t i o n : the rays a r r i v i n g at receivers located on the down-dip side of the source have only p o s i t i v e angles of emergence (with respect to the v e r t i c a l ) whereas those a r r i v i n g on the up-dip side of the source have p o s i t i v e and negative angles of emergence (see 36 Figure 16) . Consequently, when working with a sp l i t - s p r e a d common-source gather i n an earth model composed of planar dipping interfaces, PWD should be executed for both p o s i t i v e and negative angles of emergence. Having established the need to perform plane-wave decomposition for both p o s i t i v e and negative angles of emergence, i t i s now required to i d e n t i f y the o r i g i n of the angular coordinate. Below, i t i s seen that the image ray plane-wave component constitutes a natural o r i g i n from which p o s i t i v e and negative angular d i r e c t i o n s can be measured. Imacre-ray a r r i v a l . - An image ray i s defined as that ray with minimum traveltime path. Hubral (1977) shows that t h i s ray i s i d e n t i f i e d with the apex of the (t,x) r e f l e c t i o n t r a j e c t o r y . For example, the r e f l e c t i o n hyperbola (Figure 14) corresponding to the model of Figure 13 has a minimum traveltime at o f f s e t - X i , * t h i s o f f s e t i d e n t i f i e s the loc a t i o n where the image ray i s recorded. Since an image ray emerges perpendicular to the earth's surface (Hubral, 1977), i t i s used to define p o s i t i v e and negative angles of emergence of plane waves. That i s , f o r a given r e f l e c t i o n signature, the rays recorded on the up-dip side of the image-ray lo c a t i o n have negative angles of emergence, whereas those observed on the down-dip side have p o s i t i v e angles. This means that upon PWD and f o r a given r e f l e c t i o n t r a j e c t o r y , the receivers located at the l e f t side 37 of the image-ray o f f s e t w i l l transform into plane-wave seismograms for negative angles of emergence, while those receivers found at the r i g h t side of the image-ray o f f s e t w i l l y i e l d plane-wave seismograms for p o s i t i v e angles. The PWD transformation.- Here, a formal technique i s proposed to obtain plane-wave seismograms f o r p o s i t i v e and negative angles of emergence v i a the plane-wave decomposition i n t e g r a l (equation (2.2)) . At t h i s point, however, i t should be noted that because the v e l o c i t y - d i p analysis method to be developed i n the subsequent sections requires only the kinematic information of the plane-wave responses, i t i s not e s s e n t i a l to use the plane-wave decomposition i n t e g r a l to compute the plane-wave seismograms. Instead, the sl a n t -stack transformation (which i s computationally faster) can be used to compute plane-wave seismograms for p o s i t i v e and negative angles of emergence (refer to the next subsection). The plane-wave decomposition i n t e g r a l ( T r e i t e l et a l . , 1982; Cabrera and Levy, 1984; chapter 2) i s commonly used to obtain the plane-wave components of the spherical-wave seismograms. For a given angular frequency w t h i s i n t e g r a l (equation 2.2) reads as 00 n U(«,7) xS(«,x)J 0(k rx) dx (3.1) where k_ i s the r a d i a l wavenumber equal to Wsin(7)/V. 38 Since equation (3.1) assumes c y l i n d r i c a l symmetry, the associated angles of emergence given by the horizontal wavenumber k r must span only the range from 0° to 90°. In order to accommodate p o s i t i v e and negative angles of emergence ( i . e . , p o s i t i v e and negative wavenumbers), the following i s proposed: Based on the f a c t that the apex (image ray) of a r e f l e c t i o n hyperbola separates p o s i t i v e and negative angles of emergence of plane waves and also s p l i t s p o s i t i v e and negative slopes of the r e f l e c t i o n hyperbola branches, the t-x data are dip f i l t e r e d p r i o r to PWD i n order to obtain the desired r e s u l t . That i s , the common-source gather i s f i r s t taken through a frequency-wavenumber domain f i l t e r designed to r e j e c t the quadrant of p o s i t i v e u/k x values ( i . e . , p o s i t i v e time-o f f s e t slopes) and pass the negative ones. Then, the f i l t e r e d data i s decomposed v i a equation (3.1) with x=|x| to account f o r negative o f f s e t s and the resultant plane-wave seismograms are assigned to the negative quadrant of the plane-wave domain. The procedure i s then repeated to f i l l i n the opposite quadrant (see Figure 17). The slant-stack transformation.- For computational e f f i c i e n c y , throughout the remainder of t h i s work, the slant-stack (also c a l l e d T-p ) transformation w i l l be used to obtain plane-wave seismograms for p o s i t i v e and negative angles of emergence. Here, a b r i e f review of t h i s transformation i s presented. For a more complete exposition of the theory, the reader i s referred to Stoffa et a l . (1981) , T r e i t e l et a l . (1982) and Claerbout (1985). 39 The slant-stack transformation may be applied to a spl i t - s p r e a d common-source gather to obtain plane-wave seismograms f o r p o s i t i v e and negative angles of emergence. This transformation i s given by the equation > U(T,p) = S(r+px,x) dx , (3.2) where, U(T,p) i s the slant-stack seismogram, and S(t,x) i s the spherical wave seismogram. Equation (3.2) establishes that a transformed T-p seismogram may be obtained by adding a l l the values of the common-source gather f a l l i n g on the l i n e t = T + px (3.3) where t i s the t o t a l traveltime, x i s the source-receiver o f f s e t , T i s the intercept time, and p=sin(7)/V i s the ray parameter (V i s the surface v e l o c i t y ) . Unlike the plane-wave decomposition i n t e g r a l (equation 3.1) , the T-p transform can d i r e c t l y accommodate p o s i t i v e and negative o f f s e t s and angles of emergence. From a sp l i t - s p r e a d common-source gather, the r-p transform w i l l produce plane-wave components that are kinematically equivalent to those produced by the plane-wave decomposition i n t e g r a l ( T r e i t e l et a l . (1982)). Since a r e f l e c t i o n hyperbola i n the (t,x) domain i s transformed by the T-p transform into an e l l i p t i c a l t r a j e c t o r y and by plane-wave decomposition into a cosinusoid t r a j e c t o r y , the 40 kinematic equivalence between both transformations i s due to the fact that the T-p e l l i p s e can be d i r e c t l y mapped into the T-7 cosine and v i c e versa. The following paragraphs explain these statements i n d e t a i l : F i r s t , the f a m i l i a r r e f l e c t i o n hyperbolic t r a j e c t o r y 2 X 2 t 2 = t 0 + — , (3.4) V 2 where t 0 i s the two-way normal-incidence traveltime, x i s the source-receiver o f f s e t , V i s the medium v e l o c i t y and t i s the t o t a l traveltime, i s transformed by slant stack i n t o an e l l i p t i c a l t r a j e c t o r y . To see t h i s , consider the ray path between the image source I and a receiver at R on the earth's surface (Figure 18) . The t o t a l traveltime t between I and R i s s p l i t by sl a n t stack into t - T + px (3.5) where T i s the intercept time, p i s the ray parameter and x i s the source-receiver o f f s e t . From Figure 18 2Z Vt = , C O S 7 which implies 2Z t = . V C O S 7 Also, from the geometry of Figure 18, x = 2Ztan7 . 41 Substituting these l a s t two expressions into (3.5) gives 2Z = T + 2pZtan7 . VCOS7 Substitution of the ray parameter p by sin7/V y i e l d s 2Z 2Z s i n 2 7 = T + VCOS7 V COS7 Rearranging gives T = 2Z {l-sin 27} VCOS7 Z 2 COS7 (3.6) V Z I 2 -J l - s i n 2 7 V Z T = 2 -J l - p 2 V 2 F i n a l l y , squaring and rearranging t h i s l a s t equation y i e l d s the expected equation f o r an e l l i p s e P 2 + " = 1 , (3.7) T 0 2 ( V V ) 2 where T 0 i s equal to 2Z/V. Note that T 0 i s the two-way normal-incidence traveltime, and i t i s equal to the t 0 v a r i a b l e of equation (3.4) . 42 Having shown that the slant-stack transformation maps a r e f l e c t i o n hyperbola into an e l l i p s e , now i t w i l l be seen that plane-wave decomposition transforms the hyperbolic r e f l e c t i o n t r a j e c t o r y into a cosinusoid t r a j e c t o r y . The plane-wave component characterized by the ray path with angle of emergence "i has a v e r t i c a l plane-wave traveltime T ( i . e . , intercept time) equal to (see Figure 18) ID T = V z = 2 C O S 7 V or T = T 0 C O S 7 , (3.8) where as before, T 0 i s equal to t 0 , the two-way normal-incidence traveltime. The kinematic equivalence between sl a n t stack and plane-wave decomposition i s understood as the process of obtaining the cosinusoid t r a j e c t o r y given by equation (3.8) from the e l l i p t i c a l t r a j e c t o r y given by equation (3.7). However, t h i s has already been shown i n the derivation of equation (3.7) (see intermediate equation (3.6) ) . The equivalence i s possible v i a the use of the ray parameter P-43 In practice, to obtain plane-wave seismograms v i a the slant stack transformation, f i r s t a set of ray parameter values are computed v i a p=sin7/V fo r the desired range of angles of emergence and the surface v e l o c i t y V. Then, the integration (3.2) i s performed f o r a l l the set of calculated p values and the resultant seismograms are l a b e l l e d with the corresponding angles of emergence. For as long as only kinematic plane-wave information i s required (as i n the remainder of t h i s work) t h i s procedure i s much fas t e r than that of obtaining plane-wave seismograms v i a the plane-wave decomposition i n t e g r a l (equation 3.1). Method. With the help of two simple earth models, the novel v e l o c i t y - d i p analysis method i s now developed. These models provide the geometrical insight required to apply the technique to the case of an earth model composed of many planar dipping i n t e r f a c e s . The case of a h o r i z o n t a l l y layered earth model. - For a given horizontal layer with an overburden root-mean square v e l o c i t y V R M S , the usual (t,x) r e f l e c t i o n hyperbola 2 X 2 v rm s i s mapped to the plane-wave domain into the cosinusoid t r a j e c t o r y given by 44 T = T 0 C O S(7) (3.9) provided that the V r m s i s used i n the transformation performed to obtain plane-wave seismograms ( T r e i t e l et a l . . 1982; see previous section). When a wrong v e l o c i t y V i s used i n the transformation, the expected cosinusoid t r a j e c t o r y i s not obtained. To see t h i s , rewrite equation (3.9) as T - T 0 -J~i^ p*V 2 P v rm s (3.10) where p=sin7/V r m s. Using the fa c t that the ray parameter p i s conserved f o r emergence rays, an a r b i t r a r y v e l o c i t y V i s rel a t e d to V r m 8 v i a P = sin7 • r n s (3.11) As a r e s u l t , when the v e l o c i t y V i s used i n the transformation, equation (3.10) reads as T - T, "sin7•" 1 -V "I 2 2 V v rm s or 45 T — T < COS 7 rm s V V 2— V 2 v v rm i V i 2 That i s , a "distorted cosinusoid" t r a j e c t o r y i s observed. Using the above, an RMS v e l o c i t y analysis i n the plane-wave domain i s proposed. In t h i s analysis, because the plane-wave domain r e f l e c t i o n t r a j e c t o r y w i l l f a l l along a cosine only i f the v e l o c i t y V r m s i s used to l a b e l the angles of emergence, the RMS v e l o c i t y i s estimated v i a a cosinusoid-trajectory semblance-analysis search c a r r i e d out i n the plane-wave domain. In t h i s procedure, the v e l o c i t y (and thereby the angles of emergence 7) and the normal-ray traveltime T 0 vary i n a systematic manner to scan the desired portion of the v e l o c i t y spectrum. Although t h i s analysis i s equivalent to the usual (t,x) v e l o c i t y semblance analysis and no advantage i s gained i n the plane-wave domain, for v e l o c i t y models consisting of planar dipping layer interfaces the proposed plane-wave domain v e l o c i t y analysis y i e l d s s i g n i f i c a n t conceptual advantage. The case of an earth model composed of a planar dipping i n t e r f a c e . -From Diebold and Stoffa (1981), the intercept time equation corresponding to the r e f l e c t i o n from a planar dipping interface reads as, Z 8 , 1 T = — [ c o s ^ ) + c o s ( b , J ] (3.12) where, 46 T i s the intercept time f o r a given plane wave, Z S / 1 i s the thickness of the layer under the source location, W1 i s the acoustic speed i n the f i r s t layer, a : i s the take-off angle of a given plane wave, and b : i s the angle of emergence of a plane wave at the earth's surface. This i s the angle obtained v i a PWD and i t w i l l be denoted by 7 (see Figure 19 ) . In the geometry of Figure 19 the following r e l a t i o n s are observed: bj = 7 (3.13) a^ + a i — 7 — otj, where a : i s the interface dip angle. Substitution of these equations into (3.12) y i e l d s z s , 1 T = — [ C O S ( 7 - 2 a 1 ) + C O S ( 7 ) ] . V i A f t e r trigonometric manipulation, the following r e l a t i o n i s attained: 2 Z 8 / i T = — C O S ( a x ) C O S ( 7 - a x ) . (3.14) Equation (3.14) expresses the intercept time of a given plane wave as a function of i t s angle of emergence 7 and the dip angle a x . When a t = 0 ° , the kinematic plane-wave response assumes the shape of a cosine that i s symmetric around 7 = 0 ° ( i . e . , around the image ray a r r i v a l ) . However, when a 1 ^ 0 ° t h i s curve i s weighted by c o s f a j ) and s h i f t e d away from 7=0° (Wenzel and Stoffa, 1982). Indeed, from dr 2 Z 8 / 1 = — c o s f a ^ s i n f ^ - a ! ) , d7 V x i t i s seen that the cosinusoid t r a j e c t o r y has a maximum intercept time T = T N / 1 at 7=7 n / 1=a 1. That i s , the plane-wave r e f l e c t i o n signature has a peak at the normal-ray seismogram. An example of the normal ray and of the (t,x) and (T,7) r e f l e c t i o n t r a j e c t o r i e s are shown i n Figure 20. The trough of the r e f l e c t i o n t r a j e c t o r y shown i n Figure 20c gives a d i r e c t estimate of the dip angle of the r e f l e c t i n g interface RR1 ( i . e . , 7 n / 1 = a 1 ) . Because the intercept time of the normal-ray plane-wave component i s given by (replace 7 by a x i n equation (3.14)) 2 Z S , 1 r n j = — c o s ( t t l ) , (3.15) equation (3.14) may be rewritten as T " T n , l c o s ( 7 - a i ) . (3.16) Expression (3.16) constitutes the search t r a j e c t o r y used i n the proposed v e l o c i t y - d i p estimation method. Since the plane-wave r e f l e c t i o n signature w i l l f a l l along the cosinusoid t r a j e c t o r y given by equation (3.16) only i f the correct medium v e l o c i t y and interface dip angle values are used, a plane-wave domain semblance analysis search i s c a r r i e d out along the cosinusoid t r a j e c t o r y as a function 48 of v e l o c i t y and dip angle. In t h i s analysis, f i r s t a dip angle value i s selected; then, f o r a range of v e l o c i t y values, semblance i s computed along the curve defined by (3.16) and the r e s u l t i s stored i n a two-dimensional array. This array i s composed of traces that store semblance values as a function of normal-ray traveltime. Each trace corresponds to the semblance computations calculated with a d i f f e r e n t v e l o c i t y . The process i s repeated f o r a d i f f e r e n t dip angle value u n t i l the desired range of t h i s parameter i s completed. Noting that the f i n a l r e s u l t i s composed of many two-dimensional arrays corresponding to d i f f e r e n t dip angles, the following displays are formed i n order to f i n d the maximum semblance values that determine the optimal earth model parameters (v e l o c i t y and interf a c e dip angle): (a) A dip-angle p l o t consisting of traces s t o r i n g the maximum semblance values (as a function of normal-ray traveltime) found within each of the two-dimensional arrays. Each of the traces i n t h i s display i s l a b e l l e d with the dip angle value used to generate the corresponding two-dimensional array. Dip angles are selected from t h i s display. (b) A se r i e s of plo t s displaying the o r i g i n a l two-dimensional arrays that correspond to the dip angle values selected i n step (a) . Each of these p l o t s display traces l a b e l l e d with the v e l o c i t y used i s the semblance search. V e l o c i t i e s are selected from these displays. 49 In t h i s way, a data cube (velocity, dip angle, and normal-ray traveltime) i s conveniently displayed by a p a i r of 2-dimensional data displays. A l l the above process i s summarized i n the flow diagram of Figure 21. For the case of a single planar dipping in t e r f a c e , the v e l o c i t y -dip analysis method w i l l t h e o r e t i c a l l y give the exact values of the medium v e l o c i t y and interface dip angle, since no approximation i s made. On p r a c t i c a l grounds, however, the r e l i a b i l i t y of the res u l t s w i l l deteriorate as the d e f i n i t i o n of the plane-wave signatures deteriorates. The signature degradation can be caused by poor signal-to-noise r a t i o , by poor s p a t i a l sampling of the recorded wave f i e l d , or by inappropriate a c q u i s i t i o n geometry which prevented the proper recording of the plane-wave components. The case of an earth model composed of several planar dipping i n t e r f a c e s . - The v e l o c i t y - d i p analysis method proposed i n the previous subsection can also be applied to constant-velocity earth models composed of many planar dipping in t e r f a c e s . Since the ray paths do not bend at the layer boundaries and the normal-ray t r a v e l path to any given r e f l e c t i n g interface emerges at the earth 1 s surface with an angle equal to the dip angle of that interface, the v e l o c i t y -dip analysis method w i l l give exact estimates of the medium v e l o c i t i e s and interface dip angles. To apply the v e l o c i t y - d i p algorithm to an earth model composed of 50 several interfaces separating layers with d i f f e r e n t v e l o c i t i e s , the following must be assumed: Consider the case of an earth model consisting of N a r b i t r a r i l y dipping planar layer interfaces with v e l o c i t y v a r i a t i o n s characterized by I ( v j + i - v j ) l / V j « 1, j = 1,...,N. It i s assumed that the t r a v e l path of a given ray to any model inter f a c e does not d i f f e r greatly from a straight-ray path (that i s , the cumulative r e f r a c t i o n e f f e c t s along the traveled ray path are n e g l i g i b l e ) . Consequently, the angle of emergence 7 n / j associated with the normal ray r e f l e c t e d from the j t h layer i n t e r f a c e can be considered to emerge with an angle of emergence equal to the dip angle (a.j) of that interface. On the other hand, kinematically speaking, the stack of layers overlying the r e f l e c t i n g i n t e r f a c e are assumed to be replaced by a homogeneous medium characterized by the migration v e l o c i t y V m, which i s understood to be the v e l o c i t y that defines the hyperbolic t r a j e c t o r y that best describes the response of a point d i f f r a c t o r situated i n the subsurface (Berkhout, 1984) . With these assumptions, the proposed v e l o c i t y - d i p analysis algorithm may be used to estimate the migration v e l o c i t y V m and i n t e r f a c e dip angle a,j corresponding to each of the involved r e f l e c t i n g interfaces. Because the RMS v e l o c i t y can be considered to be a reasonable approximation to the migration v e l o c i t y (Berkhout, 1984), i n p r i n c i p l e the proposed v e l o c i t y - d i p estimation method w i l l also 51 estimate the RMS v e l o c i t y . However, since the work to be developed i n chapter 6 deals with the imaging of subsurface interfaces, the more conceptually-correct migration v e l o c i t y i s chosen as the parameter to be estimated i n the proposed v e l o c i t y - d i p algorithm. Recursive Estimation of Interval V e l o c i t i e s Using the v e l o c i t y - d i p analysis of the former section, estimates of the r e f l e c t o r dip angles, migration v e l o c i t i e s and normal-ray traveltimes have been obtained. Assuming that the explored earth model reasonably supports the straight-ray approximation and that the migration v e l o c i t i e s constitute a reasonable estimate of the RMS v e l o c i t i e s (Hubral andKrey, 1980; Berkhout, 1984), the v e l o c i t y - d i p analysis r e s u l t s are used to reconstruct an i n t e r v a l v e l o c i t y model. Note that the RMS v e l o c i t y i s considered i n t h i s section because the present i n t e r v a l - v e l o c i t y estimation method i s conceptually based on the t r a d i t i o n a l Dix equation (Dix, 1955). The following recursive scheme i s proposed: (a) The f i r s t RMS v e l o c i t y value V r m 8 / 1 and a l l dip angle estimates ait i = l , ...,N obtained from the v e l o c i t y - d i p analysis are considered to be reasonable representations of the f i r s t - l a y e r i n t e r v a l v e l o c i t y and the corresponding interface dip angles, respectively. 52 (b) From a modified Dix r e l a t i o n (Dix, 1955, and Figure 22 f o r a geometrical j u s t i f i c a t i o n of the following equation) ^ r i s , 2 T n , 2 V l T n # 1 C O S ( a 2 - a 1 ) n , l n, 2 cos (a2"~ot1) the second-layer i n t e r v a l v e l o c i t y i s obtained v i a ^ r u s , 2 T n , 2 V i T n / 1 C O S ( a 2 - a 1 ) n , 1 n , 2 C O S ( a 2 - a 1 ) (c) The t h i r d - l a y e r i n t e r v a l v e l o c i t y i s then calculated from the r e l a t i o n ^rms , 3 T n , 3 V, = V l T n y l C O S ( a 3 - a 1 ) - V, n, 2 n / 2 n , 1 C O S ( a 3 - a 2 ) C O s ( a 3 - a l ) n , 3 cos(a 3-a 2) In general, from the i n t e r v a l v e l o c i t i e s V j , the inte r f a c e dip angle estimates a.lf and the corresponding normal-ray traveltimes r n i , i = l , •.. ,N-1, the i n t e r v a l v e l o c i t y of the Nth layer i s computed v i a 53 V r m s,K Tn,N n, l COS ( a t ( - a 1 ) n / M - 1 n , K C O S ( a M _ a „ _ 1 ) K - 1 v . j = 2 n , j - 1 c o s ( a „ _ a j ) c o s ( a M _ a j _ 1 ) n / N - 1 n , K C O S ( a N - a H _ 1 ) In t h i s way, i n t e r v a l v e l o c i t i e s can r e c u r s i v e l y be estimated from the top to the bottom-most layer. Since t h i s estimation procedure assumes straight-ray t r a v e l paths, the attained i n t e r v a l v e l o c i t i e s are reasonable approximations to the true v e l o c i t i e s as long as v a r i a t i o n s of i n t e r v a l v e l o c i t i e s and inte r f a c e dip angles from layer to layer are not too large. In the next chapter, the straight-ray assumption w i l l be removed to develop a ray-tracing based algorithm that can be used to estimate i n t e r v a l v e l o c i t i e s and int e r f a c e dip angles more p r e c i s e l y . 54 Figure 13. An earth model consisting of a planar dipping i n t e r f a c e separating two homogeneous media. A sp l i t - s p r e a d common-source gather w i l l record the spherical wavefront r e f l e c t e d by the in t e r f a c e . 55 O F F S E T max X=0 X max Figure 14. Schematic drawing of the t-x seismograms recorded by the s p l i t -spread common-source gather of Figure 13. Note that the peak of the t -x event i s s h i f t e d to the l e f t of the source l o c a t i o n (x=0). 56 R' Figure 15. A spherical wavefront may be seen as a superposition of t r a v e l l i n g plane waves. UP-DIP PLANE-WAVE ARRIVALS R " x i X=0 • m 6 m m m • 1 1 1 V 2 ^ \ (b) R' Figure 16. Emerging plane waves, (a) PWD of the down-dip side of the gather w i l l y i e l d plane-wave components with p o s i t i v e angles of emergence, (b) PWD of the up-dip side of the gather w i l l give plane waves with p o s i t i v e and negative angles of emergence, -x^ i s the o f f s e t where the image ray emerges. PWD FOR BOTH POSITIVE AND NEGATIVE ANGLES OF EMERGENCE O F F S E T O F F S E T F-K FILTER 1 O F F S E T PWD PWD -Y Figure 17. zJL A D D Schematic representation of the steps associated with construction of plane-wave components corresponding to p o s i t i v e and negative angles of emergence. Figure 18 The t o t a l traveltime t between the source S and receiver R corresponds to the t r a v e l path IR. I i s the image source, V i s the medium v e l o c i t y , and t 0 i s the two-way normal-incidence traveltime equal to 2Z/V. 60 Figure 19. Ray geometry of a plane wave r e f l e c t e d from a planar dipping i n t e r f a c e . 61 Figure 20. (a) The normal-ray plane-wave component emerges at the receiver plane with the angle 7 n f x - a x . (b) t-x seismic record corresponding to the experiment shown i n panel (a) . The normal ray emerges at the zero-offset location, (c) The plane-wave r e f l e c t i o n response for t h i s model attains a maximum intercept time at the normal-ray plane-wave component 7 n # 1 . 62 PLANE-WAVE SEISMOGRAMS FIX DIP ANGLE PERFORM VELOCITY ANALYSIS AS A FUNCTION OF NORMAL-RAY TRAVELTIME. SEMBLANCE VALUES ARE CALCULATED ALONG THE TRAJECTORIES: r «• r„co » ( r-«|l FIND MAXIMUM SEMBLANCE VALUES IN VELOCITY SPECTRA PANEL AND FORM A DIP-ANGLE TRACE STORE FORMED TRACE IN A DIP ANGLE PANEL DISPLAY DIP ANGLE PANEL PICK DIP ANGLES DISPLAY VELOCITY .PANELS FOR CHOSEN DIP ANGLES Figure 21. PICK VELOCITIES Flow diagram i n d i c a t i n g the steps of the proposed v e l o c i t y - d i p a n a lysis i n the plane-wave domain. 63 Figure 22. Approximate ray paths f o r the normal-ray plane-wave components. The angle of emergence of each normal ray i s approximately equal to the dip angle of the corresponding r e f l e c t i n g i n t e r f a c e . 64 CHAPTER 4 RAY TRACING ALGORITHM FOR VELOCITY-DIP ESTIMATION Assuming two-dimensional earth models composed of planar, dipping interfaces separating homogeneous layers, a plane-wave domain ray-tracing algorithm w i l l be developed to estimate i n t e r v a l v e l o c i t i e s and interface dip angles. Unlike the method proposed i n chapter 3, the present method does not assume straight-ray t r a v e l paths across layer boundaries. This chapter i s divided as follows: the f i r s t section shows the fundamentals of the novel recursive technique by analysing an earth model composed of two layers. The second section extends the ap p l i c a t i o n of the method to an earth model composed of many layers. And the f i n a l section delineates some p r a c t i c a l problems encountered i n the implementation of the proposed method. Earth Model with two Planar Dipping Interfaces The problem to solve i s stated as follows: from a spl i t - s p r e a d common-shot gather, estimates of the i n t e r v a l v e l o c i t i e s and int e r f a c e dip angles of the r e f l e c t i n g layers are desired. To a t t a i n t h i s r e s u l t , i t i s assumed, f i r s t , that the probed earth model 65 i s composed of homogeneous layers separated by planar, dipping r e f l e c t i n g interfaces. Second, that the target model interfaces produce r e f l e c t i o n signatures from which traveltimes can be picked. And t h i r d , that the following processing steps have been performed: a. ) Plane-wave decomposition of the common-shot gather to produce a seri e s of plane-wave components fo r p o s i t i v e and negative angles of emergence, and b. ) Vel o c i t y - d i p analysis (refer to chapter 3) of the plane-wave domain obtained i n step (a) to produce the i n t e r v a l v e l o c i t y and interface dip angle of the f i r s t layer. Recall that t h i s analysis produces the true earth model parameters of the f i r s t layer. With the above done, the following procedure i s developed to a t t a i n the true i n t e r v a l v e l o c i t y and interface dip angle of the second layer: Figure 23 shows a ray path to the second int e r f a c e of a plane-wave with a take-off angle a x and an angle of emergence b^y. From Diebold and S t o f f a (1981) the intercept-time equation corresponding to t h i s plane wave i s given by Z g j Z g 2 T = — [cos(a :)+cos(b x)] + — — [cos(a 2)+cos(b 2)]. (4.1) V i V 2 66 In the above equation, a : i s the take-off angle of a plane wave, b x i s the angle of emergence (also denoted by 7), a 2 and b 2 are refracted angles defined i n the second layer, Z S / 1 and Z s / 2 are the thicknesses under the source l o c a t i o n of the f i r s t and second layers, respectively, and V 1 and V 2 are the f i r s t and second layer v e l o c i t i e s , r espectively (see Figure 23). The second term i n equation (4.1) gives the contribution of the second layer to the intercept time. This term w i l l be modified to e x p l i c i t l y contain the dip angle of the second i n t e r f a c e . Since the r e l a t i o n (see Figure 23) a 2 + a 2 = b 2 - a 2 (4.2) i s b a s i c a l l y the same as that i n equation (3.13), p.46, the trigonometric manipulation used to derive equation (3.14) i s applied to the second term of equation (4.1). This r e s u l t s i n equating 2 g 2 ^ ^ S 2 — — [cos(a 2)+cos(b 2) ] = — c o s ( a 2 ) c o s ( b 2 - a 2 ) , (4.3) sub s t i t u t i n g equation (4.3) into (4.1) and replacing b x by 7 to get Z s , 1 2 Z s , 2 T = — [cos(a 1)+cos(7)] + — c o s ( a 2 ) c o s ( b 2 - a 2 ) . (4.4) 67 Using t h i s equation to estimate the i n t e r v a l v e l o c i t y V 2 and interface dip angle a 2 of the second layer, the following steps are required: (a) Compute the term Z s l/V x v i a equation (3.5) , p.47. Recall that the f i r s t normal-ray traveltime r n / x , dip angle a x and layer v e l o c i t y V x required i n equation (3.15) have already been estimated v i a the v e l o c i t y - d i p analysis algorithm. (b) Select a p a i r of values ( a 2 ' ,V2') i n the desired search range of dip angle and v e l o c i t y values. For each angle of emergence 7 i f compute intercept times T ^ v i a equation (4.4). The evaluation of t h i s expression, however, requires the (see Figure 23). These quantities are determined v i a the appendix equations ( B . l ) , (B.3) and (C.4) derived i n the appendices B and C. (c) Compute the error value knowledge of the angles b 2 and a x , and of the term Z S / 2/V 2 where T j r e f e r s to T ( 7 J ) , 68 and repeat step (b) for a new p a i r of values ( a 2 1 ,V 2 1) . The quantities T 4 0 are the intercept-time values of the observed angles of emergence 7 j . (d) Search f o r those dip angle and v e l o c i t y values associated with the minimum error value E. The chosen quantities constitute the best estimates of i n t e r v a l v e l o c i t y and interface dip angle of the second layer. Earth Model with N Planar Dipping Interfaces A procedure f o r the estimation of the dip angle and the i n t e r v a l v e l o c i t y structure of two-dimensional earth models with N planar dipping interfaces i s r e a d i l y attained from the method developed i n the preceding section. In fact, the procedure i s recursive and i n general progresses from the case of N-1 interfaces to the case of N int e r f a c e s . That i s , given the estimates of the shallower N-1 dip angles and i n t e r v a l v e l o c i t i e s , the dip angle and i n t e r v a l v e l o c i t y of the underlying Nth layer can be estimated. The f i r s t and second layer parameters are estimated as explained i n the above section. For the t h i r d and subsequent layer parameters, the following steps are necessary: (a) Select a p a i r of values (a K ' ,VM ') i n the desired search range of dip angle and v e l o c i t y values of the Nth layer (in t h i s 69 case, N=3) . For each angle of emergence 7, compute intercept times T c v i a (see Appendix D) x -1 Js , 1 V *s , j T = [cosfaiJ+cosfn)] +) [cos(a<)+cos(b s)] V l ^ V i 1 j = 2 J 2 Z S , K cos(a M)cos(b M-a„) . (4.5) The evaluation of t h i s expression, however, requires the knowledge of the angles a j , b j , and of the terms Z S / j / V j , j = l , . .. ,N. These quantities are determined i n appendices E and F. (b) Compute the error value E 2 ( a \ , V H ) = ^ £ Ti " T i J 2 ' <4-6> and repeat step (a) f o r a new p a i r of values (a K',V N l) (c) Search f o r those dip angle and v e l o c i t y values associated with the minimum error value E. The chosen quantities constitute the best estimates to the Nth-interface dip angle and to the Nth-layer i n t e r v a l v e l o c i t y . 70 Comments on the Estimation of Intercept-Time Tranectories from the Observed t-x Seismograms The ray-tracing based v e l o c i t y - d i p estimation procedure described i n t h i s chapter requires the construction of a reasonably f a i t h f u l representation of the plane-wave domain. In p r i n c i p l e , t h i s can be achieved by many of the slant-stack or plane-wave decomposition techniques presently avai l a b l e (e.g., S t o f f a et a l . , 1981; Clayton and McMechan, 1981; T r e i t e l et a l . , 1982; Cabrera and Levy, 1984). However, a l i a s i n g associated with the poor s p a t i a l sampling offered by the majority of the avai l a b l e f i e l d data impose a high frequency cutoff that may cause severe loss of r e s o l u t i o n . For example, for a t y p i c a l seismic experiment with a receiver group separation of 25 m, surface wave v e l o c i t y of 2000 m/s and an assumed angle of emergence of 90°, the maximum s p a t i a l l y unaliased frequency i s given by ( T r e i t e l et a l . , 1982) V fmax = (4-7) 2AXsin7 2000 (2) (25)sin90° fmax " 4 0 HZ, which e a s i l y represents h a l f of the useful upper frequency content of the seismic data. 71 Although some remedies to t h i s loss of res o l u t i o n have been suggested (e.g., O t t o l i n i and Claerbout, 1984; Berkhout, 1984), i t seems i n e v i t a b l e that some loss of res o l u t i o n w i l l occur during the construction of the plane-wave components, whichever construction route i s used (in equation (4.7), note that as the angle of emergence increases, the maximum unaliased frequency decreases). Since the ray-tracing based v e l o c i t y - d i p estimation procedure necessitates only the estimation of intercept time t r a j e c t o r i e s f o r is o l a t e d r e f l e c t i o n events, two routes are v i a b l e f o r generating the needed (r,p) points: The f i r s t route r e l i e s d i r e c t l y on the primary r e l a t i o n s p=dt/dx = sin(7)/V, and T=t-px, as well as on the q u a l i t y of the recorded t - x seismograms. In pra c t i c e t h i s estimation route requires the following steps: (a) Identify the desired primary r e f l e c t i o n events on the t - x seismograms and se l e c t a number of representative points t j ( X j ) f o r each of the chosen events ( " i " i s the event index and " j " i s the o f f s e t index). (b) For each event " i " , c a l c u l a t e the ray parameter pj ( X j ) and the correponding intercept time T° J . Since the ray parameter i s the slope of the t - x curve, p i ( X j ) may be computed v i a a spline-function i n t e r p o l a t i o n routine applied to the picked t j ( X j ) points; while T J 0 may be computed v i a the r e l a t i o n TJ 0 = t i ( X j ) - P i ( X j ) X j . Depending on the data quality, 72 smoothing of the picked t i ( X j ) points might be needed for these computations. (c) From the calculated ray parameter values and the estimated surface P-wave speed, compute the angles of emergence 7i v i a simrj = Vpi (xj) . Although t h i s approach i s e a s i l y implemented, i t may severely be hindered by poor data qu a l i t y . A successful ray-tracing based v e l o c i t y - d i p estimation requires the c l e a r i d e n t i f i c a t i o n of a l l the target r e f l e c t i o n signatures. The second route uses the information obtained from the v e l o c i t y -dip search method presented i n chapter 3. In t h i s case, the necessary steps are the following: (a) Compute plane-wave seismograms v i a the slant-stack transformation. (b) Perform a v e l o c i t y - d i p search i n the plane-wave domain. (c) For a given r e f l e c t i o n signature N, ca l c u l a t e the intercept times T° v i a T ° - T n / K C O S ( 7 - a „ ) with s i n 7 = V» sin7° V P H D 73 i° and V P H D are the angle of emergence and v e l o c i t y used i n step (a) ; T N / N , a M and V„ are the normal-ray intercept time, dip angle and v e l o c i t y values estimated i n step (b). Since t h i s approach obtains (T 0,^) points v i a a semblance analysis c a r r i e d out i n the plane-wave domain, i t i s expected to be less s e n s i t i v e to subjective i d e n t i f i c a t i o n of major r e f l e c t i o n s . However, because both techniques may be severely hindered by poor data q u a l i t y , no e f f o r t should be spared to improve the i d e n t i f i c a t i o n and resolution of the r e f l e c t i o n signatures of i n t e r e s t before the a p p l i c a t i o n of the proposed estimation techniques. 74 Figure 23. Geometrical ray path of a plane wave r e f l e c t e d from the second int e r f a c e . 75 CHAPTER 5 EXAMPLES OF VELOCITY-DIP ESTIMATION To demonstrate the v i a b i l i t y of the proposed estimation techniques, a Kirchhoff sum algorithm ( B e r r y h i l l , 1979) i s used to generate synthetic data. These data feature sp l i t - s p r e a d common-source gathers with seismograms sampled at 4 milliseconds, a source bandwidth of 8 to 40 Hz, a receiver spacing of 25 meters, and a source function consisting of a Ricker wavelet with center frequency at 16 Hz. Since the proposed v e l o c i t y - d i p estimation algorithms use kinematic information, no e f f o r t has been made to preserve true amplitude r e l a t i o n s i n the generated model seismograms. For t h i s presentation, two basic models are considered: (a) r e f l e c t o r s with the same dip orientation, and (b) r e f l e c t o r s with c o n f l i c t i n g dip orienta t i o n . The l i m i t a t i o n s of the straight-ray assumption required by the approximate i n t e r v a l - v e l o c i t y estimation algorithm (refer to the section "Recursive estimation of the i n t e r v a l v e l o c i t i e s " , p.51), are demonstrated by considering the cases when | V i + 1 - V i | / V i « 1 and when t h i s r a t i o i s larger. 76 The Case of Small V e l o c i t y Variations Figure 25 displays the common-shot gather that corresponds to the model shown i n Figure 24. Application of the slant-stack transformation to these data resulted i n the set of plane-wave components shown i n Figure 26. Note that the peaks of the intercept-time t r a j e c t o r i e s correspond to angles of emergence that are approximately equal to the dip angles of the associated r e f l e c t i n g i n t e r f a c e s . With v e l o c i t i e s spanning the range 1000 to 4000 m/s at 50 m/s and dip angles spanning the range -40° to 40° at 5° i n t e r v a l , the v e l o c i t y - d i p analysis algorithm was applied to the data of Figure 26. Figure 27 displays the resultant dip angle panel from which dip angles ( i . e . , angles of emergence of the normal rays) as a function of normal-ray traveltime were chosen. In t h i s case, the dip angles (corresponding to the maximum semblance amplitudes) of 0°, 15° and 30° were selected. From the chosen dip angles, the respective v e l o c i t y panels (Figure 28) were displayed i n order to s e l e c t the migration v e l o c i t i e s . The f i n a l r e s u l t s are presented i n Table 1. The estimated v e l o c i t y and dip angle values were used i n the approximate i n t e r v a l - v e l o c i t y estimation algorithm (p. 51) to give the r e s u l t s shown i n Table 2. These r e s u l t s are i n agreement with the true model parameters (also shown i n Table 2) to within 10%. 77 TABLE 1 Intercept times, angles of emergence of the normal rays and migration v e l o c i t i e s obtained f o r the case of small v e l o c i t y v a r i a t i o n s and conforming int e r f a c e dips. T r a j ectory T v m I g 1 2 3 0.347 1.164 1.777 0° 15° 30° 1400 1650 1800 TABLE 2 Interval v e l o c i t y and dip angle values f o r the case of small v e l o c i t y v a r i a t i o n s and conforming int e r f a c e dips. Layer Model Estimated V e l o c i t y Dip Angle V e l o c i t y Dip Angle 1 1500 0° 1400 0° 2 1700 15° 1750 15° 3 2000 30° 2094 30° The next example represents an earth model con s i s t i n g of planar interfa c e s with p o s i t i v e and negative dip angles (Figure 29) . The corresponding (t,x) and (T,7) seismograms are displayed i n Figure 30 and 31, respectively. V e l o c i t y - d i p analysis of the data i n Figure 31 produced the estimated migration v e l o c i t i e s , i n t e r f a c e dip angles ( i . e . , angles of emergence of the normal rays) and normal-ray 78 traveltimes summarized i n Table 3. Using t h i s information i n the approximate i n t e r v a l - v e l o c i t y estimation procedure resulted i n the model parameters summarized i n Table 4. These r e s u l t s agree with the true model parameters to within 5% TABLE 3 Intercept times, angles of emergence of the normal rays and migration v e l o c i t i e s obtained for the case of small v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. Traj ectory T Vm i g 1 2 3 0. 1. 2. 782 888 507 15° -15° 0° 2000 2100 2150 TABLE 4 Interval v e l o c i t y and dip angle values f o r the case of small v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. Layer Model Estimated V e l o c i t y Dip Angle V e l o c i t y Dip Angle 1 2 3 2000 2100 2300 15° -15° 0° 2000 2188 2280 15° -15° 0° 79 The Case of Large V e l o c i t y Variations With the geometrical features of the second model presented i n the previous section, the current example exhibits l a r g e r layer-to-layer v e l o c i t y v a r i a t i o n s . These v a r i a t i o n s pose d i f f i c u l t i e s to the approximate i n t e r v a l - v e l o c i t y estimation procedure (last section of chapter 3) because of the straight-ray assumption. This example w i l l i l l u s t r a t e the performance of the ray-tracing based inversion procedure (chapter 4) to estimate the i n t e r v a l v e l o c i t i e s and i n t e r f a c e dip angles. Figures 32, 33 and 34 display the earth model, the corresponding (t,x) seismograms and the constructed plane-wave seismograms, respectively. Application of the v e l o c i t y - d i p analysis to the data of Figure 34 produced the r e s u l t s shown i n Table 5. With these r e s u l t s , the approximate i n t e r v a l - v e l o c i t y algorithm yielded the r e s u l t s of Table 6. Note that the estimated dip angle of the second in t e r f a c e d i f f e r s from the true value by 5°. Table 7 shows the earth model parameters estimated v i a the ray-t r a c i n g based inversion algorithm. These r e s u l t s were selected from the p l o t s of Figure 35, which show the degree of data f i t t i n g as a function of i n t e r v a l v e l o c i t y and interface dip angle (the f i t t i n g values were computed as the r e c i p r o c a l of the error quantities given by equation (4.6) , p. 69) . Note that the r e s o l u t i o n of the f i t t i n g values shown i n Figure 35 becomes poorer as the earth model parameter 80 estimation progresses to deeper model interfaces. This i s expected mainly because of the reduced number of independent (T,*Y) points entering into the computation of the error given by expression (4.6) . This data reduction - i s caused because the recorded plane-wave components span le s s angles of emergence as the r e f l e c t i n g interfaces becomes deeper. TABLE 5 Intercept times, angles of emergence of the normal rays and migration v e l o c i t i e s obtained f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. Trajectory T 7 v m l g 1 2 3 0.782 1.758 2.212 15° -10° 0° 2000 2250 2450 TABLE 6 Interval v e l o c i t y and dip angle values f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. Layer Model Estimated V e l o c i t y Dip Angle V e l o c i t y Dip Angle 1 2 3 2000 2400 3200 15° -15° 0° 2000 2467 3100 15° -10° 0° 81 TABLE 7 Interval v e l o c i t y and dip angle values f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. (Ray-tracing inversion algorithm). Layer Model Estimated V e l o c i t y Dip Angle V e l o c i t y Dip Angle 1 2 3 2000 2400 3200 15° -15° 0° 2000 2300 3100 15° -15° 2.5° The next example i l l u s t r a t e s the performance of the estimation algorithms on noisy synthetic data. Figure 36 shows the t-x seismograms of Figure 33 a f t e r the addition of random noise and synthetic ground r o l l , whereas Figure 37 displays the corresponding (T,7) seismograms obtained v i a the ap p l i c a t i o n of the slant-stack transformation to the seismograms of Figure 36. The degradation observed i n the computed plane-wave signature was produced by the presence of the ground-roll noise i n the input (t,x) seismograms (compare Figure 37 to 34). Ve l o c i t y - d i p analysis of the plane-wave seismograms of Figure 37 gave the r e s u l t s shown i n Tables 8. Using these r e s u l t s i n the approximated i n t e r v a l - v e l o c i t y estimation algorithm produced the r e s u l t s shown i n Table 9. Comparision of these values to those obtained i n the noise-free example (Tables 5 and 6) reveals that the 82 earth model parameter estimation was poorer i n the noisy case. The low signal-to-noise r a t i o of the input seismograms has caused more s u b j e c t i v i t y i n the i n t e r p r e t a t i o n of the dip angle and v e l o c i t y panels (for example, see Figures 38 and 39). Application of the ray-tracing based inversion algorithm to the data of Figure 37 yielded the model parameters summarized i n Table 10. As before, the earth model parameter estimation has been poorer (to within 10% f o r v e l o c i t i e s , ± 5° f o r dip angles) than i n the noise-free example (Table 7) . However, considering the low signal-to-noise r a t i o of the input seismograms, the performance of the estimation algorithms on noisy data i s considered to be reasonably good. TABLE 8 Noisy Data. Intercept times, angles of emergence of the normal rays and migration v e l o c i t i e s obtained f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. Traj ectory T 7 V v rm 8 1 2 3 0.778 1.755 2.200 15° -10° 0° 1950 2250 2500 83 TABLE 9 Noisy Data. Interval v e l o c i t y and dip angle values f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. Layer Model Estimated V e l o c i t y Dip Angle V e l o c i t y Dip Angle 1 2 3 2000 2400 3200 15° -15° 0° 1950 2503 3305 15° -10° 0° TABLE 10 Noisy Data. Interval v e l o c i t y and dip angle values f o r the case of large v e l o c i t y v a r i a t i o n s and c o n f l i c t i n g i n t e r f a c e dips. (Ray-tracing inversion algorithm). Layer Model Estimated V e l o c i t y Dip Angle V e l o c i t y Dip Angle 1 2000 15° 1950 14° 2 2400 -15° 2300 -15° 3 3200 0° 3450 5° 84 F i e l d Data The v e l o c i t y - d i p analysis algorithm i s now applied to a f i e l d data set. These data consist of two sp l i t - s p r e a d common-shot gathers with 100 traces per gather, a trace spacing of 50 m and a frequency band of 20 to 70 Hz (see Figures 40 and 41) . Trace e d i t i n g , datum correction, v e l o c i t y f i l t e r i n g , r e f r a c t i o n and residual s t a t i c s , and surface-consistent deconvolution were applied to these data. Assuming that the t-x seismograms represent the seismic r e f l e c t i o n response from an earth model composed of planar dipping r e f l e c t i n g interfaces, the v e l o c i t y - d i p analysis algorithm w i l l be used to obtain estimates of the migration v e l o c i t i e s and interface dip angles. Figures 42 and 43 display the plane-wave domains corresponding to the data of Figures 40 and 41, respectively. Plane-wave components with angles of emergence from -50° to 50° at 1.0° i n t e r v a l s and for a surface v e l o c i t y of 5500 m/s were computed using the slant-stack transformation. In Figures 42 and 43, several plane-wave r e f l e c t i o n signatures have t h e i r apices located at angles of emergence d i f f e r e n t from 0° . This indicates the presence of dipping r e f l e c t i n g i n t e r f a c e s . Figures 44 and 45 show the dip angle p l o t s obtained from the ap p l i c a t i o n of the v e l o c i t y - d i p analysis method to the data of 85 Figures 42 and 43, respectively. From these p l o t s , dip angles as a function of normal-ray traveltimes are chosen (see red dots). For example, i n Figure 44 a dip angle value of -10° i s selected at 0.881 s. Note that the dip angle value has opposite sign to those selected past 1.2 s. This indicates the presence of c o n f l i c t i n g dipping r e f l e c t i n g i n t e r f a c e s . On the other hand, i n Figure 45, since only p o s i t i v e angles are chosen, conforming dipping r e f l e c t i n g interfaces are i n f e r r e d . In Figures 44 and 45 i t should be noted the correspondence between the selected dip angle values and the l o c a t i o n of the apices of the plane-wave r e f l e c t i o n signatures. As sa i d i n chapter 3, the angle of emergence where the apex of a given r e f l e c t i o n signature i s located i s d i r e c t l y r e l a t e d to the dip angle of the corresponding r e f l e c t i n g i n t e r f a c e . Having selected dip angles, migration v e l o c i t i e s are subsequently chosen. To t h i s end, a velocity-semblance p l o t i s displayed f o r each chosen dip angle. From these p l o t s , migration v e l o c i t i e s are selected according to the maximum semblance values and the normal-ray traveltimes previously picked i n the dip-semblance p l o t s . For example, Figures 46 and 47 display the v e l o c i t y -semblance p l o t f o r the dip angle of -17° and 17°, respectively. In these p l o t s , the migration v e l o c i t i e s of 4900 m/s at 0.748 s and of 5700 m/s at 1.014 s are picked. By means of t h i s two-step parameter s e l e c t i o n ( i . e . , f i r s t , s e l e c t i o n of dip angles, and second, s e l e c t i o n of migration v e l o c i t i e s ) the true semblance cube (which 86 depends on dip angle, migration v e l o c i t y and normal-ray traveltime) i s analysed. Table 11 shows the dip angles, migration v e l o c i t i e s and normal-ray traveltimes selected from the main r e f l e c t i o n signatures of Figures 42 and 43. This information e s s e n t i a l l y gives a subsurface earth model composed of planar, dipping r e f l e c t i n g interfaces. Assuming hyperbolic t-x or cosinusoid T-7 r e f l e c t i o n t r a j e c t o r i e s , t h i s model best describes the plane-wave r e f l e c t i o n signatures observed i n Figure 42 and 43. Normal-ray traveltimes, migration v e l o c i t i e s and interface dip angles f o r the two shot-gather locations. (Velocity-dip algorithm). TABLE 11 F i e l d Data. Station 1610 Station 2015 Time V e l o c i t y Dip Angle Time V e l o c i t y Dip Angle 0.509 0.616 0.748 0.876 1.335 1.591 6000 5100 4900 4200 5200 4800 0.612 1.018 1.332 1.610 5000 5700 6200 4500 -17° -10° 17° 15° In order to have an idea of the r e l i a b i l i t y of the estimated model parameters, and i n p a r t i c u l a r of the inte r f a c e dip angles, conventional common-midpoint (CMP) processing was performed on a l l 87 the a v a i l a b l e shots to produce a stacked section which was subsequently time migrated. I t i s noted that a shot-domain dip-moveout (DMO) correction was included i n the processing sequence (a novel DMO algorithm developed by the author (Cabrera and Levy, 1989) was used) . Figure 48 shows two portions of the f i n a l stacked section located around the stations to which the data of Figures 40 and 41 correspond. Using the stacking v e l o c i t i e s , apparent time dips were converted to depth dips and the r e s u l t s tabulated i n Table 12. The computed inte r f a c e dip angles are i n reasonable agreement (to within 5°) with those obtained v i a the v e l o c i t y - d i p analysis method (compare Tables 11 and 12). The following should be kept i n mind: a. ) In conventional CMP processing, the v e l o c i t i e s needed to convert the time dips into depth dips should r e a l l y be migration v e l o c i t e s . Since these l a t t e r v e l o c i t i e s are generally estimated by the geophysicist as a percentage of the stacking v e l o c i t i e s (Yilmaz, 1987), the obtained depth dips i n general w i l l not be equal to the dip angles calculated v i a the v e l o c i t y - d i p analysis method, where dip angles and migration v e l o c i t i e s are computed d i r e c t l y from the prestack data. b. ) Both (conventional and new v e l o c i t y - d i p analysis) methods of a t t a i n i n g dip angles, v e l o c i t i e s and normal-ray traveltimes ( i . e . , an earth model structure) have the same basic underlying approximation: hyperbolic moveout t r a j e c t o r i e s . 88 c.) The conventional poststack time migration methods normally require a further approximation to account f o r l a t e r a l v e l o c i t y v a r i a t i o n s . No further approximation i s required i n the proposed v e l o c i t y - d i p analysis algorithm, since i t works on a shot by shot basis. This i s true provided that the r e f l e c t i n g interfaces are planar within the cable length. From the above, i t i s concluded that the v e l o c i t y - d i p analysis algorithm has a true p o t e n t i a l to estimate earth model parameters i n r e a l i s t i c s i t u a t i o n s . Importantly, the proposed method shows a new use of the plane-wave domain to process seismic r e f l e c t i o n data. In the next chapter, prestack migrated images w i l l be attained v i a an extension of the new v e l o c i t y - d i p analysis method. TABLE 12 F i e l d Data. Normal-ray traveltimes, stacking v e l o c i t i e s and estimated dip angles obtained from the conventionally processed stack section. Station 1610 Station 2015 Time Ve l o c i t y Dip Angle Time Ve l o c i t y Dip Angle 0.505 6000 - 3° 0.615 5900 4° 0.612 6000 3° 1.010 6400 16° 0.761 6300 -17° 1.461 7200 21° 0.852 6300 -15° 1.600 7400 22° 1.319 6300 22° 1.600 6200 20° 89 ,-1808,-1408,-1000, -6,00 , -2,08 , 208 , 688 , 18,00 , 14,08 , 18,88 3408-3888 -288 3400 --3880 Figure 24. Earth model composed of three planar interfaces separating homogeneous media. There are 80 receivers at each side of the source with a receiver spacing of 25 m; the largest v e l o c i t y jump i s V3/V2=1.18. OFFSET 90 Figure 25. Synthetic seismograms corresponding to the model shown i n Figure 24. Note the r e l a t i v e s h i f t of the apices of the lower two r e f l e c t i o n hyperbolas to the l e f t of the zero-offset trace. This indicates the presence of dipping, r e f l e c t i n g interfaces. Figure 26. Plane-wave seismograms obtained a f t e r the a p p l i c a t i o n of the slant-stack transformation to the data of Figure 25. The seismograms were computed at 1° i n t e r v a l from -80° to 80°; the surface v e l o c i t y used was 1500 m/s. The apices of each of the plane-wave r e f l e c t i o n signatures correspond to the normal-ray plane-wave components. 92 be DIP ANGLE - 1 0 o 1 0 20 30 0.6 0.8--1.0-1.2-1.4-1.6+-1.8-2.0-2.2-2.4+-2.6 2.8-] 4—f 3.0 i 40 _U Figure 27. Dip angles as a function of normal-ray traveltime. This panel shows the maximum c o r r e l a t i o n values obtained f o r each dip angle used i n the v e l o c i t y - d i p analysis search c a r r i e d on Figure 26. Note the presence of three peak c o r r e l a t i o n values at 0°, 15 and 30 . Their normal-ray traveltimes are the same as those corresponding to the apices of the respective plane-wave signatures (see Figure 26). VELOCITY 93 0. 0. 0. 1. 1. 2. 2. 2. 1000 0 1500 2000 2500 3000 3500 4000 .LL»>>^ M^k».I.IJ. 4 8 4-2--6--0--4 8 .L.l. 0. 0. 0. fe! 1 -pr I. 2. 2. 2. 1000 0 4 8 2 6 0--4--: 8 — 1500 2000 2500 3000 3500 4000 • • • • .L.l L.L 15l 0. 0. 0. 1. 1. 2. 2. 2. 1000 0 4 8 4 -2 6-f 0 44_, 8 1500 2000 2500 3000 3500 4000 i • > • • • • .. L . L L.L.:. ..... 30« Figure 28. Ve l o c i t y as a function of normal-ray traveltime f o r the dip angles of 0°, 15° and 30°. Each trace displays the c o r r e l a t i o n values obtained from the v e l o c i t y - d i p analysis search peformed on the ( T , 7) domain of Figure 26. The maximum c o r r e l a t i o n values locate the best estimate values to the migration v e l o c i t i e s . 94 -200 200 600 1000 1400 1800 2200 2680-3000-}-3400 3800 .-1800 -1400 -1000 -600 | -200 | 200 [ 600 | 1000 , 1400 | 1800 1 T I | ( "i n i — ( 1 I 1 — i — i-=—i—i Source 3 * r i r 1 1 1 1 • i • • i n n n a c : i V],=2Q 00 m/« =?mn If. / 5 • c ' i i : 1 V3 =2300 m/s ""I ?~ i -200 --200 --600 1000 1400 1800 2200 2600 -f3000 3400 3800 Figure 29. Earth model composed of three planar interfaces separating homogeneous media. There are 80 receivers at each side of the source with a receiver spacing of 25 m; the largest v e l o c i t y jump i s V 3 / V 2 = l . l . Figure 30. Synthetic seismograms corresponding to the model shown i n Figure 29. Note the r e l a t i v e s h i f t (from the zero-offset location) of the apices of the f i r s t two r e f l e c t i o n curves. This indicates the presence of r e f l e c t i n g dipping interfaces. Figure 31. Plane-wave seismograms obtained a f t e r the application of the slant-stack transformation to the t-x seismograms of Figure 30. The seismograms were computed at 1° i n t e r v a l from -80° to 80°; a surface v e l o c i t y of 2000 m/s was used to calculate the angles of emergence. The apices of each of the plane-wave signatures correspond to the normal-ray plane-wave components. 97 -200-200 600 + 1000 1400 1800 2200 2600 3000 34004 3800 -1800 [-1^00 [-1000 | -600 [ -200 t 200 [ 600 ( 1000 1 14p0 [ 1800 1 j 1 ' ' Source 1 [ J [ 1 1 1 i n n n c — 1 c i V »20( )0 r n/s v 2 = in n j ; ' =3200rr /s ! i 200 +•200 600 1000 1400 1800 2200 2600 3000 3400 + 3800 Figure 32. Earth model composed of three planar dipping interfaces featuring opposing dip angles. The largest v e l o c i t y r a t i o i s V3/V2=1.33. OFFSET -2008 -1500 -1000 -500 O 500 1000 1500 2000 Figure 33. Synthetic seismograms corresponding to the model shown i n Figure 32 Figure 34. Plane-wave seismograms obtained from the app l i c a t i o n of the slant-stack transformation to the t-x seismograms of Figure 33. A surface v e l o c i t y of 2000 m/s was used to ca l c u l a t e the angles of emergence. 100 DIP ANGLE 1 6 0 0 1 8 5 0 2 1 0 0 - -2 3 5 0 - -- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 ^ 0 2 6 0 0 2 8 5 0 3 1 0 0 -3 3 5 0 - -3 6 0 0 > CD n rH - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 2 6 0 0 J Figure 35. F i t t i n g values for the second, (a) and t h i r d (b) layer inversions done on the data of Figure 34 v i a the ray-tracing based algorithm. The maximum values locate the optimal i n t e r v a l v e l o c i t i e s and interface dip angles. Figure 36. Synthetic seismograms of Figure 33 a f t e r the addition of random noise and synthetic ground r o l l . This l a t t e r event was generated v i a a dispersive cosinusoid wave with center frequency at 16 Hz and frequency width of 20 Hz. 102 ANGLE OF EMERGENCE Figure 37. Plane-wave seismograms obtained from the ap p l i c a t i o n of the slant-stack transformation to the t-x data of Figure 36. Note the de t e r i o r a t i o n of the plane-wave signatures, i n p a r t i c u l a r the t h i r d one (compare to Figure 34) . This signal degradation was mainly caused by the presence of the grond r o l l noise i n the data of Figure 36. DIP ANGLE - 1 0 O 1 0 103 Figure 38. Dip angle panel obtained from the v e l o c i t y - d i p search c a r r i e d on the seismograms of Figure 37. High semblance values reasonably locate the dip angles of the f i r s t and second interfaces (15° and -15°, respectively) . However, the dip angle of the t h i r d r e f l e c t o r (0°) i s poorly located. Also, a high semblance value not related to any plane-wave signature has appeared at -10°, 1.8 s. The noise observed at short times and at large dip angles i s attributed to the quality degradation of the input seismograms caused by the ground r o l l noise. 104 1000 0.0-0.6 + 0.8-1.0-1.2-1.4 + 1.6--1.8 + 2.0-2.2 + 2.4 — 2.6--2.8 + 3.0-1500 2000 VELOCITY 2500 3000 3500 4000 Figure 39. Ve l o c i t y panel f o r the dip angle of 0°. Note the smearing of the semblance values around V=2500 m/s and T=2.2 S ( t h i r d layer parameters). OFFSET 105 Figure 40. F i e l d data: s p l i t - s p r e a d common-source gather at s t a t i o n 1610. Trace o f f s e t s range from -2472 m to 2455 m at 50 m i n t e r v a l . A series of conventional processing steps were applied to these data. In the marked r e f l e c t i o n s , the presence of r e f l e c t i n g dipping interfaces i s manifested by the s h i f t e d locations of the apices of the r e f l e c t i o n t r a j e c t o r i e s . Figure 41. As i n Figure 40, but at s t a t i o n 2015. ANGLE OF EMERGENCE 107 Figure 42. Plane-wave domain corresponding to the shot gather of Figure 40. Angles of emergence range from -50° to 50° at 1.0° i n t e r v a l . The surface wave v e l o c i t y of 5500 m/s was used to compute the angles of emergence. Note the presence of r e f l e c t i o n signatures with (inverted) apices at the following (T,^) locations: (0.5,0°), (0.62,2°), (0.86,-15°), (1.32,20°) and (1.58,21°). The angles of emergence of the observed t r a j e c t o r y apices are d i r e c t l y related to the dip angles of the corresponding r e f l e c t i n g interfaces. ANGLE OF EMERGENCE 108 Figure 43. Plane-wave domain corresponding to the shot gather of Figure 41. Angles of emergence range from -50° to 50° at 1.0° i n t e r v a l . A surface wave v e l o c i t y of 5500 m/s was used to compute the angles of emergence. Note the presence of r e f l e c t i o n signatures with (inverted) apices at the following (T,7) locations: (0.6,-2°), (1.07,18°), (1.3,20°) and (1.58,22°). The angles of emergence of the observed t r a j e c t o r y apices are d i r e c t l y related to the dip angles of the corresponding r e f l e c t i n g interfaces. 109 Figure 44 (next page). Dip angle as a function of normal-ray traveltime (colour panel) . This display shows the maximum semblance values obtained for each dip angle used i n the v e l o c i t y - d i p analysis search c a r r i e d on the data of Figure 42 (also shown i n the l e f t panel of t h i s f i g u r e ) . The ri g h t panel shows the maximum global semblance values as a function of normal-ray traveltime. Note the correspondence between the selected dip angles (red dots) and the l o c a t i o n of the (inverted) apices of the respective plane-wave signatures. I l l Figure 45 (next page). Dip angle as a function of normal-ray traveltime (colour panel) . This display shows the maximum semblance values obtained f o r each dip angle used i n the v e l o c i t y - d i p analysis search c a r r i e d on the data of Figure 43 (also shown i n the l e f t panel of t h i s figure) . The r i g h t panel shows the maximum global semblance values as a function of normal-ray traveltime. Note the correspondence between the selected dip angles (red dots) and the location of the (inverted) apices of the respective plane-wave signatures. 113 Figure 46 (next page). Ve l o c i t y as a function of normal-ray traveltime and f o r the dip angle of-17° (shot s t a t i o n 1610) . From the v e l o c i t y panel (colour display) the migration v e l o c i t y of 4900 m/s was selected (crosshairs l o c a t i o n ) . 0 . 0 - 4 0 - 2 0 0 2 0 4 U 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 0 0 9 0 0 0 1 0 0 0 0 Figure 47 (next page). V e l o c i t y as a function of normal-ray traveltime and f o r the dip angle of 17° (shot s t a t i o n 2015) . From the v e l o c i t y panel (colour display) the migration v e l o c i t y of 5700 m/s was selected (crosshairs l o c a t i o n ) . 0 . 0 - 4 0 - 2 0 2 0 4 0 4 0 0 0 5 0 0 0 § 0 0 0 7 0 0 0 8 0 0 0 9 0 0 0 1 0 0 0 0 2 T .>— 117 STATION Figure 48. Two portions of the conventionally-processed stack section. The locations of the shots used i n the v e l o c i t y - d i p analysis are indicated by the arrows. Using h a l f of the stacking v e l o c i t i e s of Table 12, event times of s p e c i f i c r e f l e c t i o n events were converted to depths and subsequently dip angles were calculated. CMP trace spacing i s 25 m. 118 CHAPTER 6 PLANE-WAVE DOMAIN INTERFACE IMAGING In chapter 3 a plane-wave domain method was developed to estimate migration v e l o c i t i e s and interface dip angles of simple two-dimensional earth models. From the founding concepts of that technique, a plane-wave domain imaging method i s now presented that, for a given common-shot gather, produces interpretable images of the r e f l e c t i n g earth model interfaces. Although the imaging method to be developed i s s t r i c t l y v a l i d for constant v e l o c i t y earth models composed of planar dipping interfaces, assuming straight-ray plane-wave t r a v e l paths the imaging method can be applied to earth models composed of several planar dipping interfaces separated by homogeneous layers. Since only kinematic information i s considered, the obtained subsurface images do not possess true amplitude information. The novel imaging technique i s related to e x i s t i n g migration algorithms based on ray t r a c i n g and/or extrapolation of plane waves (e.g., Gazdag, 1978; T r e i t e l e t a l . , 1982; Reshef and Kosloff, 1986). However, the proposed method has been developed e n t i r e l y from a novel i n t e r p r e t a t i o n of the plane-wave domain as i t i s related to the physical phenomenon of plane-wave propagation. I t i s hoped that the 119 founding concepts and methodology of the proposed imaging technique w i l l give further understanding of the plane-wave domain to pr a c t i s i n g geophysicists. Outline of the Method From the r e f l e c t i o n signature of a single, planar dipping interface recorded by a split-spread receiver layout, an interface image w i l l be attained v i a a series of processing steps c a r r i e d out i n the plane-wave domain. The image w i l l represent the true horizontal location of the r e f l e c t i o n points and the two-way v e r t i c a l traveltimes to them (see Figure 49) . To achieve t h i s , the following processing sequence i s proposed: (a) The r e f l e c t i o n signature recorded i n the common-offset gather i s transformed to the plane-wave domain. (b) A v e l o c i t y - d i p analysis i s performed on the plane-wave domain of step (a). (c) Downward extrapolation of the plane-wave components i s ca r r i e d out from the earth's surface to the r e f l e c t i n g i n t e r f a c e . (d) Spherical-wave reconstruction i s computed using the extrapolated plane-wave domain. (e) A mapping algorithm i s applied to the reconstructed spherical-wave f i e l d to a t t a i n the true horizontal 120 r e f l e c t i o n locations and two-way v e r t i c a l traveltimes to them. This y i e l d s the desired prestack i n t e r f a c e image. Step (a) i s achieved by the slant-stack transformation. Step (b) i s performed by using the v e l o c i t y - d i p analysis method described i n chapter 3. Step (c) i s accomplished by the ap p l i c a t i o n of time s h i f t s to each plane-wave component. Step (d) i s attained by e s s e n t i a l l y using the inverse slant-stack transformation. And the l a s t step, (e) , i s performed by using a mapping algorithm developed from simple geometrical r e l a t i o n s . The following section describes the imaging method step-by-step as i t i s applied to a simple example consisting of a r e f l e c t i o n signature (Figure 50a) obtained from an earth model composed of a single planar, dipping interface (Figure 49) . The goal i s to a t t a i n the true horizontal locations of, and the two-way v e r t i c a l traveltimes to, the r e f l e c t i o n points. Method Step fa) : Plane-wave domain.- The input t-x seismograms are transformed to the plane-wave domain by the slant-stack transformation (see Figure 50b) . In performing t h i s transformation, the following should be noted: f i r s t , since the t-x r e f l e c t i o n signature corresponds to a dipping r e f l e c t i n g i nterface, plane-wave 121 components need to be computed for p o s i t i v e and negative angles of emergence. Second, the kinematic plane-wave signature obeys the r e l a t i o n (refer to chapter 3) T = T 0 C O S ( 7 - a ) (6.1) where T i s the intercept time of the plane wave with angle of emergence 7, T 0 i s the normal-ray traveltime, and a i s the dip angle of the r e f l e c t i n g i n t e r f a c e . Equation (6.1) s p e c i f i e s the times required for the plane waves to tr a v e l the distance between the image source and the true source lo c a t i o n (see Figure 51). Step (b): V e l o c i t y - d i p a n a l y s i s . - A plane-wave domain semblance-based search along the cosinusoid t r a j e c t o r i e s given by equation (6.1) i s executed to obtain estimates of the migration v e l o c i t y , interface dip angle and two-way normal-ray traveltime. In performing t h i s analysis, the following should be kept i n mind (for d e t a i l s , r e f e r to chapter 3): (1) For the case of a single, planar dipping layer interface or several interfaces i n a constant-velocity medium, the v e l o c i t y - d i p analysis algorithm produces the true medium 122 v e l o c i t y and interface dip angle(s). (2) For the case of several planar, dipping layer interfaces separating homogeneous layers of d i f f e r e n t v e l o c i t i e s , the v e l o c i t y - d i p analysis method y i e l d s only estimates of the migration v e l o c i t i e s and interface dip angles. Straight-ray t r a v e l paths to the r e f l e c t i n g interfaces are assumed i n t h i s case. Since the v e l o c i t i e s and interface dip angles obtained i n t h i s step constitute the cornerstone of the imaging procedure, i t i s concluded that true r e f l e c t o r images for case (1) and approximate r e f l e c t o r images for case (2) w i l l be attained from the imaging algorithm. Step (c) : Plane-wave downward extrapolation. - The combined goal of t h i s step and the following i s to downward extrapolate the receivers from the horizontal earth's surface to the planar, dipping interface to produce the t-x signature that would have been recorded by receivers placed on the r e f l e c t i n g i n terface. The f i r s t part of the extrapolation step i s achieved by applying the time s h i f t s T o T s = COsCY-a) (6.2) 2 to each of the plane-wave seismograms represented by equation (6.1) . 123 The s h i f t e d seismograms w i l l obey the r e l a t i o n T = T 0 C O S(7-Cl) - T S or T o T = C O S ( 7 - a ) . (6.3) 2 To see why T s i s required to extrapolate the plane waves, i n Figure 52 note that T 0/2 i s the normal-ray traveltime between the image source and the point S' where the normal ray in t e r s e c t s the r e f l e c t i n g interface ( IS' = IS/2 ) . As a r e s u l t , T as given by equation (6.3) gives the plane-wave traveltime from I to S'. The extrapolation step, therefore, y i e l d s a seri e s of plane-wave seismograms that, i f they are transformed back to the spherical-wave domain, w i l l produce the t-x response recorded by receivers placed on the subsurface horizontal plane passing through S' ( i . e . , at depth Z s i n Figure 52). To i l l u s t r a t e the plane-wave domain extrapolation to a subsurface horizontal plane, the time s h i f t s given by equation (6.2) are applied to the (T,7) seismograms of Figure 50b. The resultant seismograms (Figure 53a) are taken through the inverse slant-stack transformation to y i e l d the t-x seismograms (Figure 53b) that would have been recorded by receivers placed on the subsurface horizontal plane at Z s = (T 0/2)Vcos ( a ) . The imaging procedure, however, requires the receivers to be placed on the r e f l e c t i n g interface and not on a subsurface horizontal plane. In order to change t h i s , the time s h i f t s of equation (6.2) are 124 s t i l l applied to the (T,7) seismograms, but p r i o r to the application of the inverse slant stack transformation, the o r i g i n of the angles of emergence of the plane-wave components i s redefined to the angle of emergence of the normal-ray plane-wave component (apex of the r e f l e c t i o n signature i n the T-7 domain). As i t w i l l be explained below, the r e d e f i n i t i o n of the o r i g i n of the angles of emergence provides the r e d e f i n i t i o n of the subsurface plane to which the plane waves are extrapolated. Step (d); Redefinition of the angles of emergence.- I t has been shown (chapter 3) that the plane-wave r e f l e c t i o n signature corresponding to an earth model composed of a planar, horizontal interface i s given by T H = T 0 H cos(7 h) (6.4) where the superscript "h" denotes a horizontal layer. T 0 H i s the normal-ray traveltime and 7 h i s the angle of emergence of a given plane wave. On the other hand, the plane-wave response corresponding to an earth model composed of a planar dipping interface follows the t r a j ectory T = T 0 cos(7-a) (6.5) where a i s the dip angle of the r e f l e c t i n g i n t e r f a c e . Importantly, the normal-ray plane-wave component follows the l i n e perpendicular to the dipping interface and, c e r t a i n l y f o r aj*0°, t h i s path does not 125 f a l l on the v e r t i c a l l i n e from which the plane-wave angles of emergence are measured. The substitution - y s h l f t = 7- a i n equation (6.5) gives T = T 0 c o s ( 7 s h i f t ) (6.6) Comparison of t h i s expression to equation (6.4) leads to the int e r p r e t a t i o n of equation (6.6) to describe the plane-wave domain tr a j e c t o r y that corresponds to a "horizontal layer earth model". This "new earth model" i s kinematically equivalent to the dipping layer earth model, since equation (6.6) s t i l l describes the plane-wave response from the dipping layer earth model ( i . e . ,the intercept times T given by equation (6.5) are i d e n t i c a l to those given by (6.6)). However, because the angle of emergence - y 8 h i f t i s now measured with respect to the normal-ray t r a v e l path (which has an angle of emergence a with respect to the v e r t i c a l ) the equivalent "horizontal layer earth model" i s composed of the dipping r e f l e c t i n g i nterface and a rotated, planar earth's surface (see Figure 54). To i l l u s t r a t e the e f f e c t of the r e d e f i n i t i o n of the o r i g i n of the angles of emergence of plane waves, Figure 55 shows the r e s u l t of applying the inverse slant-stack transformation to the downward extrapolated plane-wave domain of Figure 53a a f t e r the angles of emergence of the plane-wave components have been s h i f t e d by the value a=30° (angle of emergence of the normal ray) . In the t-x seismograms of Figure 55 the following i s noted: f i r s t , the t-x r e f l e c t i o n signature has a curvature which i s symmetrical around the 126 zero-offset trace. This happens because the o r i g i n of the o f f s e t coordinate i s located on the subsurface point where the normal-ray path inters e c t s the r e f l e c t i n g i nterface (point S' i n Figure 55b) . The i n t e r s e c t i o n point i s the l o c a t i o n of the zero-offset trace from which a l l the other o f f s e t s are measured along the dipping interface. Second, the branch of the t-x r e f l e c t i o n signature f a l l i n g on the po s i t i v e (down-dip) side of the record i s shorter than the one f a l l i n g on the opposite side (up d i p ) . This i s due to the shorter s p a t i a l range spanned by the r e f l e c t i o n points f a l l i n g on the down-dip side of the layer interface (see Figure 55b). Having attained the recording of the spherical-wave f i e l d at the r e f l e c t i o n points ( i . e . , on the r e f l e c t i n g interface) , the next and f i n a l step i s to r e f e r t h i s information to u s e r - s p e c i f i e d horizontal o f f s e t s and two-way v e r t i c a l times. Step fe) ; Spherical-wave mapping. - The seismograms reconstructed i n step (d) are defined by the s p a t i a l coordinate (hereafter named "y") measured along the dipping interface, and by the traveltime (hereafter named "t") corresponding to the ray path from the source to the extrapolated receivers (see Figure 55) . The goal of the spherical-wave mapping i s to associate to each r e f l e c t i o n signature defined by the coordinates ( y , t ) , a set of coordinates (x,t v) such that x i s the horizontal s p a t i a l coordinate measured along the earth's surface from the o r i g i n a l source location, and t v i s the two-way v e r t i c a l time from the earth's surface to the corresponding 127 r e f l e c t i o n point. With t h i s i n mind, the following two steps are required to a t t a i n the desire mapping: (1) The extrapolated spherical-wave front signature i s mapped to the horizontal s p a t i a l coordinate x v i a (see Figure 55b) x = - V T 0 s i n a / 2 + y|cosa| , (6.7) where y i s the "rotated" o f f s e t at which the seismograms were reconstructed i n step (d), and V , a and T 0 are the v e l o c i t y , dip angle and (two-way) normal-ray traveltime determined i n step (b). Using equation (6.7), the r e f l e c t i o n signature i s associated with surface positions located d i r e c t l y above the r e f l e c t i o n points. The next step i s to adjust the time to two-way v e r t i c a l time. In equation (6.7), x i s measured from the o r i g i n a l source l o c a t i o n on the earth's surface, whereas y i s measured from the normal-ray point on the r e f l e c t i n g i n t e r f a c e (points S and S' i n Figure 55b, r e s p e c t i v e l y ) . Also, x and y are p o s i t i v e to the r i g h t of S and S', whereas a i s p o s i t i v e for an interface dipping to the r i g h t of the earth model. (2) The two-way v e r t i c a l time that corresponds to the extrapolated spherical-wave f i e l d at y i s computed v i a 128 t v = 2 t 2 - (6.8) where t i s the traveltime of the r e f l e c t i o n signature obtained i s step (d) . For a given seismogram at y, t i s given by (see Figure 55b) t (6.9) To i l l u s t r a t e the above procedure, Figure 56 shows the r e s u l t of mapping the extrapolated spherical-wave f i e l d of Figure 55a v i a the use of equations (6.7) and (6.8) . As compared to the synthetic two-way v e r t i c a l traveltime section (Figure 57) corresponding to the o r i g i n a l earth model of Figure 49, the obtained interface image i s a reasonable r e s u l t . The following i s noted: f i r s t , the s p a t i a l extent of the r e f l e c t o r image i n Figure 56 i s much shorter than that of Figure 57. This happens because the former (Figure 56) represents the locations of only the interface points that were illuminated by the o r i g i n a l common-shot gather experiment (refer to Figure 49) , whereas the l a t t e r (Figure 57) represents the signature of a l l the interface points without any regards to an incident point-source wavefield. Second, because i n general the equally spaced subsurface locations y map to non-equally spaced earth's surface locations x, the resultant imaged seismograms were s p a t i a l l y interpolated to user-specified equally spaced horizontal locations x. The actual steps c a r r i e d out to obtain the image of Figure 56 from 129 the seismograms of Figure 55 were the following: (1) Norma1-moveout correction of the extrapolated (t,x) seismograms of Figure 55 v i a equation (6.9). (2) S p a t i a l mapping and i n t e r p o l a t i o n of the seismograms obtained i n step (1) from the subsurface locations y to equally spaced earth's surface locations x v i a equation (6.7). (3) Adjusting the traveltimes of the seismograms attained i n step (2) to two-way v e r t i c a l times v i a equation (6.8). Comments on the Imaging of Several Planar Dipping interfaces The plane-wave domain imaging technique presented i n the previous section i s s t r i c t l y v a l i d only for the case of a constant v e l o c i t y earth model composed of several planar dipping interfaces. For the case of an earth model composed of several planar, dipping interfaces separating homogeneous layers, the imaging technique w i l l produce approximate interface images. This i s because the determination of the imaging parameters (dip angles, normal-ray traveltimes and migration v e l o c i t i e s ) v i a the v e l o c i t y - d i p analysis method developed i n chapter 3 i s based on the assumption of straight-ray travelpaths that best account f o r the true refracted ray paths. The a p p l i c a t i o n of the imaging method to a common-shot gather 130 containing several r e f l e c t i o n signatures proceeds as follows: (a) Computation of plane-wave seismograms v i a the slant-stack transformation. (b) Application of the v e l o c i t y - d i p analysis algorithm to obtain migration v e l o c i t i e s , interface dip angles and two-way normal-ray traveltimes corresponding to each of the r e f l e c t i o n signatures. The remaining steps are applied sequentially to each plane-wave r e f l e c t i o n signature of i n t e r e s t : (c) With the earth model parameters found i n the previous step, downward extrapolation of the plane-wave components corresponding to the r e f l e c t i n g interface of i n t e r e s t . This puts the receivers on a horizontal subsurface plane. (d) Redefinition of the o r i g i n of the angles of emergence of the plane-wave components obtained i n step (c) followed by the appl i c a t i o n of inverse slant-stack transformation. This generates (t,x) seismograms corresponding to a surface shot with receivers on the dipping interface. (e) Interpolation and mapping of the reconstructed s p h e r i c a l -wave f i e l d to construct an image of the interface of in t e r e s t by l o c a t i n g the r e f l e c t i o n signature beneath user-specified surface locations and by adjusting the time to v e r t i c a l traveltime from the r e f l e c t i o n points to the earth's surface. 131 To have a succesful imaging of a given earth model, the following should be considered: (1) The imaging technique cannot handle multiple r e f l e c t i o n events (e.g., those originated by successive r e f l e c t i o n s within a layer of water). In t h i s case, the geophysicist should i d e n t i f y those events and eith e r remove them from the o r i g i n a l input record, or ignore them during the sequential plane-wave interface imaging delineated i n the above steps (c) through (e). (2) The imaging method assumes that within the cable length, the recorded r e f l e c t i o n signatures correspond to planar, dipping interfaces. This means that the proposed imaging method can be applied to image earth models composed of curved interfaces, provided that the depth to them i s large compared to the cable length so that a l o c a l planar assumption holds. (3) Since the imaging method heavily r e l i e s on the r e s u l t s of the v e l o c i t y - d i p analysis algorithm whose success depends on the q u a l i t y of the plane-wave r e f l e c t i o n signatures, no e f f o r t should be spared to optimize the f i e l d and processing parameters to maximize the plane-wave domain information. 132 Figure 49. Earth seismic experiment. The single dipping planar interface RR' i s probed by the spherical-wave f i e l d generated by the point source S. The receivers on the earth's surface span the range (x x ,x 2) . The goal of the plane-wave domain imaging technique i s , f i r s t , to obtain the true horizontal locations (x 3 to x 4) of the r e f l e c t i o n points f a l l i n g i n the segement r r ' ; and second, to obtain the two-way v e r t i c a l times from the earth's surface to the r e f l e c t i o n points. OFFSET -1500 -1000 -500 O 500 1000 1500 133 Figure 50. a.) Synthetic t-x seismograms f o r the model of Figure 49. A source Ricker wavelet with central frequency of 16 Hz, a time sampling i n t e r v a l of 0.004 s and a trace spacing of 25 m were used, b.) Plane-wave seismograms obtained from the application of the slant-stack transformation to the t-x seismograms. A surface v e l o c i t y of 1500 m/s was used to calculate the angles of emergence. The angle of emergence of the apex of the plane-wave signature i s equal to the dip angle (30°) of the r e f l e c t i n g interface. 134 Figure 51. Plane-wave decomposition produces a serie s of plane waves that may be considered to t r a v e l from the image source to the earth's surface along d i r e c t i o n s s p e c i f i e d by ray paths with angles of emergence 7. The intercept time r i s the plane-wave traveltime from the image source to the true source loca t i o n . 135 Figure 52. Plane-wave domain downward extrapolation to depth l e v e l Z s = V T 0 C O S (a)/2. Reconstruction of the spherical-wave f i e l d from the extrapolated plane waves produces the (t,x) response due to the image source and recorded by receivers placed on the horizontal plane Z s. Importantly, the receiver o f f s e t s are now measured with respect to S 1 and not with respect to S (see also Figure 53). ANGLE OF EMERGENCE 136 a -1500 -1000 -500 O 500 1000 1500 b Figure 53. a.) Plane-wave domain of Figure 50b a f t e r a p p l i c a t i o n of the time s h i f t s of equation (6.2) . b.) t-x seismograms reconstructed v i a the inverse slant-stack transformation of the (T,*Y) domain of panel (a) . Since the extrapolated receivers are closer to the image source (see Figure 52) , the signature apex i s located closer to the zero o f f s e t trace and the hyperbolic t r a j e c t o r y i s shorter, both as compared to the o r i g i n a l (t,x) event of Figure 50a. 137 Figure 54. Plane-wave ray paths to a t i l t e d receiver plane that passes through the source l o c a t i o n S. Slant-stack transformation of the s p h e r i c a l -wave f i e l d (SWF) recorded on t h i s plane w i l l produce plane-wave components that are kinematically equivalent to those obtained from the decomposition of the SWF recorded on the horizontal earth's surface. -1520 -1000 -500 o 500 1000 150C. 0. 138 0. 0. 0. I— 1. 1. 1. 1. 1. 2. Figure 55. t-x seismograms on the r e f l e c t i n g interface, a.) These seismograms were obtained from the application of the dip-incorporated inverse slant-stack transformation to the plane-wave domain of Figure 53a. The dip incorporation consisted of the r e d e f i n i t i o n of the o r i g i n of the angles of emergence to the normal-ray angle (apex location) . b.) The seismograms of panel (a) are interpreted as those recorded by receivers placed on the r e f l e c t i n g interface. Note that the s p a t i a l extent (segment Sr 1) of the down-dip r e f l e c t i o n points i s shorter that the extent (segment Sr) of the up-dip r e f l e c t i o n points. a s R b R' 139 Figure 56. Shot-domain image of the planar dipping r e f l e c t i n g interface of Figure 49. For the estimated medium v e l o c i t y of 1500 m/s, the interface image has a dip angle of approximately 32°. The depth under the trace at o f f s e t -500 m i s 682 m. As compared to the synthetic, two-way v e r t i c a l traveltime section of Figure 57, t h i s image constitutes a reasonable r e s u l t . 140 OFFSET -1500 -1000 -500 O 500 1000 1500 0.0 l i i i i i i i i i m i i i i i i i i i i i i i i i i i i i i i i m i i i l i i i i i i i i i i i i i i i i i i i ' i i i i i i i m i l 1111111111]11II1111II111111111| i Figure 57. Synthetic, two-way v e r t i c a l traveltime section corresponding to the earth model of Figure 49. 141 CHAPTER 7 EXAMPLES OF PLANE-WAVE DOMAIN INTERFACE IMAGING The plane-wave domain imaging technique developed i n chapter 6 w i l l be applied to the series of synthetic examples u t i l i z e d i n the exposition of the v e l o c i t y - d i p analysis method (chapter 5) . A f u l l synthetic seismic l i n e w i l l also be presented. Imaging of Single Common-Shot Gathers Figure 24 (p.89) displays the f i r s t earth model used f o r t h i s presentation. This model consists of three layer interfaces with the same dip dir e c t i o n s and small v e l o c i t y jumps from layer to layer. With the earth model parameters (Table 1, p.77) estimated v i a the ve l o c i t y - d i p analysis algorithm, the imaging technique was applied to the plane-wave seismograms of Figure 26 (p.91) to produce the imaged common-shot gather of Figure 58. In t h i s figure, the presence of three model interfaces i s now cl e a r . To evaluate the obtained r e s u l t s , a synthetic two-way v e r t i c a l time section corresponding to the earth model of Figure 24 was generated and displayed i n Figure 59. Comparison of Figures 58 and 59 reveals that the imaged interfaces (Figure 58) are i n very good 142 agreement with the expected temporal and s p a t i a l locations of the r e f l e c t i n g interfaces (Figure 59) . Note that the s p a t i a l extents of the images of Figure 58 are much shorter than those of the synthetic images (Figure 59). This i s due to two reasons: f i r s t , since the images of Figure 58 represent the locations of only the r e f l e c t i o n points, the interface segments that were not illuminated by the source wavefield do not appear i n the r e s u l t s of Figure 58. On the other hand, the signature of the entire model int e r f a c e appears i n Figure 59. Second, a numerical safeguard to avoid excessive wavelet stretching caused by the i n t e r p o l a t i o n and mapping of the imaged interfaces was incorporated i n the imaging algorithm. This " a n t i -stretching f i l t e r " i s s i m i l a r to the standard muting f i l t e r conventionally used i n common-midpoint processing. Using the estimated migration v e l o c i t i e s (Table 1, p.77) and the two-way v e r t i c a l times of Figure 58, depths to the r e f l e c t i n g interfaces were computed at d i f f e r e n t horizontal locations and compared to the true depths extracted from Figure 24 (p.89). The r e s u l t s , shown i n Table 13, demonstrate that the imaging algorithm was able to produce a s a t i s f a c t o r y subsurface earth model. Figure 29 (p.94) displays the second earth model used f o r t h i s presentation. This model consists of three layer interfaces with opposite dip d i r e c t i o n s and small v e l o c i t y jumps from layer to layer. With the earth model parameters (Table 3, p.78) estimated v i a the v e l o c i t y - d i p analysis algorithm, the imaging technique was applied to the plane-wave seismograms of Figure 31 (p.96) to produce the 143 imaged common-shot gather of Figure 60. In t h i s figure, the presence of three model interfaces i s now c l e a r . TABLE 13 Interface depths. Layer V m t g X Estimated Depths True Depths 1 1400 0 210 200 2 1650 -250 858 900 250 1015 1040 3 1800 -1500 936 934 -500 1530 1511 As before, to evaluate the obtained r e s u l t s , a synthetic two-way v e r t i c a l time section corresponding to the earth model of Figure 29 was generated and displayed i n Figure 61. Comparison of Figures 60 and 61 reveals that the imaged interfaces (Figure 60) are i n very good agreement with the expected temporal and s p a t i a l locations of the r e f l e c t i n g interfaces (Figure 61) . The differences i n s p a t i a l extent of the interface signatures are due to the reasons explained i n the previous example. Using the estimated migration v e l o c i t i e s (Table 3, p.78) and the two-way v e r t i c a l times of Figure 60, depths to the r e f l e c t i n g interfaces were computed at d i f f e r e n t horizontal locations and compared to the true depths extracted from Figure 29 (p.94). The r e s u l t s , shown i n Table 14, demonstrate that the imaging algorithm was able to produce a s a t i s f a c t o r y subsurface earth model with 144 c o n f l i c t i n g dipping interfaces. TABLE 14 Interface depths. Layer V m j g X Estimated Depths True Depths 1 2000 -500 610 626 275 815 834 2 2100 0 1995 1960 1000 1727 1693 3 2150 0 2628 2600 Figure 32 (p.97) displays the t h i r d earth model used for t h i s presentation. This model consists of three layer interfaces with opposite dip d i r e c t i o n s and large v e l o c i t y jumps from layer to layer. The synthetic t-x seismograms (Figure 36, p. 101) were contaminated with random and ground-roll noise. With the earth model parameters (Table 8, p.82) estimated v i a the v e l o c i t y - d i p analysis algorithm, the imaging technique was applied to the plane-wave seismograms of Figure 37 (p. 102) to produce the imaged common-shot gather of Figure 62. In t h i s figure, the presence of three model interfaces i s s t i l l c l e a r i n s p i t e of the noisy appearance of the input T-7 seismograms (Figure 37). As before, to evaluate the obtained r e s u l t s , a synthetic two-way v e r t i c a l time section corresponding to the earth model of Figure 32 was generated and displayed i n Figure 63. Comparison of Figures 62 and 63 reveals that the imaged interfaces (Figure 62) are i n 145 acceptable agreement with the expected temporal and s p a t i a l locations of the r e f l e c t i n g interfaces (Figure 63). Note that the banding observed i n the r e s u l t s of Figure 62 was caused by the way i n which the algorithm successively imaged the model interfaces, whereas the degradation of the imaged signatures (as compared to the previous noise-free results) was caused by the poor signal-to-noise r a t i o of the input T-7 seismograms. Using the estimated migration v e l o c i t i e s (Table 8, p. 82) and the two-way v e r t i c a l times of Figure 62, depths to the r e f l e c t i n g interfaces were computed at d i f f e r e n t horizontal locations and compared to the true depths extracted from Figure 32 (p.97). The r e s u l t s , shown i n Table 15, demonstrate that the imaging algorithm was able to produce a s a t i s f a c t o r y subsurface earth model with c o n f l i c t i n g dipping interfaces and noise-contaminated seismograms. TABLE 15 Interface depths. Layer V m j g X Estimated Depths True Depths 1 1950 -500 604 626 275 795 834 2 2250 0 1969 1960 1000 1733 1693 3 2500 0 2688 2600 The r e s u l t s presented up to t h i s point exemplify the f e a s i b i l i t y and p o t e n t i a l of the proposed plane-wave domain method to generate 146 common-shot gather images of an explored earth model. The imaging algorithm w i l l now be used to delineate a complete a n t i c l i n e earth model. Imaging of a Synthetic Seismic Line Figure 64 displays an a n t i c l i n e earth model that i s probed by 65 shots with a spacing of 50 m between them. The s p l i t - s p r e a d receiver layout used f o r each shot contained 60 receivers equally spaced at 25m i n t e r v a l (this produced an o f f s e t range from -1500m to 1500m). The author's implementation of the Kirchhoff wavefield extrapolation method was used to generate the synthetic seismograms. A Ricker wavelet with center frequency of 16 Hz and sampling i n t e r v a l of 0.004s constituted the source function. Figure 65a shows the a n t i c l i n e image obtained a f t e r the application of the plane-wave domain imaging algorithm to a l l the shot gathers (an enumeration of the processing steps c a r r i e d to generate the image of Figure 65a w i l l be given below). In order to evaluate the attained image, a synthetic normal-ray section and the corresponding time migrated sections were generated and displayed i n Figures 65b and 65c, respectively. Note that because the migration v e l o c i t y used to produce Figure 65c was the f i r s t layer v e l o c i t y , the time migrated section represents the true horizontal and v e r t i c a l time locations of the a n t i c l i n e signature. 147 Comparison between Figures 65a and 65c shows that the s t r u c t u r a l behaviour of the r e f l e c t i n g interfaces i s s i m i l a r . The r e l a t i v e l y poor continuity on the r i g h t side of the a n t i c l i n e signature i n Figure 65a i s at t r i b u t e d to non-optimal v e l o c i t y and interface dip angles used at these locations. In general, however, the plane-wave domain imaging algorithm has produced acceptable in t e r f a c e images. The plane-wave domain processing sequence followed to obtain the image of Figure 65a was divided i n two parts: i n the f i r s t one, v e l o c i t y and dip angle values were estimated at a few s p a t i a l locations selected along the seismic l i n e . In the second one, the shot gather images were obtained at a l l the ava i l a b l e shot locations. The required steps were the following: I Computation of the earth model parameters. * From the 65 shots available, every eighth was selected. * Plane-wave seismograms were obtained from the selected common-shot gathers. * V e l o c i t y - d i p analysis was performed on the resultant plane-wave seismograms. The estimated migration v e l o c i t i e s , i nterface dip angles and normal-ray traveltimes are shown i n Table 16. II Processing of a l l the available data. * Plane-wave seismograms were computed from each of the 148 common-shot gathers. S p a t i a l i n t e r p o l a t i o n of the earth model parameters of Table 16 was c a r r i e d out at each of the shot locations. With the earth model parameters determined i n the previous step, plane-wave domain imaging was applied to each set of plane-wave seismograms. The resultant common-shot gathers were resorted to common-receiver gathers (CRG). F i n a l l y , stacking of each CRG was performed to produce one output trace at every receiver s t a t i o n . 149 TABLE 16 Migration v e l o c i t i e s , interface dip angles and normal-ray traveltimes for the a n t i c l i n e earth model example. The f i r s t set of earth parameters corresponds to the a n t i c l i n e interface; the second set to the horizontal layer. LOCATION 800m 1200m 1600m 2000m 2400m 2800m 2800m 3200m T n 0.883 0.842 0.695 0.536 0.477 0.492 0.583 0.743 V 1850 1950 1800 1600 1700 1700 1600 1700 a 0 -5 -25 -10 0 5 15 25 T N 1.177 1.168 1.156 1.112 1.112 1.108 1.130 1.186 V 1850 1900 2000 1800 1700 1700 1700 2150 a 0 0 -5 -5 0 0 5 5 LOCATION Loc: 4000m T n 0.857 V 1800 a 0 V a 1.179 1800 0 150 OFFSET - 2 0 0 0 - 1 5 0 0 - 1 0 0 0 - 5 0 0 O 5 0 0 1000 1500 2000 Figure 58. Imaged common-shot gather corresponding to the earth model of Figure 24 (p.89). Compare to the synthetic model response of Figure 59. OFFSET Figure 59. Synthetic, two-way v e r t i c a l traveltime section corresponding to the earth model of Figure 24. 152 OFFSET -2000 -1500 -1000 -500 O 500 1000 1500 2000 Figure 60. Imaged common-shot gather corresponding to the earth model of Figure 29 (p.94). Compare to the synthetic model response of Figure 61. 153 OFFSET -2000 -1500 -1000 -500 O 500 1000 1500 2000 Figure 61. Synthetic, two-way v e r t i c a l traveltime section corresponding to the earth model of Figure 29. OFFSET 154 Figure 62. Imaged common-shot gather corresponding to the earth model of Figure 32 (p.97). Compare to the synthetic model response of Figure 63. 155 OFFSET Figure 63. Synthetic, two-way v e r t i c a l traveltime section corresponding to the earth model of Figure 32. c -156 ,0 400 800 1200 1600 2000 2400 2800 3200 36,00 4000 4400 4800 T : 1 : i 1 1 1 1 i 1 —i 1 — I 1 "• 1 r i 1— 1 •' 1 ' > i i r I t i Sources k s i -\ r 1 1 1 • T " 1 i i | "f • i V ^ O O O m/s r ' "1* V2=2500 m/s -0 400 800 1200 1600 -2000 --240E' Figure 64. A n t i c l i n e earth model. 500 1000 1500 2000 2500 3000 3500 4000 Figure 65. c a.) Image of the a n t i c l i n e earth model obtained v i a the plane-wave domain imaging algorithm, b.) Expected normal-ray section obtained v i a a Kirchhoff forward modelling program, c.) Migrated a n t i c l i n e model obtained v i a the application of a conventional constant-v e l o c i t y migration algorithm to the section of panel (b). 158 CHAPTER 8 CONCLUSIONS o The main work presented i n t h i s thesis consisted of the development of plane-wave domain algorithms to estimate migration v e l o c i t i e s and interface dip angles, and to obtain common-shot gather r e f l e c t o r images. Ad d i t i o n a l l y , two algorithms to compute plane-wave seismograms using Fourier transforms were presented. The success of a l l the proposed methods was demonstrated on a series of synthetic and f i e l d data examples. The r e s u l t s have shown the poten t i a l a p p l i c a t i o n of the proposed methods i n a seismic-data processing industry environment. The main motivation behind the present work was to show a new way to process seismic data using the plane-wave domain from common-shot gathers to f i n a l migrated sections. As an addi t i o n a l outcome, the methods presented i n t h i s work should help the exploration geophysicist to become better acquainted with the plane-wave domain. o Chapter 2 presented two algorithms to compute plane-wave seismograms v i a Fourier transforms. The algorithms consisted of the app l i c a t i o n of the double f a s t Fourier transform to the input data followed by a complex m u l t i p l i c a t i o n of e s s e n t i a l l y the Fourier representation of the Bessel function J 0 . A numerical s i n g u l a r i t y 159 was avoided by applying an a n a l y t i c a l expression that approximately accounts f o r the singular point contribution. An inverse one-dimensional fas t Fourier transform produced the desired plane-wave seismograms. A series of synthetic and f i e l d data examples showed that the algorithms of chapter 2 gave r e s u l t s comparable to those obtained v i a the true Hankel-transform based plane-wave decomposition method. However, i t i s expected that the proposed methods w i l l run very e f f i c i e n t l y i n computers with highly optimized microcode f o r fast Fourier transforms and complex vector m u l t i p l i c a t i o n s . Some possible future work on the methods of chapter 2 should include: a. ) The techniques can be extended to construct plane-wave seismograms f o r p o s i t i v e and negative angles of emergence (currently they are formulated for only p o s i t i v e angles of emergence). Since the algorithms operate i n the frequency-wavenumber domain and the angle of emergence i s d i r e c t l y related to the r a d i a l wavenumber, a dip f i l t e r i n g operation can be incorporated i n the algorithms to give plane-wave seismograms for both angular d i r e c t i o n s . As i t was shown i n chapter 3, plane-wave seismograms f o r p o s i t i v e and negative angles of emergence are required from data acquired on earth models composed of planar dipping r e f l e c t i n g interfaces. b. ) The methods can be enhanced by incorporating a n t i - a l i a s i n g 160 f i l t e r s i n the frequency-wavenumber domain. This w i l l attenuate undesirable events that show a predictable t r a j e c t o r y i n that domain. c.) Since plane-wave decomposition and spherical-wave reconstruction are related v i a a Hankel transform, an algorithm to compute spherical-wave seismograms from plane-wave components can be developed from the same theory of chapter 2. The spherical-wave reconstruction algorithm i s needed fo r cases when the plane-wave domain i s used only as a f i l t e r i n g technique, i n which case the (t,x) seismograms are the ultimate goal. Chapters 3, 4 and 5 presented the essence of t h i s research. Assuming earth models composed of planar dipping layer interfaces, two plane-wave domain approaches were proposed to estimate the earth model parameters from the information contained i n common-shot gathers: a. ) The f i r s t approach yielded migration v e l o c i t i e s and interface dip angles v i a a semblance analysis performed along cosinusoid t r a j e c t o r i e s i n the plane-wave domain. Straight-ray t r a v e l paths were assumed. b. ) The second approach gave i n t e r v a l v e l o c i t i e s and interface dip angles v i a a recursive ray-tracing algorithm applied to plane-wave traveltimes. Straight-ray t r a v e l paths were not assumed i n t h i s case. 161 Synthetic and f i e l d data examples were presented. The examples showed the a b i l i t y of the methods to produce reasonable r e s u l t s for earth models composed of planar dipping interfaces with conforming and opposing dip angle d i r e c t i o n s , and with small and large v e l o c i t y jumps from layer to layer. Acceptable r e s u l t s were also obtained i n noisy and f i e l d data examples. Since the proposed techniques work i n the plane-wave domain, t h e i r success greatly depends on the q u a l i t y of the recorded t-x seismograms. To improve the chances of success, no e f f o r t must be spared to a t t a i n the best qu a l i t y of the input r e f l e c t i o n signatures before the a p p l i c a t i o n of the proposed estimation techniques. For example, conventional processes l i k e bandpassing, deconvolution, r e f r a c t i o n and residual s t a t i c s , and v e l o c i t y f i l t e r i n g w i l l favorably precondition the data for plane-wave decomposition. This i s p a r t i c u l a r l y important for the ray-tracing based algorithm, since i n t h i s case traveltime picking i s required. As f o r possible future work, i t i s thought that the incorporation of the ray-tracing based algorithm (which does not assume s t r a i g h t -ray t r a v e l paths) i n the v e l o c i t y - d i p semblance analysis method w i l l produce a more robust earth model parameter estimation algorithm to handle cases where the earth model has non-locally planar r e f l e c t i o n interfaces, where the straight-ray approximation f a i l s , and when traveltime time picking i s very d i f f i c u l t . Lastly, chapters 6 and 7 presented a plane-wave domain imaging 162 method that, using the v e l o c i t y - d i p analysis r e s u l t s , produced common-shot gather images of the r e f l e c t i n g interfaces. The obtained images represent the horizontal surface locations of the r e f l e c t i o n points and the two-way v e r t i c a l times to them. A seri e s of synthetic examples demonstrated the performance of the proposed method to image earth models composed of l o c a l l y planar dipping interfaces with conforming and opposing dip angle di r e c t i o n s , and with small and large v e l o c i t y jumps from layer to layer. A noisy example and a f u l l synthetic seismic l i n e were also presented. Although the encouraging r e s u l t s showed the p o t e n t i a l of the imaging method as another route to e x i s t i n g time migration techniques, future work i s s t i l l necessary before the technique can be implemented i n an industry environment. Some of t h i s work should include: speeding up the algorithm, removal of the straight-ray approximation and improving the s p a t i a l i n t e r p o l a t i o n step required during the imaging. 163 REFERENCES Aki, K., and Richards, P., 1980, Quantitative seismology: W. H. Freeman and Co., San Francisco. Bath, M., 1968, Mathematical aspects of seismology: E l s e v i e r Publishing Co., Amsterdam. Benoliel, S. D., Schneider, W. A., and Sh u r t l e f f , R. N., 1987, Frequency-wavenumber approach of the r-p transform: some applications i n seismic data processing: Geophysical Prospecting, v.35, 517-538. Berkhout, A. J . , 1984, Seismic migration: imaging of acoustic energy by wave-field extrapolation, v.B: E l s e v i e r , Amsterdam. B e r r y h i l l , J . R., 1979, Wave equation datuming: Geophysics, V.44, 1329-1344. Born, M., and Wolf, E., 1980, P r i n c i p l e s of optics: Pergamon Press, Toronto. Brekhovskikh, L. M., I960, Waves i n layered media: Academic Press, New York. i 164 Brysk, H., and McCowan, D., 1986, A slant-stack procedure for point source data: Geophysics, v.51, 1370-1386. Cabrera, J . , 1983, Plane-wave decomposition and reconstruction of spherical-wave seismograms as a l i n e a r inverse problem: M.Sc. th e s i s , Dept. of Geophysics and Astronomy, Univ. of B r i t i s h Columbia, Vancouver, B.C. Cabrera, J . , and Levy, S., 1984, Stable plane-wave decomposition and spherical-wave reconstruction: applications to converted S-mode separation and trace i n t e r p o l a t i o n : Geophysics, v. 49, 1915-1932. Cabrera, J . , and Levy, S., 1989, Shot dip moveout with logarithmic transformations: Geophysics, v. 54, 1038-1041. Claerbout, J . F., 1985, Imaging the earth's i n t e r i o r : Blackwell S c i e n t i f i c Publications, Oxford. Clayton, R. W., and McMechan, G. A.*, 1981, Inversion of r e f r a c t i o n data by wave f i e l d continuation: Geophysics, v. 46, 860-868. Cohen, J . , and B l e i s t e i n , N., 1979, V e l o c i t y inversion procedure for acoustic waves: Geophysics, v.44, 1077-1085. erase, E., 1989, Robust e l a s t i c non-linear inversion of seismic waveform data: Ph. D. thesis, Univ. of Houston, Houston. Devaney, A. J . , and Sherman, G. C., 1973, Plane-wave representations fo r scalar wave f i e l d s : Soc. Ind. Appl. Math. Rev., v. 15, 765-786. Diebold, J . B., and S t o f f a, P., 1981, The t r a v e l time equation, tau-p mapping and inversion of common midpoint data: Geophysics, v. 46, 238-254. Dix, C. H., 1955, Seismic v e l o c i t i e s from surface measurements: Geophysics, v.20, 68-86. Gazdag, J . , 1978, Wave-equation migration with the phase-shift method: Geophysics, v.43, 1342-1351. Goodman, J . W., 1968, Introduction to Fourier optics: McGrawHill, San Francisco. Gradshteyn I., and Ryzhik, I., 1980, Table of in t e g r a l s , s e r i e s and products. Corrected and enlarged e d i t i o n by A. J e f f r e y : Academic Press Inc., New York. Harding, A. J . , 1985, Slowness-time mapping of near-offset r e f l e c t i o n data: Geophys. J . Roy. Astr. S o c , v.80, 463-492. 166 Hubral, P., 1977, Time migration- Some ray t h e o r e t i c a l aspects: Geophysical Prospecting, v.25, 738-745. Hubral, P., and Krey, T., 1980, Interval v e l o c i t i e s from seismic r e f l e c t i o n time measurements: Society of Exploration Geophysicists, Tulsa. Muller, G., 1971, Direct inversion of seismic observations: Z e i t s c h r i f t fur Geophysik, v.37, 225-255. O t t o l i n i , R., and Claerbout, J . F., 1984, The migration of common midpoint slant stacks: Geophysics, v.49, 237-249. Reshef, M., andKosloff, D., 1986, Migration of common-shot gathers: Geophysics, v.51, 324-331. Schultz, P. S., 1982, A method f o r d i r e c t estimation of i n t e r v a l v e l o c i t i e s : Geophysics, v.47, 1657-1671. Schultz,P., and Claerbout, J . , 1978, V e l o c i t y estimation and downward continuation by wavefront synthesis: Geophysics, v.43, 691-714. Sommerfeld, A., 1909, Uber die Ausbreitung der Wellen i n der drahtlosen Telegraphie: Ann. der Physik, v.28, 665-756. 167 Stoffa, P., Buhl, P., Diebold, J . , and Wenzel, F., 1981, Direct mapping of seismic data to the domain of intercept time and ray parameter- A plane-wave decomposition: Geophysics, v.46, 255-267. Tatham, R., Keeney, J . , and Noponen, I., 1982, Applications of the tau-p transform (slant stack) i n processing seismic r e f l e c t i o n data: Paper presented at the 52nd. Annual International SEG meeting, Dallas, Texas, 52 p. T r e i t e l , S., Gutowski, P. R., and Wagner, D. E., 1982, Plane-wave decomposition of seismograms: Geophysics, v.47, 1375-1401. Wenzel, F., and Stoffa, P. L., 1982, Seismic modelling i n the domain of intercept time and ray parameter: IEEE Transactions, v.ASSP-30, 406-422. Weyl, H., 1919, Ausbreitung elektromagnetischer Wellen uber einem ebenen L e i t e r : Ann. der Physik, v.60, 481-500. Whittaker, E. T., 1902, On the p a r t i a l d i f f e r e n t i a l equations of physics: Math. Am., v.53, 333-355. Yilmaz, O., 1987, Seismic data processing: Society of Exploration Geophysicists, Tulsa. 168 APPENDIX A BRIEF DERIVATION OF THE PLANE-WAVE DECOMPOSITION INTEGRAL A b r i e f description of the steps leading to equation (2.1) w i l l now be presented. For d e t a i l s see Aki and Richards (1980) , Devaney and Sherman (1973) and T r e i t e l et a l . (1982). Consider a point source at the o r i g i n x=0 of a Cartesian coordinate system (x,y,z) ra d i a t i n g compressional waves i n a homogeneous, i s o t r o p i c and unbounded medium. Given that the source exhibits time dependence of the form exp{-iut}, where w i s angular frequency, compressional-wave propagation may be described by the scalar displacement p o t e n t i a l <|> that s a t i s f i e s the wave equation where V i s the compressional-wave v e l o c i t y of the medium. The space-time solu t i o n of (A.l) i s (Aki and Richards, 1980) <|>(t,x) - [l/RJexp{i«(R/V-t)}, 32<j>/3<()2 - V2v2<|> - 47tV25 (x)exp(-iwt} (A.l) where x = x i + yj + zk, A <J A R = -J x 2+y 2+z 2 , and i , 3,k are the unit vectors along the coordinate axes. 169 Using Fourier transforms methods, the wavenumber-time solution of equation (A.l) i s <|>(t,k) = [47vV 2/(k 2V 2-w 2) ]exp{-iut} , where the wavenumber variable i s given by — A A A k = k x i + k v j + k 2k, and k = The r e l a t i o n between <J)(t,x) and <J)(t,k) i s given by the t r i p l e Fourier transform: <t>(t,x) = [l/R]exp{i«(R/V-t) } + 00 = [l / 2 T V 2]exp{-iwt) [l/(k 2-w 2/V 2)]exp{ik.x}dk. (A.2) u - OJ This equation represents a spherical wave <j)(t,x) t r a v e l l i n g with a constant v e l o c i t y V as a superposition of plane waves propagating with v e l o c i t y |u/k|. Because |w/k| ranges from 0 to i n f i n i t y , equation (A. 2) does not represent a true mode expansion of a spherical wave as a superposition of plane waves propagating a l l with the same medium v e l o c i t y V. To obtain plane waves t r a v e l l i n g with the same v e l o c i t y V, one of the integr a l s i n (A.2) must be evaluated. Evaluation of the k z i n t e g r a l (Aki and Richards, 1980, p.195-197, Devaney and Sherman, 1973, p.770-775) gives 170 <J)(w,x) = [l/R]exp{i«R/V} + 00 > n = [1/27T] d tl - oo [l/k z]exp{ik zz}exp{ik xx+ik yy}dk xdk y, (A.3) where k z = ±^| w 2 / v 2 - k x 2 - k 2 Note that the term exp ( - iwt) has been dropped and instead the w dependence i s e x p l i c i t l y stated i n the argument of (j). Equation (A.3) i s known as the Weyl i n t e g r a l . I t represents a spherical wave as a superposition of plane waves t r a v e l l i n g with the same v e l o c i t y V. For k z r e a l ( i . e . , f o r k x 2+k y 2 < u 2 / V 2 ) the plane waves are homogeneous and propagate along the di r e c t i o n s s p e c i f i e d by the wavenumber vector k = k xi+k yj+k zk. However, fo r k z imaginary ( i . e . , f o r k x 2 + k y 2 > w 2 / V 2 ) the plane waves are inhomogeneous, t r a v e l along d i r e c t i o n s s p e c i f i e d by k x i + k y j and have an exponential attenuation i n the z d i r e c t i o n . Transformation of (A.3) to c y l i n d r i c a l coordinates gives the plane-wave representation i n terms of the Sommerfeld i n t e g r a l : <j)(w,r,z) = [1/^| r z+z z]exp{iw^| r'+z'/V) + 00 [ l / i k z ] e x p { i k z z } k r J 0 ( k r r ) d k r , (A.4) 171 where r = •[ x 2 + y 2 k r = - J k x 2 + k y 2 = w s i n ( 7 ) / V , a n d k z = ±-J w 2 / V 2 - k r 2 = ±uCOS(7 ) / V . In these equations, 7 denotes the polar angle of the wavenumber vector k and J 0 denotes the zero-order Bessel function of the f i r s t kind. For a point source located i n a homogeneous h a l f space overlying a ho r i z o n t a l l y s t r a t i f i e d medium, equation (A.4) i s rewritten as ( T r e i t e l et a l . , 1982, Muller, 1971) + 00 n <J)(w,r,z) = U ( w , k r ) [ l / i k z ] e x p { i k z z } k r J 0 ( k r r ) dk r, (A.5) where U ( w , k r ) represents the combined e f f e c t of the plane-wave r e f l e c t i o n and transmission c o e f f i c i e n t s , and of the phase delays originated by the propagation through the medium layers. With t h i s i n mind, U ( w , k r ) i s defined as the spectra of plane-wave seismograms. The Fourier transform of the v e r t i c a l component of displacement (denoted by S(w,r,z) ) i s obtained from the displacement pot e n t i a l v i a S(w,r,z) = 3<|>(w,r,z)/az. 172 Applying t h i s r e l a t i o n to equation (A.5) gives + 00 S(«,r,z) = U(«,k r)exp{ik 2z}k rJ 0 (k rr) dk, For an observation point on the earth's surface z=0, t h i s l a s t equation reads as + 00 S(u,r) - U ( w , k r ) k r J 0 ( k r r ) dk r, (A.6) with k r = wsin(7)/V (see equation (A.4) ). Muller (1971) recognized equation (A.6) as a Fourier-Bessel transform and presented i t s formal inversion as + 00 n u ( w f k r ) = S ( u , r ) r J 0 (k rr) dr. (A.7) The above two equations are used to obtain plane-wave seismograms (via equation (A.7) ) from spherical-wave seismograms, or to reconstruct spherical-wave seismograms (via equation (A.6) ) from the plane-wave components. In chapter 2, the plane-wave decomposition i n t e g r a l (equation (A.7) ) i s solved v i a Fourier transform methods. 173 APPENDIX B COMPUTATION OF THE ANGLES a l y a 2 AND b 2 FOR THE CASE OF TWO INTERFACES In a model consisting of two layer interfaces a plane wave i s considered to take o f f from the source location, to bounce from the second interface and to emerge at the earth's surface with an angle 7. Given t h i s angle, the layer v e l o c i t i e s V : and V 2 and the interface dip angles a! anda 2, the angles a 1 # a 2 , andb 2 defined along the ray's t r a v e l path w i l l be determined (see Figure 66). F i r s t , b 2 i s found from the analysis of the ray path that connects the receivers plane to the r e f l e c t i n g i n terface. At the r e f l e c t i o n point (location TJ1 i n Figure 66) Snell's law reads as i sm(7-a 1) s i n ( b 2 ) V i v 2 Using i b 2 = b 2 — a i i n the above expression, b 2 i s obtained v i a s i n ( b 2 - a 1 ) = V i sin(7-a!). (B.l) 174 Next, from the ray geometry defined around the point P 2 i n Figure 66, a 2 i s computed from Note the s i m i l a r i t y of t h i s expression to equation (3.13), p.46, which was derived for the case of a planar dipping i n t e r f a c e . Last, the angle a x i s found from the analysis of the ray path between the r e f l e c t i o n point P 2 and the earth's surface. In t h i s case, a : i s obtained from the application of Snell's law at the point B1 i n Figure 66: that i s (B.2) r sin(a 1+a 1) = s i n ( a 2 ) , (B.3) where r a 2 = a 2 + 175 APPENDIX C DETERMINATION OF THE TERM Z S / 2/V 2 In t h i s appendix the term Z s, 2/V 2 w i l l be expressed i n terms of the traveltime of the corresponding normal-ray plane-wave component. The traveltime of the ray r e f l e c t e d normal to the second interface i s computed v i a equation (4.4), p.66: z s ,1 2 2 s ,2 r n / 2 = — - — [cos(a 1)+cos(7 I 1 / 2) ] + — - — co s ( a 2 ) c o s ( b 2 - a 2 ) . ( C l ) v t v 2 where "Yn/2 denotes the angle of emergence of the normal ray. The upward and downward r e f r a c t i o n angles are rela t e d i n the following way (refer to Figures 66 and 67): n = yn/2 (normal-ray plane wave) i b 2 = a 2 - a x , b 2 = a 2 , and a 1 = ~ 7 n , 2 ' r i a 2 = -b 2 , a 2 = -b 2 t 176 Substitution of these l a s t equations into (C.l) y i e l d s 2 Z s 1 2 Z s 2 T n # 2 = — C O S ( 7 2 ) + — C O S ( a 2 ) , V i v 2 from which 2 Z S / 1 T n , 2 - C O S ( 7 n / 2 ) 2 Z S / 2 V t V 2 C O S ( a 2 ) (C.2) In t h i s expression the quantity 2Z S t ^/V\ i s the traveltime of the ray r e f l e c t e d normal to the f i r s t i nterface. From equation (3.15), p.47, t h i s term i s given by 2 Z s , I T n , 1 V 1 C O S ( a j ) (C.3) Substitution of t h i s equation into (C.2) y i e l d s the f i n a l expression for Z S / 2 / V 2 , T n , i T n , 2 ~ C O S ( 7 n / 2 ) 2 Z S / 2 C O S ( a i ) = . (C.4) V 2 C O S ( a 2 ) 177 APPENDIX D INTERCEPT-TIME EQUATION FOR THE CASE OF N INTERFACES From Diebold and Stoffa (1981) the intercept-time equation for plane waves reads as K - 1 Z e . , Vn Z( s l V 1 s / j T = — — [cos(a 1)+cos(7) ] +) — — [cos(aj)+cos(bj) ] 1 j = 2 Z S , M [cos(a N)+cos(b H)] (D'l) where a^, b j , and Z S / j are defined i n Figure 68. The goal here i s to rewrite the l a s t term of t h i s equation as a function of the dip angle of the Nth r e f l e c t i n g i n t e r f a c e . This i s achieved by following the same sequence of steps that were used to derive equation (3.14) from (3.12), p.45. This r e s u l t s i n the expression Z s,K 2 Z 8 , K [cos(a K)+cos(b N)] = cos(a„)cos(b„-a M) . Substitution of t h i s equation into (D.l) gives 178 N - 1 Z e , Z, S l V1 8 / j T = — — [ c o s ) + c o s ( 7 ) ] +) — — [cos(aj)+cos(bj)] 1 j = 2 J 2 2 S , M C O S ( a N ) C O S ( b N - a M ) , (D.2) which i s the desired expression. 179 APPENDIX E DETERMINATION OF THE ANGLES aj AND bj For a given plane wave with an angle of emergence 7, the angles aj and bj w i l l be determined by applying a series of recursive r e l a t i o n s based on Snell's law. The angles b j , j=2,... ,N, are defined i n the upward portion of the ray path of Figure 68, i . e . , from the r e f l e c t i o n point to the earth's surface. For a given angle of emergence 7, layer v e l o c i t i e s , ... ,VN_ j , and search v e l o c i t y V H (V N ' i n the notation of the main text) the angles bj are obtained v i a the expressions shown i n Table 17. 180 TABLE 17 Determination of the upward ray angles b j . Layer Snell's Law Compute Angle 1 Angle 2 si n ( 7 - 0 4 ) s i n ( b 2 ) V, b 2 =b2+a1 s i n ( b 2 - a 2 ) s i n ( b 3 ) b 3=b 3+a 2 sin(bj-aj) s i n ( b j + 1 ) V j + 1 'j + l b j + l = b j + i + a j N-1 s i n ( b K _ 1 - a N _ 1 ) sin(b„) V N-1 bN=bK+aK_ t The angles a j , j = l , . . . ,N-1, are defined i n the downward portion of the traveled ray path. They are linked to the upward angles bj v i a the ray geometry around the r e f l e c t i o n point. S p e c i f i c a l l y , the angle a N r i s given by (see Figure 68) r 181 Once a N r i s found, a l l the angles aj are re c u r s i v e l y calculated v i a the equations shown i n Table 18. TABLE 18 Determination of the downward ray angles a } . Layer S n e l l 's Law Angle Compute 1 Angle 2 s i n ( a N _ 1 + a N N-l V N - i • _x) sin(aj) • aM- 1 • r aN- 1=aN- 1+aN- 2 • • • sin(aj+aj) j • • s i n ( a - + x) v i + i • • • • r a j = a j + a j - l sin(a 2+a 2) 2 v 2 sin(aj) v 3 a 2 r a 2 = a 2 +01 j r sinCaj+ai) sin(a2) 1 a x Vx V 2 APPENDIX F 182 DETERMINATION OF THE TERMS Z S / j / V j The term Z s / j/Vj w i l l be computed from the traveltime r n t j of the j t h normal-ray plane-wave component. For the case of a ray r e f l e c t e d normal to the j t h interface, the downward angles a k are the negative of the upward angles b k ; that i s (refer to Figures 68 and 69) * i • " b i • -Y n # J a k = ~ b k The use of these e q u a l i t i e s into equation (4.5), p.69, gives j - i 2 2 s / 1 V Z 8 ' Ic 2 2 S ' j T „ , J - c o s ( 7 n / j ) + 2 ) cos(b k) .+ C O S ( a j ) , from which 183 3 - 1 2Z e vn Z, T n , j 8 1 V 1 s k — — c o s ( 7 n <) - 2) — cos(b k) 2 Z s , j k = 2 . (F.l) Vj cos(aj) This equation i s v a l i d for any j > 3. To i l l u s t r a t e the recursive determination of the terms 2 Z S / j / V j , equation (F.l) i s expanded for the case of j=3: 2Z j j 2Z § 2 T n / 3 - — C O S ( 7 n / 3 ) - — C O S ( b 2 ) 2 Z S / 3 V x V 2 V 3 C O S ( a 3 ) However, from equation (C.2) 2 Z 8 # 1 (F.2) T n , 2 " C O S ( 7 n / 2 ) 2 Z 8 / 2 Vi V 2 C O S ( a 2 ) and from (C.3) 2 2 s , l T n , 1 Vj C O S ( a j ) Substituing these two expressions into (F.2) y i e l d s T n , l n , 3 " C O S ( 7 n # 3 ) cos ( 0 4 ) c o s ( a 3 ) n / 1 n , 2 C O S ( a 1 ) C O S ( 7 n / 2 ) C O S ( a 2 ) cos(b 2) c o s ( a 3 ) 185 Figure 66. Geometrical ray path of a plane wave r e f l e c t e d from the second interface. 186 Figure 67. Geometrical ray path of a plane wave r e f l e c t e d normal to the second in t e r f a c e . 187 Figure 68. Geometrical ray path of a plane wave r e f l e c t e d from the Nth in t e r f a c e . Figure 69. Geometrical ray path of the plane wave r e f l e c t e d normal to the Nth interface.
Thesis/Dissertation
10.14288/1.0052931
eng
Geophysics
Vancouver : University of British Columbia Library
University of British Columbia
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Graduate
Acceleration waves
Seismic waves
Velocity-dip analysis in the plane-wave domain
Text
http://hdl.handle.net/2429/30604