Copyright

by

Anubrati Mukherjee

2002

The Dissertation Committee for Anubrati Mukherjee Certifies that this is the

approved version of the following dissertation:

Seismic Data Processing in Transversely Isotropic Media: A Plane

Wave Approach

Committee:

Paul L. Stoffa, Co-Supervisor

Mrinal K. Sen, Co-Supervisor

Stephen P. Grand

Robert H. Tatham

Yosio Nakamura

Seismic Data Processing in Transversely Isotropic Media: A

Plane Wave Approach

by

Anubrati Mukherjee, B.Sc., M.Sc.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

The University of Texas at Austin

May 2002

To my parents

To Shaon

v

Acknowledgements

I would like to extend my deepest gratitude to Dr.Mrinal K.Sen for his

excellent guidance, encouragement, support and patience. His enthusiasm

motivated me to learn all that I have learnt in these past three and a half years of

graduate studies. His innovative ideas and problem solving techniques helped me

overcome many hurdles. I’m also deeply indebted to my co-supervisor Dr. Paul

L. Stoffa for his guidance, support and helpful review throughout this study. I

would like to thank Dr. Stephen P. Grand, Robert H. Tatham and Dr. Yosio

Nakamura for their invaluable advice and encouragement as committee members.

Special thanks is due to Dr Nathan Bangs for inviting to participate in the

Hydrate Ridge 3-D Oregon cruise in the summer of 2000. The cruise gave me an

opportunity to get a first hand experience of marine seismic data acquisition. I

extend my heartfelt gratitude to Dr Indrajit G. Roy for his valuable advice and

friendship. This dissertation wouldn’t have been complete without the numerous

utility softwares developed by Dr Roustam Seifoullev.

For their help with software, homework, and life in general I want to thank

my former and current fellow students at the Institute for Geophysics: Imtiaz

Ahmed, Armando Sena, Saleh Al-Salani, Abdul Aziz Alaslani, Junru Jiao, Faqi

Liu, Donna Cathro, Chengshu Wang, Jean-Paul Van Gestel, Qunling Liu, Robert

Rogers, Hongbo Lu, Carlos Huerta Lopez, Veronica Castillo, Ricardo Combellas-

Bigott, Xinxia Wu and Dhananjay Kumar. I would like to thank my friends

vi

Abhijit Chatterjee, Anirban Biswas, Swaroop Ganguly among others for making

my stay in Austin so memorable. I’ll cherish our night long weekend parties all

my life.

I also thank Mark Weiderspahn and Kevin Johnson for their computer

support, Judy Samson for her immense help with all accounting work, Charlene

Palmer and Kathy Ellins for their kind help.

My wife Shaon has been a pillar of strength during this time. Without her

love, patience and encouragement this work wouldn’t have been possible. Sincere

thanks goes to my parents for the invaluable support they have extended to me all

my life.

Finally I would like to thank Phillips Petroleum Company and British

Petroleum for their fellowship support. I would also like to acknowledge the

Student Training Fund from the Institute for Geophysics for supporting me during

the summer of 2000.

vii

Seismic Data Processing in Transversely Isotropic Media: A Plane

Wave Approach

Publication No._____________

Anubrati Mukherjee, Ph.D.

The University of Texas at Austin, 2002

Supervisors: Paul L. Stoffa and Mrinal K. Sen

Occurrences of anisotropy in the seismic data are widespread at all scales.

Thus inclusion of these anisotropic effects becomes important for obtaining

correct images and target depths. This dissertation addresses some problems

pertaining to seismic data processing in transversely isotropic media. I have

formulated an interactive traveltime analysis procedure for P-waves in delay-time,

slowness domain for wave propagation in the transversely isotropic media with a

vertical axis of symmetry (VTI). Using the assumption of weak anisotropy I

obtained a simple and physically intuitive two-term expression for vertical

slowness, which can be used in direct estimation of interval elliptic velocity and

the anisotropic parameter kappa. I have also developed a method to automatically

estimate these parameters using a non linear inversion technique called very fast

simulated annealing.

viii

Conventional ray tracing methods are difficult to apply in the VTI media.

Unavailability of vertical P wave velocity restricts us to use the time gridded

elliptic velocity and kappa as inputs for traveltime computation in offset-time

domain. However I have formulated a ray tracing technique based on the Fermat's

principle and perturbation theory. The method uses phase velocities unlike other

methods, which use group velocities. Head wave paths are not included in the

traveltime computation. Comparison with more exact Finite Difference Eikonal

solvers for both 1-D and 2-D models show small residuals.

I have used source traveltimes computed using the interval elliptic velocity

and kappa models to perform prestack split-step Fourier and Kirchhoff time

migration in the VTI media. Migration using parameters estimated from moveout

analysis and computed source traveltimes for Gulf of Mexico data show good

results. The common image gathers show increased flattening after incorporation

of anisotropic effects.

ix

Table of Contents

List of Figures ........................................................................................................xi

List of Tables......................................................................................................... xx

CHAPTER 1: INTRODUCTION ........................................................................... 1

1.1 Historical background and previous work ............................................... 1

1.2 Seismic processing concepts .................................................................... 3

1.3 Causes of Anisotropy ............................................................................... 7

1.4. Velocity surfaces, slowness surfaces and wave surfaces ...................... 9

1.5. Layer induced Anisotropy.................................................................. 11

1.6 Vertical Transverse Isotropy (VTI) ........................................................ 16

1.7 Motivation ............................................................................................. 29

CHAPTER 2: MOVEOUT ANALYSIS AND PARAMETER ESTIMATION IN TRANSVERSELY ISOTROPIC MEDIA.............................................. 38

2.1 Introduction ............................................................................................ 38

2.2 NMO in Layered Isotropic Media.......................................................... 39

2.3 NMO in VTI media ................................................................................ 41

2.4 τ-p NMO Equations for weak VTI media for quasi-P waves ................ 44

2.5 τ-p NMO equations for weak VTI media for quasi-Sv waves ............... 49

2.6 Results from Interactive analysis ........................................................... 50

2.7 Automatic estimation of Elliptic P wave velocity and anisotropic parameter ............................................................................................. 67

2.8 Summary ............................................................................................... 74

CHAPTER 3: TRAVELTIME COMPUTATION IN TRANSVERSELY ISOTROPIC MEDIA ................................................................................... 81

3.1 Introduction ............................................................................................ 81

3.2 Finite Difference Schemes ..................................................................... 82

x

3.3 Travel time computation in anisotropic media – summary of previous work ...................................................................................... 86

3.4 A New Approach................................................................................... 90

3.5 Results and Discussion........................................................................... 94

CHAPTER 4: PRE-STACK TIME MIGRATION IN TRANSVERSELY ISOTROPIC MEDIA ................................................................................. 104

4.1 Introduction .......................................................................................... 104

4.2 Anisotropic migration .......................................................................... 109

4.3 Integral formulation for migration ....................................................... 111

4.4 Implementation of Pre-stack Kirchhoff Migration in TI media ........... 113

4.5 Pre-stack Split-step Fourier migration ................................................. 115

4.6 Implementation of pre-stack Split-step Fourier migration in TI media119

4.7 Results and Discussion......................................................................... 121

CHAPTER 5: SUMMARY AND FUTURE WORK ......................................... 133

5.1 Summary .............................................................................................. 133

5.2 Future Work ......................................................................................... 138

Appendices .......................................................................................................... 140

Appendix A ................................................................................................ 140

Appendix B ................................................................................................ 143

Appendix C ................................................................................................ 146

References ........................................................................................................... 148

Vita 155

xi

List of Figures

Figure 1.1. A portion of the ray of a plane wave in a homogenous medium

with velocity V. The ray has a direction specified by the angle to

the vertical i. During the time ∆T, the ray traverses the distance

V∆T, which is decomposed into its vertical component ∆Z and

horizontal component ∆X. (Diebold et. al., 1981) ............................ 5

Figure 1.2. The traveltime plot for reflections and refractions in a three-

layer model with velocities Vj and two-ray vertical traveltimes

∆τj(0). 1.2(b) shows the tau-p mapping of the X-T data of Figure

1.2 (a). A blowup of the τ-mapping of the critical point for the

first head wave refraction HI is shown in 1.2 (c). .............................. 6

Figure1.3. Model of periodically stratified elastic medium ............................. 12

Figure 1.4. The above diagram illustrates the rotation of the coordinate axis

about an axis of symmetry which in this case is the z-axis.............. 17

Figure 1.5. For all points along the planes l$ .x is a constant. x is the position

vector, l$ is the unit normal vector to the plane. $ε is the

direction the solution advances with a phase speed c. ..................... 20

Figure 1.6. The figure graphically indicates the definitions of phase

(wavefront) angle and group ( ray) angle. (Thomsen, 1986) ........... 22

Figure 1.7. A cartoon showing a simple reflection experiment through a

homogenous VTI medium................................................................ 27

xii

Figure 1.8. Plot of PP reflection coefficient with angle of incidence for the

three classes of gas sand reflectors. The heavy solid curves are

for isotropic material properties and the light solid curves are for

average anisotropic parameters from Thomsen (1986) (i.e.,

δ=0.12, ε=0.13). (Kim et. al., 1993)................................................. 31

Figure 1.9. Reflectivity difference between TI and isotropic elastic curves

for ∆δ from top to bottom of +0.2, 0.0, -0.2, -0.4 and –0.6,

respectively. Heavy solid curves are for ∆ε=0.0 and the light

solid curves are for ∆ε=-0.3. The corresponding values of ∆δ and

∆ε for each model are indicated. Figures 1.9I(a), 1.9I(b), 1.9I(c)

are Models 1(Class1), 2 (class2), and 3 (Class 3), respectively.

(1.9J.) Reflectivity difference between TI and isotropic elastic

curves for ∆ε from top to bottom of +0.2, 0.0, -0.2, -0.4 and –0.6,

respectively. Heavy solid curves are for ∆δ=0.0 and the light

solid curves are for ∆δ=-0.3. The corresponding values of ∆δ and

∆ε for each model are indicated. Figures 1.9J(a), 1.9J(b), 1.9J(c)

are Models 1(Class1), 2 (class2), and 3 (Class 3), respectively.

(Kim et. al., 1993) ............................................................................ 32

Figure 1.10. (a) Migrated image obtained after DMO and CMO stack. Note

the presence of fault-plane reflections between 1.0 and 1.5 s. (b)

Migrated imaged obtained by pre-stack f-k migration. The fault-

plane reflections for the most part are absent. (Lynn et. al., 1991).. 35

xiii

Figure 1.11. Moveout analysis of 2 different x-t CMP gathers with and

without anisotropic correction. (Toldi et. al., 1999)......................... 36

Figure 2.1(a). τ-p curves for Dog Creek Shale model using exact equation for

vertical slowness(red), elliptic velocity isotropic model (green)

and a two term weak anisotropy model (blue). ................................ 54

Figure 2.1(b). Sensitivity of delay time to elliptic velocity and κ. Note the

trade-off between the two parameters. ............................................. 54

Figure 2.1(c). NMO corrected synthetic ? -p seismograms with best-fit

isotropic velocity model (upper curve), near p elliptic velocity

model (middle panel) and two-term weak TI model........................ 55

Figure 2.2 (a). τ-p curves for Taylor Sandstone model using exact equation for

vertical slowness (red curve), elliptic velocity isotropic model

(green curve) and a two term weak anisotropy model (blue

curve)................................................................................................ 56

Figure 2.2 (b). Sensitivity of delay times to elliptic velocity and κ. Note the

trade-off between the two parameters. ............................................. 56

Figure 2.2 (c). NMO corrected synthetic τ-p seismograms with best-fit

isotropic velocity model (upper curve), near p elliptic velocity

model (middle panel) and two-term weak TI model. Note that

with a best-fit isotropic model we are able to fit near and high

ray-parameter traces but intermediate ray-parameter traces

remain uncorrected (a diagnostic of anisotropy). A weak

anisotropy model is able to flatten the data very well...................... 57

xiv

Figure 2.3. Analysis of Gulf of Mexico data : CMP 691 in (x,t) (left panel)

and τ-p domains. .............................................................................. 58

Figure 2.4. Results from interactive τ-p velocity analysis of CMP 691: (a)

best-fit isotropic model, (b) near p elliptic velocity model, and

(c) two-term best fit TI model. ......................................................... 59

Figure 2.5. The zoomed plots of Fig 2.4(a-c) in the time window 4.4 to 4.52

sec. The target horizon is the reflection event at 4.5 sec. (a) The

best-fit isotropic model: Note the typical bulging effect. (b) near

p elliptic velocity model, and (c) TI Model. Note the excellent

improvement in the flatness of the event at 4.5s compared with

Fig (a). .............................................................................................. 60

Figure 2.6(a) Zoomed plot of the Stack section generated using the isotropic

model at the target zone. .................................................................. 61

Figure 2.6(a) Zoomed plot of the Stack section generated using the TI model

at the target zone. Note the improvement in quality of stacking

with the incorporation of κ ............................................................... 61

Figure 2.7. Stacked section obtained with the best-fit TI model on which the

elliptic velocity model is superimposed. Target zone is

highlighted with a red box............................................................... 62

xv

Figure 2.8. Stacked section obtained with the best-fit TI model on which the

anisotropic parameter κ is superimposed. Note that κ values

generally increase with depth and laterally varying; they show

significantly larger values near the target zone. Target zone is

highlighted with a red box................................................................ 63

Figure 2.9. Plot of Vsv as a function of rayparameter. The cyan curve was

generated using Thomsen’s approximate equation, the red curve

is generated using the exact equation for Vsv, blue curve using my

expression for Vsv, and the green curve plots Vsv values from

Daley and Hron’s equation. Note that up to rayparameters of 0.6

( a range realistic for all exploration purposes) my equation

shows excellent agreement with the exact result. ............................ 64

Figure 2.10(a). Comparison of exact and two-term approximate equations for

Sv-wave for Dog-Creek shale. The ray-parameter is in sec/km. ...... 65

Figure 2.10(b). Comparison of exact and two-term approximate equations Sv-

wave for Taylor Sand stone. The ray parameter is given in

sec/km............................................................................................... 65

Figure 2.11(a). NMO corrected P-S τ-p data generated using the model in

table 2.2. The correction is performed using for Vpellip and κ for

the Pwave path and only the Sv elliptic velocity for the Swave

path. .................................................................................................. 66

xvi

Figure 2.11(b). NMO corrected P-S τ-p data generated using the model in

table 2.2. The correction is performed using the TI model. Note

the improved flattening after incorporation of η.............................. 66

Figure 2.12. (a) The starting elliptic velocity bounds used for one of the CDPs

at the target zone (b) The starting kappa bounds used for one of

the CDPs at the target zone .............................................................. 75

Figure 2.13. (a) NMO corrected CDP gather from outside the target zone

using the inverted model from VFSA (b) Elliptic velocity model

fromVFSA (c) Kappa model from VFSA ........................................ 76

Figure 2.14. (a) NMO corrected CDP gather from the target zone using the

inverted model from VFSA (b) Elliptic velocity model

fromVFSA (c) Kappa model from VFSA. ....................................... 77

Figure 2.15(a). Kappa model from VFSA for the Gulf of Mexico dataset for

740 CDPs.......................................................................................... 78

Figure 2.15(b). Elliptic velocity model from VFSA for the Gulf of Mexico

dataset for 740 CDPs........................................................................ 78

Figure 2.16(a). Plot of negative correlations vs. the number of iterations for a

CDP at the target zone...................................................................... 79

Figure 2.16(b). Plot of temperatures vs. the number of iterations for a CDP at

the target zone. ................................................................................. 79

Figure 3.1. The source grid point A and the eight points in the ring

surrounding A................................................................................... 83

xvii

Figure 3.2. Traveltime computation scheme developed by Faria and Stoffa

(1993). Using the known traveltime t1 and t2 , t0 is calculated in

order to minimize the total traveltime t, from an apparent source

to the point (x,z2). ............................................................................. 88

Figure 3.3. (a) The model for a synthetic isotropic homogenous single-

layered model (b) Computed traveltimes from perturbation

approach (c) Traveltimes computed analytically (d) The

difference of (b) and (c). .................................................................. 95

Figure 3.4. (a) The model for a synthetic transversely isotropic homogenous

single-layered model (b) Computed traveltimes from

perturbation approach (c) Traveltimes computed analytically (d)

The difference of (b) and (c). ........................................................... 96

Figure 3.5. (a) Computed traveltimes using Faria et al’s method for a

homogenous isotropic model given in figure 3.3 a (b) The

difference with the analytic solution in figure 3.3 c (c) Computed

traveltimes using Faria et al’s method for a homogenous VTI

model given in figure 3.4 a (d) The difference with the analytic

solution in figure 3.4 c...................................................................... 97

Figure 3.6. (a) The elliptic velocity for a synthetic flat two-layered model

(b) kappa model (c) Computed traveltimes from perturbation

approach (d) Computed traveltimes using the Eikonal solver (e)

The difference of (c) and (d). ........................................................... 98

xviii

Figure 3.7. (a) The elliptic velocity for a synthetic dipping three-layered

model (b) kappa model (c) travel times computed using the

perturbation approach....................................................................... 99

Figure 3.8. (a) Elliptic velocity and kappa model for a 40 dipping layer

example (b) Traveltimes computed using perturbation approach

(c) Traveltimes computed using the Eikonal solver (d) Difference

plot between (b) and (c). ................................................................ 100

Figure 4.1. Principle of shot record oriented pre-stack migration. Note that

every shot record is migrated separately and then they are

summed to form the migrated section. (Berkout, 1984)................. 106

Figure 4.2. Migration principle for zero offset data recorded at z=0 .................. 112

Figure 4.3. Implementation method for Kirchhoff Migration............................. 116

Figure 4.4. Flowchart for the split-step Fourier method to migrate a single

shot gather by extrapolating the receiver wave field and using the

direct arrival times of the source wavefield to construct the

image. ............................................................................................. 120

where q is the vertical slowness. ......................................................................... 121

Figure 4.5. The Elliptic velocity and kappa model for a flat layer synthetic

test. ................................................................................................. 122

Figure 4.6. Input synthetic shot gather for the velocity and kappa model in Fig

4.5................................................................................................... 122

Figure 4.7(a). Migrated shot gather after isotropic split-step migration ............. 123

xix

Figure 4.8. Elliptic P wave velocity and kappa model for a dipping layered

synthetic experiment....................................................................... 124

Figure 4.9. Input synthetic shot gather for the velocity and kappa model in Fig

4.9................................................................................................... 124

Figure 4.10. Migrated shot gather for the dipping layer model after split-step

fourier migration using TI corrections. .......................................... 125

Figure 4.11. (a) Plot of kappa vs TWT for a location away from the target

zone. (b) CIG after isotropic split step migration (c) CIG after TI

split-step migration......................................................................... 126

Figure 4.12. (a) Plot of kappa vs TWT for a location at the target zone. (b)

CIG after isotropic split step migration (c) CIG after TI split-step

migration. ....................................................................................... 127

Figure 4.13. (a) CIG after TI Kirchhoff migration at a location away from the

target zone (b) CIG after TI Kirchhoff migration at a location

around the target zone. ................................................................... 128

Figure 4.14(a). Zoomed plot of the target zone after TI pre-stack Kirchhoff

Time migration. .............................................................................. 129

Figure 4.14(b). Zoomed plot of the target zone after Isotropic pre-stack

Kirchhoff Time migration. ............................................................. 129

xx

List of Tables

Table 1.1 Elastic parameters of Models 1,2,3 ...................................................... 30

Table 2.1: Anisotropy coefficients of Taylor sandstone and Dog Creek shale.

α0, β0, ε, δ from Thomsen(1986). .................................................... 53

Table 2.2. Model parameters for the synthetic data in figure 2.9. ........................ 64

1

CHAPTER 1: INTRODUCTION

1.1 HISTORICAL BACKGROUND AND PREVIOUS WORK

In most applications of elasticity theory to problems in petroleum

geophysics, the elastic medium is assumed to be isotropic. However many crustal

rocks are experimentally found to be anisotropic. Unlike the isotropic case, the

velocity of seismic waves varies with direction of propagation in an anisotropic

medium. The polarization of seismic waves depends not only on the type of the

wave but also on its direction of propagation. Seismic data have revealed

widespread occurrence of anisotropy in the earth at all scales of resolution.

Therefore including the effects of anisotropy in seismic processing is important

for obtaining correct images and estimating target depths. Anisotropy is not a new

subject. In the 19th century, scientists had already used some anisotropic concepts

in studies of transversely isotropic solids. G.R. Hamilton and J. McCullagh, in

1833, in independent studies used the concepts of slowness surfaces in the

geometrical description of anisotropic media. M.P. Rudzki in the late 19th century

justified the study of transverse isotropy by using the concepts of scale induced

anisotropy (seismic wavelengths much larger than the crustal structure of rocks or

constituent fine layers) and intrinsic anisotropy caused by preferred orientation of

grains. He also observed that a non-uniformly stressed solid shows birefringence

at depth in an otherwise isotropic solid. During the period of 1897 to 1899, he

derived the equations of motion for transversely isotropic solid. Nagaoka (1900)

found isotropy to be superficial based on his measurements of Young’s modulus

2

in orthogonal directions in eighty rocks. Stoneley (1949) revived geophysical

interest in anisotropy with his discussion of the effect of anisotropy on focal depth

determination of earthquakes and propagation of Raleigh waves on the surface of

the earth. Synge (1956) published a paper in which he emphasized the importance

of the velocity surface, the wave surface, and especially the slowness surface for

anisotropic wave propagation. Significant work has been done in the last few

decades, which has increased the awareness concerning seismic wave propagation

in anisotropic media. Seismic processing methods related to accurate velocity

estimates like NMO, AVO, and imaging need to consider anisotropic effects to

obtain meaningful results. Davis and Clowes (1986) showed the effects of layer

induced transverse isotropy on seismic velocities. Using seismic date from

Winona basin, Canada, they showed that high velocities obtained by seismic

refraction studies were affected by transverse isotropy of sediments. If the effects

of anisotropy were neglected, values for thickness and velocities would be over

estimated by as much as 10-15%. Banik(1984) showed that well depth misties in

the north sea were due to anisotropy resulting primarily from shales and found a

percentage anisotropy of around 15 percent. Processing techniques are being

developed to incorporate weak transverse isotropy resulting in significant

improvement in imaging and hence more accurate geologic interpretation. I will

discuss these in greater details in the proceeding chapters. In the next section I

will review some basic concepts on seismic data processing and also review the

plane wave domain.

3

1.2 SEISMIC PROCESSING CONCEPTS

Seismic data are acquired as shot gathers in a 2-D or 3-D survey. The raw

field data need to go through a series of processing steps before they are ready to

create a stacked time section or a migrated image. The domain in which data is

acquired is called the offset-time domain or the x-t domain. The equation for

reflected traveltime from a stack of layers in isotropic media is given by the

approximate equation:

2

220

2 )(

rmsv

xtxt += . (1.1)

The above equation results from the truncation of a Taylor’s series

expansion (Taner and Koehler, 1969). One can see that it describes a hyperbola;

the rms velocity, Vrms is also called the NMO or stacking velocity. The above

equation is valid strictly for small offset ranges. An estimate of the interval

velocity can be obtained from the rms velocity, Vrms,using the Dix’s equation.

Velocity analysis is performed on the data sorted into CDP gathers. By fitting the

hyperbolas using the above equation one gets an estimate of vrms. NMO results in

flattened hyperbolas, which are added to create a single stacked trace

corresponding to one CDP location. Stacked traces from all CDP’s are grouped

together to create a stacked time section.

The same process can be performed in the plane wave domain or the τ-p

domain. τ refers to the sum of the vertical slowness-thickness products (vertical

delay time) and p refers to the ray-parameter or the horizontal slowness. Let us

consider a plane wave traveling in a homogenous acoustic medium with velocity

4

V in a direction specified by i, the angle of the ray with the vertical (Fig 1.1). We

can write,

sinX V T i∆ = ∆ , and cosZ V T i∆ = ∆ . (1.2)

Traveltime can be expressed in terms of these components as (Diebold et

al, 1981):

T p X q Z∆ = ∆ + ∆ , (1.3)

where

Vi

q

Vi

p

cos

,sin

=

= (1.4)

are horizontal and vertical components of the wave slowness

( )1/ 22 21/u V p q= = + . (1.5)

The traveltime equation for a reflected or refracted wave in a structure

consisting of a stack of horizontal homogenous layers of thickness Zj, and vertical

slowness qj (Diebold et al, 1981):

1

2n

j jj

t pX q Z=

= + ∑ . (1.6)

From the above equation we can express the vertical delay time τ as,

1

2j

n

jj

q Z t pXτ=

= = −∑ . (1.7)

Since ( )2 1/ 2 1/ 22 2 21/i j jq V p u p = − = −

, (1.8)

the contribution to tau from a single layer can be written as

2/122 )(2 puZ jj −=∆τ (1.9)

5

Figure 1.1. A portion of the ray of a plane wave in a homogenous medium with velocity V. The ray has a direction specified by the angle to the vertical i. During the time ∆T, the ray traverses the distance V∆T, which is decomposed into its vertical component ∆Z and horizontal component ∆X. (Diebold et. al., 1981)

6

Figure 1.2. The traveltime plot for reflections and refractions in a three-layer model with velocities Vj and two-ray vertical traveltimes ∆τj(0). 1.2(b) shows the tau-p mapping of the X-T data of Figure 1.2 (a). A blowup of the τ-mapping of the critical point for the first head wave refraction HI is shown in 1.2 (c).

7

The above equation describes an ellipse in the τ-p plane, having semi-axial

lengths of 2Zjuj, and uj, the two-way normal traveltime and slowness of the layer.

The mapping of the space-time domain data into τ-p domain ellipses has been

shown in figure 1.2(b).

Slant stacking is a popular way of transforming a x-t gather into τ-p

domain. Analysis in the τ-p domain has numerous advantages. Firstly by fitting

tau-p curves one can estimate interval velocities from the data. The plane wave

domain is also the required domain for AVO analysis with plane wave reflection

coefficients that use phase angles. Multiples, being periodic in the plane wave

domain, are easier to eliminate. In the next chapter we will see how τ-p domain

can be efficiently used to perform NMO analysis in transversely isotropic media.

Some of the most common causes of seismic anisotropy are stated in the next

section.

1.3 CAUSES OF ANISOTROPY

A range of phenomena may cause rocks to display effective seismic

anisotropy so that the propagation of seismic waves through the rock can be

simulated by propagation through comparatively simple anisotropic models

(Crampin et. al, 1984). They are as follows:

(a) Intrinsic Anisotropy.

(i) Crystalline anisotropy: This occurs when the individual anisotropic crystals in

a crystalline solid have preferred orientations over a volume sufficiently

large to affect the transmission of seismic waves.

8

(ii) Direct stress-induced anisotropy: The elastic behavior of an initially isotropic

solid becomes anisotropic when acted upon by sufficiently large stresses.

However, the stresses required to cause observable seismic anisotropy

effects in seismic wave propagation are probably too great to cause

observable seismic anisotropy in the earth.

(iii) Lithologic anisotropy: A sedimentary solid has lithologic anisotropy when the

individual grains, which may or may not be anisotropic, are elongated or

flattened and these shapes are aligned by gravity or fluid flow when the

material was first deposited. Transverse anisotropy of clays and shales is

probably lithologic anisotropy resulting from aligned grains.

(b) Crack induced Anisotropy: When an otherwise isotropic rock contains a

distribution of inclusions, such as dry or liquid filled cracks or pores that

have preferred orientations, the resulting material will have effective

seismic anisotropy.

(c) Long-Wavelength Anisotropy: This occurs when propagation through

arrangements of isotropic layers or isotropic blocks may be simulated by

propagation through structurally simpler anisotropic solid. The best known

long-wavelength anisotropy is propagation through regular sequences of

thin isotropic layers. Propagation of seismic waves through periodic thin

layered solids can be modeled by propagation through homogenous elastic

solids with hexagonal symmetry (transverse isotropy) with five elastic

constants. This is the most common type of anisotropy and is primarily

important for exploration purposes. A stack of thin isotropic layers can be

9

equated to an equivalent homogenous transversely isotropic layer using a

weighted averaging (Backus, 1962).

1.4. VELOCITY SURFACES, SLOWNESS SURFACES AND WAVE SURFACES

The distinction between phase velocity, group velocity, and phase

slowness is important for understanding wave propagation in anisotropic media.

The solution of the elastic wave equation 2 2

2i k

ijklj l

u uC

x xtρ

∂ ∂=

∂ ∂∂, (1.10)

with the plane wave solution ( . )

0i s x t

i iu U e ωε −= , (1.11)

where

ρ is the density,

ω is the angular frequency,

s is the phase slowness, i.e., reciprocal of phase velocity

εi is the polarization vector for a given model (eigenvector),

U0 is the displacement amplitude, and Cijkl are the elastic constants,

We have the following eigenvalue equation 2 / 0ik ijkl j lv C n nδ ρ− = (1.12)

in ‘v’ for the phase velocity surface, or taking s=n/v, n.n=1,

/ 0ik ijkl j lC s sδ ρ− = (1.13)

in ‘s’ for the phase slowness surface.

The derivation of equations (1.12) and (1.13) are given in more details in

section 1.6. A velocity surface, V, can be formed by varying the phase velocity, v,

10

over all directions (phase angles) in physical space; Plane waves travel with the

phase velocity.

A slowness surface, S, is formed by varying the phase slowness, s, over

all directions or phase angles in slowness space.

The group velocity is obtained from the slowness surface. If the slowness

surface is designated by Ω(s) = 0, with s as the slowness vector, then the group

velocity is given in parametric form by

.

s

sU

s

∇ Ω=

∇ Ω$ . (1.14)

(Synge, 1956; Duff, 1960; Musgrave, 1961; Kraut, 1963).

The slowness surface handles reflection and refraction quite conveniently.

The local component of the slowness tangent to a surface is always conserved

across that surface (Snell’s law).

The wave surface, W, (group velocity surface) due to a point source at the

origin separates space into a region already reached by the resulting disturbance

and a region not yet reached by the disturbance. In a lossless medium, energy

travels with the group velocity, U. The physical location of the wave energy, x, at

a time t which was emitted by a point source at t=0 at the origin is

x Ut= . (1.15)

Some authors (Rudzki, 1898,1911; Buchwald 1959; Kraut, 1962) refer to

x as the wave surface, while other authors (Synge, 1956; Helbig, 1958; Musgrave,

1961; Hake, 1986) refer to U as the wave surface.

11

1.5. LAYER INDUCED ANISOTROPY

The purpose of this section is to investigate long wave elastic anisotropy

produced by fine horizontal layering. Fine layer refers to a layer thickness, long

enough so that the elastic properties of the medium vary appreciably over this

range of thickness. Long wave refers to seismic waves in which the distance over

which the displacements change by an appreciable fraction of their values is much

larger than the layer thickness. The variations in the medium, which have vertical

scales less than the layer thickness can be averaged out, so that the medium can be

replaced by a less wildly varying medium.

1.5.1. The Averaging Technique

Let us consider an infinite linear elastic medium made up of plane

homogenous layers. The x3 axis is perpendicular to the layering and the layering

is periodic with period H. Each period is made up of a set of N homogenous

isotropic layers with model parameters λi, µi, ρi. The compressional wave

velocity for each layer is:

i

iii ρ

µλα

2+= .

and the shear wave velocity is:

i

ii ρ

µβ = .

Let γi be another parameter defined as the ratio of the squares of shear speed βi, to

that of the compressional speed αi.

2

2

i

ii

α

βγ = , (1.16)

12

hi are the thickness weights for each of the thin layers within one period H. Thus

thickness of each layer is given by hiH. Figure 1.3 illustrates the model clearly.

Figure1.3. Model of periodically stratified elastic medium

In the following derivation I will show (following Schoenberg, 1983) that for

stress and strain fields whose scale of variation (wavelength) is much greater than

H, effective transversely isotropic moduli can be derived in terms of the elastic

model parameters µi, λi, γi. The stress-strain relations for the transverse isotropy

case ( refer to section 1.6 for further details) is given by:

−

−

=

12

31

23

33

22

11

66

44

44

331313

13116611

13661111

12

31

23

33

22

11

20000002000000200000000020002

εεεεεε

σσσσσσ

CC

CCCCCCCCCCCC

(1.17)

λi, µi,ρi,γi

H

hiH

13

If a force is applied in the x3 direction, using the conditions of low frequency

equilibrium requirements (long wavelength assumption), we can assume σ33, σ23,

σ31 to be constant across the set of layers. In the same token strains that lie in a

plane parallel to the layering, i.e., ε11, ε22, ε12 are also assumed constant. The other

strains over a full spatial period H can be written in terms of the strains of the

individual layers. Thus,

3 3 3 333 3 33 33 33

1 1 1

( ) ( ) 1 1i i

N N N

i i ii i i

u x H u xh H h

H H Hε ε ε ε

= = =

+ −= = ∆ = = ≡∑ ∑ ∑ (1.18)

where, u3 is the displacement, < > denotes the thickness-weighted average.

Similarly the average strains, ε23, ε31 can be found to be given by

23 23 31 31,ε ε ε ε= = (1.19)

Similarly,

31311

311

σσσ == ∑=

i

N

ii Hh

H (1.20)

Using the force in the x1 direction on the x1-face and the force in the x2-direction

on the x2-face the average stresses σ11 and σ22, respectively:

22221111 , σσσσ == . (1.21)

Let us consider the relations between shear stress and shear strain. In each layer

.1212

3131

2323

2

,2

,2

ii

ii

ii

i

i

i

εµσ

εµσ

εµσ

=

=

=

(1.22)

14

On averaging the above stated equations we get,

.22

,2/

,2/2/

12121212

311

31

231

232323

εµµεσσ

σµε

σµµσεε

===

=

===

−

−

(1.23)

On comparing these three equations with the last three shown in the matrix form

in eq (1.17) yields the effective transverse isotropic moduli C44 and C66 for the

layered medium as

.

,

66

1144

µ

µ

=

=−−

C

C (1.24)

The relation between the normal stress σ33 and the three normal strains ε11, ε22

and ε33 for each layer can be obtained from the matrix equation and is given by:

33221133 )2()( εµλεελσ +++= . (1.25)

which can also be written in terms of γi, µi as,

( )( )33 33 11 22 331 2 /ii i i iσ σ γ ε ε ε µ γ = = − + + (1.26)

By multiplying the above equation by γi/µi, averaging, and then dividing by <γ/µ>

gives

( ) ( ) 331

22111

33 //21 εµγεεµγγσ −− ++−= (1.27)

Comparing this equation with the third equation in (1.17) the effective elastic

moduli C33 and C13 are obtained as

15

( ) ./21

,/1

13

133

−

−

−=

=

µγγ

µγ

C

C (1.28)

Using the same idea as in (1.26) the relationship between the normal stress σ11

and the normal strains for each layer can be expressed in terms of γi, µi as,

( )( )33 33 11 22 331 2 /ii i i iσ σ ε γ ε ε µ γ = = + − + (1.29)

By substituting for ii iγεµ /33 using (1.26) and for σ33 from (1.27) we get,

( )

( ) ( ) 331

2211

121111

/21

/21422

εµγγεε

µγγγµµεµσ

−

−

+++×

−+−+=

(1.30)

Comparing this equation with the first equation of (1.17) gives

( ) 1211 /2144 −−+−= µγγγµµC (1.31)

and

66111112 22 CCCC −=−= µ . (1.32)

Thus a stack of homogenous isotropic layers can be represented by an

equivalent transversely isotropic layer whose layer properties are less wildly

varying than its constituent thin layers. The effective elastic moduli of this

equivalent transversely isotropic (TI) medium can be expressed as thickness

weighted averages of its constituent isotropic layers.

16

1.6 VERTICAL TRANSVERSE ISOTROPY (VTI)

The two most fundamental equations governing seismology are the

linearized momentum equation

,. fu +∇= τρ && (1.33)

and the constitutive relation which may be written as

klijklij C ετ = . (1.34)

For the isotropic case the stress-strain relation is given by

kljkiljlikklijklijklij C εδδδδµδλδετ )]([ ++== , (1.35)

where δ is the Kronecker delta function. The above relation in equation (1.35)

may be written as 11 11

22 22

33 33

23 23

31 31

12 12

2 0 0 02 0 0 0

2 0 0 00 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

τ ελ µ λ λτ ελ λ µ λτ ελ λ λ µτ εµτ εµτ εµ

+ + +

=

. (1.36)

Thus for the isotropic case we have only two independent constants λ and

µ, and all the elements of the 6 X 6 C matrix can be expressed in terms of these

two constants.

To understand the case of hexagonal symmetry, which correspond to

invariance of the elastic tensor matrix to rotation about one of the three coordinate

axes we need to discuss the principles of orthogonal transformation of elastic

coefficient matrix.

Let v be an arbitrary vector expressed in component form as: $ $

1 2 3v v x v y v z= + + $ . (1.37)

17

If we now rotate the coordinate system by some arbitrary angle about the z

axis to define a new coordinate system, the vector v in the new system can be

expressed as $ $' '' '

1 2 3v v x v y v z= + + $ . (1.38)

It is straightforward to show that the components vi and vi′ are related by

the following equation,

=

3

2

1''

''

3

'2

'1

1000),cos(),cos(0),cos(),cos(

vvv

yyxyyxxx

vvv

. (1.39)

x

Figure 1.4. The above diagram illustrates the rotation of the coordinate axis about an axis of symmetry which in this case is the z-axis.

Thus we can represent the transformation as, 'v Av= , (1.40)

where A is the transformation matrix. For a tensor v of order two it can be shown

that

kljlikij vAAv =' . (1.41)

z

y

y’

θ

θ

x′

18

Similarly for a tensor C of order four, we have

pqrslskrjqipijkl CAAAAC =' , (1.42)

and so on.

To derive the expressions for transformation of the matrix C under

coordinate rotations, we must look at the transformation of the stress tensor τ and

the strain tensor ε. We know that

klkljlikij MAA τττ ==' , (1.43)

where,

+++++++++

=

311222112311211322132312231322122111

311232113311311332133312133312321131

322131223123332132233322332332223121

323131333332133

232

231

222121232322223

222

221

121111131312213

212

211

222222222

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAA

M

.

(1.44)

Similarly we can obtain transformation of strain as

klkljlikij NAA εεε ==' , (1.45)

where,

+++++++++

=

311222112311211322132312231322122111

311232113311311332133312133312321131

322131223123332132233322332332223121

323131333332133

232

231

222121232322223

222

221

121111131312213

212

211

222222222

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAA

N

. (1.46)

19

Using the stress transformation,

εττ MCM ==' , (1.47)

since,

ετ C= . (1.48)

By using the strain transformation equation we obtain,

1 'Nε ε−= . (1.49)

Therefore the stress tensor in the rotated system can be written as,

' 1 'MCNτ ε−= . (1.50)

Using equations (1.44) and (1.46) it can be easily shown that N-1=MT.

Therefore the coefficient matrix C′ in the new rotated system is given by

C′=MCMT , (1.51)

where,

M=

−−

−

φφφ

φφφφ

φφφφφφ

2cos0002

2sin22sin

0cossin0000sincos0000001002sin000cossin2sin000sincos

22

22

. (1.52)

The above matrix is valid for orthogonal coordinate transformation only.

After some algebraic manipulations we get the elastic coefficient matrix for the

hexagonal symmetry as,

20

C=

−

−

66

44

44

331313

13116611

13661111

00000000000000000000020002

CC

CCCCCCCCCCCC

. (1.53)

Thus for the case of hexagonal symmetry we have only 5 independent

elastic constants namely, C11, C33, C44, C66 and C13. The above matrix may be

used in the equation of motion to yield a wave equation as given in equation

(1.10).

The most common trial solution to equation (1.10) is a plane wave

solution given in equation (1.11). Equation (1.11) can be rewritten as,

$ $0 0

.( , ) ( ) ( . )

l xu x t U f t U f s x t

cε ε= − = −

$. (1.54)

Figure 1.5. For all points along the planes l$ .x is a constant. x is the position vector, l$ is the unit normal vector to the plane. $ε is the direction the solution advances with a phase speed c.

x

ε l

21

Substituting (1.54) into the wave equation we obtain,

).().( ''0

''0 txsfUsstxsfUC jiklijkl −=− ερε , (1.55)

where, the prime indicates differentiation with respect to its argument.

Eliminating f′′ from both sides we obtain a general relationship between $ε and s

as follows

jlikijkl ssC ρεε = , (1.56)

or,

0)( =− ljlikijkl ssC ερδ , (1.57)

or,

.0).(

,0)(

=−Γ

=−Γ

ερ

ερδ

I

ljljl (1.58)

(Γ-ρI) is called the Christoffel matrix and equation (1.58) is known as the

Christoffel equation. The solutions to the Christoffel’s equation give the

eigenvector of all possible wave motions. For nontrivial solution, the determinant

of the Christoffel matrix vanishes. Acceptable slowness vectors satisfy the

equation,

0=− jlikijkl ssC ρδ . (1.59)

Equation (1.59) is same as equations (1.12) and (1.13).

At this juncture it is important to clarify the distinction between phase and

group velocity. Referring to the Fig 1.3, the wavefront is locally perpendicular to

the propagation vector k, since k points in the direction of maximum increase in

phase. The phase velocity, v, is also called the wavefront velocity as it measures

the velocity of advance of the wavefront along k(θ), where k is the propagation

22

vector. Since the wavefront is nonspherical it is clear that θ is different from φ,

the ray angle from the source point to the wavefront, which is also the direction

along which the energy propagates. V(φ) is called the group velocity.

,∧∧

+= zkxkk zx

where,

;0

;cos)(;sin)(

=

==

y

z

x

kand

kkkk

θθθθ

(1.60)

The scalar length is )(/)( 22 θωθ vkkk zx =+= where ω is the angular

frequency. The ray or the group velocity V is then given by ,

∧∧

∂∂

+∂

∂= z

kkv

xkkv

Vzx

)()(. (1.61)

Figure 1.6. The figure graphically indicates the definitions of phase (wavefront) angle and group ( ray) angle. (Thomsen, 1986)

23

φ is called the group angle and is given by,

zx kkv

kkv

∂∂

∂∂

= /))(tan( θφ

).

tan1/()

1(tan

)sincos/()cossin(

θθ

θθ

θθ

θθθ

θ

ddv

vddv

v

ddv

vddv

v

−+=

−+= (1.62)

Berryman (1979) showed that the scalar magnitude of V is given in terms

of the phase velocity magnitude by

222 )()())((

θθθφ

ddv

vV += . (1.63)

Daley and Hron (1977) give a clear derivation of the directional dependence of

the three phase velocities: 2 2

33 44 11 33

2 233 44 11 33

1( ) ( )sin ( ) ;

21

( ) ( )sin ( ) ;2

p

sv

v C C C C D

v C C C C D

ρ θ θ θ

ρ θ θ θ

= + + − +

= + + − −

and (1.64) 2 2 2

66 44( ) sin cosSHv C Cρ θ θ θ= + ,

where ρ is the density and phase angle θ is the angle between the wavefront

normal and the unique(vertical) axis. D(θ) is a algebraically complex notation in

terms of the Cij’s which become the primary obstacle to use of anisotropic models

in analyzing exploration data.

Thomsen (1986) defined three parameters for reliable measures of

anisotropy. These parameters are 11 33

33;

2C C

Cε

−≡

66 44

44;

2C C

Cγ

−≡ (1.65)

24

2 2

13 44 33 44

33 33 44

( ) ( )2 ( )

C C C CC C C

δ+ − −

≡−

The other two parameters are the vertical sound speed for P-wave, α0 and S-

wave, β0, given by,

0 33 / ;Cα ρ=

and (1.66)

0 44 / ;Cβ ρ=

Thomsen compiled laboratory data of these anisotropic parameters for a

host of sedimentary rocks. It is observed that the anisotropy for most common

sedimentary rocks is from weak to moderate range(<0.2). He defined this as

weak elastic anisotropy. Under this assumption the equations for the phase

velocities become much simpler as now we retain only the linear terms in ε,δ, and

γ from the Taylor series expansions. For the case of weak anisotropy the phase

velocities derived by Thomsen (1986) are, 2 2 4

0

2 200

0

( ) (1 sin cos sin ),

( ) [1 ( )sin cos ],

p

SV

v

v

θ α δ θ θ ε θ

αθ β ε δ θ θ

β

= + +

= + −

and (1.67) 2

0( ) (1 sin ).SHv θ β γ θ= +

It can be observed from the expression of the P-wave phase velocity that

the near vertical propagation is dominated by δ whereas the near horizontal

propagation is dominated by ε. The parameter ε can also be written as

25

0

0)2/(

α

απε

−= pv

(1.68)

ε is in fact the fractional difference between the vertical and horizontal P

wave velocities and is often referred to as the “anisotropy of the rock”. Since ε is

usually > 0 the horizontal P velocity is normally greater than the vertical P-

velocity. Banik (1987) described δ as a effective anisotropic parameter in TI

media. In a P-Sv vertical plane, he defined two anisotropic parameters εp and εs as

follows

εp = (αh − α0)/α0

εs=(β45−β0)/β0 (1.69)

αh is the P-wave phase velocity in the horizontal direction

β45 is the SV wave phase velocity at an angle of 45 degrees to the axis of

symmetry.

α0 is the vertical P wave phase velocity and

β0 is the vertical Sv wave phase velocity.

He termed εp, as the P-wave anisotropy and εs, as the S-wave anisotropy.

However, P-wave anisotropy depends both on εp and εs. εs describes the

deviation of phase velocity surface from ellipticity. P and S- wave phase

velocities in terms of these parameters are as follows:

α(θ) = α0[1 + (εp − 4εs βs2/α0

2) sin2θ cos2θ + εpsin4θ] (1.70)

β(θ) = β0[1 + 4εs sin2θ cos2θ] (1.71)

On comparing with the P-wave phase-velocity expression in

Thomsen(1986), Banik (1987) obtained :

26

δ = εp − 4εs β02/α0

2 . (1.72)

The above expression suggests an intuitive physical meaning of δ. It

describes the relative competitiveness between the P-wave anisotropy and the Sv-

wave anisotropy.

For the quasi-P-wave, the group velocity is given as

])(21

1)[()( 22 θ

θφ∂

∂+= p

ppp

v

vvV , (1.73)

which is quadratic in terms of anisotropic parameters. Therefore if this term is

neglected for the case of linear approximation

)()( θφ pp vV = . (1.74)

Similarly for other wave types we get,

( ) ( );

( ) ( );SV SV

SH SH

V v

V v

φ θφ θ

==

(1.75)

The above equations, however, do not mean that the phase velocity is

equal to the group velocity. What it says is that if we calculate the phase angle for

a particular group angle using the linear approximate relation between phase and

group angles then the above equations can be used to find the ray or group

velocity.

The linear relation between group and phase angle is given by,

])(

1cossin1

1[tantanθθθθ

θφddv

v+= . (1.76)

The traveltime t is given as 222

22)0(

2)(

)(

+

=

xV

tV

τφφ , (1.77)

27

where τ is the vertical traveltime. Solving for t2 we have

+

=

)0()()0(

)(2

22

22

V

xVV

t τφ

φ . (1.78)

Moveout Velocity

x

Figure 1.7. A cartoon showing a simple reflection experiment through a homogenous VTI medium.

The function plots along a curved line in the t2-x2 plane. The slope of this line is

−=

φ

φφ

φ

φ 2

2

22

2

sin

)()(

cos21

)(

1

d

dVVVdx

dt. (1.79)

The normal-moveout velocity is defined as the initial slope of this line:

−=

=

>− φ

φ222

2

02 sin

)()0(

21

)0(

1lim

1

d

dVVVdx

dt

V xNMO. (1.80)

The second term on the right is not zero. Hence it is clear from the

equation that, even in the limit of small x offsets, with all velocities near V(0), the

resulting moveout velocity is not the vertical velocity V(0). For P-wave we have,

δα 21)( 0 +=PVNMO . (1.81)

For SV waves,

φ

V(φ)t/2 Homogenous Transversely isotropic Medium

28

2/1

20

20

0 )(21)(

−+= δε

β

αβSVVNMO , (1.82)

and for SH-waves

)2/(21)( 0 πγβ SHNMO VSHV =+= . (1.83)

For weak anisotropy, the above equations reduce to

[ ])1()(

)(/1)(

)1()(

0

220

0

γβδεβαβ

δα

+=−+=

+=

SHV

SHV

PV

NMO

NMO

NMO

. (1.84)

From the expression for the P-wave NMO velocity it can be observed that

the departure of VNMO/α0 from unity is related to the anisotropic parameter δ.

Horizontal stress

Using the relation

3,2,1,,3

1

3

1

== ∑∑==

jiC kll

ijklk

ij εσ , (1.85)

the expressions for the vertical stress σ33 and the horizontal stress σ11 are given as

σ33 = C31ε11 + C32ε22 + C33ε33,

and

σ11 = C11ε11 + C12ε22 + C13ε33. (1.86)

29

In the isotropic case, the ratio of the elastic moduli can be expressed as:

σ11/σ33=C13/C33=1-2β2/α2, (1.87)

where α and β are the velocities of P-wave and S-wave, respectively.

In the anisotropic case, the corresponding expression can be written as: 1/ 22 2 2 2 2 2

11 33 13 33 0 0 0 0 0 0/ / (1 2 / ) (1 / ) 1 2 /(1 / ) 1C Cσ σ β α β α δ β α = = − + − + − −

.(1.88)

For weak anisotropy, the above equation reduces to:

σ11/σ33=C13/C33=(1-2β02/α0

2)+δ . (1.89)

Careful investigation of the above equation reveals that the anisotropic

correction is simply given by the anisotropic parameter δ. In a typical case β2/α2 ≈

0.5, so that the first term of the above equation is also 0.5. δ values from the rock

samples show that it is not always negligible in comparison to 0.5. Thus, it may

result in significant errors in the values of the horizontal stress if an isotropic

model is considered.

1.7 MOTIVATION

Transverse Isotropy is the most common type of anisotropy observed in

geologic formations. Geophysicists have studied the effects of transverse isotropy

on field seismic data extensively in the last few decades. In this section, I present

a few cases regarding the influence of transverse isotropy, which motivated me to

choose this area of reasearch.

(i) Effect of transverse isotropy on P-wave AVO for gas sands:

Rutherford and Williams(1989) classified three types of gas sands. Class I

is characterized by a strong positive normal incidence reflection coefficient,

30

followed by a decrease in amplitude with increase in incidence angle. Class II has

a very small normal incident P-wave reflection coefficient, and reflection

amplitudes decrease with offset. Class III starts with a strong negative normal

incidence P-wave reflection coefficient and becomes increasingly negative with

increasing incidence angle. Class III corresponds to the classical bright spot

reflection. The effect of transverse isotropy was studied by Kim et. al. (1993) for

a model where shale overlies these three types of gas sands. They defined the

AVO effect as | R(θ) – R(0) |, where R(θ) is the reflection coefficient at an angle

of incidence θ , while R(0) is the normal incidence reflection coefficient. They

compared the AVO effects for the isotropic and the transverse isotropic cases for

these different models for different angles of incidence.

Model1 Model2 Model3 Parameter Shale Sand Shale Sand Shale sand

α (km/s) 3.3 4.2 2.96 3.49 2.73 2.02 β (km/s) 1.7 2.7 1.38 2.29 1.24 1.23 ρ (g/cm3) 2.35 2.49 2.43 2.14 2.35 2.13

Table 1.1 Elastic parameters of Models 1,2,3

The Thomsen parameters ε and δ which have been defined in section 1.6

were varied to study the influence of these parameters on AVO. The models they

considered for their tests are given in Table 1.1.

They made the following observations from the tests:

31

Figure 1.8. Plot of PP reflection coefficient with angle of incidence for the three classes of gas sand reflectors. The heavy solid curves are for isotropic material properties and the light solid curves are for average anisotropic parameters from Thomsen (1986) (i.e., δ=0.12, ε=0.13). (Kim et. al., 1993)

32

(I) (J)

Figure 1.9. Reflectivity difference between TI and isotropic elastic curves for ∆δ from top to bottom of +0.2, 0.0, -0.2, -0.4 and –0.6, respectively. Heavy solid curves are for ∆ε=0.0 and the light solid curves are for ∆ε=-0.3. The corresponding values of ∆δ and ∆ε for each model are indicated. Figures 1.9I(a), 1.9I(b), 1.9I(c) are Models 1(Class1), 2 (class2), and 3 (Class 3), respectively. (1.9J.) Reflectivity difference between TI and isotropic elastic curves for ∆ε from top to bottom of +0.2, 0.0, -0.2, -0.4 and –0.6, respectively. Heavy solid curves are for ∆δ=0.0 and the light solid curves are for ∆δ=-0.3. The corresponding values of ∆δ and ∆ε for each model are indicated. Figures 1.9J(a), 1.9J(b), 1.9J(c) are Models 1(Class1), 2 (class2), and 3 (Class 3), respectively. (Kim et. al., 1993)

33

(a) The combined effect due to both isotropic and anisotropic components for

expected values of δ and ε in TI medium is an increase of AVO effect

with incidence angle and zero-offset reflectivity.

(b) Both anisotropy parameters ∆δ and ∆ε between the upper and lower strata,

amplify the AVO effect. At angles of incidence below approximately 20 to

30 degrees, ∆δ is the most significant factor controlling AVO, while ∆ε

dominates above that angle range.

Figure 1.8 shows the elastic reflection coefficient variation with angles of

incidence for the three classes of gas sand reflectors. Figure (1.9I) and (1.9J)

shows the plot of reflectivity difference between TI and isotropic elastic curves

which clearly shows the influence of ∆δ and ∆ε for different range of incidence

angles.

ii) Effect of Anisotropy on imaging:

Lynn et al (1991) reported contrasting imaging results of a Gulf of Mexico dataset

from two different processing sequences: 1) DMO followed by CMP stack and

post-stack time migration, 2) pre-stack f-k time migration. The former routinely

images fault plane reflections better than the later. There can be four reasons

attributed to this imaging differences. 1) the effect of three-dimensionality on

two-dimensional imaging procedures. 2) effect of ray bending resulting due to

vertical velocity variation. 3) effect of geometry errors and/or cable feathering. 4)

Amplitude vs. Offset show significant anomalies for anisotropic shale

overlying gas sands.

34

transverse isotropy. When the first three causes where tested on synthetic data the

two processing schemes didn’t show much difference. However DMO-stack-

migration yielded superior results for transverse isotropy. The seismic sections

after these two types of processing are shown in figure (1.10). DMO used a higher

stacking velocity, which is very close to the primary velocity function for TI

medium for about 10 % anisotropy (Gulf of Mexico). So the events stack well in

the first case for DMO and post-stack time migration. If anisotropy is greater even

DMO fails to image complicated structures like fault planes and the processing

scheme needs to be modified to handle the anisotropy.

Another example may be cited from the field data from Angola, West

Africa reported by Alkhaliffah et al(1999). They did a moveout analysis using

(a) Hyperbolic moveout using Vnmo(0) (near offset curvature),

(b) Hyperbolic moveout using vstack ( best fit hyperbola over the full range of

offsets),

(c) Non hyperbolic moveout using Vnmo(0) and η(anisotropy parameter defined

in chapter 2.0).

The results are shown in figure 1.11. It is clear from the figure that it was

possible to flatten the events only after both Vnmo(0) and η was used.

Imaging is severely affected by anisotropy.

35

(a)

(b)

Figure 1.10. (a) Migrated image obtained after DMO and CMO stack. Note the presence of fault-plane reflections between 1.0 and 1.5 s. (b) Migrated imaged obtained by pre-stack f-k migration. The fault-plane reflections for the most part are absent. (Lynn et. al., 1991)

36

-

Figure 1.11. Moveout analysis of 2 different x-t CMP gathers with and without anisotropic correction. (Toldi et. al., 1999)

37

The above case studies clearly demonstrate that the error can be

significant if we use isotropic propagation model for a dataset affected

sufficiently by anisotropy. I plan to develop reliable techniques to incorporate

anisotropy into the existing processing schemes so that we are able to obtain

correct and better-resolved images of complicated structures from the seismic

data. In the next chapter I will discuss the basic concepts of NMO velocity

analysis and review some existing work that has been done to perform NMO

velocity analysis in anisotropic media. I will also give details about my approach

to estimate interval P wave velocity and the anisotropic parameter κ from the τ-p

transformed CMP gathers. In Chapter 3 a fast and efficient traveltime

computation scheme has been discussed which calculates plane wave travel-times

using the estimated parameters from the moveout analysis. These estimated

parameters and the computed traveltimes were used to perform pre-stack time

migration, the details of which are covered in chapter 4.

Normal moveout correction is affected due the non hyperbolic moveout resulting from anisotropy..

38

CHAPTER 2: MOVEOUT ANALYSIS AND PARAMETER ESTIMATION IN TRANSVERSELY ISOTROPIC MEDIA

2.1 INTRODUCTION

As stated in the first chapter many crustal rocks of interest in exploration

geophysics are either inherently anisotropic or behave as anisotropic materials

when the formations are sampled by seismic waves. Anisotropic wave

propagation is manifested in seismic data as anomalies in traveltimes, amplitudes

and waveforms. Subsurface rock layers are assumed to be isotropic in most

seismic processing methods. Isotropic assumptions are generally valid only for

reflections within small angles of incidence. Large offset and multi-component

recordings in seismic exploration provide data sets that are more appropriate for

studying anisotropic earth models. Anisotropy has generally been ignored

because of the additional complexity it introduces to the analysis of seismic data.

Studies of seismic anisotropy carried out in the last decade, however, have shown

that tremendous improvements can be made when velocity anisotropy is

incorporated into, and accounted for during seismic processing. For example,

crosswell seismic tomographic images are enhanced when velocity anisotropy is

considered (e.g., Chapman and Pratt 1992) and anisotropy can be a good indicator

of lithology (Byun et al. 1989).

In seismic exploration applications, ignoring the effects of anisotropy may

result in misties of seismic lines and erroneous estimates of target depths. Since

the pioneering work of Thomsen (1986), several attempts have been reported in

39

which anisotropic effects were included in seismic data processing (e.g., Tsvankin

and Thomsen 1994, 1995; Alkhalifah and Larner 1994; Alkhalifah 1997, etc.).

Most of these studies have attempted to find an extension of the standard rms

(root mean square) velocity travel time equation that includes the effect of

anisotropy in the offset-time domain. Here I propose to use the plane wave

domain for the processing of seismic data. The plane wave domain offers several

advantages. For example, no assumption needs to be made about the nature of

anisotropy (weak or strong), interval parameters can be estimated exactly and

τ − p or the vertical delay time equations for isotropic media are easily extended

to anisotropic media without any need for approximations. In the next section the

Normal Move-Out equations for the anisotropic media are developed.

2.2 NMO IN LAYERED ISOTROPIC MEDIA

For a horizontally stratified isotropic earth model, Taner and Koehler

(1969) derived the following travel time equation ( ) ,2 ∑=

n

nn xcxt (2.1)

where x is the offset, c0 = t(0), c1 =1

vrms2 , c2 and c3 are complicated functions.

For small offset approximations, equation (2.1) can be truncated to obtain

( ) ( ) )(0 42

222 xO

v

xtxt

rms

++= , (2.2)

where t(0) is two-way normal time and vrms is the rms velocity. By fitting the

above equation to the reflection travel times, one obtains the rms velocities, which

are then converted to interval velocities using Dix’s equation (Dix, 1955). This

40

procedure avoids the ray tracing required to compute exact reflection traveltimes.

Nonetheless it introduces errors in the velocity model estimate (Stoffa, Diebold

and Buhl 1982).

In the plane wave or delay time and slowness (τ − p) domain, the delay

time as a function of horizontal slowness is given by ( ) ( ) ( ) ,up dnp h q p q pτ = ∆ + (2.3)

where ∆h is the layer thickness, qup and qdn are the upgoing and downgoing

vertical slownesses respectively. In an isotropic medium they are equal and are

given by

,1 22

pv

qqq dnup −=== (2.4)

where v is velocity of the layer. τ − p curves for a multi-layered earth model can

be computed simply by summing the delay times through individual layers at each

ray parameter. That is, τ( p) = 2 ∆hi qi

i∑ . (2.5)

For future reference to anisotropic media, it is important to point out that

upgoing and downgoing slownesses in isotropic media have the same magnitude

and therefore we have the factor of 2 in equation (2.5).

Unlike equation (2.2), equation (2.5) makes no approximation to delay

time representation, makes no use of any rms parameter and delay times can be

computed without any numerical ray tracing (Stoffa et al. 1981). Thus by fitting

ellipses in the τ − p domain, one can obtain interval velocities. However, the

seismic data must have dense spatial sampling and large offset coverage for

41

meaningful transformation of the data to the plane wave domain. This

requirement is usually satisfied in modern seismic surveys.

2.3 NMO IN VTI MEDIA

2.3.1 Offset-time domain

Except for the work of Hake (1986) and Graebner (1991) for surface

seismics and Schmitt and Kebaili (1993) and Miller and Spencer (1994) for VSP,

most work on NMO analysis in VTI media have made use of an extension of the

NMO equation in isotropic media to VTI by including a fourth order non-

hyperbolic term (Hake et al. 1984; Thomsen 1986; Tsvankin and Thomsen 1994,

1995). Byun and Corrigan (1990) used a modified non-hyperbolic formula for P-

wave moveout in weakly anisotropic media. Berge (1991) tied the degree of

nonhyperbolic moveout for the SV-wave to the curvature of the wavefront near

the vertical. Analogous to equation (2.2), Hake et al. (1984) derived a three term

Taylor series for t2 − x2 for Quasi-P (hereinafter referred to as P) and Quasi-SV

(hereinafter referred to as SV) for layered transversely isotropic media. They

expressed the coefficients in terms of the anisotropy coefficients. An important

distinction between isotropic and anisotropic media is that one needs to

distinguish between phase velocity and group velocity in anisotropic media even

in the absence of attenuation. Namely, one needs to deal with group velocity

surfaces for analysis in the offset-time domain. Tsvankin and Thomsen (1994)

pointed out that since it is not possible to derive a concise analytic expression for

travel time curves using group velocities without assuming weak or elliptical

42

anisotropy, Taylor’s series expansion of t2 similar to equation (2.2) is useful for

VTI media. They derived the following non-hyperbolic equation for VTI media

,44

220

2 K+++= xAxAAtT (2.6)

where

( )

20 0 2 2

0

1, ,

1 2A t A

α δ= =

+

4 2 20

2

nmoA

t v

η= for small offset approximation

and 4 2 2 2 20

2

(1 2 )nmo nmo

Av t v x

η

η=

+ +

for long offsets.

( )(1 2 )ε δ

ηδ

−=

+

where t0 is the two-way normal time. At this point we make the following

remarks:

Equation (2.6) is an approximation of traveltime in a VTI medium.

Tsvankin and Thomsen (1994) have, however, shown that the approximation is

fairly accurate for many known anisotropic rocks.

In order to estimate the anisotropy parameters using least square fitting of

equation (2.6) to the observed travel time data, one essentially solves for two

model parameters, namely A2 and A4 (two way time is held constant and assumed

known). These two parameters are functions of the fundamental anisotropy

parameters that describe a medium. Thus it is not possible to derive independent

estimates of vertical velocity, ε and δ. Tsvankin and Thomsen (1994) showed

that one could uniquely estimate a combination of 0α and δ and so on. The non-

hyperbolic term (term containing A4) may be required even in the case of

isotropic media to model travel times at large offsets.

43

2.3.2 Plane-wave domain

In the plane wave domain, the basic procedure for analysis in VTI media

remains the same as that in isotropic media as described in the previous section.

We can make use of cylindrical symmetry and deal with radial (or horizontal)

slowness only. Consequently, the upgoing and downgoing vertical slownesses are

equal. They are, however, no longer given by equation (2.4) but need to be

obtained from the solutions of the following equation (e.g., Kennett 1984, p. 70)

2 21 1 2 3

14

2pq K K K K = − −

, (2.7)

where 21, KK and 3K are functions of the elastic coefficients 11 33 44 13, , ,c c c c

and the density, ρ, given by

( )213 44 211 44

133 44 44 33 44 33

2112

33 33

23

44

2 ,

,

.

c cc cK p

c c c c c c

cK p

c c

K pc

ρ ρ

ρ

ρ

+ = + − + −

= −

= −

( 2.8)

The relationship between the elastic coefficients and parameters

α0 ,β0 , ε ,δ and γ are given in Thomsen (1986).

In the plane wave domain, we can make use of equation (2.3) to compute

the delay time. Note, however, that for P waves in transversely isotropic media

)()( pqpq dnup = and thus we can use equation (2.5) even for the TI media; the

44

only difference is that we need to make use of equation (2.7) to evaluate the

vertical slowness. Equation (2.7) can be evaluated exactly for each layer and

requires no approximation. However, sensitivity of delay-time (or vertical

slowness) to different model parameters is not clearly observable. In the case of

elliptical anisotropy, the τ-p equation is given by 1/ 222

i1

( ) 2 1 .nl

iih

i

hp pτ α

α0=

= −

∑ (2.9)

where, hi is the layer thickness, α0i is the vertical P wave velocity, and αh

i is the

horizontal P wave velocity for the ith layer.

The above equation is exact for elliptical anisotropy and we clearly

observe that the vertical velocity term factors out completely, which gets divided

into the layer thickness. That is, 1/ 222

01

( ) 2 1nl

i ih

ip pτ τ α

=

= −

∑ , (2.10)

where, 0iτ is the two way normal time. Thus the vertical velocity cannot be

derived from P-wave delay time data and the NMO is controlled entirely by the

horizontal velocity - a fairly well known result (e.g, Thomsen 1986).

2.4 τ-P NMO EQUATIONS FOR WEAK VTI MEDIA FOR QUASI-P WAVES

The vertical slowness can be computed exactly for each layer in a

transversely isotropic medium for a given horizontal slowness. However the exact

equation (equation 2.7) is complicated and not physically intuitive. Moreover,

since all the five coefficients are not sensitive to plane wave delay times, reliable

estimates cannot be obtained by curve fitting. The coefficients by themselves are

not negligible in the second or higher order in the case of weak TI. As a result, the

45

special case of weak TI does not simplify any of the equations. For this reason, I

decided to map the equations in terms of the Thomsen’s parameters.

Parameterizing the equations in terms of the anisotropic parameters ε and δ,

which are algebraic combinations of the elastic coefficients is advantageous.

Since, for the case of weak TI we can neglect 2nd and higher order terms involving

these two parameters, the resulting equations become significantly simpler and

physically intuitive. Here I derive some simple forms for some special cases such

as elliptic anisotropy and weak transverse isotropy starting with the equations for

phase velocity (based on Cohen, 1997) shown in appendix B.

Elliptic Anisotropy: In the case of elliptical anisotropy, the expression for

vertical slowness can be derived using the relation

( )2 2

2

1q p

v p= − , (2.11)

where, v(p) is the phase velocity.

From equations (B8) and (2.11) vertical slowness is obtained as

( ) ( )2220

2 11

hppq α

α−= . (2.12)

where,

( ) )21(20 δαα +=ph (2.13)

The choice of ( )phα to represent the right hand side is not arbitrary – it

turns out that it represents the horizontal velocity in an elliptically anisotropic

medium (Alkhalifah and Tsvankin 1995). We can demonstrate this as follows. In

an elliptically anisotropic medium

46

( )

2 2 2 20

2 2 220

1

11

h

h

q p

q p

α α

αα

+ =

= −

(2.14)

This is the same as our equation (2.12) above. Note that in the (τ-p)

expression for each layer the vertical velocity gets divided into the layer thickness

resulting in the vertical two way time for the layer. This would be true for each

layer with elliptic anisotropy. In an elliptically anisotropic medium the (τ-p)

curves are elliptical (as in the case of isotropic media), the moveout is determined

completely by the horizontal velocity. The vertical velocity is used in depth to

time conversion. Thus one can never estimate the vertical velocity unless the

depth is known.

Weak Transverse Isotropy:

Using equation (B9), we can express the vertical slowness as

( )( )

−

−++= z

zzq 222

0

2

1

11

δεδα (2.15)

where, 2 20z pα= .

Now by expanding the denominator in powers of z and retaining terms containing

only up to first power of z (i.e, we assume that the p values are small) and δ I

obtained

( ) ( )[ ]δεαδαα

−−+−≈ 40

420

220

2 22111

ppq . (2.16)

In equation (2.16) we substitute

47

( )

[ ]2220

2

20

2

11

21

elel

el

pq αα

δαα

−=

+= , (2.17)

to obtain ( )

−−≈ 2

20

422 2

1el

elq

pqq

δεα. (2.18)

Note that for small ray-parameters or for isotropic media ( δε = =0 )

equation (2.18) reduces to the expression for vertical slowness in the isotropic

media. Thus in this case of weak transverse isotropy the first term in equation

(2.18) contributes to the elliptic anisotropy and the second term is the nonelliptic

correction at large ray-parameters. Equation (2.18) is rewritten to factor out the

vertical velocity to obtain

( )

−−≈

2

40

42

20

2 21

1

elel

q

pqq

δεα

α, (2.19)

where,

( )2 2 21el elq p α= − .

From (2.17) and (2.19) we have

−≈

2

442

20

2 21

1

el

elel

q

pqq

κα

α, (2.20)

where

)21()21()(2 δ

ηδδε

κ+

=+

−== . (2.21)

0 1 2elα α δ= + . (2.22)

Note that since δ is generally very small for weak TI media, ηκ ≈ .

48

Using equation (2.20) we have the following equation for delay time in

weak VTI media:

( ) ( )∑=

ι0

−−−=

NL

iiel

ieli

eli

p

pp

hp

1

2/1

22

442/122

1

2112)(

α

καα

ατ

, (2.23)

or,

( ) ( )

1/ 24 41/ 22 2

2 21

2( ) 2 1 1

1

NL i ii i el0 el i

i el

pp p

p

α κτ τ α

α=

= − − −

∑, (2.24)

where i0τ is the two-way normal time, i

elα is the elliptic velocity, iκ is the

anisotropy parameter for layer i, and NL is the total number of layers. Note that

for isotropic and elliptic anisotropy cases, equation (2.24) is simply a sum of

ellipses. The anisotropy parameter κ introduces the non-elliptic part to the

moveout, which would be significant at mid to large ray parameters. At this

juncture I would like to point out that κ alone may not be sufficient to fit the non-

elliptic moveout at large ray parameters. Equation (2.24) is analogous to the (x,t)

travel time equation derived by Alkhalifah (1997). Note, however, that this

expression does not make use of any rms parameters. Interactive parameter

estimation using equation (2.24) was performed both on synthetic and real field

data from the Gulf of Mexico. The results from the interactive analysis will be

discussed in section 2.7. I have also formulated a automatic parameter estimation

technique using a optimization tool called very fast simulated annealing (VFSA).

The methodology and results will be discussed in section 2.8.

49

2.5 τ-P NMO EQUATIONS FOR WEAK VTI MEDIA FOR QUASI-SV WAVES

Like the case of quasi-P waves exact equations for vertical slowness and

delay times can be derived for the case of vertically polarized S waves. I derived a

simple equation for the special case of weak transverse isotropy for SV phase

velocity as a function of ray parameter. The derivation is given in details in

appendix C.

The simple looking expression for the phase velocity vsv as a function of

ray parameter is given below,

( ) ( )0 2 2 4 40 02

01svv p p

αβ ε δ β β

β

= + − −

. (2.25)

Since 220 pz α= , I can express the vertical slowness as

( )

22 0

2 2 220 02020

1 1

1 2( )

q z

z z

β

β αβε δ ε δ

α

= − + − + −

. (2.26)

Now I expand the denominator in powers of z and retain terms containing

only up to first power of z (i.e, we assume that the p values are small), ε and δ to

obtain 1/ 2

2 2 4

20

211

el

el elel

pq q

q

ηα ββ

= +

(2.27)

where ( )

0

20

0 20

2 2

0

1 2

1 2

11 .

el

el

elelq p

α α δ

αβ β ε δ

β

ββ

= +

= + −

= −

50

Substituting the above expression for vertical slowness equation for delay

time for the Sv case can be written as:

1/ 2

2 2 4

0 22

( ) 1

el

el elel

pp q

q

ηα βτ τ

= +

. (2.28)

A careful inspection of the above equation suggests that we have an

expression similar to that for the quasi P wave. However the delay time for Sv

depends on both P and Sv elliptic velocities. Also the anisotropic parameter is

slightly different from κ for the P-wave case. Since interval estimates of both κ

and η can be interactively estimated from the PP, Sv-Sv or P-Sv tau-p data

respectively the above equation will be very useful to perform multicomponent

anisotropic analysis of seismic data. In section 2.7 I will present results from

interactive analysis on some synthetic data sets.

2.6 RESULTS FROM INTERACTIVE ANALYSIS

Equations (2.24) and (2.28) were used to design an interactive parameter

estimation scheme from the plane wave data. Interval elliptic P wave velocity,

elα and the anisotropic parameter κ can be estimated in a top down fashion

interactively form the data. The results from the interactive analysis are discussed

here. Figures 2.1(a) and 2.2(a) show τ-p curves for a Dog Creek Shale and Taylor

Sandstone whose properties are listed in table 2.1 (taken from Thomsen 1986).

The top curve in each one of the figures is computed by using the exact equation

for vertical slowness (equation. 2.7). The lower curve is computed with elliptic

51

velocity alone and the middle curve is computed using both elα and κ using

equation (2.24). Note that for weak transversely isotropic media, the τ-p moveout

is predicted quite well by the approximate equation given in equation (2.24).

Clearly the effect of κ is observable at large ray parameter values (p>0.2 sec/km

for Dog Creek Shale and p>0.14 sec/km for Taylor Sandstone). The sensitivity of

vertical delay time calculations to elα and κ for the Dog Creek Shale and Taylor

Sandstone models are shown in Figures 2.1(b) and 2.2(b) as contour plots of rms

delay time residuals for a large suite of elα and κ values. Notice that within

some small region, there exists trade-off between these two parameters. In other

words, within some small range in the parameter space, elα can be increased and

κ can be decreased (or vice-versa) to match the moveout. For one layer

examples such as those shown in Figures 2.1 and 2.2, the τ-p development may

not appear very interesting since even in the (x,t) domain one can demonstrate

such features. Note, however, that unlike the (x,t) rms velocity analysis, we need

to make no further approximations for the multi-layer case. At this stage, it is

worthwhile to summarize the principal advantages of using the τ-p method. They

are as follows:

Unlike the (x,t) method, the higher order or non-elliptic term in the τ-p

equation is non-ambiguous. Under 1D assumption, the non-elliptic term, if

required to fit the data, must be due to anisotropy.

The 2-term approximate equation for weak anisotropy is not required due

to any computational or theoretical limitations. The exact form of the equation

can be evaluated equally well. The two-term representation, however, brings out

52

the physics well and demonstrates that for weakly anisotropic media, vertical

velocity cannot be resolved from P wave travel time alone. This is, however, not

necessarily true for general (strong) VTI case.

Unlike the (x,t) velocity analysis method, the τ-p curves for each layer can be

fitted in a top down fashion resulting in direct estimates of interval elα and

κ values. Figures 2.3 – 2.6 shows real data example for a Gulf of Mexico

dataset. In Figure 2.5, which shows a zoomed plot of the NMO corrected tau-p

gather at the target zone after three types of correction. One that uses both elliptic

P-wave velocity and κ shows enhanced flattening. The zoomed plot (Figure 2.6)

of the stack section around the target illustrates the enhancement of stacked events

after TI nmo. There was no real data dataset to test the converted wave (P-Sv)

nmo. However some synthetic tests were performed. From Figure 2.9 we can see

that our expression of vsv(p) shows excellent agreement with the exact equation

for ray parameter range up to 0.6 sec/km. Sv τ-p curves are compared with the

exact expression for delay time for Dog Creek Shale and Taylor Sandstone

models. Notice that the two-term approximate equation does a good job in

predicting the moveout. Some P-Sv synthetic data examples (Fig. 2.11) are

presented for a data generated using the model in table 2.2. The residual moveout

using βellip is well corrected using both βellip and η. Apart from the interactive

method I have also formulated an automatic parameter estimation technique using

VFSA. The next section talks in length about the methodology. The results using

the automatic estimation method for the Gulf of Mexico dataset are also discussed

in the section.

53

Taylor sandstone Dog Creek shale

0α (km/s) 3.368 1.875

0β (km/s) 1.829 0.826

ε 0.110 0.225

δ -0.035 0.1

t0 (s) 1.781 3.2

thickness (km) 3.00 3.0

Table 2.1: Anisotropy coefficients of Taylor sandstone and Dog Creek shale. α0, β0, ε, δ from Thomsen(1986).

54

Figure 2.1(a). τ-p curves for Dog Creek Shale model using exact equation for vertical slowness(red), elliptic velocity isotropic model (green) and a two term weak anisotropy model (blue).

Figure 2.1(b). Sensitivity of delay time to elliptic velocity and κ. Note the trade-off between the two parameters.

55

Figure 2.1(c). NMO corrected synthetic ? -p seismograms with best-fit isotropic velocity model (upper curve), near p elliptic velocity model (middle panel) and two-term weak TI model.

56

Figure 2.2 (a). τ-p curves for Taylor Sandstone model using exact equation for vertical slowness (red curve), elliptic velocity isotropic model (green curve) and a two term weak anisotropy model (blue curve).

Figure 2.2 (b). Sensitivity of delay times to elliptic velocity and κ. Note the trade-off between the two parameters.

57

Figure 2.2 (c). NMO corrected synthetic τ-p seismograms with best-fit isotropic velocity model (upper curve), near p elliptic velocity model (middle panel) and two-term weak TI model. Note that with a best-fit isotropic model we are able to fit near and high ray-parameter traces but intermediate ray-parameter traces remain uncorrected (a diagnostic of anisotropy). A weak anisotropy model is able to flatten the data very well.

58

Figure 2.3. Analysis of Gulf of Mexico data : CMP 691 in (x,t) (left panel) and τ-p domains.

59

(a)

(b)

(c)

Figure 2.4. Results from interactive τ-p velocity analysis of CMP 691: (a) best-fit isotropic model, (b) near p elliptic velocity model, and (c) two-term best fit TI model.

60

(a)

(b)

(c)

Figure 2.5. The zoomed plots of Fig 2.4(a-c) in the time window 4.4 to 4.52 sec. The target horizon is the reflection event at 4.5 sec. (a) The best-fit isotropic model: Note the typical bulging effect. (b) near p elliptic velocity model, and (c) TI Model. Note the excellent improvement in the flatness of the event at 4.5s compared with Fig (a).

61

Figure 2.6(a) Zoomed plot of the Stack section generated using the isotropic model at the target zone.

Figure 2.6(a) Zoomed plot of the Stack section generated using the TI model at the target zone. Note the improvement in quality of stacking with the incorporation of κ

62

1.480

1.531

1.582

1.633

1.685

1.736

1.787

1.838

1.890

1.941

1.992

2.043

2.094

2.146

2.197

2.248

2.2992.00

2.50

3.00

3.50

4.00

4.50

5.00

Figure 2.7. Stacked section obtained with the best-fit TI model on which the elliptic velocity model is superimposed. Target zone is highlighted with a red box.

T I ME IN S E C

100 200 300 400 500 600 700 800 900 1000 CDP #’s à

63

0.000

0.014

0.028

0.041

0.055

0.069

0.083

0.096

0.110

0.124

0.138

0.151

0.165

0.179

0.193

0.206

0.2202.00

2.50

3.00

3.50

4.00

4.50

5.00

Figure 2.8. Stacked section obtained with the best-fit TI model on which the anisotropic parameter κ is superimposed. Note that κ values generally increase with depth and laterally varying; they show significantly larger values near the target zone. Target zone is highlighted with a red box.

100 200 300 400 500 600 700 800 900 1000 CDP #’s à

T I ME I N SEC

64

Figure 2.9. Plot of Vsv as a function of rayparameter. The cyan curve was generated using Thomsen’s approximate equation, the red curve is generated using the exact equation for Vsv, blue curve using my expression for Vsv, and the green curve plots Vsv values from Daley and Hron’s equation. Note that up to rayparameters of 0.6 ( a range realistic for all exploration purposes) my equation shows excellent agreement with the exact result.

Vp(km/s) Vsv(km/s) κ η

2.35 1.00 0.08 1.1

Table 2.2. Model parameters for the synthetic data in figure 2.9.

65

Figure 2.10(a). Comparison of exact and two-term approximate equations for Sv-wave for Dog-Creek shale. The ray-parameter is in sec/km.

Figure 2.10(b). Comparison of exact and two-term approximate equations Sv-wave for Taylor Sand stone. The ray parameter is given in sec/km.

66

Figure 2.11(a). NMO corrected P-S τ-p data generated using the model in table 2.2. The correction is performed using for Vpellip and κ for the Pwave path and only the Sv elliptic velocity for the Swave path.

Figure 2.11(b). NMO corrected P-S τ-p data generated using the model in table 2.2. The correction is performed using the TI model. Note the improved flattening after incorporation of η.

67

2.7 AUTOMATIC ESTIMATION OF ELLIPTIC P WAVE VELOCITY AND ANISOTROPIC PARAMETER

In the previous sections I have presented a technique to interactively

estimate the elliptic P wave velocity, αel and the anisotropic parameter κ in the

plane wave domain. While interactive estimation of parameters is very accurate

and gives the user more control, it is very laborious, especially when one needs to

process a large dataset. Here I present a technique to automatically estimate these

parameters from the data. Techniques to estimate velocities for the isotropic

media are well known. The most well known technique uses a scan type velocity

analysis where coherencies for a suite of velocities are calculated (Taner and

Koehler, 1969; Schneider, 1971; Schneider, 1984). The correlation is displayed as

a function of stacking velocity and two way time. The higher correlation values

correspond to the best fit to the hyperbolas. The table of estimated stacking

velocities can then be converted to interval velocity using the Dix’s equation. One

of the pitfalls of the scan type velocity analysis on the x-t domain data is that it

only searches for the rms velocities assuming straight ray paths giving the best fit

to the reflection hyperbolas. The interval velocity obtained from the rms estimate

using the Dix formula can be significantly different from the true layer velocity.

Analysis in the τ-p domain enables one to directly estimate interval velocities

from the data. The philosophy of scan type velocity analysis was extended to the

τ-p domain by Schultz (1982). He used a layer stripping approach to estimate the

interval velocities, where the layers where stripped in a top-down fashion. Stoffa

et al (1980) presented a method to interactively estimate interval velocities.

Simmons (1994) implemented a method reported by Schneider and Backus

68

(1968). A gather is first NMO corrected using a global velocity model (e.g. a

series of gradient functions in time or depth), and the residual NMO is then

related to the perturbation of the velocity with respect to the model. Reflection

tomography approach has been used to estimate background velocities (Bishop,

1985).

Considerable work has been done in the area of seismic anisotropy to

estimate parameters by inversion of seismic data. Alkhalifah and Tsvankin (1995)

showed that P-wave NMO velocity for dipping reflectors in homogenous VTI

media depends just on the zero-dip value Vnmo(0) and the anisotropic parameter η.

They designed an inversion procedure to estimate η and the NMO velocity as a

function of ray parameter using moveout velocities for two different dips. Toldi

et. al. (1998) presented two techniques to automatically estimate η and Vnmo. One

of the methods was based on Dix type inversion (layer stripping) for interval

values of η and Vnmo(0). To stabilize the procedure, each output curve was

generated from inversion of a collection of 25 CMPs around the output point. This

resulted in significant signal enhancement. In another approach they inverted for

the velocity as well as the Thomsen parameters ε and δ for the 3-D case. They

used measured picks of Vnmo(0) and η, along with 3-D time-migrated maps and

depth picks in wells as inputs into the inversion program. The inversion iteratively

models these effects through ray-trace based modeling and data fitting. At the

same time their program also evaluates the value of δ that will be required to

make the computed depths agree with the depth provided from the well picks.

69

From the estimate of η and δ, the other parameter ε can be determined. They

presented some very encouraging results from a dataset in West Africa.

Xia et al (1998) a novel method to estimate background velocities for 1-D

earth models in isotropic media using NMO as the criterion for automatic

parameter estimation. They used Very Fast Simulated annealing (VFSA) as the

nonlinear inversion tool. I propose to extend their method to estimate the

parameters αel and κ from the τ-p data. In the next section I will describe the

linear and non-linear inversion techniques. Section 2.7.2 describes the model

parameterization scheme. The results obtained for the Gulf of Mexico dataset are

shown in section 2.7.3.

2.7.1 Non-Linear Vs. Linearized Inversion

When the data and the model are related linearly, linearized inversion

schemes are used to estimate model parameters. Ideally a linearized inversion is a

one step inversion procedure and hence is very fast. Even iterative linear

inversion often converges after no more than 5 iterations, which means very few

forward calculations are needed. But pure linear problems are extremely hard to

find and most geophysical problems are quasi-linear or nonlinear in nature. For

quasi-linear problems the data and the model are related through the Frechet

derivatives, which are evaluated either analytically or numerically. Data noises

are incorporated as data covariance. Incorporation of a priori information aids in

faster and accurate convergence. Posterior model covariance and model resolution

are quantitative measures of the inversion results.

70

The misfit function, which measures the differences between the observed

data and the predicted data, is typically characterized by one global minimum and

many local minima for seismic data. The linearized inversion method may get

trapped in one of the local minima if the starting model is not close to the global

minimum.

Monte Carlo methods, which are among the popular nonlinear inversion

methods, randomly sample the model space, and estimate the uncertainty in the

model estimation. Being truly random, each trial model does not care about a

good or a bad model. As a result Monte Carlo methods are computationally

expensive.

Simulated annealing, which is a variant of the Monte Carlo method, uses a

guided search method. It was first proposed by Kirkpatrick, Gelatte, and Veccchi

(1983). Simulated annealing (SA) is a global optimization technique that

simulates the crystallization process from a melt, which essentially means going

from a disordered to an ordered system. In Boltzmann annealing, a random point

in model space is selected and the energy E0 or misfit is calculated. The new

model is accepted unconditionally if the energy or misfit associated with the new

point E1 is lower, i.e., if E1 < E0 . If the new point has a higher misfit, then it is

accepted with probability exp (-(E1 - E0)/T) where T is a temperature like factor

that controls the likelihood of accepting the step in model space. This acceptance

criterion is known as Metropolis criterion. The probability of accepting a step is

always greater than zero; hence the algorithm can climb out of a local minimum.

The temperature is usually slowly decreased as the algorithm progresses to reduce

71

the probability of accepting a bad step to zero as the global minimum is reached.

The classical Metropolis algorithm (Metropolis et al., 1953: Sen and Stoffa,1991)

draws perturbation to a current model from a flat distribution in a user-defined

search window, and therefore many moves are rejected.

The particular scheme used here is called very fast simulated annealing

(VFSA). It was proposed by Ingber (1989). VFSA has two major differences from

its variant SA.

(1) Instead of drawing a new model from a temperature independent flat

distribution, it is now drawn from a temperature dependent Cauchy like

distribution centered around the current model. As a result a larger

sampling of model space is possible at the early stages of inversion when

the temperature is high. Also in VFSA each model parameter can have its

own cooling schedule and model space sampling scheme.

(2) The second difference lies with the cooling scheme. In VFSA an

exponential type of cooling scheme is used instead of linear or logarithmic

schemes used for SA. Since exponential function decreases faster than

linear or logarithmic functions, this type of cooling scheme adds to the

speed of the algorithm.

2.7.2 Model Parameterization and Error Function

Once the inversion method was decided upon a good model

parameterization scheme was necessary. I chose a simple ramp type of model

parameterization scheme to give the starting model and the search bounds for

inversion. I used two different slopes for both velocity and kappa models. The

72

point for the break in the slope was chosen depending on the CDP position within

the line. While the same bound for the elliptic velocities were used for the whole

line, the bounds for kappa were chosen depending on whether the CDP as near the

target zone or away from it. From CDPs at and near the target zone wider bounds

for the kappa values were used. The model bound for one of the CDPs at the

target is shown in figure (2.12).

The calculation of the error was based on cross-correlation of each of the

NMO corrected traces with a pilot trace. The pilot trace was created by the

following equation,

1( , )

( )

np

ipx ip jt

xpilot jtnp

==

∑, (2.43)

where x(ip,jt) is the nmo corrected trace, np is the total number of traces

and nsamp is the total number of samples in each trace. The pilot trace is then

normalized once again by dividing it by the maximum amplitude, i.e.,

max( )xpilot

xpilotxpilot

= . (2.44)

Finally as mentioned earlier the accuracy of the NMO correction is

determined and is tested by examining the values obtained from the

crosscorrelation. The correlation value is calculated using the following equation

given by,

( )( )* 1

** *

xpilot xxcor

npxpilot xpilot x x

= −

∑∑ ∑

. (2.45)

So the model returning the smallest value of crosscorrelation after all the

VFSA runs is saved as the final model.

73

2.7.3 Results from automatic parameter estimation

I have presented the results obtained automatic parameter estimation using VFSA.

Figure (2.13) shows the NMO corrected τ-p CDP gather and the elliptic velocity

and κ obtained from the inversion for a CDP outside the target zone for the Gulf

of Mexico dataset. Figure (2.14) shows the results obtained for a CDP in the

target zone. The event at the target zone is well flattened. The model parameters

obtained from the inversion generally agree with the ones obtained from

interactive analysis. However there are a few differences. From figures (2.13) and

(2.14) the oscillatory nature for the velocity and kappa estimates can be observed

as compared to the smooth step like estimates from the interactive analysis. This

results from the fact that for inversion the sampling interval was taken as the layer

thickness. Since our inversion algorithm is based on traveltime fitting it was hard

to constrain the zones with no reflected events. Inversion was run on the whole

line. Gridded model parameters obtained from the inversion have been presented

in figure (2.15). The high kappa values around the target show the good

agreement of inverted results with that obtained from interactive analysis. Figure

(2.16a) presents plot of negative correlations vs. the number of iterations for a

CDP around the target zone. Increase in correlation with increase in the iteration

number can be seen from the plot. In figure (2.16 b) the gradual cooling scheme

has been presented. In spite of the oscillatory nature of the estimates there is much

to be gained from the automatic parameter estimation technique. It saves one from

the laborious process of interactive analysis for every CDP gather. However note

that a reliable parameter estimate through inversion needs sufficient a priori

74

information and tight bounds for model parameter values. One way would be to

estimate parameters interactively for a set of few CDPs spanning across the entire

dataset. This provides a reconnaissance idea of the trend of variation in the model

parameter values and helps the user to choose suitable bounds at different parts of

the dataset. From figure (2.15) we can observe a sharp jump in kappa values at

around cdp # 600 as opposed to a gradual increase from the interactive analysis.

This sharp change results primarily due to the fact that we have introduced a sharp

change in the search bounds for kappa from that CDP location. Some error may

also be introduced in the inversion results due to the tradeoff between the elliptic

P wave velocities and κ. One can overcome this to some extend by using a

constrained optimization algorithm.

2.8 SUMMARY

The existence of seismic anisotropy is widespread, as is evidenced by travel time,

amplitude and waveform anomalies that cannot be accounted for by isotropic or

even laterally heterogeneous earth models. Anisotropy causes misties in

reflection lines and wrong estimates of target depths. Because of this, several

attempts have recently been made to include anisotropy parameters in seismic

processing. Several standard processing algorithms have been modified to include

anisotropic effects. For example, the standard NMO equation in the offset-time

domain has been modified to include a non-hyperbolic term to account for

anisotropy. In this chapter I have revisited the parameter estimation problem in

anisotropic media using P-wave and Sv-wave travel time data in the plane wave

domain. The plane wave domain is the most natural way to study anisotropy.

75

(a) (b)

Figure 2.12. (a) The starting elliptic velocity bounds used for one of the CDPs at the target zone (b) The starting kappa bounds used for one of the CDPs at the target zone

76

(a ) (b) (c)

Figure 2.13. (a) NMO corrected CDP gather from outside the target zone using the inverted model from VFSA (b) Elliptic velocity model fromVFSA (c) Kappa model from VFSA

T I M E IN S E C

Ray Parameters in sec/km

77

(a) (b) (c)

Figure 2.14. (a) NMO corrected CDP gather from the target zone using the inverted model from VFSA (b) Elliptic velocity model fromVFSA (c) Kappa model from VFSA.

T I M E IN S E C

Ray Parameters in sec/km

78

(a)

Figure 2.15(a). Kappa model from VFSA for the Gulf of Mexico dataset for 740 CDPs.

(b)

Figure 2.15(b). Elliptic velocity model from VFSA for the Gulf of Mexico dataset for 740 CDPs.

79

(a)

Figure 2.16(a). Plot of negative correlations vs. the number of iterations for a CDP at the target zone.

(b)

Figure 2.16(b). Plot of temperatures vs. the number of iterations for a CDP at the target zone.

80

Vertical slowness can be computed exactly (either analytically or numerically)

which is required for the calculation of delay time as a function of ray-parameter.

However the equations are complicated and offer little physical insight. As such

estimation of parameters are extremely difficult from the data. For weak TI media

we derived an simple equation for vertical slowness that includes a non-elliptic

term for large ray-parameters both for the P-wave and Sv wave cases. By

matching delay time curves, one can obtain interval parameters. Similar analysis

in the offset-time domain will result in rms parameters that need to be converted

to interval parameters using Dix-type formulae, introducing additional levels of

approximation. I also note that the plane wave domain is the required domain for

AVO analysis since it requires least squares fitting of linearized reflection

coefficients that are given as a function of plane wave angle. The τ − p analysis

as described here for the VTI case can be easily extended to a more general type

of anisotropy and the effect of dipping layers can also be incorporated. The results

from the analysis described here can be used directly in the full waveform

inversion of the seismograms resulting in the estimates of uncertainties in

different layer anisotropy parameters. Finally the automatic parameter estimation

method presented in the last part of the chapter can be mode more robust by

incorporating sophisticated inversion tricks like regularization to further constrain

the algorithm.

81

CHAPTER 3: TRAVELTIME COMPUTATION IN TRANSVERSELY ISOTROPIC MEDIA

3.1 INTRODUCTION

Travel time calculations play an important role in many aspects of seismic

data processing and imaging. For example, Kirchhoff methods of migration and

modeling seismic data require calculation of Green’s functions, which depend on

the traveltimes from survey points on the surface and depth points in the velocity

model. There are a variety of methods available to compute transit times for

seismic waves. If the medium is uniform and homogenous, the direct path of

seismic waves is the straight line from source to the receiver, and the traveltime is

easy to calculate. For complicated media, one often requires computationally

expensive schemes to find accurate travel times. Usually some form of ray tracing

is required to determine travel times in media that vary laterally as well as

vertically.

Ray tracing is based on the concept that seismic energy of infinitely high

frequency follows a trajectory determined by the ray tracing equations, which

physically describe the continuity of seismic energy until it is refracted by

velocity variations. In “shooting” methods of ray tracing, a fan of rays is shot

from one point in the direction of the target. The correct path and traveltime to

connect the two points may be approached with successively more accurate

guesses (Cerveny et. al., 1977). “Bending” methods of ray tracing start with an

initial guess for the ray path. The ray path is bent by the perturbation method until

82

it satisfies a minimum travel-time criterion (Thurber, 1987). There are several

problems associated with ray tracing methods. First, for strong laterally varying

velocities, there can be several paths connecting the two points of interest.

Second, even for a smooth medium there may be shadow zones because a small

change in take-off angle may result in a large change in the ray path. Shooting

methods of ray tracing often have trouble finding the correct rays in the shadow

zones. Bending methods do give an answer in shadows but the ray path may

correspond to a local minimum. Due to these problems ray tracing methods over

the last decade have sometimes been replaced with the direct computation of

traveltimes. One of the popular methods of directly computing travel times is by

obtaining a finite-difference solution to the Eikonal equation. Here travel times

from a specified source location are calculated for all grid points, thus avoiding

many of the problems associated with shadow zones. In the following section, I

will discuss some of the finite difference schemes.

3.2 FINITE DIFFERENCE SCHEMES

To avoid problems associated with ray tracing methods, direct travel time

computation techniques were developed. Vidale (1988, 1990) proposed a finite-

difference traveltime computation method that calculates the first event’s arrival

at each grid point from a source of seismic energy. The source of seismic waves is

assumed to be at grid point A (figure 3.1). The timing process is initiated by

assigning point A the traveltime of zero. The four points adjacent to the point A,

labeled B1 though B4 in the figure (3.1), are given the traveltimes

)(2 ABii ssh

t += ,

83

Figure 3.1. The source grid point A and the eight points in the ring surrounding A.

where h is the mesh spacing, sA is the slowness at the point A, and sBi is the

slowness at the grid point Bi being timed. The propagation of two-dimensional

geometric rays and therefore the propagation of two-dimensional wavefronts is

guided by the Eikonal equation of ray tracing

222

),( zxszt

xt

=

∂∂

+

∂∂ , (3.1)

that relates the gradient of the travel time to the velocity structure. The coordinate

axes are x and z, and s is the slowness. The two differential terms can be

approximated using finite differences as

)(21

3120 tttthx

t−−+=

∂∂ (3.2a)

C2

B3

C3

B2

A

C1

B1

C4 B4

84

and

)(21

3210 tttthz

t−−+=

∂∂ , (3.2b)

where, t0, t1, t2 and t3 are the traveltime to the points A, B1, B2, and C1 from the

origin. The origin can be chosen at any point in the grid. Substituting equations

3.2(a) and 3.2(b) into equation (3.1) we get, 2

122

03 )()(2 tthstt −−+= . (3.3)

Equation (3.3) gives the travel time to point C1 using the travel times from

the origin (source) to point A, B1, B2, in a plane wave approximation. Point A

does not need to be the source point for this equation to apply. In Vidale’s scheme

the time at the fourth corner of a square can be obtained from the times at the

other three corners using equation (3.3). First, the times of the four corners are

found from the times of their neighbors. Solution will progress by solving rings of

increasing radius around the source point. One of the drawbacks of Vidale’s

method is that it fails for media with high velocity contrast. Also his algorithm is

not vectorizable. Qin et al. (1992) presented an algorithm which is very similar to

the algorithm presented by Vidale (1988). But the algorithm performs the wave

front extrapolation calculation in a way that is more analogous to actual wave

propagation.

Van Trier and Symes (1991) proposed an alternative traveltime method

based on the viscosity solution of the Eikonal equation. The method solves a

conservation law that describes changes in the gradient components of the

traveltime field. They used the Eikonal equation in polar coordinates (r, θ),

),(1 2

22θ

θrs

trr

t=

∂∂

+

∂∂ . (3.4)

85

The conserved-flux function in polar coordinates was defined as,

2

22)(

r

usuF −= , (3.5)

where θ∂

∂=

tu . Thus the Eikonal equation can know be written as,

)(uFrt

=

∂∂ .

By taking a derivative with respect to θ we get,

θ∂∂

=∂∂ )(uF

ru , (3.6)

which is the conservation law used for the finite difference scheme. To

understand the conservation law lets take an integral of u in equation (3.6) over a

small angle range [a,b].

))(())((),( auFbuFdrur

b

a

−=∂∂

∫ θθ . (3.7)

Thus the rate of change of u over [a,b] equals the net flux F[u(b)]-F[u(a)].

If one arranges a zero net flux, for example, by demanding that F[u(b)]-F[u(a)]=0,

then the total amount of u in [a,b] is conserved, hence the name “conservation

law”. Van Trier and Symes gave an upwind finite difference scheme to solve for

equation (3.6) using the following equation,

)])1(,([)],([(1),())1(,(

rnjuFrnjuFr

rnjurnju∆−∆−∆∆

∆=

∆∆∆−∆+∆

θθθ

θθ . (3.8)

Using the above equation, u can be obtained for all the grid points. After

computing u, traveltimes can be obtained from an integration of rt

∂∂ over r with a

simple trapezoidal rule. rt

∂∂ is obtained from the Eikonal equation (3.4). Though

86

this method is vectorizable, like Vidale’s method it fails for media with high

velocity contrasts.

3.3 TRAVEL TIME COMPUTATION IN ANISOTROPIC MEDIA – SUMMARY OF PREVIOUS WORK

Considerable work has been done in the last decade to develop techniques

for computing travel times in anisotropic medium. Dellinger (1991) presented an

extension of Van Trier and Symes’s method to transversely isotropic media. He

derived the flux terms for the TI case as,

2)( 2

++−−= BBAuFp (3.9)

for qP waves and

2)( 2

++−= BBAuFsv , (3.10)

for qSV, where

3344

24411

44411 )1)((4

CCuCCuCC

A++−−

= (3.11)

and

4433

44332

44132

1311213 )(2

CCCCuCCuCCC

B+++−

= . (3.12)

He used the same upwind finite difference scheme described in (3.8).

Faria and Stoffa (1994) extended the method proposed by Schneider Jr. et

al. (1992) to transversely isotropic media. In figure (3.2) the traveltime to a point

87

(x,z2) is given by the traveltime from the source to point (xa, z0) plus the

traveltime from point (xa, z0) to point (x,z2). Assuming a point source, the

traveltime at points (xa, z1) and (xa, z2) are given by: 2/12

12

1 )( zxSt aa += , (3.13)

and 2/12

22

2 )( zxSt aa += , (3.14)

where Sa corresponds to the average slowness from the apparent source to point

(xa,z1) and point (xa, z2). Using (3.13) and (3.14) they obtained t0 as

)( 21

20

221

20 zzStt a ++= . (3.15)

Equation (3.15) can be rewritten as a function of an angle:

+

∆−+= 2

1

2

222

120 tan

zx

zStt a ϕ, (3.16)

while the total traveltime to point (x,z2) is given by,

ϕφ

cos)(

0xS

tt∆

+= , (3.17)

where S(φ) is the group slowness, φ is the group angle and ϕ is the angle between

the vertical and the ray-path that ranges from 0 to 90 degrees in each individual

calculation. The traveltime at grid point (xa,z2) is calculated by minimizing

equation (3.17). To get the minimum one need to solve the equation

0=ϕd

dt , (3.18)

to obtain the propagation angle. For group velocities they used the approximate

equation for group velocity from Byun et. al., (1989) given by

φφφ 43

221

2 coscos)( aaag p −+=− (3.19)

88

Figure 3.2. Traveltime computation scheme developed by Faria and Stoffa (1993). Using the known traveltime t1 and t2 , t0 is calculated in order to minimize the total traveltime t, from an apparent source to the point (x,z2).

(Xa, Z1)

(Xa, Z2) (X, Z2)

(Xa, Z0)

t1

t2

t0

ϕ

t

89

where a1, a2, a3 are functions of the anisotropic parameters (α, β, ε, δ).They can be

solved by using the group velocity expressions at 0, 90 and 45 degrees. For

traveltime mapping they used two different schemes and repeated their

calculations for each grid point several time to ensure that the true minimum

could be obtained for the derivative in equation 3.18.

Alkhalifah et al (2001) derived an Eikonal equation for the time domain.

In his 2000 paper he derived an Eikonal equation for the VTI media using an

acoustic assumption, which is physically unrealistic. However the equation results

in realistic traveltime values. It is given as:

121)21(2

22

22

2 =

∂∂

−

∂∂

+

∂∂

+xt

zt

xt

nmovnmo ηααηα . (3.20)

Since αnmo and η are the only parameters that can be obtained from time

processing in VTI media they tried to extend the traveltime computation to a new

coordinate system in space and time from the conventional space and depth

domain. To account for a laterally inhomogenous medium they assumed the

media to be laterally factorized to remove the dependence of the Eikonal equation

on the vertical P wave velocity. For the condition of lateral factorization to be

valid they assumed that the ratio of the P wave nmo velocity and the vertical P

wave velocity is laterally constant. To transform the Eikonal equation from depth

to time coordinate, they replaced x with x and used the chain rule to represent the

derivatives as

στ∂

∂+

∂∂

=∂∂ t

x

txt , (3.21)

where

90

ξξα

τσ d

xxxzx

v

z

),(2

),(0∫ ∂

∂=

∂∂

= , (3.22)

and

τα ∂∂

=∂∂ tzt

v

2 . (3.23)

On substituting equations (3.22) and (3.23) into equation (3.20) we get the

Eikonal equation in terms of the new derivatives as

1214)21(2

222

2 =

∂∂

+∂∂

−

∂∂

+

∂∂

+∂∂

+ στ

ηατ

στ

ηαt

xttt

xt . (3.24)

Using αα kv = (laterally factorized assumption) they obtained

ττα

τασ

τ

dx

x

xzx

∂∂

−= ∫),(

),(1

),(0

. (3.25)

Therefore the dependence on the vertical P wave velocity vanishes.The Eikonal

equation can be used to compute seismic traveltimes in laterally factorized

inhomogenous media without the need to estimate the vertical P wave velocities.

Even though the above method is simple there are some shortcomings associated

with it. Firstly the Eikonal equation in (x, z) was derived with an acoustic

assumption, which has no physical validity for VTI media. Secondly departure

from lateral factorization might result in significant errors in the calculated

traveltimes.

3.4 A NEW APPROACH

Conventional ray tracing techniques are difficult to apply in the VTI

media without the knowledge of vertical P-wave velocities and depth gridded

91

anisotropic parameter estimates from well logs. As discussed in chapter 2 the

parameters that can be estimated are the elliptic P-wave velocities (αel) and the

anisotropic parameter (κ) gridded in space and time. A finite difference solution

to Alkhalifah’s Eikonal equation may be used by assuming that the parameter η is

approximately equal to κ but then we need to use group velocities and compute

group angles.

I developed a method to compute travel times, which will efficiently use

the parameters estimated from the moveout analysis (Chapter 2) and compute

direct arrival times in an offset time grid. The method is based on Fermat’s

principle and perturbation theory, which accounts for weak lateral heterogeneity.

In this method, point source plane wave traveltimes are computed using the

parameters stated earlier. By doing so, we are able to compute the times using

only the phase velocities.

ALGORITHM

The objective of the traveltime computation method is to compute plane

wave traveltime between two points using a space-time gridded elliptic velocity

and kappa model. As a first step, an array of ray parameters are generated within a

range of a minimum (usually zero) and a maximum value of ray parameters. To

calculate the maximum value of ray parameter, the following relation between

offset x, and the delay time τ is used: d

xdpτ

= − . (3.26)

The isotropic expression for the delay time is used. On performing the

differentiation we obtain,

92

, (3.27)

where α is the P wave velocity, p is the ray parameter and τ0 is the delay time at

the ray parameter value of zero. To obtain the maximum value of ray parameter

the maximum value of the offset, x is substituted to obtain,

22max

420

maxmax

αατ x

xp

+= . (3.28)

Model parameter values, for the first time level, are used to calculate pmax.

The ray parameters generated correspond to the phase angles for the rays

propagating into the underlying medium from a point source. The number of ray

parameters (npmax) to be used need to be provided at the beginning of the

computation and its value depends on the maximum offset in the grid. To avoid

the problem of numerical shadow zones, the interval between the ray parameters

are divided into two groups. For the near normal angles, the ray parameter

interval ∆p is chosen to be relatively big for “np” values of p. To determine ∆p a

minimum value for offset separation ∆xmin is chosen. Using the value of ∆xmin,

∆p can be defined using equation (3.27). Using the maximum offset the user

would like to use with coarse ray parameter spacing and from equation (3.27) the

value for np can be obtained. As the propagation angles increase, a small increase

in the ray parameter p results in wide separation in the offsets and as a result one

runs into the problem of numerical shadow zones. The next set of ray parameter

interval dp is calculated using the following formula, max ( 1)*

maxp np p

dpnp np

− − ∆=

−. (3.29)

)1( 22

20

α

ατ

p

px

−=

93

The above formula ensures that we have a very small p interval, as a result

of which numerical shadow zones are significantly reduced. For most cases, the

entire range, npmax values of p, will not be required. The size of the ray

parameter array is determined by the first value of the calculated offset, which

exceeds the maximum offset xmax.

Once the ray parameters values to be used are determined, plane wave

delay times are calculated using the expression for the delay time as a function of

elliptic P wave velocity αel and the anisotropic parameter κ.

( ) ( )

1/ 24 41/ 22 2

2 21

2( ) 2 1 1

1

NL i ii i el0 el i

i el

pp p

p

α κτ τ α

α=

= − − −

∑ .

The point source traveltime is calculated using the equation,

t px τ= + . (3.30)

The offset x can be computed using an analytic expression, which is

obtained using equation (3.26). The corresponding offset for each ray parameter

can be determined by using the values of interval elliptic P wave velocities and κ.

Effects of weak lateral heterogeneity are incorporated into the traveltime

computation algorithm by adding a perturbation term to the computed traveltimes.

The path length ∆s is calculated using the equation,

elts α=∆ . (3.31)

The time perturbation can be obtained from equation (3.27) by multiplying

the path length with the perturbation in elliptic velocities. Therefore the corrected

traveltime is calculated using the following equation:

elcorr stt α∆∆+= /. . (3.32)

94

Ideally the group velocities and their perturbations should have been used.

However one assumption is made here. The perturbation in elliptic P wave

velocity is close to that in group velocities for weak lateral heterogeneity; this is

realistic for most real earth problems. The calculated plane wave traveltimes and

the ray parameters are then interpolated to the grid points using a simple linear

interpolation. The ray parameters are also interpolated linearly.

Since ray parameters do not change across the interface (for background

laterally homogenous model) the interpolated values of the ray parameters are

used to re-shoot into the underlying layer and the offsets and the plane wave

traveltimes can be calculated in the same fashion. The above steps are when

repeated for all the time levels and give us plane wave traveltimes in an offset-

time grid.

3.5 RESULTS AND DISCUSSION

Figures 3.3 and 3.4 present a comparison between the analytically

computed traveltimes with the traveltimes computed using the perturbation

approach for homogenous isotropic and transversely isotropic models

respectively. The difference plot for the isotropic model in figure 3.3(d) show

very small residuals of the order of 10-4. For the homogenous transversely

isotropic model, the residual is observed to be growing diagonally away from the

source. For the analytically computed traveltimes, the grid points are directly

joined with the source and the times are computed by the dividing the distance by

the group velocities. The interpolation at each time level for the perturbation

95

(a)

(b) (c) (d)

Figure 3.3. (a) The model for a synthetic isotropic homogenous single-layered model (b) Computed traveltimes from perturbation approach (c) Traveltimes computed analytically (d) The difference of (b) and (c).

P-wave velocity =1.875 km/sec Thickness = 1.0 km

T I M E In S E C

T I M E In S E C

T I M E In S E C

Distance in km à Distance in km à Distance in km à

96

(a)

(b) (c) (d)

Figure 3.4. (a) The model for a synthetic transversely isotropic homogenous single-layered model (b) Computed traveltimes from perturbation approach (c) Traveltimes computed analytically (d) The difference of (b) and (c).

Vertical P-wave velocity =1.875 km/sec ε=0.225 δ=0.1 Thickness = 1.0 km

T I M E In S E C

T I M E In S E C

T I M E In S E C

Distance in km à Distance in km à Distance in km à

97

(a) (b)

(c) (d)

Figure 3.5. (a) Computed traveltimes using Faria et al’s method for a homogenous isotropic model given in figure 3.3 a (b) The difference with the analytic solution in figure 3.3 c (c) Computed traveltimes using Faria et al’s method for a homogenous VTI model given in figure 3.4 a (d) The difference with the analytic solution in figure 3.4 c.

T I M E In S E C

T I M E In S E C

T I M E In S E C

T I M E In S E C

Distance in km à Distance in km à

Distance in km à Distance in km à

98

(c) (d) (e)

Figure 3.6. (a) The elliptic velocity for a synthetic flat two-layered model (b) kappa model (c) Computed traveltimes from perturbation approach (d) Computed traveltimes using the Eikonal solver (e) The difference of (c) and (d).

1.5 km/s

1.78 km/s

0

0.086

Head waves

(a) (b)

T I M E In S E C

T I M E In S E C

Distance in km à Distance in km à Distance in km à

99

(c)

Figure 3.7. (a) The elliptic velocity for a synthetic dipping three-layered model (b) kappa model (c) travel times computed using the perturbation approach.

1.5 km/s

1.8 km/s

1.75 km/s

0

0.086

0

(a) (b)

T I M E In S E C

Distance in km à

100

(a)

(b) (c) (d)

Figure 3.8. (a) Elliptic velocity and kappa model for a 40 dipping layer example (b) Traveltimes computed using perturbation approach (c) Traveltimes computed using the Eikonal solver (d) Difference plot between (b) and (c).

αel =1.5 km/sec κ = 0.0

αel =1.75 km/sec κ = 0.086

Head waves

T I M E In S E C

T I M E In S E C

T I M E In S E C

Distance in km à Distance in km à Distance in km à

101

(a)

(b) Figure 3.9. (a) Estimated Elliptic velocity for the Gulf of Mexico data (b) Estimated kappa (c) Travel-time computed using the perturbation approach for a shot location around the target.

T I ME I N S

T I ME I N S

CDP #’s à

CDP #’s à

(c)

102

approach results in an increase in error as we go deeper into the layer. One of the

reasons for small residuals in the isotropic case is due to the fact that exact

equations are used to compute the delay times. For the TI case, the two term

approximate equation was used for computing the delay times. Figure 3.5 shows

the computed traveltimes using Faria et al’s method for the same homogenous

isotropic and transversely isotropic models given in figures 3.3 (a) and 3.4(a)

respectively. The difference with the analytically computed travetimes are mostly

similar to those obtained from the perturbation approach as can be seen in the

residual plots shown in figures 3.5 (b) and (d). Figures 3.6 (a) and (b) show the

elliptic P wave velocity and the anisotropic parameter κ model for a synthetic flat

layer model. The traveltimes calculated using the perturbation approach is shown

in fig 3.6(c). Traveltimes computed using Faria et al’s algorithm are given in

figure 3.6(d). Figure 3.6(e) shows the difference plot of the traveltimes computed

using the two methods. Since my perturbation approach does not model head

waves, they can be seen clearly as large residuals in the difference plot. Other

than that the maximum difference is 0.005 s. The traveltimes calculated from the

finite difference method uses depth gridded anisotropic parameters and velocities

and are scaled to the time grid. Since the Eikonal solver is a grid based approach

it tends to smooth the traveltimes across a layer interface. The perturbation

approach being layer-based succeeds in capturing the contrasts more successfully.

Figure (3.7a) shows the elliptic P wave velocity and the anisotropic

parameter κ model for a synthetic dipping three-layer model. Only the middle

layer is anisotropic and the top and the bottom layers are isotropic. Figure (3.7c)

103

shows the computed traveltimes using the perturbation approach. The dipping

interface has been drawn on the traveltime contour plot. The algorithm tends to

smooth out the traveltime contrast across the interface for this model because the

dip is steep and the perturbation is unable to correct for such a strong lateral

heterogeneity. However for small dip angles the method will be accurate. Figure

(3.8) shows the result obtained for a 40 dipping two layer model. As in the flat

layer model, the upper layer is isotropic while the lower layer is transversely

isotropic with a κ value of 0.086. The difference plot in figure (3.8 d) shows small

residuals.

Figures (3.9 a) and (3.9b) show the elliptic velocity and kappa models

estimated from the Gulf of Mexico dataset. A subset of the estimated parameter

values was picked to compute traveltimes from one of the shot locations near the

target zone. The computed traveltime contour for the shot location is shown in

figure (3.9c). Chapter 4 discusses in further detail how these traveltimes can be

used to perform pre-stack time migration.

104

CHAPTER 4: PRE-STACK TIME MIGRATION IN TRANSVERSELY ISOTROPIC MEDIA

4.1 INTRODUCTION

The geometry of subsurface formations is mapped by recording the times

required for seismic waves to return to the surface after reflecting from interfaces

between formations of differing physical properties. The recorded seismic data

need to be suitably processed to reveal the true subsurface image. Some of the key

steps of CDP processing have been discussed earlier. Even though it is a fast and

simple way of obtaining a seismic section, the method has its pitfalls.

Sedimentary basins can be highly folded and faulted. Moreover presence of

cracks and fractures in the subsurface introduces additional complexities. A

stacked time section in the above cases will show distorted and displaced images.

Migration is a process to remove the distortions and displacements by projecting

the wave motion backward to their true subsurface positions. It helps to reveal the

true nature of the subsurface by moving dipping reflectors to their true subsurface

locations and also collapsing diffractions. Hence migration improves spatial

resolution.

Initial attempts to perform migration used a stacked or a zero-offset

section as input. A zero offset section is equivalent to data acquired using

coincident sources and receivers. It uses an rms velocity model obtained from

stacking velocity analysis. However one can also use interval velocities to

perform post stack migration. The advantage of migrating a stacked section is that

105

in many cases it can position reflected events in their true locations and collapse

diffractions successfully. Since the sources and receivers are coincident in the

zero-offset section, they can be downward continued simultaneously. Thus the

process is computationally very fast. Post-stack migration fails in the presence of

steep dips and strong lateral heterogeneity. To handle such situations pre-stack

migration algorithms were developed. It uses certain forms of pre-stack data such

as the cdp or the shot gathers as input. Figure 4.1 illustrates the basic principle of

individual shot gather migration, where each shot is migrated separately and then

combined to form the migrated depth or time section. There are practical

problems associated with migration before stack because the data volume

increases significantly. This makes them less desirable for routine use. However

much is to be gained from pre-stack migration. Apart from being a superior

imaging tool, it can also be suitably used for interval velocity analysis. There are

many cases where even 2-D pre-stack migration is not sufficient to obtain the

correct image due to the 3-D complexity of the subsurface geology. In such cases

one needs to perform 3-D pre-stack migration, which can easily become

computationally intractable.

There are numerous techniques to perform migration. In the early years,

migration was performed by superposition of semicircles. The scheme consists of

mapping an amplitude at a sample in input (x,t) space of an unmigrated time

section onto a semicircle in the output (x,z) space. The migrated section is formed

as a superposition of many semicircles. Another method, the diffraction

summation method, was the first computer implementation of migration. In this

106

Figure 4.1. Principle of shot record oriented pre-stack migration. Note that every shot record is migrated separately and then they are summed to form the migrated section. (Berkout, 1984)

107

method amplitudes are summed in the (x,t) space along a diffraction curve that

corresponds to Huygen’s secondary source at each point in the (x,z) space. The

result of this summation is then mapped to that point in the (x,z) space. The

trajectory of the hyperbola is governed by the velocity function. Schneider (1978)

identified the diffraction stack migration as the Kirchhoff integral solution of the

scalar wave equation. The Kirchhoff integral solution, also known as Kirchhoff

migration is fully applicable to pre-stack data. I will discuss the formulation

given by Schneider in more details in a later section.

Claerbout (1970) presented a finite difference (FD) solution for the scalar

wave equation. This method is popularly known as implicit finite difference

migration. He used the paraxial wave equation given by 2

1/ 22 2

( , , )(1 ) ( , , )x x

xP k z k

i u P k zz u

ωω ω

ω∂

= −∂

(4.1)

where, P(kx,z,ω) is the wavefield, kx is the horizontal wavenumber, ω is the

frequency, u is the slowness and z is the depth location. The square root term is

approximated using the continued fraction expansion. A transformation to a time-

retarded coordinate system allows the wave field to be stationary. By substituting

this time retarded wave field into 4.1, he separated the down going wave field into

two terms, the diffraction term and a thin lens term. So solving the diffraction

term using the average slowness, downward continues the stationary wave field.

Then lateral slowness changes are taken into account using the thin lens term.

Cascaded migration schemes were introduced by Larner and Beasly(1987)

and Cambois (1991). They showed that instead of using a higher order

approximation for the continued fraction expansion, a smaller degree

108

approximation cascaded enough times will result in a same level of accuracy and

will at the same time be computationally efficient.

Another finite difference migration scheme is known as reverse time

migration. McMechan (1982,1983), Baysal et al. (1983) and Whitmore (1983)

have contributed much to the development of this method. It differs from the

implicit finite difference migrations not only in the way it solves the wave

equation but also in the way the observed wave field is extrapolated. The

derivatives in the 2D acoustic scalar wave equation are implemented in a second

order time and fourth order space explicit finite difference scheme. The reverse

time migration operator is given by

( )

( ) ( )

( ) ( )

( 1, , ) 2 2.5( ) ( , , ) ( 1, , )

16 ( , 1, ) ( , 1, ) ( , 2, ) ( , 2, )12

16 ( , , 1) ( , , 1) ( , , 2) ( , , 2)12

x z

x

z

P n m j A A P n m j P n m j

AP n m j P n m j P n m j P n m j

AP n m j P n m j P n m j P n m j

− = − + − + +

− + + − − + +

+ − + + − − + +

(4.2)

In the above equation P(n,m,j) corresponds to the wave field at tn at the

mth grid location along x and jth grid location along z. So we can see that the

wavefield at a future time tn-1 is derived from the wave field at seven neighboring

grid points at the present time tn and one at a past time tn+1. Ax is a function of

space, time sample intervals and slowness u. Az is a funtion of z, time sample

interval and the slowness u. To obtain the migrated depth section the operator is

marched backward in time and at time equal to zero, which is the imaging

condition, the depth migrated image is obtained.

Stolt (1978) and Gazdag (1978) showed that Claerbout’s (1970) FD

migration can be applied in the frequency-wavenumber domain. This method

109

became very popular because of the computational advantages in the Fourier

transformed domain. But this method is strictly valid only for 1D velocity models.

Stoffa at al. (1990) presented a method called split-step Fourier migration that

takes advantage of frequency-wavenumber domain (f-k) computation for the

mean slowness for each depth interval. A correction for the variable slowness is

then added in the f-x domain. Split-Step Fourier migration will be discussed in

further detail in a later section.

4.2 ANISOTROPIC MIGRATION

The success of seismic migration methods in providing correct depth

images is directly dependent on the accuracy of the subsurface velocity function.

In an anisotropic medium, propagation velocities are a function of propagation

angles. Consequently plane waves traveling in one direction can go faster than

those traveling in other directions. Neglecting the effect of anisotropy will thus

result in improper focusing and hence erroneous images and target depths.

Uren et al. (1990) presented an expression for an anisotropic migrated

section where they used the plane wave solution to the scalar wave equation and

used anisotropic phase velocities in place of isotropic velocities. Their method is

basically an extension of Gazdag and Stolt’s F-K method to the anisotropic

domain. Their expression for the migrated section is given by 2 ( )( , ) ( , ) x zi k x k z

x z x zz

dP x z P k k e dk dk

dkπω += ∫∫ , (4.3)

where, ω is a known function of kx and kz and the derivative is computed

numerically. Equation (4.3) allows for both elliptic and transverse isotropy.

110

However it assumes the medium to be homogenous , i.e., velocity field despite

being anisotropic can have no spatial variations.

Sena and Toksöz (1993) outlined a procedure to perform Kirchhoff

migration in anisotropic media based on a new tensor representation for

azimuthally isotropic media. This method, different from the phase shift method

allows for a laterally heterogenous medium. Analytic forms of the traveltimes and

ray amplitudes were used to compute the Green’s function. They gave an

expression for the pseudo reflection coefficient, R, defined as the ratio of the back

propagated field from the receivers to the forward propagated field from the

source. The image I(x,z) is obtained by setting t=0 or by applying the imaging

condition, i.e.,

( , ) ( , , 0)I x z R x z t= = . (4.4)

Faria (1993) slightly modified Sena and Toksöz approach by incorporating

his traveltime computation algorithm, (discussed in chapter 3) into the migration.

Kitchenside (1991) presented an extension of the phase-shift migration

method for transversely isotropic media. He substituted the usual isotropic

dispersion relationship by the appropriate dispersion relation for the transversely

isotropic media.

Alkhalifah (2000) proposed a pre-stack phase shift migration method,

which migrates separate offsets instead of the whole pre-stack data. Since pre-

stack phase shift migration is implemented by evaluating the offset-wavenumber

integral using the stationary-phase method, he calculated the stationary point prior

to applying the phase shift.

111

Gonzalez et al. (1991) used an anelliptic dispersion relation in pre-stack f-

k migration to compensate for the effect of transverse isotropy. They showed that

the dip-dependency of imaging velocities can be minimized by incorporating their

velocity characterization into pre-stack f-k migration.

4.3 INTEGRAL FORMULATION FOR MIGRATION

Schneider (1978) posed the migration as a boundary value problem, which

led to an integral or summation algorithm in either two or three dimensions.

Solution of the scalar wave equation based on Green’s theorem is given by,

0 0 0 0 0 01

( , ) ( , ) ( , )4

P r t dt dS G P r t P r t Gn nπ

∂ ∂ = − ∂ ∂ ∫ ∫ , (4.5)

where, r denotes the spatial coordinates, P(r0,t0) is the wave field recorded at the

surface, G is the Green’s function and n is the outward normal vector to the

surface. Since P(r0,t0) in equation (4.5) is equated to the observed seismic data, we

require that G=0 on the surface in order to eliminate the gradient of P. A Green’s

function having the desired properties at the free surface consists of a point source

at r0 and its negative image at r0’ or

'

0 0

0 0 '

( ) ( )( , | , )

R Rt t t t

V VG r t r tR R

δ δ− − − −= − , (4.6)

where 2 2 2

0 0 0( ) ( ) ( )R z z x x y y= − + − + − ,

and ' 2 2 2

0 0 0( ) ( ) ( )R z z x x y y= + + − + − .

V is velocity of the medium. Other choices of G are also possible.

Substituting equation (4.6) into equation (4.5) and simplifying we get the

112

following integral representation of the wave field P(r,t) at any point in the image

space in terms of observations of the wavefield P(r0,t0) on the surface,

0

0 0 0 00

1( , ) . ( , )

2

Rt t

VP r t dt dA P r tz R

δ

π

− − ∂ =∂

∫ ∫ . (4.7)

DOWNWARD EXTRAPOLATION

RVR

trPdA

ztrP

+

∂∂

−= ∫,

1),(

0

0π z

IMAGING PRINCIPLE – EXTRAPOLATE RECEIVERS FOR ALL Z>0

AT t=0

RVR

yxPdxdy

ztrP

∂∂

−= ∫,0,,

21

),(π = 3D MIGRATION

Figure 4.2. Migration principle for zero offset data recorded at z=0

P(x,y,0,t)

P(x,y,z,t)

113

The above equation is a rigorous statement of Huygen’s principle and is

commonly called the Kirchhoff integral. By interchanging the z0 derivative with a

z derivative, which may be taken outside the integral we get,

0

0

,1

( , )

RP r t

VP r t dAz Rπ

− ∂ = −∂ ∫ . (4.8)

Equation (4.8) forms the basis of Kirchhoff migration. Equation (4.7) can

also be written symbolically as a three dimensional convolution,

00

( )1( , , , ) ( , , , )*

2

rt

VP x y z t P x y z tz r

δ

π

± ∂= ∂

, (4.9)

which translates the observed wavefield from one z-plane to another. Migration

using equation 4.7 is illustrated clearly in the figure 4.2, which also highlights

how the imaging condition is applied.

4.4 IMPLEMENTATION OF PRE-STACK KIRCHHOFF MIGRATION IN TI MEDIA

I performed the Kirchhoff migration in the x-t domain or the offset-time

domain. The input data was organized as shot gathers. For pre-stack migration

each shot gather is migrated independently to produce the partial image. The

summation of all partial images produces the final pre-stack migrated section.

There are several advantages to Kirchhoff migration. It is intuitive and is

reasonably accurate as long as one can compute the traveltime tables accurately.

Different source receiver geometries and target oriented imaging can also be

handled easily by this method.

114

There are two steps necessary to implement Kirchhoff migration –

traveltime computation and imaging. The traveltime computation can be

accomplished in different ways. For wave propagation in isotropic media the most

popular methods are finite difference solutions of the Eikonal equation (Vidale,

1990) and ray tracing. These methods become difficult to apply in the anisotropic

media as vertical wave velocities are extremely difficult to estimate. Also the

directional dependence of the velocities adds further complications. Considerable

work has been done in the field of anisotropic traveltime computation. These

methods have been discussed in some details in the previous chapter. Here I use a

traveltime computation algorithm I developed for the TI media, which uses the

time gridded elliptic P wave velocities and the anisotropic parameter kappa. The

kinematic accuracy of the migration depends on the correct traveltime

computation. Imaging is performed by mapping the amplitude of each trace into

the image space. This technique is basically equivalent to migration using the

Kirchhoff integral discussed in the previous section. Use of a different traveltime

computation method, which incorporates the anisotropic model parameters,

implicitly modifies the effective Green’s function used.

Figure 4.3 shows a schematic diagram illustrating the technique. A shot is

located at s, a receiver at r and a scatterer is located at x in the model respectively.

The gridded traveltime tables for the shot and receiver are required in the first

step. If the shot and receiver are coincident then only one traveltime table is

needed. We would like to image the grid point x. The traveltime t(s,x) from s to x

is found from the shot traveltime table and the traveltime t(r,x) from x to r is

115

found from the receiver traveltime table. The total time t(s,r) is the summation of

t(s,x) and t(r,x). As a next step we look at the trace produced by this shot and

recorded by this receiver and extract the amplitude corresponding to the total time

t(s,r). The amplitude is put in the grid location x in the image space. The

procedure is repeated for all grid points of the image space for all shots and for all

receivers. The stacking of all partial images produces the final image.

4.5 PRE-STACK SPLIT-STEP FOURIER MIGRATION

Stoffa et al. (1990) first introduced the split-step Fourier migration method

as an application to migrate seismic data. This method is an extension of

Gazdag’s phase shift migration method to account for lateral velocity variations.

The solution of the wave equation for this method is obtained in the frequency-

wave number domain. The 2D acoustic wave equation is given by 2

2 22( , , ) ( , ) ( , , ) 0P x z t u x z P x z t

t

∂∇ − =

∂, (4.10)

where, P is the pressure and u is the slowness.

The split-step Fourier migration is based on decomposing the laterally

varying slowness function, u(x,z), into two components; a mean slowness

component )(zu that is vertically varying, and a laterally varying component

∆u(x,z). That is

( , ) ( ) ( , )u x z u z u x z= + ∆ . (4.11)

Substituting this slowness function into equation (4.10) and double Fourier

transforming over time and space coordinates, the wave equation is obtained in

the frequency-wave number domain as

116

Figure 4.3. Implementation method for Kirchhoff Migration.

117

2 22 2

2( , , )

( ) ( , , ) ( , , )xx x x

P k zu z k P k z S k z

z

ωω ω ω

∂+ − = −

∂, (4.12)

where, after ignoring the slowness perturbation contribution we have

),,(),()(2),,( 2 ωωω zxPzxuzuzxS ∆= . (4.13)

The solution to equation 4.12 is given as (Stoffa et al., 1990) '

1 1(' '

1)

( , , ) ( , , ) ( , , )2

n z nz

n

z ik z zik z

x n x n xzz

eP k z P k z e S k z dz

ikω ω ω

+ +−∆

+ = − ∫ , (4.14)

where,

222 )( xz kzuk −= ω . (4.15)

After evaluating the integral in equation (4.14) and rearranging it as given

in Appendix A, the wavefield extrapolation operators for the split-step Fourier

migration algorithm are obtained as follows zik

xxzezkPzzkP ∆=∆ ),,(),,,( ωω , (4.16a)

and zxui

nn ezzxPzxP ∆∆+ ∆= )(

1 ),,,(),,( ωωω , (4.16b)

where P(x,zn,∆z,ω) is the inverse spatial Fourier transform of ),,,( ωzzkP x ∆ . If

both the equations are combined, then

( ) 11( , , ) ( , , )zik zi u x z

n x x nP x z e F e P k zωω ω∆∆ ∆ −+ = , (4.17)

where Fx-1 represents the inverse spatial Fourier transform.

From the above expression, we can see that there are two distinct phase

shifts applied to the recorded wave field. The first phase shift given in (4.16a) is

118

based on the reference wave number, kz. this phase-shift downward continues the

data across each migration interval ∆z using the reference or mean slowness, )(zu ,

in the frequency-wave number domain. Equation (4.16b) describes the second

phase-shift, which is based on the perturbation term or laterally varying slowness

component, ∆u(x,z), and is applied in the frequency-space domain to account for

lateral velocity variations.

For the pre-stack case, both the receiver and the source wave fields are

downward continued using the downward continuation kernel discussed earlier.

To obtain the migrated image for a particular depth location, M(x,zn+1), the

downward continued source and receiver wave fields are cross correlated and then

summed over all frequencies of interest (Berkhout, 1984), i.e. *

1 1 1( , ) ( , , ) ( , , )n s n r nM x z P x z P x zω

ω ω+ + += ∑ . (4.18)

Another approach to obtain the migrated image for an individual shot

gather using the split-step algorithm is by computing the direct wave arrival time,

ts(x,zn), from the source to each subsurface location using any traveltime

computation algorithm and then applying these times, ts(x,zn) as an additive phase

term to the downward continued receiver wave field, Pr(x,zn+1,ω) for each depth

level before the summation over the frequencies. The imaging equation for this

case is given by, 1( , )

1 1( , ( , , )s ni t x zn r nM x z e P x zω

ωω+−

+ += ∑ . (4.19)

This method eliminated the need for extrapolating the source wavefield

with the split-step method. Figure 4.4 gives a flowchart describing the pre-stack

119

split-step Fourier migration algorithm by using the direct arrival times of the

source wave field.

4.6 IMPLEMENTATION OF PRE-STACK SPLIT-STEP FOURIER MIGRATION IN TI MEDIA

To apply the split-step Fourier migration method in the TI media we need

to make a few modifications to account for the anisotropy. At this point I would

like to mention that since we do not have a depth gridded model parameter

estimate we would have to restrict ourselves to time migration. However since we

have time gridded interval model parameter we can downward continue the wave

fields over interval times. So with the knowledge of vertical P wave velocities it

basically mean a scaling manipulation to convert from time to depth migration.

The phase term zik ze ∆ needs to be modified so that the estimated interval

parameters from moveout analysis, i.e., elliptic P wave velocity and κ can be used

to perform the migration. The vertical wave number kz is given by 2 2 2z xk k k= − ,

where,

kvω

= ,

and

xk pω= ,

v is the phase velocity and p is the ray parameter.

Therefore 2 2 2

21

( )zk pv

ω= − , (4.20)

or zk qω= , (4.21)

120

Pre-compute the direct arrival times from the source position ts( x,z) FFT( t → ω) the receiver wave field Pr ( x,z 0,t ) ⇒ Pr(x,z = 0,ω ) FFT ( x → kx ) the receiver wave field Pr ( x,z = 0,ω) ⇒ Pr ( kx,z = 0,ω ) Apply the first phase-shift to the receiver wave field zik

xrxrzezkPzzkP ∆=∆ ),,(),,,( ωω

Inverse FFT (kx → x) Pr(x,z,∆z,ω) Apply the second phase-shift zui

rr ezzxPzzxP ∆∆∆=∆+ ωωω ),,,(),,(

Apply a phase delay based on the pre-computed times and

sum over all the frequencies to form the image

),,(),( ),( ωω

ω zzxPezzxM rzzxti s ∆+=∆+ ∑ ∆+−

Figure 4.4. Flowchart for the split-step Fourier method to migrate a single shot gather by extrapolating the receiver wave field and using the direct arrival times of the source wavefield to construct the image.

Repeat for each depth level

121

where q is the vertical slowness.

Using the expression for kz the phase term kz∆z can be written as

zk z ω τ∆ = ∆ . (4.22)

Thus, the downward continuation phase term can now be written as

( , , , ) ( , , ) ix xP k t t P k t e ω τω ω ∆∆ = . (4.23)

The above equation can be conveniently applied to perform pre-stack time

migration in transversely isotropic media. The change in delay time, which is a

function of the interval elliptic P-wave velocity and the anisotropic parameter κ,

can be calculated using the delay time equation (2.24) derived in chapter 2.

Background value of the elliptic P-wave velocity is used to downward continue

the wave field for every time step. To account for lateral changes in velocity, only

the lateral variability of elliptic P-wave velocity is considered. Assuming that the

lateral change in κ is negligible, its effect on the phase term is ignored.

4.7 RESULTS AND DISCUSSION

Figure 4.6 shows a synthetic shot gather generated using a finite difference

modeling code for a model with elliptic P wave velocity and κ shown in figure

(4.5). Figure (4.7a) is the migrated output from pre-stack split-step Fourier

migration using only the isotropic parameter and fig (4.7b) shows the output after

incorporating the TI correction. From the plot of the anisotropic layer interface

from the migrated outputs we can see that the TI migration does a much better job

in imaging the anisotropic flat layer. A dipping layer synthetic shot gather is

shown in fig (4.9) using the elliptic P-wave velocity and kappa model in fig (4.8).

Fig (4.10) is the migrated output after pre-stack split-step Fourier migration. It

122

Figure 4.5. The Elliptic velocity and kappa model for a flat layer synthetic test.

Figure 4.6. Input synthetic shot gather for the velocity and kappa model in Fig 4.5

V=1.5 km/s, κ=0

V=1.65 km/s, κ=0.088 V=1.75 km/s, κ=0

1.0

2.0

3.0

4.0

5.0

0.0 1.0 2.0 3.0

Distance in km

T I M E I N S

123

Figure 4.7(a). Migrated shot gather after isotropic split-step migration

Figure 4.7(b). Migrated shot gather after TI split-step migration

0.0

1.0

2.0

0.0 1.0 2.0 3.0

0.0

1.0

2.0

0.0 1.0 2.0 3.0

Distance in km

Distance in km

T I M E I N S

T I M E I N S

124

Elliptic velocity Kappa

Figure 4.8. Elliptic P wave velocity and kappa model for a dipping layered synthetic experiment.

Figure 4.9. Input synthetic shot gather for the velocity and kappa model in Fig 4.9

1.5 km/s

1.81km/s

1.75 km/s

0

0.086

0

0.0 1.0 2.0 3.0

0.0

1.0

2.0

3.0

4.0

5.0

Distance in km

T I M E I N S

125

Figure 4.10. Migrated shot gather for the dipping layer model after split-step fourier migration using TI corrections.

0.0 1.0 2.0 3.0

1.0

2.0

3.0

4.0

5.0

T I M E I N S

Distance in km

126

(a) (b) (c)

Figure 4.11. (a) Plot of kappa vs TWT for a location away from the target zone. (b) CIG after isotropic split step migration (c) CIG after TI split-step migration.

T I M E IN S

T I ME I N S

T I M E IN S

Kappa à

127

(a) (b) (c)

Figure 4.12. (a) Plot of kappa vs TWT for a location at the target zone. (b) CIG after isotropic split step migration (c) CIG after TI split-step migration.

Target Zone

T I ME I N S

T I ME I N S

T I ME I N S

Kappa à

128

(a) (b)

Figure 4.13. (a) CIG after TI Kirchhoff migration at a location away from the target zone (b) CIG after TI Kirchhoff migration at a location around the target zone.

T I ME I N S

T I ME I N S

129

Figure 4.14(a). Zoomed plot of the target zone after TI pre-stack Kirchhoff Time migration.

Figure 4.14(b). Zoomed plot of the target zone after Isotropic pre-stack Kirchhoff Time migration.

T I ME I N S

600 650 700 750 800 850 CDP à

600 650 700 750 800 850 CDP à

T I ME I N S

130

Figure 4.15. The Pre-stack Time migrated stack after Split-step Fourier Migration.

Target Zone

131

does a good job in imaging both the dipping and anisotropic flat layer interface.

The migration was tested on the Gulf of Mexico dataset provided by the Shell Oil

Company. (Fig 4.11a) is a plot of κ vs. two-way traveltime at a shot location away

from the target zone. We can see that the values of κ are very small (< 0.09) for

this location. As such there is not much difference in the migrated CIGs after

isotropic and TI migration as shown in figure (4.11b) and (4.11c) respectively.

For a CIG gather at a location on the target zone (fig 4.12) we can see significant

differences in the flattening of the events at the target after isotropic and TI

migration. This agrees very well with the kappa values at this shot location, which

shows high P wave anisotropy (~ 0.2). The target is highlighted with a box around

the events in fig (4.12). A pre-stack migrated section of the GOM data is

presented in figure (4.15).

The Kirchhoff migration results were also encouraging. A CIG away from

the target zone (fig 4.13a) and a CIG at the target zone (fig 4.13b) have been

presented. Both show good flattening of the events after pre-stack migration. A

zoomed plot of the pre-stack migrated sections around the target zone is presented

in figure 4.14. Figure 4.14(a) and (b) shows the sections after TI and isotropic

migration respectively. We can see that the target zone has imaged very nicely

after Kirchhoff migration using the TI parameters namely the time gridded elliptic

P-wave velocity and kappa whereas the isotropic migration is unable to

coherently image some of the events. The improvement in the imaged section is

especially noticeable in the time range of 4.5 to 5.0 seconds. Finally the pre-stack

132

time migrated section after split-step Fourier TI migration is presented in figure

4.15.

The migration after incorporation of the TI corrections result in

improvement in the flattening of the CIG gathers. The improvement in the

flattening of event in the CIG gathers indicate that these corrections can be even

more important for areas having more severe anisotropy. The cause of transverse

isotropy for the Gulf of Mexico investigated in this dissertation is unknown. No

information was available regarding the stratigraphy of the area or for that matter

the lithologic ditribution. However presence of shale around the target zone may

be the cause for anisotropy. It might also have resulted due to overburden pressure

around that zone. One interesting feature observed in the estimated results from

both interactive and automatic analysis discussed in chapter 2 is that the high

values for kappa are centered on the target zone and dies down to low values

outside it. I would suggest that the anisotropy probably extends beyond that zone.

The lack of any strong reflected events outside the target zone comes in the way

of detecting them from the seismic data.

The methods discussed above can easily be extended to perform converted

wave pre-stack migration. Estimation of both κ and η will also give us a starting

model to perform pre-stack depth migration. The migrations algorithms discussed

in this chapter only perform time migration. However since we are only using

interval parameters extending this to depth migration is trivial given that we can

estimate vertical wave velocities from PP and PS data.

133

CHAPTER 5: SUMMARY AND FUTURE WORK

5.1 SUMMARY

Seismic wave propagation in most cases is assumed to be isotropic for

reasons of simplicity. However incorporation of anisotropic effects have resulted

significant improvements in the area of seismic data processing. Over the past few

decades a considerable amount of work has been done in this area, which has

succeeded in enhancing peoples understanding for wave propagation in

anisotropic media. In this dissertation I have developed a new approach for

processing seismic data in the plane wave domain for transversely isotropic

media. The plane wave domain is the natural domain for analyzing anisotropic

wave propagation. Since anisotropy is an angle dependent phenomenon the

effects on seismic data are more obvious when the data are transformed to the

plane wave domain. The equation for travel times in the x-t domain is often

represented by a two-term expression, which results from the truncation of a

Taylor’s series expansion. However for analysis in the x-t domain one needs to

introduce a fourth or higher order terms to model travel times at large offsets. A

common practice has been to interpret the necessity of the fourth or higher order

terms to model travel times, as due to anisotropy. It is important to understand

that these higher order terms may also be necessary for the isotropic case to model

travel times at large offsets. This ambiguity is resolved in the τ-p domain, as

under 1-D assumption the non-elliptic move-out in the data at large ray

parameters must be due to anisotropy. The processing flow I developed includes a

134

parameter estimation technique, travel time computation using the estimated

parameters and finally, pre-stack time migration using the computed travel times

and the offset-time gridded parameters.

I have derived two-term equations for the delay time τ as a function of ray

parameters for compressional and shear wave propagation in transversely

isotropic media. This equation forms the backbone of the parameter estimation

technique. In the x-t domain one is limited to modeling travel times using a

truncated Taylor’s series. On the contrary in the τ-p domain it is possible to derive

exact equations for any kind of media without the loss of any generality. However

even in the τ-p domain these equations can be very complicated for the

anisotropic media and hence estimation of parameters becomes difficult using

these equations. So, I derived a simple looking two term equation, which even

though being an approximate equation is very accurate for small to mid ray

parameter ranges. τ-p curves computed using the two-term equation were

compared with those obtained using the exact equation for the Dog Creek shale

and Taylor’s sandstone models. The comparison shows that the derived equation

models the delay time curves accurately for ray parameter ranges important for

exploration seismics. Apart from being simple, this two-term expression for the P-

and S-waves also helps in understanding the physics better. For the P wave case

the delay time is a function of the elliptic P wave velocity αel and an anisotropic

parameter κ, which is a combination of the two anisotropy parameters ε and δ,

defined by Thomsen. For shear waves the delay time τ is a function of the elliptic

P wave velocity αel, elliptic S wave velocity βel and the anisotropic parameter η

135

which is again a combination of the two anisotropy parameters ε and δ. Even

though the parameter η is different from κ, for very small values of δ the

difference between them is negligible. These equations can be used to

interactively estimate the above stated parameters from the τ-p data. The fitting of

the τ-p data can be broken up into two parts. For the P-P data at small to mid ray

parameters only the elliptic P wave velocity is used to model the elliptic delay-

time moveout and to fit the non-elliptic moveout at higher ray parameters the

fourth order term, which is a function of anisotropy parameters κ and αel, is

introduced. An improvement in flattening after introduction of the fourth order

anisotropic term can be seen clearly in both synthetic and real data examples.

Synthetic two layer examples for the Dog Creek shale and Taylor’s sandstone

clearly demonstrate this point. I have interactively estimated the parameters αel

and κ for a 2-D line in the Gulf of Mexico. Higher values of κ were required

around the target zone to flatten the reflected events. Using the two-term equation

to model the delay times of vertically polarized shear waves one can estimate the

elliptic S wave velocity βel and η from the data using prior knowledge on αel

estimated from the P-wave data. One synthetic example for PS NMO has been

presented. In addition to the interactive parameter estimation I have also presented

a technique to automatically estimate model parameters from PP data using a non-

linear inversion technique called VFSA (Very Fast Simulated Annealing). VFSA,

which is an extension of the well known global optimization technique called

simulated annealing, is a very efficient and fast optimization scheme. It has been

used very effectively for the isotropic case to estimate background velocities for

136

1-D earth models. My approach presented in this dissertation is an extension of

the work done for the isotropic case to the VTI media. Assuming locally 1-D

earth models, I simply used NMO as the criterion for the inversion. Cross-

correlation was used to estimate errors from the NMO correction. Bounds for the

model parameters, αel and κ, were chosen based on some apriori investigation on

a set of CDP gathers. The estimated gridded model parameters are presented for

the Gulf of Mexico data. The parameters obtained after inversion agree well with

the values obtained from the interactive analysis.

As a second step in the processing flow, I have developed an efficient

traveltime computation algorithm based on the Fermat’s principle and

perturbation theory. This algorithm uses the parameters estimated in the moveout

analysis and computes plane wave traveltimes in an offset time grid. Plane wave

traveltimes are first computed for an array of ray parameters and then they are

interpolated to the grid points using a simple linear interpolation. Before the

interpolation is performed a correction is made to the traveltimes using the

perturbation theory to account for weak lateral heterogeneities. To avoid the

problem of shadow zones the ray parameter generation is broken up into two

steps. For the near normal angles, the ray parameter interval, ∆p, is chosen to be

relatively big for a range of value for the ray parameters. The number of ray

parameters to be used can be calculated. For higher propagation angles, very fine

ray parameter spacing is used ensuring good ray coverage. Head waves were not

considered in the method. Effects of weak lateral heterogeneity are incorporated

into the traveltime computation algorithm by adding a perturbation term to the

137

computed traveltimes. The time perturbation is obtained by multiplying the path

length with the perturbation in elliptic velocities. Ideally the group velocities and

their perturbations should have been used. However since the perturbation in

elliptic P wave velocity is close to that in group velocities for weak lateral

heterogeneity; this is realistic for most real earth problems. Comparison of the

computed traveltimes with those using the Eikonal solver shows small residuals.

Pre-stack time migration using the estimated parameters and the computed

traveltimes, constitutes the last step in the processing flow. I have implemented

two migration algorithms for the transversely isotropic media for the PP case:

split-step Fourier; and, Kirchhoff migration. The main challenge for

implementing the Kirchhoff migration was to compute traveltimes using the

parameters estimated from the moveout analysis. Since perturbation approach

efficiently computes the traveltimes in an offset-time grid implementation of the

Kirchhoff migration was straightforward. To perform the split-step Fourier

migration a modification of the phase term was made. The product of the vertical

wavenumber and the depth step can be represented as a product of the frequency

and change in delay times in the phase term. As a result I could use the derived

two-term equation for the delay time to downward propagate the receiver

wavefield for every time step. To account for the source phase term I used the

traveltimes from the source to each of the grid points. To correct for the lateral

heterogeneities a phase term, which is a function of the perturbations in the

elliptic P wave velocities, is used. A phase term, which uses the perturbation in

the velocities as well as the anisotropy term, κ, would have resulted in a more

138

accurate correction. However since perturbations in κ are negligible for most part

of the dataset only perturbations in the elliptic P wave velocity was used. The

manner in which the pre-stack time migration is performed is similar to depth

migration. Since I have used only interval parameters to downward propagate the

wavefields, conversion from time to depth is reduced to a problem of scaling. A

significant improvement in the flattening of the common image gathers can be

seen after incorporation of anisotropy.

5.2 FUTURE WORK

Although I have designed the methods to correct for weak lateral

heterogeneities, extension of the traveltime computation and the migration

algorithms to strong laterally varying media the perturbation terms need to be

modified. Theoretical development in a direction, which uses the perturbations in

the elastic coefficients, will succeed in modeling the lateral heterogeneities more

accurately.

I did not have an opportunity to work with real field data for converted

waves. However extension of the processing flow can be made to converted wave

data easily. Since I have already derived a two-term equation for the S-wave,

parameter estimation, traveltime computation and migration should be

straightforward for the P-S case. Joint inversion of PP and PS data can be made to

estimate the elastic coefficients. This will help reduce uncertainty in the estimates

of parameters which one faces using the PP data alone. Moreover vertical P wave

and S wave velocities can be obtained from a joint inversion, which can

139

subsequently be used for depth migration. The automatic parameter estimation

technique presented in chapter can also be easily extended to the PS case.

I did not develop any standardized model-updating tool for anisotropic

migration. However one might use tomographic inversion as a tool to update the

velocity and anisotropic parameter models. After each pass of pre-stack time

migration the non-flatness of the common image gathers can be used as an input

for the inversion. The traveltime computation algorithm I have developed, can be

used as a forward modeling tool for this inversion. Algorithms have been

formulated to perform residual migration velocity analysis in the plane wave

domain for the isotropic case. The same idea can be extended to the anisotropic

domain to update velocity models as well as the anisotropic parameters.

In my dissertation I have focused only on the transversely isotropic case.

The methods discussed can be extended to handle other complicated cases like the

azimuthally anisotropic media.

140

Appendices

APPENDIX A

Derivation of the split-step Fourier Migration Method

In this appendix I derive the split-step Fourier migration method as

outlined by Stoffa et. al.(1990). I will start with equation (4.14) given by, '

1 1(' '

1)

( , , ) ( , , ) ( , , )2

n z nz

n

z ik z zik z

x n x n xzz

eP k z P k z e S k z dz

ikω ω ω

+ +−∆

+ = − ∫ (A1)

where,

222 )( xz kzuk −= ω , (A2)

and

° % °2 ' ' '( , , ) 2 ( , ) ( , , )x x x x xS k z u u k k z P k z dkω ω ω∞

−∞

= ∆ −∫ . (A3)

Let, '

1 1(' ')

( , , )2

n z n

n

z ik z z

xzz

eI S k z dz

ikω

+ +−= ∫ (A4)

On substituting kz into the integral we get,

( ) ( )

( ) ( )1/ 2 '

1

1

1 /

2 ' ' ' ' ' '1/ 22

2 , , ,2 1 ( / )

x n

n

n

i u k u z zz

x x x xzx

eI u dz dk u k k z P k z

i u k u

ω ω

ω ωω ω

+

+

− − ∞

−∞

= ∆ − −

∫ ∫

Expanding the denominator, keeping the first term and bringing the

exponential term inside the integral for small ∆z, we obtain

141

( )' '1

10( )' ' ' ' ' ', ( , , )

nnz

n

zik z z

x x x xz

I i dz dk u k k z P k z eω ω ε+

+∞ −

−∞

= − ∆ − +∫ ∫ (A5)

where

( )[ ] 2/1/1'

0 uxkzk ω−=

Now a wavefield at a depth level “z+d” can be written as ,

01( , , , ) ( , , ) zik dx xP k z d P k z eω ω= . (A6)

So, substituting the above equation into (A1) and neglecting the error

term, ε, we have 1

' ' ' ' ' '1( , ) ( , , , ),

n

n

z

x x x x nz

I i dz dk u k k z P k z dω ω+ ∞

+−∞

= − ∆ −∫ ∫ (A7)

where dn+1(z’)=zn+1-z’. Substituting the simplified integral term in equation

(A7) into the solution, we obtain 1

' ' ' ' ' '11 1( , , ) ( , , , ) ( , ) ( , , , )

n

n

z

x n x n x x x x nz

P k z P k z z i dz dk u k k z P k z dω ω ω ω+ ∞

+ +−∞

= ∆ + ∆ −∫ ∫ .

By taking the inverse Fourier transform of the above equation to transform

from kx to x space we get,

1

' ' '1 1 1( , , ) ( , , , ) ( , ) ( , , , )

n

n

z

n n nz

P x z P x z z i dz u x z P x z dω ω ω ω+

+ += ∆ + ∆∫ . (A8)

With the assumption of very small ∆z and using the Trapezoidal rule to

evaluate the integral we get,

142

[ ]1 1 1 1 1( , , ) ( , , , ) ( , ) ( , , ) ( , ) ( , , , )2n n n n n nz

P x z P x z z i u x z P x z u x z P x z zω ω ω ω ω+ + +∆

= ∆ + ∆ + ∆ ∆

, (A9)

Upon rearranging we get,

1 1 1( , , ) 1 ( , ) ( , , , ) 1 ( , )2 2n n n n

i iP x z u x z z P x z z u x z z

ω ωω ω+ +

− ∆ ∆ = ∆ + ∆ ∆ . (A10)

By making the assumption that the slowness at the top and bottom of the

depth interval ∆z as equal the above equation can be further simplified to,

1 1

1 ( )2( , , ) ( , , , )

1 ( )2

n n

iu x z

P x z P x z zi

u x z

ω

ω ωω+

+ ∆ ∆ = ∆ − ∆ ∆

. (A11)

If one uses a polar coordinate form equation (A11) can be further

simplified to, ( )

1 1( , , ) ( , , , ) i u x zn nP x z P x z z e ωω ω ∆ ∆

+ = ∆ . (A12)

Recall that the inverse Fourier transform of P1(x,zn,∆z,ω) is

01( , , , ) ( , , ) zik zx xP k z z P k z eω ω

∆∆ = . (A13)

Equation (A13) can be used to downward continue the wavefield from

depth level z to level z+∆z using the reference wavenumber, kz0. Equation (A12)

applies the second phase shift to account for the lateral variations in velocity.

143

APPENDIX B

Equations for P wave phase velocities for weak VTI media

In this appendix I will derive simplified expressions for P wave phase

velocities as a function of ray parameter using the approximation of weak TI. The

following derivation is based on the work by Cohen(1997). For a homogenous

transversely isotropic media, introduction of a plane wave solution in the

equations of motion allows us to obtain the phase velocities of P-, SV- and SH-

waves as functions of the five elastic coefficients, the density and the direction of

propagation. As given by Stoneley (1949), the P wave phase velocity is: 1/ 2( ) [( ) / 2 ] ,V a bθ ρ= + (B1)

where ( ) ( ) ( )

2 2

1/ 22 22 2 2 2

11 33 44 66 13

sin cos 2

sin cos 4 sin cos

, , , ,

a A C

b A L C L F L

A C C C L C N C F C

θ θ

θ θ θ θ

= + +

= − − − + +

= = = = =

and ρ is the density of the medium.

Using (B1) the P wave phase velocity can be written as

.cossin)(4]cos)(

sin)[(cos)(sin)()(22/1222

441322

4433

24411

24433

24411

2

θθθ

θθθθρ

CCCC

CCCCCCV

++−

−−++++=

(B2)

By introducing the horizontal slowness p and vertical slowness q, (B2) can

be written as

.)(4])(

)[()()(22/1222

441322

4433

24411

24433

24411

qpCCqCC

pCCqCCpCC

++−

−−++++=ρ

(B3)

where,

144

( )

( )

22

2

22

2

sin,

cos

pV

qV

θθ

θθ

=

=

Equation (B3) can be expressed as a quadratic in q2, the solution of which

is given in equation (2.7). Equation (B3), given by Cohen(1997), can be written in

terms of Thomsen’s parameters as, 22222222

20

]2)([)2(4)22()2(2

pqpfpqffpfqf εδεα

+−+++−++−= , (B4)

where,

20

201

α

β−≡f .

α0 and β0 are the vertical P- and S-wave velocities. Equation (B4) is

simplified to obtain an equation quadratic in the P wave phase velocity 2pV given

by,

0)1()2(2)(2)(21 40

220

42 =−+−−−+−− ααδεδε fVfzfVzf pp , (B5)

where, 220 pz α= .

The solution for V2(p) from equation (B6) is of the form (Cohen, 1997),

,2

)( 20

2C

BApV

+= α (B6)

where,

.)(221

,)())(1(24)2(24

,)(21

2

222

zfzC

zfffzffkfB

zffA

δεε

δεδεδε

δε

−−−=

−+−−+−−−=

−−−=

Equation (B6) is no simpler than the one that can be obtained from the

solution of equation (B2). However, as mentioned earlier, parameterization in

145

terms of the Thomsen’s parameters helps us to obtain considerably simpler

equations for the special cases of elliptic anisotropy and weak transverse isotropy.

Since I’m interested in weak VTI media, I chose to use equation (B8) for future

derivations. Cohen’s equations for the cases of elliptic anisotropy and weak TI are

given below.

Elliptic anisotropy

For elliptical anisotropy, δ=ε. As a result the constants A, B, and C reduce to

.21,])1(2[

,)1(222

zCzffB

zffA

δδ

δ

−=−+=

−−−=

so that

)21()(

202

zpV

δα−

= . (B8)

Weak transverse isotropy

The limit of weak transverse isotropy implies retaining only linear terms in δ and

ε (Thomsen, 1986). In this case, equation (2.17), as given by

Cohen(1997),reduces to

( ) ( ) 20 1v p z zα δ ε δ= + + − (B9)

The above equation is equivalent to Thomsen’s(1986) expression for P

wave velocity in terms of phase angle.

146

APPENDIX C

Equations for Sv wave phase velocities for weak VTI media

In this appendix I will derive a simple equation for the special case of

weak transverse isotropy starting with the expression for Sv phase velocity given

by White(1983). He gave the following form for Vsv:

( ) ( ) 1/ 2/ 2

vsV a bθ ρ = − (C1)

where ( ) ( ) ( )

2 2

1/ 22 22 2 2 2

11 33 44 66 13

sin cos 2

sin cos 4 sin cos

, , , ,

a A C

b A L C L F L

A C C C L C N C F C

θ θ

θ θ θ θ

= + +

= − − − + +

= = = = =

ρ is the density of the medium.

Expanding the above equation we get, ( ) ( )

( ) ( )

2 2 211 44 33 44

1/ 222 2 2 2 211 44 33 44 13 44

2 sin cos

sin cos 4( ) sin cos

SvV C C C C

C C C C C C

ρ θ θ

θ θ θ θ

= + + + −

− − − + +

(C2)

which can be written as,

2 211 44 33 44

1/ 222 2 2 2 211 44 33 44 13 44

2 ( ) ( )

( ) ( ) 4( )

C C p C C m

C C p C C m C C p m

ρ = + + + −

+ − − + +

(C3)

where m2=1/Vsv2 – p2

On introducing Thomsen’s notations and inducting the vertical P and Sv

velocities we get, ( ) ( ) 2 4 2 2 2

0 0(1 2 2 ) 2 2 (1 ) 0v svsz f z V f z f V fε ε δ α ε δ δα− − − + − − + + − = (C4)

147

where z = α02p2 and f = 1 - β0

2/α02, α0 is the vertical P wave velocity, and

β0 is the vertical SV velocity.

The solution of the above equation is

( )( )

20

2 2 2

2

22 2

4 2 4 2 (1 )( ) ( )

1 2 2 ( )

vSA B

VC

A f f z

B f f f z f f f z

C z f z

α

ε δ

ε δ ε δ ε δ

ε ε δ

−=

= − − −

= − − − + − − + −

= − − −

(C5)

I introduced the above expressions in the equation for Vsv and simplified it

using the Mathematica software. For the simplification I retained only terms for

the first power in z and the anisotropic parameters ε and δ. Finally I obtained a

simple looking expression for the Vsv phase velocity as a function of ray

parameter, which is given below,

( ) ( )0 2 2 4 40 02

01vVs p p

αβ ε δ β β

β

= + − −

(C6)

148

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155

Vita

Anubrati Mukherjee was born in Kharagpur, India, on September 8,1973,

to Ratna Mukhopadhyay and Madhujit Mukhopadhyay. After completing his high

school in Kendriya Vidyalaya, Kharagpur, India in 1991, he entered the

Department of Geology and Geophysics at the Indian Institute of Technology,

Kharagpur, India. He graduated with the Master of Science degree in Exploration

Geophysics in 1996. After his graduation, he was employed as Software Engineer

at Satyam Computer Services Ltd., Secunderabad, India. In August 1998, he

enrolled in the PhD program in the Department of Geological Sciences, The

University of Texas at Austin. He spent the summer of 2001 in Occidental Oil and

Gas Corporation at Bakersfield, California. Upon graduation, he will join

Schlumberger in its HRT division.

Permanent address: 2501 Lake Austin Blvd, # L104, Austin, Texas 78703.

This dissertation was typed by the author.