Frequency domain seismic forward modelling: A tool for waveform ...

Frequency domain seismic forward modelling:
A tool for waveform inversion
I Stekl
Department of Geology, Royal School of Mines, Imperial College London
Submitted for a degree of Doctor of Philosophy and
Diploma of Imperial College
September 22, 1997

Abstract
Modelling the propagation of seismic waves, and thereby predicting the response
at seismic receivers is crucial in order to interpret, or formally invert data
from seismic experiments. Commonly used seismic waveform modelling techniques
become impractical when one has to simulate datasets involving a large number of
sources.
The multiple source problem can be eciently solved by frequency domain
forward modelling.
Futhermore, viscous attenuation is easy to incorporate into
frequency-domain methods. Once the frequency domain equations are discretized,
the solution (at each given frequency) is implicit in the solution of an extremely
large matrix equation. The essential problem is to ensure the structural sparsity of
the matrix and to take full advantage of this. The sparsity of the matrix is best
handled by the nested-dissection method described by George and Liu (1981).
Ihave analysed and extended the visco-acoustic rotated nite dierence scheme
developed by Joetal.(1996). Ihave shown that these operators are optimal: if the
nested dissection method is used, nothing can be gained by higher order operators.
Several modelling and waveeld inversion examples using the scheme are desribed
that demonstrate the eciency of optimised frequency domain modelling scheme. A
waveeld inversion example proves that frequency domain modelling, when used as
an integral part of the inversion procedure, can generate an accurate, high quality
image quickly and eciently. A pre-processing technique for waveeld inversion is
developed and the eects of the pre-processing on the image and on the convergence
are analyzed. The need for an elastic scheme is identied.
To meet the need for an elastic sheme, I have further extended the rotated
operator method to the visco-elastic case. This extension leads to a high accuracy
sheme. The visco-elastic scheme is used to predict and identify the presence of shear
waves on a real data example.
1

Acknowledgements
I would like to gratefully acknowledge the assistance and encouragement of
my supervisor, Gerhard Pratt, during the course of this work. Gerhard's suggestions
and inuence have taken the nal result of this thesis at least one step further than
Ihaveinitially expected.
I am grateful to Prof. M Worthington and to the Imperial College borehole
consortium for the founds and the data provided. Also I would like toacknowledge
Dr Albert and NAGRA for founding of the inversion part of the research and the
supply of the appropriate data set. I would like to acknowledge the help from the
Overseas Research Award Scheme. I would also like to give special thanks to the
members of the geophysics group at Imperial College for their comradeship and
encouragement. Thanks to Paul Williamson, Claudia, Zhong-Min, Graham, Mike,
Peter R., Claire, John, Michel, Kevin, Miguel, Martijn, Jo, Hamish, Pui, Paul D.,
Kerry, Anna, Marcus, Richard, Eric, Yanghua, Jeremy, Patricia, Richard, Heraldo
and George for creating an amiable and supportive working environment at Imperial
College.
Many thanks to my friends Momo, Neven, Branka, Sandra and all others for
the great time we had together. I am grateful to my relatives here in London for
the support in the last ve years I have had from them. I would like to dedicate
this thesis to my parents.
2

Contents
Abstract 1
Acknowledgements 2
List of Figures 7
Chapter 1 Introduction 16
1.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2 The signicance of the frequency-space domain . . . . . . . . . . . . . 22
1.3 Forward modelling in the frequency-space domain . . . . . . . . . . 24
1.4 Fourier transforms and frequency domain modelling . . . . . . . . . 28
1.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.2 Sampling and the Sampling Theorem . . . . . . . . . . . . . . 29
1.4.3 Anti time-aliasing . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.4 Reduced time and the Fourier transform shifting property . . 31
1.5 Overview of chapters in this thesis . . . . . . . . . . . . . . . . . . . . 33
Chapter 2 Solving frequency domain wave equations: Numerical Considerations
35
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Solving linear equation systems: bottlenecks . . . . . . . . . . . . . . 36
2.3 Solving linear equation systems with multiple right hand sides . . . . 38
2.4 Matrix \ll in" and ordering schemes . . . . . . . . . . . . . . . . . . 41
3

2.5 Nested dissection ordering . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Operators and memory requirements . . . . . . . . . . . . . . . . . . 49
2.7 Comparison of band and nested dissection ordering . . . . . . . . . . 52
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 3 Visco-acoustic frequency domain acoustic forward modelling
using rotated nite dierence operators 58
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Forward modelling using rotated nite dierence operators . . . . . . 60
3.2.1 Second order frequency-domain seismic modelling . . . . . . . 60
3.2.2 The rotated operator concept . . . . . . . . . . . . . . . . . . 61
3.2.3 Finite dierence scheme in homogeneous media . . . . . . . . 63
3.2.4 Lumped and consistent matrix terms . . . . . . . . . . . . . . 64
3.2.5 Determination of optimal coecients . . . . . . . . . . . . . . 64
3.2.6 Discussion of savings with rotated operators . . . . . . . . . . 67
3.2.7 Extension to the heterogenous case . . . . . . . . . . . . . . . 68
3.3 Improvements acheived by rotated nite dierence operators . . . . . 70
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter 4 Frequency domain waveeld inversion example 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Site description: Grimsel Rock Labaratory . . . . . . . . . . . . . . . 79
4.3 Waveeld inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Waveeld inversion theory . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.1 Ecient calculation of the gradient direction . . . . . . . . . 89
4.5 Processing of third party synthetic data . . . . . . . . . . . . . . . . . 91
4.5.1 Travel time tomography . . . . . . . . . . . . . . . . . . . . . 94
4.5.2 Full waveeld inversion . . . . . . . . . . . . . . . . . . . . . 94
4.5.3 Full waveeld inversion of trace-normalised data . . . . . . . . 95
4

4.5.4 Comparison of travel time and full waveeld inversion methods 96
4.6 Inversion of real eld data . . . . . . . . . . . . . . . . . . . . . . . . 97
4.6.1 Initial full waveeld inversion . . . . . . . . . . . . . . . . . . 98
4.6.2 Regularization tests . . . . . . . . . . . . . . . . . . . . . . . . 102
4.7 Isotropic results: Evaluation and verication . . . . . . . . . . . . . 102
4.7.1 Discussion of isotropic results . . . . . . . . . . . . . . . . . . 107
4.8 Anisotropic inversion of the eld data . . . . . . . . . . . . . . . . . . 110
4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Chapter 5
Visco-elastic frequency domain seismic forward modelling119
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Visco-elastic modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.1 Rotated nite dierences: Computational stars . . . . . . . . 122
5.2.2 Rotated nite dierences: Operators . . . . . . . . . . . . . . 125
5.2.3 Consistent and lumped mass terms . . . . . . . . . . . . . . . 128
5.2.4 Heterogeneous formulation . . . . . . . . . . . . . . . . . . . . 129
5.3 Numerical errors and optimization . . . . . . . . . . . . . . . . . . . . 132
5.3.1 Determination of optimal coecients . . . . . . . . . . . . . . 132
5.3.2 Numerical dispersion . . . . . . . . . . . . . . . . . . . . . . . 134
5.3.3 Modelling in uids . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4 Elastic modelling example . . . . . . . . . . . . . . . . . . . . . . . 141
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Chapter 6 Conclusions and further work 148
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.1.1 Matrix solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1.2 Rotated nite dierence operators . . . . . . . . . . . . . . . . 150
6.1.3 Visco-elastic forward modelling . . . . . . . . . . . . . . . . . 150
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6.1.4 Waveeld inversion . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.1 Developments in seismic modelling . . . . . . . . . . . . . . . 152
6.2.2 Developments in waveeld inversion . . . . . . . . . . . . . . . 155
Bibliography 159
Appendix A Dispersion analysis for visco-elastic modelling 170
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List of Figures
1.1 A Discrete representation of the forward modelling problem. The
representation is schematic; the assumption of two dimensions is not
required at this stage, nor is this ordering of the node points necessary.
The waveeld (either a scalar or a vector quantity) is sampled at each
of the n x n z node points. . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 Nested dissection versus sequentially ordered matrix a),b) before LU
decomposition, and c),d) equivalent L matrix after LU decomposition
(George and Liu,1981). Only non-zero elements are shown in each
case. a) Matrix S for a sequentially ordered grid. b) Matrix S for a
grid ordered using nested dissection. c) L part of the LU decomposed
matrix S for case a) (memory required is O(n 3 )). d) L part of the LU
decomposed matrix S for case b) (memory required is O(n 2 log(n))).
The memory required to store matrix for a realistic value of n on
gure d) is signicantly lower than the one required for the matrix
on gure c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Two-way dissected nite dierence grid. The two way dissector, 5 (in
black) is the last part of the grid to be ordered. . . . . . . . . . . . . 44
7

2.3 Two waydissected matrix S ~
= L ~
U ~
. During LU decomposition the
values for L i;j and U i;j are lled in at the corresponding locations used
by S i;j . L i;j and U i;j denotes possible non-zero elements in matrices
L ~
and U ~
respectively after LU decomposition while 0 denotes zero
elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4 All possible subgrid (S(n; 2);S(n; 3) and S(n; 4)) situations arising
during nested dissection.
The thick black borders represent neighbouring
dissectors from previous dissections in the recursion. . . . . . 45
2.5 Enlarged L 5;5 part of the twoway dissected matrix. Non zero elements
are in grey. White space represents logical zero elements. . . . . . . . 46
2.6 Fourth order nite dierence computational star. The symbol identies
those grid points coupled to the central grid point. . . . . . . . 49
2.7 Memory requirements comparison for n x = 6:25 n z in case of band
and nested dissection ordering.
The required mesh size represents
the model size necessary to perform acoustic modelling of a wide
angle experiment with 10 Hz data and a model 350 km by 48 km.
The minimum P wave velocity is 2.8 km/s. . . . . . . . . . . . . . . 55
2.8 CPU time versus number of grid points for the case in which n x = n z ,
computed on Digital Alpha 3000/300 workstation. . . . . . . . . . . 56
3.1 Finite dierence operators for acoustic frequency domain seismic modelling
in two coordinate systems.
The symbol indicates that the
model parameter is used at the corresponding grid point. a) Finite
dierence operator in the original coordinate system. b) Finite dierence
operator in the rotated coordinate system. c) The combination
of both schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8

3.2 Numerical dispersion curves for frequency domain acoustic forward
modelling using ordinary second order nite dierence operators. a)
Phase velocity dispersion. b) Group velocity dispersion. . . . . . . . 61
3.3 Numerical dispersion curves for frequency domain acoustic forward
modelling using rotated nite dierence operators. a) Phase velocity
dispersion. b) Group velocity dispersion. . . . . . . . . . . . . . . . 65
3.4 Dierence between the numerical velocity produced with and without
the additonal coecient, d. a) Dierence in group velocity. b)
Dierence in phase velocity. See text for detail explanation. . . . . . 67
3.5 a) Model used for wide angle forward modelling, from McCarthy et al.
(1991). b) c) and d) The shaded regions depict the size of the models
that one could simulate without nested dissection and/or rotated
nite dierences if the same equipment were used. . . . . . . . . . . 71
3.6 a) Synthetic data section from the model on gure 3.5. b) Common
shot gather from the eld data. c) One of the models suggested by
McCarthy et al. (1991) showing the ray paths used in their modelling
approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.7 Time slices generated by forward modelling true the model on Figure
3.5(a) at 5, 10, 15, 20, 25 and 30 seconds. . . . . . . . . . . . . . . . 74
4.1 Grimsel Pass areal photo. . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Inside of the Grimsel Rock labaratory tunnel. . . . . . . . . . . . . . 80
9

4.3 Two representative source gathers of VSP data from Field 2, astrue
amplitude displays. a) A VSP source gather with large oset. The
spurious variation of amplitude from trace to trace is evident, as
is the consistency of alternate traces.
The data were recorded in
two passes, with intermediate traces recorded during a later, \in-ll"
survey. b) A near oset VSP source gather, on which the dramatic
change in amplitude with receiver depth is evident. These variations
in amplitudes cannot be modelled using the 2D acoustic method.
In order to invert these data I apply a normalization to each trace
separately. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 A representative common receiver gather of the Field 2 data, following
windowing and trace normalization. The receiver was in borehole 3.
The rst portion of the gather was recorded with sources in borehole 2,
and thus represents a portion of the cross borehole data. The second
section was recorded with sources in the tunnel, and thus represents
a portion of the VSP data. The data have been windowed and tracenormalized.
The random static shifts in the cross borehole data, and
the systematic static shifts in the VSP data are evident. The labels
indicate the VSP source groups that were identied, in order to solve
for the source consistent static shifts. . . . . . . . . . . . . . . . . . 83
4.5 Map of the Field 2 study area at the Grimsel Test Site. The seismic
data were acquired using the tunnel and boreholes BOUS85.002 and
BOUS85.003 (\boreholes 2 and 3").
The remaining boreholes are
exploratory boreholes in which velocity information is available and
is used to test the waveform images. The scale of this map is 1:1000,
a representative square area 160m 160m is shown for reference. . . 84
10

4.6 Comparison of the travel time tomography result and the full wave-
eld inversion from the third party synthetic elastic wave data. a)
True velocity model used in elastic forward waveform modelling, b)
traveltime tomographic image formed from the picked synthetic data,
c) acoustic waveeld inversion of the elastic synthetic data, without
trace normalization, d) acoustic waveeld inversion with tracenormalization.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.7 Starting model for waveeld inversions of the eld data (from anisotropic
velocity tomography). . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.8 Preliminary full waveeld inversion image using non normalized crosshole
part of the data only. . . . . . . . . . . . . . . . . . . . . . . . . 100
4.9 Preliminary full waveeld inversion image using non normalized VSP
part of the data only. Short oset VSP data are excluded due to large
amplitude variations. . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.10 Preliminary full waveeld inversion image using non normalized Field
2 data, including both crosshole and VSP sections of the data. Short
oset data are excluded due to large amplitude variations. . . . . . . 101
4.11 Isotropic full waveeld inversion results with various values of smoothing
parameter increasing from 0 (top left corner) to 100 (bottom right
corner). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.12 Trade o curve showing RMS roughness vs RMS residuals for a suite
of smoothing parameters. . . . . . . . . . . . . . . . . . . . . . . . . 104
4.13 Final isotropic full waveeld inversion result. . . . . . . . . . . . . . 104
4.14 Frequency domain eld data at 800Hz. Please see the text for a
full description of this gure. The grey scale is a relative amplitude,
from the maximum negative values through to the maximum positive
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11

4.15 Frequency domain modelled (predicted) data at 800Hz. See text for
a full description of this gure. The grey scale is a relative amplitude,
from the maximum negative values through to the maximum positive
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.16 Dierence between eld and modelled data at 800Hz. See text for a
full description of this gure. The grey scale is a relative amplitude,
from the maximum negative values through to the maximum positive
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.17 Inverted source signatures. These signatures were extracted as an
integral part of the waveeld inversion scheme.
The similarity of
the VSP source signatures, apart from the known static shifts, gives
credence to the robustness of the inversion scheme. . . . . . . . . . . 107
4.18 Isotropic inversion of synthetic data set from a homogeneous, anisotropic
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.19 Data residuals for the waveform inversion runs on the acoustic syntetic
elliptically anisotropic (2 percent) data by assuming: a) Isotropic
data (underestimated level of anisotropy) b) 2 percent elliptical anisotropy
(correct value) c) 4 percent eliptical anisotropy (overestimated value). 110
4.20 Anisotropic full waveeld inversion results with 0, 1, 2 and 3% elliptical
anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.21 RMS residuals for each test anisotropy level. . . . . . . . . . . . . . 113
4.22 Final full waveeld inversion image using 2% elliptical anisotropy. . . 114
4.23 Frequency domain dierence eld (i.e., data residuals) at 800 Hz from
the anisotropic inversion. See text for a full description of this gure.
The grey scale is a relative amplitude, from the maximum negative
values through to the maximum positive values. . . . . . . . . . . . 115
4.24 Final waveeld inversion images from both Fields 1 and 2, using 2%
elliptical anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12

4.25 Final waveeld inversion images from both Fields 1 and 2, using 2%
elliptical anisotropy (colour version). . . . . . . . . . . . . . . . . . . 117
5.1 Computational stars for frequency domain elastic modelling. These
stars indicate the coupling of the components of the displacement
eld at the central node to displacements at the nearest neighbors.
a) The ordinary, second order computational star, b) a possible rotated
computational star, and c) a minimal, rotated computational
star. The symbol, represents the coupling of the same displacement
components, and also represents the only non-zero terms required in
acoustic modelling. The symbol, symbol represents the coupling between
perpendicular displacement components. The star in c) does
not use additional points over the star in a), but introduces additional
coupling between components not present in the original star. . . . . 121
5.2 Optimal values of coecients, a (the fraction of the ordinary second
order scheme) and b (the fraction of the consistent mass matrix),
plotted as a function of the Poisson's ratio, . The optimal value of
coecient b is relatively insensitive to the value of . The optimal
value of coecient a decreases for high values of , and becomes 0
for the uid case, in which case only the rotated scheme is used. . . 134
13

5.3 Numerical dispersion of the new scheme for a Poisson ratio = 0:33,
depicting normalized numerical velocity curves for compressional and
shear phase velocities (top tworows) and group velocities (bottom two
rows). Results are presented for the standard second order scheme
(left column) and the new, combined scheme (right column).
The
dispersion curves are plotted against the shear wavenumber in grid
point units, i.e., the reciprocal of the number of grid points per shear
wavelength, G s . See text for the meaning of the symbols used on the
vertical axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4 Numerical dispersion for a Poisson ratio =0:4, depicting normalized
numerical velocity curves for compressional and shear phase velocities
(top two rows) and group velocities (bottom two rows). Results are
presented for both the standard second order scheme (left column)
and the new, combined scheme (right column). The dispersion curves
are plotted against the shear wavenumber in grid point units, i.e., the
reciprocal of the number of grid points per shear wavelength, G s . See
text for the meaning of the symbols used on the vertical axes. . . . . 136
5.5 Compressional wave dispersion in uids for the new, rotated scheme.
In the uid case I use only the rotated scheme, with no component of
the original, unrotated scheme (a = 0). a) Normalized compressional
phase velocities. b) Normalized compressional group velocities. . . . 139
5.6 P-wave velocity model for the Imperial College crosshole experiment.
The model was obtained using acoustic fullwave inversion (Pratt at
al. 1995). Data from the experiment, and modelled data for this
velocity structure, are shown in Figures 5.7 and 5.8 . . . . . . . . . . 143
14

5.7 a) A representative common source gather from the crosshole data
collected at the Imperial College test site. The signal to noise ratio is
high, and the rst arrival waveforms are clear and coherent. At late
times, incoherent, large amplitude arrivals dominate. b) Predicted
common source data using acoustic forward modelling in the velocity
structure shown in Figure 5.6. The rst arrival traveltimes and waveforms
match well with the observed data, but the large amplitude,
late arrivals are not predicted with the acoustic method. . . . . . . . 144
5.8 Predicted common source data using the new visco-elastic modelling
results. a) Horizontal displacement component. b) Vertical displacement
component. The horizontal component shows rst arrival times
and waveforms that are similar to the acoustic modelling results, and
some high amplitude arrivals at late times. The vertical component
shows high amplitude arrivals similar to those observed on the real
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
15

Chapter 1
Introduction
Modelling the propagation of seismic waves, and thereby predicting the response
at seismic receivers is a crucial step in the interpretion, or the formal inversion
of data from seismic experiments. Seismic modelling is thus an important tool in
geological hypothesis testing.
As seismic experiments become increasingly more
sophisticated and complete, we naturally seek to model the seismic response of increasingly
realistic media. To model complete, wide band, seismic wave behaviour,
in a heterogeneous, porous, and visco-elastic medium, numerical modeling of the full
visco-elastic wave equation would seem desirable. Ideally one would like to include,
if possible, 3 dimensions (3-D), general anisotropy and arbitrary visco-elasticity.
While formulations for 3-D anisotropic media are possible (Mora, 1989a; Carcione
et al., 1992), the memory and cpu time requirements for realistic model sizes currently
still prevents the production use of such methods, especially for multi-source
problems. Nevertheless, there has been a historical progression toward the practical
use of ever more general methods (Alterman and Karal Jr, 1968; Kelly et al., 1975;
Gazdag, 1981; Dablain, 1986; Holdberg, 1987; Virieux, 1986a; Dai et al., 1995),
both for one-dimensional (1-D) two-dimensional (2-D) earth models.
There are two major approaches used in seismic modelling: The rst of these
is asymptotic ray theory (e.g. Cerveny et al. (1982) or Chapman (1985)), a technique
that can oer insight into the nature of various arrivals in the seismic record but
16

that may fail to adequately modelthewaveforms in complex media. Asymptotic ray
theory assumes a high-frequency wave behaviour; this puts certain constraints on the
model complexity asa function of the lowest wavelength. If velocity discontinuities
are reached, explicit boundary conditions must be applied in order to divide the ray
into reected and transmitted (i.e., refracted) rays, each of which are further traced
through the model. The high frequency restriction limits the use of the technique
to simple models with relatively few data phases, usually specied in advance. The
second group of modelling methods comprise the numerical methods based on partial
dierential or integro-dierential wave equations, without the use of a high frequency
approximation. These methods are usually formulated as nite dierence or nite
element problems. Such wave equation methods equation guarantee the simulation
of all possible phases (within the assumptions built into the initial wave equation).
The generation of mode conversions, reections and refractions is not determined
by the choice of input parameters (as in asymptotic ray theory), but is instead
an integral feature of the modelling itself. [An exception to this are the numerical
methods of (Madariaga, 1984), based on matrix propagator methods. However these
methods are usually only available for 1-D models]. As a result, relating the phases
in the seismic record to individual features in the model may not be straightforward
in complex models.
Wave equation methods can be further sub-divided intoanumber of classes,
depending on the domain in which the initial wave equation is solved.
Possible
choices of domain include any combination of time/frequency, space/wavenumber
or other domains, such as the , p transform domain. Each domain has its own
advantages and disadvantages. For 2-D earth models, time domain methods have
dominated the literature.
In contrast, this thesis will be largely concerned with
numerical modelling in the frequency-space domain. The primary reason for this is
that the modelling algorithm is tightly coupled to a formal method for the automatic,
frequency-space domain inversion of seismic waveform data. The results obtained by
17

Pratt (1995) and Song (1995) showed the great potential of frequency-space domain
waveeld inversion. Unfortunately, the size of the geological experiments in which
these techniques could potentially be applied were limited by the ineciency of the
forward modelling technique. The overall objective of the project described in this
thesis was to develop improved forward modelling methods and to incorporate these
into frequency-space inverse methods, thereby increasing the maximum model size
that can be handled in these methods.
1.1 Historical overview
Seismologists began using wave equation based numerical methods in the late
1960's. Most of this inital work was based on nite dierence techniques. Numerous
discrete solutions for the second order wave equation in homogeneous regions by use
of explicit time integration were published (Alterman and Kornfeld, 1968; Alterman
and Karal Jr, 1968; Alterman and Rotenberg, 1969; Alterman and Loewenthal,
1970). In Alterman's work, a homogeneous wave equation formulation was used and
interfaces were treated using explicit boundary conditions. This early work was only
of limited value due to the limited computational resources available at the time,
and due to the limitations on model complexity due to the necessity of treating
each interlayer boundary explicitly.
Nevertheless, these experiments produced a
deep theoretical understanding of wave propagation in homogeneous and two layer
media, and proved that a numerical methods were useful in the innitely many earth
modesl for which no analytical solution is available.
Today exploration geophysicists attempt to model much more complex, realworld
media that include irregular boundaries and laterally varying model parameters
in all directions. In order to predict the response in such cases, the interlayer
boundary conditions had to be built implicitly into the modelling scheme. It become
common practice in the mid 1970's to use a heterogeneous wave equation formulation
18

(Boore, 1972; Kelly et al., 1975). With these modelling techniques, the simulation
of complex media became possible, although due to the simple, low accuracy nite
dierence formulations and the still limited computational resources, realistically
sized models were still out of reach.
As methods became more capable, a number of other development directions
were explored. These included a switch from the earliest, 1 - D earth models
(Abramovici and Alterman, 1965), to 2-D models (Alterman and Karal Jr, 1968),
and nally to 3-D models (Johnson, 1984; Reshef et al., 1988a; Reshef et al., 1988b;
Mora, 1989a). A fundamental limitation of Cartesian 2-D methods is that they do
not accurately simulate the phase and amplitude of eld seismic data, even within
the assumption of a 2-D earth model. Bleistein (1986) suggested a \2.5-D" method
for correcting for phase and amplitude data from point sources using ray trace parameters
later reformulated by Randall (1991) as a nite dierence formulation;
Song and Williamson (1995) suggested a wavenumber transform approach for accounting
for these corrections with nite dierence modelling methods in frequency
domain.
In addition to the progress made in the last decades in extending the dimensionality
of numerical wave equation modelling methods, researchers have also
attempted to model increasingly general physical phenomena.
Methods for the
acoustic wave equation (Michell, 1969; Gazdag, 1981; Virieux, 1986b; Reshef et
al., 1988a; Song and Williamson, 1995), the elastic wave equation (Alterman and
Karal Jr, 1968; Virieux, 1986a; Pratt, 1990a), the visco-elastic wave equation (Kjartansson,
1979; Emmerich and Korn, 1987; Robertsson et al., 1994), the anistropic
wave equation (Mora, 1989a; Carcione et al., 1992; Carcione, 1995) and the poroelastic
wave equation (Zhu and McMechan, 1991; Dai et al., 1995) have all been
developed. While one knows that the earth is 3-D, porous and anisotropic, in production
modelling and inversion choices and compromises must be made. Even if
it were possible to incorporate all these physical eects, dening appropriate model
19

parameters is a daunting task.
Extracting a detailed P-wave velocity eld from
reection seismic data is already dicult (Al-Chalabi, 1994); the full extraction of
heterogeneous, complex-valued, anisotropic visco-elastic parameters in detail would
seem impossible. It is perhaps obvious that one must always simplify the model in
order to be able to represent the essence of the recorded data without unnecessary
overparameterization. This decision naturally depends on the modelling objectives;
in some cases the arrival times may be a sucient data representation. In other case
waveform data will be required. If the data do not contain a signicant amount of
the S-wave energy, or if the S-wave phases are not used in the interpretation, the
acoustic assumption may be sucient. However, if S-wave phases are important,
then additional considerations regarding the physical parameters (e.g., the source
mechanism, anisotropy, polarization and borehole eects) often become important.
The second order elastic wave equation is analytically equivalent to the coupled,
rst order, elasto-dynamic equations. However, the two formulations of the
wave equation lead to dierent numerical solutions. Numerically stable, dierencing
expressions are much easier to formulate for rst order partial dierential equations
than for second order equations.
However, the model parameters must then be
dened on two, separate, \staggered" grids. The denition of the model itself becomes
ambiguous at intermediate points on the grid. Madariaga (1976) developed
the rst, staggered grid, nite-dierence method for the elasto-dynamic wave equation
formulation. This formulation became dominant (Virieux, 1986a; Dai et al.,
1995) for time domain schemes, due to the fact that it was the only known scheme
which enabled the simulation of elastic waves in models with liquid-solid interfaces
(obviously a critical facility in exploration studies (see, for example Kerner (1990))).
Some simplications are possible for certain kind of experiments by using a
one-way wave equation (Claerbout, 1970). The one-way wave equation is primarily
used due to the high computational cost of simulating the full wave equation. The
method can predict a transmitted wave eld; it is possible to simulate scattered wave
20

elds by explicitly dening each back scattered event, but reverberations and surface
waves travelling perpendicular to the paraxial direction cannot be modelled at all.
Even with this serious disadvantage, the approach has been very popular as a migration
algorithm (Claerbout and Doherty, 1972; Loewenthal et al., 1976; Berkhout
and Van Wulten Palthe, 1979; Berkhout, 1985), since in post-stack migration the
propagation is required in only one direction (down), and the computational costs
are much lower than full wave equation modelling. Full wave equation methods,
however have been used extensively in migration from the late 70's (Hemon, 1978;
Beysal et al., 1983; Loewenthal and Mufti, 1983).
In many disciplines the nite element formulation is the primary choice of
numerical method. However, seismic modelling the nite element method has never
taken over from nite dierences as a main stream technique. Although the earliest
papers on seismic modelling used the nite element method, (Smith, 1974),
the essential diculty remains with us today: The lack of a mesh generator which
will utilise the full advantage of nite elements, distorting the grid where possible
and still providing a sucient number of node points for accurate wave equation
modelling. There is another reason why nite element seismic modelling is not used
more often: Since wave propagation problems demand that the model be sampled
at a very ne scale, using exact, irregular boundaries will not signicantly aect the
result. In most practical cases the knowledge of the model itself is only known at a
relatively long scale length, much coarser than the model parametrization, so that
exact boundaries cannot be dened. Finite elements may have certain advantages
in the case of theoretical, simple models in which only a limited number of homogenous
regions represent the model and an exact solution is required, but in applied
cases where the model is highly heterogeneous and the shape of the \homogeneous"
elements is not known the main advantages of nite elements appear to be of little
use. The main nite element work is still on square (rectangular) grids. In this
case there is no particular advantage of using either the nite dierence or the nite
21

element methods.
1.2 The signicance of the frequency-space domain
It is clear from the review given above that frequency domain methods are
less common than time domain methods in seismic wave propagation modelling. An
early exception was (Lysmer and Drake, 1972), and the fundamental advantages
of frequency domain modelling (especially for multi-source inverse problems) was
clearly pointed out by Marfurt (1984a; 1984b), by Marfurt and Shin (1989) and by
Pratt (1989a).
Time domain methods are suitable if the full time domain seismic section
for a single source, or for a small number of sources is required.
On the other
hand, frequency domain methods are ecient in cases in which a limited number of
single frequency data are required, or in cases in which a time response for a large
number of sources is required. As I will show in this thesis, in these circumstances a
frequency domain approach can produce results at a fraction of the computational
costs required by time domain schemes.
Recently, much attention has been focussed on the modelling of seismic waves
in models that include visco-elastic losses. Because the attenuation due to viscoelastic
losses is thought to be related to time dependent creep and relaxation effects
(see for example Kjartansson (1979)) that lead to integral terms in the wave
equation, special techniques are required to include these eects into time domain
numerical simulations (Emmerich and Korn, 1987; Carcione et al., 1988; Robertsson
et al., 1994). A solution to the diculty of the representation of the integral
terms in the time dependent wave equation is to transform the equations into the
frequency domain, and model the resultant Helmholtz type equations, in which case
frequency dependent attenuation can be easily represented by complex valued elastic
parameters (Muller, 1983), without any additional computational eort.
22

The advantages of frequency domain schemes are utilised to the full extent
in waveeld inversion approaches, in which only a few frequencies may be required
(Pratt et al., 1995; Song et al., 1995); in this thesis I will deal exclusively with
frequency-space domain methods and their application to inverse problems | hence
the sub-title of this thesis, \A tool for waveeld inversion". In order to eciently
invert eld data one needs an accurate and fast forward modelling technique. Computation
time and accuracy normally tradeo against each other: Accuracy can
usually be achieved by using very ne discretization grids, but this is costly in terms
of computational resources. Sophisticated numerical methods achieve accuracy by
optimized design of the numerical method, allowing the number of grid points to be
reduced (for a given level of accuracy).
In order to improve the performance of any forward modelling approach, the
limitations of the particular scheme must be well understood. These limitations include
those introduced by the original choice of the underlying wave equation (i.e.,
are we modelling acoustic or elastic waves, are we using one, two or three dimensions,
are we accounting for viscous eects, etc), and those limitations caused by
the numerical approximations (i.e., are the numerical methods suciently accurate).
This understanding of the modelling method is important in order not to misinterpret
the results, in order to build better models, and in order to be able to choose
appropriate modelling techniques and modelling parameters. Simple frequency domain
modelling codes, such as the one developed by Pratt (1989b) (in use when I
started the project), are not suciently accurate to be able to handle large surveys.
The software engineering problems associated with frequency domain forward
modelling are rather dierent from those associated with time domain methods.
Once the frequency domain equations are discretized, the solution (at a given
frequency) is implicit in the solution of an extremely large matrix equation. The
essential problem is to control the sparsity pattern of the matrix (itself controlled
by the spatial extent of the dierencing operators), and to take full advantage of
23

this sparsity. As in time domain methods, however, the overiding concern is to limit
the number of grid points per wavelength that are required. Thus, for frequency
domain methods, it is critical to optimize the accuracy of the numerical operators,
while also minimizing the spatial extent of these operators.
In time domain forward modelling, accuracy can be achieved through the use
of high order spatial operators; for large problems this is crucial. In frequency domain
modelling, high order spatial nite dierence operators lead to large increases
in computational costs that are not compensated for by the gain in accuracy (see
chapters 2 and 3 for a full explanation). Instead we must seek other methods by
which to increase the accuracy (and thereby reduce the costs).
Two major improvements were implemented: The rst of these involved modications
to speed up the numerical aspects of the matrix solver; the second involved
modications to the numerical operators themselves. Together these two modications
lead to a dramatic improvement in the computation times for acoustic wave
equation modelling and inversion.
These improvements were made use of in the
inversion of a large, transmission seismic survey in which the use of extended parameter
tests, involving a large number of modelling runs, was made possible by
the improvements. During the inversion of the eld data, it became clear that the
acoustic method needed to be replaced by an elastic method; in the nal stage of
this project I developed extensions of the modelling methods to the elastic wave
equation, paving the way for future elastic waveeld inversion methods.
1.3 Forward modelling in the frequency-space domain
In this section the fundamental equations for seismic modelling in the frequencyspace
domain are presented, and some of the basic considerations for frequencydomain
methods are reviewed. I begin with the assumption that a particular wave
equation has been selected for modelling purposes, and we have already discretized
24

the partial dierential equations for numerical modelling. The discretized equations
for the time domain acoustic or elastic wave equations using either a nite dierence
or a nite element approach can be written as
M ~
~u(t)+ K ~
~u(t)= ~ f(t) (1.1)
(see for example Marfurt, 1984a), where ~u(t) is the discretized waveeld (i.e., the
pressure, or the displacement) arranged as a column vector, M is the mass matrix,
~
K is the stiness matrix and ~ f(t) are the source terms, also arranged as a column
~
vector.
Equation (1.1) can be approached in either the time domain or in the frequency
domain.
From this point on, this thesis is concerned with the frequency
domain method for solving these problems. Taking the temporal Fourier transform
of equation (1.1) yields
where
u(!) =
Z 1
,1
K ~
u(!) , ! 2 M ~
u(!) =f(!) (1.2)
~u(t)e ,i!t dt and f(!) =
Z 1
,1
~f(t)e ,i!t dt (1.3)
are Fourier transforms. If viscous damping is included, equation (1.2) becomes
K (!) u(!)+i! C (!) u(!) , ! 2 M (!) u(!) =f(!) (1.4)
~ ~ ~
where C (!) is the damping matrix. Details of the nite-element and nite-dierence
~
approaches can be found in many textbooks (Zienkijevic, 1977; Bathe and Wilson,
1976). In Chapter 3 and Chapter 5 I will give explicit formulas for the matrix coecients
for both the acoustic and elastic wave equations. The mass, stiness and
damping matrices are computed by forming a discrete representation of the underlying
partial dierential equations and the physical parameters (for example, the
seismic velocities, the bulk density and the attenuation parameters). For simplicity
I rewrite equation (1.4) as
S ~
(!) u = f or u = S ~ ,1 (!) f (1.5)
25

where the complex \impedance" matrix, S ~
, is given by S ~
(!) = K ~
(!),! 2 M ~
(!)+
i! C ~
(!). I shall refer to any modelling approach based on equation (1.5) as \Frequency
domain modelling". Frequency domain modelling is an implicit nite dierence
method (Marfurt, 1984a); the second, explicit form shown in equation (1.5) is
only representational, as it is not generally possible (or desirable) to actually invert
the very large impedance matrix S ~
. Equation (1.5) is often solved using matrix
factorisation methods, such as LU decomposition (Press et al., 1992; George and
Liu, 1981; Pratt and Worthington, 1990).
If LU decomposition is used to solve
equation (1.5), the matrix factors can be re-used to solve the forward problem for
any new source vector, f extremely eciently. This point is especially important in
the iterative solution of the inverse problem, in which many forward solutions, for
real sources and \virtual" sources will be required at each iteration. It is critical
to use ordering schemes that allow maximum advantage to be taken of the sparsity
of both S ~
and its LU factorisation; nested dissection (George and Liu, 1981)
is such a method. Later in this thesis I will explain this method and discuss the
computational aspects that may aect the eciency.
Inowintroduce a specic discretization, depicted in Figure 1.1, in which the
waveeld is to be computed at n x n z nodal points on a regular grid (the grid is
2 dimensional for illustration purposes, but could be 1, 2 or 3 dimensional). The
model can be thought of as being specied at each of these node points.
The waveeld vector, u and the source vector, f are (n x n z ) 1 column
vectors; the complex impedance matrix, S is an (n x n z ) (n x n z ) matrix. All
~
quantities except the model parameters can take on complex values.
Note that,
although we will treat equation (1.5) as if it describes forward modelling for a single
source position, additional source locations can be incorporated simply by increasing
the number of elements in u by n x n z for each additional source; S ~
and S ~
,1
then have block diagonal structures, in which each diagonal block is an identical
submatrix.
We could also feed in additional frequency components in the same
26

Figure 1.1: A Discrete representation of the forward modelling problem. The representation
is schematic; the assumption of two dimensions is not required at this
stage, nor is this ordering of the node points necessary. The waveeld (either a
scalar or a vector quantity) is sampled at each of the n x n z node points.
manner, although the diagonal block submatrices of S ~
are then no longer identical.
The same comment applies to the 2:5 , D method of Song and Williamson (1995),
in which a new diagonal block would be generated for each wavenumber considered.
By examining the solutions to equation (1.5) when the components of the
source vector, f i are replaced by a Kronecker delta, ij , it is clear that the columns
of S ~ ,1 must contain the discrete approximations to the Green's functions. Thus,
h
S ,1 = g
(1)
g (2)
~
::: g (nxnz) i
; (1.6)
where the column vectors g (j) approximate the discretized Green's function for an
impulse at the jth node. If the original physical problem is exactly reciprocal with
,1
respect to an interchange of source and receiver elements, then both S and S ~ ~
must be symmetric (not Hermitian) matrices. [In implementation S is often not
~
perfectly symmetric when certain (unphysical) absorbing boundary conditions are
implemented (Pratt and Worthington, 1990). This does not cause any problems].
27

1.4 Fourier transforms and frequency domain modelling
An understanding of Fourier transforms and their properties is important for
frequency domain modelling. The continuous Fourier transform is dened in equation
(1.3); for numerical computations this transform and its inverse are discretized,
leading to a Discrete Fourier Transform (DFT), and its optimized implementation,
the Fast Fourier Transform (FFT). The Fourier transform in exploration seismology
is most commonly used to transform time domain data into the frequency domain,
in order to apply a particular lter, following which an inverse Fourier transform is
applied to bring the data back into the time domain. However, in frequency domain
modelling the rst step is not needed. We generate the components of the DFT of
the data directly; if time domain results are required we obtain these by the inverse
DFT, in which case sucient sampling in the frequency domain is required. Often,
as we shall see, when solving the inverse problem we never need the time domain
data, and we need not be as concerned with sampling criteria. The following subsections
will show the Fourier transform properties and explain the implications for
frequency domain modelling.
1.4.1 Theory
The Fourier transform, in essence, decomposes or separates a waveform or
function into sinusoids of dierent frequencies, which sum to yield the original waveform.
In frequency domain modelling we use a monofrequency component of the
source to produce a monofrequency response at the receiver points. By performing
an inverse Fourier transform of the monofrequency responses we are able to produce
a required time domain response at the receiver locations. The forward and inverse
DFT pairs for a time series h and a frequency series H are dened as (Hatton et al.,
1986)
H k = 1 N
N,1
X
r=0
h r e ,i2kr=N ; (1.7)
28

for k = 0; 1;:::;N ,1 and
h r =
N,1
X
k=0
H k e i2kr=N ; (1.8)
for r =0;1;:::;N,1, where r is a time sample index, k is a frequency domain sample
index, H k is the k-th Fourier transform coecient, h r is the time series. Provided
each representation is complete (the time series or the frequency components), each
series can be uniquely recovered from the other, using these formulas.
1.4.2 Sampling and the Sampling Theorem
As we actually work with a discrete representation h n = h(t n ). The function
h(t) is said to be band limited if its Fourier transform H(f) = 0 for jfj > f c
where f c is a nite \critical" frequency. In seismic case all signals are band limited
due to a limited source spectrum, and are almost always treated with an analogue
\anti-alias" lter to ensure this property before sampling.
The sampling theorem states that a band-limited function h(t) is completely
specied by the sampled values f n (t n ), provided that the sampling interval, t
satises
f c 1 N y
= 1
2t
(1.9)
The frequency N y is known as the Nyquist frequency for the given sampling interval
t.
Beyond the Nyquist frequency, the periodicity and the conjugate symmetry
of the DFT (for a real valued time series) causes the highest frequencies to be
\wrapped" around the frequency axis and to be aliased as lower frequencies. This
theorem is important if we are attempting to construct an un-aliased frequency
spectrum from a time series.
A similar sampling theorem is relevant ifweare trying to reconstruct a time
series from a limited number of samples of the frequency spectrum, as is the case for
frequency domain modelling. We must sample the frequency spectrum suciently
29

in order to unambiguously reconstruct the time series for the required length of
time. If we again assume the time series is real valued, and make use of the resultant
conjugate symmetry in the frequency spectrum, then the time series up to a
maximum time, t max is completely specied by the sampled values, provided that
the frequency sampling interval, f satises
f 1
t max
: (1.10)
This formula assumes that the time series is completely causal, i.e., that the time
series is equal to zero for all negative values of time. If this is not the case, then
an additional factor of two must incorporated into the denominator. Provided the
frequency sampling criteria is met, the time function may be reconstructed with any
desired time sampling, t until the maximum time, t max .
If the frequency sampling criteria above is not satised, then the periodicityof
the inverse DFT causes the time samples for times greater than t max to be wrapped
around the time axis, and to appear as if these were early time samples (i.e., these
are aliased in time). Thus it is important that the model be designed in such a
fashion as to prevent the simulation of any arrivals later than t max . Naturally this
is not always possible; fortunately a trick exists that can be made use of to inhibit
time aliased signals.
1.4.3 Anti time-aliasing
The technique for anti time-aliasing frequency domain modelling results has
been described by (Subhashis and Frazer, 1987).
Due to the periodicity in any
Fourier series, the inverse DFT returns not a non periodic h(t), but periodic
1X
n=,1
h(t + nt max ): (1.11)
Thus, a time series which is non-zero for times greater than t max will be corrupted.
To prevent this we can compute F (!+i) instead of F (!) where is an appropriate,
30

small real number. This computation is easily implemented in frequency domain
modelling, and it has the advantage of yielding, after the inverse DFT, the time
function
1X
n=,1
h(t + nt max )e ,(t+ntmax) : (1.12)
Thus, by using a complex value for the frequency, the time function has been eectively
multiplied by a decaying exponential function. Each successive alias component
of the time function is multiplied by a smaller value. To recover an approximation
of the original, desired function we multiply this result by e t and produce
the result
1X
n=,1
h(t + nt max )e ,ntmax : (1.13)
For the orginal, unaliased component (n = 0), the original signal is recovered. For
all positive values of n, the signal is attenuated by an ever smaller factor { the
aliased signal is still there, but it is attenuated. The method fails if the time series
has non-zero values for negative times (n

If the forward Fourier transform is dened by H(!) =
1 ,1 h(t)e i!t dt, then
Z 1
,1 h(t , x=c)e,i!t dt =
Z 1
,1 h()e,i!(t+x=c) d
= e ,i!x=c Z 1
,1
h()e ,i! d
= e ,i!x=c H(!): (1.15)
The quantity x=c is a simple time shift, and thus we make use of the shifting property
of the Fourier transform (in which the dummy integration variable is changed from
t to ). Equation (1.15) shows that we can move the time window of interest by
multiplying the frequency domain result by e ,i!x=c before performing the inverse
DFT. This is an important result in frequency domain modelling, since we often
need to simulate the seismic time domain response at far oset receivers in large
experiments.
Let us take for example a source-receiver oset of 300 km. In this case the
full wave propagation time can be up to 30 to 40 seconds, while the required signal
may be only a few seconds long. Due to the shifting property above, we have the
option in frequency domain modelling to compute only a ve second time window,
from, e.g., 35 to 40 seconds. We do this by setting the time shift equal to 35 seconds
in equation (1.15). A ve second window can be completely represented using a
frequency sample interval of f =1=5=0:2 Hz. This should be compared with the
frequency sample interval required for the full 40 second record, f =1=40 = 0:025
Hz. The number of samples required for a given maximum frequency is reduced by
87.5%. In the time domain, it would be necessary to simulate the full 40 seconds in
order to generate the same, nal, ve seconds of useful data.
In this way we may directly calculate the time domain data in reduced time,
in order to decrease the number of frequencies required for forward modelling. If
reduced time output is used in conjunction with anti time-aliasing, it is important
to ensure that the rst output time sample occurs before the rst data arrival. This
is becaues non-zero signal arriving before the desired time window begins will be
32

amplied instead of attenuated.
1.5 Overview of chapters in this thesis
This thesis begins with a discussion of the matrix solver used to generate
solutions to the frequency domain nite dierence matrix, equation (1.5). If the
matrix solver is inecient, no matter how good the nite dierence formulation
is, the costs involved will be prohibitive.
In Chapter 2 I will initially dene the
requirements expected from ecient matrix solvers. I will provide an analysis of
the structure of the matrix, and the manner in which this structure aects the
matrix solver in general, and the eects that various nite dierence operators will
have on this structure.
This will enable guidelines to be set for the appropriate
nite dierence operators in order that the computational costs can be kept low.
The technique of nested dissection, which optimises the initial sparsity pattern will
be described and quantitative estimates of the computation times and the storage
requirements will be given.
After analysing the general problem of the matrix solver, I will move on to
a specic nite dierence technique for visco-acoustic media in Chapter 3. I will
use the rotated nite dierence operators suggested by Jo et al. (1996). I will
present and analyze these operators, and then extend them to the heterogeneous
case. I will then discuss the combined use of the nested dissection method and the
implementation of the rotated nite dierence operators and prove that the scheme
is optimal, and that no improvement will be achieved by the use of higher order
spatial operators. Chapter 3 concludes with an example (based on a real wide-angle
experiment) that demonstrates the visco-acoustic modelling described and analyzed
in these initial chapters.
In Chapter 4 the application of the frequency domain visco-acoustic modelling
scheme as a tool for waveform inversion will be presented. The example presented is
33

ased on data setfrom an underground laboratory in a crystalline rocks. The data
suer from signicant noise problems. I will describe a pre-processing ow used to
deal with these data problems. A way of determining the correct parameters based
on the level of data residuals to obtain an optimal image will be presented. Potential
anisotropy eect on the image will be discussed with the procedure, base on data
residuals, for minimizing the imaging artefacts when present.
A complete set of
tests for selecting the inversion parameters is made possible by the improvements in
eciency presented in Chapters 2 and 3. I will show evidence for spatial variation
of the anisotropy; an eect that cannot be properly modelled or inverted using the
visco-acoustic method.
As a result of the conclusions of Chapter 4, in Chapter 5 I develop a viscoelastic
modelling scheme, as a rst step toward the development of a fully anisotropic,
visco-elastic modelling and inversion scheme. In developing these scheme, I begin
by dening the rotated nite dierence operators required for the visco-elastic wave
equation. A full description of the visco-elastic scheme, including a dispersion analysis,
will be presented. An analytical proof that the scheme can work in the uid case
will be given. As an example a cross-borehole data set from the Imperial College
test site will be shown and compared with the visco-acoustic modelling results.
In Chapter 6 I summarize the developments presented in the thesis and
present my conclusions. For some of the models I present in the thesis, a reduction
of over 90% in computational requirements, in comparison with the original simple
modelling techniques, have been acheived, in both the visco-acoustic and the
visco-elastic modelling cases. This has been achieved through the use of a fully integrated
approach, in which I concentrated on all aspects of the modelling procedure
| optimization of each individual aspect of modelling technique separately is not
enough. The thesis concludes with several indications as to where possible further
work could be concentrated, to allow the extension of these results to more complex
data examples.
34

Chapter 2
Solving frequency domain wave equations:
Numerical Considerations
2.1 Introduction
Seismic forward modelling can be formulated as a time domain inital value
problem or as a frequency domain boundary value problem (see Chapter 1 equation
(1.5)). Explicit initial value problems do not require a large amount of memory to
run, however the amount of computational time can be signicant if the number of
time steps or the number of sources is large. The numerical solution of boundary
value problems involve solving a large (usually sparse) system of linear equations
(i.e., the matrix S ~
in equation (1.5)). The cost of solving the system increases dramatically
as the number of equations increases. To perform full matrix inversion,
or Gaussian elimination on a large system of linear equations requires a signicant
amount of memory and CPU time. However, for sparse systems, savings can be
obtained by exploiting the sparsity, and further savings are realized when a large
number of right hand sides are involved (representing additional sources in the seismic
modelling case). The utility of dealing with multiple right hand sides is critical
in seismic inverse problems, in which only a limited number of frequencies for a
large number of sources may be required (Pratt and Worthington, 1990). This is
35

therefore one of the main applications of frequency-domain seismic forward modelling.
To solve a large system of linear equations eciently one has to consider the
detailed numerical properties of the problem and use them to the full extent tokeep
overheads as low aspossible.
In this chapter I will consider the characteristics of the frequency domain
seismic forward modelling problem in the case of multiple source experiments, and
develop the appropriate matrix description. I will then move on to a consideration
of the characteristics of the nite dierence operator required to generate solutions
at minimum computational cost. These characteristics will be utilized in chapters 3
and and 5todevelop optimal operators.
2.2 Solving linear equation systems: bottlenecks
As shown in Chapter 1, frequency domain forward modelling requires a solution
to a system of linear equations (see equation 1.5). In 2-D, to nd the solution
(the eld vector u), one has to solve a linear system of n x n z
equations with
n x n z unknowns (where n x and n z are the number of grid points in the x and
z directions in the model).
If the problem is elastic, each eld component is a
two-component vector, and the total number of equations is doubled.
Although
conceptually straightforward, the computational costs involved in Gaussian elimination
or matrix inversion can become prohibitive when the problem size (n x
n z
)
increases due to a cubic (O(n x n z min(n x ;n z ))) growth in memory requirements.
In order to decrease the computational costs involved one has to consider the properties
of the matrix
S ~
and of the underlying physical problem, before developing
an appropriate matrix solver.
The requirements I am going to consider in this chapter include the following:
The problem must be ecently solved for mutiple right hand sides (multiple sources),
the matrix solver must be computationally ecient and use a minimum of physical
36

(RAM) memory, andtheunderlying numerical approximation must betuned tothe
matrix solver for minimum overall computational costs. These problems must all be
considered together, as the each choice at each stage aects the choice, at the next
stage.
Iterative solvers are usually considered to be the best way of solving positive
denite linear systems (for a denition of positive denite equations see for example
George and Liu (1981)). The main advantage of iterative matrix solvers is that full
advantage can be taken of the initial matrix sparsity. As a result, the amount of
memory required is small (of the order of n x n z ). The problem with frequencydomain
forward modelling of the seismic problem is that the matrices arising from
the nite dierence equations are not always positive denite.
For example, the
absorbing boundary conditions often used are not physical - they are used only
because we are attempting to model innite media by using a nite model. As a
result, due to ill conditioning, iterative methods either do not converge, or converge
too slowly to be considered appropriate to solve the system. The other problem with
iterative solvers is that they are not suitable for systems with multiple right hand
sides. The computational costs for iterative solvers increases in linear proportion to
the number of right hand sides. In the problems I am going consider, the number
of right-hand sides can be signicant. Direct methods, which are able to solve the
problem eciently for multiple right hand sides, are therefore more eective than
the iterative ones in this case.
Direct methods for solving linear equation systems require signicantphysical
memory.
Matrices produced by nite-dierence (or nite-elements) methods are
always sparse, but the sparsity pattern is not preserved by most direct matrix solvers.
The sparsity pattern of the initial matrix depends on the nite dierence operator
used and on the grid ordering used. In this chapter I will initially concentrate on
the eect of grid ordering, and then move on to consider the eect of the size (i.e.,
the order) of the nite dierence operator on the memory requirements.
37

2.3 Solving linear equation systems with multiple right hand
sides
An eective direct method for solving a system of linear equations with the
multiple right hand sides is LU decomposition, which transforms a system :
S ~
u = f (2.1)
into the system
L ~
U ~
u = f; (2.2)
where matrices L ~
and
U ~
are lower and upper triangular matrices. LU decomposition
inevitably destroys some of the sparsity of the original sparse matrix through
matrix \ll in" (not a big problem if the matrix is dense); in section 2.5 I discuss how
this ll in is minimised. The solution can then be eciently obtained by performing
the following set of Gaussian eliminations:
L ~
u 0 = f (2.3)
(forward reduction) and
U ~
u = u 0 (2.4)
(back substitution). Due to the fact that L ~
and U ~
are triangular this procedure is
simple and there is no additional ll in suered by these eliminations. The number
of operations is in direct proportion to the number of non-zero elements in L and
~
U . If an additional result is required for a new right hand side vector, f 0 , then the
~
same cheap forward and back substitution procedure can be repeated with f 0
as the
right hand side in the equation (2.3), using the original LU factors.
Matrices generated from nite dierence (or nite element) equations are
usually well structured if simple grid ordering is used. I will concentrate initially on
the simple row ordering of the nodes shown on Figure (1.1). I will refer to this later
38

as sequential grid ordering. Sequential ordering just involves starting for example,
in the top left corner of the grid and numbering the grid points in the rst row
sequentially up to n x (where n x is number of grid points in x direction). We then
move to the next row, repeat the procedure, and continue in this manner until we
run out of grid points. Imagine our problem is dened on a grid of n x by n z nodes:
If each node is coupled only to it's immediate neighbours (as in nite dierence
equations arising from second order nite dierence operators), the initial matrix
S ~
will only have non-zero elements on the main diagonal, on the two neighbouring
sub-diagonals, and on two sub-diagonal bands at a distance of n z diagonals away
from the main diagonal.
In general any nite dierence operator will produce a
symmetric sparsity pattern in the initial matrix
S ~
. This does not imply that the
matrix itself will be symmetric. This depends on the boundary conditions and on
the type of nite dierence operators used.
Now let us examine the way in which LU decomposition can be performed.
The algorithm is relatively simple (more details can be found, for example, in Peres
et al. (1992)): Let i;j be elements of the starting matrix S ~
, i;j be elements of
matrix L ~
and i;j be elements of matrix U ~
. The algorithm proceeds as follows.
Set i;i =1
For each j =1;2;:::;N carry out the following two procedures:
First, for i =1;:::;j
i;j = i;j ,
Second, for i = j +1;:::;N
Xi,1
k=1
i;k k;j : (2.5)
0
1
i;j = 1
j,1
X
@ i;j , i;k k;j
A : (2.6)
j;j
k=1
Once the element a i;j is used the value is not required any more, so the same memory
location can be used to store the corresponding i;j or i;j . Values i;j and i;j
39

are always calculated by the time they are needed to calculate next values. The
diagonal unity elements i;i =1need not be stored at all. From this description of
the algorithm one can see that all the elements between the rst physical non-zero in
the lower triangular part of S and the main diagonal on the same row will become
~
non-zero elements in L , while all the elements from the rst physical non-zero in the
~
upper triangular part of S and the main diagonal on the same column will become
~
non-zero elements in
U ~
matrix. Elements outside this band remain logically zero
and need not be stored.
Sequential ordering of the grid is the most natural way of grid ordering and
it is the easiest to implement. The simple matrix structure ts well into ordinary
array variables available in almost any programming language, and no overhead is
needed to describe the matrix structure. However, as I will show sequential grid
ordering requires too much memory in comparison with alternative grid ordering
schemes.
It is relatively easy to predict the number of non-zero elements in the matrices
L ~
and U ~
in this case. They have approximately rectangular regions which are lled
in with non-zero elements. The number of matrix rows is n x n z and the bandwidth is
n z . Thus the number of elements in L ~
and U ~
matrix is approximately 2n x n z (n z +1)
in the case of sequential row ordering. In the case of n x = n z = n the memory
required is the order of n 3 , or O(n 3 ), where n is number of grid points along one
edge.
One can thus see that care has to be taken whether the row or column
ordering is used, due to the fact that one of the grid dimensions inuences the
memory required by O(n 2 ) while the other is only O(n). The memory capacity of
commonly available systems is of the order of 1GB 10 9 B (where B means bytes).
If one can store a complex number in 8B then the maximum (square) problem size
will be of order 400 by 400 grid points. If one assumes 10 grid points per wavelength
this will imply that 40 wavelengths in both directions will be the maximum model
size.
40

2.4 Matrix \ll in" and ordering schemes
Here I will show that the same system of linear equations can produce high ll
in or no ll in at all depending on the grid ordering used. An example (see George
and Liu, 1981 ) will be used to show this extreme case. Consider the following two
matrices, both representing the same equation system:
Case 1
2
S =
~ 6
4
4 1 2 .5 2
1 .5 0 0 0
2 0 3 0 0
.5 0 0 .625 0
2 0 0 0 16
3
7
5
Case 2
2
S =
~ 6
4
16 0 0 0 2
0 .625 0 0 .5
0 0 3 0 2
0 0 0 .5 1
2 .5 2 1 4
3
7
5
After performing LU decomposition on these matrices the following matrices are
obtained:
Case 1
2
S =
~ 6
4
2 .5 1 .25 1
.5 .5 -1 -.25 -1
1 -1 1 -.5 -2
.25 -.25 -.5 .5 -3
1 -1 -2 -3 1
3
7
5
41

Case 2
2
S =
~ 6
4
4 0 0 0 .5
0 .791 0 0 .632
0 0 1.73 0 1.15
0 0 0 .707 1.41
.5 .632 1.15 1.41 .129
3
7
5
Case 2 needs 13 memory locations to store the non-zero results of LU decomposition,
while case 1 needs 25 memory locations (almost twice as much). The
linear systems are exactly the same, except that the variables have been reordered.
The clear conclusion from this example is that care in the ordering of equations can
keep the computational costs and memory requirements low. Now I will move on
to show the way in which I will reorder the nite dierence grid nodes so that the
resulting matrix suers the minimal ll in.
2.5 Nested dissection ordering
In this section I will discuss the optimal way of transforming the system
into a system
S ~
u = f (2.7)
( P ~
S ~
P ~
t
)( P ~
u)= P ~
f; (2.8)
where the matrix P ~
is a permutation operator which will transform the matrix S ~
in such a manner as to ensure that the L ~
and U ~
matrices have the lowest number
of non-zero elements. The grid reordering I will use is known as \nested dissection"
and is explained in detail by George and Liu (1981). The equivalent matrices (before
and after LU decomposition) for a sequentially ordered grid, and for a grid ordered
using nested dissection are shown on Figure 2.1. The same approach for grid ordering
is used by Marfurt et al. (1987) to decrease memory requirements for frequencydomain
seismic forward modelling.
42

(a)
(b)
(c)
(d)
Figure 2.1: Nested dissection versus sequentially ordered matrix a),b) before LU
decomposition, and c),d) equivalent L matrix after LU decomposition (George and
Liu,1981). Only non-zero elements are shown in each case. a) Matrix S for a sequentially
ordered grid. b) Matrix S for a grid ordered using nested dissection. c) L part
of the LU decomposed matrix S for case a) (memory required is O(n 3 )). d) L part
of the LU decomposed matrix S for case b) (memory required is O(n 2 log(n))). The
memory required to store matrix for a realistic value of n on gure d) is signicantly
lower than the one required for the matrix on gure c).
43

Figure 2.2: Two-way dissected nite dierence grid. The two way dissector, 5 (in
black) is the last part of the grid to be ordered.
Let us assume initially that n x = n z = n (i.e., that the grid is square).
The grid is then dissected into four quarters so that there are approximately n 2 =4
elements in each part of the dissected grid. Each of the four sections are are coupled
only through the dissectors and within themself (see Figure 2.2). The minimal twoway
dissector has to have at least approximately 2n elements. For the moment I
will assume that it is possible to nd a dissector of this size (this is actually the case
if second order nite dierences are used). The nested dissection recipe for ordering
the elements is to rst number the elements in each block of the nite dierence
grid, and then to number the elements in the dissector. The matrices S ~
, L ~
and U ~
are shown schematically on Figure 2.3. This procedure is called two-way dissection.
If one continues with the procedure recursively on all parts of the dissected matrix
the result is called \nested dissection".
Now let us consider the memory requirements necessary to store the non-zero
elements of the matrices when performing LU decomposition on the n by n grid
ordered by nested dissection. From Figure 2.3 one can see that the memory can be
divided into ve parts. The nal part, L 5;5 is the memory necessary to perform LU
decomposition on the dissector itself, while the remaining four parts L 5;i and L i;i are
44

U 11
0
U 15
L 11
U 55
U 22
U 25
L 22
L 33
L 44
U 33
0
U 35
U 44
L 15 L 25 L 35 L 45
L 55
U 45
Figure 2.3: Two way dissected matrix S ~
= L ~
U ~
. During LU decomposition the
values for L i;j and U i;j are lled in at the corresponding locations used by S i;j . L i;j
and U i;j denotes possible non-zero elements in matrices L ~
and U ~
respectively after
LU decomposition while 0 denotes zero elements.
the amounts necessary to perform LU decomposition on the n 2
by n 2 grids (L i;i), plus
L 5;i which comes from the coupling between the elements within each subgrid and
the elements within the dissector. In the rst dissection one can write the memory
requirements as:
S(n; 0) = 4S(n=2; 2) + D(n; 0) (2.9)
where S(i; j) represents memory requirement for the subgrid of size i bordered by
n
n
n
n
n
n
S(n,2) S(n,3) S(n,4)
Figure 2.4: All possible subgrid (S(n; 2);S(n; 3) and S(n; 4)) situations arising during
nested dissection. The thick black borders represent neighbouring dissectors
from previous dissections in the recursion.
45

(n/2*n/2)/2
L 55
(n/2*n/2)/2
n*n/2
n*n/2
(n*n)/2
Figure 2.5: Enlarged L 5;5 part of the two way dissected matrix. Non zero elements
are in grey. White space represents logical zero elements.
dissectors at j sides (Figure 2.4) (L i;i +L 5;i in Figure 2.3), while D(i; j) is the memory
required to perform LU decomposition on the dissector itself, which is coupled to
j parts of the other dissectors. By continuing the dissection one will nd that only
two more situations can occur: S(n; 3) and S(n; 4) as shown on Figure 2.4. So we
obtain the following equations, together with equation 2.9:
S(n; 2) = S(n=2; 2)+2S(n=2; 3) + S(n=2; 4) + D(n; 2) (2.10)
S(n; 3) = 2S(n=2; 3) + S(n=2; 4) + D(n; 3) (2.11)
S(n; 4) = 4S(n=2; 4) + D(n; 4) (2.12)
From here on I will concentrate on the memory necessary to store only the L ~
part of the matrix; the full amount is just twice the values I will derive. I will start
with D(n; 0) = L 5;5 from Figure 2.3. If the dissectors are ordered sequentially the
enlarged part L 5;5 of the matrix L ~
from Figure 2.3 will look like the one shown on
Figure 2.5. Here I consider the worst possible case in which all the last n elements
inatwoway dissector are coupled to each other, and that both n=2 sized dissectors
are related to all n elements in the n sized dissector. With these considerations one
46

can write directly from the Figure 2.5:
D(n; 0) n 2 =2+2(n=2) 2 =2+2 n n 1
=n 2
2 2 +1 4 +1 = 7 4 n2 : (2.13)
In a similar manner the following equations can be derived:
D(n; 2) 19 4 n2 (2.14)
D(n; 3) 25 4 n2 (2.15)
D(n; 4) 31 4 n2 (2.16)
and equation 2.12 can be expanded in the following form using 2.16:
S(n; 4) 31 31
4 n2 +4
4 (n=2)2 +4S(n=4; 4) =
31
4 n2 (1+1)+16S(n=4; 4) =
:::= 31
X
4 n2 log 2 (n)
i=1
1 = 31
4 n2 log 2 (n): (2.17)
Substituting this into the equations (2.9) to (2.11) and using (2.13) to (2.15) the
following expressions can be obtained:
S(n; 3) 31 4 n2 log 2
(n)+O(n 2 ) (2.18)
S(n; 2) 31 4 n2 log 2
(n)+O(n 2 ) (2.19)
S(n; 0) 31 4 n2 log 2
(n)+O(n 2 ) (2.20)
which gives us a total memory requirementof 31
2 n2 log 2
(n) for the matrices L ~
and U ~
together. George and Liu (1981) have shown that the theoretical minimal memory
requirements to perform the LU decomposition on an n by n grid is of the same
order of magnitude, so that nested dissection can therefore be assumed to be an
\optimal" grid ordering to within, at least, an order of magnitude. George and Liu
(1981) also showed that nested dissection gives an optimal number of operations
((n; 0))
(n; 0) 829
84 n3 ; (2.21)
47

(to within anorder of magnitude) necessary to perform the LU decomposition.
The amount of CPU time required to solve the system for each right hand
side is again of order of n 2 log 2
n (i.e. of the order of the number of elements in the
LU decomposed matrix). This amount of CPU time can easily be less then for the
iterative matrix solver where one needs at least n 2 operations per iteration and for a
large n the number of iterations will almost certainly be greater than log 2
(n). From
this observation we see that for the numerical problems with the multiple right hand
sides the nely tuned direct matrix solver may perform better than the iterative one.
There is an additional computational cost for LU decomposition that has not
yet been mentioned. A certain amount of CPU time (a signicant one) is needed
to generate the nested dissection ordering. The algorithm complexity necessary to
dissect the matrix is of the order of O(n 4 ), however it is well worth the eort, as
I will show in the following chapters, to generate nested dissection ordering. The
same ordering can of course be used for all runs with any model of the same size. A
second hidden cost is that the sparsity pattern of the LU decomposed matrix is far
from simple, and a suitable pointing algorithm is required to track this sparsity. The
memory requirements for this algorithm are the same as for the non-zero elements of
the matrix, so that there is a linear increase in memory required. A certain amount
of computing time is also lost during factorization on searching through the matrix
structure to nd given matrix locations. In the case of sequential row and column
access (as in LU decomposition) this is negligible.
It is important to point out that I have not been limited to a particular partial
dierential equation while working with nested dissection: The implementation
depends only on the structure of the matrix. From now on all the developments
on nite dierence methods for frequency domain seismic forward modelling will assume
that the LU decomposition will be performed on the grid ordered by the nested
dissection and that the properties of the nite-dierence scheme will be adjusted to
take the full advantage of the nested dissection ordering.
48

Figure 2.6: Fourth order nite dierence computational star. The symbol identies
those grid points coupled to the central grid point.
2.6 Operators and memory requirements
The memory requirement predictions in the previous section have assumed
that one can nd a two way dissector of size 2n on an n by n grid. This assumption
is valid provided second order nite dierence operators are used. If a fourth order
nite dierence operator is used, this involves coupling of grid points at distances
of 2 x and 2 z as shown on Figure 2.6. The minimal two way dissector size then
increases to 4n. At rst sight this does not look like a big increase. However the
memory requirements for nested dissection are highly dependent on the size of this
dissector.
If the dissector size is increased to 4n from 2n, how big will the impact be on
the required memory? Let us return to Figure 2.5. In this case n becomes 2n so:
49

D 4 (n; 0) (2n) 2 =2 + 2(2n=2) 2 =2+2 2n 2n 2
= n 2 (2+1+4)
= 7n 2 (2.22)
In a similar manner the following equations can be derived:
D 4 (n; 2)=23n 2 (2.23)
D 4 (n; 3)=31n 2 (2.24)
D 4 (n; 4) = 39n 2 : (2.25)
Substituting these into the equations (2.9) to (2.12) for S(n; i) the required memory
to perform LU decomposition in this case will be:
39
S 4 (n; 4) = 4S 4 (n=2; 4) + D 4 (n; 4) = 39n +4 2 4 n2 +4S(n=4; 4)
(2.26)
39n 2 log 2 n (2.27)
and similarly
S 4 (n; 3) = 39n 2 log 2 n + O(n 2 ) (2.28)
S 4 (n; 2) = 39n 2 log 2 n + O(n 2 ) (2.29)
S 4 (n; 0) = 39n 2 log 2 n + O(n 2 ) (2.30)
where S 4 (n; i) is the equivalent ofS(n; i) if the fourth order nite dierence scheme
is used.
Equations (2.20) and (2.30) show the memory required to perform LU decomposition
on an n by n grid if second and fourth order nite dierence operators
are used, respectively. However, the use of higher order nite dierence operators
reduces the required grid size (for a given accuracy).
I will now show what the
decrease in the number of grid points in one direction would have to be in order to
50

educe the memory required to perform LU decomposition. To show this one has
to solve the following equation
S(n 2 ; 0) = S 4 (n 4 ; 0); (2.31)
for n 4 = kn 2 where k is the factor by which we have to reduce the number of grid
points in one direction in order to at least equal the second order scheme with respect
to the required memory. Here S 4 (n 4 ; 0) represents the memory required to perform
LU decomposition on the n 2 by 4 n2 (or 4 k2 n 2 by 2 k2 n 2 2
) matrix generated by
using 4th order nite dierence operator and S(n 2 ; 0) (as dened in equation 2.20)
represents the memory required to perform the LU decomposition of the n 2 2
by n 2 2
matrix generated by using second order nite dierence operators.
If we equate
equations 2.20 and 2.30:
39n 2 4
log 2
(n 4 )+O(n 2 4)= 31 4 n2 2
log 2
(n 2 )+O(n 2 2);
then
39k 2 n 2 2 log 2(n 2 )+O(n 2 2 )=31 4 n2 2 log 2(n 2 )+O(n 2 2 ):
This equality can be approximately expressed by discarding O(n 2 2) terms as:
39k 2 n 2 log 2 2(n 2 ) = 31
4 n2 log 2 2(n 2 )
31
k 2 n 2 2
log 2
(n 2 ) =
39 4 n2 2
log 2
(n 2 )
k 2 =
31
39 4
k = :445 (2.32)
This result shows that one would need to reduce the number of grid points per
wavelength by more than 50% in order to justify the use of higher order nite
dierence operators in a nested dissection ordered grid.
In the case of sequential ordering a much smaller improvement will justify
the higher degree operators due to the n 3 dependency of the memory requirements:
51

The memory required to perform LU decomposition on sequential n 4 by n 4 grid if
the 4th order nite dierence operators are used is:
S 4 (n) =4n 3 4
(2.33)
If this is compared with the second order scheme there is only a linear increase so:
4n 3 4 =2(n 2) 3 (2.34)
n 4
n 2
=( 1 2 )1 3 =:7937: (2.35)
This shows that, for the sequential ordering scheme, a reduction of only 21% in
number of grid points per wavelength will justify the use of a higher order nite
dierence scheme. Nevertheless, the overall cost will be much higher than for the
equivalent nested dissection scheme.
As a comparison, for time domain schemes the best results are expected by
using a staggered grid, with a fourth order nite dierence operator in space and
a second order operator in time, as pointed out by Sei (1994a). This choice comes
from the CPU time requirements versus accuracy for the time domain approach.
The reason for this comparison between the second and the fourth order
nite dierence schemes will become clear in the following chapters, when I will
show that it is possible to develop second order nite dierence operators which will
require only 4 grid points per wavelength to achieve high accuracy. Due to the fact
that the theoretical limit for any nite dierence operator is two grid points per
wavelength, I consider that no gains will be achieved if higher order nite dierence
approximations are used.
In any case, the use of local nite dierence operators
improves performance in heterogenous media ( Ozdenvar and McMechan, 1996).
2.7 Comparison of band and nested dissection ordering
In this section the advantage of using nested dissection to perform frequency
domain forward modelling with realistic models will be demonstrated. Here I will
52

consider rst a set of parameters for a realistic crosshole data set, using as an
example a cross-borehole experiment described by Pratt and Sams (1996). In that
experiment the following parameters apply:
Source and receiver array length: 100 m
Borehole separation: 100 m
Minimum P-wave velocity: 2.5 km/s
Data frequency: 1 kHz
If one assumes that the required accuracy can be achieved by using 10 grid points per
wavelength (which is consistent with the ordinary second order frequency domain
seismic modelling scheme accuracy), then the required nite dierence grid would
have to be 400 grid points by 400 grid points, with x = z = :25m. If we need
8 bytes to store a complex number, then a band ordered scheme will require in
order of 1000 MB to store the LU decomposed matrix, whereas a nested dissection
scheme will require 100 MB. This demonstrates dramatically the need for nested
dissection methods. The situation becomes even more critical with larger and more
general seismic experiments. In the following chapter I will show that far less than
10 grid points per wavelength are actually required. If we use 4 grid points per
wavelength, we will require only 13.5 MB to store the matrix. This represents an
overall reduction of 98:65% from the initial gure of 1000 MB.
The previous considerations were based on square models with n x = n z .
However, useful geological models are not always square. Geophysical experiments
usually have larger distances in one direction. The savings introduced by nested
dissection are the highest in the n x = n z
case and much less for a models where
n x >> n z or n x

are of theorder of hundereds of kilometers (see (Holbrook et al., 1992)). In the case
of wide angle experiments one records not only the reections from the impedance
contrasts beneath the source, but also refracted arrivals which travel deep into the
earth (up to 30 to 50 km) and turn back to the surface. The results for this numerical
experiment are shown on Figure 2.7.
For this aspect ratio, we found out that a
nested dissection ordering allowed a grid with 5000 by 800 grid points to t within
2 GB of memory. A sequentially ordered grid of the same size would require 50
GB. All predictions assume that acoustic frequency domain forward modelling can
be done with four grid points per wavelength, as will be explained in the following
chapter. The frequency domain numerical simulation of such models (with hundreds
of wavelengths propagation distance between the sources and receivers) would not be
possible if simple grid ordering were used. Simulations of such experiments require
huge computational resources (mainly CPU time) even for the time domain based
schemes. This kind of experiment involves a large number of sources, and late arrival
times, which consequently makes a time domain approach too expensive even with
the fastest available computer resources.
For example, if we assume a machine capable of one gigaop (where one
gigaop is equal to one billion oating point operations per second) the following
prediction is obtained: Let a model be dened by six hundred thousand grid points,
and let ten oating point operations be required per grid point for one time step.
We further assume that the maximal time step is 0:0001 second, that the maximal
required simulation time is 40 seconds and that the number of sources is 150. The
approximate CPU time under these conditions will be four days. For comparison,
a similar computation in the frequency domain can be carried out within ten hours
on Digital alpha 600/333 workstation (104 megaops) with 256 MB of RAM, if the
data are generated in reduced time (see the example in the following chapter). On a
one gigaop machine this calculation would take only one hour. Most importantly,
if additional sources responses were required, these could be computed in a trivial
54

1x10 11
Memory (BYTES)
1x10 10
1x10 9
1x10 8
Sequential Ordering
Nested dissection
2GB
Actual Mesh Size 5000X800
50GB
1x10 7
500 1000 2000 5000 10000
n x
Figure 2.7: Memory requirements comparison for n x = 6:25 n z in case of band
and nested dissection ordering. The required mesh size represents the model size
necessary to perform acoustic modelling of a wide angle experiment with 10 Hz data
and a model 350 km by 48 km. The minimum P wave velocity is 2.8 km/s.
amount of extra time.
The diagram on Figure 2.7 shows that in order to work with 10 Hz data with
a 350 km wide model and the depths in order of 48km (using 4 grid points per
wavelength) and minimum P wave velocity of 2:8 km/s one would need a machine
with approximately 2 GB of memory. Such machines are available these days at
the top end of the workstation market. One can see that without frequency domain
methods, nested dissection, current workstation resources cannot tackle experiments
of this size in production time scales.
Figure 2.8 illustrates the CPU times on Digital Alpha 3000/300 workstation
for the two ordering schemes. For small models, nested dissection performs worse
(due to a computation overhead imposed by an irregular matrix structure), but when
n x is greater than 200, CPU times are lower for the nested dissection case. However,
it is important to point out that the main consideration in frequency domain forward
modelling is the memory requirements; the CPU time is usually low. The elapsed
time is dominated by a disk input and output due to the large amount of seismic
data being computed. In our numerical tests, the nested dissection matrix solver
did not need more than 15 minutes per frequency, even for models with the grid
55

10000
1000
Time (s)
Sequential ordering
Nested dissection ordering
100
10
100 200 500
Grid points (n x or n z )
Figure 2.8: CPU time versus number of grid points for the case in which n x = n z ,
computed on Digital Alpha 3000/300 workstation.
approaching 1,000,000 grid points. To perform nested dissection ordering on a grid
with 1; 000; 000 grid points requires two days.
2.8 Conclusions
For frequency domain seismic modelling using nite dierences, direct matrix
solvers are the method of choice, due in part to poor conditioning of the matrices.
Direct solvers have further advantages over iterative solvers if the linear systems are
to be solved for multiple right hand sides. The LU decomposition matrix solver is
the most apropriate. Ihaveshown that the amount of ll in suered by the matrix
during LU decomposition depends strongly on the grid ordering. Nested dissection
is an optimal grid ordering, but requires that the nite dierence operator be as
local as possible in order to keep the ll in as small as posible.
In order to justify using higher order operators, one would have to achieve
an improvement in accuracy sucient to allow a greater than 50% reduction in the
number of grid points per wavelength. As I will show in the following chapter it
is more eective to keep the nite dierence operator small and accurate by using
56

otated nite dierence operators.
57

Chapter 3
Visco-acoustic frequency domain acoustic forward
modelling using rotated nite dierence operators
3.1 Introduction
Forward modelling of the scalar wave equation in the frequency domain was
introduced by Lysmer and Drake (1972), extended by Marfurt (1984b), and applied
to seismic imaging by Pratt (1989b; 1990). Modelling in the frequency domain is
computationally more demanding than time domain based schemes if a time domain
result is required for only a limited number of sources. The advantage of frequency
domain seismic modelling is realized in multi-source experiments, and in frequencydomain
waveeld inversion in particular, in which only limited number of frequencies
from a large number of sources are needed.
In Chapter 2 I concentrated on the minimization of the computational costs
for a xed matrix size (and a xed dierence operator) on the ll in suered by the
matrix. In this chapter I will concentrate on the minimizing the size of the initial
matrix (a function of the grid size), by improving the accuracy of the nite dierence
operators.
This chapter begins with an overview of the nite dierence scheme developed
by Jo et al. (1996) (itself an extension of a result by Cole (1994)) based on rotated
58

nite dierence operators. As pointed out in the previous chapter, the size of the
nite dierence operator is one of the crucial factors inuencing the total memory
required to perform frequency-domain forward modelling. If higher order (larger)
nite dierence operators are used, the result is more accurate and thus a smaller
grid is needed. However, a direct matrix solver then becames more expensive. The
main problem is to nd a balance between the following two objectives:
i) to use as small an operator as possible, and
ii) to obtain as accurate result as possible.
Both these objectives must be balanced to minimize the overall cost. Although high
order nite dierence operators can be easily implemented in the frequency domain,
I will show that this leads to an unacceptable increase in computational costs ( in
particular, in memory requirements).
Jo et al. (1996) showed that by using more than one second order nite difference
operator for the same partial derivatives it is possible to develop a scheme
which is comparable in accuracy to higher order schemes without signicantly increasing
computational costs. In this chapter I will review the scheme proposed by
Jo et al. (1996), and extend it to the heterogenous case. I will further discuss some
of the parameters introduced by Jo et al. and evaluate their eect on the overall
scheme.
As I am not working on the boundary conditions a formulation is from
Pratt (1989b) in all examples.
59

Figure 3.1: Finite dierence operators for acoustic frequency domain seismic modelling
in two coordinate systems. The symbol indicates that the model parameter
is used at the corresponding grid point. a) Finite dierence operator in the original
coordinate system. b) Finite dierence operator in the rotated coordinate system.
c) The combination of both schemes.
3.2 Forward modelling using rotated nite dierence operators
3.2.1 Second order frequency-domain seismic modelling
The visco-acoustic, constant density frequency-domain wave equation in homogeneous
isotropic (source-free) media can be written in the following form:
r 2 P + !2
P =0; (3.1)
2
v
where P is the pressure wave eld, ! is the angular frequency and v is the velocity.
Because we pose the problem in the frequency domain, we may allow for viscous
eects by using complex valued velocities if we wish. By using second order nitedierence
approximations one can obtain the following nite dierence equation:
P m+1;n , 2P m;n + P m,1;n
2 x
+ P m;n+1 , 2P m;n + P m;n,1
2 z
+ !2
v 2 P m;n =0; (3.2)
where P m;n represents the pressure of waveeld at the discrete location (m; n) (see
Figure 3.1(a)) within the grid while x = z = is the grid spacing (grid point
interval in x and z direction). Using this simple equation, one can solve the wave
60

V /V ph
1.03
V/V gr
1.03
1.02
1.02
1.01
1.01
1.00
1.00
0.99
0.99
0.98
0.97
1/G
0.05 0.1 0.15 0.2 0.25
0.98
0.97
1/G
0.05 0.1 0.15 0.2 0.25
(a)
(b)
Figure 3.2: Numerical dispersion curves for frequency domain acoustic forward modelling
using ordinary second order nite dierence operators. a) Phase velocity dispersion.
b) Group velocity dispersion.
propagation problem numerically. However, as in Chapter 2, we will see that the
simplest solution is not always the best.
The usual way of describing the numerical accuracy of a particular scheme is
to plot the normalized velocity as a function of number of grid points per wavelength.
The normalized velocity isusually expressed by the ratio of the numerical velocity,
bv, over the analytical velocity, v. The numerical result can be derived by applying a
plane wave solution into the nite dierence equation 3.2 (see for example (Marfurt,
1984a)). Figure 3.2 shows that for this simple second order scheme one needs over
ten grid points per wavelength in order to keep dispersion errors small (under 3%).
3.2.2 The rotated operator concept
For a particular physical problem we will normally choose an orthogonal coordinate
system in which to pose the equations and solve the problem. If the physical
problem is described by a partial dierential equation in a Cartesian coordinate system,
then the same solution should be obtained in all Cartesian coordinate systems.
In the analytical case the solution will not depend on the coordinate system used.
61

However, analytical solutions do not exist for most realistic cases, so numerical solutions
are required. In this case the choice of the coordinate system will aect the
solution. A numerical solution is only an approximation, and the accuracy of the
approximation usually has an angular dependence, so that the result depends on
the orientation of the coordinate system.
In the case of plane wave propagation through homogeneous media, one would
usually choose a coordinate system congruent with the direction of the wave propagation.
For a single plane wave one can always develop a numerical scheme (or
adjust the coordinate system) to produce an accurate result using low order nite
dierence operators and a low number of grid points per wavelength. However, if
waves can propagate in all directions (in complex models) how can one minimise
the errors that arise due to the choice of the coordinate system?
The solution utilized by Jo et al. (1996) is to use more than one Cartesian
coordinate system, without including any points except nearest neighbours. In the
2D case there are two possible coordinate systems (see Figure 3.1). We may pose
the numerical problem in each of these coordinate systems and attempt form a
combined solution. On Figure 3.1(a) the nite dierence operator used in original
coordinate system for the 2D acoustic wave equation is shown. In this operator,
values from only ve grid points are used. Figure 3.1(b) shows the same operator
in the rotated coordinate system.
This operator uses values from four new grid
points.
A combination of the two operators (Figure 3.1(c)) uses values from all
nine neighbouring grid points. In terms of memory requirements for direct matrix
solvers (including a nested dissection one), there is virtually no extra cost associated
with using the additional four grid points in the operator. The same is true for the
CPU time involved. The main advantage of this approach is that it is possible to
solve only one combined linear system of the same size and average the solutions
implicitly during the calculation.
Note that the procedure is specied for a grid with square elements ( x =
62

z ).
A similar procedure can be applied in the case of a rectangular grid, but
the rotated coordinate system is then no longer Cartesian, and the appropriate
wave equation formulation must be used. Furthermore, a scheme developed for a
rectangular grid would work correctly only for the x = z ratio for which the scheme
is developed.
3.2.3 Finite dierence scheme in homogeneous media
Here I will apply the rotated nite dierence operators concept to equation
3.1. If a second order nite dierence formula is developed using a rotated grid (see
Figure 3.1(b)) one can write:
P m+1;n+1 , 2P m;n + P m,1;n,1
2 2 x
+ P m,1;n+1 , 2P m;n + P m+1;n,1
2 2 z
+ !2
v 2 P m;n =0: (3.3)
The factor 2 2 (as opposed to 2 in equation 3.2) comes from the increase in the
grid point distance, in this case, from to p 2. A linear combination of the two
schemes can be expressed by simple addition and multiplication of (3.2) by a and
(3.3) by (1 , a) as:
aA +(1,a)B+ !2
v 2P m;n =0 (3.4)
where A is the part of (3.2) consisting of nite dierence approximations for the
Laplacian term in equation (3.1) while B is the equivalent part of equation (3.3).
If equation (3.4) is expressed in a nite dierence form it is easy to see that in fact
only one system of linear equations need be solved. The the size of the resulting
matrix is almost the same as that required for the single coordinate system alone
(see Chapter 2).
For example, if sequential grid ordering is used, the additional
points used as shown on Figure 3.1(c) will generate four additional diagonals in the
matrix next to exsisting diagonals, and the LU decomposed matrix will have only
two more diagonals. This will add 2 n x n z elements (negligible in comparison with
the total number of elements, 2 n x n z min(n x ;n z ) for realistic n x and n z ).
63

3.2.4 Lumped and consistent matrix terms
The second improvement introduced by Joetal.(1996) focused on the algebraic
part of the acoustic wave equation, and is based on an approach used in the
nite element method (Zienkijevic, 1977): The algebraic part of equation (3.1) is
approximated by averaging the solution from neighbouring points. In nite-element
terminology this is called a lumped matrix approach. For homogeneous media this
approach results in the following replacement in the equations:
! 2
v 2 P m;n ) !2
v 2 bP m;n + !2
v 2 c(P m+1;n + P m,1;n + P m;n+1 + P m;n,1 )+
! 2
v 2d(P m+1;n+1 + P m+1;n,1 + P m,1;n+1 + P m,1;n,1 ); (3.5)
where
b +4c+4d=1: (3.6)
This approach then combined with the approach indicated in the equation (3.4).
3.2.5 Determination of optimal coecients
Although any choice of values for the coecients a, b, c and d (satisfying
equation (3.6)) will produce a possible numerical solution for the acoustic wave
propagation problem in homogenous media, to obtain the most accurate solution
for the problem optimal values for coecients a, b, c and d must be found. Note
that due to equation (3.6) only two of the cocients b, c and d are independent.
This can be posed as a minimization problem in which the errors in the
solution are minimized as a function of the coecients. The minimization problem
can be set up in more than one way, depending on what is actually minimized. The
function to be minimized chosen by Jo et al. (1996) is :
F (a; b; c) =
Z :5
Z =4
0 0
2
bv ph (a; b; c; g; )
, 1!
d dg (3.7)
v
64

V/V ph
1.03
V/V gr
1.03
1.02
1.02
1.01
1.01
1.00
1.00
0.99
0.99
0.98
0.97
0.0
1/G
0.05 0.1 0.15 0.2 0.25
0.98
0.97
1/G
0.05 0.1 0.15 0.2 0.25
(a)
(b)
Figure 3.3: Numerical dispersion curves for frequency domain acoustic forward modelling
using rotated nite dierence operators. a) Phase velocity dispersion. b)
Group velocity dispersion.
where g = 1=G, G is the number of grid points per wavelength, is the wave
propagation angle, bv ph (a; b; c; g) is the numerical phase velocity while v is the exact
velocity. The coecient d is not used since d = (1 , b , 4c)=4 (see equation
3.6). Expressions for the numerical phase velocity bv ph (a; b; c; ; g) and group velocity
bv gr (a; b; c; ; g) can be derived by applying a plane wave solution into the nal
nite dierence equation (for details see Jo et al. (1996)). Jo et al. suggested the
following optimal values:
a = :5461
b = :6248
c = :09381
d = :1297 10 ,5 (3.8)
The value for the coecient d, in equation (3.5), suggested by Jo et al. (1996) is
negligible. Thus it would appear that the following equation can be used for the
parameter estimation:
65

+4c=1 (3.9)
which makes the minimization problem one dimension smaller. The slight non-zero
value obtained by Jo et al. (1996) is quite likely due to the minimization procedure;
the coecient d may be set to zero without any noticable deteriation of the result. I
found that if d is increased by any signicant amount the normalized velocity starts
to oscillate strongly as a function of G. For some extreme values of the coecient d
the resulting velocity becomes complex valued. To demonstrate that d can eectively
be set equal to zero, I have reproduced group and phase velocity dispersion curves
for the case:
a = :5461
b = :6248
c = 1 (1 , b)
4
d = 0: (3.10)
Figure 3.4 shows the functions:
D gr;ph (%) = 1 ,
!
bv(a; b; c; d; ; g)
100
bv(a; b; (1 , b)=4; 0;;g)
where bv(a; b; c; d; ; g) is the numerical group or phase velocity asa function of Jo's
coecients a, b, c and d and propagation angle . The dierent curves depict various
wave propagation angles in isotropic homogenous media. The maximal dierence in
numerical velocity introduced by setting coecient d equal to 0 is less than :004%,
which is negligible in comparison with the errors we are dealing with. A similar
low level of discrepancy is found if the coecient c is kept with the value suggested
by Jo et al., and the coecient b set equal to b = 1 , 4c.
For this reason the
additional coecent, d will not be used here, nor will I use the equivalent parameter
in the elastic case (see Chapter 5 for the visco-elastic forward modelling scheme).
66

D gr(%)
D
ph(%)
0.004
0.003
0.002
0.001
0.000
-0.001
-0.002
-0.003
-0.004
0.05 0.1 0.15 0.2 0.25
(a)
1/G
0.004
0.003
0.002
0.001
0.000
-0.001
-0.002
-0.003
-0.004
1/G
0.05 0.1 0.15 0.2 0.25
(b)
Figure 3.4: Dierence between the numerical velocity produced with and without
the additonal coecient, d. a) Dierence in group velocity. b) Dierence in phase
velocity. See text for detail explanation.
The minimization problem thus reduces to a problem with two unknowns a and b.
This reduces the search space, and it is possible to plot the minimzation result as
a surface for various values of coecients a and b (while coecient c is a function
of a and b see equation 3.10). This helps avoid local minima in the optimisation
problem.
3.2.6 Discussion of savings with rotated operators
The dispersion curves for the set of parameters dened in the equation 3.8
are shown on Figure 3.3.
The results show that numerical errors in the phase
velocity of less than 1% can be acheived with 4 grid points per wavelength, with
errors of up to 3% for the group velocity for the same value of G. In comparison
with the ordinary second order nite dierence schemes (see Figure 3.2) for the
same problem, this shows a saving of more than 60% in the number of grid points
required per wavelength for the same accuracy. The computational costs are almost
the same as for the ordinary scheme (for the same number of grid points). Recalling
67

that the required memory is afunction of n 2 log 2
(n) (from equation (2.20), where n
is the number of nodes on one side of a square grid), we can produce the following
exact results for the savings in memory obtained if the same accuracy is required in
both cases, by using :4n instead of n for the rotated operators case:
M new
M old
=
4
n 2
4n log2
10 10
n 2 log 2 n
< 16 2+log 2 n , 3
100 log 2 n
!
= :16 1 , 1
log 2 n
< :16; (3.11)
where M is required memory. This shows that the saving in memory obtained for
the square model is at least 84%. The savings are slightly more for smaller grids
than for larger grids. With regard to CPU time, one can use the following equation
(from the equation 2.21):
which shows a CPU time saving of over 90%.
CPU new
CPU old
= :43 n 3
n 3 = :4 3 =0:064; (3.12)
3.2.7 Extension to the heterogenous case
written:
The 2-D visco-acoustic wave equation in heterogenous isotropic media can be
@
@x
!
1 @P
+ @ (x; z) @x @z
!
1 @P 1
+
(x; z) @z K(x; z) !2 P =0; (3.13)
where (x; z) is the 2D density function and K(x; z) is the bulk modules (in general
complex valued). In this case one can still apply the rotated nite dierence formulation,
but the appropriate partial derivatives for the model parameters (K and )
will have to be used. There is only one missing nite dierence operator required:
68

@
@u
!
1 @P
(u; v) @u
(3.14)
This problem was solved by Kelly (1975) in the case of the original coordinate system
using the operator:
@
@x
!
1 @P
(x; z) @x
1
m+
1
2
;n
where m
1
2 ;n = 1 2 ( m;n + m1;n )
[P m+1;n , P m;n ] , 1
m, 1
2
;n
2 x
[P m;n , P m,1;n ]
; (3.15)
The same approach can be reformulated in the rotated coordinate system by
substituting x = x 0 , m = m 0 and n = n 0 :
@ 1
@x 0 (x 0 ;z 0 )
where m
1
2 ;n 1 2
@=@z 0
!
@P
@x 0
=
partial derivatives.
1
m
0 +
1
2
;n 0 hP m
0 +1;n
0 ,P m
0 ;n
0
1
m+
1
2
;n+ 1 2
i
,
1
m
0 , 1
2
;n 0
2 x 0
[P m+1;n+1 ,P m;n ], 1
m,
1 ;n, 1 2 2
2 2 x
h
P m
0 ;n
0 ,P m 0 ,1;n 0 i
[P m;n ,P m,1;n,1 ]
; (3.16)
= 1 2 ( m;n + m1;n1 ). Equivalent equations can be derived for the
In the case of lumped and consistent mass matrix terms equation (3.5) can
be used, but the bulk modulus has to be distributed as well. If the coecient d is
set to zero the one obtains the replacement formula:
1
1
1
K !2 P m;n ) ! 2 [b P m;n + c( P m+1;n
K m;n K m+1;n
+ 1
K m,1;n
P m,1;n +
1
K m;n+1
P m;n+1 +
1
K m;n,1
P m;n,1 )]: (3.17)
Substituting equations (3.15), (3.16) and (3.17) into (3.13), together with the equivalent
equations for @=@z and @=@z 0
denes the heterogenous nite dierence formulation.
The use of the heterogenous wave equation gives much more accurate results
(when compared with use of the homogeneous wave equation with explicit boundary
conditions) in the case of realistic geological models ( Ozdenvar and McMechan,
69

1996). However quantitativeevaluation ofthe accuracy is dicult, although possible
(Sei and Symes, 1994b). Tests we have run on a number of models have shown that
the values for the coecients a, b and c obtained for the homogenous case may be
used in the heterogenous formulation sucessfully, even in highly heterogenous media
(see Pratt et al. (1995)).
3.3 Improvements acheived by rotated nite dierence operators
In this section I will show the real improvements produced by the introduction
of the nested dissection grid ordering and rotated nite dierence operators.
Dealing with orders of magnitude and numbers of grid points per wavelength does
not depict the achievements visually. Here I show the frequency domain, seismic
forward modelling of a realistic wide angle experiment. The model used is taken
from McCarthy et al. (1991), simplied as in Pratt et al. (1996).
The metamorphic core complex belt in southeast California and western Arizona
is a NW-SE trending zone of unusually large Tertiary extension and uplift.
Three seismic refraction/wide angle reection proles were acquired and analyzed
by McCarthy et al. (1991) as a part of of the U.S. Geological Survey's Pacic to
Arizona Crustal Experiment. The seismic data were of excellent quality, and a large
number of phases were observed and interpreted. A prominent midcrustal reection
was indentied between 10 and 20km depth. Some non-horizontal features on the
crust-mantle boundary can be observed on the data. The acqusition geometry consists
of hundreds of sources on all proles spaced at 500 m intervals; the data were
recorded at 250 m intervals. The proles are between 250 km and 400 km long. The
data recorded evidence of structures at more that 30 km depths.
The model used in this synthetic study (see Figure 3.5(a)) consists of most
of the features observed on the three data sections presented by McCarthy et al.
70

Figure 3.5: a) Model used for wide angle forward modelling, from McCarthy et
al. (1991). b) c) and d) The shaded regions depict the size of the models that one
could simulate without nested dissection and/or rotated nite dierences if the same
equipment were used.
71

(1991). The low velocity regions (1.5 km/s)from the top 500 mofthemodelare not
incorporated, in order to make the simulation easier. The model topography is at,
although the actual site is in a mountainous region. The dominant frequencies in the
real data are as large as 10 Hz, however I have used a maximum frequency of 10 Hz
(with a dominant frequency of 3.3 Hz), due to the lack of processing power available.
The grid used represents 250 km by 38 km (2000 by 320 grid points) with a grid
spacing of 125 m. Expressed in wavelengths this is 500 80 minimal wavelengths.
This results in a linear system with 640 ; 000 complex (or 1 ; 280 ; 000 real) linear
equations. The whole computation was carried out on a DEC Alpha 600/333 with
512 MB of RAM. This workstation conguration will be standard very shortly and
more powerfull equipment is already available on the market. The model is close
to our current limit for frequency domain forward modelling, but memory prices
continue to be reduced and machines are increasingly congured with more and more
memory. The computational times are acceptable for this model size, approximately
30 minutes per frequency using 240 sources, which would result in a total time of less
than one day to invert the data set of this size in the frequency domain (assuming
four iterations per frequency, for four frequencies). The time required to produce
the time domain response for all 240 sources (128 frequencies for 256 time samples)
was under two days.
The main portion of the time was spent in the disk input
and output. This is largely due to an inecient implementation for time domain
output, since we read and write all the time domain data after each frequency, which
required a signicant amount of the total time. We utilized the ability of frequency
domain forward modelling to produce the data directly in reduced time (see Section
1.4 for an explanation), so that less frequencies were required.
Figure 3.6 shows a resulting synthetic common shot gather in reduced time
for a shot located at the top left corner of the model, and a similar section of real
data from McCarthy et al. (1991). Many phases from the eld data, such as midcrustal
reections, moho reections, and the head wave, can be observed on the
72

Reduced time (T-x/6.0) Reduced time (T-x/6.0)
4
2
0
4
2
0
0
P P mc
50
PP m
100
OFFSET (km)
PP lc
(a)
P g
P mc
P n
150
4
2
0
4
2
0
0
50
100
OFFSET (km)
(b)
150
0
Depth (km)
10
20
30
mc
lc
Moho
50 100 150 200
OFFSET (km)
(c)
Figure 3.6: a) Synthetic data section from the model on gure 3.5. b) Common
shot gather from the eld data. c) One of the models suggested by McCarthy et al.
(1991) showing the ray paths used in their modelling approach.
73

a) Time slice at 5s
b) Time slice at 10s
c) Time slice at 15s
d) Time slice at 20s
e) Time slice at 25s
f) Time slice at 30s
Figure 3.7: Time slices generated by forward modelling true the model on Figure
3.5(a) at 5, 10, 15, 20, 25 and 30 seconds.
74

synthetic section. It is also possible to see weak phases in the synthetic data that
are not visible on the eld data. Those phases are diractions from discontinuities
on the reectors. The time slices at 5, 10, 15, 20, 25 and 30 s (Figure 3.7) produced
by the forward modelling code clearly show the formation of a head wave on the
Moho. It is possible to see the diractions from the model discontinuities on some
of these time slices.
In order to depict the improvements acheived by using nested dissection
method (see Chapter 2) and the rotated nite dierence operators (this chapter),
Figures 3.5(b), (c) and (d) show the model from the Figure 3.5(a) with rectangles
covering the size of the regions that could be modelled using the frequency domain
technique, using only some or none of these improvements.
It is clear that it is
not feasible to predict the response of a realisticly sized wide angle model without
nested dissection and without rotated nite dierences. Without our improvements,
the largest acceptable model will corespond to a maximal source receiver distance
of 50 wavelengths. Introducing either nested dissection or rotated nite dierence
operators increases this to 100 or 150 wavelengths.
The model used here represents
500 wavelengths in oset and 80 wavelengths in depth. The total increase in
the size of the model in Figure 3.5 a) is not just the sum of the improvements on
Figures b) and c). A certain improvement comes from the interaction between the
two techniques. This shows the importance of simultaneously developing both the
nite dierence operator and the matrix solver. If we were to try to simulate the
smallest model (gure 3.5(d)), but with our improvements, the required memory is
reduced to 25 MB, which represents savings of over 95% (from 512 MB). Seen from
this perspective, the improvements have reduced the memory requirements down to
that normally available on a small personal computer.
By generating the time domain data in reduced time directly it is possible
to minimise the number of frequencies required to produce time domain output: I
needed to model only 5 seconds of reduced time output.
In comparison, a time
75

domain approach would have to generate the full time domain simulation for 35 to
40 seconds, using a small time A step (due to the highest velocity of 8.5 km/s in
the model). If one multiplies this eort by the number of sources involved (240) the
simulation is seen to be completely impractical.
This same model was used earlier by Pratt et al. (1996) to show the feasibillity
of the waveeld inversion on wide angle synthetic data. Although the frequencies
used in that simulation (up to 2 Hz) were less than realistic wide angle data frequencies,
machines available on the market today will be able to perform frequency
domain forward modelling at the more realistic frequencies used in this chapter.
3.4 Conclusions
In this chapter I have reviewed the development of the rotated nite dierence
operators that allow one to signicantly reduce the number of grid points per
wavelength for second order schemes. Ihave pointed out that not all the coecients
introduced by Jo et al (1996) are useful and that the elimination of one of them
does not aect the result in a measurable way. If this coecient is not used, the
minimization problem becomes a 2D search, and can be carried out graphically.
This approach will be of use in a later chapter in which the rotated nite dierence
operators are developed for elastic forward modelling.
Ihave further shown the extension of the rotated nite dierence operators to
the heterogenous case and I have shown that by using both rotated nite dierence
and nested dissection grid ordering it is possible to solve a realistic, large scale
problem. Taking into account the whole solution procedure while working on the
matrix solver puts certain constraints on the method in question.
76

Chapter 4
Frequency domain waveeld inversion example
4.1 Introduction
Computer modelling is used in many engineering disciplines for product development
and testing. However, in exploration geophysics the main problem in not
to model the data but to try to nd the model which \ts" the data collected at the
site. This is a reverse engineering problem and in geophysics it is usually referred
as inversion. By transforming the data into a geological model the target area can
be better understood and exploited. Ideally one would like to determine the exact
position, size and geometry of the target. This is not an easy problem. In order
to transform from the data space into the appropriate model space it is necessary
to have a good and fast seismic modelling algorithm with which the comparison
between the real data and the synthetic data can be made, and with which updates
to the model can be computed. It is critical that the main data phases from the
eld data can be reproduced. In this sense seismic inversion is closely dependent on
seismic modelling.
Traveltime tomography, is a standard processing technique for certain kinds
of seismic experiments due to its eciency and robustness. Tomographic approaches
using seismic travel times have been used for a long time to generate images of geological
regions (Dines and Lytle, 1979; Peterson et al., 1985; Dyer and Worthington,
77

1988). Reviews have been provided by Worthington (1984), Bording et al. (1987)
and Wong et al. (1987). In traveltime tomography, ray based methods are usually
used to predict the travel times, and form the required matrices. Waveeld inversion,
as opposed to travel time tomography, attempts to t the waveform data instead of
the travel times only. Waveeld inversion is a computationally expensive procedure.
It relies on ecency of forward modelling to quickly predict the synthetic responses
through the model. Simulating waveforms requires more resources than simulating
the arrival times only. Tomographic datasets require a large number of sources in
order to acheive the required data coverage. As pointed out earlier the frequency
domain forward modelling can deal with large number of sources eciently.
The improvements described in the previous chapters have been incorporated
into the forward modelling part of a waveeld inversion routine in order to
signicantly increase the speed of the procedure. This enables multiple runs with
weighting constraint parameters to be evaluated and the correct constraints selected
and used to produce the optimal output result. In this Chapter I will present the
results obtained by waveeld inversion of a transmission data set recorded at the
Grimsel Rock Laboratoty in Switzerland. The data set is an unusual one from the
acquisition point of view. The full data set can be devided into the almost horizontal
cross bore-hole data set and the two almost horizontal multiple oset VSP data
sets recorded by applying the sources in between the two bore-holes used for the
cross bore-hole survey and applying the receivers into the bore-holes. This acquisition
geometry enabled excellent data coverage in a large part of the area. We see
waveeld inversion as one of the main applications of the frequency domain forward
modelling.
Resolution limitations of the traveltime methods (Williamson and Worthington,
1993) have lead to attempts to t not only the arrival times but the seismic
waveforms (eg, Devaney, (1984)). In the last decade, due to an increase in computational
power, seismic waveeld inversion has become a feasible approach (Gauthier
78

et al., 1986; Kolb et al., 1986; Zhou et al., 1985; Song et al., 1995; Pratt et al.,
1995).
Waveeld inversion was introduced by Lailly (1984) and is a non linear problem.
The aim is to build the model which will t the wave forms in the data. This
approach is much more physical then the travel time tomography due to the fact
that the travel times can be over ted (with a rough model with a lot of nodes it may
be possible to reduce travel time residuals to zero), while it is imposible to t certain
forms of noise in the wave forms, as those are a physical phenomena. For example
if the random noise is present in the data and one attempts to invert the data using
the waveeld inversion the underlying wave equation can never reproduce rapidly
varying noise in the data however it can t some \source generated noise". The main
dierence between waveeld inversion and traveltime tomography is in the nature
of the data. The travel-times are not a directly recorded parameter: They include
subjective information introduced during traveltime picking. In some cases it may
be dicult to pick consistent rst break travel time due to a signicant amount of
noise in the data. Even in the cases where there is no signicant noise problem, the
consistency of the picks may be systematically aected by human factors. On the
other hand, the data used in waveeld inversion are a directly measured physical
property. There is no subjective transformation involved in the processing which
will aect the data. The only error in the input data is the error introduced by the
eld equipment. Provided we can simulate the right waveeld we should be able to
use the full undistorted data in inversion.
4.2 Site description: Grimsel Rock Labaratory
The Grimsel Rock Laboratory is located in SW Switzerland in the Aar Massif.
The site is owned and operated by NAGRA, the Swiss national cooperative for
the disposal of radioactive waste. The laboratory is an underground test site located
79

Figure 4.1: Grimsel Pass areal photo.
Figure 4.2: Inside of the Grimsel Rock labaratory tunnel.
80

eneath the Grimsel pass (see Figure 4.1) in anunderground tunnel (aphotograph
of the tunnel interior is shown on Figure 4.2). The purpose of the site is to provide
an in-situ, controlled location for the testing of rock characterization methods, with
the ultimate objective being the application of techniques at a long term site for
the storage of radioactive waste. In this chapter I will present the re-processing of a
tomographic data set acquired in 1985 (Gelbke et al., 1989). The test site is located
within granitic rocks with a few mac dike intrusions, and a number of predominantly
vertical fracture zones. A series of approximatly horizontal boreholes where
used to deploy sources and receivers in the conguration shown on Figure 4.5. The
data quality was quite high, with noise-free records and clean rst arrivals, however
the data set suers from relatively large static shifts and signicant, unexplained,
amplitude variations (representative data sections are shown on Figure 4.4 and Figure
4.3). Similar data problems are observed by Gelbke etal.(1989) and Song and
Worthington (Song and Worthington, 1995).
The project aim was to test the utility of tomographic images as a tool for
detection of fractures capable of transmitting uids in nuclear waste depositories.
Various tomographics techniques were tested at the site and compared. The techniques
included radar tomography, dierential radar tomography and traveltime
tomography (isotropic and anisotropic). Here I investigate the waveeld inversion
approach to the tomographics data.
The Field 2 region at the Grimsel Test Site is shown schematically in Figure
4.5. It comprises a horizontal panel, bounded on two sides by horizontal boreholes,
and on a third side by the underground access tunnel (the bottom of the Figure
4.5). The boreholes dip approximately 15 degrees downwards from the tunnel. A
number of other small boreholes traverse the region, the projection of onto the
source-receiver plane of those boreholes is also shown in Figure 4.5. The Field 2
seismic survey consisted of locating sources in the tunnel and recording two \oset
VSP" datasets with receivers in both boreholes, and locating sources in one of the
81

5 10 15 20 25 30 35 40 45 50 55 60
Waveform problem due to "in fill" survey
0.0
0.0
0.01
0.01
0.02
0.02
0.03
0.03
Time (s)
0.04
0.05
0.06
0.04
0.05
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
Receiver no
(a)
0.0
0.0
0.01
0.01
0.02
0.02
0.03
0.03
Time (s)
0.04
0.05
0.06
0.04
0.05
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
10 20 30 40 50 60 70 80 90 100 110 120
Receiver no
(b)
0.1
Figure 4.3: Two representative source gathers of VSP data from Field 2, as true
amplitude displays. a) A VSP source gather with large oset. The spurious variation
of amplitude from trace to trace is evident, as is the consistency of alternate traces.
The data were recorded in two passes, with intermediate traces recorded during a
later, \in-ll" survey. b) A near oset VSP source gather, on which the dramatic
change in amplitude with receiver depth is evident. These variations in amplitudes
cannot be modelled using the 2D acoustic method. In order to invert these data I
apply a normalization to each trace separately.
82

Sources from 1 to 121
0
VSP1
5
10
VSP2
15
VSP3
Time (ms)
20
25
Crosshole
VSP4
30
35
40
45
50
Figure 4.4: A representative common receiver gather of the Field 2 data, following
windowing and trace normalization. The receiver was in borehole 3. The rst
portion of the gather was recorded with sources in borehole 2, and thus represents
a portion of the cross borehole data. The second section was recorded with sources
in the tunnel, and thus represents a portion of the VSP data. The data have been
windowed and trace-normalized. The random static shifts in the cross borehole data,
and the systematic static shifts in the VSP data are evident. The labels indicate the
VSP source groups that were identied, in order to solve for the source consistent
static shifts.
83

160m
BOUS 85.003
BOBK 85.004
BOBK 85.008
FBX 95.002
N
BOUS 85.002
Tunnel
160m
Figure 4.5: Map of the Field 2 study area at the Grimsel Test Site. The seismic
data were acquired using the tunnel and boreholes BOUS85.002 and BOUS85.003
(\boreholes 2 and 3"). The remaining boreholes are exploratory boreholes in which
velocity information is available and is used to test the waveform images. The
scale of this map is 1:1000, a representative square area 160m 160m is shown for
reference.
boreholes and recording cross-borehole data in the other borehole.
A number of
other small boreholes traverse the region, the projection of onto the source-receiver
plane of those boreholes is also shown in Figure 4.5.
4.3 Waveeld inversion
The idea of waveeld inversion (which attempts to t the complete arrival
waveeld) follows on from the results obtained by tting the travel times through
tomography. Initially, waveeld research was focused on development of migration
84

algorithms. Conventional migration techniques attempt tofocus scattered waves at
their point of origin (McMechan, 1983; Hu et al., 1988). From this starting point,
work was extended to inversion techniques which produce quantitative information
on the physical parameter of the medium (Devaney, 1984; Gauthier et al., 1986).
Lailly (1984) and Tarantola (1984) laid the foundations for waveform inversion
by posing the problem as a least-squares optimisation and showing how to
eciently calculate the gradient of the objective function. The analytic form of the
Frechet derivative of waveeld data with respect to changes in the model parameters
is given by the Born approximation, formulated as an integral solution to the
wave equation. This method attracted a lot of interest (Mora, 1987; Mora, 1989b;
Beydoun and Mendes, 1989). The nonlinearity of the problem can be overcome by
iterative procedures. The general nature of the approach enabled its implementation
with various forward modeling approaches. Gauthier et al. (1986) demonstrated the
application of Tarantola's idea to synthetic acoustic data using a time-domain nite
dierence modelling algorithm. Gauthier et al. commented on the computational
complexity of the problem due to slow convergence and the expense in a multisource
conguration. Pratt and Worthington (1990) applied Tarantola's idea using
frequency domain nite dierence modelling in 2D and overcame the problem of multiple
sources. They showed that only a limited number of frequencies are required
in some experimental geometries, particularly for the cross-borehole conguration.
4.4 Waveeld inversion theory
Waveform inversion in general will require many solutions to systems of equations
of the form of equation (1.5). The iterative approaches to solving the non-linear
problem assumes the following:
One has access to n experimental observations, u (0)
at a subset of grid
points corresponding to receiver locations (for convinience of problem formulation I
85

will assume that rst n r n x n z (where n x n z isnumber of grid points in the model)
grid points are receiver locations)
An initial model exists that lies within range of a global minimum in the
objective function
Synthetic data u generated using this initial model that are representative
of the real data
For the purpose of the inversion the solution to the forward problem (equation
(1.5)) can be written schematically as:
u = S ~
,1
f (4.1)
where S ~
is in general a complex \impedance matrix", f is a source term and u is a
column vector of length n representing the eld variable. The residual error, u is
dened as the dierence between the initial model response and the observed data
at the receiver locations. Thus
u i = u i , u (0)
i ; i =(1;2;:::;n r ) (4.2)
where the subscript i represents the receiver number, n r is a number of receivers
and the subscripted quantities are the individual components of u; u (0) , and u.
As is common in many inverse problems, we seek to minimize the l 2 norm of
the data residuals. Thus we minimize the \objective" function
E(p) = 1 2 ut u ; (4.3)
where p is the vector corresponding to the discretization of the physical parameters.
In equation (4.3) the superscript t represents the ordinary matrix transpose and the
superscript represents the complex conjugete, introduced to ensure the objective
function is a true (real valued) norm for complex valued data.
One method which may be used to calculate the update of the model at each
iteration is the gradient method. The gradient method is a recipe for reducing the
86

l 2 norm (8) by iteratively updating the parameter vector according to
p (k+1) = p (k) , (k) r p E (k) ; (4.4)
where k is an iteration number, and is a scalar step length chosen to minimize the
l 2 norm in the direction given by the gradient ofE(p). The gradient of the objective
function represents the direction in which the objective function is changing fastest.
Thus, the objective function can always be reduced by pursuing such a direction.
Although the optimal step length can be computed for linear problems, the
step length in non-linear problems must generally be sought using line search techniques.
The iteration in equation (4.4) is performed until some suitable stopping
criteria is reached. The convergence rate of the gradient method is generally quite
slow, especially in the early iterations. Convergence can be improved by adopting a
conjugate gradient approach (see for example Mora, 1988), which does not require
any signicant additional computations.
One may evaluate the gradient direction by taking partial derivatives of equation
(4.3) with respect to the inversion parameters, p
r p E = @E
@p = Re n J~ t u o (4.5)
where Re fxg denotes the real part of x. I assume there are m model parameters,
so that p is a column vector of length m, and
J ~ t is the transpose of the n r m
Frechet derivative matrix, J ~
, the elements of which are given by
J ij = @u i
@p j
i =(1;2;:::;n r ); j =(1;2;:::;m): (4.6)
One can see from equation (4.5) that the elements of J ~
are not explicitly required
in the gradient method, all that is required is to be able to compute the action of
J ~ t on the vector u .
Computation of the step length, required in the equation 4.4, is straightforward.
For linear forward problems the step length is given by the followin equation:
= jr pEj 2
J ~
r p Ej 2 (4.7)
87

where jxj represents represents the Euclidean lenth of the vector x. For non-linear
forward problems, the step length must be found using line search techniques along
the direction opposite to the gradient (this is the case for seismic waveform inversion).
The gradient vector, required in euation (4.5), can be eciently computed
through additional frequency domain forward modelling steps. To show this, I rst
augment the m n r matrix J ~
with the additional terms required to dene partial
derivatives at all node points, not just at the receiver locations, to obtain a new
m n x n z matrix c J ~
. One can write an equation similar to the equation (4.5)
r p E = Re
cJ~t c u
; (4.8)
where c u is the data residual vector, of length n r , augmented with n x n z , n r zero
values to produce a new vector of length n x n z . Explicitly, equation (4.8) represents
2
6
4
3
@E
@p 1
@E
@p 2
.
7
5
@E
@p m
=
=
2
6
4
@u 1
@p 1
@u 1
@p 2
:::
.
@u n
@p 1
.
@u n
@p 2
:::
@u n+1
@p 1
@u n+1
@p 2
:::
.
@u nxnz
@p 1
@u
@p 1
.
@u nxnz
@p 2
:::
@u
@p 2
:::
. .. .
. .. .
3t 2
@u 1
@p m
@u n
@p m
@u n+1
@p m
7
5
6
4
@u nxnz
@p m
2
t @u
@p m
6
6
4
u 1
.
u n
0
.
0
3
7
5
u 1
.
u n
0
.
0
3
7
5
: (4.9)
An expression for any of the partial derivatives in equation (4.9) in terms
of the forward modelling matrix equation (1.5) can now be obtained by taking the
88

partial derivative of both sides of equation (1.5) with respect to the ith parameter
p i :
or
where
@ u
S = , @ S ~ u
~ @p i @p i
@ u
@p i
= S ~
,1
g (i) (4.10)
g (i) = , @ S ~
@p i
u: (4.11)
By analogy with equation (1.5), the partial derivatives in equation (4.10) are
the solution to a new forward modelling problem, one in which the term on the right
hand side plays the part of a \virtual" n x n z 1 source vector, g (i) . Perturbing the ith
parameter by an amount p i will yield a perturbation in the seismic waveeld with
values given by the solution to the forward problem in equation (4.10) multiplied
by p i . The virtual source represents the interaction (or scattering) of the predicted
(or background) waveeld, u with the parameter p i . I will therefore refer to @u=@p i
as the \partial derivative waveeld from the ith node". As shown in equation (4.9),
each column of
J ~
contains a partial derivative waveeld from a single physical
parameter; there are m such columns. Where the inversion parameters consist of
the values of a single physical parameter at the node points (the \point collocation"
scheme), there will be m = n x n z columns and J ~
is a square matrix.
4.4.1 Ecient calculation of the gradient direction
Since I could generate an equation similar to equation (4.10) for any choice of
i, I can represent all the partial derivatives simultaneously by the matrix equation
c J~ =
@u
@p 1
@u
@p 2
:::
@u
@p m
,1
= S ~
g (1) g (2) ::: g (m)
(4.12)
89

or
c J~ = S ~
,1
G ~
(4.13)
where
F ~
is a n x n z m matrix, the columns of which are the virtual source terms
for each of the m physical parameters.Equation (4.13) gives an explicit formula for
the Frechet derivative matrix, J ~
(being the rst n n x n z rows of b J). Computation
of the elements of J using equation (4.13) would require m forward propagation
problems to be solved, in addition to the one required to compute the virtual sources
using equation (4.11). However, in order to compute the gradient using equations
(4.5) or (4.8) it is not necessary to compute the elements of J explicitly. Substituting
(4.13) into (4.8) I obtain
r p E = Re cJ~
t
n
bu = Re G~
t vo ; (4.14)
where
v =
h i t
S~
,1
bu (4.15)
or
v = S ~
,1
bu (4.16)
(by symmetry of the impedence matrix), which only requires one additional forward
problem to be solved. Thus the gradient is calculated in two steps: i) The \backpropagated"
eld, v, is computed by solving a forward problem with the source terms
replaced by the conjugate predicted waveeld (time reversed) and ii) The backpropagated
eld is multiplied by the conjugate (time reversed) sources generated by the
original predicted waveeld u.
It is informative to use equations (4.11) and (4.14) to express the i-th component
of the gradient vector as
(r p E) i
= Re
(u (i)t "
@ S~
t
@p i
#
)
v
(4.17)
90

from which itisevident thatwhere @S
@p i
consists ofhighly local non-zero values near
or at the ith row, as it will for the point collocation scheme, the gradient can be
computed by a scaled multiplication of forward and backpropagated waveelds. This
is the description usually given for the computation of the gradient vector, and it is
clearly closely related to some reverse time migration algorithms, and to Claerbout's
(1976) U/D imaging principle.
4.5 Processing of third party synthetic data
In this section I show the application of frequency domain modelling as a part
of the frequency domain waveform inversion technique. Before inverting the eld
data an extensive study was carried out using a full elastic, 2D synthetic dataset
generated by Prof. Korn of Leipzig University using a time domain nite dierence
method. The velocity model used for this numerical experiment was provided by
NAGRA, and is shown in Figure 4.6 (a). This model is intended to represent some
of the expected geological features at the Grimsel Test Site, and the source-receiver
geometry mimics that of Field 2. The large, low velocity zone in the lower right
hand section of the model represents the known presence of lampophyre dykes that
intersect the tunnel wall, and the thin, dipping features represent the known fracture
directions at the site. There is a low velocity zone situated at the top of the model.
This zone lies within a region with poor coverage, and will serve to illustrate the
image degradation of features not well covered by the data. The eld geometry is
bounded by borehole 2 at the right side of the gure, borehole 3 on the left hand
side of Figure 4.6 (a) and the access tunnel at the bottom of the low velocity zone
close to the bottom of the gure.
In order to process these synthetic data, in preparation for the processing to
be used for the real Field 2 data, the following pre-processing steps were undertaken:
i) Project the two-component geophone data onto a local coordinate system de-
91

Figure 4.6: Comparison of the travel time tomography result and the full wave-
eld inversion from the third party synthetic elastic wave data. a) True velocity
model used in elastic forward waveform modelling, b) traveltime tomographic image
formed from the picked synthetic data, c) acoustic waveeld inversion of the elastic
synthetic data, without trace normalization, d) acoustic waveeld inversion with
trace-normalization.
92

ned by straight ray paths.
ii) Window the projected waveform rst arrivals in time using an exponentially
tapered time window 15 ms wide, starting 5 ms before the picked arrival time.
iii) Trace normalise the windowed data to remove spurious trace-to-trace amplitude
variations.
iv) Use travel-time tomography to produce a starting model for waveeld inversion.
In the following paragraphs I summarize the reasoning behind the application of
each of these steps:
Data projection is used to transform the two component displacement data
into single component data. This is required since the inversion software models only
acoustic, compressional waves, and hence requires data that represent equivalent
pressure eld variations. By geometrically projecting the two components onto the
straight ray direction I enhance the compressional waves and partly eliminate the
shear waves. This step was largely successful in eliminating most of the shear wave
energy on the synthetic elastic data.
Data windowing should ensure that only the rst arrival, transmission waveforms
are in the data. Transmission data are more suitable for waveeld inversion
than the reections. Windowing also serves to exclude remaining shear wave energy
from the data.
Trace normalisation is not generally necessary, however, the amplitude variations
in the eld data make this step essential when processing the real Field 2
data. I therefore include this step with the synthetic data in order to assess the
eect, detrimental or otherwise, on the inversion scheme. The data were collected
in two passes, the original survey and the inll survey. The trace to trace consistency
is not high. The traces seem to be consistently oset by a small time shift
and the amplitude diers for more than an order of magnitude from trace to trace
(see Figure 4.3 and Figure 4.4).
93

In order to initialize the waveeld inversion scheme, it is necessary to begin
with an adequate starting model. This model should be capable of describing the
time domain data to within a half of the dominant period, in order to avoid tting
the wrong cycle of the waveforms. The lower the frequency, the less accurate the
starting model need be, however all real data are band limited, and thus a certain
accuracy is required of the starting model. In the real data the lowest frequencies
are corrupted by an unacceptable amount of noise. I therefore choose to generate an
accurate initial model using traveltime tomography, and proceed with the waveeld
inversion using the higher frequencies.
4.5.1 Travel time tomography
All arrival times in the full synthetic dataset were picked, and used to form
a velocity image using travel-time tomography. The procedure used for travel-time
tomography has been described Pratt and Chapman (1992) and Chapman and Pratt
(1992). The anisotropic travel time tomography at the Grimsel test site is decribed
in a report by Pessoa and Worthington (1995). Although that report describes the
use of anisotropic velocity tomography, on the synthetic dataset here I have used
only isotropic travel-time tomography, since an isotropic forward modelling scheme
is used to generate the data. The tomographic result is shown on Figure 4.6 (b).
Some of the features are recovered, but the thin low velocity layers do not appear
in the image. There is a severe imaging problem with the low velocity layer on the
top of the model due to poor coverage.
4.5.2 Full waveeld inversion
I carried out waveeld inversion of the projected synthetic data to show the
advantage of using the waveeld information, instead of travel-times only, and to
verify the processing approach for the eld data. The result of waveeld inversion
for the synthetic data is shown on Figure 4.6 (c). These images were formed using
94

6frequency components ofthedata: 200, 300, 500, 700, 800 and 1000 Hz. Each frequency
componentwas used for a maximum of 5 iterations before moving to the next
frequency, using the current image as a starting model for the next frequency. The -
nal frequency was used for 10 iterations. The individual frequency components were
iterated upon until convergence, dened as the point beyond which the algorithm
could no longer reduce the mist function. Following the amplitude normalization
of the data (see next section), this occurred typically within 2 or 3 iterations. The
same iteration strategy was followed for all subsequent images. In Figure 4.6 (c) it is
evident that there is some improvement with respect to the traveltime image shown
in Figure 4.6 (b). In particular, the exact geometry of the low velocity \dyke" at
the bottom right is better resolved, and there is a subtle improvement in the geometry
of most of the features. Moreover, the magnitudes of the velocity values are
closer to the \true" velocity values. Nevertheless, the image is largely comparable in
resolution to the traveltime image, although it is true that the use of full waveform
data excludes systematic errors introduced by manual travel-time picking.
4.5.3 Full waveeld inversion of trace-normalised data
In order to investigate whether the inaccurate amplitude simulation of the
acoustic inversion method is adversely aected by the elastic wave amplitudes in
the synthetic data, and furthermore to verify completely the approach for processing
eld data (see below), I carried out waveeld inversion of trace-normalised synthetic
data. This was necessary on the eld data due to high amplitude variations {
here I attempt to verify the normalization as a pre-processing step. As the image in
Figure 4.6 (d) shows some important features are better recovered than from the nonnormalised
data. While inverting these data I found that the convergence rate was
higher for normalised data set. This shows that the trace-normalisation can be used
as a preconditioning technique in a waveform inversion. The result conrms that the
main source of information in transmission data is in the waveform itself and not in
95

the trace-to-trace amplitude variations. It will be appreciated that amplitude preprocessing
was important even in this synthetic example as elastic wave amplitudes
are aected in a dierent manner from acoustic amplitudes.
4.5.4 Comparison of travel time and full waveeld inversion methods
In this section I will compare the results from the travel time and the full
waveeld inversions carried out on the synthetic elastic data. On Figure 4.6 I depict
the model used for forward modelling, the traveltime result and the two full waveeld
results, on raw synthetic data and on trace normalised synthetic data. This gure is
presented to show on a single gure the advantages of using the high resolution of the
waveeld inversion technique. Smaller anomalies, completely overlooked by traveltime
tomography, are completely recovered by waveeld inversion. All the anomalies
lie at the correct positions in the region with good coverage. It is, however, possible
to obtain false anomalies in the regions with poor coverage (at the top of the model),
where the traveltime result has generated a low velocity anomaly. This anomaly is
transfered into the full waveeld result by using the traveltime tomogram as an
initial guess.
This can be avoided in synthetic studies, in which, in most cases,
one can start the waveeld inversion from a homogeneous model. However, when
working with real data it is usually impossible to use suciently low frequency data,
so that a better initial guess is required.
In conclusion, once the data have been trace-normalized, waveeld inversion
produces images in which the low velocity anomalies are much better resolved than
on the traveltime tomographic image, and the velocity values are closer to the ones
in the model shown on Figure 4.6 (a). It is important to point out that this is a
signicant test of the method, as the data were generated by a third party, using
an elastic wave simulation. The inversion software uses an acoustic wave method,
which ignores elastic eects, but the images justify the use of this approximation.
96

4.6 Inversion of real eld data
Having successfully demonstrated the waveeld inversion technique on third
party elastic synthetic data, and having veried much of the pre-processing techniques
required, I now turn my attention to the real Field 2 data. The most signicant
problem with the real data (originally identied in the report by Song and
Worthington (1995)) was the trace-to-trace amplitude variation. An example of this
amplitude variation is shown in Figure 4.3. As I shall show in this section, this problem
has been entirely solved by trace-normalisation of the data. The other problem
visible on gure 4.3 (a) is a trace to trace waveform change. This is due to the data
acqusition. The data were collected in two attempts: The original survey and the
inll survey. The acquisition equipment had a dierent characteristic so the trace to
trace consistancy is not high. The traces seem to consistently oset by a small time
shift. The same behaviour can be observed on the trace normalised VSP common
shot gathers. I have decided not to try to account for this problem.
The pre-processing ow for the real data, with one exception, was identical
to the pre-processing used for the elastic synthetic data. The full procedure was:
i) Project the two-component geophone data onto a local coordinate system de-
ned by straight raypaths.
ii) Window the projected waveform rst arrivals in time using an exponentially
tapered time window 15 ms wide, starting 5 ms before the picked arrival time.
iii) Trace normalise the windowed data to remove spurious trace-to-trace amplitude
variations.
iv) Use travel-time tomography to produce a starting model for waveeld inversion.
v) Separate the unknown source behaviour into ve distinct physical \groups".
Four individual groups were used for the VSP data, and one additional group
was used to represent all of the crosshole data.
Figure 4.4 shows in which
97

manner these groups were identied.
The additional step here, not used with the synthetic data, was the manner
in which the unknown source behaviour was separated into ve distinct groups and
solved for. For the synthetic data I solved for the source behaviour, but I treated the
entire data as if it came from a single physical source. The eld data are known to
contain signicant source-consistent static time shifts (as commented on by Gelbke
et al, (1989)). An example of these static time shifts is shown in Figure 4.4.
The source-consistent static time shifts were included into the inverse problem
by using 4 separate VSP source \groups" for the eld data, and solving for 4 separate
source functions. Using more than one source group does not signicantly eect the
uniqueness of the inversion approach, but it is essential that these source-consistent
errors are accounted for. There are also random source and receiver static shifts on
the cross-borehole data, that I do not account for. The random nature of these latter
problems causes a decrease in the signal to noise level of the nal images (see next
section), but does not cause a signicant systematic deterioration of the images.
4.6.1 Initial full waveeld inversion
I begin the discussion of the results from the eld data by showing the initial
results that were obtained before the complete pre-processing ow described in the
previous section was worked out. In this section I will also study the cross-borehole
and VSP components of the data separately. In all cases I begin from a starting
model obtained from anisotropic velocity tomography, as described by M. Pessoa
and M.H. Worthington in their 1995 report.
This tomogram, after some simple
smoothing, is shown in Figure 4.7.
Ihave carried out tests to study the image quality if only a subsection of the
data set is used. The result if only the cross-borehole component of the data is used
is shown on Figure 4.8. The result is contaminated by strong velocity variations
apparently originating at the borehole source-receiver locations. From this result
98

km/s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
Figure 4.7: Starting model for waveeld inversions of the eld data (from anisotropic
velocity tomography).
I may conclude that condence in these cross-borehole data cannot be high. The
problem appears to be linked with the inconsistent source coupling in the borehole
and the random static shifts described in the previous section.
This section of
the data is much noisier than the VSP section. In contrast, the result from VSP
component of the data is shown on Figure 4.9. This result is less contaminated, and
much closer to the expected geology at the site. These results show that imaging
each subset of the data is not sucient on its own. However, the use of the whole
data set should improve the result considerably.
Figure 4.10 shows the inversion result using both cross borehole and VSP
sections of the Field 2 data. The image shows some signicant improvements when
compared with the individual images in Figures 4.8 and 4.9. However, there is still
a strong noise component to these images that is apparently related to individual
source and receiver locations. These noise patterns seem to propagate into the image
and obscure the geological features. Ihave traced these noise features to the strong
99

km/s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
Figure 4.8: Preliminary full waveeld inversion image using non normalized crosshole
part of the data only.
km/s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
Figure 4.9: Preliminary full waveeld inversion image using non normalized VSP
part of the data only. Short oset VSP data are excluded due to large amplitude
variations.
100

km/s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
Figure 4.10: Preliminary full waveeld inversion image using non normalized Field
2 data, including both crosshole and VSP sections of the data. Short oset data are
excluded due to large amplitude variations.
and spurious trace-to-trace amplitude variations pointed out in Figure 4.3. This led
to the decision to apply a trace normalization factor to each time domain trace after
windowing and before extracting the various frequency components.
A further decision was made to attempt to control remaining noise in the
images by applying a constraint on the roughness of the solutions. This constraint
is similar to the constraint used by Pessoa and Worthington in their 1995 report on
traveltime tomography. The objective is to form images that contain no unnecessary
structure | the only structure that should appear in the images is structure is
required to t the data. The eect of this additional constraint is explored in the
next section.
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4.6.2 Regularization tests
From this point on I depict images obtained from the data following the
full pre-processing scheme, including the trace-normalization of the data.
In the
previous section I described the use of an additional constraint ontheroughness of
the solution (I term this a \smoothing constraint"). From the pre-processed data
I have generated a series of full waveeld inversion results with various levels of
smoothing parameters. The resulting images are shown on Figure 4.11. In order
to select an appropriate regularization level, I also computed the RMS residuals,
and RMS roughness for each of these images, and plotted these against each other
(Figure 4.12). As Pratt and Chapman have advocated for travel-time tomography
in the past, I select an image that simultaneously ts the data as well as possible
(low residuals) and is as smooth as possible (low roughness). I seek a \knee point"
on the tradeo curve, which, in this case indicates a smoothing parameter of close to
15. The full waveeld image shown on Figure 4.13 is my nal isotropic result, using
a regularization level of 15, as determined from the previous gures. This image
is already an important improvement on the starting model, however it appears to
suer from a strong variation in background velocities from the left side of the image
to the right. In the next sections I will further evaluate this image by studying the
residuals, and I propose that this eect is caused by the low level anisotropy present
at the test site.
4.7 Isotropic results: Evaluation and verication
In order to evaluate the isotropic result, I produced Figures 4.14 to 4.16,
which represent respectively the eld data, the predicted data following the waveeld
inversion and nally, the dierences, or residuals following the inversion.
These
plots are somewhat unconventional: Each pixel in these gures represents the real
part of the complex-valued, single frequency waveeld at 800 Hz, recorded by a
102

0 5 10 15
20 25 30 35
40 45 50 100
km/s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
Figure 4.11: Isotropic full waveeld inversion results with various values of smoothing
parameter increasing from 0 (top left corner) to 100 (bottom right corner).
103

RMS Roughness
0.8
0.7
0.6
0.5
5
10
15
20
25
30
3540
4550
0.4
0.3
0.0002 0.0004 0.0006 0.0008 0.0010 0.0012
RMS Residuals
Figure 4.12: Trade o curve showing RMS roughness vs RMS residuals for a suite
of smoothing parameters.
km/s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
Figure 4.13: Final isotropic full waveeld inversion result.
104

Figure 4.14: Frequency domain eld data at 800Hz. Please see the text for a full
description of this gure. The grey scale is a relative amplitude, from the maximum
negative values through to the maximum positive values.
single source-receiver pair. The horizontal axis represents the receiver number (with
receiver 1 at the left hand edge), the vertical axis represents the source number (with
source 61 at the top edge). The data divides naturally into 3 sections: The crosshole
data (top left quadrant), and the two VSP datasets (bottom two quadrants).
As this is not a common representation, it is useful to explain the regular features
one can (and should) observe: If this were a homogeneous media one would expect
to see a set of linear features parallel to the main diagonal in the cross hole part of
the survey, and circular patterns in the two quadrants representing two VSP data
sets. These patterns are indeed visible in the data, Figure 4.14, in spite of the fact
that this is not a perfectly homogeneous region. The patterns may also be compared
with the synthetic waveelds predicted in the nal isotropic image, Figure 4.15. On
both these gures the source-consistent static shifts can be observed in the VSP data
sets as horizontal lines on the gures. If I had been successful in predicting the data
with the inversion result, the dierences between these gures would be small, but
more importantly would not show any systematic patterns. However, Figure 4.16
shows that much of the systematic patterns in the data remain unaccounted for.
This indicates a failure to explain some of the main features in the data. As I will
show in the following sections of this chapter, this is likely to be due to anisotropy.
It is also importanttoverify the method used to account for source-consistent
105

Figure 4.15: Frequency domain modelled (predicted) data at 800Hz. See text for
a full description of this gure. The grey scale is a relative amplitude, from the
maximum negative values through to the maximum positive values.
Figure 4.16: Dierence between eld and modelled data at 800Hz. See text for a full
description of this gure. The grey scale is a relative amplitude, from the maximum
negative values through to the maximum positive values.
106

-5
-2.5
0
2.5
5
7.5
10
12.5
15.
Crosshole VSP1 VSP2 VSP3 VSP4
-5
-2.5
0
2.5
Time (ms)
5.
7.5
10.
12.5
15.
17.5
20.
22.5
Time (ms)
17.5
20.
22.5
Figure 4.17: Inverted source signatures. These signatures were extracted as an
integral part of the waveeld inversion scheme. The similarity of the VSP source
signatures, apart from the known static shifts, gives credence to the robustness of
the inversion scheme.
static shifts. As described above, to account for these static shifts, I divided the
VSP data into 4 source \groups", each assumed to have a separate source behaviour
(recall, these groups were identied on Figure 4.4). I also included a fth group
to collectively represent all crosshole sources.
In order to evaluate this approach
I display the resultant (inverted) time domain source signatures (shown on Figure
4.17). Each of these signatures was estimated independently from the data alone {
it is reassuring that the waveforms of the VSP source signatures are consistent, and
that most of the dierences are due only to time shifts. This consistency tends to
verify the approach.
4.7.1 Discussion of isotropic results
We have seen that the isotropic results show a large variation in velocities
from the left hand edge of the images to the right hand edge. We have also seen
that the data residuals show that much of the data variation remains unexplained
107

y the best isotropic results. In all studies of Field 2 using travel-time tomography
it has proven necessary to account for a small level of anisotropy (Gelbke et al.,
1989; Pessoa and Worthington, 1995). I believe that the variation in velocities in
the images and the remaining residual levels in the data are both best explained by
the seismic anisotropy of the rocks.
The anisotropy at the Grimsel Test Site is expected to be relatively low.
Previous estimates (Pessoa and Worthington, 1995) from the seismic traveltimes
have shown an overall level from 1% to 3 %, with a slow axis dipping 45 o from the
top right corner to the bottom left corner. From the results of Chapter 3, I would
expect that the velocity errors in the modelling code are of the order of 1%, and
that the inversion errors will be at least an additional few percent. If the errors of
the method are of the same order as the anisotropy level, can the anisotropy aect
the images so strongly? The answer may lie in the systematic distribution of the
ray directions in the data. The main ray directions in the VSP data sets are, in this
case, almost exactly matched with slow and fast velocity axes. As the VSP data
primarily recorded low and high velocities this had to be compensated in the image
regions which where covered by a single part of the VSP data.
The eect of anisotropy on the wave form images has not been examined in
detail primarily due to the expense of anisotropic forward modelling. However some
experiments with homogenous elliptical anisotropy have been published (Pratt et
al., 1995) but only if the amount of anisotropy is high (in the example used by Pratt
et al. (1995) the amount of anisotropy was of the order of 20%, much larger than the
maximum expected numerical errors). At Grimsel, in homogenous crystalline rocks,
the anisotropy level is expected to be low and we did not expect any signicant
artifacts on the image from anisotropy. However as shown in previous section the
data residuals for the nal image are coherent and the image suers from signicant
left right velocity distribution.
In order to test the possible eect of low anisotropy, using the acqusition
108

km/s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
Figure 4.18: Isotropic inversion of synthetic data set from a homogeneous,
anisotropic model.
geometry at Grimsel, I generated a synthetic, homogeneous, elliptically anisotropic
model (with 3% anisotropy and the slow axis dipping 45 o from the top right corner
to the bottom left one) by shrinking the model in the fast velocity direction by
3% and using the exact Field 2 source receiver conguration. Using this anisotropic
model, I generated a full waveeld dataset using the isotropic frequency domain nite
dierence modelling as described in previous chapter. The homogeneous (isotropic)
velocity that was perturbed was V p = 5:2 km=s. I then inverted these data using
the isotropic inversion scheme. The result, shown on Figure 4.18, suers from the
same left-rightvelocity distribution problem as the isotropic images computed using
the real data. The synthetic inversion result is correct in the central region where
I have coverage from both the VSP datasets and from the cross-hole data sets. In
the regions covered by only a single VSP data set the image compensates for the
mismatch bycreating a alow (or high) velocity anomaly.
As an additional test I have modelled and inverted the 2% elliptically anisotropic
109

a) b) c)
Figure 4.19: Data residuals for the waveform inversion runs on the acoustic syntetic
elliptically anisotropic (2 percent) data by assuming: a) Isotropic data (underestimated
level of anisotropy) b) 2 percent elliptical anisotropy (correct value) c) 4
percent eliptical anisotropy (overestimated value).
synthetic data from the test model (Figure 4.7) and examined the data residuals for
various levels of assumed anisotropy. The data residuals for the isotropic assumption,
the correct 2 percent elliptical anisotropy result and the overestimated elliptical
anisotropy of 4 percent are shown on Figure 4.19. The gure shows that i) The data
residuals are coherent when the incorrect amount of anisotropy is used and ii) The
amplitude of data residuals in the correct case is the smallest (Thus an objective
determination of the correct image is to use the level of data residuals) In the cases
where incorrect anisotropic assumptions are made (the isotropic case and the 3%
anisotropic case) the residuals are similar in apperance to residuals from the isotropic
waveform inversion of the Field 2 data on Figure 4.16. This tends to conrm that
the nal isotropic image suers from unacounted anisotropy.
The amplitude of data residuals in the correct case is the smallest. Thus
an objective determination of the correct image is to use the level of data residuals.
4.8 Anisotropic inversion of the eld data
In order to compensate for the strong anisotropy eect evident from the
initial isotropic inversions, I have carried out inversion of the eld data by assuming
constant level of elliptical anisotropy of 1, 2 and 3% by shrinking the model (and
110

the acquisition geometry) bythe same percentage in the high velocity direction. In
each case the slow axis was chosen as in the previous synthetic study and consistent
with the orientation used in most of the traveltime tomography studies at the site,
i.e., dipping 45 o from the top right corner to the bottom left corner. The images are
shown on Figure 4.20. There is a signicant dierence between the images (especially
in the top corners). A high velocity at the top left corner of the Figure 4.20 (a) has
become the low velocity zone on the gure 4.20 (d). The opposite transformation
has occurred in the top right corner. The top corners are the main regions covered
by a single VSP data set only. However, it is not clear from these images which
is the correct background level of anisotropy. In order to aid the selection of this
parameter, I also computed the RMS residuals for each of these images. The result
is shown on Figure 4.21. The diagram shows that a level of 3% anisotropy gives
residuals that are as far from the solution as the isotropic result is, and that the
optimal result will have 1:8 , 1:9% anisotropy.
Figure 4.22 show the nal anisotropic result, obtained by assuming 2% elliptical
anisotropy. The left-right velocity distribution has largely disappeared. In
order to verify this image I also show the data residual eld from this image on Figure
4.23. The data residuals no longer display the strong systematic distributions
observed in the isotropic case (see gure 4.16). Instead the data residuals are more
nearly randomly distributed.
Finally, I now include the result from the area directly to the right of Field
2 (known as Field 1). The acqusition geometry is similar to the Field 2 geometry,
although the boreholes are only 70m accros. The data were inverted independantly
of the eld 2 data, using the same processing sequence. The nal image is shown
next to the nal result for eld 2 on gure 4.25 and shows agreement on the common
borehole (borehole 2). The level of anisotropy found from the Field 1 data was the
same as for Field 2 (i.e. approximately 2 percent). This seems to verify the Field 2
result and the approach used for data processing. The consistency of the anisotropy
111

Figure 4.20: Anisotropic full waveeld inversion results with 0, 1, 2 and 3% elliptical
anisotropy.
112

-8
7.0x10
Data residuals
-8
6.8x10
-8
6.6x10
-8
6.4x10
-8
6.2x10
0 1 2 3
% Anisotropy
Figure 4.21: RMS residuals for each test anisotropy level.
estimation achieved by the waveeld inversion points out that it may be possible
to give anestimate of the anisotropy level by waveeld inversion or even invert the
data by using the anisotropic waveeld inversion to obtain a detailed anisotropic
model.
4.9 Conclusions
In this Chapter I have shown the potential of frequency domain modelling as
atoolinwaveeld inversion, and I have demonstrated the ability ofwaveeld inversion
to yield high resolution images. A speedup of several orders of magnitude has
been acheived during the course of the project. The whole computation takes approximately
10 minutes per frequency, including ve iterations on a Digital 600/333
workstation and requires only 40MB of RAM. Five to six frequencies are usually suf-
cient so the full computation takes about 60 minutes (in comparison with 700MB
of RAM and about ve days required before). The speed increase enabled multiple
runs with various smoothing values and anisotropy levels. If less ecient modelling
113

km/s
4.80
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
5.25
5.30
5.35
5.40
Figure 4.22: Final full waveeld inversion image using 2% elliptical anisotropy.
114

Figure 4.23: Frequency domain dierence eld (i.e., data residuals) at 800 Hz from
the anisotropic inversion. See text for a full description of this gure. The grey scale
is a relative amplitude, from the maximum negative values through to the maximum
positive values.
techniques were used this amount of testing would not be possible. From a computational
point of view this problem size (14; 400 traces, 40 by 40wavelengths across,
grid size of 160 by 160 grid points, 120 sources and 120 receivers) may be solved on
a fast pentium based personal computer with enugh RAM (40MB) in a reasonable
time (under 1day).
From the inversion point of view the following conclusions may be drawn.
High resolution waveeld images in a controlled test using synthetic elastic data can
be achieved. This proves that if the underlying physics is sucient representation of
the data one can expect the correct result. It is possible to pre-process the data to
cope with large amplitude variations in a data, inconsistent trace to trace variations
and signicant time static shift problems. This shows that the common eld data
problems can be overcame. It is possible to produce high resolution reliable and
interpretable images from the eld data. The following problems have been veried:
we are unable to work directly with the two component displacement data, it is
necessary to use only rst arrival waveeld in order to overcame the S wave arrivals
in the data (the underlying physics is not good enough) and even low level anisotropy
115

Figure 4.24: Final waveeld inversion images from both Fields 1 and 2, using 2%
elliptical anisotropy.
116

Figure 4.25: Final waveeld inversion images from both Fields 1 and 2, using 2%
elliptical anisotropy (colour version).
117

can eect theimages (the wrong theory once more). The problems accounted for in
this Chapter have lead to the development of the elastic frequency domain modelling
scheme in the next Chapter in order to build the waveeld inversion procedure on
top of it which may overcame some of the problems seen in this example.
118

Chapter 5
Visco-elastic frequency domain seismic forward
modelling
5.1 Introduction
In this Chapter I will extend the rotated nite dierence operators introduced
in Chapter 3 to the visco-elastic wave equation. As seen in Chapter 4, waveeld
inversion based on the acoustic forward modelling can not work with the eld data
directly.
A certain degree of preprocessing was required, including projection of
the two component data and amplitude normalisation.
In order to improve on
generality and accuracy of the model (and the resulting data), I have extended the
improvements of the modelling scheme to the elastic wave equation.
This chapter will describe the method I have developed and implemented to
improve the accuracy and eciency of two-dimensional isotropic, frequency domain
visco-elastic seismic modelling. The new scheme uses the same grid points in the
computational star as a standard second order dierencing scheme, thus conserving
numerical bandwidth and sparsity. The new scheme allows a signicant reduction
in the size of the numerical mesh, from 15 grid points per smallest wavelength in
the model to 4 grid points per wavelength, dramatically reducing the computational
costs.
119

In essence, our new scheme is the extension to the visco-elastic case of the
ideas discussed in Chapter 3 for visco-acoustic modelling. The method involved
the introduction of two new numerical operators, the rst in a rotated coordinate
frame, and the second using a lumped mass term, both of which are combined
with standard second order numerical operators in an optimal manner to minimize
numerical errors.
I will begin with a development of the required dierencing operators. Because
the visco-elastic waveeld (i.e., the displacement) is a vector quantity, the
rotation of the coordinate frame presents additional diculties (when compared
with the visco-acoustic case).
Moreover, since the 2-D dierencing operators are
nine point stars (as opposed to the ve point stars required for acoustic modelling),
the rotated operators must be modied to minimize their spatial extent. I show how
the required rotation can be achieved on the original nine point star. The second
modication discussed in Chapter 3, the use of a lumped mass term, is dealt with
in a straightforward manner.
Following the development of the required operators, I then show how the new
schemes are to be combined with the original, ordinary second order operators, to
yield a scheme which is optimized to have minimum numerical errors. The dispersion
analysis required for this optimization is given in the appendix of this thesis. I then
analyze the numerical errors in the proposed scheme, and compare these with the
errors for the standard second order scheme. I also show that, in contrast to the
standard second order scheme, the new scheme correctly predicts a zero numerical
shear velocity in uids.
Finally, the modelling scheme is used to generate synthetic crosshole data
from a model representative of the geological section at a near-surface test site.
Boundary condition formulation is the same as in Pratt (1990b). The modelling
results are used to demonstrate a possible relationship between strong, late arrivals
in these crosshole data and the generation of mode converted shear waves in the
120

(a) (b) (c)
Figure 5.1: Computational stars for frequency domain elastic modelling. These stars
indicate the coupling of the components of the displacement eld at the central node
to displacements at the nearest neighbors. a) The ordinary, second order computational
star, b) a possible rotated computational star, and c) a minimal, rotated
computational star. The symbol, represents the coupling of the same displacement
components, and also represents the only non-zero terms required in acoustic
modelling. The symbol, symbol represents the coupling between perpendicular
displacement components. The star in c) does not use additional points over the
star in a), but introduces additional coupling between components not present in
the original star.
highly layered, attenuating sediments at the site.
5.2 Visco-elastic modelling
In this section I will fully develop the method I have developed for nite
dierencing the 2-D, frequency domain, homogeneous, elastic wave equation. I will
comment at the end of this section on the extension to the heterogeneous wave
equation.
As with most other works in nite dierence methods, I will use the
homogeneous formulation to analyze the numerical errors, and I obtain a scheme
that minimizes these errors.
The 2-D, second order, frequency domain, visco-elastic wave equation in a
homogeneous, isotropic and source free media consists of the two coupled equations:
! 2 u +(+2) @2 u
@x + u
2 @2 @z +(+) @2 v
=0 2 @x@z
(5.1)
! 2 v +(+2) @2 v
@z + v
2 @2 @x +(+) @2 u
=0; 2 @x@z
(5.2)
121

where ! = 2f is the angular frequency, is the density, and and are Lame
parameters. In order to be able to simulate visco-elasticity these Lame parameters
will, in general, be frequency dependent and complex-valued. The waveeld
variables, u and v are, respectively, the horizontal and vertical components of the
Fourier transformed displacements.
5.2.1 Rotated nite dierences: Computational stars
The numerical error of a regular grid nite dierencing scheme for equations
(5.1) and (5.2) will depend on the wave propagation angle (an eect termed \numerical
anisotropy"). This is because the distance between two discrete grid points
is not the same in every direction. Usually, propagation will be most accurate in
directions parallel to the coordinate axes. The solution suggested by Cole (1994)
and by Jo et al.
(1996) for scalar wave equations is to use two separate coordinate
systems, one rotated with respect to the other, on the same discrete numerical
mesh. A linear combination of the two results will, we hope, compensate for some
of the numerical anisotropy. The aim is to minimize the numerical anisotropy, while
retaining the existing grid and keeping the computational star as small as possible.
The basic approach for developing a rotated nite dierence coordinate system
is best understood with reference to Figure 5.1, in which the computational
dierencing stars used to approximate the local partial derivatives on the grid are
depicted. In the scheme devised by Jo et al., (1996) for acoustic wave propagation,
the ve point dierence star for second order nite dierencing of the acoustic wave
equation was rotated by 45 , expanded, and overlayed on the original grid (see Figure
5.1(a) and 5.1(b)). This introduces four additional node points into the star,
turning the combined computational star from a ve point star into a nine point star.
For the elastic wave equation, applying standard second order nite dierencing to
equations (5.1) and (5.2) results in a nine point computational star (e.g., Pratt,
1989) (Figure 5.1(a)). At rst sight, it would appear that the same technique can
122

e simply extended to the elastic waveequation by using a rotation and expansion
of the computational star, resulting in the new star seen in Figure 5.1(b). Unfortunately,
this star is not useful for forward modelling. In order to understand this, it
is necessary to understand the manner in which the nite dierence approximation
of the continuous equations (5.1) and (5.2) is actually solved.
In general, wave equations such as (5.1) and (5.2) can be represented by:
L(!) u(r) =f(r) (5.3)
where L(!) is the appropriate, frequency dependent, linear partial dierential operator,
u(r) is the eld variable (in this case the displacement, a continuous, 2
component, vector eld), and f(r) is a source term (zero everywhere except at the
location of the source). This equation, together with the boundary conditions must
be satised everywhere. In 2-D, when equation (5.3) is approximated numerically,
by nite dierences using a grid of n n nodes, this yields the matrix equation
S ~
u = f; (5.4)
where S ~
is a 2n 2 2n 2 complex valued matrix approximating the partial dierential
operator L(!), u is now a 2n 2 -vector representing the two components of the
displacement eld at all n 2 node points, and f is a similar 2n 2 -vector representing
the source terms (the equation (5.4 is the same as the equation (1.5) but we just
have two times as much eld variables).
The matrix S represents a signicant storage requirement. The requirements
~
are largely determined by the sparsityof S , and by the manner in which this sparsity
~
is maintained in any solution method. In order to take advantage of the fact that
additional sources involve only a change in the right hand side vector, s, I use direct
solution methods as described in Chapter 2. Direct solvers are also required because,
if articial absorbing boundary conditions are used, S ~
will be non-symmetric and
non-denite (precluding iterative solvers that require positive deniteness).
It is
123

dicult to formulate direct solvers for arbitrarily sparse matrices, however it is
simple to restrict computations to only those matrix elements which lie within the
numerical bandwidth of the matrix. Better schemes can be developed by making
use of optimal ordering schemes; I have discussed these in Chapter 2.
The size of the computational star directly determines the eective numerical
bandwidth of the dierencing matrix
S ~
. The bigger the star, the wider the
bandwidth of non-zero elements in the matrix.
The optimal bandwidth for the
visco-elastic case is obtained by using a nine point star. The inclusion of any additional
points will increase the bandwidth severely. If this increase is balanced by a
corresponding decrease in the number of grid points required per wavelength, then
this is acceptable. When one uses an optimal storage scheme (nested dissection from
Chapter 2) for the matrix, the 13 point dierencing stars must be accurate enough
to allow more than a 50% reduction in the number of grid points per wavelength in
comparison with the 9 point dierencing star (see Chapter 2 for details). As I shall
show, the new scheme I present requires of the order of 4 grid points per wavelength
using a 9 point dierencing star for reasonable accuracy. Since one can never subsample
the waveeld below the Nyquist criterion of two grid points per wavelength,
there is no benet in using a dierencing star larger than 9 points. The exception
to this may befor cases in which one requires an extremely accurate scheme.
Therefore, the rotated star in Figure 5.1(b) is not an acceptable computational
scheme. These considerations lead to the new choice of a nite dierence star
in the rotated coordinate frame, shown in Figure 5.1(c). This computational star
does not require the use of any new grid nodes. The implication of this is that there
will be no increase in the bandwidth of the dierencing matrix, and the increase
in computational cost and in storage requirements over the ordinary second order
scheme will be negligible. Having described the design of the optimal dierencing
star, I now proceed in the next section to derive the exact form of the required
operators.
124

5.2.2 Rotated nite dierences: Operators
Solutions to the 2-D, visco-elastic wave equation, represented by the partial
dierential equations (5.1) and (5.2) should naturally be independent ofany rotation
of the coordinate system in which they are expressed. However, numerical solutions
are approximations of the exact solution, and usually diverge from the actual solution
in a manner that depends on the coordinate system.
If there is more than one
approximate solution for a particular problem, then a linear combination of them
may bea more accurate approximate solution for the same problem.
Inowintroduce a scheme which works with the original Cartesian coordinate
system (x; z) and a new system (x 0 ;z 0 ) rotated by 45 o (see Figure (5.1)). I will
assume from hereon that the equations for both coordinate systems will will be
discretized on the same discretization mesh, with sample intervals x = z .
The relationship between the displacements u,v in the original coordinate system,
and u 0 ,v 0 in the new coordinate system is given by:
u = 1 p
2
(u 0 , v 0 ) v = 1 p
2
(u 0 + v 0 ) (5.5)
u 0 = 1 p
2
(u + v) v 0 = 1 p
2
(v , u): (5.6)
Equations (5.1) and (5.2) in the new coordinate system are:
! 2 u 0 +(+2) @2 u 0
@x + u 0
0 2 @2 @z +(+) @2 v 0
0 2
@x 0 @z 0 =0 (5.7)
! 2 v 0 +(+2) @2 v 0
@z + v 0
0 2 @2 @x +(+) @2 u 0
=0; 0 2
@x 0 @z 0 (5.8)
where x 0 and z 0 are the new coordinate directions. If we subtract and add equations
(5.7) and (5.8), we obtain:
! 2 u 0 , v 0 +
( +2)
+ ( + )
! !
@ 2 u 0
@x , @2 v 0
+ @2 u 0
0 2
@z 0 2
@z , @2 v 0
0 2
@x 0 2
!
@ 2 v 0
, @2 u 0
=0 (5.9)
@x 0 @z 0 @x 0 @z 0
125

! 2 u 0 + v 0 +
( +2)
+ ( + )
@ u
@x + @ v
0 2
@z 0 2
+ @ u
@z 0 2
+ @ v
@x 0 2
!
@ 2 v 0
+ @2 u 0
=0: (5.10)
@x 0 @z 0 @x 0 @z 0
Dividing by p 2, and recalling the transformations (5.5) and (5.6), the resulting
system is:
! 2 u + 1 2
! 2 v + 1 2
"
( +2)
+ ( + )
"
( +2)
+ ( + )
@ 2 u
@x , 2 @2 u
0 2
@x 0 @z 0
@ 2 v
@x 0 2 , @2 v
@z 0 2
!#
@ 2 v
@x +2
@2 v
0 2
@x 0 @z 0
@ 2 u
@x , @2 u
0 2
@z 0 2
!#
!
+ @2 u
+ @2 u
@z 0 2
@z +2 @2 u
0 2
@x 0 @z 0
!
+ @2 u
@x 0 2
=0 (5.11)
+ @2 v
@z 0 2
!
+ @2 v
@z 0 2 , 2 @2 v
@x 0 @z 0
+ @2 v
@x 0 2
=0: (5.12)
This procedure transforms the eld variables from the rotated coordinate system
into the original coordinate system, but leaves the coordinate axes themselves in the
rotated frame of reference. The equations (5.11) and (5.12) are the elastic equivalent
of equation (3.2) from Chapter 3. They represent the wave equation expressed as a
nite dierence equation in the rotated coordinate system, using the original eld
variables. This is required in order to be able to combine the resulting numerical
solutions with numerical solutions to the original system.
We now have two partial dierential equation systems: In the original coordinate
system
!
! 2 u + A 1 =0 (5.13)
! 2 v + B 1 =0; (5.14)
(where A 1 and B 1 are the partial dierential parts of equations (5.1) and (5.2)). In
the new (rotated) coordinate system
! 2 u + A 2 =0 (5.15)
! 2 v + B 2 =0; (5.16)
126

(where A 2 and B 2 are the partial dierential parts ofequations (5.11) and (5.12)). I
also have described the dierencing operators that will be used to approximate each
of these two systems. The resulting two numerical systems will each havenumerical
errors, but these errors will dier, and the numerical anisotropy for the two systems
will be dierent.
We can write a linear combination of the two systems as:
! 2 u + aA 1 +(1,a)A 2 =0 (5.17)
! 2 v + aB 1 +(1,a)B 2 =0; (5.18)
and, by varying the coecient a, we obtain a whole family of results. Once again the
exuations (5.17) and (5.18) are the elastic equivalent of the equation (3.4) for the
elastic case. There are no limitations in the selection of the value of the coecient,
a, as long as the value is real, although Jo at al. 1996 suggest a search in the region
0 a 1 for practical purposes. The optimal value of coecient a must then be
sought to maximize the accuracy of the solution, for all propagation directions. In
other words, we seek to combine the two solutions in order to minimize the numerical
anisotropy.
Adequate second order nite dierence approximations for partial derivatives
for equations (5.1), (5.2), (5.11) and (5.12) in each coordinate system can be found
in Kelly at al. (1975), and are unchanged in this approach. For completeness I shall
give the dierence formulas required for the rotated scheme. The approximations
used for the non-mixed partial derivatives in the 45 o coordinate system are:
@ 2 v
@x 0 2
@ 2 v
@z 0 2
!
!
m;n
m;n
v m+1;n+1 , 2v m;n + v m,1;n,1
2 2 (5.19)
v m,1;n+1 , 2v m;n + v m+1;n,1
2 2 ; (5.20)
where is the grid spacing in x and z directions, and m, n are discrete grid point coordinates.
In order to better visualize the computations implied by equations (5.19)
and (5.20), and similar equations to follow, I will present these as computational
127

\stars":
!
@ 2
1 0 0 1
0 -2 0
@x 02 2 2 1 0 0
!
@ 2
1 1 0 0
0 -2 0
@z 02 2 2 0 0 1
; (5.21)
The mixed nite dierence term in the rotated frame of reference, using the
star shown in Figure 5.1(c), is given by
or
@ 2 v
@x 0 @z 0 !m;n
v m,1;n + v m+1;n , v m;n+1 , v m;n,1
2 2 (5.22)
!
@ 2
1 0 -1 0
1 0 1
@x 0 @z 0 2 2 0 -1 0
: (5.23)
5.2.3 Consistent and lumped mass terms
The previous discussion targeted the dierential parts of the equations. This
led to a scheme to minimize the amount of numerical anisotropy. In order to minimize
the overall numerical dispersion, I now concentrate on the algebraic terms,
! 2 v and ! 2 u in equations (5.17) and (5.18). These terms are normally approximated
by using the value of the density, and the eld variable u or v at each local
node point. This is known as a consistent formulation. An alternative formulation,
known in nite element methods as a lumped formulation, is obtained by using an
interpolation of the eld values from the nearest node points, where the interpolation
is weighted by the local mass (density) (Zienkijevic, 1977). If we combine the consistent
and lumped mass methods by aweighted average, the required replacement
terms for homogeneous media (with constant ) become
and
! 2 v m;n ) ! 2 2
(1 , b)
b v m;n + ! (v m+1;n + v m,1;n + v m;n+1 + v m;n,1 ) ; (5.24)
4
(1 , b)
!u m;n ) ! b u m;n + ! (u m+1;n + u m,1;n + u m;n+1 + u m;n,1 ) ; (5.25)
4
128

where the coecient b, as with the combined rotated schemes, is chosen to minimize
the numerical errors (the equations (5.24) and (5.25) are the equivalent of the
equation (3.5) in the acoustic case). Here I have only used the values from the ve
point star, and I have ignored the values from the corners of the nine point star. As
I have shown in Chapter 3, the value of the third coecient (related to the corner
points of the 9 point star) is always close to zero and can be set to zero without
any visible eect on the nal result. At the same time this makes the minimisation
problem 2-D and helps avoid local minima.
The nal dierencing scheme for the 2-D, homogeneous, visco-elastic wave
equation is obtained by combining the nite dierence approximations for equations
(5.17) and (5.18), with equations (5.24) and (5.25). The complete scheme is given
in the appendix, as equations (A-1) and (A-2).
We now have a total scheme that i) minimizes numerical anisotropy (by an
appropriate choice of the weighting factor, a) and ii) minimizes overall numerical
dispersion (by an appropriate choice of the weighting factor b). All that remains is
to determine the optimal values of the two weighting parameters, a and b. These
parameters are determined by searching for values that provide a minimum of numerical
anisotropy and numerical dispersion over the range of expected values of
velocity. In the next section I describe the manner in which the optimal selection
of parameters a and b is made. It should be noted that the two coecients, a and
b, are not independent, and must determined simultaneously. Before proceeding to
the optimization scheme, I will comment briey on the scheme for the heterogeneous
wave equation.
5.2.4 Heterogeneous formulation
The approach used in the previous three sections for the homogeneous viscoelastic
wave equation can also be applied to the equivalent wave equation for heterogeneous
media, in which the elastic constants, and , and the density are
129

free to vary from one node point tothe next.
The partial dierential equations for visco-elastic wave propagation in a heterogeneous,
2-D media are:
! 2 u + @
@x
"
@u
@x + @v
@z
!
# "
+2 @u + @
@x @z
!#
@v
@x + @u =0 (5.26)
@z
and
! 2 v + @ @z
"
! # "
@u
@x + @v +2 @v + @
@z @z @x
!#
@v
@x + @u =0: (5.27)
@z
In an analogous manner to the approach used for homogeneous media, I
substitute u,v,x and z with u 0 ,v 0 ,x 0 and z 0 to obtain equations in a 45 o rotated
coordinate system, following which I apply similar manipulations to those used in
equations (5.7), (5.8), (5.9) and (5.10), to obtain a new, mixed system of partial
dierential equations in heterogeneous media:
( "
@ @u
! 2 u + a
@x @x + @v
#
@u +2 + @ @z @x @z
( "
(1 , a) @ @u
+ @v + @v , @u
2 @x 0 @x 0
@x 0
@z
"
0
@ @v
@z 0 @x 0
"
@ @u
@z 0 @x 0
+ @v
@x 0
+ @v
@z 0
@v
"
@
@x 0
@x 0
"
@v
@x + @u
@z
@u +2
@z 0 @x 0
, @u
@x 0
+ @u
@z 0
, @u
@z 0
+2
@v
@z 0
, @u
@x 0
+ @u
@z 0
#) +
+ @v
@x 0 # +
+ @v
@z 0 # ,
, @u
@z 0 # ,
+ @v
@z 0 #) =0 (5.28)
and
(
@
! 2 v + a
@z
(1 , a)
2
"
(
@
@x 0 "
"
@ @u
@z 0 @x 0
#
"
+ @ @v
@x
@u
@x + @v
+2 @v
@z @z
@x + @u
@z
@u
+ @v + @v , @u @u
+2
@x 0
@x 0
@z 0
@z 0 @x
"
0
@ @v
, @u + @u
@z 0 @x 0
@x 0
@z 0
+ @v + @v , @u @v
+2
@x 0
@z 0
@z 0 @z
"
0
@ @v
, @u + @u
@x 0 @x 0
@x 0
@z 0
#) +
+ @v
@x 0 # +
+ @v
@z 0 # +
, @u
@z 0 # +
+ @v
@z 0 #) =0: (5.29)
130

in which, asbefore, a is aweighting term used to control therelative importance of
the two coordinate systems used in this mixed equation.
Equations (5.28) and (5.29) must then be nite dierenced.
Once again,
I use the dierencing stars given by (Kelly et al., 1975) to produce the required
operators. As an illustration I present the four required dierence operators for the
rotated coordinate system as nite dierence stars, using the symbol to represent
either of the Lame parameters, or :
!
0 0 + +
@
@x 0 @
@x 0 1
0 ,(
2 + + 2 + , , ) 0
, , 0 0
; (5.30)
!
, + 0 0
@
@z 0 @
@z 0 1
0 ,(
2 + 2 , + , ) 0
+
; (5.31)
0 0 + ,
and
!
0 , + +
0
@
@x 0 @
@z 0 1
2 , 2 , 0 + +
0 , , , 0
!
0 ,, + 0
@
@z 0 @
@x 0 1
2 + 2 , 0 , +
0 , + , 0
; (5.32)
; (5.33)
where the parameters () at intermediate grid points are given by
= m
1
;n 1 : (5.34)
2 2
These four stars specify all required nite dierence operators for the rotated coordinate
system; the remaining operators for the original coordinate system are
unchanged from Pratt (1990a). The approach used for the lumped and consistent
mass terms is introduced in exactly the same manner as for the homogeneous case
131

(see equations 5.24 and 5.25), except that the density value must also be averaged
from the neighboring node points, along with the eld variables.
The nal system for modelling the 2-D, heterogeneous, visco-elastic wave
equation is thus fully specied, apart from the unspecied weighting parameters,
a, the relative amount of the original, unrotated second order scheme, and b, the
relative amount of the consistent mass term with respect to the lumped mass term.
These weighting parameters are obtained by returning to the homogeneous formulation,
and choosing values that provide minimum numerical errors.
5.3 Numerical errors and optimization
5.3.1 Determination of optimal coecients
As discussed in the previous section, in order to fully specify the new differencing
scheme, I now must determine values for the weighting coecients a in
equations (5.17), (5.18), (5.28) and (5.29), and b in equations (5.24) and (5.25). In
order to minimize the errors, I must be able to predict the numerical errors for a
particular choice of a and b. The numerical errors are are predicted in a standard
fashion by assuming a plane wave solution for the homogeneous scheme (given in the
appendix as equations (A-1) and (A-2)), and solving the resultant system for the
numerical compressional and shear wave velocities. The required analysis is given
fully in the appendix. The nal equations depend on a, b, , K and , where is
the Poisson ratio of the elastic medium, K is the wavenumber in grid point units
(i.e., K =1=G where G is the number of grid points per wavelength), and is the
propagation angle relative tothegrid axes.
The method applied for determining the coecients is as follows: I search for
a set of values for a and b, using a representative value of for the elastic medium,
such that a given mist function is minimized. The mist function is designed to
measure the aggregate mist of the error in the numerical velocities over a range of
132

possible values of K (governed by the range in true velocities in the medium), and
over a range of propagation angles, . Formally I minimize
where
and
F (a; b; ) =
Z :5
Z =4
0
0
max n F p (a; b; ;K;);F s (a; b; ;K;) o d dK (5.35)
F p (a; b; ;K;)=
1,bv p g
(a; b; ;R K; )
v pg
2
F s (a; b; ;K;)=
1,bv s g
(a; b; ;K;)
v sg
2
(5.36)
: (5.37)
In equations (5.36) and (5.37), K is the number of grid points per shear wavelength,
q
R = (0:5 , )=(1 , ) isthev s =v p ratio in the medium, v pg and v sg are the (true)
compressional and shear wave group velocities, and bv pg and bv sg are the numerical
group velocities, for which explicit expressions in terms of the variables (a; b; ;K;)
are given in the appendix.
For a given value of there are only 2 unknown parameters. It is therefore
possible to evaluate the function, F (a; b; ), for a reasonable range of values of a and
b, and plot this function as a surface. The optimal values can then be estimated,
and the procedure can be repeated on a tighter interval near the optimal point.
This was the procedure used to determine a and b for the examples that follow.
The coecients a and b will in general depend on the Poisson ratio used in the
model. If this is expected to vary widely, one could include an integral over possible
Poisson ratios in the mist function.
In Figure (5.2) the optimum values of the
parameters as a function of the Possion ratio are shown. While the optimal value of
the ratio between consistent and lumped mass matrix methods, b is relatively stable,
it is evident that for large Poisson ratios (i.e., for near uids) the weighting of the
unrotated scheme, a required for minimal numerical errors approaches zero. This
is consistent with the expectation that the ordinary second order scheme cannot
handle near uids (Stephen, 1983; Virieux, 1986a; Kerner, 1990). I shall return to
the uid case in a later section, when I show that the scheme predicts no numerical
133

1
1
0.8
0.8
a
0.6
0.4
b
0.6
0.4
0.2
0.2
0 0.1 0.2 0.3 0.4 0.5
σ
0 0.1 0.2 0.3 0.4 0.5
σ
Figure 5.2: Optimal values of coecients, a (the fraction of the ordinary second order
scheme) and b (the fraction of the consistent mass matrix), plotted as a function of
the Poisson's ratio, . The optimal value of coecient b is relatively insensitive to
the value of . The optimal value of coecient a decreases for high values of , and
becomes 0 for the uid case, in which case only the rotated scheme is used.
dispersion for shear waves in uids, a necessary condition for being able to model
liquid-solid interfaces.
5.3.2 Numerical dispersion
In this section I present some representative dispersion analyses for the combined
scheme presented in this Chapter. The analysis is generated using the choice
of weighting parameters a and b depicted in Figure 5.2, and the dispersion analysis
given in the Appendix A. Figures 5.3 and 5.4 depict the normalized numerical phase
and group velocities (for both compressional and for shear waves), using the standard
second order scheme (left column), and the new, optimally combined scheme
(right column). A value of 1.0 for the normalized velocity represents an error free
numerical result; in all gures this is achieved when K =1=G, the wavenumber in
grid point units, is zero.
From Figure 5.3 it is evident that the original second order scheme yielded
good, isotropic and undispersed results for the compressional wave phase velocities,
and very poor phase and group results for shear waves. The original scheme also
134

Numerical dispersion curves for σ=.33
Old scheme
Combined scheme
1.1
1.05
v Pph /v Pph
0.95
0.9
1.1
1.05
v Sph /v Sph
1.0
0.95
0.9
1.1
1.05
v Pgr /v Pgr
1.0
0.95
0.9
1.1
1.05
1.0
0.95
0.9
v Sgr /v Sgr
1.0
0.05 0.1 0.15 0.2 0.25
1/Gs
0.05 0.1 0.15 0.2 0.25
1/Gs
0.05 0.1 0.15 0.2 0.25
1/Gs
0.05 0.1 0.15 0.2 0.25
1/Gs
1.1
1.05
v Pph /v Pph
1.0
v Sph /v Sph
v Pgr /v Pgr
0.95
0.9
1.1
1.05
1.0
0.95
0.9
1.1
1.05
1.0
0.95
0.9
1.1
1.05
v Sgr /v Sgr
1.0
0.95
0.9
0.05 0.1 0.15 0.2 0.25
1/Gs
0.05 0.1 0.15 0.2 0.25
1/Gs
0.05 0.1 0.15 0.2 0.25
1/Gs
0.05 0.1 0.15 0.2 0.25
1/Gs
Legend: Propagation angle (degrees)
0 11.25 22.5 45
Figure 5.3: Numerical dispersion of the new scheme for a Poisson ratio = 0:33,
depicting normalized numerical velocity curves for compressional and shear phase velocities
(top tworows) and group velocities (bottom tworows). Results are presented
for the standard second order scheme (left column) and the new, combined scheme
(right column). The dispersion curves are plotted against the shear wavenumber in
grid point units, i.e., the reciprocal of the number of grid points per shear wavelength,
G s . See text for the meaning of the symbols used on the vertical axes.
135

Numerical dispersion curves for σ=.4
Old scheme
Combined scheme
1.1
1.1
1.05
1.05
v Pph /v Pph
1.0
v Pph /v Pph
1.0
0.95
0.95
0.9
0.05 0.1 0.15 0.2 0.25
1/Gs
0.9
0.05 0.1 0.15 0.2 0.25
1/Gs
1.1
1.1
1.05
1.05
v Sph /v Sph
1.0
0.95
0.9
0.05 0.1 0.15 0.2 0.25
1/Gs
v Sph /v Sph
1.0
0.95
0.9
0.05 0.1 0.15 0.2 0.25
1/Gs
1.1
1.05
1.1
1.05
v Pgr /v Pgr
1.0
0.95
v Pgr /v Pgr
1.0
0.95
0.9
0.05 0.1 0.15 0.2 0.25
1/Gs
0.9
0.05 0.1 0.15 0.2 0.25
1/Gs
1.1
1.05
v Sgr /v Sgr
1.0
0.95
0.9
1.1
1.05
v Sgr /v Sgr
1.0
0.95
0.9
0.05 0.1 0.15 0.2 0.25
0.05 0.1 0.15 0.2 0.25
1/Gs
1/Gs
Legend: Propagation angle (degrees)
0 11.25 22.5 45
Figure 5.4: Numerical dispersion for a Poisson ratio =0:4, depicting normalized
numerical velocity curves for compressional and shear phase velocities (top tworows)
and group velocities (bottom two rows). Results are presented for both the standard
second order scheme (left column) and the new, combined scheme (right column).
The dispersion curves are plotted against the shear wavenumber in grid point units,
i.e., the reciprocal of the number of grid points per shear wavelength, G s . See text
for the meaning of the symbols used on the vertical axes.
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yielded a nearly isotropic result for compressional wave phase and group velocities,
but contained errors of more than 8% at 4 grid points per wavelength (0.25 on
the horizontal axes) for the compressional wave group velocity. The new combined
scheme introduces a small amount of anisotropy into the compressional wave results,
and slightly decreases the group velocity dispersion for P waves, but yields a
much improved shear wave performance. Figure 5.4, in which the Poisson ratio has
been increased to 0:4, is similar to Figure 5.3, except that the shear wave dispersion
is much more severe for the standard second order scheme. Again, the combined
scheme introduces a small amount of anisotropy into the compressional wave velocities,
but decreases the overall velocity dispersion for compressional waves and yields
a dramatically improved shear wave performance. In contrast to the standard second
order scheme, the new scheme is thus able to cope well with a range of Poisson
ratios.
5.3.3 Modelling in uids
It is well known that standard second order schemes generate innite dispersion
for shear waves when used to simulate propagation in liquid layers (Bamberger
et al., 1980; Stephen, 1983). This problem has eectively been solved for
time domain modelling by Virieux (1986a) and others (Kerner, 1990), in which the
elasto-dynamic system, rather than the wave equation, is simulated on a staggered
numerical grid.
Let us consider the performance of the new scheme in uid layers. In Figure
5.2 I show that the optimal value of the parameter a (the fraction of the standard
scheme) approaches zero as the Poisson ratio approaches 0:5. This is a direct consequence
of the behaviour of the standard scheme at large Poisson ratios. Therefore,
in pure uids, I will have to use the rotated scheme only (a=0). I now show analytically
that the rotated scheme predicts the true shear wave behaviour in uids
(v s =0:0):
137

Ifollow closely the dispersion analysis given in the appendix. Equation (A-5)
predicts the normalized, numerical shear wave group velocity. This equation cannot
be used where the true shear velocity, v s is zero, as v s appears on the denominator
of equation (A-5). However, a similar expression for the non-normalized, numerical
shear velocity can be obtained:
bv Sp =
v p
2K
vu q
u
t 1 , 2 1 , 4 2 3
; (5.38)
2 3
where the coecients are the same as those dened for equation (A-5). Inserting
a =0into the coecients I nd that, for R = 0 (because v s = 0),
1 =[,1 + cos x
, cos z
+ cos x
cos z
] ;
2 =[,1,cos x
+ cos z
+ cos x
cos z
],
3 = b + (1,b)
2
(cos x
+ cos z
), 1 = 3 ( 1 + 2 ),
and 2 = 2 1 , sin 2 x
sin 2 z
where x = 2K cos , z = 2K sin and K =
=2 (the wavenumber in gridpoint units). Inserting these simplied coecients
back into equation(5.38) yields
bv Sp =
v p
2K
vu
u
q t( 1 + 2 ) , ( 1 + 2 ) 2 , 4 2
; (5.39)
2
Further algebraic manipulation reveals that 2 = 0, from which it is then obvious
that v s = 0 (for all values of K).
Thus the new, rotated scheme, used on its own gives the exact shear wave
group velocity (v s = 0) in a uid. An exact, constant, numerical phase velocity
for all wavenumbers implies that the numerical group velocity (bv g =@!=@) is also
exact.
It is interesting to note that the rotated scheme generates a dierencing
scheme that has certain similarities with the staggered grid dierencing schemes
used by Virieux, (1986a) and Kerner, (1990) to solve the uid layer problem.
The compressional wave group and phase dispersion in a uid for the new
scheme are depicted in Figure 5.5. These results show that the numerical dispersion
for compressional wave in uids can be signicant with the new scheme. However,
138

(a)
(b)
1.1
1.1
1.05
1.05
v Pph /v Pph 1.0
v Pgr /v Pgr 1.0
0.95
0.95
0.9
0.9
0.05 0.1 0.15 0.2 0.25
0.05 0.1 0.15 0.2 0.25
1/G P
1/G P
Legend: Propagation angle (degrees)
0 11.25 22.5 45
Figure 5.5: Compressional wave dispersion in uids for the new, rotated scheme.
In the uid case I use only the rotated scheme, with no component of the original,
unrotated scheme (a = 0). a) Normalized compressional phase velocities. b)
Normalized compressional group velocities.
errors of less than 5% can be achieved with 7-8 grid points per wave length.
5.3.4 Discussion
From Figures 5.3 to 5.5 it is evident that the new, combined scheme performs
better than the standard second order scheme for a wide range of Poisson ratios
(naturally, the optimized scheme can be no worse than a single scheme).
When
compared with the optimal frequency domain (nite element) scheme given by Marfurt
(1984a), the new combined scheme provides better accuracy in cases where is
greater than approximately 0.3. For values of less than 0:3 the combined scheme
gives comparable results to the optimal nite element scheme, with a slightly higher
numerical anisotropy.
By using a linear combination of two schemes I distorted the isotropic nature
of the original scheme for modelling compressional waves, and thereby gained
a higher accuracy for shear waves. For most Poisson ratios, the new rotated scheme
models shear waves better than compressional waves, while the original scheme gives
139

the opposite results. The fraction of each scheme (the a coecient) appears to control
the relative accuracy for compressional and shear waves, and thus acts as a
\tradeo factor". The other parameter (the b coecient) controls the overall dispersion
for a given linear combination of both the standard and rotated schemes. It
is therefore clear that there is room for customizing the scheme in certain situations:
By choosing critical data phases, and choosing alternative values of the parameters
a and b, itwould be possible to obtain a scheme which requires even less grid points
per wavelength and obtain the same, or better accuracy for the particular wave type.
If the model contains a range of Poisson ratios, then there may be inaccuracies
in regions in which the Poisson ratio diers largely from that used in the selection of
the parameters a and b. Fortunately, from Figure 5.2, the value of the parameters are
relatively stable over a range of Poisson ratios. Only at large values of Poisson ratios
is there an indication of a need to adjust these parameters { in heterogeneous models
it may then be necessary to adjust these locally. A local variation of coecients for
variable Poisson ratios was also suggested for the nite element method by Marfurt
(1984a). The eect of using variable coecients within the same model is not clear,
although initial numerical tests Ihaverun look promising.
One eect of the use of a rotated dierencing scheme is that any interface,
dipping or at, will be treated in the model as a \staircase", due to the fact that
at least one scheme is not aligned with the interface.
This leads to regular, low
intensity grid diractions on the interfaces. Such grid diractions are common for
dipping layers with standard schemes, and I do not consider the presence of these
diractions for at interfaces to be a serious drawback. Muir et al. (1992) and
Zeng and West (1996) pointed out a simple schemes based on eective media for
minimizing these eects.
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5.4 Elastic modelling example
Verication of the new modelling scheme has been carried out in several standard
models. Rather than showing these obvious results here, I now demonstrate
the new scheme using the class of complex medium for which the scheme was designed:
I use the scheme to study crosshole seismic data from a layered and faulted
sedimentary sequence, in which a high level of attenuation is known to exist. The
data come from the Imperial College test site at Whitchester in Northern England.
The site consists of cyclically layered, interbedded mudstones, sandstones and carbonates
(as described in more detail elsewhere (Pratt and Sams, 1996; Neep et al.,
1996)).
The crosshole data were acquired between two boreholes at the test site penetrating
the top 220 m of the sequence, and separated by 75m. The acquisition was
carried out using a clamped piezo-electric transmitter and hydrophone receivers.
The transmitter was driven with a pseudo-random binary signal at a central frequency
of 400Hz.
The source was positioned in the rst borehole (the left hand
side of the following gures), and recordings were made from each source position
at receivers positioned every 2m between 17m and 217m in the second borehole (on
the right hand side of the gures).
The only structural feature in the section is a small, steep, right-dipping normal
fault crossing Borehole 1 at 120m with a vertical displacement of approximately
10m. The largest velocity contrasts are provided by two high velocity carbonate layers
beds at depths of 145m and 170m, with thickness of 5m and 10m respectively.
Figure 5.6 is a grey scale representation of the P-wave velocity variations used
in a forward model of wave propagation at the site showing the location of the normal
fault. This model is the result of an earlier study in full waveeld inversion (Pratt
et al., 1995), using an acoustic inversion routine as explained in Chapter 4, after the
method described by Song et al (1994). The high velocity carbonates can be easily
141

identied; it is also possible to identify the location of the normal fault from the
truncations of these carbonates towards the left of the image. Figure 5.7(a) depicts
the observed seismic data from a representative common source gather, and Figure
5.7(a) depicts the result of acoustic forward modelling in the velocity model shown in
Figure 5.6. The acoustic modelling succeeds in reproducing the arrival times nearly
exactly, and in predicting much of the character of the waveform within the rst 5
ms of the rst arrival. There is, however, little correspondence between the predicted
and observed waveelds at late time. The modelling has failed to generate the large
amplitude, incoherent events observed 10 to 20 ms following the rst arrivals.
In order to study the remaining discrepancies between the observed data and
the predicted data, I built a fully visco-elastic model from the P wave velocities in
Figure 5.6 using the following assumptions: First, the S-wave velocities are assumed
to be everywhere 50% of the P-wave velocities (i.e., I assumed a Poisson ratio of
= 0:33). Next, since the rocks at the site are known to be highly attenuating
(Neep et al., 1996), I incorporated inelastic attenuation by assuming that the P and
S quality factors were each constant over frequencys, and homogeneous. I selected
a quality factor of Q p =50for the P waves and Q s =20for the S waves (Neep et
al., 1996). Appropriate complex valued Lame parameters for this elastic model were
separately computed at each frequency, after Muller (1983). Finally, I modelled the
source by using a horizontal point force introduced into the numerical mesh at the
source location.
It should be noted that the site exhibits signicant elastic anisotropy (as
reported by Pratt and Sams (1996)), with P-wave velocities 20% faster in the horizontal
direction than in the vertical direction. Although the modelling scheme has
not been extended to the anisotropic case (an extension to simple transverse isotropy
would be feasible but has not yet been carried out), I eected a simulation of the
anisotropy by compressing the horizontal distances in the model by 20%, thus creating
the kinematic equivalent of an elliptically anisotropic media with a vertical
142

Distance
from BH1 (m)
Depth (m)
0 15 30 45 60 75
130
140
150
160
170
180
190
200
210
Fault
km/s
4.4
4.2
4.0
3.8
3.6
3.4
3.2
3.0
2.8
2.6
Figure 5.6: P-wave velocity model for the Imperial College crosshole experiment.
The model was obtained using acoustic fullwave inversion (Pratt at al. 1995). Data
from the experiment, and modelled data for this velocity structure, are shown in
Figures 5.7 and 5.8 .
symmetry axis.
This is consistent with the manner in which the anisotropy was
simulated by Shipp and Pratt (1995), and most importantly, predicts the correct
traveltimes.
The results of visco-elastic modelling using the new scheme are shown on
Figures 5.8(a), and 5.8(b)). The modelling synthesizes the horizontal and vertical
components of displacement.
No direct comparison of these data with the borehole
pressure measured by the hydrophones in the eld can easily be made: The
relationship is complicated and highly dependent on a number of poorly controlled
variables (Peng et al., 1993). However, a qualitative comparison can be made: The
horizontal component shows rst arrival times and waveforms that are similar to
the acoustic modelling results, and some high amplitude arrivals at late times. The
vertical component shows high amplitude, incoherent arrivals similar to those observed
on the real data. There is no exact match between these late arrivals and the
143

Real data
Acoustic modelling results
0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1
Receiver depth (km)
0.0
0.0
0.01
0.01
0.02
0.03
0.04
0.05
Source
0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1
Receiver depth (km)
0.02
Time (s)
0.03
0.04
0.05
0.0
0.01
0.02
0.03
0.0
0.04
0.05
0.01
0.02
Time (s)
0.03
0.04
0.05
(a)
(b)
Figure 5.7: a) A representative common source gather from the crosshole data collected
at the Imperial College test site. The signal to noise ratio is high, and the
rst arrival waveforms are clear and coherent. At late times, incoherent, large amplitude
arrivals dominate. b) Predicted common source data using acoustic forward
modelling in the velocity structure shown in Figure 5.6. The rst arrival traveltimes
and waveforms match well with the observed data, but the large amplitude, late
arrivals are not predicted with the acoustic method.
144

Elastic modelling horizontal component
Elastic modelling vertical component
Source
0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1
Receiver depth (km)
0.0
0.0
0.01
0.01
0.02
0.03
0.04
0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1
Receiver depth (km)
0.02
0.05
Time (s)
0.03
0.04
0.05
0.0
0.0
0.01
0.02
0.03
0.01
0.04
0.05
0.02
Time (s)
0.03
0.04
0.05
(a)
(b)
Figure 5.8: Predicted common source data using the new visco-elastic modelling results.
a) Horizontal displacement component. b) Vertical displacement component.
The horizontal component shows rst arrival times and waveforms that are similar
to the acoustic modelling results, and some high amplitude arrivals at late times.
The vertical component shows high amplitude arrivals similar to those observed on
the real data.
145

observed data. However, even with the simple assumptions I have made in building
the visco-elastic model from the P-wave velocity model, I managed to create
synthetic visco-elastic data that look more like the data collected at the site than
the synthetic acoustic wave data. The late arrivals thus appear to be related to the
mode conversion of P wave energy into shear wave energy within the heterogeneous
model. One may speculate as to whether a better match in the times of the late
arrivals could be achieved by adjusting the Poisson ratios in the model, or even to
ask whether a formal visco-elastic inversion of the data from this experiment could
be attempted.
In comparison with the visco-acoustic modelling (see Chapter 3) for the viscoelastic
one needs twice as big grid (due to = :33). If the required memory for the
visco-acoustic case is n 2 log 2
n (see Chapter 2) the memory required in visco-elastic
case, for = :33, can be written as 4(2n) 2 log 2
(2n) 16n 2 log 2
n. The rst factor
4 comes from the fact that we need a 2 2 matrix to solve atwo component vector
at each point instead of a single value in the scalar visco-acoustic case. The factor
2, in 2n instead of n, comes from the double grid size. The calculation shows that
16 times more memory is required in the visco-elastic case, although some small
extra overhead in memory will e required.
The actual required memory in this
case was 250 MB for a grid size of 310 258 grid points (the linear system with
approximately 160,000 variables) as opposed to 15 MB in the visco-acoustic case
(on a 155 129 grid and a linear system with approximately 20,000 variables). If
the old second order scheme were used, without nested dissection, the required grid
size would be 1162 967, which would require 130 GB of RAM; the memory saving
for this case is thus about 99.9%. For the Whitchester model, including 2 sources
and 51 frequencies, the total CPU time for the visco-elastic scheme was about two
hours and forty minutes. The total CPU time for the visco-acoustic scheme is about
6 minutes. This increase in CPU time of 32 times can be calculated theoretically
146

using the equation (2.21) and assuming a double grid size ( = :33):
CPU elastic =CP U acoustic = k 4(2n)3
kn 3 = 4 (2) 3 =32 (5.40)
The factor 4 in the elastic CPU time comes from the fact that a 2 2 matrix is a
single element in the matrix (vector instead of scalar value).
5.5 Conclusion
In this Chapter I have shown that it is possible to dramatically improve on
standard second order nite dierence schemes for visco-elasticity without increasing
computational costs. It would appear that former limitations on second order
schemes were due to the shape of the dierencing operators; by reshaping these
operators one can use models with high values of Poisson's ratio in a manner not
previously possible with frequency domain schemes. This has been achieved by extending
the grid rotation technique proposed by Cole (1994) and Jo et al. (1996) to
the visco-elastic case. The technique would appear to be quite generally useful, and
worthy of testing in other applications of the nite dierence method. A substantial
increase in accuracy is achieved with little or no increase in computational costs.
I would expect signicant improvements in 3-D, due to possibility of combining
rotation in each Cartesian plane with the original scheme.
Ihave shown analytically the improvements in accuracy for homogeneous media,
and I have formally proven that the scheme predicts the correct shear wave behaviour
in uid layers. Using my numerical scheme I was able to successfully model
crosshole eld data from a highly heterogeneous sedimentary environment known
to be anisotropic and strongly attenuating. To do this I made several simplifying
assumptions (a constant Poisson's ratio, a homogeneous, constant Q attenuation,
a homogeneous, elliptical anisotropy, and simple, point force source mechanisms).
Nevertheless, I was able to generate a synthetic data set qualitatively consistent
with the eld data.
147

Chapter 6
Conclusions and further work
6.1 Conclusions
The primary objective of the research described in this thesis was to develop
and implement a sequence of improvements in numerical seismic modelling that
would allow ecient simulation of large scale, multi-source seismic surveys, and to
apply the resultant method to a number of test problems. A secondary objective
of the research was to use the resulting modelling code as the basis for a waveeld
inversion method, and to test the inversion method on a representative data set.
It was decided early on that the method of choice to meet these objectives was
the frequency domain nite dierence method. Although Marfurt (1984) pointed
out the potential of frequency domain nite dierences more than a decade ago,
little use has been made of his suggestion since, although an elementary version
of this approach has been used successfully for waveeld inversion for several years
(Pratt et al., 1995; Pratt et al., 1996; Song et al., 1995). The details of the modelling
method used in these studies were given by Pratt (1990); the method was a relatively
unsophisticated implementation of simple, second order approach and was not useful
for large problems.
The research proceeded by rst developing and implementing a nested dissection
method for solving the matrix equations in frequency domain nite dierences,
148

then developing and implementing a rotated operator approach for reducing the
number of grid points required for visco-acoustic modelling. The combination of
these two techniques led to signicant increases in numerical eciency. Once these
improvements were in place, the waveeld inversion software was updated to include
these and an extensive study of a real data set was carried out using the new methods.
This led to the conclusion that a visco-elastic approach was required. The nal
chapter of this thesis describes the development of the nested dissection and rotated
operator approaches to the visco-elastic forward modelling problem. This makes the
future development of a visco-elastic waveeld inversion procedure possible.
6.1.1 Matrix solvers
It is a primary conclusion of this project that in order to retain the potential
advantages of frequency domain seismic modelling (to eciently solve the multiple
source problem), one has to use a direct matrix solver. Although there it may be
possible to solve single source, monofrequency problems by using an iterative matrix
solver, the computational cost involved in solving realistic, multiple source problems
will inevitably, Ibelieve, involve the use of optimised direct matrix solvers.
Large computational savings can be achieved if appropriate care is taken with
the initial grid ordering. Nested dissection is an optimal solution to the grid ordering
problem. The memory requirements can be cut down from an n 3 requirement (for
sequential ordering) to an n 2 log 2
(n) requirement, where n is the number of grid
points in one direction in a square model.
If a realistic value of n is used (n
300), the savings in memory requirements can be over 70% just by using the nested
dissection instead of the ordinary grid ordering.
Regardless of the method used to solve the matrix equations, the size of the
dierence operator has to be kept as small as possible, for frequency domain nite
dierence methods. This is because, when nested dissection is used, the increases
in memory requirements due to the larger dierence operators are unlikely to be
149

compensated for by the accuracy gained by using a higher order of dierence operator.
Ihave shown that the number of grid points per wavelength would have tobe
decreased by more than 50% in order to justify the use of the higher order dierence
operators if the grid is ordered by the nested dissection. Although in some cases this
may beachieved, for the level of accuracy one would normally require this actually
would involve sampling at less than the Nyquist criterion.
6.1.2 Rotated nite dierence operators
I have shown that the introduction of rotated nite dierence operators and
lumped mass terms can increase accuracy without any signicant increase in computing
costs for both acoustic and elastic methods. This is a technique that has
very general potential application, to a wide range of nite dierence methods. For
the acoustic scheme, this step, in conjunction with the nested dissection method for
grid ordering, has reduced the memory requirements by 96:4%, for a given velocity
model of a realistic size (30 30 wavelengths).
6.1.3 Visco-elastic forward modelling
By developing the rotated operator and lumped mass methods and applying
them to the visco-elastic problem, I was able to achieve even greater increases in
computational eciency. Due to a reduction from 15 grid points to 4 grid points per
wavelength, and the use of the nested dissection implementation, the full memory
saving (for the elastic scheme) is 99%, for a given velocity model of a realistic size
(50 50 wavelengths). I anticipate that this development, implemented on the
appropriate hardware, will allow the routine production of time domain 2D full
multiple source pre-stack data for realistic, 2D data problems in the near future.
Currently we can solve a visco-elastic model with 250 50 S-wavelengths using a
machine with 512 MB of RAM, and produce results for hundreds of sources within
three to four days.
150

The visco-elasticscheme Ihave presented istheoretically capableofmodelling
the eect of uid layers on seismic wave propagation. I was not jet able to solve
all the aspects of the problem, which require that the boundary conditions and the
source denition be properly handled for uid layers, however it is anticipated that
these problems can also be overcome.
6.1.4 Waveeld inversion
Having developed the necessary forward modelling code, the routines were
used to improve the eciency of an existing seismic waveeld inversion method
(Song, 1994). This allowed a large tomographic data set (from the Grimsel Rock
Laboratory) to be eectively modelled and inverted within a fraction of the time
required using the original code, and using a fraction of the memory requirements
(5% of the originally required memory).
The application of waveeld inversion to the data from the Grimsel Rock Laboratory
showed clearly the advantages of waveeld inversion over simple, traveltime
methods. These advantages were rst conrmed on a synthetic data set (generated
by a third party), demonstrating the potential resolution advantages of the waveeld
approach. In order to invert the eld data, a large number of tests with variable
smoothing constraints and variable levels of anisotropy were run. It was in the ecient
computation of these test results that the fast forward modelling routines were
particularly useful. Such tests would have been too expensive without the improvements
introduced by a nested dissection and the rotated nite dierence operators.
The results eectively prove the utility of the frequency domain approach as a basis
for the production waveeld inversion of multiple source seismic transmission data.
The data example in Chapter 4 showed the manner in which the inversion
parameters, specically the smoothing constraints and the anisotropy level, can be
estimated from the results of a set of parameter tests, using the level of data mist
and the solution roughness as a guide in the selection of the parameters (after Pratt
151

(1992)).
I have also shown how sensitive the nal images can be to even a low level
of seismic anisotropy. I therefore conclude that the inclusion of anisotropy in the
waveeld inversion may beextremely important step to take in the near future.
6.2 Future work
There are clear avenues for future research into the techniques that have
been developed in this thesis. These topics can be divided into two main topics:
i) Developments in the modelling methods and ii) developments in the waveeld
inversion techniques.
6.2.1 Developments in seismic modelling
Simple improvements on the existing codes
There are a number of possible simple improvements in the existing modelling
codes that will improve the modelling speed. Currently we require that the
sources and the receivers be located exactly at grid point locations. Instead we can
interpolate the waveeld in order to nd the receiver responses at intermediate grid
positions. The source can be described over a small region, allowing the eective
location of the source to also be interpolated to intermediate grid points. To do this
we may use the formulation suggested by Alterman and Aboudi (1970). This will
enable us to progressively increase the grid size with the increase in frequency and
reduce the computational time even further.
A second important improvement will be to optimise the generation of the
time domain output in the codes. The current codes update the time domain output
trace by trace after each forward modelling step is nished. Although this does not
initially seem to be a time consuming task it can represent a bottleneck in the
computations. For example if we generate 3 GB of synthetic seismic data, we may
152

need to run 200 frequencies during the modelling. This implies that we have to read
and write 1200 GB of data during the computation. The maximal disk I/O speed on
fast wide SCSI II disks is 20 MB/s. Thus 17 hours would be wasted on unnecessary
disk I/O operations. We should just save the required frequency domain data at
the receiver positions after each frequency/source step and perform the inverse FFT
at the end of the modelling run. This would signicantly improve performance if a
large amount of time domain data is required as output. There is a demand for a
modelling code which can generate realistic 2D (or 2.5D) pre-stack data sets. The
code is well suited for simulating full 2D eld experiments for a variety of purposes,
such as processing and acquisition testing.
Currently, the main problem with the elastic scheme is to redene the source
description mechanism, which eectively prevents us from positioning sources within
uid layers.
The solution to this problem is still under investigation. The other
necessary improvement in the elastic code is to improve on the current absorbing
boundary conditions. Currently we use the one way wave equation (Clayton and
Enquist, 1985; Pratt, 1990b) which cannot cope with high values of Poisson ratio.
One easy way to improve this is to use sponge boundary conditions (Cerjan et al.,
1985; Shin, 1995). However this is not an ideal solution. Sponge boundary conditions
require an increase in the model size to accommodate the absorbing boundary. As we
have seen in the case of frequency domain modelling, the main problem is to reduce
the model size as much as possible so this increase will not be welcome. There is
a possibility that the rotated nite dierence based approach could be extended to
the boundary conditions. This may improve the absorbing boundary, since schemes
based on the rotated operators are more stable for the high Poisson ratios. A linear
combination of the absorbing boundary conditions based on rotated operators and
ordinary operators (as we use for the full wave equation) may reduce reections from
the edges of the model.
153

Extensions to more complex cases
The extensions of the rotated nite dierence frequency domain techniques
to anisotropic media is the next step to be taken. We have seen the eect of the
low level anisotropy on the waveeld images in Chapter 4.
In order to improve
the quality of the synthetics and the waveeld images we will have to simulate
anisotropy. Extension to TI anisotropic case (for example like Tsingas et al. (1990))
can be relatively easily implemented, but the required accuracy will depend on the
nature and the level of the anisotropy. In order to simulate a low level of anisotropy
we will have to take great care of numerical accuracy and numerical anisotropy.
Higher order nite dierence operators may in fact perform better in this case, as
we will require extremely low numerical anisotropy and high accuracy; I do expect
that fourth order in space will be sucient. If we use fourth order operators, we
will be able to use at least four second order and two fourth order schemes in a
combined operator. There is a possibility that we may be able to dene additional
second order operators.
With more degrees of freedom in search of the optimal
coecients, one could hope to nd the scheme which will need not more than ve
grid points per shortest wavelength. This accuracy would be sucient to enable us
to run realistic 2D exploration models on existing top-range workstations.
Full 3D, production anisotropic modelling is still beyond us. The formulation
of a frequency domain 3D scheme is however straightforward, and I would expect
to be able to run small 3D examples (of the order of tens of wavelengths in all
directions) within two to three years. However we will have to wait for about ve
years from then to model full, realistic 3D surveys (using the acoustic wave equation
to begin with). These predictions assume that the amount of available memory on
workstations eectively doubles every year or two (as it has for last fteen years).
Accuracy in 3D modelling should not be a problem since we can utilise at least
four rotated coordinate systems without increasing the nite dierence operator
154

size. With such a number of possible second order schemes one would expect to
achieve high accuracy. In the meantime I would expect that the implementation
of the rotated nite dierence techniques to 3D time domain seismic modelling will
produce a computationally inexpensive solution (in comparison with the existing
schemes). Low order, high accuracy operators will enable the use of coarse grids
with large time steps required while the CPU time per grid point/time step will
be low. If the time domain nite dierence computation is performed only in the
regions close to the wave fronts the CPU time can be further reduced (this is similar
to the reduced time idea in frequency domain).
In this way we may be able to
improve the speed for 3D modelling. The small spatial extent of nite dierence
operators will enable easy utilisation of parallel computer architectures.
6.2.2 Developments in waveeld inversion
In Chapter 5 I have presented an ecient visco-elastic modelling technique.
The next step will be to implement aninversion algorithm which will use it. In principle
the existing inversion code can be extended to the elastic case. The potential
benets are improved imaging and the recovery of additional elastic parameters. We
mayeven be able to obtain high resolution images of the Poisson ratio, an interesting
parameter for the oil industry. The problem in elastic inversion will be to simulate
(and invert) the correct source signature together with the source mechanism. In
the acoustic case the only available source mechanism is a P-wave source, with a
circular radiation pattern. In the elastic case the waveforms can vary dramatically
as a result of the source type used. We haveto adjust the source amplitudes of the
source generated P and Swaves, and we may haveto use complex synthetic source
mechanisms to reproduce the observed far eld source behaviour. This may be the
main secret of a successful elastic inversion of a eld data. In the elastic waveeld
inversion case the correct 3D source behaviour may bemore important than in the
acoustic case so the extension of the elastic scheme to 2.5D may be required.
155

Although waveeld inversions have been used for quite some time, little is
known about the appropriate data processing sequences. The data processing required
for waveeld inversion is dierent from the processing required for conventional
purposes. Any processing step which may inuence the waveforms (even a
simple bandpass lter) may eect on the nal result. The processing example shown
in Chapter 4 may not work on other datasets. We have had success with rst arrival
windowing on many eld data sets. This is due to the relatively simple acoustic assumption
used in the inversion procedure, and to the fact that the main diraction
information is contained in the rst arrival. The longer the time window the more
likely it is that important S-wave phases and conversions may be included in the
data; windowing eectively excluded non P-wave events. When we extend the inversion
algorithm to the elastic case the images will be improved as will resolution but
we will not wish to use restrictive windowing as a pre-processing approach. Elastic
inversion may cope better with complex waveforms, but other signal generated noise
(for example tube waves) can generate the undesired image artefacts. The use of a
longer time window may imply the use more frequencies in the inversion (to improve
frequency domain sampling) but there is a possibility of more local minima.
We will have to nd the appropriate processing which will remove such signal
generated noise from the data, without adversely aecting the waveforms. Additional
problems can be expected if a recorded amplitudes are aected by eg coupling
problems (as in the example in Chapter 4).
There is an outstanding question of appropriate data weighting. Ihave shown
in Chapter 4 that a distortion of the recorded signal amplitudes can additionally
help with convergence and the resolution of the images.
It remains to nd an
appropriate way of working with the data amplitudes in productive way.
In the
transmission surveys we have had usually only small amplitude ranges in the data
(with the exception of the example from Chapter 4). If the technique evolves into
one which will also work on reection data sets, the main data amplitudes will be
156

in the direct arrivals. We are usually very interested in the reections from the oil
reservoirs, which are much weaker. The deeper the reector from which the data
comes from the weaker is the signal going to be. Thus the dierence between the
modelled signal and the eld signal will be small in comparison with the direct
arrival. If we do not take the amplitude decay with depth into account the resulting
image will be dominated by the information from the part of the signal with the
highest amplitude. To prevent this we will have to think the way of scaling the
gradient vector with the depth in order to enhance the information from the deep
weak arrivals. One way will be to use the reciprocal of the waveeld to multiply the
gradient vector so we can enhance the deep signal.
Although we need large grids to model the data eectively in order to prevent
numerical artefacts in the inversion procedure, we cannot resolve the model at very
ne scales (below seismic resolution). Inverting for all parameters in the model is
not necessary nor desirable, as it involves higher computational costs and additional
potential convergence to local minima. Alternatively, we can use a more sparsely
varying (inversion) model parameters at the level of the seismic resolution (or even
more sparsely initially to prevent convergence to a local minima). The idea comes
from Williamson (1990) and Bunks et al. (1995). The ideas of using certain parts of
the signal spectrum at the time are inherent part of the frequency domain waveeld
inversion; the only problem is to use the correct model parametrisation at each
frequency.
This approach will reduce the computational costs of the frequency
domain waveeld inversion even further and reduce the possibility of convergence to
the local minimum of the mist function.
Although the methods described in the thesis are all 2D, the results show how
little information we still use from the data with conventional techniques, and the
improvements we can expect once we start using more information from the data
in the industry. As exact reservoir positioning becomes more and more important,
more accurate (but expensive) techniques may be considered, even in 2D, in order
157

to nd more dicult targets.
An increase in the data quantity will not help if
the data processing is too simplied. With extensions to the more complex cases
(eg elastic, anisotropic) we may expect to produce detailed, and quantitative depth
images which may be related to the site geology and which will help predicting
parameters of great importance, such as the Poisson ratio, the fracture orientation,
etc.
Hopefully the improvements in acquisition such as the development of new
sources capable of generating higher frequency data, the development of see bottom
cables (recording shear waves), and recording longer osets will increase the data
resolution and quality and will produce data which are more suited to the inversion
techniques.
The application of the methods I have shown are not limited to the examples
covered in this thesis. The implementation of the rotated nite dierence operators
for the time domain based nite dierence methods may prove to be the easiest way
to reduce the cost of 3D seismic modelling. Historically, geophysicist have learned
from medical science how to perform tomography. Similarly, awider audience may
discover applications of the modelling and inversion techniques described in this
thesis to similar problems in other disciplines.
158

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169

Appendix A
Dispersion analysis for visco-elastic modelling
If we take the second order nite dierence equations generated from equations
(5.17) and (5.18), and use the combined consistent and lumped mass formulations
for the density weighted term in the visco-elastic wave equation (5.24 and
5.25), We obtain the following scheme for homogeneous media:
"
(1 , b)
#
! 2 b u + u + + u , + u + + u , +
4
"( +2) u+ ,2u+u ,
a
(1 , a) 1 2
"
( +2)
"
! 2 b v +
a
(1 , a) 1 2
"
"
#
+ u +,2u+u ,
+(+) v+ +,v ,+v + ,,v , , +
+
2 2 4 2
!
u + +,2u+u , ,
, u+ , u , , u + + u ,
+ u+ , , 2u + u , +
+
2 2 2 2 2 2
!
u+ +
, 2u + u , ,
+ u+ , u , , u + + u ,
+ u+ , , 2u + u , +
+
2 2 2 2 2
!#
2
v + +
, v , + ( + )
+ v, , , v , +
=0; (A-1)
2 2
(1 , b)
4
v + + v , + v + + v ,
# +
#
( +2) v +,2v+v ,
+ v+ ,2v+v ,
+(+) u+ +,u + , +u, ,,u , +
+
2 2 4 2 !
v +
( +2)
+,2v+v ,
, + v+ , v , , v + + v ,
+ v+ , , 2v + v , +
+
2 2 2 2 2 2
( + )
!
v+ +
, 2v + v ,
, , v+ , v , , v + + v ,
+ v+ , , 2v + v , +
+
2 2 2 2 2 2
!#
u + +
, u + , + u, , , u , +
=0; (A-2)
2 2
170

where isthegrid pointinterval, u = u m;n , u + = u m+1;n , u = u m,1;n , u + = u m;n+1 ,
u , = u m:n,1 , u + + = u m+1;n+1, u , , = u m,1;n,1, u + , = u m+1;n,1, u , + = u m,1;n+1 and the
equivalent for v +;,
+;,.
By substituting a vector plane wave solution
0 1 0
B
@ u C
A = B
v
@ U V
1
C
A e,i r ;
(A-3)
where = ( x ; z ) is the wave vector and r = (x; z) is the position vector, into
equations (A-1) and (A-2), one obtains a homogeneous linear system of two equations
with two unknowns (U and V ). The determinant of this homogeneous system
must equal zero, leading to a quadratic equation in ! in terms of = jj. The
two solutions of this determinant represent the numerical compressional and shear
wave modes. By using the relations for the group velocity, v g = !
and for the phase
velocity, v p = @!
@
I obtain the numerical group and phase velocities, bv Pp, bv Pg , bv Sp
and bv Sg . Finally, normalized numerical velocities are obtained by dividing by the
exact values.
The nal expressions depend on K = =2 (the wavenumber in
gridpoint units, i.e., the inverse of G, the number of gridpoints per wavelength),
(the propagation angle), R (the v s =v p ratio in the homogeneous medium), and a and
b (the weighting factors of the rotated and lumped mass schemes):
vu q
bv Pp
= 1 u
t 1 + 2 1 , 4 2 3
; (A-4)
v Pp 2K 2 3
bv Sp
v Sp
=
1
R 2K
bv Pg
v Pg
= 1
2
bv Sg
v Sg
= 1
R 2
vu
u
t 1 ,
vu
u
@ t 1 +
@K
vu
u
@ t 1 ,
@K
q
2 1 , 4 2 3
; (A-5)
2 3
q
2 1 , 4 2 3
; (A-6)
2 3
q
2 1 , 4 2 3
2 3
; (A-7)
where 1 = a [,2+2cos x
]+(1,a)[,1 + cos x
, cos z
+ cos x
cos z
];
2 = a [,2+2cos z
]+(1,a)[,1,cos x
+ cos z
+ cos x
cos z
],
3 = b + (1,b)
2
(cos x
+ cos z
), 1 = 3 ( 1 + 2 )(1 + R 2 ),
171

and 2 = ( 1 R 2 + 2 )( 1 + 2 R 2 ) , (R 4 , 2R 2 +1)sin 2 x
sin 2 z
. Inthe computation
of these coecients, x = cos =2K cos and z = sin =2K sin
are the wavevector components in grid point units. The v s =v p ratio, R is related to
the Poisson ratio, by
R 2 =
+2 = 0:5,
1,
(A-8)
172