repositories.lib.utexas.edu...v acknowledgements i would like to extend my deepest gratitude to...
TRANSCRIPT
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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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.