Fundamentals of Seismic Tomography

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GEOPHYSICAL MONOGRAPH SERIES

David V. Fitterman, Series Editor

Larry R. Lines, Volume Editor

NUMBER 6

FUNDAMENTALS OF

SEISMIC TOMOGRAPHY

By Tien-when Lo and Philip L. Inderwiesen

SOCIETY OF EXPLORATION GEOPHYSICISTS

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Lo, Tien-when, 1957- Fundamentals of seismic tomography /by Tien-when Lo and

Philip L. Inderwiesen. p. cm. (Geophysical monograph series;no. 6)

Includes bibliographical references and index. ISBN 978-1-56080-028-6:$22.00

1. Seismic tomography. I. Inderwiesen, Philip L., 1953- II. Title. III. Series.

QE538.5.L6 1994 551.2' 2' 0287--dc20 94-23818

CIP

ISBN 978-0-931830-56-3 (Series) ISBN 978-1-56080-028-6 (Volume)

Society of Exploration Geophysicists P.O. Box 702740

Tulsa, OK 74170-2740

¸ 1994 by the Society of Exploration Geophysicists All rights reserved. This book or portions hereof may not be reproduced in any form without permission in writing from the publisher.

Published 1994

Reprinted 2000 Reprinted 2004 Reprinted 2006 Reprinted 2008

Printed in the United States of America

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Contents

Preface vii

I Introduction

1.1 The Concept of Seismic Tomography ............. 1 1.2 Applications ........................... 3 1.3 Ray vs. Diffraction Tomography ............... 5 1.4 Suggestions for Further Reading ............... 6

2 Seismic Ray Tomography 9 2.1 Introduction ........................... 9

2.2 Transform Methods ....................... 10

2.2.1 Projection Slice Theorem ............... 10 2.2.2 Direct-Transform Ray Tomography .......... 16 2.2.3 Backprojection Ray Tomography ........... 20

2.3 Series Expansion Methods ................... 22 2.3.1 The Forward Modeling Problem ........... 23 2.3.2 Kaczmarz' Method ................... 26

2.3.3 ART and SIRT ..................... 33

2.4 Summary ............................ 39 2.5 Suggestions for Further Reading ............... 42

3 Seismic Diffraction Tomography 45 3.1 Introduction ........................... 45

3.2 Acoustic Wave Scattering ................... 46 3.2.1 The Lippmann-Schwinger Equation ......... 47 3.2.2 The Born Approximation ............... 51 3.2.3 The Rytov Approximation ............... 52 3.2.4 Born vs. Rytov Approximation ............ 56

3.3 Generalized Projection Slice Theorem ............ 58 3.3.1 Crosswell Configuration ................ 59

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3.3.2 Vertical Seismic Profile Configuration ........ 72 3.3.3 Surface Reflection Configuration ........... 78

3.4 Acoustic Diffraction Tomography ............... 82 3.4.1 Direct-Transform Diffraction Tomography ...... 84 3.4.2 Backpropagation Diffraction Tomography ...... 87

3.5 Summary ............................ 90 3.6 Suggestions for Further Reading ............... •3

4 Case Studies 95

4.1 Introduction ........................... 95

4.2 Steam-Flood EOR Operation ................. 95 4.2.1 Crosswell Seismic Data Acquisition .......... 96 4.2.2 Traveltime Parameter Measurements ......... 99

4.2.3 Image Reconstruction ................. 108 4.2.4 Tomogram Interpretation ............... 111

4.3 Imaging a Fault System .................... 125 4.3.1 Crosswell Seismic Data Acquisition .......... 125 4.3.2 Traveltime Parameter Measurements ......... 130

4.3.3 Image Reconstruction ................. 133 4.3.4 Tomogram Interpretation ............... 135

4.4 Imaging Salt Sills ........................ 137 4.4.1 Assumptions and Preprocessing ............ 137 4.4.2 Data Acquisition .................... 139 4.4.3 Diffraction Tomography Processing .......... 141 4.4.4 Tomogram Interpretation ............... 146

4.5 Suggestions for Further Reading ............... 150

A Frequency and Wavenumber 153 A.1 Frequency ............................ 153 A.2 Wavenumber .......................... 154

B The Fourier Transform 157 B.1 Fourier Series .......................... 158

B.2 Exponential Fourier Series ................... 159 B.3 Fourier Transform- Continuous f(x) ............. 160 B.4 Fourier Transform- Sampled f(x) ............... 161 B.5 Uses of Fourier Transforms .................. 162

C Green's Function 167

C.1 Filter Theory .......................... 168 C.2 PDE's as Linear Operators .................. 170 C.3 Green's Function Example ................... 173

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C.4 Suggestions for Further Reading ............... 174

INDEX 175

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Preface

Motivated by the successful implementation of medical tomography in the early 1980s, geophysicists and production engineers have attempted analogous methods using seismic energy for hydrocarbon exploration, reservoir characterization, and production engineering. The theoretical methods and field techniques employed are broadly classified as "seismic tomography" and were fundamentally developed in the late 1980s. Today, seismic tomography is conducted on a commercial basis with its theory anchored on a solid base, and its strengths and limitations known. Field applications have demonstrated that seismic tomography can provide valuable services in upstream operations, such as mapping subsurface structures, delineating reservoirs, and monitoring enhanced oil recovery processes.

Our book develops the fundamentals of seismic tomography at the level of a tutorial or practical guide. Considerable effort has gone into making the book self-contained so that any reader who has had calculus can easily follow the material. References for further reading on specific topics are given at the end of each chapter. We give a short statement following each reference detailing its significance as a supplement to this book. In doing so we hope the reader will not feel the references must be read to fully understand a given concept. We use appendices to review physical terminology and mathematics required to understand the theoretical presentations.

We present various tomographic methods in a logical and straightforward manner. Unlike many other books on tomography, we use standard notation for variables which span the various methods, enabling the reader to easily contrast differences. Also, mathematical steps glossed-over by most research articles are filled-in for our readers. Sometimes we deviate

from well-known derivations to provide a deeper physical understanding. However, for completeness, our derivations are followed-up with references to the "well-known" derivations at the end of the chapter. In addition, we discuss the limitations of seismic tomography and illustrate successes and pitfalls with case studies. Our ultimate intent is that after reading this presentation, the reader will exhibit both a greater understanding and appreciation for seismic tomography articles presented in the literature.

Chapter I is introductory and summarizes the development of seismic tomography and describes how this new technology can benefit the oil industry at both the exploration and producing stages. Chapters 2 and 3 are tutorials on the theoretical fundamentals of seismic ray tomography and seismic diffraction tomography, respectively. Chapter 4 presents the

vii

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data acquisition, processing, and tomogram interpretation for three seismic tomography case studies. Each case study has its own unique data acquisition, data processing, and interpretation challenges. They provide useful insight into designing and conducting future tomography studies.

The authors thank Texaco for releasing data and results for the case studies published in Chapter 4. We also acknowledge the release of the McKittrick data by Texaco's joint partner, Chevron, for that project. Both Eike Rietsch and Bob Tatham of Texaco have encouraged the authors to pursue this project and have provided support during its progress. We also acknowledge our fellow borehole seismology team members at Texaco: Danny Melton, Don Howlett, Ron Jackson, and Stan Zimmer for their contributions to this field throughout the past few years. In addition, David Fitterman, the SEG monographs series editor, and Larry Lines, the volume editor for this book, have been patient with our progress and have provided valuable guidance. Finally, the authors thank Texaco for permission to publish this book.

Philip L. Inderwiesen Tien-when Lo

E•tP Technology Department Texaco Inc.

Houston, Texas

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Chapter I

Introduction

1.1 The Concept of Seismic Tomography

We define tomography as an imaging technique which generates a cross- sectional picture (a tomogram) of an object by utilizing the object's response to the nondestructive, probing energy of an external source. Seismic tomography makes use of sources that generate seismic waves which probe a geological target of interest.

Figure l(a) is an example configuration for crosswell seismic tomography. A seismic source is placed in one well and a seismic receiver system in a nearby well. Seismic waves generated at a source position (solid dot) probe a target containing a heavy oil reservoir situated between the two wells. The reservoir 's response to the seismic energy is recorded by detectors (open circles) deployed at different depths in the receiver well. The reservoir is probed in many directions by recording seismic energy with the same receiver configuration for different source locations. Thus, we obtain a network of seismic raypaths which travel through the reservoir.

The measured response of the reservoir to the seismic wave is called the projection data. Tomography image reconstruction methods operate on the projection data to create a tomogram such as the one in Figure l(b). In this case we used projection data consisting of direct-arrival traveltimes and seismic ray tomography to obtain a P-wave velocity tomogram. Gen- erally, different colors or shades of gray in a tomogram represent lithology with different properties. The high P-wave velocities (dark gray/black) in the tomogram in Figure l(b) are associated with reservoir rock of high oil saturation.

Seismic tomography has a solid theoretical foundation. Many seismic

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2 CHAPTER 1. INTRODUCTION

Crosswell Seismic /'-wave Velocity Con fig urat ion Tom ogram

o

o

source

receiver

heavy oil

(a) (b)

F•G. 1. (a) Geometry for crosswell seismic tomography example. P-wave energy traveling along raypaths probe the geological target. (b) P-wave velocity tomogram reconstructed from observed traveltime data. Different shades of gray correspond to different P-wave velocities.

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1.2. APPLICATIONS 3

tomography techniques have close ties to more familiar seismic imaging methods such as traveltime inversion, Kirchhoff migration, and Born inversion. For example, seismic ray tomography used to determine lithologic velocity is essentially a form of traveltime inversion and seismic diffraction tomography is closely related to Born inversion and seismic migration. Thus, seismic tomography may actually be more familiar to you at this point than you might think since it is just another aspect of the subsurface imaging techniques geophysicists have been using for years.

1.2 Applications

Seismic tomography is applicable to a wide range of problems in the oil industry, ranging from exploration to development to production. The case studies presented in Chapter 4 demonstrate that seismic tomography can complement conventional seismic methods and provide unique, previously unavailable subsurface information. Tomography applied to surface seismic data can generate subsurface velocity models for exploration problems. These velocity models can in turn be used as soft information in the geostatistical interpolation of well-log data between wells.

Seismic tomography applied to development and production problems is generally implemented by a crosswell configuration, as shown in Fig- ure l(a). Figure 2 illustrates the benefit of crosswell seismic tomography for reservoir characterization over conventional reservoir characterization

tools. Figure 2(a) represents the true geology between two wells in a producing field where the producing formation is a tar sand layer overlaid by a thinner, less permeable bed (shaded interval). The heavy oil in such tar sands is somewhat immobile unless heated using the enhanced oil recovery technique of steam flooding. A production engineer planning to steam flood a tar sand interval needs to know whether the less permeable bed is capable of confining the steam to the tar sand. In our cartoon the less permeable bed is breached by a small fault.

Well logging is a conventional reservoir characterization tool that provides information about the reservoir only a small distance from the borehole as depicted in Figure 2(b). Thus, no hard geological information about the unprobed reservoir between the wells can be extracted from conventional well-log data. The well-log data will only show that the low permeability layer exists between 500 and 600 feet in well A and between 400 and 500 feet in well B. Based upon the relative formation dips in each well, the engineer may decide the low permeability layer is continuous and interpret the well-log data using linear interpolation as shown in Figure 2(c). The small fault is therefore not detected and unexpected steam flood results will

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CHAPTER 1. INTRODUCTION

(a) true geology

steam-'

A B

200 ft

4OO ft

600 ft

8OO ff

(b) well logging

A B

(c) well logging interpre[ation

A B

I logging tool I source

D receiver

(d) tomography

A B

(e) _tomography Interpretati__on

A B

FIG. 2. (a) True geology we wish to know. (b) Well logs sample only a short distance into the reservoir, requiring some type of interpolation between wells as depicted in (c). (d) Crosswell seismic records the earth's response to seismic energy between wells thereby permitting an image reconstruction of the geology as shown in (e).

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1.3. RAY V$. DIFFRACTION TOMOGRAPHY 5

occur.

On the other hand, crosswell seismic tomography can directly probe the reservoir between the two wells as shown in Figure 2(d). A downhole seismic source in well A generates seismic waves powerful enough to travel through the reservoir and to be recorded by sensitive detectors in well B. Af- ter applying tomography processing to the projection data, a tomographic interpretation like the one in Figure 2(e) might be obtained. Thus, the engineer will be aware of the small fault and can make the necessary alterations to the steam flood operation. Although just a cartoon, Figure 2 illustrates how crosswell seismic tomography is a more reliable tool for delineating the reservoir between wells than any interpolation method between well logs. However, crosswell seismic tomography becomes an even more significant tool for reservoir characterization when used in conjunction with well-log information and core data as is demonstrated in Chapter 4.

1.3 Ray rs. Diffraction Tomography

To do seismic tomography we must model the seismic wave traveling through the subsurface. Both ray and diffraction theoretical models are available to us for describing seismic wave phenomena. Which model we use depends upon the relative sizes of the seismic wavelength and the target we wish to image. A judicial choice of theoretical model for a given seismic wave and target becomes important to the success of the seismic tomography application.

If the target's size is much larger than the seismic wavelength, then we may model the propagation of seismic waves as rays using ray theory. This is similar to using geometrical optics to describe light wave propagation through lenses. Seismic tomography based upon the ray theoretical model is discussed in Chapter 2 under the title Seismic Ray Tomography. We subdivide the topic into "transform methods" and "series expansion methods." The transform methods are commonly used in medical tomography experiments while the series expansion methods see much use in seismic tomography applications. Currently seismic ray tomography is very popular because it is simple to implement under a variety of situations, is computationally fast, and gives good results.

When the size of the target is comparable to the seismic wavelength, then we model the propagation of seismic waves as scattered energy using diffraction theory. Such a target scatters the seismic wave in many directions and only diffraction theory can properly model this response. Seismic tomography based upon the diffraction theoretical model is discussed in Chapter 3 under the title Seismic Diffraction Tomography. As

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6 CHAPTER 1. INTRODUCTION

you will see, seismic diffraction tomography presented in its simplest form requires restrictions on the source-receiver geometry, ignores multiple scattering of energy, places limits on the sizes and velocity contrasts of targets, and is computationally intensive. Because of these restrictions the method is currently applied only to a few select situations. However, re- cent developments, which we list under "suggestions for further reading" in Chapter 3, are overcoming some of these restrictions. Our presentation of seismic diffraction tornography in its simplest form should give you a solid foundation for understanding and appreciating these developments.

Several case studies are presented in Chapter 4 to illustrate the application of theory to various situations. We emphasize the need to assimilate as much data as possible from other sources, such as from well logs and core samples in the crosswell tomography examples. Only by integrating all information available with the tomogram can one make an optimum assessment about the reservoir.

' As a final note, we have crisply divided the application of seismic tomography into ray and diffraction tomography, depending upon the relative sizes of the seismic wavelength and target. However, in reality the probing seismic wave is usually a broad-band signal consisting of a large range of wavelengths, and the subsurface contains potential targets with relative sizes ranging from small to large. Thus, this suggests a blend of seismic ray tornography and seismic diffraction tornography be used to optirnally image all possible targets. Although interesting, we pursue this possibility no further as it is more of a research matter at this point in time. In this book we will concentrate on presenting the fundamentals of seismic tomography.

1.4 Suggestions for Further Reading Aki, K., and Richards, P., 1980, Quantitative seismology: The-

ory and methods, Vol. II: W. H. Freeman & Co. Section 13.3.5 o.f Chapter 13 presents a classification scheme based upon seismic wavelength and target size which will give you a good idea when to use ray theory or diffraction theory.

Anderson, D. L., and Dziewonski, A.M., 1984, Seismic tomography: Scientific American, October, 60-68. Popular arlicle on seismic ray tomography applied to imaging the earth's mantle.

Lines, L., 1991, Applications of tomography to borehole and reflection seismology: Geophysics: The Leading Edge, 10, 11-17. Overview of seismic ray tomography applications.

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1.4. SUGGESTIONS FOR FURTHER READING 7

Menke, W., 1984, Geophysical data analysis: Discrete inverse theory: Academic Press, Inc. Our book addresses only those topics in inverse theory required to understand the basics in seismic tomography. Menke's book provides a good introduction to inverse theory.

Tarantola, A., 1987, Inverse problem theory: Methods for data fitting and model parameter estimation: Elsevier. A com- prehensive book on inverse theory which includes many prob-

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Chapter 2

Seismic Ray Tomography

2.1 Introduction

We begin the study of seismic tomography with image reconstruction methods based on ray theory. We assume that the source produces seismic wave energy with wavelengths much smaller than the size of the inhomogeneities encountered in the medium. Only when this assumption is obeyed can the propagation of the seismic wave energy be properly modeled by rays. Otherwise, the seismic diffraction tomography in Chapter 3 must be applied to solve the problem.

Two groups of image reconstruction methods exist for doing seismic ray tomography. The transform methods in Section 2.2 comprise the first group. Applications of transform methods have their roots in astronomical and medical imaging problems. They are very limiting as far as seismic imaging problems are concerned since straight raypath propagation and full-scan aperture are generally assumed. However, the transform methods make an excellent introduction to the principles of tomography because of their simplicity and serve as a bridge between applications of tomography in other fields with applications in seismology. Also, the development of seismic diffraction tomography has a close relationship with the transform methods. The series expansion methods in Section 2.3 comprise the second group of image reconstruction methods. Out of all the methods presented in this book the series expansion methods presently see the most use in seismic tomography. Therefore, a large part of Chapter 2 is spent addressing the series expansion methods.

Before proceeding further one should have a good grasp of the Fourier transform concepts to understand the material in Section 2.2. Appendix B

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10 CHAPTER 2. SEISMIC RAY TOMOGRAPttY

.---i'""• O x-ray transmitter - r

FIG. 3. Setup for a medical tomography experiment. X-ray scans are taken in different directions about a person's head by rotating the transmitter- detector assembly.

presents a review of the Fourier transform.

2.2 Transform Methods

The projection slice theorem is presented first in this section since it provides the theoretical foundation for the transform methods. Then, two transform methods are derived from the projection slice theorem' direct- transform ray tomography and backprojection ray tomography.

2.2.1 Projection Slice Theorem

The derivation of the projection slice theorem is illustrated by a typical medical tomography experiment. Figure 3 shows the setup for medical tomography. A donut-shaped x-ray transmitter-detector assembly surrounds the target, a person's head in this example. X-ray intensity is measured for a fixed orientation of the assembly. Then the assembly is rotated about the person so that x-rays pass through the head in a different direction. The experiment is completed when the person's head is scanned in all directions. The objective of the transform methods is to use attenuation information from the measured x-ray intensities to reconstruct a cross-sectional image

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2.2. 'fRANSFORM METHODS 11

of the person's head in the z - z plane which contains the transmitters and detectors. Thus, a brain tumor which attenuates x-rays differently than normal tissues may be readily "seen" by the radiologist.

Figure 4 depicts a cross-section of a person's head. The varying contrasts within the target represent nonuniform x-ray attenuation associated with a tumor, normal tissues, and the skull. For the purpose of deriving the projection slice theorem, we define the model function M(z, z) • as the spatial distribution of the attenuation. In general, the model function represents the unknown distribution in space of some physical property of the target medium which affects the propagating energy in some observable manner. A typical model function used in seismic tomography is the reciprocal compressional-wave velocity, or slowness, which has a direct influence on the observed traveltime of the propagating energy.

The projection slice theorem requires that observations of the propagating energy be taken along a given projection which is perpendicular to the raypaths. Figure 4 illustrates a single projection in the medical tomography experiment. X-rays emitted by the transmitters travel along the parallel rays and are recorded by detectors positioned along the u-axis. The rotated spatial coordinate system (u,v) is introduced to describe all of the possible orientations for the transmitter-detector assembly about the target. The v-axis is defined parallel to the direction of x-ray propagation and the u- axis, defined perpendicular to the v-axis, is the direction along which the x-ray intensity is measured. If the u- v coordinate system shares the same origin as the z- z coordinate system, then the relationship between the two coordinate systems when one is rotated through an angle 0 relative to the other is

z sin 0 cos 0 u ] (1) ß

For a given ray in Figure 4 we define P(u, O) as the decimal percent drop in x-ray intensity,

P(u, o) = [.'o - 0)l/o,

where I(u,O) is the intensity measured by the detector at (u,O) and Io is the x-ray intensity at the transmitter. We refer to P(u, O) as the data

1 Other literature on tomography might refer to the model function as an image function or as an object function. We chose "model function" to be consistent with inverse problem terminology.

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12 CHAPTER 2. SEISMIC RAY TOMOGRAPHY

v

x

X-RAY

SOURCES

P(., O)

Z

U

FIG. 4. Cross section of a person's head where varying contrasts represent nonuniform x-ray attenuation. The projection P(u, t•) is the decimal- percent drop in x-ray intensity measured along the rotated coordinate axis, u. The u-axis is perpendicular to the v-axis which always parallels the x-ray propagation direction. The model function M(x, z) provides a numerical value for the attenuation and is an unknown which must be determined

from the observed projections P(u, •).

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2.2. TRANSFORM METHODS 13

function 2. If the attenuation is small over the raypath, then data function P(u, O) is linearly related to the attenuation M(z, y) as the line integral,

e(,,, o) - (2) ay

taken over the raypath. s Note that each observation of the data function a P(u, O) provides an empirical solution to equation (2) along the given raypath without actually knowing the model function M(x, z).

If there is negligible x-ray attenuation outside the target, then equation (2) gives the same measured projection for any transmitter-detector separation as long as each transmitter and detector remains outside of the target. We use this assumption to rewrite equation (9•) with infinite limits; a mathematical step which will be taken advantage of later in this section. Thus,

0) - f:: z)a. (a) To obtain a simpler and more meaningful relationship between the

model function M(x, z) and the data function P(u, 0), we transform each into the Fourier domain. The 2-D Fourier transform of the model function

M(z, z) is

117I(k•, k,.) - f;: ];: M(x, z)e-J(k•x + k,.z)dxdz ' (4) where k• and kz are spatial frequencies along the x- and z-axes, respectively. Spatial frequency, or, wavenumber, is defined as k = 2•r/A where is wavelength. Figure 5 represents the 2-D Fourier transform 's amplitude spectrum • of a hypothetical model function M(x, z). Note that if •(k•,

2 P(u, O) is a projection in the ray tomo•aphy problem, but is c•ed a data ]unction in inv•e theory te•nology. The, we c• the v•iable P a "data f•ction" for the sine re.on we c•ed M(x,y) the "model f•ctioff' e•Ser. However, we represent the data f•ction with the v•iable P to re.rid you that the me•ed data •e projections.

awe o•y consider the 5ne• inv•se problem in t•s book. Thus, the data f•ction will •waya be •ne•ly related to the model f•ction, even if • approximation is req•red to force the •ne• relatio•p. The •amption of am• x-ray attenuation is req•red for the x-ray tomo•aphy problem.

•Although the actuM observation is the x-ray intensity l(u,O) in t•a c•e, we wi• frequently refer to "the observation of the data f•ction" since it is in•rectly obt•ned from the observed data.

•Although the 2-D Fo•ier tryfore of the model f•ction •(x,z) h• both mp•- tude •d ph•e spectra, we represent the 2-D Fo•ier tryafore M(kx,k•) with o•y the mpftude spect• component, desi•ated • ]•(k,, k,) I.

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K

FIG. 5. The essence of the projection slice theorem is represented. The 2-D Fourier transform of a hypothetical model function M(x, z) produces the amplitude spectrum I M(ks, k,) [. The contours in the ks - k, plane connect equal values of amplitude. The amplitude spectrum [ P(fi, d) [ from the 1-D Fourier transform of the data function P(u, t•) represents a slice of [ J17/(ks, k,) [ along the fl-axis in the ks - k, plane.

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is known, then the unknown model function M(z, z) can be found by the 2-D inverse Fourier transform

M(x,z) = 1 fj: fj: 11•'I(k• k,)eJ(k•x + k,Z)dk•dk, (5) 4•r2 , ß

Now let •5(f•, 0) represent the 1-D Fourier transform of the data function P(u, O) along the u-axis, shown in Figure 4. The 1-D Fourier transform is written

0) - f/: where f/ is spatial frequency along the u-axis. Substituting equation (3) into equation (6) gives

We now wish to put equation (7) entirely in terms of z and z. The variable • is replaced with z and z using the inverse of equation (1) given by

v - sin 0 cos 0 z '

Using equation (8) and replacing dvdu with dzdz in equation (7) we get,

•5(f•, O) -- f:: f:: M(x, z)e-Jf•( x cos 0 +z sin O)dxdz = f:: f:: M(x,z)e-j[(f•cosO)x + (ftsinO)z]dxdz ' (9)

Comparing the integrands of equation (9) and equation (4) we see that equation (9) is simply the 2-D Fourier transform of M(x, z) where kx and k, are restricted to the Q-axis by setting

k• = •cos0, and

k, = f/sin0. (10)

This relationship is evident in Figure 5. Substituting equation (10) into equation (9) we write

•5(fi, O) - /:: /:: M(x,z)e-J(k:•x + k,Z)dxdz, (11)

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16 C•APTER 2. SEISMIC RAY TOMOGRAPHY

where ks and kz are defined by equation (10). Comparing the integrands of equations (11) and (4) shows that we have achieved a simple relationship between the data function P(u, O) and the model function M(x, z) in the spatial frequency domain,

P(•, O) - A7/(k,,, kz). (12)

In words, equation (12) states that the 1-D Fourier transform of the projection represented by the data function P(f],_0) is equal to one slice of the 2-D Fourier transform of the model function M(kz, k•) defined on the loci: kz = f] cos 0 and k, = f]sin 0. Equation (12) is called the projection slice theorem.

The projection slice theorem gives only one slice of the model function per projection as shown in Figure 5. We will now show how many projections at different angles of 0 are used to reconstruct the entire model function via the projection slice theorem. The two techniques presented are the direct-transform ray tomography method and the backprojection ray tomography method. In Chapter 3 we will define an analogous theorem for the reconstruction methods in diffraction tomography called the generalized projection slice theorem.

2.2.2 Direct-Transform Ray Tomography

Direct-transform ray tomography utilizes the projection slice theorem in a straightforward manner. We showed in the previous section that the application of the projection slice theorem to the 1-D Fourier transform of a single projection represented by the data function /5(12, 0) determines only one slice of the model function A•(k• - 12 cos 0, k• - f] sin 0). Figure 5 illustrates such a slice through the model function. To recover the entire model function, the target must be probed from many different directions.

Figure 6 shows three directions along which x-rays probe the head of our make-believe patient. The observed data functions for these three projections are P(u, 0•), P(u, 02), and P(u, 0a). After applying the projection slice theorem to the 1-D Fourier transforms of these data functions, we obtain the three slices through the model function's amplitude spectrum shown in Figure 7. Now the 2-D Fourier transform of the model function M(k•, k,) is better defined than by the single slice shown in Figure 5, but is still inadequate for image reconstruction. We must probe the target with x-rays from all directions letting 0 range from 0 degrees to 180 degrees. Only then will the k•- kz plane in Figure 7 be completely covered by slices M(f] cos0, f]sin0), where 0 ranges from 0 degrees to 180 degrees. After such an experiment, the complete 2-D Fourier transform of the unknown model function M(x, z) is determined along radial lines in the k•-k, plane.

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•X

Z

0 2

FXG. 6. The cranium of our make-believe patient is probed with x-rays in three different directions: 0•, 09., and 03. Application of the projection slice theorem to the data functions resulting from thesc projections yields the slices depicted in Figure 7.

To obtain M(z, z) from .g/(f• cos O, f• sin 0), an algorithm employing the direct-transform ray tomography method must first interpolate the data from a polar grid (f•cosO, f•sinO) onto a Cartesian grid (k•,,kz) in the k• - kz plane, or

AT/(f•cosO, f•sinO) in,•r__•ot.,• A•(k•,kz). (13) One must exercise caution in performing the interpolation since large errors introduced by the operation could obscure the true solution.

Lastly, a 2-D inverse Fourier transform of M(k•,,k;•) is performed to obtain M(x, z),

M(z z) = I '• .•l(k• k,)eJ(k•z 4-k•Z)dk•,dk ' (14) , 4•r 2 ,• ,• ' ß

Thus, the image reconstruction is completed and the technician may give the tomogram of the patient's head to the radiologist for interpretation.

The direct-transform ray tomography method is easily summarized in five steps:

Step 1: Acquire the data function P(u, O) of the target with the projection direction, 0, ranging from 0 degrees to 180 de-

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• sin61 ) M(•cosO 3 , • sin63 )1

K

K z

F•(•. 7. Plot showing slices through the unknown model function's amplitude spectrum I M(k=, k•) I found by applying the projection slice theorem to the data functions found for the x-ray projection directions 01, 0•, and 03 shown in Figure 6.

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grees. Remember that the unknown model function M(z, z) corresponding to the target represents a physical property which affects propagating energy in some manner (e.g., attenuation which affects propagating x-ray intensity in medical tomography or slowness which affects seismic wave traveltimes in seismic tomography). Thus, a data function is just the line integral of the model function along each ray, or

Step 2: Perform a 1-D Fourier transform along the u-axis for each data function given by

P(n, o) - f:: o)-J Step 3: Use the projection slice theorem to obtain slices of the

2-D Fourier transform of the model function. Each slice is

defined by

cos0, iqsin0) -- J5(i2,0).

Step 4: Convert the 2-D Fourier transform of the model function in the k• -kz plane from a polar grid (f• cos 0, f•sin 0) to a Cartesian grid (k•, kz),

Step 5: Perform a 2-D inverse Fourier transform on M(k•,, k•) to obtain M(x,z), the reconstructed image of the target. The inverse transform is given by

47i -2

The direct-transform ray tomography method would be quick to implement if it were not for the fourth step above requiring interpolation of the model function in the frequency domain. In the next section we present backprojection ray tomography which obviates the need for interpolation resulting in a faster and more accurate algorithm.

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CHAPTER 2. SEISMIC RAY TOMOGRAPtIY

2.2.3 Backprojection Ray Tomography

Backprojection • ray tomography uses the same data acquisition scheme as the direct-transform ray tomography: record the data function P(u, O) by experimentally measuring the line integral of the unknown mode] function M(:r,z) along different raypaths or projections. The difference between backprojection ray tomography and direct-transform ray tomography is how we compute the model function from the data function.

To derive the backprojection ray tomography method, we first write down the 2-D inverse Fourier transform of the model function •(k•, k,),

•(• z) = • • •(• •)•j(•+•z)•• (•5) ' 4•2 ' ß

Next we make a change of variables in equation (15) by replacing k• with • cos0 and k• with •sin 0, and by changing the integration from dk•dk, to ] • l d•dO. • This gives,

M(x,z) = 4• 2 M(•cos0,•sin0) • ej•(xcosO + zsin 0) I•ld•dO. (1•)

Integration with respect to 0 in equation (16) can be rewritten • two integrals,

1 • • (• cos 0, • sin 0) M(r,z) = 4=• • ej•(x cos 0 + z sin 0) I•ld•dO

• ' •[• •o•(0 + •) • •in(0 + •)] +• ,

• •j•[• •o•(0 + •) + z•in(0 + •)] I• [ •0. (17) Using the fundamental trigonometric angl•sum relations, cos(0 + •) = - cos 0 and sin(0 + •) = - sin 0, we may rewrite the second model function

•The te• "b•mjection" imp•es the inve•e problem where we st•t with the pr• jection •d •lve for the model f•ction. Here the projections •e t•en Mong raypat•. • Section 3.4.2 the me•ed projectio• of scattered energy •e described by the wave equation •d we •e • •Mogo• te•, "backpropagation."

•The ch•ge • inte•ation is •Mogo• to going from C•tesi• coorSnates to pol• coor•n•es to compute the •ea of a •sk. For a •sk we replace dxdz by rdOdr, where r is the r•M •st•ce from the •sk's center. The absolute vMue of • is t•en in o•

c•e to prese•e the si• of the •fferentiM •ea when we will shortly •ow • to t•e on negative vMu•.

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on the right-hand side in equation (17) as,

.g/[f•cos(0 + •r),f•sin(0 + •r)] = .tfl(-f• cos O, -f• sin O). (18)

The second set of integrals on the right-hand side in equation (17) is rewritten by replacing the model function with equation (18), applying the angle- sum relations used in obtaining equation (18) to the exponential, setting f• - -f• and dfl - -dfl, and reversing the direction of integration with respect to fl. With these operations equation (17) is written,

M(x z) = 1 • •(fl cos 0, fl sin 0) , 4•2

x ej•(x cos 0 + z sin 0)

•(• cos •, • sin •) +• • •j•(• •o• • • • •i. •) I • I ••.

Combining the integrals with respect to the variable • we get,

' 4• '

• d•(• •o• • • • •i. •) I • I ••. (•)

Using the projection slice theorem, we replace M(• cos•, •sin •) in equa-

struction formula ß

, 4•2 '

We can summarize the backprojection ray tomography reconstruction method in just three steps:

Step 1: Data acquisition. Let the model hnction M(x, z) represent the unknown parameter (such • seismic wave slowness) at position (x, z). Experimentally determine the line integral of the model function along each ray which yields a set of data hnctions (such • seismic wave traveltime),

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22 CHAPTER 2. SEISMIC RAY TOMOGRAPttY

Step 2: Take the 1-D Fourier transform of each data function along the u axis.

P(fl, O) - /:: P(u, O)e-Jfl u du. Step 3: Use the backprojection formula equation (20) to com-

pute the unknown model function M(x, z), or

= 4•.2 '

x + [ I

Unlike direct-transform ray tomography, backprojection ray tomography does not require a 2-D interpolation in the wavenumber domain, and therefore, is in general faster and more accurate than direct-transform ray tomography.

It should be mentioned that most commercial CAT (Computerized Ax- ial Tomography) scanners use the backprojection ray tomography or its modification as their image reconstruction algorithm.

2.3 Series Expansion Methods

Series expansion methods comprise a group of computation algorithms which, like the transform methods, determine the model function M(x, z) of the target area. However, unlike the transform methods, these algorithms easily allow curved raypath trajectories through the target area and are therefore well suited for applications in seismic tomography. As before, we restrict the discussion to a 2-D problem so that the model function M(z, z) is determined in a plane which cuts through the target and contains all of the sources and receivers.

Our discussion of the series expansion methods is divided up into three subsections. Section 2.3.1 presents the forward modeling problem which permits us to predict the tomography data in terms of a system of linear equations which explicitly contain an estimate of the true model function. Section 2.3.2 shows how the true model function is determined using Kacz- marz' method. The method devised by Kaczmarz in 1937 is iterative and determines an approximate solution to the true model function. Drawing an analogy, the Kaczmarz method is to the series expansion method as the

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2.3. SERIES EXPANSION METHODS 23

projection slice theorem is to the transform methods. We exploit Kacz- marz' method in Section 2.3.3 to derive two series expansion algorithms: the algebraic reconstruction technique (ART) and the simultaneous iterative reconstruction technique (SIRT).

2.3.1 The Forward Modeling Problem

As will be shown shortly, the series expansion method iteratively updates an estimated model function M e't so that it converges toward a true model function M true. The updates are found by comparing the observed data function polo with a predicted data function ppre. Forward modeling is required to determine the predicted data function and is the subject of this section.

Equation (2) in Section 2.2 defines the experimental process for the transform methods as the line integral of the function M(a:,z) along a straight raypath in the v-axis direction. According to the above notation equation (2) could be written,

Pøbø(u, O) -- i Mt"u•(z' z)dv' ay

We did not require a forward modeling procedure in Section 2.2 because the raypaths were straight and the projection slice theorem could be employed directly to determine the true model function Mtrue(x, z).

For the series expansion methods we wish to include curved raypaths. Equation (2) is easily transformed to accommodate curved raypaths by rewriting the model function in terms of a position vector r. Thus, for a given source-receiver pair the line integral of the model function M(r) over the raypath is

pobo = f• MtrU•(r)dr, ay

where the observed projection given by the data function polo represents the measured line integral (observed tomography data) and MtrUe(r) is the true model function which remains to be determined. The last equation is used to formulate the forward modeling by setting

P- i M(r)dr, (21) ay

where P is now the predicted data function and M(r) is the estimated

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24 CHAPTER 2. SEISMIC RAY TOMOGRAPtIY

M I M 2 M 3 M 4 M 5 M 6

M 7 M8 M9 M10 Mll M12

M13 M14 M15 M16 M17 M18

M 19 M 20 M 21 M 22 M 23 M 24

FIG. $. The series expansion methods use a discrete model function Mj, j - 1,..., J, where Mj is the average value of the continuous model function M(r) within the jth cell. Here J - 24.

model function s . Thus, forward modeling is defined as determining the predicted data function from the line integral along the raypath through a known, but estimated, model function.

Just as was done with the transform methods, the model function in the series expansion ray tomography is discretized to allow computation by digital computer. Figure 8 shows an image area of a target divided into many small cells. Each cell is assigned the average value of the physical parameter (e.g., x-ray attenuation, slowness, etc.) represented by the continuous model function M(r) within that cell. The model function in Figure 8 is divided into 24 cells and is written discretely as M•, where j - 1,..., 24. Thus, Mj represents the average value of M(r) within the jth cell.

Figure 9 depicts a single ray traveling through the discretized model function. Equation (21) is rewritten in discrete form, to describe ray travel through the discrete model function, as

J

P- j=l

where Mj is the estimated model function for the jth cell, ,5' i is the raypath

SHere we will symbolize the predicted data function as P and the estimated model function a• M for brevity. These symbols will be changed to ppre and M est, respectively, in the following section.

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• r

Z 7 6 ..•_1.. •• source

receiver • •._•..1•.. ••15 M16 M17 IV116

M 19 M 20 M 21 M 22 M 23 M 24

FIG. 9. Ray travel through a discrete model function. The resulting data function, determined by the line integral through the discrete model function, is defined by equation (22).

length of the ray within the jth cell, and J is the total number of cells in the gridded target. The example in Figure 9 has J - 24 cells, but the ray penetrates only seven cells (j - 12, 11, 10, 16, 15, 14, and 13). To keep equation (22) consistent with equation (21) we set Sj - 0 for all cells not penetrated by the ray. After all, the ray's path length Sj for the jth cell is obviously zero if the ray did not traverse that cell.

Figure 9 shows 17 cells for which we don't have information because the single raypath did not traverse them. By adding more sources and receivers around the unknown target region, different rays sample the 17 unsampled cells in addition to some of the cells already sampled. The addition of extra rays is depicted in Figure 10. Now all of the cells are interrogated by this network of rays.

We must modify the index notation of equation (22) to include a projection value for every ray. If Pi represents the projection, or line integral, predicted for the ith ray, then equation (22) is rewritten,

J

Pi = E Mj $ij, for/- 1,...,1, (23)

where I is the total number of rays, $ij is the path length of the ith ray through the jth cell, and, as before, Mj is the discrete estimate of the model function for the jth cell and J is the total number of cells. Equation (23) is

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ray I

ray 2

!

ray I

source

receiver

FIG. 10. Generally, a single ray does not provide information on all of the model function's cells. However, by using more source and receiver locations around the target all of the cells can eventually be sufficiently interrogated. I rays were found sufficient here.

the formulation of the "forward modeling problem" used in series expansion ray tomography.

Equation (23) can effectively model the data acquisition process if we let the projections Pi, i - 1,..., I, be the observed data (i.e., traveltime or decimal percent decrease in x-ray intensity) and the model function j: 1,..., J, be the true, but unknown model function, or

J

Pi ø•' = • M]"•'*Sii, for i- 1,...,I. (24) j=l

Kaczmarz' method provides the theoretical framework for indirectly solving equation (24) for the true model function.

2.3.2 Kaczmarz' Method

In this section we introduce Kaczmarz' method to indirectly solve

equation (24) for the true model function M] ru*, j = 1,..., J, which is the tomogram. As stated in the previous section, forward modeling is required to determine the true model function. Thus, before proceeding we will reformulate equation (23) into a matrix form to simplify the mathematical discussion. Since equation (23) is discrete its elements are easily put into matrices. In matrix form equation (23) becomes,

P - SM, (25)

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where the predicted projections in data vector P are,

P1

P - . ,

Pl

the discrete estimated model function values in model vector M are,

ml

m - . ,

Ma

and the raypath lengths for I rays and J cells in S are,

S• S•. ... S• S•.• S• ... S•

s - . ß

S•i S•2 -..

Note that S in equation (25) can be thought of as a linear operator that operates on the estimated model vector M producing the predicted data vector P.

We could also formulate equation (24) in matrix form as

po•, = SM '•. (29)

Although we will not directly solve equation (29), we would want to determine the true model vector M true given pob• and S. The problem becomes one of finding a generalized inverse operator S-•.9 Then we could apply the generalized inverse operator S-• to both sides of equation (29) to determine the true model vector, or

S-ap oh, = S-aSM '•e

m M true .

Theoretically the last equation is true, but in practice it is very often difficult to determine S-a for two reasons. First, S is usually quite large and

9We write S-g rather tha•n S -1 as in usual matrix notation since S-g need not be

squaxe and because S-9S is not always the identity matrix.

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sparse, which makes computation of S-• costly. Second, S is usually "ill conditioned" which makes computation of S-• very unstable. iø

Kaczmarz' method circumvents the problems associated with the inversion of a large and sparse matrix and provides an efficient means for determining an approximate solution to equation (29) using an iterative procedure. Figure 11 presents a flowchart outlining the method in which there are three basic steps to the iterative part of the algorithm. An initial estimate of the model vector M init is input to the iterative loop of the algorithm and serves as the first "current estimate" M •st of the true solution M true. For now we assume that the initial model vector M init is known.

However, more will be said on the selection of the initial model vector M in the case histories presented in Chapter 4.

With the current estimate of the model vector M est known, the first step is to use the forward modeling problem defined by equation (25) to determine a predicted data vector ppre. This step is carried out by applying the linear operator S (determined by some ray tracing technique of personal choice) defined by equation (28) to the estimated model vector M •s•,

ppr• = SM•t. (30)

In the second step the predicted data vector PPr• is compared with the observed data vector pob, by taking the difference between the two. A small difference or good agreement between the predicted and observed data vectors implies good agreement between the estimated model vector M • and the true model vector M t•"•. Thus, if the difference is smaller than a specified tolerance, then the current estimate of the model vector M •t is output as the solution to equation (29) in the final step of the algorithm. The selection of a suitable tolerance for the difference is discussed with the

case histories in Chapter 4. The third step of the iterative portion of Kaczmarz' method comes into

play when the difference between the predicted and observed data vectors is larger than the specified tolerance. This important step essentially makes use of the difference information, pob• _ pp,.e, to update the current estimated model vector M • with a new estimate of the model vector

M("ew) • which hopefully is closer to the true model vector M •r"•. This third step is written in equation form as

M ("•w)•t = M e'• + A/M, (31) fori - 1,...,I,

Ill conditioned means small changes in S produce large changes in the model function true or in S-g however you wish to look at it

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Step 1

Initial estimate

Minit

currenf estimafe est

Step 2

Step 3

predicted observed data vector data vector

P pre pObS

FIG. 11. Flow chart for Kaczmarz' method. M i"i' is the initial estimate

of the model vector; M e'• is the current updated estimate of the model vector; ppre is the predicted data vector from the forward modeling given by equation (30); and pobo is the observed data vector. M eø• is iteratively updated until ppre matches pobo to within a specified tolerance.

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3O CttAPTER 2. SEISMIC RAY TOMOGRAPttY

where AiM represents the incremental update to the current estimate of the model vector and superscript i means applying equation (31) when the ith row of the pob• and ppre vectors are compared. The new estimate M (new)e•t is then taken as the current estimate for the next iteration. As

you will soon see, equation (31) brings the current model estimate toward the true solution, at least in theory.

The method of computing AiM in equation (31) is obviously a crucial 'method. Kaczmarz' method computes factor in the success of Kaczmarz

AiM with the equation'

where

AiM1 A i M2

- . , (32) ß

AiMj

pi .øb s __ piP r e -

(33)

Note that the summation in the numerator is just the predicted data vector ppre found in the forward modeling in equation (30).

Now we will geometrically derive equation (33) and show the convergence of Kaczmarz' method through a simple example. Let two rays (I = 2) travel through a two-cell model (J = 2) so that the vector equation (29) for the problem can be written as

p•,b, = Sll M1 + S12M2, for ray 1, and (34) p•b, __ S21 M1 + S22M2, for ray 2, (35)

where P•'b• and P•*• are observed data and M1 and M2 are unknown TM. Remember that $i1 is just the ith ray's path length through the jth cell and is generally known from the forward modeling. We plot equations (34) and (35)in Figure 12 as lines (hyperplanes) on a 2-D space with axes M1 and M2 .12 The solution to the equations occurs at point X where the two

• • Here M• and M2 are unknown and therefore defined as independent variables. Only when the solution is found are they referred to as M[ •ue and M• •ue as in equation (29).

•2If I = 3 and J = 3, then we would be looking for the intersection point of three planes in a 3-D model space. For situations where J > 3, equation (29) represents a J-dimensionM space and we would look for the solution at the intersection of I = J hyperplanes where a hyperplane has J - 1 dimensions. Note that we must have at least I = J hyperplanes to solve equation (29) and generally I > J.

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M 2

o\

H

obs

M2 = P1 ,,....__/ S12

• obs

12M2

FIG. 12. For two rays (I = 2) in a two-cell model (d = 2), possible values for M• and M2 are defined by [he two lines (hyperplanes). p•,b, and are [he observed data from [he two rays. Point X represents [he desired model values which lie at the intersection of the two hyperplanes. A• M• and A•M2 for ray 1 are geometrically derived so that point B is the projection of point A onto the hyperplane defined by equation (34). The result of this geometrical derivation is equation (33).

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32 CHAPTER 2. SEISMIC RAY TOMOGRAPItY

lines (hyperplanes) intersect. Since the solution at point X with coordinates (M•"•'e,M• "•'e) is un-

known, we must start with an initial estimate at point A given by the coordinates (M1 A, M• A). This initial estimate becomes the current estimate as shown in the flow chart for Kaczmarz' method in Figure 11. The geometrical step in Figure 12 is to find the perpendicular projection of point A onto the hyperplane •:• defined by equation (34) at point B. Mathematically this step is given by,

s = M+AM,and (36)

where A•M• and A•M2 are the corrections sought geometrically and defined by equation (33) for i- 1 and j- 1,2.

The first geometrical relationship to note in Figure 12 is the similarity of triangles AABC, AFED, and AGEH. Using these similarities we can immediately write

AIM• = AB cosc• = DFcos c•

A • M2 = AB sin c• = DF sin c•

= EF cos 2 c•

__GH • = EF and

'•E:Z ' = EF cos c• sin c•

(38)

--GH EH

= EFG---•- • . (39) Our task now is to determine EF, GH/•--•, and EH/GE in equa-

tions (38) and (39). On the line segment EF, point E is located at (M• = P•'b'/sI•, M2 = 0). Point F is the intersection with the M•-axis of a line which is parallel to equation (34) and contains the point (MIA,M•). The equation for this line is

S11M• A + S• 2M• A - S• • M1 + S1 :• (40)

From equation (40) we determine the coordinate of point F along the M•- axis as (M• - (SI•M1A + S•2M•A)/S•I, M2 - 0). Thus, the length of the line segment EF is given by

1 (41)

laAlthough equations (34) and (35) both represent lines, we will continue to refer to them as hyperplanes since that is what they are called when J > 3.

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The ratio GH/GE is simply,

GH

GE

ß (42)

Similarly, EH/GE is given by,

EH P•ø/Szz +

= S12 . (43) +

results in the model corrections due to the first ray,

p•b• I:•p re •_--' i and (44) AiM1 -- •'! 1 S121 .•. S122 p•bs I•Pr ß - -• (45) AiM2 - $1• $•---•+$• ,

where ,rre _ SzzMi • + $z•M• t Equations (44) and (45) are the same as a I ß

equation (33) when i = 1 and j = 1,2. Thus, we see that equation (33) simply determines the projection of a model estimate onto one of the hyperplanes defined by equation (29).

Carrying this example one step further, we can determine point I in Fig- ure 12, the projection of point B onto hyperplane 2 defined by equation (35) for the second ray using the indices i - 2, j - 1, 2 in equation (33). If we alternate projections of the model estimates between the two hyperplanes, then the updated model estimates (step 3 in Figure 11) must converge on point X as depicted in Figure 13. Thus, Kaczmarz' method will converge to the solution of equation (29).

2.3.3 ART and SIRT

The algebraic reconstruction technique (ART) and the simultaneous iterative reconstruction technique (SIRT) are the two common implemen- tations of Kaczmarz' method in seismic ray tomography. This section describes the basic features of both algorithms.

ART is a computational algorithm for solving equation (29) that directly uses Kaczmarz' method. Thus, the ART algorithm is comprised of the

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M 2

I Hyperplane 2

Hyperplane I

:-M I

FIG. 13. By applying equation (31) to alternating hyperplanes, the model estimate of (M•, M2), starting at point A, converges towards the solution for equations (34) and (35) at point X. The iterative updating of the model estimate corresponds to the loop in Kaczmarz' method shown in the flow chart in Figure 11.

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steps shown in Figure 11. We first set the estimated model function M• 't, j- 1,..., o r , to the initial model estimate Mj "i' j - 1, or Then the following three steps are iterated cyclically from one hyperplane to the next until the observed data p/ob, matches the predicted data pfre, for i= 1,...,I.

Step 1: Conduct forward modeling (ray tracing) for the ith ray using equation (23) or equation (25), restated for reference here as,

J

Only one ray is traced out of a total of I rays since we are determining the projection of the current model estimate onto only one hyperplane. Note that if our model consists of slownesses, then the predicted data P•'re are calculated traveltimes from the forward modeling and p/oh, are observed traveltimes.

Step 2: Subtract the predicted ith ray data p•0•, from the observed ith data p/o•,, and use equation (33) to find corrections for all of the J cells comprising the model function estimate TM,

= ' Step 3: Apply the corrections to the model estimate recom-

mended by the ith ray to all or cells,

SIRT differs from ART in that all I rays are traced through the model so that all AiMj corrections determined for the I hyperplanes are known.

14Note that the model adjustment depends upon the discrepancy between the predicted aaad observed data values and the raypath length through the cells for the ith ray.

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36 CHAPTER 2. SEISMIC RAY TOMOGRAPtIY

Then an average of AiMj with respect to index i is taken for each model cell j - I ..., J, to get new model estimates , ,j-1,...,J. As with ART, the model estimates M[ •t are updated until the predicted data Pf•e compares favorably with the observed data p/oh,, i- 1,..., I.

After setting the current model function estimate equal to the initial model function, or Mf øt - M'. i"it for j = I J the following three steps are iterated to update the model estimates:

Step1: Conduct forward modeling (ray tracing) using equation (23)or equation (25),

for all raysi -

Step 2: Find the correction for each cell by examining the rays cut through that cell and averaging the corrections recommended by each ray. This operation is defined for the jth cell by,

I

1 •AiM 1 = W,..= I BlObS _ 1

- i=1

forj - 1,...,J.

- EsS= s,s 1 • (46)

The weight Wj is the number of rays intersecting the jth cell or some other suitable ray density weight used to obtain an average correction AMj.

Step 3: Determine the new model estimate from the average model corrections AMj, or

M« "ew)e" = M;" +AM./, j - 1,...,J. Figure 14 illustrates how equation (46) makes SIRT different from ART.

As in Figure 12 we use only two rays (or I - 2 hyperplanes) and two model cells (or J - 2 model space) in order to visualize the problem. The ART algorithm is shown in Figure 14(a) and the SIRT algorithm in Figure 14(b).

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M2 M2

E . •

D C

A B

Hyperplane I

(a) (b)

FIG. 14. Comparison of ART and SIRT algorithms for the two-ray (or I - 2 hyperplanes) and two-cell model (or J - 2 model space) example given in Figure 12' a) Convergence of model estimates for the ART algorithm starting with an initial model estimate at point A. b) Convergence of model estimates for the SIRT algorithm starting with an initial model estimate at point A. The iteratively updated model estimates, determined from the average corrections AMj in equation (46), are along the solid line defined by points A, B, C, D, and E. Each estimated point is the average of the same letter's primed and double primed projection points located on hyperplanes I and 2, respectively.

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38 CHAPTER 2. SEISMIC RAY T¸M¸GRAPHY

The ART algorithm in Figure 14(a) finds the solution by alternately projecting the current model estimate onto each hyperplane. The model estimate moves along the solid line from the initial model estimate at A to B, to C, to D, and so on. On the other hand, the SIRT algorithm finds point A's projections on both hyperplanes, points B t and B •t, then moves the initial estimate from point A to point B, the midpoint between B t and Btt. For the next iteration the SIRT algorithm finds point B's projections on both hyperplanes, points C t and C", then moves the current estimate from point B to the midpoint between C t and C t•, or point C. Starting with the initial model estimate at point A, SIRT converges towards the solution along the solid line from point A to B, to C, to D, to E, etc. If a true model solution exists and is unique, then both ART and SIRT will converge to that solution. However, one should note that the convergence of ART depends upon the ordering of the hyperplane projections while the convergence of SIRT does not. You may see this by projecting point A onto hyperplane 2 first in Figure 14(a).

The ART and SIRT methods, as we have stated, are intended to solve linear systems of equations like those represented by equation (29) which explicitly relate the model function to the data function. But just because equation (29) explicitly relates the model function to the data function in a linear form does not imply a linear relationship for all types of model and data functions. Take for instance a model function of slowness and

a data function of observed direct-arrival traveltime. Equation (29) does not provide a linear relationship in this case because the raypat. h lengths in S are also dependent upon the slownesses defined in the model function. Thus, we do not know the true raypath lengths in S until the true slowness field is also known.

To solve the nonlinear problem in practice we compute estimated raypath lengths using the estimated slownesses in the model function and use the estimated raypath lengths in the ART or SIRT algorithm. This is called an iterative linear approach to solving a nonlinear problem. We can use Figure 12 to visualize what happens to the hyperplanes when solving a nonlinear problem by an iterative linear approach. The two hyperplanes shown in Figure 12 are the true hyperplanes when we know the raypath lengths in S. When we use estimated raypath lengths in S the estimated hyperplanes will not be coincident with the true hyperplanes. The resulting projection will be different from the projection shown in Figure 12 as point A is projected onto an estimated hyperplane. Each time we update the model function slownesses, using either the ART or SIRT method, the new estimated hyperplanes will be located differently in the model space since we also have new estimated raypath lengths in S. What we hope happens is that the estimated hyperplanes will not be wildly repositioned to a new

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2.4. SUMMARY 39

location each time the model slownesses are updated. Then as we iterate further towards the true model function slownesses the estimated hyperplanes should become more coincident with the true hyperplanes since the estimated raypath lengths will be approaching the true raypath lengths. Thus, we can get convergence to a solution even though the problem is nonlinear.

2.4 Summary

1. Seismic ray tomography attempts to solve the inverse problem formulated by the line integral equation,

p __ •.y M(r)dr, taken over the raypath. P is called the data function and represents the observed data. M(r) is called the model function and represents the spatial distribution of some physical property of the medium which affects the propagating energy in some observable manner. The model function is unknown and the goal of seismic ray tomography is to determine an estimated model function M e•t of the true model function M tr•e.

2. Transform methods in seismic ray tomography are of limited use since straight raypaths and full scan apertures are generally assumed. How- ever, they serve to introduce the tomography concept and terminology, and provide insight into seismic diffraction tomography presented in Chapter 3.

3. The projection slice theorem is the basis for the transform methods. The theorem states that the 1-D Fourier transform of the data func-

tion/5(f•, 0) provides a slice of information in the k• - k, wavenumber plane of the model function A74(k•,kz) defined on the loci' k• = f• cos0 and k, - f•sin 0 as shown in Figures 4 and 5. Equation (12) defines the projection slice theorem as,

4. Direct-transform ray tomography applies the projection slice theorem to many projections of the data function/5(f•, 0• for 0 degrees _( 0 _( 180 degrees. The result is the model function M(f•cosO, f•sinO) defined on a polar grid. Interpolation of the model on the polar grid onto a rectangular ks - k, grid is required to take the 2-D inverse Fourier

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transform of the model function which yields M(x, z). Interpolation error may distort the resulting estimated model function.

5. Backprojection ray tomography is another transform method which utilizes the projection slice theorem. However, by making a change of variables we were able to do the image reconstruction without the interpolation step required by direct-transform ray tomography. The reconstruction formula given by equation (20) is

M(x, z) = 4•r• , x ej•(xcosO + zsin0) ]•]dC2da.

Backprojection ray tomography or its modification is used as the image reconstruction algorithm in computerized axial tomography because it is both accurate and fast.

6. Series expansion methods are the most frequently used seismic tomography methods. The model function is divided up into small cells where each cell is assigned an average value of the continuous model function within that cell. Thus, the ith observation of the data function is related to the discrete model function by the equation,

J

Pi ø•' -- Z$ijM]•"•,i- l,...,I, j=l

where I is the total number of rays or observations, J is the total number of cells in the discrete model function, and $ij is the path length of the ith ray in the jth cell. The inverse problem is to determine an estimated model function M• '• of the true model function M] rue given the observed data function Pi ø•s.

7. Kaczmarz' method iteratively solves the system of equations defined for the series expansion methods for the estimated model function Mf s•. The iterative part of the algorithm consists of three steps as shown in Figure 11 and an initial estimate of the model function Y init must be input.

Step i requires that the current estimate of the model function be used in forward modeling (i.e., ray tracing) to get a predicted data function P/P• for the ith ray. The forward modeling is defined by the equation,

J

j=l

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2.4. SUMMARY 41

Step 2 compares the predicted data function pfre with the observed data function Pi øb•. If the observed and predicted data functions agree to within a specified error, then the estimated mode] function Mf • is taken a.s a good estimate of the true model function M] rue. Step 3 updates the current estimate of the model function M• • if the observed and predicted data functions do not compare favorably. The update correction is determined by projecting the current estimate of the model function onto the hyperplane defined by the ith ray. The correction is given by,

pfbz __ pfre 1

The corrections to the model estimate recommended by the ith ray are applied to all J cells,

j - 1,...,J.

Then, the updated estimated model function M« "ew)e• becomes the current estimated model function back in step 1.

8. The arithmetic reconstruction technique (AP•T) is a series expansion method which directly uses Kaczmarz' algorithm.

9. The simultaneous iterative reconstruction technique (SIP•T) uses a modified Kaczmarz' method. Instead of updating the model after tracing each ray in the forward modeling step a.s is done in AP•T, all rays are traced through the current estimated model and a model correction found for each ray. Then the corrections to each model cell are averaged according to the equation,

I

i=1

forj - 1,...,J.

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42

2.5

CHAPTER 2. SEISMIC RAY TOMOGRAPHY

The weight Wj is the number of rays intersecting the jth cell or some other suitable ray density weight used to obtain an average correction AM 1. The average correction is applied to the current estimated model function to get an updated or new estimated model function.

Suggestions for Further Reading

Anderson, A. H., and Kak, A. C., 1982, Digital ray tracing in two-dimensional refractive fields: J. Acoust. Soc. Am., 72, 1593-1606. Addresses ray tracing through a gridded refractive- index model commonly used in ultrasound computerized tomography.

Berryman, J. G., 1990, Stable iterative reconstruction algorithm for nonlinear traveltime tomography: Inverse Problems, 6, 21-42. Discussion of the nonlinear problem associated with traveltime tomography.

Bishop, T. N., Bube, K. P., Cutler, R. T., Langan, R. T., Love, P. L., Resnick, J. R., Shuey, R. T., Spintiler, D. A., and Wyld, H. W., 1985, Tomographic determination of velocity and depth in laterally varying media: Geophysics, 50,903- 923. Traveltime tomography applied to reflection seisinology.

Chapman, C. H., and Pratt, R. G., 1992, Traveltime tomography in anisotropic media- I. Theory: Geophys. J. Int., 109, 1-19. Expands traveltime tornography applications from isotropic media to anisotropic media, a subject which has much current interest. A companion paper immediately follows this paper on the applications of the theory.

Dines, K. A., and Lytle, R. J., 1979, Computerized geophysical tomography: Proc. IEEE, 67, 1065-1073. First application of ray tornography to subsurface imaging. Discusses both ART and SIRT.

Langan, R. T., Lerche, I., and Cutler, R. T., 1985, Tracing of rays through heterogeneous media: An accurate and efficient procedure: Geophysics, 50, 1456-1465. We do not elaborate on how to determine the raypath lengths through a gridded velocity model for the matrix S. This reference is more than adequate in addressing the problem as applied to direct arrivals in seismic ray tomography.

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Moser, T. J., 1991, Shortest path calculation of seismic rays: Geophysics, 56, 59-67. Utilizes network theory to get raypaths and traveltimes of first arrivals. Very robust for defining the matrix S, whether the first arrival is a direct arrival or a head wave.

Peterson, J., Paulsson, B., and McEvilly, T., 1985, Application of algebraic reconstruction techniques to crosshole seismic data: Geophysics, 50, 1566-1580. ART applied to crosswell seismic data.

Phillips, W. S., and Fehler, M. C., 1991, Traveltime tomography: A comparison of popular methods: Geophysics, 56, 1639-1649. Comparison of various linear inversion methods.

Vidale, J. E., 1988, Finite-difference calculation of traveltimes: Bull. Sets. Soc. Am., 78, 2062-2076. Approximates the eikonal equation using finite differences to compute traveltimes for first arrivals.

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Chapter 3

Seismic Diffraction

Tomography

3.1 Introduction

Seismic diffraction tomography is useful for reconstructing images of subsurface inhomogeneities which fall into two categories. The first category includes inhomogeneities that are smaller in size than the seismic wavelength and have a large velocity contrast with respect to the surrounding medium. Imaging these inhomogeneities with the seismic ray tomography methods presented in Chapter 2 is generally out of the question. The second category includes inhomogeneities that are much larger in size than the seismic wavelength and have a very small velocity contrast with the surrounding medium. Although seismic ray tomography is valid for imaging these inhomogeneities, it works best when the velocity contrasts are large. Note that both categories of inhomogeneity are capable of producing mea- surable scattered wavefields of similar power. The large velocity contrast of the first category inhomogeneity offsets its small size while the large size of the second category inhomogeneity makes up for its small velocity contrast.

The outline for this chapter closely parallels that of Chapter 2. First, in Section 3.2 we review acoustic wave scattering theory and derive two independent linear relationships between data functions representing scattered energy and the model function M(r). The model function M(r) used in this chapter is a measure of the velocity perturbation caused by an inhomogeneity at vector position r from a constant background velocity. Second, using either of the linear relationships between a data function and the model function M(r), the generalized projection slice theorem is derived

45

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46 CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPHY

in Section 3.3 which serves as the foundation for the image reconstruction algorithms used in seismic diffraction tomography. Finally, two seismic diffraction tomography image reconstruction algorithms are presented in Section 3.4' direct-transform diffraction tomography and backpropagation diffraction tomography.

Before proceeding further one should have a good understanding of the following mathematical concepts: frequency and wavenumber (Ap- pendix A); Fourier transform (Appendix B); Dirac delta function and Green's function (Appendix C). Also, since we present seismic diffraction tomography for inhomogeneities embedded in a constant velocity medium, the implementation of diffraction tomography is actually very similar to the transform methods for ray tomography. Thus, a review of Section 2.2 on ray tomography's transform methods might be of some benefit.

3.2 Acoustic Wave Scattering

The propagation of an acoustic wavefield P(r,t) through a medium consisting of a variable velocity C(r) and constant density is modeled by the acoustic wave equation,

1 O2P(r,t) _ 0 (47) V•P(r,t) C•(r) at • - , where r is a vector position within the model and t is time. The Laplacian operator X7 • is defined in terms of the vector operator •7 which in the Cartesian coordinate system is given by

v - 0 + where •, j, and • are mutually orthogonal unit vectors.

We use the Helmholtz form of the acoustic wave equation to describe acoustic wave scattering. The Helmholtz acoustic wave equation, found by taking the temporal Fourier transform of equation (47), is

VaP(r,w)+ k•(r,w)P(r,w) = 0. (48)

The variable k(r, w) is the magnitude of the wavenumber at position r and is defined by

k(r,w) = C(r)' (49) Note that equation (48) depends upon the value set for angular frequency w. Henceforth, for the sake of brevity, we will write both P(r,w) and k(r,w)

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3.2. ACOUSTIC WAVE SCATTERING 47

as P(r) and k(r) with the understanding that there is a dependence on angular frequency w.

Two nonlinear integral-equation solutions for the scalar Helmholtz wave equation defined by equation (48) are derived in this section. The Lippmann- Schwinger integral equation is one such solution which, because of its prominence in quantum mechanical scattering, is presented by itself in Section 3.2.1. The Lippmann-Schwinger equation nonlinearly relates the data function P0(r), called the scattered wavefield, to the model function M(r). The Born approximation in Section 3.2.2 linearizes the Lippmann- Schwinger equation. Another nonlinear integral-equation solution is formulated in Section 3.2.3 using exponentials. Although the definition of the model function M(r) remains the same, the data function is different from the scattered wavefield P0(r). The Rytov approximation is used to linearize this nonlinear integral-equation solution. Except for the data functions, both linearized integral-equation solutions are identical in form. Section 3.2.4 provides a comparison between the Born and Rytov linear integral solutions. Either solution enables us to derive the generalized projection slice theorem in Section 3.3 which serves as the foundation for the diffraction tomography image reconstruction algorithms in Section 3.4.

3.2.1 The Lippmann-Schwinger Equation

We begin the formulation of the acoustic wave scattering problem with Figure 15. The acoustic wave velocity is represented by C(r), where r is the vector position of a point within the model. The shaded region in Figure 15 depicts an inhomogeneity imbedded in an otherwise homogeneous medium. The acoustic velocity of the inhomogeneity varies spatially and can be thought of as a velocity perturbation from the constant background velocity Co of the homogeneous medium.

An incident wavefield Pi(r) is initiated by an acoustic source and propagates outward in the homogeneous medium. No scattering of the incident wavefield takes place until the inhomogeneity is reached. At that point any velocity contrast as a result of the inhomogeneity causes the creation of a second wavefield called the scattered wavefield P•(r). Each point in the inhomogeneity may be considered as a secondary source of seismic acoustic energy. Note that once acoustic energy is scattered from one inhomogeneity, then that scattered energy may be scattered again from another inhomogeneity which leads to higher order sources of seismic acoustic energy. As you will see, we ignore multiple scatterings in Section 3.2.2 to linearize the Lippmann-Schwinger equation and assume that the scattered wavefield P•(r) arises only from scattering the incident wavefield P•(r) from the source as depicted in Figure 15. Therefore, for layered media we as-

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48 CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPIIY

Receiver Source

C(r) = C o

FIG. 15. Acoustic wave scattering problem. The incident wave Pi(r) propagates from the source at the constant background velocity Co. The velocity inhomogeneity, depicted by the shaded area, acts as a secondary source and scatters the incident wavefield. The scattered wavefield Pj(r) travels away from the inhomogeneous region at the background velocity Co where it is recorded by the receiver.

sume that multiple reflections are negligible when compared to primary reflections.

The wavefield recorded by a receiver in the model consists of both the incident wavefield Pi(r) and the scattered wavefield P,(r) which we call the total wavefield Pt(r) , or

P•(r) - Pi(r) -[- P,(r). (50)

For a constant density model, equation (48) describes the propagation of the total wavefield Pt(r) through an inhomogeneous medium, or

[V '• + k•(r)]Pt(r) = O. (51)

At this point we reformulate k2(r) in equation (51) as a perturbation to a constant ko • for the homogeneous background medium where the magnitude of ko is given by

=

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We begin by writing

- +

[ k•(r) 1]. (53)

Substituting equations (49) and (52) into the bracketed term on the right- hand side of equation (53) for k(r) and ko, respectively, gives

ke(r) [ C• 1] - ko •+k• C•.(r)

[ ½;] = ko 2-ko 2 1 C2(r ) . (54)

If we define the model function M(r) as the bracketed term in equation (54), then we get the desired reformulation of k2(r),

k2(r) - k• - ko•M(r), (55)

where M(r) is defined by

M(r)- 1 C• C•(r ) . (56) The model function M(r) in equation (55) defines a perturbation to an otherwise constant ko • of the background medium. When C(r) - Go in equation (56), M(r) - 0 and there is no perturbation of ko • (i.e., by equation (55), k•(r)-

We now want to establish a relationship between the scattered wavefield P,(r) and the model function M(r) using equation (51). First, equations (50) and (55) are substituted into equation (51) for Pt(r) and k2(r), respectively, giving

[•7 • + k•- k•M(r)][Pi(r)+ Po(r)] - O. (57)

We rearrange equation (57) so that terms involving sources of scattered energy t are on the right-hand side. Thus,

[V 2 + ko2]Pi(r)+ [V 2 + ko•]Po(r) - ko2M(r)[Pi(r)+ Po(r)]. (58)

Note that the term ko•M(r)[Pi(r) + P,(r)] is the source of the scattered wavefield P0(r) and, as previously stated, depends upon both the incident

1 Terms containing the model function M(r) are sources of scattered energy.

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5O CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPItY

wavefield and scattered wavefield at r. The left-hand side of equation (58) describes the propagation of the incident wavefield Pi(r) and the scattered wavefield P,(r) in the background medium, both of which travel at a constant velocity Co.

The incident wavefield Pi(r) is generated by a source in the homogeneous background and contains no scattered energy. Therefore, the incident wavefield travels through the model at the background acoustic velocity Co and contributes to the scattered wavefield through the scatter source term in equation (58) when M(r) •- 0. Thus, the Helmholtz acoustic wave equation for a constant velocity medium describes the propagation of the incident wavefield Pi(r) and is given by

[V '2+ko•]Pi(r) - O. (59)

Equation (59) permits us to reduce equation (58) to

[v + - (60)

Equation (60) describes the propagation of the scattered wavefield at the background velocity when inhomogeneities occur which scatter both the incident wavefield Pi(r) and existing scattered energy Ps(r). Note that if no inhomogeneities occur, then M(r) = 0 and the right-hand side of equation (60) is zero, implying there is no source for the scattered wavefield and P•(r) = 0.

Solving equation (60) directly for P,(r) is difficult. A simple approach is to formulate an integral solution using the properties of the Green's function developed in Appendix C. For the Helmholtz equation the Green's function is the response of the differential equation to a negative impulse source function. 2 Thus, the Green's function becomes the solution to equation (60) if we replace the source term with a negative impulse source function -5(r- r t) • or

[V 2 + ko•]a(r [r') - -5(r - r'). (61)

The Green's function G(r Jr') gives the solution at position r for a negative impulse at r' which corresponds to the location of a point scatterer.

The solution to equation (61) for a 2-D space containing an infinite-line scatterer at r' and infinite-line receiver at r, where both lines are perpendicular to the plane containing r • and r, is

a(r I r') -- j Ho(X)(ko I r- r' l) (6:2) 4 '

• Traditionally the Green's function solution for the Helmholtz equation is determined for a negative source density on the right-haxtd side, or X7 2 P + k s P = -p. In order for the Dirac delta function to represent a source density impulse, the sign in front of the Dirac delta function must also be negative.

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where Ho (•) is the zero-order Hankel function of the first kind. In a 3-D space containing point scatterers and field points, the solution to equation (61) is

G(rlr') = 4•rl r-r' I ' (63) With the Green's function for equation (61) known, the solution to

equation (60) is found by multiplying the Green's function by the negative of the source term in equation (60) and integrating over all space where M(r •) •- 0, or 3

P,(r) - -ko • / G(r [ r')M(r')[P•(r') + P•(r')]dr'. (64) Equation (64) is called the Lippmann-Schwinger equation, which is the desired integral solution for the acoustic wave-scattering problem. We point out in Appendix C that such an integral is analogous to obtaining an output from a filter system when the impulse response of the filter (i.e., the Green's function) and the input (i.e., source term from equation (60)) are known. Equation (64) is just a convolution integral, which is easily seen if the Green's function from either equation (62) or (63) are substituted in for

I,?). The Lippmann-Schwinger equation nonlinearly relates the model func-

tion M(r) to the data function (scattered wavefield) P,(r). The nonlinearity is a result of the scattered wavefield P,(r) inside the integrand of equation (64) whose value depends on the model function M(r). Because of this nonlinearity, it is difficult to use equation (64) to perform either forward modeling (compute P,(r) from M(r)) without resorting to computationally extensive approaches such as finite difference methods or to derive diffraction tomography image reconstruction algorithms (compute M(r) from P,(r)). One way to get around this problem is to linearize equation (64) by making a simplifying approximation called the Born approximation.

3.2.2 The Born Approximation

The Born approximation linearizes equation (64) by assuming that the scattered wavefield P,(r) is much weaker than the incident wavefield Pi(r),

3In this book we use special integration notation which should not be confused with a line integral. Equation (64) could be either an integration over a plane (2-D space) or a volme (3-D space) depending upon which Green's function is used. Thus, to remain general, if we are integrating over a plane, then f dr' • f dx'dz'; and if we are integrating over a voltune, then f dr' ::• f dx'dy'dz'.

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or

P,(r) • P•(r). (65)

If the condition given by equation (65) is true, then the Born approximation states that

Pi(r) + P,(r) • Pi(r). (66)

The Lippmann-Schwinger equation is linearized by substituting equation (66) into equation (64),

P,(r) • -ko 2 / G(r [ r•)M(r•)Pi(r•)dr •. (67) Note that the integrand no longer contains the scattered wavefield P•(r) and the data function P•(r) and mode] function M(r) are now linearly related. If the primary acoustic source is a negative impulse located at vector position r,, then, by the definition of a Green's function, we can directly represent the incident wavefield P•(r') in equation (67) by a Green's function,

Pi(r') : G(r' [ rs). (68)

If a pressure-sensitive receiver (e.g., hydrophone) is located at position r = rp, then by substituting equation (68) into equation (67) for P•(r') we find,

P,(r,,rp) • --k• 2 / M(r')G(r' l r,)G(r p I r')dr', (69) where P,(r,, rp) is the scattered wavefield observed at position rp when the negative impulse source is located at position r•. Both Green's functions are defined by either equation (62) or (63).

Equation (69) is the Lippmann-Schwinger equation linearized by the Born approximation. This equation establishes the linear relationship between the data function (scattered wavefield) h(r,,rp) and the unknown model function M(r) required by the diffraction tomography problem. Note that since we exploited the Born approximation in deriving equation (69) that the model function M(r) must be a weak scatterer. Only then will the scattered wavefield be much weaker than the incident wavefield and the

condition specified by equation (65) be satisfied.

3.2.3 The Rytov Approximation

In this section we derive a nonlinear integral-equation solution to equation (48) using exponentials. Although the model function M(r) is defined

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the same as for the Lippmann-Schwinger solution, the data function representing the observed scattered wavefield is different. The Rytov approximation establishes a linear relationship between the data function and the model function. The resulting linearized integral solution strongly resembles equation (69) for the Born approximation.

We begin the derivation by returning to equation (51) which describes the propagation of the total wavefield P•(r) through a constant density, variable-velocity medium. Using equation (55) and equation (56) as the definitions for k2(r) and the model function M(r), respectively, equation (51) is rewritten as

Iv • + • - •4(•)]P,(•) - 0. (70)

Except for setting Pt(r) = Pi(r)+ P,(r), equation (70) is the same as equation (57). We deviate from the earlier derivation of the Lippmann- Schwinger equation by representing the total wavefield with the exponential equation,

P•(r)- e•,(r), (71) where •b•(r) is called the "complex total phase function." Note that, as in previous sections, variables which are a function of the position vector r carry an implied frequency dependence.

We wish to substitute equation (71) into equation (70) to obtain a differential equation in terms of •,(r). The Laplacian of P,(r) must be taken to achieve the this result. We start with

v•P,(•) = v. [vP,(•)].

Substituting equation (71) in for Pt(r) and performing the differential operations gives,

v•P,(•) - v. b•,(•)v•,(•)], = e•'(•)v-v•,(•) + Ve•'(•). v•,(•), = •,(•)v•,(•) + •,(•)v•,(•). v•,(•), = e•'(r)[v•,(r)+ V•,(r). V•,(r)]. (72)

Equations (71) and (72) are substituted into equation (70) for V2P,(r) and P,(r) giving,

•,(•)[v•,(•) + v•,(•). v•,(•)] + [• - •(•)1•,(•) - 0. (73)

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Dividing through by eqb'(r), equation (73) becomes, V•6 (r) + [V6t(r) V6t(r)] + ko • ko•M(r) - 0, (74)

which is the desired differential equation in terms of •bt(r). At this point we introduce the incident wavefield Pi(r) expressed

Pi(r)- e0i(r), (75) where $•(r) is called the "complex incident ph•e function." The complex incident ph•e function •i(r) is related to the complex total ph•e function •t(r) through the "complex ph•e difference function," defined by

0a(r) = 0t(r)- 0,(r). (76)

Since the total wavefield P•(r) varies from the incident wavefield P•(r) only when scattering occurs, we can surmise that the complex ph•e difference function $a(r) is a means of accounting for scattered energy.

We continue the derivation by replacing 0,(r)in equation (74) by •a(r) • defined by equation (76). Carrying this operation out yields a differential equation in terms of 0,(r) and

V•0i(r) + V•0•(r) + [V0i(r). V0i(r)] + 2[V0i(r). +[V$•(r). V&a(r)] + k• - k•M(r) - 0. (77)

The terms in equation (77) are rearranged in a form which will prove convenient later on,

= + (78) The terms inside the square brackets on the left-hand side of equa-

tion (78) are all related to the incident wavefield and have a sum equal to zero. This is e•ily shown true by using equation (59) which describes the propagation of the incident wavefield Pi(r) through the background medium. Equation (75) defining Pi(r)is substituted into equation (59)to get the differential equation in terms of •i(r). Following the same procedure which gave the Laplacian of Pt(r) in equation (72), the Laplacian of

V•P•(r) - e0•(r)[v•0•(r) + V0•(r). V0,(r)]. (79) Substituting equations (79) and (75)into equation (59) for V•P•(r) and P•(r), respectively, gives

e•i(r)[v2•i(r) + V•i(r)-V•i(r)] + k•e•i(r) - O. (80)

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Dividing equation (80) by e•i(r) results in an equation which demonstrates that the bracketed terms in equation (78) have a zero sum, or

V2•i(r) + V•i(r).V•i(r) + ko • - O. (81)

By equation (81), equation (78)is rewritten •

•v•,(•). v•(•) + v•(•) - -v•(•). v•(•) + ••(•). (8•)

Equation (82) provides a crucial relationship which we will use later on in this derivation. Now we turn our attention to the important product P•(r)•(r) which will become the data function for this derivation or a me•ure of the scattered wavefield. For purposes of the derivation the Laplacian is taken of this product. Performing the differentiation we have,

V•[P,(•)•,(•)] - V. [•,(•)VP,(•) + P,(•)V•,(•)] = •,(•)V•(•) + •V•(•). V•,(•)

+P,(r)V2•d(r). (83)

Rearranging the terms in equation (83) and making use of the fact that V•P•(r)- -k•P•(r) by equation (59), we get the following,

2VP,(r). V•(r)+ P,(r)V•(r) - V•[Pi(r)•(r)]- •(r)V•P,(r) = V2[pi(r)•d(r)] + •d(r)k 2 oP•(•) = [V • + k•]P,(r)•d(r). (84)

Switching the left- and right-hand sides of equation (84) and using the definition of P•(r) given by equation (75), we can write the following,

Iv • + •]P,(•)o,(•) - •v•,(•). v•,(•) + P,(•)v•o,(•)

= •(•)vo•(•). v•,(•) + p•(•)v•o,(•) = •P,(•)v•,(•). v•,(•) + P•(•)v•,(•) = p,(•)[•v•,(•). vo,(•) + V•d(•)]. (SS)

The quantity inside the square brackets on the right-hand side of equation (85) is defined by the earlier "crucial" relationship given by equation (82). Substituting equation (82) into equation (85) gives,

IV • + •]P,(•)o,(•) - -P,(•)[v•,(•). vo,(•)- •(•)]. (86)

Just • w• done for equation (60), equation (86) can be solved for P•(r)•d(r) in terms of an integral equation by exploiting the properties of the Green's function. The resulting solution is

P,(•)o,(•) - f P,W)[vo,(•'). v•,(•')- •(•')]a(• I •')d•', (87)

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where G(r Ir') is a Green's function given by either equation (62) or equation (63).

The data function P•(r)da(r) is nonlinearly related to the model function m(r) since da(r) appears inside the integrand of equation (87). Both forward modeling (computing P•(r)•a(r) from M(r)) and image reconstruction (computing m(r) from P•(r)&a(r)) are difficult because of this nonlinearity. At this point in the derivation the Rytov approximation is formulated to remedy this situation. The Rytov approximation linearizes equation (87) by assuming the condition V•a(r) <( 1. When V•a(r)is small the quantity X7•a(r'). V•a(r') in equation (87) can be neglected and we write the approximation,

- f (ss) As with the Born approximation, the incident wavefield Pi(r') can be

represented by a Green's function,

P•(r') = G(r'}r,), (89)

for a negative impulse source at vector position rs. If a receiver is located at position r - rr, then by substituting equation (89) into equation (88) we have,

P•(r•,rv)•Sa(r,,rv) • -ko • f m(r')G(r' l r,)O(r v I r')dr'. (90) The Green's functions are satisfied by either equation (62) or equation (63). However, to use equation (90) properly, the gradient of the phase difference function V•Sa(r), must be small as required by the Rytov approximation.

3.2.4 Born vs. Rytov Approximation

Except for the data functions, the linearization of the Lippmann- Schwinger equation using the Born approximation in equation (69) is identical to that of the Rytov approximation in equation (90). Here we demonstrate that the data function Pi(r,,rr)•Sa(r,,rr) associated with the Ry- tov approximation reduces to the data function associated with the Born approximation, the scattered wavefield P•(rs, rp), when the complex phase difference function d•(rs, rr) is small. Next, we show that forcing to be small is the same as requiring a weak scattered wavefield P,(r,,rr) for the Born approximation. Finally, we state the limitations for applying either the Born or Rytov approximation.

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First we wish to show that

Pi(r0,rr)•ba(r0,rr) • P0(r0,rr), (91)

when 4a(r0, r•) is small. This result is achieved by using the Taylor series expansion of e4a(r0, r•) which is given by,

•*•(•0,•) - 1 + ,•(•, •)+ *](•"•) + ' 2] 3] +.... (92) When •(r•, r•) • 1, we can neglect the second and higher order terms in the Taylor series expansion and write

•(•,,•) • •(•',•) - 1. (•) The approximation of the data function •sociated with the Rytov approximation is found by multiplying equation (93) by the incident wavefield Pi(r•, rp), giving,

P,(r,,rr)•(r, ,rr) • Pi(r•,rr)[e•(r•'rr) - 1]. (94) This approximation is e•ily shown to be the scattered wavefield P•(r•, rr) by using the relationships, Pi(r,, rr) - e•i(r•, rr), P•(r•, rr) - e•'( r•, rr) and P•(r,, rr) - P•(r•, rr)-Pi(r,, rr) , from equations (75), (71), and (50), respectively. Hence, equation (94) is rewritten:

•,(•. •),,(•. •) • •,(•. •)[•,(•. •) - 1]

• •,(•,, r•) _ •,(•,, r•) • •,(r•, •) - •,(r•, r•) • •(r•, r•). (•)

Thus, the data functions for equations (69) and (90) are approximately the same when •(r•, r r) is small.

Now we wish to show that setting •(r•,rr) • 1 is the same • the Born approximation requirement of a weak scattered field (i.e., P•(r,, rr) • P•(r•, rr) ). We begin by explicitly writing the total and incident wavefields

and

P•(r)- A,(r)eJ•b,(r), (96)

Pi(r) - Ai(r)eJ•bi(r), (97)

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respectively, where ;bt(r) and ;b,(r) are the real phases and At(r) and A,(r) are the amplitudes. Equation (76) gives the definition of the complex phase difference function as

4a(r) = 4,(r)- 4,(r). (98)

Solving equations (71) and (75) for dr(r) and di(r), then substituting the results into the last equation gives

da(r) = lnPt(r) - lnP•(r). (99)

Using the explicit relations for Pt(r) and Pi(r) given by equations (96) and (97), equation (99) becomes,

da(r) - In [At(r)1 q- j[•bt(r)- •bi(r)] (100) '

The complex phase difference function rid(r) in equation (100)is small when In ta,(r)/Z,(r)l << I and [;bt(r)- ;b•(r)] << 1. This is the comparable to requiring that the difference between Pt(r) •na g(r) be small for the Born approximation. The In ta,(r)/a,(•)l term in equation (100) demonstrates that the Born approximation is a weak scattering approximation. Also, the accumulative phase difference, [;bt(r)- ;b•(r)l , may become significant if the region where M(r) • 0 is large relative to the seismic wavelength. Thus, in addition to requiring weak scatterers, the size of the inhomogeneities may become an important factor to consider when using the Born approximation.

On the other hand, the Rytov approximation does not require rid(r) to be small, only that the gradient of rid(r) be small (i.e., the change of rid(r) within a wavelength be small). The Rytov approximation requires only a smooth model function and places no restrictions on the strength of the scatterers or their size. Thus, the Rytov approximation is a smooth scattering or smooth perturbation approximation.

3.3 Genera'ized Projection Slice Theorem

Either equation (69) or (90) in Section 3.2 define a linear integral relationship between a data function representing scattered energy and the model function M(r). In this section we simplify this linear integral relationship by taking the spatial Fourier transform of the data function over the source and receiver profiles and by taking the 2-D spatial Fourier transform of the model function. The result is the generalized projection slice

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3.3. GENERALIZED PROJECTION SLICE THEOREM 59

theorem which serves as the theoretical foundation for the seismic diffrac-

tion tomography image reconstruction algorithms presented in Section 3.4. We let P(r,, rr) for the remaining part of this chapter represent the data

function for either P, (r,, rr) of the Born approximation or Pi(rs, rr)•b,(rs, rr) of the Rytov approximation. Thus, the generic equation,

-}o (xox) defines the linear integral relationship between the data function and model function given by either equation (69) or (90).. Also, since most source and receiver geometries are for 2-D problems, we will exclusively use the Green's function defined by equation (62).

We find it instructive to derive the generalized projection slice theorem for three typical source-receiver configurations: the crosswell profile, the vertical seismic profile or VSP, and the surface reflection profile. One may justify this approach by noting that the spatial Fourier transform of the data function P(r•, rr) must be taken over the source and receiver profiles. However, by the end of this section the reader will find that the theorem is actually independent of the source-receiver configuration.

3.3.1 Crosswell Configuration

We begin the derivation of the generalized projection slice theorem for the crosswell configuration by replacing the position vectors in equation (101) with coordinates from the x-z plane. Figure 16 shows a crosswell configuration with source locations represented by solid circles and receiver locations identified by open circles. We assume both source and receiver wells are vertical so that the x-coordinates of all sources and receivers are

given by the constants, ds and dr, respectively. The z-coordinates of the source and receiver locations are z• and z r. Thus, the position vectors for the source location r• and receiver location r r' can be expressed in terms of their respective coordinates (d•, z•) and (dr, zr). Substituting these coordinates into equation (101) for the position vectors we get,

x O(z, z I ds, zs)O(d v, z v [ z, z)dxdz, (102)

for the crosswell configuration. The Green's function G(z,z I a,,•,) i, equation (102)is chosen to

represent a cylindrical wave emitted from an infinite line source located a• the coordinates (d,, z,). Equation (62) is the appropriate Green's function

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ds

z dp

..X

Source

Receiver

FIG. 16. Source-receiver geometry for the crosswell tomography experiment. The source locations are represented by solid circles at the constant horizontal location d,. The receivers are represented by open circles at the constant horizontal location d r . The orientation of the line sources and line receivers is perpendicular to the page. We restrict our reconstruction of the model function to d, < z < dr.

for this problem. We rewrite the position vectors in the Green's function in terms of x-z coordinates using (x, z) for r, (d,, %) for r', and I r - r ' I- Vf(x -d•) 2 + (z- %)2. Making these substitutions in equation (62) gives,

J G(x,z I d,,z,) - •Ho(•)(koV/(X - d,)• + (z - z,)• ). (103)

The Green's function in equation (103) is awkward to use because of the zero-order Hankel function Ho (•). The Green's function represents a cylindrical wave at the point (x, z) propagating away from a line source at (d,, z,) with vector wavenumber ko. Fortunately, it turns out that the zero- order Hankel function can be mathematically thought of as the summation (integration) at (x, z) of an infinite number of plane waves whose wavefronts are tangent to the cylindrical wavefront. This plane wave decomposition of the cylindrical wave is defined by

+ -

1 /_• 1 eJ[kl(z-z,)+?llx-d, I]dkl •r oo 7• (104)

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where the wavenumber for a given plane wave for the crosswell configuration has the components k• along the z-axis and 7• along the x-axis. 4 The absolute value of (x -ds) is required because 71 - +v/ko • -k•, which is always greater than zero. The absolute valueof x-ds insures 7•(x-do) > O, regardless of the value of x.

Figure 17 illustrates the concept behind equation (104). Plane wave •1 is perpendicular to the cylindrical wavefront at (x,z). This plane wave has the same vector wavenumber ko as the cylindrical wavefront at (x, z) and has the wavenumber components (7•,kl). As previously mentioned we require the • component for all plane waves be positive, defined by 7• - +v/ko 2 -k•. This definition of 7• ensures that all plane waves are tangent to the cylindrical wavefront with a wavenumber magnitude of ko. The direction of propagation for plane wave #1 is a = tan -• (7•/k•). Plane wave #2 is traveling in the +x direction with vector wavenumber ko•: and components (7• = ko, k• = 0). Note that the summation of plane wave #1 by equation (104) at (x,z) is at the onset of the plane wave while the summation of plane wave #2 at (x,z) occurs later on in its wave train. The waveform of any plane wave will be summed by equation (104) at a later point in its wave train if the plane wave is tangent to the cylindrical wavefront at a point other than (x, z).

Now we apply plane wave decomposition to the Green's function in equation (103). This is achieved by substituting the plane wave decomposition of the zero-order Hankel function defined by equation (104) into equation (103) giving,

+ -

where the absolute value can be dropped by requiring z > ds. Note that this Green's function is adequate for the crosswell configuration shown in Figure 16 since we are using equation (102) to image only between the source line and receiver line or d0 < z < dv. We are effectively saying M(z, z) = 0 outside of the imaging area.

The Green's function G(dv,z r [ z,z) in equation (102) is chosen to represent a cylindrical wave recorded at (dv,zv) scattered by an infinite line scatterer located at (z,z). As with the line source, equation (62) is the appropriate Green's function for the line scatterer. After substituting

4 In derivations of the generalized projection slice theorem for different source-receiver configurations, the kl component is always taken along the line of sources and -yl is perpendic,,lar to k• in a right-handed coordinate system sense. Here the source line just happens to be along the z-axis. The reason for doing this will become clear in a few pages.

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#2

#1

z

FIG. 17. Plane wave decomposition of a cylindrical wavefront at (x,z). Two plane waves, labeled #1 and •2 out of an infinite number of plane waves tangent to the cylindrical wavefront, illustrate the concept behind equation (104).

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coordinates for position vectors in equation (62) and performing a plane wave decomposition on the Green's function we get,

G(dr, z• [ x, z ) - 4•----•• fj l eJ[k•(zr - z) -l- ?•(dr - x)]dk• (106) for x ( dr. Here a given plane wave has the wavenumber components k• along the z-axis (parallel to the receiver line) and 7• along the x-axis (perpendicular to the receiver line), analogous to the k x and 7• reference directions with respect to the source line. The direction of propagation for the scattered plane wave is tan -• (72/k2). As with equation (105), this Green's function for the scattered wavefield is adequate since the imaging area has the limits d0 < x < dp.

One final relationship is required before we continue the derivation. We must write down the equation for taking the Fourier transform of the data function P(d,,zo; dr, zr) in equation (102) along the source line (z,) and the receiver line (z r). This relationship is found by first taking the Fourier transform of P(d,, z,; dr, z r) with respect to z, followed by a second Fourier transform with respect to z r which produces the double integral Fourier transform,

x e-jk•z• e-Jkrzr dz•dzr ' (107) where k• and k r are the wavenumbers of the Fourier transforms along the source line and the receiver line, respectively. Note that the directions taken for kl and k2 in the two Grecn's functions are oriented the same as ks and kr, a "lucky" coincidence as far as the derivation is concerned.

Now we may resume the derivation of the generalized projection slice theorem for the crosswell configuration by obtaining an integral relationship between the data function Po(d•,k•; dr, kr) and the model function M(x, z). We begin by substituting the linear integral relationship between the data function P(d•, z•; dr, zr) and the model function M(x, z), defined by equation (102) into equation (107). This substitution gives the relationship,

2 M(x,z) -k o

x G(x, z [ d,, zo)½-Jk, z, dzo

x [ (os)

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64 CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPttY

between the data function ]5(d,, k,; dr, kr) and the model function M(x, z). Note that we have changed the order of integration in equation (108).

Next, we substitute the Green's functions from equations (105) and (106) into equation (108), regroup terms with similar integration variables, and interchange the order of integration for the two innermost integrals over each Green's function, giving the equation,

•5(d, ko' d•,, Iq,) = k• 1 • M(x z} (109) ' ' 4 '

{•• 1 eJ[-•2z+'2(dp-z)]• • -J(•p-'2)Zpdzpd,2}dxdz. The integrals in equation (109) with respect to the variables z, and

zr are easily evaluated in terms of Dirac delta functions as shown in Ap- pendix B. The following are the integral solutions: 5

:'ø e-j(k• + k,)z, dz, (110)

and

•ø e-j(k •, - ka)zp dzp 2•rS(kr - k,). (111)

Substituting the integral solutions given by equations (110) and (111) into equation (109) and evaluating the integrals with respect to the variables k, and k= (remembering that 7• - ko • - k• and 7• - ko • - k•) gives,

; - -- M(z,z) ' 4 %% •

where

(112)

'rr - Vko = -/%= (113) and

7, - 7ko • - k, •. (114) 5 We chose kl and ks in the same direction as ks and kp, respectively, so that simpli-

fying relationships llke equations (110) and (111) could be used.

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Both 7p and % are wavenumber components perpendicular to the wavenumber components kp and k0 which parallel the receiver and source lines, respectively 6 .

The double integral in equation (112_) is just a 2-D Fourier transform of the model function M(x, z) defining M(k:,, k,) along circular arcs in the k• - k, plane, the placement of the arc depending upon k0 and k• ?. At this point we must make a brief digression to further explain this concept. Figure 18(a) shows four down-hole source positions labeled $•, $2, and $4 in one well and four down-hole receiver positions labeled R•, Rs, and R4 in another well. A line diffractor lies at the location (x, z). If ] represents the direction of a plane wave propagating from a source to the point (x, z), then by equation (105) the associated wavenumber components (7•, k•) can be written,

koi - 7•:• + k•., (115)

where • and •. are unit vectors in the positive x-direction and z-direction, respectively. Now, as a mathematical abstraction, let • point in the opposite direction of a plane wave traveling away from the source to the point (x, z), or • - -{. By equation (115) and from the fact that kx - -ks and 7• - 7: by equation (110), we may write the components of koõ as

(116)

Figure 18(a) shows the unit vector g for each of the four sources labeled with subscripts as gx, •2, •a, and g4.

Now let •) represent the direction of a plane wave propagating from the point (x, z) toward one of the receivers in Figure 18(a). By equation (106) and by the fact that equation (111) defines k•. = k r and 72 = 7r, we may write the components of ko•) as

(117)

Figure 18(a) shows the unit vector • for each of the four receivers labeled with subscripts as •)x, •2, •)a, and •4.

Figure 18(b) shows the wavenumber domain representation of the source wavenumber vectors ko•x, ko.q2, ko.qa, and ko.•,•. Similarly, Figure 18(c) shows the wavenumber domain representation of the plane waves traveling from the point (x,z) to the four receivers as the vectors ko•l, ko•.,

6Refer back to equation (107) for the definitions of k r axtd k•. 7 Equation (112) is analogous to equation (12) defining the projection slice theorem

for ray tomography. Instead of arcs, the projection slice theorem defines straight lines in the k: - k: plane as shown in Figure 5.

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66 CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPtIY

•x

s1 R1

ß p Zs2 ̂ • •^ R2 ^ 2

S4 R4

(a)

Kz K z

(b) (c)

FIG. 18. The wavenumber components (k,,%) and (kr,7r) in equation (112) depend upon the source and receiver locations, respectively, relative to the line diffractor in the model. (a) Four sources (solid dots) and four receivers (open dots) are shown. The unit vectors radiate from a line diffractor and point toward their respective source or receiver. The unit vectors for the sources point opposite to the unit vectors of the incident waves. (b) The wavenumber vectors for each source are shown on the k• - kz plane as terminating along a dashed circle with radius ko. If sources are deployed from +oo to -oo in the borehole direction, then the entire dashed circle terminus is defined. Equation (116) defines the components (k•, %) for these wavenumber vectors. (c) The wavenumber vectors for each receiver are shown on the k•- kz plane as terminating along a dashed circle with radius ko. As for the source, the entire dashed circle terminus is defined if receivers are placed from +c• to -c•.

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ß ,ks k P

•K x p,i •- Kx ß

ß

o

Kz

(a) (b)

F•c. 19. Wavenumber vector components for the crosswell configuration. (a) The components of a source wavenumber vector kog are shown as projections onto the k•-axis and kz-axis as defined by equation (116). (b) Sim- ilarly, the components of a receiver wavenumber vector kot5 are shown as defined by equation (117).

kof)3, and koO4. If the sources can be deployed from +c• to -c• along the source well, then the wavenumber domain representation of the source wavenumber vectors ko.• are vectors pointing to the dashed semicircle in Figure 18(b). Likewise, if the receivers can be deployed from +c• to -c• in the receiver well, then the wavenumber domain representation of the plane waves traveling to these receivers are vectors pointing to the dashed semicircle in Figure 18(c).

The components of ko•, and ko• defined by equations (116) and (117) are shown graphically in Figures 19(a) and 19(b), respectively. At this point we add the two vectors kof, and koO which gives the vector equation,

- = (7r -7s)• + (k r + k•)•.. (118)

The dot product between ko(g + •) and position vector r pointing to the line diffractor at (x, z) in the model is

ko( + . ,: - + . + = k•z +

= (7r - 70) x 4-(k r 4- ko)z. (119)

At this point we note that equation (119) multiplied by -j forms the terms

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for the exponentials in the integrand of equation (112). Substituting equation (119) into equation (112) results in the more interpretable equation,

tS(d•, k•; dp, kp)

(120)

where/•/(k•, kz) is the 2-D Fourier transform of the model function M(z, z), or

- +

= + (121)

Using the last result, equation (112) may be written more briefly as

P(do ko dp, kp) - kø2 eJ(7pdp - %d,) ~ ; - -- M[ko(f, + f>)]. (122) ' 4 %7p

The integration limits in either this equation or in equation (120) must obey the spatial limitations set by equations (105) and (106). That is, the imaging area is restricted to ds < x < dp. We do this by setting M(z, z) to zero in the integrand when the integration is outside of the imaging area, which is the same as restricting the integration to only the imaging area.

Equation (122) establishes a relationship between /5(ds, ks;dp, kr), the double integral Fourier transform of the data function along the source and receiver profiles, and g•[ko(• + •)], the 2-D Fourier transform of the model function for the diffraction tomography problem. Consider the crosswell configuration in Figure 20(a). The unit vector • points toward the single source from the line scatterer located at (x, z) while receivers deployed from +cx> to -cx> along the receiver well cause the unit vector • to have the range within the dashed semicircle. The locus of the quantity ko(• + •) in the wavenumber domain is a semicircle with radius ko centered on the point (-ko, 0), as shown in Figure 20(b).

Note that the coverage in the wavenumber domain changes as the source is moved. For example, Figure 20(c) shows the unit vector • for the source at -cx>. Since the receivers are at the same locations as in Figure 20(a) the coverage of • remains the same. The locus of the quantity ko(.• q- •) in the wavenumber domain is shown in Figure 20(d) as a semicircle with radius

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,.

source - -

(a)

ß ource at

- O0

', ,'

(c)

•o• .ource at

+oo

{e)

K z Kz

(b) (d)

gx gx

Kz

FI6. 20. Coverage provided the model function M(k=, k,) by three different source locations relative to the receiver borehole with receivers ranging in depth from +cx> to -cx>. The wavenumber vector ko• is shown in (a), (c), and (e) for sources at the same depth as the line scatterer, at -cxv, and at +oo, respectively. The termini for the receiver wavenumber vectors are shown as the dashed circles. For the fixed receiver locations, the solid circle arcs in (b), (d), and (f) are the coverages provided of the model function 37/(k=, kz) by the source locations in (a), (c), and (d), respectively. The solid arcs are a result of the vector sum, koi + kolS, as defined in equation (122) on the right-hand side.

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70 CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPItY

ko centered on the point (0,-ko). Figure 20(e) shows the unit vector • for the source at -]-•x) and the same receiver coverage. Now the locus of the quantity ko(• + •) in the wavenumber domain is the semicircle centered on (0, +ko) shown in Figure 20(f).

We can use equation (122) in the diffraction tomography problem to compute the Fourier transform of the unknown model function M on the semicircular loci defined by ko(• + •)in Figures 20(_b), 20(d), and 20(f). The loci defined by ko(O + •) in the evaluation of M are called the slice of the 2-D Fourier transform of the model function. The data function

P(d,, z,; dr, zr) representing the observed scattered wavefield along the receiver line is also called the projection. Equation (122) links the slices of M with the projections, resulting in its name, the generalized projection slice theorem. The theorem states that 15,(d,, k,; dr, kr) , the double integral Fourier transform of the data function along the source and receiver lines, is equal to M[ko(g + •))], the 2-D Fourier transform of the model function evaluated along the semicircular slice, multiplied by the quantity k• eJ(7pdr -

We may extend the situation shown in Figure 20 by deploying many sources from +½x• to -½x• in the source well as shown in Figure 21(a). Using more sources results in more slices in the wavenumber domain as illustrated

in Figure 21(b). The range of all possible koõ is depicted by the dashed line. We use equation (122) to compute the model function AYl[ko(õ + •)] along the solid slices in the k, - kz plane. For such a multisource- multireceiver configuration, the 2-D Fourier transform of the model function can be recovered only within the portions of the two disk-shaped regions of the k• - kz plane covered by the solid semicircles.

Most multisource-multireceiver configurations provide only a partial coverage of M(ka,,kz) leaving parts of the model uncertain. Informa- tion from slices on the entire k•- k• plane is required to uniquely determine M(x, z), which is computed from the inverse Fourier transform of M(k•,, k•). Most of the information provided by the crosswell configuration in Figure 21 is in the direction of the vertical wavenumber kz. Only longer wavelength information (smaller wavenumber values) about the model is provided in the horizontal wavenumber direction k•. Thus, with the crosswell configuration we expect good vertical resolution and poor horizontal resolution. Also, the resulting model is nonunique because we may arbitrar- ily define a model spectrum to "fill-in" the undefined parts of/l•/(k•, k,).

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•x

ß 8ource

0 receiver

()

()

()

()

)

locus of ko 8

•o•.. o• • o (• + •)

(a) (b)

Fla. 21. (a) The situation in Figure 20 is expanded by placing sources in depth from +oo to -oo. (b) The dashed circle is the terminus of the vector ko• for the sources in (a). The solid circles are the resulting vector sum ko(• q- I5). Now the model function/f4(k=, k,) is well defined within the two circular regions of solid line coverage using the generalized projection slice theorem for the crosswell configuration defined by equation (122).

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z

ds

dp

. source

o receiver

FIG. 22. Source-receiver geometry for the VSP experiment. our reconstruction of the model function to x < d r and z > d,.

We restrict

3.3.2 Vertical Seismic Profile Configuration

Now we derive the generalized projection slice theorem for the vertical seismic profile (VSP) source-receiver configuration in Figure 22. Sources (solid dots) are deployed along a line parallel to the x-axis at a constant vertical location z, = d,. Receivers (open dots) are deployed inside a receiver well parallel to the z-axis at a constant horizontal location x r = dr. Thus, the position vectors for the source location r• and receiver location r r can be expressed in terms of their respective coordinates (x•,d•) and (dr, zr). We shall assume that the domain of the model function M(x, z) is restricted to x < d r and z > d•. The derivation for the VSP configuration follows closely the derivation of the crosswell configuration of the last section. In fact, we end up with the same form of the generalized projection slice theorem given by equation (122), but with its application based on the VSP geometry.

Substituting the VSP coordinates into equation (101) for the position

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vectors rs and r r gives

P(zs,ds;dr, zr) - -ko • M(z,z) (123)

Again we choose equation (62) to represent the Green's functions in a 2- D medium so that the acoustic sources are infinite line sources, the diffractors in the model are infinite line diffractors, and the receivers are infinite line receivers. Thus, we are once again dealing with cylindrical acoustic waves in a 2-D medium.

The development of the Green's functions here closely follows the discussion of the previous section and will not be presented in detail. The Green's function G(z,z I z,,d,) for the infinite line source at (z,,d,)is found by substituting the appropriate coordinates into equation (62) for position vectors and performing a plane wave decomposition on the zer• order Hankel function H• •) of the first kind. The resulting integral form of the Green's function is

J ff •eJ[k•(x- x,) + 7•(z - d,)]dk•, (124) G(x, z l x•, d,) = 4• • 7• for z > ds. Note that, as before, the wavenumber component k• is taken along the source profile which for the VSP geometry is in the direction of the x-axis. The wavenumber component 7x is along the z-axis and is defined as 7• - v/ko • - k•. A similar development of the Green's function G(dr, z r I z, z) for an infinite line diffractor at (z, z) whose energy is recorded by a receiver located at (dr, zr) gives the equation,

j f: 1 eJ[k•.(z r _ z) + - z)lak2, (125)

for z < dr. Here the wavenumber component kg. is taken along the receiver profile which for the VSP geometry is in the direction of the positive z-axis. The wavenumber 79. is in the positive x-axis direction and is defined as

Finally, we must write down the equation for taking the Fourier transform of the data function P(xs, ds; d r, zs) in equation (123) along the source line (xs) and the receiver line (zr). This relationship is found by first taking the Fourier transform of P(xo, ds; d r, zs) with respect to zs followed by a second Fourier transform with respect to z r which produces the double integral Fourier transform,

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P ( •: , , d , ; ct,, , •:,, ) /_:,o /_• P(z,, d,; dp, zp) x e -jk'x' e-jkpzp dxsdzp, (126)

where k, and kp are the wavenumbers of the Fourier transform along the source line and the receiver line, respectively. Note that k• is in the same direction as k, and k= is in the same direction as

We continue the derivation of the generalized projection slice theorem for the VSP configuration by substituting equation (123) into equation (126) to get a relationship between the data function P(k,,d,; dr, and the model function M(x, z). Carrying-out this substitution gives the equation,

where we have changed the order of integration. Next we substitute the Green's functions defined by equations (124)

and (125) into equation (127), regroup terms with similar integration variables, and interchange the order of integration for the two innermost integrals over each Green's function, yielding the equation,

•5(k, d,; dp, kp) = k• 1 /_• /_ © ' 4 (2a')' oo M(a:, z) (128)

{/_ } • • d[-•z + •(d.- •)] •-i(• - •)Z•z.d• d•dz

The integrals in equation (128) with respect to the variables x• and z r are easily evaluated in terms of Dirac delta functions as

and

'ø e-J(k• + k,)Xsdx ' (129)

_=ø e-J(k p - k2)zp dz r 2•r5(kp- k:•). (130)

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Substituting equations (129) and (130)into equation (128) and evaluating the integrals with respect to the variables kl and k2 (remembering that • - •o • -•i •na • - •o • -•) give•,

ß = -- M(•, •) ' ' 4 7•7p • •

• •-J(•. +7•)•-J(•-%)•&, (•s•)

where 7, and 7p are defined by equations (113) and (114). Remember that the wavenumber components k, and kp parallel the source and receiver profiles, respectively.

As we saw in the previous section, the exponential terms in the integrand of equation (131) may be rewritten • the dot product of the vector sum,

•o(• + •) - (-•i - 7•) + (7•i + •) : (•.i- %•) + (•i + %•) = (•. +•)i+(%-%)• : k•+k•, (•3a)

with the position vector r pointing to (x,z). The unit vector • points in the opposite direction of a plane wave traveling from a source to (x, z) and the unit vector • points in the direction of a plane wave propagating kom (x, z) to a receiver. The components of ko• and ko• are shown graphically in Figures 23(a) and 23(b), respectively. Sources are deployed kom +• to -• along the source line parallel to the x-axis and receivers are deployed kom +• to -• along the receiver line parallel to the z-axis. This geometry results in vectors pointing to the d•hed semicircles in Figure 23.

Performing the dot product between equation (132) for the VSP configuration and the position vector r gives,

•o(a + •).• - (•i + •). (•a + •) = k• x + k• z

= (k, + 7p)x + (kp - 7,)z. (133)

Note that equation (133) multiplied by -j is just the term in the exponential of the integrand in equation (131). Substituting equation (133) into equation (131) results in the equation,

ß - • M(•, •)

x e-J(k•x + k•z)dxdz

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: :Kx ks

K z

kp

(a) (b)

FIG. 23. Wavenumber vector components for the VSP configuration as defined by equation (132). (a) The components of a source wavenumber vector kog are shown as projections onto the k•-axis and kz-axis; and likewise, (b) the components of a receiver wavenumber vector koI3.

(134)

where 37/(k•, kz) is the 2-D Fourier transform of the model function M(x, z) taken over the image area restricted to x < d r and z > d,. Note that equation (•a4) is the same as equation (122) for the crosswell configuration except for the directions over which the double integral Fourier transforms of the data function P(x,, z,; zr, zp) are taken. We require that the transforms be taken along the source and receiver profiles, which for the VSP configuration are different from the crosswell configuration.

Equation (134) is the generalized projection slice theorem for the VSP configuration. It states that P,(k,,d,;dp, kr), the double integral Fourier transform of the data function, taken along the source line and the receiver line, is equal to M[ko(,• + 15)], the 2-D Fourier transform of the model function, evaluated along the semicircular slice multiplied by the quantity

-Y %% ' Figure 24(a) shows sources and receivers deployed from +oo to -oo

along their respective profiles for the VSP configuration. Figure 24(b) is a representation of the model coverage provided by the VSP configuration

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z

K z

^

ß source locus of ko 8

o receiver locus of/• • ( • + I• )

(a) (b)

FIG. 24. (a) Sources and receivers are extended from +cx> to -cx> along their respective profiles. (b) The dashed circle is the terminus of the vector ko• for the sources in (a). The solid circles are the resulting vector sum ko(• +/3). The model function iff(k•,k,) is well defined within the zone of solid line coverage using the generalized projection slice theorem for the VSP configuration defined by equation (134).

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0 .X

ß source

o receiver

FIG. 25. Source-receiver geometry for the surface reflection experiment, where in the marine data acquisition case the depths are measured from sea level. We restrict our reconstruction of the model function for the

geometry shown here to z • dp.

where the model reconstruction defined by equation (134) is restricted to points x • dp and z • d,. The range of all possible ko• is depicted by the dashed line in Figure 24(b). We use equation (134) to compute the model function ]l•[ko(• + •)] along the solid slices in the k• - kz plane. Note that the VSP configuration provides a different coverage of ]17f(kx, kz) than the crosswell configuration in Figure 21. Thus, we should expect different estimated model functions M(x, z) from each configuration even though the true model is the same. As with the crosswell configuration, resolution varies with direction and the resulting model is nonunique.

3.3.3 Surface Reflection Configuration

The last source-receiver geometry we will derive the generalized projection slice theorem for is the surface reflection configuration in Figure 25. The sources (solid dots) are deployed along a line parallel to the x-axis at a constant vertical location z• - d•. Similarly, the receivers (open dots) are deployed along a line parallel to the x-axis at a constant vertical location zp = dp. Thus, the position vectors for the source location r• and receiver location rp can be written in terms of their respective coordinates (xs, ds) and (xp,dp). We shall restrict the domain of the model to depths below the deeper of the two, d• or dp. As with the previous two configurations, the derivation here generates a similar form for the generalized projection slice theorem, but with its application based on the surface reflection geometry.

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We begin by substituting the surface reflection coordinates for the position vectors r• and r r into equation (101) which gives

x z I 4, I z)aaz.

We assume a 2-D medium and choose equation (62) to represent the Green's functions for the infinite-line acoustic sources, infinite line diffractors, and the infinite line receivers. Thus, all acoustic waves will be cylindrical.

The Green's function G(x,z I x•, d•) for the source is the same as the VSP's Green's function defined in equation (124) since both configurations have the same source geometry. Thus,

J /5 1 eJ[k•(x- x )+ ?•(z- d•)ldk•, (136) for z > d,. The wavenumber component k• is taken along the source profile which is in the direction of the x-axis. The wavenumber component 7• is along the z-axis and is defined 7• - V/ko 2 - k• 2.

Since the receiver geometry is similar to the source geometry we use the same form of the Green's function as for the sources giving

3__'/5 1 eJ[k9.(x r _ x) -72(d r - Z)]dk 2 (137) where k2 is the wavenumber component parallel to the receiver line and 72 is along the z-axis, where 72 - V/ko • -k•. Note that -?2(d r - z) is used since we removed the absolute value in the plane wave decomposition of the Hankel function for z > dr.

Proceeding with the derivation, we write down the equation for taking the Fourier transform of the data function P(x•, d•; x r, dr) in equation (135) along the source line (x•) and receiver line (xr). The relationship is found by first taking the Fourier transform of P(x•, d•; xr, dr) with respect to xs, followed by a second Fourier transform with respect to x r which produces the double integral Fourier transform,

P(zo,d•;zr,

x e-Jz•koe-Jxrkrdx•dxr, (138)

where k, and k r are the wavenumbers of the Fourier transforms along the source line and the receiver line, respectively. As with previous configurations, k• is in the same direction as k, and k2 is in the same direction as k r ß

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8O CItAPTER 3. SEISMIC DIFFRACTION TOMOGRAPttY

Next, we substitute equation (135) into equation (138), giving

P(k,,d,;kp, dr) - -k• M(x,z) (139)

x •(•, • I •., a.) •-j•'•'

x 6(•:,,, 4, [ •:, z) •-j•'"

where we have changed the order of integration. Next, we substitute the Green's functions defined by equations (136)

and (137) into equation (139), regroup terms with similar integration variables, and interchange the order of integration for the two innermost integrals over each Green's function, yielding the equation,

15(k ' d,;kr, dr ) = k•o 1 • M(z z) (140) ' 4 (2•r)' '

{f_• 1 •'It" + '•'(' - •')1 j'_" -'(•' + t')' } x -- e sdzsdk•

x -- e -j(kr - k=)zPdzrdk= dzdz.

The integrals in equation (140), with respect to the variables x, and are easily evaluated in terms of Dirac delta functions as

øø e-j(k• + k,)Z,dx ' 27rS(k• + k,), (141)

and

j e-j(kp - k:•)%, 2•rS(k•, - k2). (142)

Substituting equations (141) and (142)into equation (140) and evaluating the integrals with respect to the variables k• and k= (remembering that 7• - ko • -k• and 7• - ko • -k•) gives,

ko • ej(-%dp - -/,d,) oo M(x, z) 4 7,% oo oo

(143)

where 7, and 7p are perpendicular to k, and kp.

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ß t•s ':-Kx t•p •Kx

•z •:z

(a) (b)

FIG. 26. Wavenumber vector components for the surface reflection configuration as defined by equation (144). The dashed semicircles are the termini of the wavenumber vectors for sources and receivers positioned from -oo to +cx• along the x-axis. (a) The components of a source wavenumber vector kog are shown as projections onto the kx-axis and kz-axis; and likewise, (b) the components of a receiver wavenumber vector koP.

As we saw for the earlier configurations, the exponential terms in the integrand of equation (143) may be rewritten as the dot product of the vector sum,

•o(• + •) - (-• - 7•.) + (•.• - 7•.•.) = (•0• - •0•.) + (• - •.) = (•o + •)• + (--r• --r•)• = kx• + kz•, (144)

with the position vector r pointing to (x,z). The unit vector .q points in the opposite direction of a plane wave traveling from a source to (x, z) and the unit vector f) points in the direction of a plane wave propagating from (x, z) to a receiver. The components of koõ and ko•) are shown graphically in Figures 26(a) and 26(b), respectively. Sources are deployed from +oo to -c• along the source line parallel to the x-axis and receivers are deployed from +c• to -oo along the receiver line also parallel to the x-axis. This geometry results in vectors pointing to the dashed semicircles in Figure 26.

Performing the dot product between equation (144) and the position vector r gives,

•o(• + 0).,, - (• + •,•)-(• + z•)

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= k•z + kzz

= (k, + kr)z + (-7, -7r) z. (145)

Note that equation (145) multiplied by -j is just the term in the exponential of the integrand in equation (143). Substituting equation (145) into equation (143) results in the equation,

_ ko • ej(-7rdr - %d,) oo M(x, z) 4 %%0 oo

x e-J(k•x + k•z)dxdz

- %d,) _ = _

4

• •j(-•dp - •) . = -- U[ko(• + •)],

4 (146)

where 217/(k•, k.) is the 2-D Fourier transform of the model function M(x, z) taken over the image area restricted to the greater of z > d, or z >dp.

Equation (146) is the generalized projection slice theorem for the surface reflection configuration. It states that ['•(k,, d•; kr, dp), the double integral Fourier transform of the data function, taken along the source line and the receiver line, is equal to 20[ko(S + iO)l, the 2-D Fourier transform of the model function, evaluated along the semicircular slice multiplied by the

quantity --• eJ(-7pdp - 7sds) %%0 ' Figure 27(a)shows sources and receivers deployed from

along their respective profiles for the surface reflection configuration. The range of all possible ko• is depicted by the dashed line in Figure 27(b). We use equation (14{5) to compute M[ko(g + •)] along the solid slices in the k: - kz plane. The surface reflection configuration provides a different coverage of M(kz, k,) than the crosswell and VSP configurations and therefore we should expect different estimated models from each configuration. Once again, each configuration provides a different resolution and degree of nonuniqueness depending upon how the configuration "fills-in" the model spectrum and how much of the model spectrum the configuration leaves undefined.

3.4 Acoustic Diffraction Tomography

The generalized projection slice theorem was derived in the previous section for the crosswell, VSP, and surface reflection configurations which

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3.4. ACOUSTIC DIFFRACTION TOMOGRAPttY 83

• x

source

receiver ........ locus of/• o s

locus of/• o (s + p)

(a) (b)

F•G. 27. (a) Sources and receivers at the same depth are extended from +c• to -c• in the direction of the x-axis. (b) The dashed circle is the terminus of the vector koõ for the sources in (a). The solid circles are the resulting vector sam ko(g+I5). The model function A7/(kx, k,) is well defined within the zone of solid line coverage on the -k• half plane.

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$4 CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPtIY

correspond to equations (122), (134), and (146), respectively. For each configuration the theorem establishes a relationship between the double Fourier transform of the data function s taken along the source and receiver profiles and the 2-D spatial Fourier transform of the model function. Further analysis of the generalized projection slice theorem shows that each source location defines the model function along a circular arc in the ks - k• domain, as depicted in Figures 21, 24, and 25 for each source-receiver configuration.

To obtain the model function M(x, z) we must take the inverse Fourier transform of.•/[ko(•+•)] which is not readily carried-out since AY/[ko(•+•)] is defined by circular arcs. Thus, in this section we develop two image reconstruction algorithms to handle this problem called the direct-transform diffraction tomography and the backpropagation diffraction tomography. The direct transform method involves estimating the values of •[ko(•, + P)] on the k•- k• coordinate plane, which is analogous to the direct-transform ray tomography method presented in Section 2.2.2. On the other hand, the backpropagation method takes a more elegant mathematical approach to achieve the same result. The backpropagation method presented here is analogous to the backprojection ray tomography presented in Section 2.2.3.

3.4.1 Direct-Transform Diffraction Tomography

The generalized projection slice theorem enables us to compute the 2-D Fourier transform of the model function M[ko(• + •)] from s•tte•ea wavefield data. Therefore, the unknown model function M(•, z) can be obtained as long as we can perform an inverse Fourier transform on AYl[ko(• + •)]. However, most inverse Fourier transform algorithms use Cartesian coordinates • which means they can handle the inverse Fourier transform from ?l•f(k•,k•) to M(z,z), but not from ?l•f[ko(• + •)j to M(z,z). Fig- ure 28 illustrates the difference between the loci of M[ko(• + •)l (solid dot•) •nd the loci of •(k,, k•) (open dot•) for a crosswell configuration. Clearly the loci of the two representations of AY/ do not coincide and we cannot directly make use of the inverse Fourier transform.

An obvious, brute force solution is to estimate values of M(k•, k•) on a rectangular kz - k• grid from values of AY/[(• + •)] obtained along circular arcs using the generalized projection slice theorem. As mentioned in Sec- tion 2.2.2, one must exercise caution in performing an interpolation since the operation may introduce error which can obscure the true solution. However, once the values of M are found on a k• - k• rectangular grid, we can find the unknown model function M(x, z) through a 2-D inverse

s Keep in mind that in diffraction tomography the data function consists of measured projections of scattered energy.

•Namely, (x,z) in the space domain and (k•, k,) in the wavenumber domain.

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3.4. ACOUSTIC DIFFRACTION TOMOGRAPHY 85

o o o o

o o

o o

K, x

o o

o o

o o o

0 Iocu• of (k,•, k•.) grid

FIG. 28. The generalized projection slice theorem defines the model function M(k•,, kz) at the solid dots along circular arcs in the ks- kz plane. To get the estimated model function M(x, z) the 2-D inverse Fourier transform must be taken of M(kr,k,) defined at points (open dots) on a Cartesian grid. Direct-transform diffraction tomography defines the open dots from the solid dots by interpolation.

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Fourier transform. This is called direct-transform diffraction tomography and can be summarized in five steps'

Step 1: Acquire tomography data by probing the target with acoustic waves and record the scattered wavefield informa-

tion from the target, represented by the data function P(x, z).

Step 2: Take the double Fourier transform of the data function P(x, z) along the source and receiver lines, which can be represented by

,of_,o ' = t, + ' (147)

Here is and i r are the distances along the source and receiver lines, respectively. Equation (147) corresponds to equation (107) for the crosswell configuration when is - zs and i r = zr; to equation (126)for the VSP configuration when I• -- x• and i r = zr; and to equation (138) for the surface reflection configuration when l• = x• and l r - x r.

Step 3: Compute the 2-D Fourier transform of the unknown model function A7l[ko(k 4- •)] along circular arcs using the generalized projection slice theorem,

ll•l [ k o ( • -t- •)] = 47o%, - j ( :l: 'y p d p 4- 'y , d , ) •5 ( k k p ) . k• ø e ,, (148)

Equation (148) corresponds to equation (122) for the crosswell configuration, equation (134) for the VSP configuration, and equation (146) for the surface reflection configuration.

Step 4: Perform a 2-D interpolation on the values of M from the generalized projection slice theorem to determine AT/on the rectangular k,- kz grid,

(149)

Step 5: Take the 2-D inverse Fourier transform of M(k•,kz) to obtain M (x, z),

M(z z) = I " 1171(k• k,)e j(k•z + k'Z)dk•dk, (150)

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3.4.2 Backpropagation Diffraction Tomography

We saw that direct-transform diffraction tomography estimated values of the model function •[ko(õ + •))] on a rectangular k, - k, grid enabling us to take th• inverse 2-D Fourier transform of • to get the estimated model function M(x, z). The interpolation step required is computationally slow and the possibility of introducing errors into the interpolated model function is very real. In this section we introduce the more mathematically elegant method of backpropagation diffraction tomography. With this method the required coordinate transformation from M[ko(• + •)] defined on semicircles to M(kx, k,) defined on a rectangular grid is performed while taking the 2-D inverse Fourier transform thereby obviating the need for interpolation.

The derivation begins with the 2-D inverse Fourier transform of the model function M(k•, k•) given by

1 • •(k• k,)eJ(k•z + k,Z)dk•dk, (151) M(x, z) = 4• • ß ß , . Since •[ko(• + •)] is defined in terms of the wavenumbers (k•, k•), we will perform a change of variables from (k•, k,) to (k•, kr) on the right-hand side of equation (151). This will permit us to directly take the 2-D inverse Fourier transform of the model function •[ko(• + •)] without having to perform an interpolation • required by the direct-transform diffraction tomography. Equation (151) in the form for this change of variables is

M(x z) = 1 • •[ko(• + •)]eJ[k•(k•, kr)z + k,(ks, kr)z ] ' 4•2 •

x I I (152)

where •[ko(i + •)] is the model function from the generalized projection slice theorem, k•(k,,kr) and k,(k,,kr) are the wavenumbers expressed in terms of the variables (k•, kr) , and J(kr, k, [ k•, kr) is the JacobJan relat- ing dk, dk r with dk•dk,. The JacobJan can be expressed in terms of the determinant,

J(k• k, ] k,,kr) - '

Ok:Ok, Ok•Ok:

= Ok: Ok r Ok r Ok,' (153) The form of equation (152) is • far • we can go without selecting a par-

ticular source-receiver configuration. In the following derivations we shall

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complete the change of variables for the three source-receiver configurations presented in the previous section. The resulting equation for each configuration is in a form which permits use of a 2-D inverse Fourier transform algorithm and the result is backpropagation diffraction tomography. Crosswell Configuration.--The model function for the crosswell configuration is defined by the generalized projection slice theorem in equation (122) which may be rewritten as

.l17f[ko(• + f))] = 4%7r -j(%,d r - %d•)P(d• k• d r kr) (154) ko • e , ; , . From equation (119) we see that the coordinate system transformation between (k•, kz) and (ks, kr)is given by

Substituting this coordinate system transformation into equation (153) gives the Jacobian for the crosswell configuration,

J(k•,kz I k,,kr) = k,%,+k•,%. (156)

Substituting equations (154), (155), and (156)into equation (152) gives,

M(z z) = 1 /• /_• 4%7r -j(7rdr_%d•) , 4•.2 •' •' •-o• e - + +

x [ ks 7r + krYs [ dks dkp,

and rearranging terms gives the equation for backpropagation diffraction tomography for the crosswell configuration,

(157)

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VSP Configuration.-- The model function for the vertical seismic profile configuration is defined by the generalized projection slice theorem in equation (134) which may be rewritten as

•[•o(• + •)] = 4,0,• _j(,• _,,•,)p(•, • • •) (•S8) ko • e , ; , . From equation (133) we see that the coordinate system transformation between (k•, k•) and (k•, kv)is given by

k•(k,, k•) = k, + 7•

= •+•-•,•na

= kv - •k• - k•. (159) Substituting this coordinate system transformation into equation (153) gives the Jacobian for the VSP configuration,

Substituting equations (158), (159) and (160)into equation (152) gives,

M(x, z) = 4• • ß ß • e

x [k•k v+%7v ]dk•dkv, and rearranging terms gives the equation for backpropagation diffraction tomography for the VSP configuration,

M(•, •) = • ß ß k• '

x eJ[(k, + 7v)x + (kv - 7,)z]ak, akv. (161) Surface Reflection Configuration.-- The model function for the surface fiection configuration is defined by the generalized projection slice theorem in equation (146) which may be rewritten •

•[ko(• + P)] = 47•7• e-J(-7•d• - 7•d•)p(k• d,' k•,dp). (162)

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9O CHAPTER 3. SEISMIC DIFFRACTION TOMOGRAPHY

From equation (144) we see that the coordinate system transformation between (k•, kz) and (ks, kp) is given by

-- ks + k r, and = -% - 7r

(163)

Substituting this coordinate system transformation into equation (153) gives the JacobJan for the surface reflection configuration,

s(•, • I k•, •) = kr•'• - k•'•r. (164)

Substituting equations (162), (163) and (164)into equation (152) gives,

and upon rearranging terms we get the equation for performing backpropagation diffraction tomography for a surface reflection configuration,

1 f_•/: I •'r,-/•,•'• Ip(/• •)

x eJ[(ks + kp)x + (-% - 7p)Z]dk,dk•, ' (165)

Equations (157), (161), and (165) are used in backpropagation diffraction tomography image reconstruction algorithms for the crosswell, VSP, and surface reflection configurations, respectively.

3.5 Summary

1. The model used for acoustic wave scattering was the Helmholtz form of the acoustic wave equation for a constant-density medium of variable velocity C(r) given by

IV • + •:=(•.)]•,(•.) - o,

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3.5. SUMMARY 91

where Pt represents the total wavefield which contains both the incident wavefield from the source and scattered wavefields from inho-

mogeneities embedded in a constant-velocity background medium.

2. We treat the inhomogeneities as secondary sources and write the acoustic wave equation as the inhomogeneous differential wave equation,

IV + -

where ko is the constant wavenumber magnitude of the background medium and M(r) is the model function for diffraction tomography defined as a perturbation from the constant background velocity Co given by,

. We found two integral equation solutions to our formulation of the acoustic wave scattering problem. The first was found by letting the total wavefield Pt(r) be the sum of the incident wavefield Pi(r) and scattered wavefield P,(r). We formulated a Green's function integral solution to the inhomogeneous differential wave equation in terms of the scattered wavefield P, resulting in the nonlinear Lippmann- Schwinger equation. We linearized the Lippmann-Schwinger equation by assuming the Born Approximation (i.e., P•(r) << Pi(r)) resulting in a linear relationship between the data function Ps (r) and the model function M (r),

I I r')dr',

where P.(r.,rp) becomes the data function observed at position rp when the negative impulse source is located at position r•. The Green's functions are defined by either equation (62) or (63).

4. The second integral equation solution was found by representing the total wavefield Pt(r) by the exponential equation eqSt(r), where the complex total phase function qbt(r) was set equal to the sum of the complex incident phase function q5i(r) and the complex phase difference function qba(r). After lengthy manipulations of these phases, formulating a Green's function solution to the resulting inhomogeneous wave equation which operated on the quantity Pi(r)c)a(r), and

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applying the Rytov approximation (i.e., Vdd(r) << 1), we found,

P•(r,,r•)•a(r,,r•) • -ko •/M(r')G(r' I r,)G(r• I r')dr'. Here the data function is P•(r,, r•)qba(r,, r•), instead of P,(r), which is linearly related to the model function M(r).

5. The Born approximation is valid for weak scatterers of limited size while the Rytov approximation requires only a smooth model function.

6. Letting P(r,,r•) represent the data function for either the Born or Rytov approximation integral solutions, we simplified the integral equation solutions to the generalized projection slice theorem for three sourcereceiver configurations. The generalized projection slice the• rem for the crosswell configuration is

P(d, k, dp, kp) k• eJ(%dp- 7, d,) _ ß = -- ' ' 4

This equation establishes a relationship between P(d•, k•; d•, kp), the double integral Fourier transform of the data function along the source and receiver profiles in the crosswell configuration, and M[ko(• + •)], the 2-D Fourier transform of the model function along curves in the k• - k, plane. The curves are defined by the sum of the vector ko• which points from the point (•, z) in the model to the source and the terminus of the vector ko• which points from (•, z) to each receiver.

7. The generalized projection slice theorem found for the VSP configuration is

ß = -- + ' ' 4

8. The generalized pro•ection slice theorem found for the surMce reflection configuration is

P(k, d,' kp, dp) = k• eJ(-7p½ - 7, d,) •[ko(g + •)1. • • 4 7s 7p

9. The coverage of •(kz, kz) in the kz - kz plane is dependent upon the sourcereceiver configuration and always incomplete to some degree. This results in a nonuniquely estimated model M(x, z) that h• in- herently poor resolution in directions •sociated with the incomplete coverage in the kz - k, plane.

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10. In Section 3.4 the generalized projection slice theorem could not be directly applied to estimating M(z,z). This occurs because the 2- D Fourier transform requires data points on a rectangular grid in the kx- k; plane and the generalized projection slice theorem defines the data points on semicircles. The direct-transform diffraction tomography just performs an interpolation to solve the grid differences, or

before taking the inverse 2-D Fourier transform to get the image M(x, z). However, the potential for interpolation error showing up in M(x, z) is very real.

11. The backpropagation diffraction tomography method accomplishes the same result as the direct-transform method by replacing the (kx, k,) variables in the 2-D inverse Fourier transform by (k0,kr), the variables used in the generalized projection slice theorem. The resulting coordinate transformation is dependent upon the source-receiver geometry, but we can find M(z, z) directly through a 2-D inverse Fourier transform without the need for interpolation.

3.6 Suggestions for Further Reading Devaney, A. J., 1984, Geophysical diffraction tomography: IEEE

trans., ClE-22, 3-13. First proposal of seismic diffraction tomography.

Esmersoy, C., Oristaglio, M. L., and Levy, B.C., 1985, Multi- dimensional Born velocity inversion' single wideband point source: J. Acoust. Soc. Am., 78, 1052-1057. Reformu- lares the Born inversion problem so that the data function becomes the field extrapolated through the wave equation. A variable background velocity is permitted.

Morse, P.M., and Feshbach, H., 1953, Methods of theoretical physics: McGraw-Hill. We just throw equation (10•), the plane-wave decomposition of a cylindrical wave, right at you without proof. Section 7.2 of this reference derives the equation in detail with the resulting equation found on page 8œ3.

•zbeck, A., and Levy, B.C., 1991, Simultaneous linearized inversion of velocity and density profiles for multidimen- sional acoustic media: J. Acoust. Soc. Am., 89, 1737-1748.

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This paper includes the formulation for simultaneously reconstructing both the velocity and density using the Born approximation.

Pratt, R. G., and Worthington, M. H., 1990, Inverse theory applied to multi-source cross-hole tomography. Part 1: Acous- tic wave-equation method' Geophys. Prosp., 38, 287-310. We have presented diffraction tomography in terms of a homogeneous background medium with velocity Co. This paper presents a nonlinear inversion technique in the frequency- space domain which essentially permits inhomogeneous acoustic background media. A companion paper treats the same subject with respect to elastic media.

Rajan, S. D., and Frisk, G. V., 1989, A comparison between the Born and Rytov approximations for the inverse backscatter- ing problem: Geophysics, 54, 864-871. Includes numerical comparisons of the two approximations.

Wu, R. S., and ToksSz, M. N., 1987, Diffraction tomography and multi-source holography applied to seismic imaging: Geo- physics, 52, 11-25. Modified Devaney's plane wave seismic diffraction tomography using line sources and formulated the backpropagation reconstruction algorithms for crosswell, VSP, and surface seismic geometries.

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Chapter 4

Case St udies

4.1 Introduction

The purpose of this chapter is two-fold. First, through case studies, we illustrate the procedures for implementing the theory presented in Chap- ters 2 and 3. Second, the case studies highlight some of the benefits of using seismic tomography in the oil industry.

The first two case studies utilize crosswell seismic data in conjunction with the simultaneous iterative reconstruction technique (SIP•T) presented in Chapter 2 on seismic ray tomography. The first case study addresses the production problem of monitoring the progress of a steam-flood enhanced oil recovery (EOR) program. The second case study involves more of a development problem in which the structural interpretation of a fault- controlled reservoir must be better understood for in-fill drilling. The third case study uses the seismic diffraction tomography presented in Chapter 3 to image two salt sills using marine surface seismic data. We selected this problem to illustrate the seismic diffraction tomography methodology and limitations rather than to solve an exploration problem.

4.2 Steam-Flood EOR Operation

Some reservoirs encountered by the oil industry contain petroleum re- sembling a heavy, tar-like substance rather than a low-viscosity fluid. To produce such reservoirs, steam is injected into the reservoir with the intent of heating-up the petroleum making it less viscous so that it may flow. A steam-flood enhanced oil recovery (EOR) operation is expensive and may be adversely affected by reservoir inhomogeneities which channel

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96 CHAPTER 4. CASE STUDIES

steam away from parts of the reservoir the production engineer wishes to heat. This case study demonstrates how crosswell seismic ray tomography can identify reservoir inhomogeneities before initiating steam flooding and monitor the EOR operation during steam injection.

The Potter B1 tar-sand reservoir in the Midway Sunset Field, California, produces heavy oil using steam-flood EOR operations. We conducted two crosswell seismic experiments at a steam injection site in the Midway Sunset Field with the intent of aiding decision making by the production engineers on the steam-flood EOR operation. A presteam injection crosswell seismic survey was run to tomographically image reservoir inhomogeneities which might affect the operation and to provide a "base-line" P-wave velocity tomogram for comparison with a later poststeam injection P-wave tomogram. The P-wave velocity in this tar-sand reservoir drops dramatically when the reservoir is heated. Thus, a second poststeam injection crosswell seismic survey provides a P-wave velocity tomogram which readily identi- fies the heated parts of the reservoir when compared with the "base-line" tomogram. The production engineer may alter the chosen steam injection program if steam by-passes parts of the reservoir.

4.2.1 Crosswell Seismic Data Acquisition

The data acquisition geometry is depicted by the cross-section and map view in Figure 29. The seismic source was located in the temperature observation well T02 which is deviated 99 ft to the south. The production well 183 served as the receiver well and is offset 283 ft from the TO2 well at

the surface and 184 ft at the TD of well TO2. The steam injector well is situated downdip from the TO2-183 profile (dip direction is to the NNE) and the steam is expected to travel updip where it will intersect the crosswell profile.

Bolt Technology Corporation's downhole air gun shown in Figure 30 was used as the seismic source in this experiment. The air gun has an 80 cubic inch chamber and is pressurized to 2000 psi above the ambient pressure in the borehole. Geosource's three-component geophone VSP tool was used to detect the seismic energy. A DSS-10 system recorded the seismic signals at a sample interval of 0.5 ms with a one second record length. The downhole air gun was fired at 40 depth stations from 1360 ft to 1750 ft at 10 ft intervals, while the VSP tool was set at fixed station depths. The source was fired four times at each station so that better signal-to-noise ratio could be attained in the stacked signal. The 40 source stations were repeated for each of the 40 receiver stations, which ranged from 1630 ft to 1240 ft at 10 ft intervals, giving a total of 1600 unique source-receiver pair locations.

The common-receiver gather in Figure 31 was recorded by the vertical

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4.2. STEAM-FLOOD EOR OPERATION 97

Midway Sunset Crosswell Experiment Cross-section

TO2 183 - 0 283 ft

- 200

- 400

_ 600

- 800

- 1000 Potter A

- 1200

B1 - 1400

- 1600

Antelope Shale - 1800

- 2000 <-- 18411

Depth

Map View

TO2

99 ft i "• 400 DIp I•i : + Steam Injector

TI• ß 134R

184 11"'-.. .

ß

183

FIG. 29. Well configuration for the Midway Sunset crosswell experiment shown in cross-section and map view. Solid circles in the map view indicate well surface locations. The air gun source was placed in the temperature observation well TO2 which is deviated 99 ft along the dashed line. A three-component VSP tool was located in the vertically drilled production well 183. The dotted line indicates the profile defined by connecting the total depth (TD) positions of the two wells which corresponds to a 184 ft horizontal offset.

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98 CHAPTER4. CASE STUDIES

FIG. 30. Bolt employees are shown preparing an 80 cubic inch air gun at the TO2 well. The umbilical to the air gun contains a pressure line and a wireline. Photo courtesy of Don Howlett, Texaco.

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component of the VSP tool fixed at a depth of 1500 ft while the source moved through its 40 stations. The P-wave first arrivals are easily identified and have a dominant frequency of approximately 200 Hz. The large amplitude event immediately following the P-wave first arrival is the S-wave first arrival. Time zero occurs at 60 ms on the record which was monitored

by a pressure sensor mounted near the air gun. The large amplitude events trailing the S-wave arrival are from upward traveling tube waves.

Steam injection commenced shortly after completion of the presteam injection crosswell seismic survey. The steam flood operation continued for approximately one year before we ran the poststeam injection crosswell seismic survey. The second survey was acquired along the same TO2-183 profile with data acquisition parameters similar to the first survey. Tomographic images were processed using traveltime ray tomography for both data sets in an attempt to provide the production engineers with information on the performance of the steam flood operation.

4.2.2 Traveltime Parameter Measurements

The crosswell seismic data in this case study were processed using the simultaneous iterative reconstruction technique (SIRT) described in Sec- tion 2.3.3. A three-step recipe was given in that section for reconstructing tomographic images using SiRT. Here we explain how the traveltime parameters used in the SIRT recipe were measured, first for a P-wave velocity tomogram and then for an S-wave velocity tomogram.

The traveltime parameters required by SIRT are the source and receiver locations and measured traveltimes. The first step in determining source and receiver locations is to run a borehole survey. At Midway Sunset we used GyroData's tool to obtain the surface locations (x-y positions) for various depths (z-positions) in both the T02 and 183 wells. With the borehole geometries known all that remains is to provide good measured depth values to the source and receiver. Depth gauges on the vehicles used to deploy the source and receiver in the boreholes provided the measured depth values in this experiment. The starting position of the source was recalibrated each time a source pass was completed. We have since attached collar locators to the source and receiver to calibrate the measured depth estimates. Gamma ray logging tools can be used in place of collar locators to achieve similar calibrations.

The most critical and tedious step of ray tomography processing is selecting the traveltimes from the observed data. The observed data, given by p/ob• for the ith source-receiver pair in equation (46), should be selected from data preprocessed for maximum signal-to-noise ratio. To provide this optimum environment for picking P-wave traveltimes, the Midway Sunset

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P -Wave

$ -Wave

Time (ms)

Tube Wave

Common-Receiver Gather (1500 ft) Source Depth (ft)

1400 1500 1600 1700

100- , • ,

, ;

2oo- I: I

I I I.

300 - .i ii i ' ,

_• , , 400 -

F•(•. 31. Common-receiver gather for the VSP tool at a depth of 1500 ft. The 40 traces correspond to the source stations.

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Source Well Receiver Well

Low Velocity

High Velocity

Direct arrlval's ray path ....

Head wave's ray path

FIc. 32. This cartoon depicts the raypaths for a direct arrival with traveltime Td and a head wave with traveltime Th • Td.

data required only a minimum-phase trapezoidal bandpass frequency filter of 50-100-200-300 Hz.

Generally one selects first arrivals on crosswell seismic data when P- wave traveltimes are desired. That is, the first arrival catches our eye as being the first significant signal. However, in many cases the first arrival is not a direct arrival, but a head wave which travels along an interface as shown by the solid raypath in Figure 32. The head-wave traveltime pick creates a problem when the forward modeling method used in step I of the SIRT algorithm determines only direct arrival raypaths, such as the one depicted in Figure 32. Obviously, a computed raypath giving the predicted traveltime p•re in equation (46) which is not consistent with the event's observed traveltime p/oh, will cause wrong cells in the model to be updated at incorrect velocities.

The remedy for this dilemma can take two avenues. The avenue with the

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potholes is to stick with the ray tracing scheme which models only direct arrivals. Here we are forced into trying to identify when the first arrival is a head wave and then, whether we should attempt to pick a later, more noisy, event as the direct arrival or to make no traveltime pick at all. This can be a very frustrating process when the subsurface is prone to many head waves.

The second avenue is to use a ray tracing scheme which models the first arrival's raypath, whether it be a direct arrival or head wave. Some of these schemes are referenced at the end of Chapter 2. A possible problem with these schemes is that a raypath may be found which gives a first arrival corresponding to an event with negligible amplitude. Thus, we end up selecting the wrong observed traveltime once again. Our experience shows that when head waves are prevalent, this second avenue seems to be the more successful of the two and the least frustrating. However, for the Midway Sunset data we ray traced for predicted direct-arrival traveltimes since head waves were not a problem.

The direct arrival versus first arrival problem is an important consideration in selecting measured traveltimes. However, another concern just as important is the radiation patterns of the source and receivers, since these determine which polarity one should pick. The air gun is an explosive-type source with a positive (outwards) particle motion in the first half-cycle of the signal with greatest strength directed horizontally as shown by the radiation pattern in Figure 33. For a constant velocity medium the rays between source and receivers are straight. Figure 33 shows the vertical (V) and radial (R) geophone components. For a positive particle motion the vertical component produces a positive kick when the wave strikes the geophone from above, and the horizontal component produces a positive kick as long as the wave is traveling radially outwards.

The vertical component of the geophone remains fixed along the borehole axis so that any change of polarity in the signal can be attributed to the up or down propagation of the P-wave. If the radial component were indeed fixed at all receiver levels, then the polarity would always be positive except for under very extreme subsurface geologic conditions. However, the two horizontal components in a VSP tool are not fixed and rotate randomly throughout the survey. The good news is that the P-wave's first half-cycle should always be positive on the radial component. Thus, we can numer- ically rotate the horizontal components so that we always have a positive first kick on the radial component. The even better news is that with the air gun source the receiver remains fixed over each source pass (common- receiver gather). This means we need to determine the numerical rotation only once per receiver level. For the Midway Sunset survey we only were required to determine 40 rotation angles for the entire survey! To summa-

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Air gun Source / Geophone Receiver

Source Receiver V R

P-Wave

FIG. 33. This cartoon depicts the P-wave radiation pattern with a positive polarity during the first half-cycle of motion (explosive source). The vertically (V) and radially (R) recorded first half-cycles are shown on the right as flags with the expected polarities and signal strengths represented. The vertical component changes polarity with receiver depth while the radial component's polarity remains positive.

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rize Figure 33, we must keep in mind that the polarity will change on the vertical component, but not on the radial component. Thus, its best to cross-reference one component with the other during traveltime picking.

The Bolt air gun also emits an SV-wave (vertically polarized S-wave). The radiation pattern for the SV-wave is shown in Figure 34 and resembles a four-leaf clover. The particle motion polarities of the first half-cycle are shown as small arrows perpendicular to the raypaths. For the rays shown, positive polarity corresponds to an arrow pointing in the counter-clockwise direction and negative polarity corresponds to an arrow pointing in the clockwise direction. The resulting first half-cycles of the S-wave are shown as flags for the vertically (V) and radially (R) recorded signals. Similar to the P-wave, the $V-wave exhibits a phase reversal on the vertical component as the receiver changes depth from above the source to below the source. The radial component does not change polarity. Thus, we must be cautious in selecting traveltimes for the S-wave arrivals, making sure to pick the correct polarity. Again, selecting the correct polarity is best done by viewing both components while traveltime picking.

The next step in obtaining quality traveltime picks is to insure consistent picks for the selected event (direct arrival here) over the entire crosswell data set. With surface seismic data we tie the selected event around loops defined by intersecting strike and dip lines. An analogous technique may be applied to crosswell data provided that the same source and receiver stations are used throughout the survey, or at least over large portions of the survey.

The Midway Sunset data acquisition program called for common-receiver gathers as depicted by the cross-section in Figure 35(a). The source was moved through its set of fixed stations as the receiver remained fixed. These locations plot on a vertical line at receiver depth R4 in the source-receiver depth plane on the right in Figure 35(a). Locations of other common- receiver gathers are represented by vertical lines corresponding to other fixed receiver stations. If we keep the same source stations for each receiver (which we did), then the data can be sorted into common-source gathers as shown in Figure 35(b). The source-receiver depth locations plot along horizontal lines for common-source gathers, such as the one at source depth S3. Finally, Figure 35(c) shows that common-offset gathers are also possible where the offset is a depth difference between source and receiver.

Figure 36 indicates how the common-receiver and common-source gathers are used to tie time picks around loops. The sources and receivers corresponding to straight rays within the box in the cross-section on the left define a square loop in the source-receiver depth plane on the right. We begin with the common-source gather at station S2 and select traveltimes from receiver stations R2 to R5. Next we take the common-receiver gather

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Air gun Source ! Geophone Receiver

Source Receiver V R

Sl/-Wave • o o

FIG. 34. This cartoon depicts the $V-wave radiation pattern with the up- going $V-waves exhibiting a positive polarity on the first half-cycle while the downgoing SV-waves exhibit a negative polarity. The vertically (V) and radially (R) recorded first half-cycles are shown on the right as flags with the expected polarities and signal strengths represented. The vertical component changes polarity with receiver depth while the radial component 's polarity remains negative. No S-wave signal is received directly across from the source since the radiation pattern is zero horizontally.

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(b)

(c)

Cross-section

SOURCE

WELL

RRCEIVER

WELL

Commor•-Receiver - RI

Gather

R6

S1-

$2.

Depths $4,

$5,

$6

Source / Receiver Depth Plane Receiver Deptl•

RI R2 R.3 R4 RS R6 •

I

I

I

I

I

Cross-section

SOURCE

WELL

RECEIVER

WELL

Sl Common-Source

:'_'___ _:. s4 ' C C .. -.-•

Source / Receiver Depth Plane

RI Sl

R2 S2.

R.3 83

R4 Deptlm 84

R5 85.

R6 86.

Cross-section

SOURCK

WELL

SI

S2

S5

S4

S6

RECEIVER

WELL

Gommon-Off• Gather _-•, R• SI-

R2

R.3

R4 Dept!• S4-

R5 85-

R6 S6-

Source / Receiver Depth Plane Receiver Dept!•

FIG. 35. Possible crosswell seismic gathers shown by raypath correlations on the left and in the source-receiver depth plane on the right for: (a) receiver, (b)source, and (c) offset gathers.

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Tying Data Using Source / Receiver Gathers

Cross-section

souac•

WELL WELL

S1 R1

S2 ......... R2

S3 R3

S4 R4

S• R•

S6 R6

Source / Receiver Depth Plane

RI

S3 •

Depths

S6 •

Rccclvcr Dcptlu

R2 R3 R4 R• R6

!

I

i

I

I <-

FIo. 36. The sources and receivers within the range of the box of the cross-section are used to tie traveltime picks around the loop shown in the source-receiver depth plane. Two source gathers and two receiver gathers are required to complete one loop.

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108 CHAPTER4. CASE STUDIES

at R5 and pick traveltimes off of traces corresponding to source stations S2 to S5. Then the common-source gather at S5 is used and time picks taken from receiver stations R5 to R2. Finally, the loop is completed by picking times for traces S5 to S2 on the common-receiver gather at R2. Since the S2 trace on common-receiver gather R2 is the same trace R2 on common-source gather S2, the traveltime picks must be the same. If the picks appear to be different, then a mispick must have been made somewhere along the loop and should be rectified before you continue. All picks in the data set can and should be tied in this manner to provide consistency. Generally, on the first pass, loops are made as large as the data quality will allow with in-fill loops checked later. Besides providing consistent picks, this method also lets one identify poor data regions in the source-receiver depth plane and to develop strategies for tying time picks around "not useable" data regions.

Finally, a recommended practice is to plot the computed traveltimes from the converged image reconstruction on the data records with the observed traveltime picks. Since the image reconstruction is based upon both good and bad picks, many times the bad picks will "stand out" when compared with the computed traveltimes; assuming most of your traveltime picks are good. Also, traveltime picks left out of the inversion process because of uncertainty are more clearly identified by the computed traveltimes permitting you to add them to the inversion process.

4.2.3 Image Reconstruction

With the P-wave direct arrival traveltimes selected and the associ-

ated source/receiver locations determined, we proceed by establishing the gridded model which represents the initial model function Mj "it, where j = 1,..., J and J is the total number of cells in the gridded model. We chose square cells 5 ft on a side giving 42 cells horizontally and 84 cells vertically for a total of J = 3528 cells in the model. •

Figure 37 shows the initial estimate of the P-wave velocity profile between wells TO2 and 183. We chose a two-layer initial model since the velocity contrast between the Potter B1 tar-sand and Antelope shale in Figure 29 was significant and the average layer velocities and interface dip were known from well logs in the TO2 and 183 wells. The model in Fig-

1 We originally started with 10 ft square cells because the source and receiver spacings were 10 ft. Five foot square cells worked just a.s well, but with more resolution. Lesser size cells did not work well. Note that all of the tomograms shown in this section are interpolated to 1 ] ft square cells for display.

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1400

1450

Depth (ft)

1500

1550

1600

1650

Initial Guess

TO2 Offset (ft) 200 150 100 50

'['

183

0

Velocity (fUs) ..':.- .... 6650

...

.......... 6900

.•!::?:ii:•ii::?:i!ii[ 6970 '::;;:;:::::.::;

'•:•:•;•:•:•:•*• 7015 -:::.>:+•-'..::•

•:;•?• 7070

i•':J-'-"•';•:: 7120 ;/•i=iii• 7176 .::::::::::'./.::.:

724O

7350

7550

8100

FIG. 37. Two-layer model used as the initial estimate in the SIl•T algorithm.

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11o CHAPTER 4. CASE STUDIES

ure 37 is the reciprocal 2 of the initial model function Mj '•i' which becomes the first model estimate Mf •t in the SIRT algorithm.

Step I of the SIRT algorithm is to perform ray tracing through the estimated model Mf st for each source-receiver pair with an observed traveltime, Pi ø•'s. As mentioned earlier, we use a "Snell's Law" type of ray tracing for this data set, which models only direct arrivals. Thus, since we selected 1600 observed traveltimes, we end up with 1600 predicted direct- arrival traveltimes p•,re, where the subscript i is the index for the ith source- receiver pair. Besides giving the predicted direct-arrival traveltimes, the ray tracing also provides the raypath length through each cell as required by equation (46) in step 2 of the SIRT algorithm. The raypath length for the ith ray in the jth model cell is represented by the variable

Step 2 of the SIRT algorithm uses equation (46) to determine the model corrections AM 1 to the estimated model Mf st. The predicted direct-arrival traveltimes p•,re and raypath lengths $ij from step 1 along with the associated observed traveltimes Pi ø•'s are used to compute each term in the summation in equation (46). The weight I/V 1 for the jth cell is simply the number of rays which intersect the jth cell, frequently called the ray density. Ray density is easily determined during ray tracing when computing the raypath length

Once the corrections AM 1 for all J = 3528 model cells have been determined in step 2, we apply those corrections to the estimated model function

M[ st in step 3, giving a new estimated model function MJ new)est. However, because of the nonlinear nature of seismic ray tomography we apply only a fraction of the update given by equation (46), say 60 percent. This prevents instabilities and is most important during the first few iterations when the corrections can be quite large. The model updates are also smoothed with a small spatial filter before they are applied to help reduce instabilities. These extra steps are necessary because in seismic ray tomography SIRT is a linear inversion scheme used in an iterative manner to solve a nonlinear

problem, as discussed in Chapter 2. Steps I through 3 are repeated using the previous iteration's new model

estimate M« n•w)•t as the current estimated model Mf st. The steps are iterated until we are satisfied that the estimated model has converged to an acceptable solution. Figures 38 through 42 show the estimated P-wave models in terms of velocity for the 1st, 5th, 10th, 20th, and 44th iterations of the SIRT algorithm, respectively. The first 10 iterations show the development of large scale features in the P-wave tomograrns such as the high

:ZRemember that the SIRT algorithm uses slowness for the model function Mj instead of velocity Vj, where for the jth cell, the two are related by Mj = 1/Vj. We display Vj because most people are more familiar looking at velocity rather than slowness.

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velocity feature at 1475 ft and low velocity feature at 1550 ft near the TO2 well. After 10 iterations, only small-scale adjustments are made to the large features defined in the early iterations.

There is not a lot of difference between the 20th iteration and the 44th

iteration P-wave tomograms. A natural question to ask is, "When do we stop the iterations?" One method used to answer this question involves plotting traveltime residuals as a function of iteration, as shown in Fig- ure 43 for the P-wave data. We see that the traveltime residual decreases

significantly during the first 10 iterations in which large-scale features on the P-wave tomogram are determined. After the 10th iteration the traveltime residual curve begins to toe-out as small-scale features are added to the large-scale features. By the 44th iteration the traveltime residual curve had approached the horizontal, implying further iterations could not improve the convergence and the P-wave tomogram in Figure 42 is the best possible estimate of the true velocity profile. This method assumes that as the computed traveltimes approach the observed traveltimes, the estimated model approaches the true model.

Observed traveltimes for shear wave direct arrivals were also selected.

However, because the S-waves are imbedded in other arrivals we could select only 745 of the 1600 possible S-wave arrivals. Utilizing a similar procedure as for the P-wave tomogram, we constructed the S-wave tomogram in Fig- ure 44. The noisy appearance of the S-wave tomogram is a direct result of the small number of rays utilized by the SIRT method.

Approximately one year after the presteam injection crosswell seismic experiment we ran a poststeam injection experiment to evaluate the success of the EOR steam flood program. We processed the second crosswell seismic data set for a poststeam injection P-wave tomogram using the same processing procedures as for the presteam injection P-wave tomogram in Figure 42. Figure 45 compares the poststeam injection tomogram with the presteam injection tomogram, both sharing the same velocity scale. Heated parts of the reservoir resulted in a reduced P-wave velocity as expected based upon core study results.

4.2.4 Tomogram Interpretation

The presteam injection P-wave tomogram in Figure 42 is used to delin- eate reservoir inhomogeneities. We made the lithology/porosity interpretation shown in Figure 46 assuming a direct correlation between the P-wave velocity and lithology/porosity determined from cores. The Potter A sand, Potter B1 sand, and Antelope shale are delineated. The Potter B1 sand is the reservoir in which steam was to be injected. Using core information we interpreted the high and low velocity zones within the Potter B1 sand

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Depth (ft)

1400

1450

1500

1550

1600

1650

TO2

200

Iteration I

150

Offset (ft) 100 50

183

0

Velocity (ft/s)

6650 . ß

:': ..... s•oo

............ '6970

ß :•:•.:•:•: 7015

ß •!•-'-'•...:i•:• 7070 .:.>•:.:.-.'•.-:::

ß •:•[•:-..:•i 7120

•:....:.....•.•:.:• 7176 .:::::•::-...-:

7240

7350

7550

8100

FIG. 38. P-wave velocity tomogram after 1st iteration.

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1400

1450

Depth

(ft)

1500

1550

1600

1650

Iteration 5

TO2 Offset (ft) 200 150 100 50

183

0

Velocity (fUs)

:?:iii•iii::.': 6650

:':':"•!! 7015 !•:..._.•. • 7070 !-"'i!=•i 'i 7120

7176

7240

7350

7550

8100

_

FIC. 39. P-wave velocity tomogram after 5th iteration.

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114 CHAPTER4. CASESTUDIES

1400

1450

Depth

(ft)

1500

1550

1600

TO2

200

Iteration 10

150

Offset (ft) 100 50

183

0

Velocity (fUs)

.i.:!i!i::::iiii:: 6650 . :.....:.: :.:..

': ............ '6900 '.:::::::::::::::;: .'::•:::::::::::.:

.........

'::..::•::.':::::.:

:iiiiii?:?:?:ii?:ii 6970

.....•......• 7OlS •?//. 7070 ::.:•;•... -. 7120

•i•! •':'•:• 7176 7240

7350

7550

8100

FIG. 40. P-wave velocity tomogram after 10th iteration.

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1400

1450

Depth

(ft)

1500

1550

1600

1650

Iteration 20

,i'O2 Offset (ft) 200 150 100 5O

183

0

Velocity (fus) .::i:'•!:•:!:... 6650

..::::::•i¾:: .... 6900 .........

.........

................... 6970 .:.:.:.:.:.:..-.:.: :.::::.'::::::::;:;:

................. 7015

•:...•:.: 7070 ?::.•: 7120

-:i:::::•:.-.'•'..-:-'•:: 724O

7350

7550

8100


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1400

TO2

2OO

Iteration 44

150

Offset (ft) lOO 50

1450

Depth

(ft)

1500

•i!i•i!':."i.?iL::::•?,:.•i!:!.. ' ' ..

1600

1650

183

0

Velocity (fUs)

'i:!:i:i:i:i:i:!:!: :.:.:.:.:.:.:.:.:.: ..........

ii.•! 7015 ::• /:: :•-.;.•:.•.' 7070

• •."-'."";• 7120

•:i:•:•:•:'::•,• 7176 7240

735O

7550

8100


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Traveltime

Residual

(xl(• 4 s 2)

Iteration

0 10 2O 3O 4O

Traveltime Residual I

= •, • i:•iire obs i= 'Pi ) 2

FIG. 43. Plot of traveltime residuals as a function of iteration for P-wave

velocity tomograms. Traveltime residual is defined as the sum of the squares of all differences between predicted and observed traveltimes.

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1400

1450

Depth (ft)

1500

1550

1600

1650

TO2

200

Iteration 50

Offset (ft) 150 100 50

183

0

Velocity (fUs)

2634 ......

•:•'•.i-'!!•:'

•:'•:•*•"•": 3041

iiiiii!1111!111i 3113

:'• '3242 , .......

i.11:...i::.• 32• ....:.:..-..-.:..-.

3385

3508

3645

3898

4300

FIG. 44. S-wave tomogram processed in a similar fashion as the P-wave tomogram in Figure 42.

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Presteam Injection Tomogram Oei•h TO2 Offset (ft) 183

(ft) 200 150 100 5•) 0 i

::::::::::::::::::::::::::::::::::::.:.:..'.::

-:i::i::?..-:.. :- .......•.... •....¾...!...,..•:• ............. :.:•.•.,?.,.. .......... ..

1500 -

1550 _ ' ..... ..]i•.!i•i:i:i:i:i:i:i:[3•":' _

ß -.-.-• •,.-,.,.L...-.'-'.'-'.

Poststeam Injection Tomogram Dei•h TO2 Off#t (ft) 183

•) 2o0 •so •oQ so o ! ! I i i

• .:: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ===================== :::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::,: .. :::::::::::::::::::::::::: 1400 :.: :::::::::::::::::::::::::::::::':".--.• :-:-:-:-:-:.:' ============================================= '.:.: ?'•::!:i:i:i:i:!:i:!:i:!:i:i:i:!:. :. :•-.•!i::::::' ::!:::!:!:i::::':":"!":'"' :":'::i:i:!:!:!.'.:.':

•.i:•;.:=•.?.•i•.,..%..•.;:!:!;:!ii::.;..!... ' •.... ..... •04• :....,•.•½.,x.•;:::::::,,:.:.::::.::::::..: . ß 1450 ::::::::::::::::::::::!.•:::!:::!:: 6293 ::::::::::::::::::::::::: :::.:.;,:.:.:.:.. ß . :..

i.:•.•.'..i::!:!i!i!•ii:i::..' ..... ." i:i:::i:i:i".• S705 1500 :i .".:?.!:!:i:i ..... • ..... . .... :: ....... .:.:.:.:.:.:.:.:.:.:.>;.:. 6936

i• x.•!ii: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:::•i.•::=!:•:!: 7200 :.:.',•,•.:.:.:.:.:.:.:.:.;.:.:.:.;...;.:.;.:.:.:.:.;.:.:.:...:.: ;.:.:.:.:.;.:.:•:.',-.•: '•:::.':.;:.•: ::::. ================================================================= -1.•.:.:..:::::..-.. '"::: -::•:,'•-• .......... '. ........ ::.'•.-•.•-'..:.: ...... :.•,.,.:•,•- 1550 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :2::•::.:•i:.,x.'

:-.:,-.:. -:.-.-...:.:..............-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.....-..S; '.-.x.¾4::.... 790 I !::•..'•i.';:.>-."-!'-::!:!:!:!:!:!::: :': ....... "::!:i:!:!:!:::i..x%.•i•:-.:..•:!. :::::::::::::::::::::::::: ..... ::::::::::::::::::::::::::::::::: 6000

Velocity

F](•. 45. •'he presLeeLm injecLion P-weLve LomogreLm on Lhe lefL forms Lhe baseline LomogreLm for observing reservoir changes as eL resulL of sLeeLm injecLion. The posLs[eeLm injection P-wave [omogreLm on the right Laken one year leLter indirectly indiceLtes a significeLnt portion of the reservoir was healed. P-wave velociL¾ in Lhe heeLv¾ oil reservoir rock is reduced when Lhe reservoir is healed.

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Lithology/Porosity Interpretation

1400

TO2 Offset (ft) 183 200 150 100 50 0

Velocity (rt/s)

6650

ß 6900 1450

.....

::i.!:i:i:!.i:!:

1500 ii•."a•'.:i•11• 7070 Depth

(ft) '-':•l•i•..*.." 7120 1550 .•i!•! 7176 7240

735O

lSOO ',i',i',i',i',iii',i?, 7sso 8100

1650

FIG. 46. Lithology/porosity interpretation of the presteam injection P- wave tomogram. The Potter A sand, Potter B1 sand, and Antelope shale are shown. Core data showed that high and low velocity zones within the Potter B1 sand correspond to low and high porosities, respectively.

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as low and high porosities, respectively. Thus, the reservoir is certainly not homogeneous as far as porosity is concerned. At the top of the Pot- ter B1 sand are high velocity clay stringers. The production engineers rely on these clay stringers to confine steam to the Potter B1 sand. However, the clay stringers do not appear continuous, and actually drop in velocity towards the center of the tomogram indicating a possible breach through which steam may flow across into the Potter A sand. Thus, the production engineer might anticipate some steam loss into the Potter A sand.

We can also estimate a porosity tomogram of the Potter B1 tar sand reservoir using the P-wave velocity tomogram and Wyllie's time-average equation in the form,

where V! is the fluid velocity, V,• is the matrix velocity, and V is the P-wave tomogram velocity. The measured average matrix velocity and pore-fluid velocity for the Potter B1 sand are Vm = 10000 ft/s and V I = 4500 ft/s, respectively. The P-wave tomogram in Figure 42 provides the values for V, giving the porosity tomogram shown in Figure 47. This porosity tomogram is valid only for the Potter B1 tar sand for which the average values of Vm and V! were determined. The porosity values for the Potter B1 sand range from 33 percent to 44 percent according to the porosity tomogram. These porosity values are higher than the porosities determined from core samples which have high values of 32 percent. This discrepancy implies that the Wyllie's time-average equation does not take into account all petrophysical properties which affect the P-wave velocity, such as clay content. However, we do believe that the relative porosity information for the Potter B1 sand is meaningful. Such porosity information is helpful to engineers who need to model the production of a reservoir.

The second objective of the Midway Sunset field tomography project was to monitor the EOR steam-injection project. To monitor the steam flood progress in the Potter B1 reservoir using seismic tomography, we must know the effect of heated Potter B1 reservoir rock on seismic velocity. Cores from a well located 300 ft east of the 183 well provided Potter B1 sand samples for which P-wave velocities could be measured in the laboratory at various temperatures. Figure 48 displays the laboratory measured P-wave velocities for a Potter B1 sand core at temperatures of 25øC, 55øC, 90øC, and 125øC, under confining pressures of 500 psi, 1000 psi, 1500 psi, and 2000 psi. Clearly, we will expect the P-wave velocity to decrease as the temperature of the reservoir increases as a result of steam injection.

The core results provide us with the temperature-velocity relationship required to interpret the presteam and poststeam injection tomograms in

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1400

1450

Depth

(ft)

TO2

200

Porosity Tomogram Offset (ft) 183

150 100 50 0

..:::.-:

:..-:.:

1550 .•,.

1600

1650

Porosity

20

25

30

33

35

38

41

45

FIG. 47. Porosity tomogram determined from the P-wave velocity to- toogram in Figure 42 and Wyllie's time-average equation. The lithology]porosity interpretation from Figure 46 is superimposed for comparison.

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Fwave Velocity vs. Temperature 8500

8000

75OO

Velocity

7OOO

6500

6000 20 40 60 80 100 120

Temperature (deg C)

2000 psi

15o0 psi

1000 psi

500 psi

FIc. 48. Measured P-wave velocities for a Potter B1 sand core at temperatures of 25øC, 55øC, 90øC, and 125øC, under confining pressures of 500 psi, 1000 psi, 1500 psi, and 2000 psi.

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Sonic Log at TO2 Well

1400

1500

Depth (ft)

1600

Tomogram-Sonic Log Comparison

400

5O0

600

•-•Smoothed Sonic Log

P-wave Tomogram

6000 7500 10000 12500 7000 7500 8000 8500

Velocity (if/s) Velocity (if/s)

FI(•. 49. The original sonic log from the TO2 well is shown on the left. On the right, a smoothed version of the sonic log is compared with a tomogram profile parallel to and offset 15 ft from the TO2 well in Figure 42.

Figure 45. The seismic P-wave velocity over parts of the Potter B1 tar sand dropped by 15 percent to 20 percent after one year of steam injection. We also see that the clay stringers in the upper Potter B1 sand restrict the steam flood to the Potter B1 down dip. However, further up dip the clay stringers are breached by the steam and some steam is lost to the Potter A sand, which exhibits a similar velocity-temperature relationship as the Pot- ter B1 tar sand reservoir. Figure 45 indicates that the Potter B1 sand was not uniformly heated and gave the production engineers information for modifying the steam injection project at this site.

Finally, we checked the reliability of the presteam injection tomogram by comparing a tomogram profile taken about 15 ft in and parallel to the TO2 well in Figure 42 with the sonic log in the TO2 well. Figure 49 shows the original sonic log velocity on the left and provides a comparison of the smoothed sonic velocity log with the tomogram profile on the right. The sonic log and tomogram profile are in good agreement as to overall trend. Velocity differences corresponding to the clay stringers may be attributed to averaging-out of the high velocities by the tomography inversion. The Potter B1 reservoir, corresponding to depths below 1500 ft, show the sonic

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4.3. IMAGING A FAULT SYSTEM 125

log and tomogram profile agree to within 5 percent. Thus, analysis of the tomograms in this portion of the Potter B1 tar sand are made with some justified confidence. Since similar data acquisition and processing techniques were applied for the poststeam injection tomogram, we also assume this reliability analysis extends to the poststeam injection results as well.

We make the following conclusions regarding crosswell seismic tomography based upon this case study' 1) Crosswell tomography as a reservoir characterization tool is useful for determining reservoir lithology and porosity inhomogeneities which is not possible with only well log information; and 2) Crosswell tomography is useful for monitoring nonuniform heating of reservoir rock between wells as a result of steam flooding.

4.3 Imaging a Fault System

The McKittrick Field in California is located near the Midway Sunset Field and produces from the Potter sand, a massive unconsolidated con- glomerate with permeabilities ranging from 1 to 10 darcys. Figure 50 shows the well-log based reservoir geology interpretation prior to running crosswell seismic tomography between the 806 and 429 wells. The McKittrick Thrust placed Miocene age diatomite over Pleistocene Tulare sand. A subthrust fault developed subsequent to the McKittrick Thrust which intersects the 806 well at a depth of 800 ft.

The Potter sand reservoir contains a heavy oil which is subject to gravity drainage. Based on data from the 806 well, the subthrust is believed to act as a sealing fault which prevents further downward migration of the heavy oil. The well data show that the Potter sand above 800 ft is saturated with

up to 50 percent oil while the Potter sand below 800 ft is desaturated with less than 30 percent oil. The subthrust lies at the boundary between these two zones.

The estimated oil reserves in this reservoir and optimum development of the field depend upon proper positioning of the subthrust. Thus, the objective of this crosswell tomography study is to image the faults associated with the reservoir and define the boundaries of the saturated and

desaturated Potter sand.

4.3.1 Crosswell Seismic Data Acquisition

The clamped-vibrator source shown in Figure 51 was provided by Chevron for this experiment. The source is coupled to the borehole wall through the clamp located at the top of the tool. One of the smaller lines

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Well-log Based Interpretation wel 8O6

315 ff

D!atomite

well 429

Oft

20O ft

McKIttrlck

. ..•• _ 40Oft

_ 60Oft

_ 8OO ft

1 ooo ft

FIG. 50. The reservoir geology interpretation before the crosswell seismic survey was run between the 806 and 429 wells. The interpretation was based mainly upon well-log data and some surface geology.

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q

FIG. 51. Chevron's clamped-vibrator source in preparation for deployment. The primary components from top to bottom are the clamp for source-to- borehole coupling, the hydraulic servovalve and actuator module, and the reaction mass. Photo courtesy of Don Howlett, Texaco.

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Source

Data Acquisition Geometry well 806 well 429

315 ft :

-- 2O0 ft

40O ff

-- $00ft

Oft

.___ Shallowest Receiver

Deepest Receiver

-- 80Oft

FIG. 52. Data acquisition geometry involved keeping a 90-degree receiver aperture bisected by the horizontal through the source. The aperture was formed using thirty-three receiver levels spaced at 20 ft intervals about each source location. The shallowest and deepest receiver positions for the source at a depth of 400 ft is depicted.

provides air pressure to the clamp for its activation and deactivation. The other small line is a wire line which both supports the tool and provides elec- tricity to the hydraulic servovalve and actuator which control the sweeping action of the vibrator. A pump at the surface pressurized hydraulic fluid which flows to and from the source through the two larger diameter hoses. This pressure drives a hydraulic cylinder which is attached to a 50 pound reaction mass located at the bottom of the tool. An axial driving motion imparts a vertical stress on the borehole wall at the clamp which introduces the seismic energy into the formation.

Figure 52 shows the data acquisition geometry. The clamped vibrator was deployed in the 806 well while a three-component geophone receiver

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Time

5O

lOO

150

200

Common-Source Gather (660 ft) (ms) Receiver Depth (ft)

1100 1000 900 800 700 600 500 400 300 200

FIG. 53. A common-source gather from the radial component for a source depth of 660 ft and receiver depths ranging from 160 ft to 1150 ft at 10 ft intervals.

was deployed in the 429 well. Thirty-seven common-source gathers were collected with the source ranging in depth from 200 ft to 920 ft at 20 ft intervals. Common-source gathers were collected, rather than common- receiver gathers, because of the unwieldy nature of the four lines attached to the source which had to be clamped together at regular intervals as the source was lowered into the well. For most source levels 33 receiver levels

were recorded at 20 ft intervals in such a manner that a 90-degree angle receiver aperture bisected by the horizontal at the source was acquired, as shown in Figure 52.

A linear sweep was applied to the vibrator from 10 Hz to 360 Hz with a sweep length of 14 s and a listen time of 2 s. Four sweeps per source level were required to get a good signal-to-noise ratio. Figure 53 shows a common-source record for the clamped vibrator at a depth of 660 ft and receiver stations ranging from 160 ft to 1150 ft at 10 ft intervals. The record is from the radial component and shows both first-arrival and reflected P- wave events.

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4.3.2 Traveltime Parameter Measurements

The trove]time p•r•meter measurements follow • procedure similar to the one for the Midway Sunset Field. Chevron provided deswept data with the horizontal components decomposed into r•dial and transverse components. We applied a zero-phase bandpass frequency filter to further increase the signal-to-noise ratio for optimum traveltime picking. Because of the "blocky" nature of the subsurface we chose to select direct-arrivM P-wave tr•veltimes, similar to wh•t we did with the Midway Sunset dat•.

The only exceptionM difference has to do with what polarity to chose for the direct-arrival traveltime. The clamped vibrator source has a radiation pattern significantly different from the air gun. Figure 54 shows the P- w•ve r•di•tion pattern for the clamped vibrator source in a homogeneous medium with the signal recorded by geophone receivers with vertical (V) and radiM (R) components. s The clamped vibrator is essentially a vertically directed dipole which has P-wave motion directed towards the source in the lower lobe while, at the same instant, the P-wave motion in the upper lobe is •way from the source. For a homogeneous medium this radiation pattern results in no polarity change on the vertical geophone component while polarity changes on the radial component. Theoretically, no P-wave energy should be observed at the same depth as the clamped-vibrator source in a homogeneous medium.

Figure 55 depicts the S-wave radiation pa•tern for •he clamped vibrator which may be compared with the S-w•ve r•diafion pattern for the air gun source in Figure 34. The clamped vibrator should produce excellent S-w•ves for use in crosswell tomography since the S-wave radiation pattern's maximum strength is directed horizontally. As with Figure 54, in a homogeneous medium we expect no polarity change of the direct arrival on the vertical component while a polarity change is expected on the radial component.

At first glance the dipole nature of the damped vibrator appears to produce records which should be easy to interpret. After all, there are no polarity reversals seen on the verticM component for either P-wave or S-wave direct arrivals in Figures 54 or 55, respectively. However, now introduce • velocity field such that a P-wave raypath initially traveling upwards gets refracted so that it is • downgoing ray at the receiver. The P-wave event you are selecting appears to shift • cycle on the vertical component. Thus, you must make a decision: 1) to pick on the reversed polarity because you suspect the ray turned up-to-down (or down-to-up), or 2) to pick the same polarity at the shifted time because you suspect a velocity change. For complicated subsurface velocities this decision process is tedious and

3 Compaxe Figure 54 for the clamped vibrator with Figure 33 for the •ir gun.

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P-Wave

Clamped Vibrator Source ! Geophone Receiver

Source Receiver V R

' -I-

o o

FIG. 54. This cartoon depicts the clamped vibrator's P-wave radiation pattern where particle motion in the upper lobe is away from the source while in the lower lobe the particle motion is towards the source at the same instant (dipole source). The vertically (V) and radially (R) recorded signals are shown on the right as flags with the expected polarities and signal strengths represented. The vertical component remains negative with receiver depth while the radial component's polarity changes sign.

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Clamped Vibrator Source / Geophone Receiver

Source Receiver V R

S-Wave

FIG. 55. This cartoon depicts the •qV-wave radiation pattern for the clamped vibrator with all $Vowaves exhibiting the same polarity as indicated by the small arrows perpendicular to the raypaths. The vertically (V) and radially (R) recorded signals are shown on the right as flags with the expected polarities and signal strengths represented.

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requires the use of properly oriented horizontal components. The horizontal geophone components are free to take on random direc-

tions when the receiver tool is moved from level to level. Frequently the relative amplitudes on the horizontal components are used for numerical rotation to get the radial and transverse components desired in traveltime picking. However, the polarity of the incoming P-wave or S-wave must be assumed to select the proper rotation. Figures 54 and 55 show that both P- wave and S-wave polarities change on the horizontal component with depth, thus complicating the rotation analysis process. Add a complex subsurface and the process of orienting horizontal components based on relative amplitudes becomes nontrivial. The best bet is to run an orientation device with the receiver package when using a clamped-vibrator source. Properly oriented horizontal components are a definite help in picking both P-wave and S-wave arrivals from a dipole source which travel through a complex medium.

Also, since the clamped vibrator required common-source data acquisition, every source-receiver pair in the survey required a unique rotation analysis to orient the horizontal components. A common-receiver data acquisition technique is more desirable if an orientation device is not available. Then, one only needs to perform one rotation analysis per receiver level since the receiver remains fixed for all source levels as was done with

the Midway Sunset data.

4.3.3 Image Reconstruction

We used the same image reconstruction method here as for the Mid- way Sunset data in the previous section. The initial model function Mj ni• had square cells 10 ft on a side which is half of the source-to-source and receiver-to-receiver spacings. Ten-foot square cells required 32 horizontal cells and 60 vertical cells for a total of J = 1920 cells in the model. The

initial model function values were given a constant velocity of 5300 ft/s. Just as with the Midway Sunset image reconstruction, step i of the

SIRT algorithm utilized a "Snell's Law" type of ray tracing to model the direct arrivals for both raypath and computed traveltime. Fifty iterations were required for the estimated model to converge to the true model. The resulting P-wave velocity tomogram is shown in Figure 56, resampled to 2.5 ft square cells for display.

One part of the image reconstruction methodology not mentioned in the previous section is how to select a color (or gray) scale. One school of thought is to use fixed intervals of velocity by assigning many colors (gray levels) at one time which highlight all features including artifacts. Although this technique may be great for quality control purposes, it is

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!•4 CHAPTER 4. CASE STUDIES

/'-Wave Velocity Tomogram Well 806 Offset (ft) Well 429

0 100 200 300

300 .......................... 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:i:i.'!:i:'•' :':':' ....'•.... L•:'•:• ..... :i:•:::::i.%-'.•:•.'•:.•.•s::'.!.•.?.:'. .............

..; ....:. ........ ..:.::..::..!:-...:.;::.:.

700 . Saturated Potter "--- .............. ' "-..': .'.

.!?,!•' . ....

Velocity (ft/s) 4000

:i:::":?:i::'•:'

ß .•,'.:: 5546

................ 5794

.. ....... 6500 ß "•':' 7206

9000

FIG. 56. P-wave velocity tomogram reconstruction of the McKittrick Thrust and subthrust.

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not very useful for displaying the results for which the tomographic data were acquired. Instead, one should highlight the tomogram's features for which we have an interest. This generally requires that the person doing the image reconstruction either possess, or work with another person who possesses, some interpretation skills.

With this study we knew that the diatomite, Tulare sand, saturated Pot- ter sand, and desaturated Potter sand were targets of interest. Well data provided the information that the McKittrick Thrust lies at the boundary of the diatomite and Tulare sand while the subthrust at a depth of 800 ft in the 806 well separates the saturated Potter sand from the unsaturated Potter sand. Using this combination of well information with what the P- wave velocity tomogram was telling us, we chose a gray scale to highlight the desired features. Only seven shades of gray were needed to fulfill the objectives for which this tomogram was run. The resulting tomogram display is in reality a lateral extension of the well-log information obtained at the 806-well. If a velocity scale at fixed intervals had been used, the well information correlation with the tomogram would not be apparent and the resulting tomogram would be less of a benefit to the end user, or maybe even confusing. 4


The important objective of this tomogram was to image the saturated Potter sand sealed-off from further gravity drainage by the subthrust intersecting well 806 at 800 ft. Figure 57 is our interpretation of the tomogram in Figure 56, which when compared with the well-log based interpretation in Figure 50, shows that the lateral extent of the saturated Potter sand is much longer than previously thought and does not extend as far in the vertical direction. This interpretation is based upon the correlation that, for the Potter sand, seismic P-wave velocity increases as oil saturation increases.

The tomogram in Figure 56 delineates the McKittrick Thrust which placed the Miocene age diatomite above the Pleistocene age Tulare. The tomogram also suggests the presence of two smaller scale subthrusts which penetrated the McKittrick Thrust fault plane after it was in place. The two subthrusts appear to have throws as small as 40 ft.

Overall, the crosswell seismic tomogram showed a more complex fault system than previously thought and substantially redefined the boundaries of the saturated Potter sand. Such information which determines and ver-

4Even though the tomogram in Figure 56 was resstapled to 2.5 ft square cells for display, the image remains "sharp" because only seven gray levels were used in the display.

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Tomogram-Based Interpretation well 806

315 ft

Dlatomlte

Tulare Sand

McKIttrlck Thrust

well 429

Oft

200 ft

400 ft

600 ft

800 ft

1000 ft

FIG. 57. Our interpretation of the P-wave velocity tomogram in Figure 56.

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4.4. IMAGING SALT SILLS 137

ifies the reservoir configuration enables the development engineers to opti- mally develop the field.

4.4 Imaging Salt Sills

We chose a shallow salt sill problem to illustrate the seismic diffraction tomography imaging technique using the Born approximation presented in Chapter 3. 5 The imaging technique requires a constant velocity medium with imbedded objects of finite extent which scatter seismic energy. This requirement, along with other simplifications, is only partially honored by the salt sill problem. Thus, the resulting tomogram is instructive as to the limitations of the method as presented in this book. Extensions of the methodology to complex situations are discussed in references listed at the end of Chapter 3.

4.4.1 Assumptions and Preprocessing

This case study uses a marine seismic data set collected over two shallow salt sills as depicted in Figure 58. Application of the seismic diffraction tomography methodology using the Born approximation in Chapter 3 requires making some assumptions and preprocessing the data.

We require the object under investigation to have a finite extent. Fig- ure 15 shows a finite-extent object which has a velocity perturbation C(r) confined within the gray area. The region surrounding the gray area is the background medium with constant velocity Co. The finite-extent object in this case study consists of the shallow salt sills and the surrounding sedimentary layers. The overlying sea water is taken as the constant velocity background medium.

Collectively, the salt sills and surrounding sediments do not satisfy the assumptions of the Born approximation as stated at the end of Section 3.2.4. However, the scatterers comprising the water bottom do meet the require- ments of the Born approximation and should image properly. The velocity contrast between the salt and sediments probably violate the weak scattering approximation. The salt sill tops are close enough to the water layer that they can be considered part of a "finite-extent" object and, with the exception of possibly violating the weak scattering approximation, should image properly. The salt sill bases, especially the large salt sill base, are

SFinding the data function Pi(r)qbd(r) for the Rytov approximation (Section 3.2.3) is considerably more involved than determining the data function Ps(r) for the Born approximation (Section 3.2.2). Thus, we use only the Born approximation in this caae study.

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Salt Sill Problem Surface

5000 ft/s 4000 It

Water Bottom • ......... -....... ..... •.:.....:.:.:.:.:.:.:.:.:.:.:.:.: •

FIG. 58. The two salt sills and surrounding sedimentary layers form the finite-extent object which have velocities represented by C(r). The velocity of the water layer is (7• - 5000 ft/s.

comprised of scatterers quite removed from the constant background water layer. The accumulative phase difference between the total and incident wavefields (see Section 3.2.4) is likely to be quite large for these scatterers and a proper velocity image for the salt sill bases is unlikely.

Besides the assumptions stated above, we also assume: (1) constant density in all media, (2) 2-D wave propagation, and (3) multiple free data. We know beforehand that none of our assumptions is one-hundred percent valid. However, we continue so that we may see the result and learn.

The wavefield recorded by a source-receiver pair in a marine seismic record represents the total wavefield Pt(r), which consists of the incident wavefield Pi(r) and scattered wavefield P•(r) as defined in equation (50). Figure 59 shows a source-receiver pair for the marine seismic data case. The recorded incident wavefield travels directly between the source and receiver while the recorded scattered wavefield travels over a considerably longer raypath to get to the receiver. Thus, the scattered wavefield arrives later than the incident wavefield because of the large water bottom depth.

The scattered wavefield P,(rs, rp) is the data function used in diffraction tomography when equation (101) is represented through the Born approximation. Thus, a required preprocessing is to extract the scattered wavefield from the marine seismic records. The water bottom in this case

study is at a depth of 4000 ft or greater. The time lag between the recorded incident wave and the wavefield scattered from the water bottom at the near

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Incident Wavefield vs. Scattered Wavefield Raypath

Source Receiver

• Surface

%, ,/' C O = 5000 IVs 4000 It • / Scattered Wave

\ o.o. ..............

:i•:i:•:•:i:i*i:•i:ii•iii•i•iii?:iii!i::iiiiiii::i?:•ii::i:•!::iii::iiii:•-:•i!i::i::i•:::i::i•i::•ii::!::::::?:?:::::•:::::::•::::::::::•i•?:iiiii•i•i!i•iii•i!i!:• ............... ?:i::iiii? ................ '"'!::11"'•i•?:•, ------- -

FIc. 59.. The raypath for the incident wavefield will always be much shorter than any raypath for the scattered wavefield because of the large water layer thickness.

source-receiver offset is a large 1.6 seconds. Thus, the data function is simply determined by muting the incident wavefield from the total wavefield for each marine seismic record.

The diffraction tomography method also utilizes infinite-line sources and scatterers as defined by the Green's function in equation (62). Infinite-line sources have a cylindrical divergence in which amplitude decreases in a constant velocity media by r-«, where r is distance traveled. However, the observed data are acquired in a 3-D medium with point sources and scatterers which have a spherical divergence in which amplitude decreases by r -1. Thus, to make the observed data "mimic" a cylindrical divergence we multiply the data on each trace by r« = (Cot)« (Co is the constant background velocity) before performing the tomographic processing. As an alternative we could have applied diffraction tomography using the 3-D Green's function in equation (63) in a 3-D model which has a 2-D geometry. Thus, the sources and scatterers would exhibit spherical divergence in our calculations, but at the cost of significantly greater computation time.

4.4.2 Data Acquisition

The data for this case study came from a preexisting 2-D marine seismic survey. The experimental geometry in Figure 60 shows the marine streamer

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Data Acquisition Geometry

246 ;[ 824 ;[

Re. vets

I 2

Next Source Receivers

................ _V V V V I 2 3 120

v v 3 120

FI,_q. 60. Experimental geometry for the marine seismic survey.

cable with 120 hydrophones spaced at 82 ft intervals with a near offset of 824 ft. The air gun source was fired at 246 ft intervals.

We chose backpropagation diffraction tomography for the surface reflection configuration given in Section 3.4.2 to do the image reconstruction. Equation (165) shows that we must take the Fourier transform of the data function P(rs, rp) along the source and receiver profiles to get /5(k,, kp). Since we require that the spatial frequency content for the receiver profile be the same as the source profile, only every third hydrophone trace was used. Thus, the effective receiver spacing was the same as the source spacing of 246 ft. We used 271 sources out of the survey line to provide an adequate aperture for imaging the salt sills.

Five representative source records taken at equal source point (SP) intervals from along the marine seismic survey are shown in Figure 61. Source records $P 823 and $P 995 are taken from over the smaller and larger salt sills, respectively. Source record $P 1081 lies at the edge of the larger salt sill. Several events of interest can be identified on source record $P 995 at the near offsets. The water bottom reflection is found at 1.80 sec-

onds while the top-of-salt and bottom-of-salt reflections lie at 2.36 seconds and 2.92 seconds, respectively. Most of the reflections between 3.60 seconds and 5.30 seconds are identified as multiples and even more multiples are found beyond 5.80 seconds.

Figure 27 shows the hypothetical coverage of a model function in the k= - kz plane for the surface reflection configuration if sources and receivers

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Time (s)

•/ SP 1167 , 't'. I •'• ß •:• i:.. "ß. !1tl•..'.•..•'- I!, . ' ß ' ß .•-' >• '1 F ,' ß .•-•'. : ,i•:•....•>.-

4 '•":•"'.: '• '• .... .. . :: .•_ , :-.":,•>•

• • . •....•:•'•.• .- ß •-• .•-..

..... •-:•. •.•.•

Marine Seismic Records 081 SP 995 SP 909 SP 823

F ..... !' '"•' 2- 'i• "! • i• '•.•'-•.ff•--:•"

.... ,,.>•-.•';•.. ß . .

,•.•••..-._..-..•.-.•- !• •>-.•-<•-";'•'--:.-- I-'"'•/':•-:-'.-.-•"'"--: : "- -" -• •' •..,•./:,.:,,-..• .' . •

1::::'•/'"-.:L•?-': ':'

;..• •..•.. ......•

F•o. 61. Five marine seismic records extracted at equal source point intervals from the marine survey.

could be placed along the surface in both directions out to infinity. In this study we are limited to a streamer cable towed to one side of the seismic source as shown at the top of Figure 62. The resulting coverage of the model function in the kx - kz plane for scatterers located at a depth of 8000 ft is shown as solid arcs. Each solid arc corresponds to a separate source and streamer location. The dashed line represents the locus of ko.q. The associated dotted radial lines point to the source locations used to construct the coverage. Obviously we can expect less resolution and more nonuniqueness for the imaged model function using a streamer cable than from the ideal situation in Figure 27.

4.4.3 Diffraction Tomography Processing

Figure 63 illustrates the flow chart for implementing backpropagation diffraction tomography for a surface reflection configuration as given by equation (165). The total wavefield P•(xo,xr,t ) was recorded on the field records, where t is traveltime, and x• and x r are the distances from a fixed reference to the source and receiver as illustrated in Figure 64. The 2- D Cartesian coordinate system in Figure 64 has its origin at sea level with

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sourcs

! streamer cable

locus of k o s

locus of k o ( s +'• )

F•c•. 62. Coverage in the k= - k• plane for the model function at 8000 ft depth using the source and streamer cable configuration from this study. The solid arcs represent the coverage.

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Diffraction Tomography Procedure

Step I

Pt(Xs,Xp, t)

Mute direct arrivals

Step 2

Ps(x s , Xp, t)

Fourier transform along time axis, t

Step 3 Fourier transform along x s and Xp

Step 4 Backpropagation, equation (3.119)

•(x,z)

FIG. 63. Flow chart for implementing backpropagation diffraction tomography in the salt sill case study.

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Cartesian Coordinate System

• •Y •7 •7 X' 0 source receivers

FIG. 64. Cartesian coordinate system for the salt sill case study. The source and receiver offsets, x• and xp, are measured from a common origin.

the z-direction positive downwards. Both source and receiver depths were set to sea level for implementation of backpropagation diffraction tomography.

Step I (in Figure 63) is the preprocessing step discussed in the previous section in which the recorded incident wavefield Pi(xs,xr,t ) was muted from the total wavefield. Thus, after step 1 the seismic records are assumed to contain only the scattered wavefield P•(xs,xp,t), which is the desired data function for the Born approximation.

In Step 2 we take a 1-D Fourier transform of each trace of the scattered wavefield P•(x•,xp,t) with respect to time, t. Thus, we transform the data function from its space-time domain representation P•(x•, xp, t) to the space-frequency domain representation P•(x•,xp,w), where w represents the angular frequency.

Figure 65 shows the data volumes before and after this 1-D Fourier transform. Before the Fourier transform the x•-axis, xr-axis , and /-axis span the data volume. After the 1-D Fourier transform the data volume is spanned by the x•-axis, xp-axis, and the w-axis. We have effectively reduced the original scattered wavefield into many single-frequency scattered wavefield data sets as depicted in Figure 65. We see that each single-frequency scattered wavefield data set is represented by a plane perpendicular to the w-axis within the P•(x•,xp,w) data volume. The backpropagation diffraction tomography formula in equation (165) can be applied to one or all of

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x

Ps (X s , Xp , t)

Space-time domain representation

Fourier transform along ,I

x esxs'xp' $

Space-frequency domain representation

Single frequency scattered wavefield data

FI(•. 65. The Fourier transform of the scattered wavefield Po(zo,zr,l ) is taken with respect to time t resulting in single-frequency scattered wavefield data Po(z,, zr,• ).

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these single frequency data sets in reconstructing an image. • Step 3 is to apply a 2-D Fourier transform to P•(x•, xp,w) along the

x j-axis and the xp-axis. This operation transforms the scattered wavefield Pj(x•,•cp,w) in the space-frequency domain to the scattered wavefield Po(ko,kp,w) in the wavenumber-frequency domain, which is required by equation (165). The 2-D Fourier transform is expressed by equation (138) which defines k• and kp as the wavenumbers of the Fourier transform along the source line and receiver line, respectively. Note that the depth values of the source d• and receiver d r are set to zero in equation (138) as indicated in Figure 64. We apply equation (138) to P•(x•,xp,w) one frequency w at a time.

Step 4 is to actually apply equation (165) to the transformed data function or scattered wavefield, P•(k•, kp,w). Equation (165) is numeri- cally evaluated at each point on the x - z plane to get the model function M(x, z). A separate model function M(x, z)is determined each time equation (165) is applied to a different angular frequency w. Again, both d• and dp are set to zero. The wavenumber ko in the background medium is set equal to w/Co, where Co is the water velocity (5000 ft/s) and w is the frequency. The wavenumbers k• and kp range from-ko to ko. For each (k•, pair, the corresponding vertical wavenumbers, % and 7p, are computed by

7• - V/ko • - k] and 'rp - y/ko • - kp •, respectively. We obtained a sequence of model functions M(x, z) by applying equa-

tion (165) at frequencies from 12 Hz to 26 Hz. These model functions were then stacked to form one multifrequency model function. The model function was then converted to velocity C(x, z) using equation (56) and the resulting diffraction tomogram is shown in Figure 66.


The velocity tomogram in Figure 66 clearly depicts the water bottom and the presence of two salt sills along the survey line. The larger salt sill is about 31,800 ft wide while the smaller salt sill is about 8,860 ft wide. An apparent 20 percent velocity variation exists within each salt sill. The layer boundaries of the sedimentary layers surrounding the salt sills are not resolved and the gray scale was selected to emphasize the salt sills and water bottom for analysis.

For comparison sake, the same data set was prestack depth migrated, with the results shown in Figure 67. The velocity model for the depth

•Equation (165) does not show an explicit frequency dependence. That is because we stopped writing the angular frequency dependence of the acoustic wavefield and wavenumber after equation (49) in Chapter 3 for the "sake of brevity."

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Salt Sill Diffraction Tomogram

1219 819 0 ' ,,, t ............ I ............ • ............. • ............ t ............

-

Water Layer : ß ...::-;.;.:.;.;.:::.:.:.:.:.:.:.:.:.:-L-'-'-'..-'.-'.;.-'.',;.. :.:..'.......:.:-:-:-;.:.:-:-:.:.:.:."•:;::::-.'-:.:.: .-" :. ...:.:..•-:-:.:.:?•:;::..'::-:-:-:.: '..;-:-;.:-:..'.:-.'•'- - -.'-:-.'-.'-'. .... .: ;. :-:.:.:.;.:.:-:-:-:-:.:-:..'.:-:, ...... :::::::::::::::::::::::::::::::: ..................... :: ........ : ................ ::: ....... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::•.....`• "•.- -.'•....'••i• - :::::::::::::::::::::::::::::::::::::::: i:!:i:!:i:i:!:i:!:!:!:!:!:!:!:i:•.)k.%•i:..•::$:::::.:.:.....•:::!:i:!:!:!:!:!:!:!:i:!:!:!:i:!:!:!:!.i:i:::!:i:!:::!:!:!:!:!:!::•!•:::..i:!:!:i:i•.:.•:i::.:::2i:!:i:i•2!..!.•:!:i:!:!:i:!.•... :"-,•....'-:•:•......•...•..'.•.•:..-:.'c•'::::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:!:!: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::."."::: ".'::i :::::: :: ::::::::::::::::::::::::::-':.'" .•

Depth (kft)

Surface Location (S. P.) 1119 1019 919 719

(ft/s)

1

10000 ft

Velocity Scale 5000

8198

11347

12271

12649

13027

13992

15000

Fio. 66. The diffraction tomogram showing velocities for the water bottom and salt sills along the survey line.

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Prestack Depth Migrated Section Surface Location (S. P.)

1219 1119 1019 919 819 719 0---• ......... t • I • •

"' ' .... S "' "'-' ' ' '"•' "•'•;'•-"•-'•-'• .... ............. ALT SlLL•..•,•.•.:".-' -. ..... •'- .... • •- -•-:' ....... •• -•.•••••• •, ........ .•.• ..•.•.•• .•-•• •-• •-••

Depth (•) 10000 ft

FIG. 67. The prestack depth migrated section corresponding to the velocity tomogram in Figure 66. Processing and display courtesy of Guy Purnell, Texaco.

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migration was derived using an interactive prestack migration focusing- analysis approach. The salt sills are labeled and their boundaries correspond to the strong reflection events surrounding the labels. The water bottom is also clearly visible.

The water bottoms defined in both the diffraction tomogram and the depth-migrated section are in good agreement as to location and shape. The salt sill tops, which are visible on the seismic section, are not well defined in the tomogram. This is possibly because of the low frequencies (12 to 26 Hz) used to construct the tomogram, which may not provide the needed resolution. However, the widths of the salt sills are comparable.

The shapes of the salt sill bases are in very poor agreement. Obviously the depth-migrated section portrays a more correct picture than the tomogram for the salt sill bases. The invalid assumption of a finite-extent object for the deeper salt sill bases is causing their images to be both mispositioned and degraded on the tomogram. This can be shown using the first line of equation (146) in a simple computation.

First, we focus our attention on an infinitesimally small part of the base of salt reflector where the salt-sediment interface is horizontal. This

isolates a single scatterer of the incident wavefield at location (zo, zo), which we define in terms of a model function as M(z, z) = 6(z- zo)6(z - zo), where the deltas are Dirac delta functions. Next we place the source and receiver directly above the scatterer with the coordinates (x0 = Xo, do = O) and (ze = zo, d e = 0), respectively. In doing so we have insured that the scattered energy from the selected scatterer and the specular reflection from the horizontal interface approximately share the same vertical raypath and traveltime.

Second, given the set-up from the last paragraph we must evaluate the other parameters which go into equation (146). For vertical raypaths Figure 26 indicates that the source wavenumber vector has components (7, = ko, ko = 0) and the receiver wavenumber vector has components (7e - ko, ke = 0). Substituting these components into equation (144) gives the wavenumber components kr = 0 and k• = -2ko required by equation (146).

Third, we substitute the model function for the selected scatterer along with the information derived in the previous paragraph into equation (146) and integrate. The result is

- I (2), e(t,o - o, o; t,,, - o, o) - where 2Zo is the round-trip raypath length between the selected scatterer and the zero-offset source-receiver pair. We restate the last equation by

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substituting in equation (52) for k0 which gives,

- o. o; - o. 0) lej(2zo/C'o ) ß

Here 2zo/Co is just the traveltime to of the associated diffraction event (or specular reflection event) on the seismic trace, or to = 2zo/Co. It states that the incident energy travels to the selected scatterer and the scattered energy travels to the receiver at the constant background velocity Co. For a finite-extent object this propagation model holds, but the salt sill bases obviously cannot be considered as part of a finite-extent object since they are so distant from the water layer (our constant velocity background). Replacing the propagation velocity of salt with that of water cannot yield a properly imaged model function as we will now demonstrate.

SP 995 in Figure 61 shows the specular reflection from the base of the large salt sill at time to = 2.92 seconds. Even though Figure 67 shows some reflector dip at SP 995, we will assume the scattered energy's traveltime and the specular reflection 's traveltime are so close as to be the same and that the raypaths are nearly vertical. Using to = 2.92 s and Co = 5000 ft/s we get an image depth on the tomogram of Zo = 7300 ft. The depth-migrated section in Figure 67 shows the large salt sill's base at SP 995 at a depth of 10000 ft, a 2700 ft difference which is easily greater than can be accounted for by reflector dip. The diffraction tomogram in Figure 66 shows a high velocity anomaly near zo = 7300 ft which is probably the image location of the large salt sill's base. Thus, this example demonstrates that scatterers outside of an acceptable finite-extent object are mispositioned and result in a deteriorated image.

Finally, we ask, "What can be done to get a correct tomographic image?" The immediate answer is to use a variable background velocity instead of the constant velocity assumed in Chapter 3. We would want a variable background velocity which doesn't scatter a significant amount of energy and is capable of properly imaging the model function in space. Then the higher frequency perturbations to the variable background velocity would be smaller in size and magnitude than for a constant background velocity, making the Born approximation more acceptable. Several of the references at the end of Chapter 3 address the application of a variable background velocity.

4.5 Suggestions for Further Reading Gibson, Jr., R. L., 1994, Radiation from seismic sources in cased

and cemented boreholes: Geophysics, 59, 518-533. Theo-

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retical investigation into the radiated energy by volumetric (air gun), radial stress, and axial stress (clamped vibrator) sources located in cased and cemented boreholes. The volu-

metric source is the only source which does not have a nonlinear frequency dependent radiation pattern.

Howlett, D. L., 1991, Comparison of borehole seismic sources under consistent field conditions: Expanded Abstracts for the Society of Exploration Geophysicists' Sixty-first Annual International Meeting and Exposition, Nov. 10-14, Houston, Texas. Compares data from explosive, clamped-vibrator, air glun, and cylindrical-bender-crgstal sources taken at Tezaco 's geophysical test facility in Humble, Texas.

Meredith, J. A., ToksSz, M. N., and Cheng, C. H., 1993, Sec- ondary shear waves from source boreholes: Geophys. Prosp., 41,287-312. Borehole sources submerged within a liquid can create tube waves. When the tube wave velocity is greater than the formarion's S-wave velocity, a reach wave (converted S-wave) is generated by the tube wave which may in- terfere with other events recorded in the receiver well.

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Appendix A

Frequency and Wave numb er

Frequency and wavenumber are two basic terms used extensively in this book to describe wave propagation. This appendix is intended to clarify their definitions.

A.1 Frequency

Assume a source emits the sinusoidal signal in Figure A.1. The particle displacement B(t) of the medium at the source is shown as a function of time, t. The signal repeats itself every 4 ms with the same amplitude after a lapse of time T called the "period." Another measure of the signal's cycle is the frequency f which gives the number of cycles the signal goes through per unit time, or

1

f = •, (A-l) where the appropriate unit is hertz, abbreviated "Hz", which stands for "cycles per second."

Many times we will represent a sinusoidal signal in terms of the projection of a rotating vector onto some axis. Thus, we might represent the particle displacement in Figure A.1 by the equation,

B(t)- [B[sin(-•t-•b), (A-2) 153

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154 APPENDIX A. FREQUENCY AND WAVENUMBER

I B(t) ß

I I L*

FIG. A.1. Particle displacement B(t) is plotted as a function of time, t, for a sinusoidal signal with a period T- 4 ms. The associated frequency is, f - 1/T- 250 Hz. Angular frequency is defined as co - 2•rf - 2•r/T.

where I B I is the magnitude of the rotating vector B(t) and •b is the phase of the signal at time t - 0. • Since the vector B(t) rotates through 2•r radians every period T, we call 2•r/T in equation (A-2) the "angular frequency" which is written as2

co - •- = 2•'f. (A-3) In Appendix B you will see that actual signals can be thought of as a

sum of many sinusoidal signals of different frequency, where each can have a different magnitude and initial phase. The Fourier transform is used to decompose an actual signal into its angular frequency components I B(w) I and •b(w).

A.2 Wavenumber

When we defined angular frequency co the observation point of the signal was kept fixed in space (at the source in Figure A.1) and we studied the signal's properties in time. In this section we arrive at similar concepts by observing the signal throughout space at a fixed time. Figure A.2 shows a sinusoidal signal traveling in either the +X of-X direction with a wavelength, ,•, the distance required for the waveform to repeat itself.

1 The phase of the signal in Figure A.1 is zero at time t = 0. 2 Both w and f are commonly referred to as "frequency." The context of the equation

will tell you which type of frequency.

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A.2. WAVENUMBER 155

I B(x) I I L ø

FIG. A.2. Particle displacement B(x) is plotted as a function of location, x, for a sinusoidal signal traveling either in the +X or -X direction with a wavelength of A = 4 m. The associated wavenumber is, 1/A• - .25/m. The angular wavenumber is defined as k• -- -l-2•r/A where the sign chosen is decided by the direction of wave propagation.

Wavenumber is analogous to frequency f, but is defined in terms of the periodic distance of the signal A as q-1/A. The sign chosen for the wavenumber is decided by the direction of wave propagation. Most of the time we work with angular wavenumber,

2•r

k• - +-•-. (A-q) We generally refer to the angular wavenumber as just the wavenumber and let the context of the equation define the type of wavenumber.

Unlike time, wave propagation in space can be in many directions and actually must be defined in terms of a vector instead of a scalar. In higher order dimensions of space equation (A-4) must be written as a vector,

27r,, k = Tg, (A-5)

where • is a unit vector pointing in the direction of propagation. Fig- ure A.3(a) shows a sinusoidal wave propagating along the •, direction in a 2-D space with wavelength X. We plot the wavenumber for this wave in the k• - k, plane in Figure A.3(b). If ko = 2•r/A represents the magnitude of the wave's wavenumber, then k in equation (A-5) can be decomposed into its vector components,

k - kog - k•,i+k,i, (A-6)

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156 APPENDIX A. FREQUENCY AND WAVENUMBER

(a) (b)

FI•. A.3. (a) A sinusoidal wave propagates along the •; direction in a 2- D space with wavelength A. (b) The wavenumber of the propagating wave in Figure A.3(a)is k = (2•r/A)•;. The components of the wavenumber are k• and k•, both of positive value in this case.

where the values of k', and kz have signs determined by the direction of wave propagation; both positive in Figure A.3(b).

Lastly, a single frequency wave will travel one wavelength in one period. We may determine a special velocity of propagation for this wave called the "phase velocity" defined by,

= = I kl = (A-7) Note that the phase velocity is defined for a single frequency. If a signal is composed of sinusoids of many different frequencies, then the velocity of propagation of that signal may not be the same as the phase velocity, but a different velocity called the group velocity, defined as

V = Ok,, •: + • •' (A-S) In this book we use o., - Ck so that V - C and we assume no dispersion.

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Appendix B

The Fourier Transform

Seismic tomography frequently utilizes Fourier transforms of functions. Therefore, an understanding of the concept is important to understand the material presented in this book. This appendix is intended to refresh your memory of the Fourier transform. You might want to review Appendix A since we make use of the concepts of frequency and wavenumber in our discussion.

If h(t) is a function of time, then we represent its temporal Fourier transform by,

FT[h(t)] :• •(w), (B-i)

where w is the temporal frequency. The inverse Fourier transform is represented by,

We refer to h(t) and h(w) as a Fourier-transform pair. Similarly, we may take a spatial Fourier transform of a function g(x), where x is a spatial coordinate, and represent the operation by,

FT[g(x)] • •(k•), (B-3)

where k• is the spatial frequency, or angular wavenumber •, along the x- direction. As before, the inverse Fourier transform operation is represented by,

rT-•[O(k=)] • g(x). (B-4) • Most of the time the n•es temporM frequency •d •g• waven•ber •e short-

ened to just frequency •d wavenmber, respectively.

157

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158 APPENDIX B. THE FOURIER TRANSFORM

Before presenting the actual Fourier transform equations used to compute the operations implied by the above equations, we give a short review of the Fourier series and exponential Fourier series to provide a transition; since most everyone is familiar with the Fourier series. Then, the Fourier transform of continuous functions is discussed followed by the Fourier transform of sampled functions. The last section gives some special transforms used in this book.

B.1 Fourier Series

A periodic function f(x) of length 2L can be written in terms of a series of cosines and sines provided it contains a finite number of discontinuities and a finite number of maximum and minimum values. The series is called a Fourier series and is represented by,

The series coefficients ao, an, and b• are given by,

1/_L = - f(x)dx, (B-6) ao L •;

1/_ œ = -- f(z) cos dx (B-7) a, L L \ L ' and,

- fix)sin (-•-/dx (B-8) respectively. Equation (B-6) shows that ao/2 is just the average value of f(x) over the interval I-L, L], commonly called the DC shift. Equation (B- 7) gives the same definition as equation (B-6) for ao when n = 0 and will be used henceforth. Also, equation (B-8) requires bo = 0 for any f(x).

For our purposes, equations (B-5), (B-7), and (B-8) are best rewritten in terms of spatial frequency, or wavenumber 2, defined as 2•r radians per wavelength. The longest wavelength for f(x) is 2L; determined by

2Note that we could equally as well present this discussion in terms of a time coordinate t and its frequency w .

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B.2. EXPONENTIAL FOURIER SERIES 159

setting n = 1 in equation (B-5). The corresponding wavenumber or fundamental frequency is 7r/L. Higher valued wavenumbers (shorter wavelengths) are found by multiplying the fundamental frequency by n where n = 1, 2, 3, ...;, n = i gives the fundamental frequency. Thus, we now write the wavenumber along the x-axis as kn-- n7r/L and the Fourier series in terms of wavenumber as,

f(x) - -•-+ anosknx+bnsinknx

where the series coefficients anand bn are defined by,

- cos (B-10) (In • L and,

- f(x)sinknxdx. (B-11)

B.2 Exponential Fourier Series Here we rewrite the Fourier series of the previous section in terms of

exponentials, thus getting us one step closer to the Fourier transform which also uses exponentials. The key equation permitting this step is Euler's formula,

eJknx = cosknx +jsinknx, (B-12)

where j - x/Z-1. We recognize that the cosine is an even function and the sine is an odd function and use equation (B-12) to write,

ejkn x + e-jkn x cosknx = -, and (B-13)

2

ejkn x _ e-jkn x sinknx = , respectively. (B-14)

Substituting equations (B-13) and (B-14) into equation (B-9) and re- grouping the terms with respect to the sign of the exponential gives,

-jknx ß (B- 15)

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Equations (B-10) and (B-11) show that a, - a_, and b, - -b_,, respectively. Applying these relationships to the second series on the right-hand side in equation (B-15) and redefining the summation from n - -1 to -c• gives the exponential Fourier series,

f(x) - Z c,•eJk,-,x, (n-16)

where c,., - (a•- jb•)/2 and k• - n•/L. Substituting equations (B- 10) and (B-11) for a• and b• in the definition of c•, along with using equation (B-12), gives the integral equation for the series coe•cients ca,

-- f(z)e-Jk•xdz, (B-17) C• • L for n - 0,•1,•2, •3,..., •c•.

B.3 Fourier Transform- Continuous f(x) The Fourier transform is intended to operate on nonperiodic functions

over an infinite range. Thus, we can no longer restrict x to the range I-L, L] as defined for the exponential Fourier series. However, equations (B- 16) and (B-17) can be utilized in extending z to infinite limits. First, we substitute equation (B-17)into equation (B-16) resulting in,

or

= .•(k,•)eJk'• x __1 where 2L'

_t, _jk,., X,dx • f(:rt)e . L

The frequency interval between successive k• is given by,

Ak k•+• k• (n + 1)• •r •r We rewrite this equation in a convenient form,

1 Ak

2L 2•r

(B-18)

(B-19)

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B.4. FOURIER TRANSFORM- SAMPLED F(X) 161

Taking the limit of equation (B-20) as L--• c• allows us to write,

lim (•L) - dk (B-21) œ--,oo 271'

and to write kn as a continuous variable k. Now, taking the limits of equations (B-18) and (B-19) as L -• oo and using the result from equation (B- 21) yields the Fourier transform equations for a nonperiodic continuous function,

I I•(k)eJkzdk (B-22) f(:) - oo ' and

i>(k) - f(x)e-Jkx dx. (B-2a)

Equations (B-22) and (B-23) enable us to compute one of the Fourier transform pairs given the other. Equation (B-22) is the inverse Fourier transform operation represented by equation (B-4) and equation (B-23) is the Fourier transform operation represented by equation (B-3).

B.4 Fourier Transform- Sampled f(x) The use of digital computers to carry out computations requires that

we sample the continuous function f(x). We may take a sample from the continuous function f(x) at intervals of Ax giving the discrete samples: ß .., f-2, f- •, f0, f•, f2,. ß., where the subscripts indicate the sample number. If the sample number is n, then the position of the sample is x - nAx.

However, a sampled function can no longer contain wavenumbers (or frequencies) out to 4-00 as demonstrated by the integral limits for the continuous function f(x) in equation (B-22). The shortest wavelength represented by a sampled function is 2Ax which gives the highest possible wavenumber (or frequency),

knyq = Az' (B-24) referred to as the Nyquist frequency. a Note that this also applies to tem- porally sampled functions.

a The Nyquist frequency has a wavelength or period which is sampled by two points; or we may state that the shortest wavelength or period in a sampled signal must contain at least two sample points.

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In sections of this book where we are referring to a sampled function, you will see the limits on the inverse Fourier transform equations written as :t:•r. Equation (B-24) shows that these frequency limits correspond to a unit sample interval, Az = 1. A unit sample interval is commonly used in digital computations to avoid multiplying by Az, which becomes nothing more than a scale factor. Thus, you will see the Fourier transform equations written as 4

1 p(k)eJkXdk (B-25) f(x)- • , , and

P(k) - f(x)e-Jkx dz, (B-26)

when f(x) and •(k) are sampled functions. Again, the clue to f(x) being sampled is the noninfinite frequency limits. Of course, in actual computer computations we usually do not use equations (B-25) and (B-26) directly. Instead, the Fast Fourier Transform (FFT) method is utilized.

B.5 Uses of Fourier Transforms

From equaLion (B-26) we see that F(k) is generally a complex valued function, called the complex spectrum, which may be written as,

P(k) - •Re{P(k)} + jlm{P(k)}, (B-27) where Re and Im designates the operation of taking the real and imaginary parts of P(k), respectively. Figure B.1 shows the location of one point of the complex spectrum in the complex plane; so called because the real part of the complex function is plotted on one axis and the associated imaginary part on the other axis. The figure demonstrates that the complex spectrum can be also described in terms of polar coordinates. In polar form we can write the complex spectrum as,

P(•) - Ip(•)leJ•(•), where I P(•) I is called the amplitude spectrum given by,

]#(•)] - V/ee{#(•)} •+t,•{#(•)}•,

(s-28)

4Mar•y times geophysicists cha•nge the sign in the exponential of the forward and inverse Fourier transforms when the transform involves time. This is done so that we stay consistent with the physics, that is, a wave traveling in the +x direction is described

by A(w, kx)eJ(kxx- wt) and not A(w, kx)eJ( kxx + wt).

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B.5. USES OF FOURIER TRANSFORMS 163

Complex Plane

Im {F(k)}l (Re {.•(k)} , lm {•(k)} ) >

Re {F(k)}

FIG. B.1. Complex plane with an arrow directed to the point (Re{17'(k)},Im{17'(k)}). The point in polar coordinates is represented by the magnitude I F(k) I defined by equation (B-29) and the phase (I)(k) defined by equation (B-30). The magnitude is called the amplitude spectrum and the phase is called the phase spectrum when each is plotted separately as a function of k.

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and (I)(k) is called the phase spectrum given by,

(I)(k)- tan -1 Im{F(k)} (B-30) • ß

The amplitude spectrum gives the magnitude of a particular sinusoid at frequency k while the phase spectrum gives its shift in space or time. Most often we are interested in the frequency content of a signal and a plot of the amplitude spectrum serves this purpose.

Throughout this book we work in tl•e Fourier domain when it simplifies derivations. In addition to the above concepts of amplitude and phase spectra, we encounter three other applications of the Fourier transform' the 2-D Fourier transform, the Fourier transform of the time derivative, and the Fourier transform of eJkox.

Suppose we wish to take the Fourier transform of a function f(x,z) with respect to the spatial variables x and z. This operation is called a 2-D Fourier transform. We begin by taking the Fourier transform of jr(x, z) with respect to x, or

P(k•, z) - f(z, z)e-Jk•zdz, (B-31)

where the wavenumber along the x-axis is denoted by ks. Next we apply a Fourier transform to equation (B-31) with respect to z, or

•(k,,,kz) - •(k,:,z)e-JkzZdz, (B-32)

where the wavenumber along the z-axis is denoted by kz. Substituting equation (B-31) into equation (B-32) gives the definition of a 2-D Fourier transform,

•(k•, k,) - f(x, z)e-J(k•x + k•Z)dxdz ' (B-33)

Similarly, the inverse 2-D Fourier transform may be defined as,

f(x,z) 471-2 /_•o /_• •(k•, k•)eJ(k•x + k•Z)dk•dk• ' (B-34) Time derivatives are simpler in the Fourier domain which is one reason

many differential equations are solved in the frequency domain. Look at the inverse Fourier transform,

1 f'(w)e-Jwt dw, (B-35) =

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B.5. USES OF FOURIER TRANSFORMS 165

where t is time and w is frequency. The first time derivative of equation (B- 35) is,

ef(t) dt • /_•o _ -J•t = -• j.•(•)• a•,

= FT-•[-j•P(•)]. (B-36)

A second-order derivative is found by taking a time derivative of equation (B-36) or,

dt 2

(B-37)

Thus, an nth order derivative is defined,

dtn rr-•[(-i•)-p(•)]. (B-38)

We simply multiply P(w) by the appropriate factor s (-jw)" and take the inverse Fourier transform.

The last item to note is the Fourier transform of the exponential e-J kox, which comes up many times in diffraction tomography. We determine this in a roundabout way. We use the Dirac delta function defined in Appendix C in this derivation. Take the inverse Fourier transform of .b(k)- 5(k - ko), where the Dirac delta function is located at wavenumber ko in the spatial frequency domain. By equation (B-22) we find that,

f(x) = -- 5(k- ko)eJkxdk, 2•r

= l,j•o•. 2•r

(B-39)

Thus, the Fourier transform of eJ kox must be 2•rS(k- ko).

SUse (+jkx) n and (+jk,) n for spatial derivatives.

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Appendix C

Greens Function

Equation (60) in Chapter 3 on seismic diffraction tomography describes the propagation of the scattered wavefield P0(r) at a constant background velocity when inhomogeneities scatter both the incident wavefield Pi(r) and existing scattered energy P0(r). We rewrite equation (60) here for your reference as

Iv • + •o•]•,(•) - •o•U(•)[•,(•)+ •,(•)1. (C-l)

Through the use of Green's functions we end up with an integral solution to this equation given by equation (64), the Lippmann-Schwinger equation, which we rewrite here for your reference as,

?o(,,) - -• f •(,, I •")•(,?)[?,(,?) + •0(•')]a•'. (c-•) G(rl rt) is the Green's function and the integral is taken over a plane in 2-D space or a volume in 3-D space.

The intent of this appendix is to provide you with an intuitive feeling for Green's functions. With this understanding you will know how to immediately write down an integral solution to any inhomogeneous differential equation with constant coefficients, such as the solution to the partial differential equation (C-l) given by the integral equation (C-2). We do NOT go over techniques for determining the Green's function G(r I r•) which are readily found in many texts with chapters devoted to the subject, such as referenced at the end of this appendix. However, an example problem is worked later on to give you some idea of how one may solve for G(r I r•) .

The approach we take for conveying the ideas of Green's functions is through the concepts involved with filter theory. We do this for two reasons:

167

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168 APPENDIX C. GREEN'S FUNCTION

LINEAR

OPERATOR

G t

FIG. C.1. Gt is the impulse response of the linear operator given the input discrete impulse 50 at time sample t - 0.

(1) its easy to see, and (2) many of you are already familiar with filter theory, especially the concept of convolution. We will look at the discretely sampled case first and then extrapolate to the analog case.

C.1 Filter Theory

In filter theory we can input a discrete impulse 5• of unit height at sample t into a linear operator or system which gives an output, called the impulse response Gr. Figure C.1 shows a plausible impulse response of a linear operator when a unit impulse 5o is input at sample ! - 0. • A linear system is time invariant when the impulse response G• is found at the same time lag relative to the time sample of the input impulse. For example, Figure C.1 shows an input impulse 50 with an impulse response Gt starting at time sample 0. If we had used an impulse 5•, then the impulse response G• would have begun at time sample 5 which may be written as Gt-•.

The impulse response G• in Figure C.1 can also be scaled. We may multiply the unit impulse input 5o by a scalar f0 which results in the impulse response G•-0 being scaled by f0 as shown in Figure C.2(a). The t-0 in the subscript of G•-0 indicates the delay of the impulse response Gt because of a delay in the input. Here no delay occurs and the output foG• is the response to the input fo5o, where f0 ---1.

We can scale the impulse input 51 by a scalar fl - 1 which results in the linear operator response flG•-i - G•-i in Figure C.2(b). The subscript t- I indicates that G• is delayed by one time unit in accordance with the time invariance principle stated earlier. Similarly, Figure C.2(c) shows the impulse input 52 scaled by f2 - -1/2 results in a linear operator response

1 We assume a unit interval, At = 1, between samples.

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C. 1. FILTER THEORY ' 169

-1 •o o

1•1

0 !

-1/2 õ 2

01--1 ft llU-

OPERATOI•.

(a)

LINEAR

OPERATOI•

•)

LINEAR

OPF•ATO•

(c)

LINEA•

OPF•ATOI•

(d)

fog t.o

t

+

1Gt.1

o t

+

f2 Gt. 2

!øi Pt

FIG. C.2. Illustration of the principle of superposition for a linear operator. The input consists of a series of scaled, time-shifted impulses given by ft -- j•050-}'f151 -}' j•252, where fo = -1 in (a), fl ---- 1 in (b), and/2 = -1/2 in (c). The resulting output to f• is the sum of the responses in (a), (b), and (c): P• - foG,-o 4-fxG,_x + f2G,-:• shown in (d). This is just an example of convolution of an input signal ft with an impulse response of a linear filter

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Figure C.2(d) is the sum of the inputs and outputs from Figures C.2(a- c). The discretized input is thought of as a sum of scaled impulses given by,

1)5:•. ft -- fo•o q- fl•l q- f2•2 -- (-1)50 + (1)51 q- (-• (C-3)

The linear operator's response to the input function ft is given by,

---- foGt-o q- flGt-1 q- f2Gt-2

= (-1)G,_0 q- (1)G,_l q- (-• (c-4)

Equations (C-3) and (CL4) combined illustrate the principle ofsuperposition for a linear operator. That is, the same output results whether a linear operator acts on the entire input as depicted in Figure C.2(d), or on each individual component of the input as shown in Figures C.2(a-c) and the results summed.

Generalizing equation (C-4) we have,

This equation is the well-known convolution equation for two discretized signals. Here the convolution is between the input function .It and the impulse response of the linear operator Gt which gives the response of the linear operator Pt. If the above functions are continuous, then the sum in equation (C-5) becomes an integral or

P(t) - • f(t')G(t - t')dt', (C-6) where dr' is explicitly written in place of At' in equation (C-5) which was assumed equal to one.

C.2 PDE's as Linear Operators

The partial differential equation (C-l) can be cast in the same light as the linear operator in filter theory. The source term, ko•M(r)[Pi(r)+ P,(r)], is analogous to the input to the linear operator. The linear operator is the bracketed part of the function on the left-hand side, IX72 + ko•], and the solution to the differential equation, P,(r), can be thought of as the output from the linear operator.

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C.2. PDE'S AS LINEAR OPERATORS 171

-6 (r-r') • •2 + k 2 ] • G(rlr' o I ) 2 ] G(rlr') - -5 (r-r') [•2+k o

FIG. C.3. The Green's function is the impulse response of a linear differential equation operator to a source term (input) given by a negative Dirac delta function. Compare this with the impulse response in filter theory depicted in Figure C.1.

As in the previous section, the first step is to determine the impulse response of the linear operator to an impulsive input. For the partial differential equation we use a negative Dirac delta function 2 -5(r- r •) for the impulsive input and the Green's function G(r- r •) for the impulse response to the negative Dirac delta function. Thus, using the analogies in the previous paragraph, equation (C-1) is rewritten as,

IV 2 + ko•lG(r I r') - -5(r-r'), (C-7)

to get the impulse response, or Green's function, G(r I r') = •(r- r'). Note that the space vector r is the independent variable here instead of- time used in the previous section on filter theory. Figure C.3 summarizes the analogy made between the differential equation operator and the filter theory operator in Figure C.1.

The Green's function solution G(r I r •) to equation (C-7) is the impulse response to the negative Dirac delta function. Just as with filter theory, we may weight the negative Dirac delta function, say -f(r')5(r- r'), on the right-hand side of equation (C-7) which results in the solution (output) f(r')G(r Jr'). Using the principle of superposition shown in Figure C.2, the resulting response is just the integration (sum) over all output response

2The Dirac delta function 6(x) has ax• infiaitesi• width, infinite height, and an

area equal to one at x = 0. It is defined as f_+• S(x)dx = 1. When combined with another function, f_+• !(z)a(z- Xo) = l(Zo), where the Dirac delta function is located at 3• '-' 3• o.

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components from the linear operator, or

: - fr, (c-s) where the negative sign in front of the integral is present because the Green's function is the impulse response to a negative Dirac delta function. Equa- tion (C-8) is similar to equation (C-6) except for the negative sign and the independent variable.

Setting f(r') in equation (C-8) to the right-hand side of equation (C-l),

-

becomes the source term (input to the linear operator) in equation (C- 8) and we get the integral solution given by equation (C-2). Thus, the integral solution to equation (C-l) is just the negative of the convolution of the source term (input) f(r) with the Green's function (impulse response), analogous to filter theory. The r- r • part of the Green's function (see equations (C-9) and (C-10) below) gives the "lag" in the impulse response as a result of the weighted negative Dirac delta function being located at a position other than r • = 0. This is a space-invariant property equivalent to the time-invariant property of the linear operator in the previous section on filter theory.

Chapter 3 on seismic diffraction tomography makes extensive use of Green's functions. However, close inspection reveals that only two Green's functions are actually ever mentioned, both are solutions to equation (C-7). The 2-D solution to equation (C-7) is exclusively used in Chapter 3 and is given by,

J Ho(1)(ko [r r' a(rl') - - I), where Ho (1) is the zero-order Hankel function of the first kind. Here the negative Dirac delta function represents an infinite-line seismic source or an infinite-line scatterer at r' which causes a cylindrically shaped seismic disturbance that is determined at an infinite-line field location r. Both the

infinite-line source and infinite-line field location are perpendicular to the plane representing the 2-D space.

A 3-D space solution to equation (C-7) is given by,

ejkolr-r'l G(rlr') = 4•rlr-r'] ' (C-10)

This equation is not used in Chapter 3 since most data acquisition schemes, such as crosswell seismic, are geared to 2-D space or cross-sectional studies

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C. 3. GREEN'S FUNCTION EXAMPLE 173

rather than volumetric investigations. The 3-D space solution represents point seismic sources or point scatterers at r t which cause a spherical seismic disturbance that is determined at a point field location r.

C.3 Green's Function Example

In this section we find the integral solution of a simple 1-D differential equation to reinforce the ideas presented in the last section. The method of determining the Green's function here is just one example of many and we refer you to the references at the end of this appendix for other methods.

The differential equation is,

[d-••- a2]p(x) - f(x), for -cx> < x < cx>. (C-11) The first step is to find the impulse response, or Green's function, G(z) to a negative Dirac delta function source term located at x - 0. The above equation is rewritten, a

dx - aV']G(x) - -5(x). (C-12) We solve for G(x) by taking the Fourier transform of each term in equation (C-12) as discussed in Appendix B which gives,

[(jk) 2 - aV']0(k) - -1, (C-13)

where k is the wavenumber and j - v/Z'-•. Solving for (•(k) gives the Green's function in the wavenumber domain,

1 (C-14) = + Taking the inverse Fourier transform of the last equation gives the Green's function for a negative impulsive source term at x - 0,

G(x) - 2a (C- 15) If the negative Dirac delta function were located at x - x', then the Green's function would be written with a spatial lag,

G(x I x') = 2a ' (C-16) 3Assume no lag, x t = O, in the Dirac delta function at this point, and introduce

the lag when doing the convolution (sum of weighted output from the linear operator) between f(x) and G(x).

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174 APPENDIX C. GREEN 'S FUNCTION

which is used in the integral solution in equation (C-17). We know the integral solution to equation (C-12) is just the negative of

the convolution between the Green's function G(x) and the source function f(x) or,

p(x) = -/-+2 f(x')G(z I x')dz ', (C-17) where the x - x' in the Green's function is just the spatial lag as a result of the weighted negative Dirac delta function being located at x = x'. Substituting equation (C-16) into equation (C-17) gives the integral solution to our example,

2a dx'. (C-18)

We can now solve for p(x) by integrating equation (C-18) once the source term f(z) is defined in equation ((3-11). As a quick check try a negative impulsive source located at x'= 0 given by f(x')= -5(x•). Your solution should be equation (C-15).

C.4 Suggestions for Further Reading The following references cover the computation of Green's functions in

detail.

Arfken, G., 1970, Mathematical methods for physicists, 2nd edi- tion: Academic Press, Inc.

Courant, R., and Hilbert, D., 1953, Methods of mathematical physics: John Wiley and Sons (Interscience).

Morse, P.M., and Feshbach, H., 1953, Methods of theoretical physics: McGraw-Hill.

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Index

acoustic wave equation 46 Helmholtz form 46

acoustic wave field 46

algebraic reconstruction technique 33 amplitude spectrum 162 angular frequency 154 ari thmetic reconstruction technique (see

Aa) ART 33, 41

compared with SIRT 37 nonlinear aspect 38

background velocity C'o 48 backprojection ray tomography 20, 40

CAT scan 22

reconstruction formula 21

sununary 21 backpropagatlon diffraction tomography

87

crosswell configuration 88 Jacobian 87

procedure for 143 surface reflection configuration 90 vsp configuration 89

borehole survey 99 Born Approximation 51

Rytov approximation comparison 56

cell size 108

color scale 133

complex spectrum 162 polar form 162

computerized axial tomography 22 convolution equation 170

continuous form 170

crosswell seismic

advantage over well logs 4 benefit of 3

crosswell seismic gathers 106

data acquisition crosswell configuration 2

data function 11

observed pobs related to true model function

M true 23

predicted P or prre related to estimated model func-

tion M or M est 23 vector form 27

DC shift 158

diffraction tomography 5 2-D Green's function 50

3-D Green's function 51

constant density assumed 48 data function

Born approximation 52, 138 Rytov approximation 56

generic data/model relationship 59 illustrated 137

incident wavefield Pi(r) 47 model function 45, 49 scattered wavefield Ps(r) 47 total wavefield

Born approximation 48 Rytov approximation 53

use with ray tomography 6 wavelength consideration 5 when to use 45

Dirac delta function 171

direct arrival 101, 101 direct-traxmform diffraction tomography

85

steps for implementing 86 direct-transform ray tomography 16, 17,

39

Enhanced oil recovery 95 EOR 95

estimated model vector 28 Euler's formula 159

175

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176 INDEX

first arrival 101

forward modeling defined 24

direct arrival 102

discrete formulation 25 first as'rival 102

formulated in a continuous domain 23

matrix form 26

need for 23

raypath lengths 27 Fourier series 158

exponential form 160 in terms of wavenumber 159

Fourier transform

2-D 164

nonperiodic continuous functions 161

of e-J køx 165 operation symbols 157 representation 161 sampled function 162 time derivatives 164

transform pairs 157 frequency 153

generalized projection slice theorem 58 crosswell configuration 68 statement of 70

surface reflection configuration 82 vsp configuration 76

gray scale 133 Green's function 171

2-D solution to wave equation 172 3-D solution to wave equation 172 example 173

group velocity 156

head wave 101 hertz 153

hyperplane 30

image function 11 impulse response 168 incident wavefield 47

integration notation 51

Jacobian 87

crosswell configuration 88 surface reflection configuration 90 vsp configuration 89

Kaczmarz' method 26, 40

advantages of 28 flow chart 29

Laplacian operator 46 linear inverse problem 13

generalized inverse operator 27

ill-conditioned matrix 28

linear system principle of superposition 170 scaling 168 time invariance 168

Lippmann-Schwinger equation 51 Born approximation 51 linearized 52

nonlinearity 51 lithology interpretation

of P-wave tomogram 120

magnitude 154 McKittrick Field 125

medical tomography configuration 10 data function 11

Midway Sunset Field 96 model function 11, 49

cell size 108

diffraction tomography 45, 49 discrete 24

estimated M or M est

related to predicted data function P or ppre 23

incremental update A iM• ' 30 initial estimate 28, 108, 109 new estimate M(new) est 28 .true M true

related to observed data function pobs 23

vector form 27

nonunique 70 Nyquist frequency 161

object function 11 observed data vector 28

partial differential equation linear operator 170

period 153 phase 154 phase spectrum 164 phase velocity 156

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INDEX 177

plane wave decomposition 60 porosity tomograrn 122 predicted data vector 28, 28 projection 13 projection data '1 projection slice theorem

defined 16

essence of 14

ray density 110 ray tomography 5

assumptions 9 backprojection 21 direct-transform method 17

wavelength 9 wavelength consideration 5 with diffraction tomography 6

ray tracing direct arrival 102

first arrival 102

SIRT information 110

receiver

orienting horizontal components air gun source 102 clamped vibrator 133

reservoir interpa•tation based on tomogram a•ad well in-

formation 136

based on well logs 126 resolution 70

Rytov approximation 56 Born approximation comparison 56 complex incident phase function

complex phase difference function

complex total phase function 53 Rytov-data function 55

note on data function 137

salt sill problem 137 diffraction tomograxn 147 prestack depth migrated section 148

scattered wavefield 47

observed 52

seismic data

downhole air gun 100 downhole clamped vibrator 129 marine surface seismic 141

seismic ray tomography 39 seismic tomography 1

diffraction 5

ray 5 series expansion method 9, 22, 40

estimated model function, M ½st 23 forward modeling 26 Kaczmarz' method 26

observed data function, pobs 23 predicted data function, prr½ 23 true model function, M true 23

simultaneous iterative reconstruction tech-

nique (see SIRT) SIRT 35, 41, 99

compared with ART 37 example 110 handling nonlinear aspect 110 nonlinear aspect 38 ray density weight 36, 42 terminating iterations 111 traveltime residual 117

slowness 11

source

air gun 98 downhole clamped vibrator 127

source radiation pattern P-wave

air gun 103 clamped vibrator 131

S-wave

air gun 105 clamped vibrator 132

spatial frequency 13 steam flood enhanced oil recovery 95

tomogram 1 base line 96

lithology interpretation 120 McKittrick thrust 134

meaning of colors 1 porosity estimation 122 presteam vs. poststeam injection

119

core study 121,123 reliability check 124 selecting a scale 133 true model function 26

tomography 1 seismic 1

total wavefield

Born approximation 48 Rytov approximation 53

transform methods 9, 39 traveltime parameters 99

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17 8 INDEX

traveltime pick consistency crosswell seismic data 107

mispick 108 poor data 108 surface seismic data 104

traveltime picking 99 direct m'rival 102

first arrival 102

polarity change air gun (P- wave) 103 air gun (S-wave) 105 claxnped vibrator (P-wave) 131 clamped vibrator (S-wave) 132

radiation pattern 102 tying crosswell data 107 using computed traveltimes 108

traveltime residual 111, 117 true model vector 28

velocity perturbation 47

wavelength 154 wavenumber 13, 155

angular 155 vector components 155

well logging disadvantages 3

Wyllie's time average equation 121

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