Sirgue, Laurent, 2003. Inversion de la forme d'onde dans le ...

Thèse de Doctorat de l’Université Paris XI
présentée par
Laurent Sirgue
pour obtenir le titre de
Docteur de l’Université PARIS XI
Spécialité: Sciences de la Terre
Inversion de la forme d’onde dans le
domaine fréquentiel de données sismiques
grands offsets
soutenue le 3 juillet 2003, devant le jury composé de:
Patrick Lailly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examinateur
Gilles Lambaré . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Rapporteur
Raùl Madariaga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directeur de thèse
Gerhard Pratt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Co-directeur
Jean Virieux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapporteur
Hermann Zeyen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Examinateur
Thèse préparée aux Laboratoires de Géologie de
l’Ecole Normale Supérieure de Paris et de
Queen’s University, Canada

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c○
Copyright 2003, Laurent SIRGUE

Abstract
The standard imaging approach in exploration seismology relies on a decomposition by spatial
scales: the determination of the low wavenumbers of the velocity field is followed by the reconstruction
of the high wavenumbers. In this standard approach, intermediate spatial wavenumbers
are ignored, partly because they are known to have little effect on the data for standard
near-normal incidence geometries. However, for models presenting a complex structure, the
recovery of the high wavenumbers may be significantly improved by the determination of intermediate
wavenumbers. These can potentially be recovered by local, non-linear waveform
inversion of wide-angle data. However, waveform inversion schemes based on the computation
of the gradient of the misfit function are limited by the non-linearity of the inverse problem,
which is in turn governed by the minimum frequency in the data and the starting velocity model.
It is critical to invert from the lowest to the highest frequencies of the seismic data. For very
low temporal frequencies (< 7 Hz), the problem is reasonably linear so that waveform inversion
may be applied using a starting model obtained from traveltime tomography. The frequency
domain is particularly advantageous as the inversion from the low to the high frequencies is
then very efficient.
It is possible to discretise the frequencies with a much larger sampling interval than that
dictated by the sampling theorem and still obtain an imaging result that does not suffer from
aliasing (wrap-around) in the depth domain. The number of input frequencies can be reduced
when a range of offsets is available; this creates a redundancy of information in the wavenumber
coverage of the target. In order to optimise the use of this information, I define a new discretisation
strategy that depends on the maximum effective offset present in the surface seismic survey.
This strategy is defined by deriving an analytical prediction of the wavenumber coverage of a
1-D target, for a single frequency and for a given acquisition geometry. This analytic expression
is obtained using the results of diffraction tomography, and is adapted to formulate an expression
for the gradient image in waveform inversion. The main idea of the selection strategy is:
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the larger the range of offsets, the fewer frequencies are required. Validation tests are carried
out on a 1-D velocity model that show the efficiency of frequency domain when a very limited
number of frequencies are adequately chosen. In these tests, the frequency domain performs
as well as time domain inversion at a much smaller computation cost. This strategy remains
efficient in 2D structures. This is demonstrated by the very good results that are obtained in the
2D Marmousi model from a wide-angle seismic acquisition survey, using only 3 frequencies.
Real seismic data do not contain very low frequencies and waveform inversion at higher frequencies
are likely to fail due to convergence into a local minimum. Preconditioning techniques
must hence be applied in order to enhance the efficacy of waveform inversion starting from realistic
frequencies. Because the high wavenumbers dominate the gradient image, the latter must
be smoothed to insure the proper reconstruction of lower wavenumbers. The determination
of the low wavenumbers is delicate as these correspond to the most non-linear components of
the model. These non-linearities may be mitigated by applying adequate preconditioning on the
data residuals that focus on the inversion of the early arrivals. A numerical test carried out on an
extended version of the 2D Marmousi model, in which a dense wide-angle survey is modelled,
shows that the proposed preconditioning strategy significantly improves the result of waveform
inversion. This test nevertheless also shows, that the starting model remains an important aspect
of waveform inversion.
The potential of waveform inversion in solving a given imaging problem may be evaluated
by defining the requirements of the starting model that ensure the success of the inversion. To
this end, I develop a linearity study that analyses the evolution of the misfit function with respect
to increasing degree of smoothness of the true model.
Such study is carried out on a 1D velocity model representing the sub-basalt imaging problem.
The analysis leads to the conclusion that waveform inversion may be used for the recovery
of the intra-basalt velocities, although the determination of the overburden sediments is more
difficult. Due to the presence of the basalt layer, only a migration-like image (i.e., the high
wavenumbers) can be obtained for the sub-basalt sediments. This study also emphasises the
importance of the low frequencies, as the requirements of the starting model are much more
demanding for higher frequencies.

Résumé de thèse
L’approche standard en imagerie sismique repose sur une décomposition par échelle du modèle
de vitesse : la détermination des bas nombres d’ondes est suivie par une reconstruction des
hauts nombres d’ondes. Dans une telle approche, les nombres d’ondes intermédiaires sont
ignorés, en partie car ils ont peu d’effets sur les données sismiques classiques courts offsets.
Cependant, pour des modèles présentant une structure complexe, la détermination des hauts
nombres d’ondes peut être améliorée de manière significative par l’apport des nombres d’ondes
intermédiaires. Ces derniers peuvent être déterminés par l’inversion non-linéaire de la forme
d’onde de données sismiques grands angles qui est, par ailleurs, limitée par la non-linéarité du
problème inverse. La non-linéarité est gouvernée par la fréquence minimum dans les données
et le modèle de vitesse initial.
Pour les trés basses fréquences, inférieures à 7 Hz, le problème est raisonnablement linéaire
pour appliquer l’inversion de la forme d’onde à partir d’un modèle de départ déterminé par inversion
tomographique des temps de trajets. Il est donc très important d’inverser des fréquences
les plus basses présentes dans les données. Le domaine fréquentiel est alors très efficace pour
inverser des basses vers les hautes fréquences.
De plus, il est possible de discrétiser les fréquences avec un pas d’échantillonnage plus grand
que celui dicté par le théorème d’échantillonnage tout en conservant un résultat en imagerie qui
ne contient pas d’aliasing (recouvrement) en profondeur. Le nombre de fréquences peut être
réduit lorsqu’une gamme d’offset est disponible, ce qui crée une redondance d’information de
la couverture spectral de l’objet à reconstruire. Afin d’optimiser l’utilisation de cette information,
une stratégie de discrétisation des fréquences est définit qui dépend de l’offset maximum
effectif présent dans l’acquisition sismique. Cette stratégie est définie en dérivant une formule
analytique de la couverture spectrale d’un objet uni-dimensionnel, pour une fréquence et une
acquisition données. Cette expression analytique est obtenue à partir des résultats de la tomographie
de diffraction qui utilise l’approximation d’onde plane en milieu homogène. Cette
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formulation est adaptée afin d’être applicable au vecteur gradient de l’inversion de la forme
d’onde. L’idée principale de la stratégie de sélection est la suivante : plus la gamme d’offset est
grande, moins de fréquences sont nécessaires. La méthode proposée utilise l’effet bien connu
d’étirement de l’ondelette (stretch) qui apparait lors d’une correction NMO ou en migration. Du
fait des similarités entre le vecteur gradient et la première itération et l’opérateur de migration,
cette stratégie peut facilement être implémentée en migration.
Des tests de validations sont effectués sur un modèle de vitesse 1-D et montrent l’efficacité
du domaine fréquentiel quand un nombre limité de fréquences est choisi de manière adéquate.
Lors de ces tests, le domaine fréquentiel donne un résultat comparable au domaine temps à
un coût de calcul informatique bien inférieur. Cette stratégie reste efficace pour les modèles
2D : de trés bons résultats sont obtenus pour le modèle Marmousi 2-D à partir de données
sismiques grands angles de type OBC, en utilisant uniquement 3 fréquences ( la fréquence de
départ utilisée est 5 Hz).
Ces résultats montrent l’intérêt du domaine fréquentiel car en inversion de la forme d’onde,
comme en migration, les coûts de calculs informatiques sont directement proportionnels au
nombre de fréquences utilisées.
Les données sismiques réelles ne contiennent malheureusement pas de très basses fréquences.
Des techniques de pré conditionnement doivent alors être appliquées afin d’améliorer l’efficacité
de l’inversion à partir de fréquences réalistes (~ 7 Hz).
Une décomposition en valeurs singulières de la matrice des dérivées de Fréchet pour une
seule fréquence montre que les hauts nombres d’ondes correspondent aux vecteurs propres associés
aux grandes valeurs propres. Le vecteur gradient est donc dominé par les hauts nombres
d’ondes et l’importance de ces derniers doit donc être diminuée. Ceci peut être réalisé en lissant
le vecteur gradient afin d’assurer une bonne reconstruction des bas nombres d’ondes. La détermination
des bas nombres d’ondes reste délicate car ils correspondent aux composantes du modèle
les plus non-linéaires par rapport aux données. La linéarité peut cependant être améliorée
en appliquant un pré conditionnement des résidus des données qui accentu l’importance de
l’information contenue dans les premières arrivées. Cette approche est utile car les premières
arrivées correspondent à la composante des données la plus linéaire.
Une inversion à 7 Hz est effectuée sur une version étendue du modèle Marmousi 2D, dans
lequel une acquisition grand angle est modélisée. Cette expérience montre que les méthodes de
pré conditionnement améliorent considérablement le résultat de l’inversion de la forme d’onde.
Ces tests montrent également que le modèle de vitesse initial reste un aspect fondamental de

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l’inversion de la forme d’onde.
Le modèle de départ est un aspect fondamental de l’inversion de la forme d’onde. Le potentiel
d’utilisation de l’inversion de la forme d’onde pour un problème d’imagerie donnée peut
être évalué en définissant les critères du modèle de départ qui assurent le succés de l’inversion.
A cette fin, une étude de linéarité est effectuée qui repose sur l’analyse de l’évolution de la
fonction coût par rapport au degré de lissage du modèle vrai.
Une telle étude est appliquée à un modèle de vitesse 1-D représentant le problème de
l’imagerie sous basalte. L’analyse de linéarité montre que l’inversion de la forme d’onde peut
être utilisée pour déterminer les vitesses de la couche de basalte bien que la détermination des
sédiments supérieurs au basalte soit plus difficile. Du fait de la présence de la couche de basalte,
seule une image de type migration (hauts nombres d’ondes) peut être obtenue pour les sédiments
inférieurs au basalte. En effet, la présence du basalte empêche l’illumination du sous basalte
par des données grands angles. Cette étude met également l’accent sur l’importance des basses
fréquences compte tenu du fait que les critères du modèle de départ sont plus restrictifs pour les
hautes fréquences.

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Ackowledgments
First and foremost, I would like to thank Gerhard Pratt for his assistance throughout the thesis
project. I am most grateful to him for welcoming me into his laboratory, here at Queen’s
University, Kingston, Canada, where most of the research for the project was conducted. His
advise and insight on the topic of waveform inversion was invaluable and greatly appreciated.
Also, I want to offer him my sincerest thanks you for accepting the burden of correcting the
spelling and grammar, as well as commenting on the manuscript.
Secondly, I am most thankful to Georges Pascal who welcomed me as a Ph.D. student into
the laboratoire de géologie de l’Ecole Normale Supérieure de Paris. I wish him all the best for
the future.
I would also like to thank the members of the jury for agreeing to review my work, especially
Raùl Madariaga for assuming the task of thesis advisor for my defense.
In addition, I would like to offer a special thank you to François Audebert and Philippe Herrmann
from CGG for funding my project through the CIFRE program as well as for providing
me with the means to carry out my research in Canada. Likewise, I am most grateful to Paul
Williamson for my 4 months stay at the Geosciences Research Center of Total in London.
These acknowledgments would not be complete without thanking my friends who throughout
my PhD, made my life in Kingston so enjoyable and memorable. Thanks to Stéphane
and Doug for their friendship and entertaining coffee breaks, to Graham for our technical discussions
and his kindness. Many thanks to ZZ Rob and Alfredo for being such easy going
housemates. To Marc, thanks for all the fun times we had. Thanks to Nadine for her patience
and her generosity.
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Contents
1 Introduction 1
1.1 Determination of the macro-model . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Determination of the high wavenumbers by migration . . . . . . . . . . . . . . 4
1.3 Quantitative migration and linearised inversion . . . . . . . . . . . . . . . . . 5
1.4 Migration/Inversion and non-linearity . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Non-linear waveform inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Non-linear inversion of wide-angle data . . . . . . . . . . . . . . . . . . . . . 8
1.7 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Summary of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Inverse theory 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Inversion of linear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Singular value decomposition . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Local methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Inversion of non-linear problems . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.1 Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Linearised inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.3 Non-linear gradient method . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.4 Importance of the starting model . . . . . . . . . . . . . . . . . . . . . 37
2.3.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Frequency domain waveform inversion . . . . . . . . . . . . . . . . . . . . . 43
2.4.1 The forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.2 The inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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CONTENTS
3 A strategy for selecting temporal frequencies 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Gradient vector and wavenumber illumination . . . . . . . . . . . . . . . . . . 55
3.2.1 Analysis of the gradient within the Born approximation . . . . . . . . . 56
3.3 The 1-D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.1 Wavenumber illumination for the 1-D case . . . . . . . . . . . . . . . 60
3.4 Strategy for choosing frequencies for 1-D imaging . . . . . . . . . . . . . . . 62
3.5 Numerical test I: 1-D models . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5.1 The synthetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.2 Validation of the plane wave predictions . . . . . . . . . . . . . . . . . 65
3.5.3 A remark on “image stretch” . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.4 1-D waveform inversion . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5.5 Sensitivity to noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Numerical test II: 2-D models . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.1 Marmousi velocity model . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.2 Determination of the starting model by traveltime inversion . . . . . . 77
3.6.3 Waveform inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.7.1 Efficiency of the strategy for 2-D heterogeneous media . . . . . . . . . 85
3.7.2 Practical implementation of the frequency selection strategy . . . . . . 87
3.7.3 The equivalence between gradient images and migration . . . . . . . . 87
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Waveform inversion starting from realistic frequencies 91
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 SVD of the Fréchet derivative matrix . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.1 1-D medium in homogeneous background . . . . . . . . . . . . . . . 93
4.2.2 1-D medium with a constant vertical velocity gradient . . . . . . . . . 100
4.3 Non-linearity of the waveform inverse problem . . . . . . . . . . . . . . . . . 108
4.3.1 The effect of cycle skipping . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.2 Analytic linearity study of the 3D homogeneous Green’s function . . . 108
4.3.3 Non-linearity and wavenumber domain . . . . . . . . . . . . . . . . . 110
4.3.4 Non-linearity and offset . . . . . . . . . . . . . . . . . . . . . . . . . 112

CONTENTS
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4.4 Tools for the mitigation of non-linearities . . . . . . . . . . . . . . . . . . . . 114
4.4.1 Gradient preconditioning by wavenumber filtering . . . . . . . . . . . 114
4.4.2 Time damping of the data residuals . . . . . . . . . . . . . . . . . . . 116
4.4.3 f-k filtering of the data residuals . . . . . . . . . . . . . . . . . . . . . 122
4.4.4 Offset windowing of the data residuals . . . . . . . . . . . . . . . . . . 124
4.4.5 Model parameterisation . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5 Definition of a strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.6.1 1-D velocity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.6.2 The 2-D extended Marmousi model . . . . . . . . . . . . . . . . . . . 128
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5 Waveform inversion and starting models: the sub-basalt imaging problem 135
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 The sub-basalt imaging problem . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2.1 Transmission loss at the top basalt interface and multiples . . . . . . . 136
5.2.2 Interface scattering at the top of the basalt . . . . . . . . . . . . . . . 138
5.2.3 Body wave scattering within the basalt . . . . . . . . . . . . . . . . . . 138
5.2.4 Sub-basalt imaging and waveform inversion . . . . . . . . . . . . . . . 138
5.3 The 1-D sub-basalt numerical experiment . . . . . . . . . . . . . . . . . . . . 139
5.4 Inversion starting from a “correct” macro model . . . . . . . . . . . . . . . . . 141
5.4.1 Misfit function and model smoothness . . . . . . . . . . . . . . . . . . 141
5.4.2 The top basalt and bottom basalt interfaces . . . . . . . . . . . . . . . 141
5.5 Wavefield separation by layer stripping . . . . . . . . . . . . . . . . . . . . . . 143
5.5.1 The sediment layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5.2 The basalt layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5.3 The sub-basalt layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.6 Waveform inversion and starting model requirements . . . . . . . . . . . . . . 146
5.6.1 Frequency dependence of the starting model requirements . . . . . . . 146
5.6.2 Waveform inversion at 7 Hz . . . . . . . . . . . . . . . . . . . . . . . 148
5.7 Discussion: starting model and standard methods . . . . . . . . . . . . . . . . 151
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

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CONTENTS
6 Conclusions 155
6.1 Wavenumber, frequency and offset . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2 Waveform inversion and starting frequency . . . . . . . . . . . . . . . . . . . 157
6.3 Waveform inversion and starting model . . . . . . . . . . . . . . . . . . . . . 158
6.4 Towards the waveform inversion of real data . . . . . . . . . . . . . . . . . . . 158
A Image stretch and NMO stretch 161
Bibliography 163

List of Figures
2.1 The Inverse and the Forward Problem in waveform inversion. . . . . . . . . . . 14
2.2 Illustration of the Singular Value Decomposition. . . . . . . . . . . . . . . . . 17
2.3 Scheme of the Gauss-Newton method applied to linear problem. . . . . . . . . 22
2.4 Scheme of gradient method applied to linear problem. . . . . . . . . . . . . . . 24
2.5 Construction of the gradient vector from eigenvectors and eigenvalues of the
Hessian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Example of a 2-D, linear inverse problem. . . . . . . . . . . . . . . . . . . . . 30
2.7 Local methods applied to the 2-D, linear inverse problem. . . . . . . . . . . . 31
2.8 Evolution of the misfit function with iterations using local methods. . . . . . . 32
2.9 Scheme of linearised inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.10 Example of a non-linear inverse problem. . . . . . . . . . . . . . . . . . . . . 38
2.11 Results of local methods with an accurate starting model. . . . . . . . . . . . . 40
2.12 Results with a starting model where the quadratic assumption fails. . . . . . . . 41
2.13 Results with a starting model is close to a local minimum. . . . . . . . . . . . . 42
3.1 Wavenumber illumination for a single frequency and source/receiver pair. . . . 57
3.2 Wavenumber illumination for different incident/scattering angle. . . . . . . . . 58
3.3 The 1-D basic scattering experiment. . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Illustration of the frequency discretisation strategy. . . . . . . . . . . . . . . . 64
3.5 Velocity model after Freudenreich and Singh, 2000. . . . . . . . . . . . . . . 66
3.6 Illustration of the computation of the 5 Hz gradient. . . . . . . . . . . . . . . . 67
3.7 A set of amplitude normalized gradient images. . . . . . . . . . . . . . . . . . 69
3.8 Frequency sequence generated by equation (3.18). . . . . . . . . . . . . . . . . 71
3.9 Sequential frequency domain inversion results. . . . . . . . . . . . . . . . . . 72
3.10 Time-like waveform inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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LIST OF FIGURES
3.11 Imapact of noise on the reconstruction of the 1-D target. . . . . . . . . . . . . 75
3.12 Marmousi wide angle synthetic data. . . . . . . . . . . . . . . . . . . . . . . . 77
3.13 Example of the finite difference solver of the eikonal. . . . . . . . . . . . . . . 78
3.14 Evolution of the traveltime misfit function with respect to smoothness of the
true Marmousi model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.15 First arrival traveltime inversion. . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.16 Waveform inversion in Marmousi. . . . . . . . . . . . . . . . . . . . . . . . . 82
3.17 Waveform inversion results in the Marmousi model at progressively higher frequencies.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.18 Velocity profiles at three locations in the Marmousi model. . . . . . . . . . . . 84
3.19 Data residuals in the Marmousi model. . . . . . . . . . . . . . . . . . . . . . . 86
4.1 SVD at 3 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 SVD at 5 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 SVD at 10 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 1-D velocity model with a constant vertical velocity gradient. . . . . . . . . . . 100
4.5 SVD at 3 Hz in the model with a vertical velocity gradient. . . . . . . . . . . . 101
4.6 SVD at 5 Hz in the model with a vertical velocity gradient.. . . . . . . . . . . . 102
4.7 SVD at 10 Hz in the model with a vertical velocity gradient.. . . . . . . . . . . 103
4.8 SVD at 3 Hz in model with velocity gradient. Slowness parameterisation. . . . 105
4.9 SVD at 5 Hz in model with velocity gradient. Slowness parameterisation. . . . 106
4.10 SVD at 10 Hz in model with velocity gradient. Slowness parameterisation. . . . 107
4.11 Illustration of the cycle skipping problem. . . . . . . . . . . . . . . . . . . . . 109
4.12 Misfit function of the linearization of the 3D homogeneous Green’s function . . 111
4.13 The 1-D Marmousi velocity model. . . . . . . . . . . . . . . . . . . . . . . . 112
4.14 Non-linearity and wavenumbers. . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.15 Non-linearity and offset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.16 2-D low-pass wavenumber filter of the gradient vector. . . . . . . . . . . . . . 116
4.17 1-D inversion with gradient preconditioning by wavenumber filtering. . . . . . 117
4.18 Finite difference modelling with different value of τ. . . . . . . . . . . . . . . 120
4.19 Misfit function of the far offset at 7 Hz and time damping. . . . . . . . . . . . 121
4.20 1-D velocity model containing a shallow and deep heterogeneity. . . . . . . . . 123
4.21 Preconditioning of the data residuals using an f-k filter. . . . . . . . . . . . . . 123

LIST OF FIGURES
xvii
4.22 Standard waveform inversion at 7 Hz in the 1-D Marmousi model. . . . . . . . 126
4.23 1-D waveform inversion at 7 Hz with preconditioning of the data residuals and
gradient vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.24 2D extended Marmousi model. . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.25 Standard waveform inversion at 7 Hz. . . . . . . . . . . . . . . . . . . . . . . 129
4.26 Preconditioned Inversion at 7 Hz starting from the extended “FAST” model. . . 130
4.27 Velocity profiles at 3 locations in the model. . . . . . . . . . . . . . . . . . . . 131
4.28 Shot gathers at 3 locations in the true model. . . . . . . . . . . . . . . . . . . . 132
4.29 Inversion at 7 Hz starting from an improved macro model. . . . . . . . . . . . 133
5.1 Converted waves in sub-basalt imaging. . . . . . . . . . . . . . . . . . . . . . 137
5.2 The sub-basalt synthetic data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 Inversion starting from an accurate macro-model. . . . . . . . . . . . . . . . . 142
5.4 Evolution of the misfit function as the entire model is decreasingly smoothed. . 143
5.5 Evolution of the misfit function as the sediment and top basalt interface are
decreasingly smoothed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.6 Illustration of the idealised layer stripping strategy . . . . . . . . . . . . . . . . 145
5.7 Evolution of the misfit function as the sediment layer is decreasingly smoothed. 146
5.8 Evolution of the misfit function as the basalt layer is decreasingly smoothed. . 147
5.9 Evolution of the misfit function as the sub-basalt layer is decreasingly smoothed. 147
5.10 Adequate velocity models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.11 Waveform inversion at 7 Hz when for a wrong and adequate starting models. . . 150
5.12 Wavefield in the sediments in time in the true model and inversion results with
the bad starting model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

xviii
LIST OF FIGURES

List of Tables
2.1 Condition of existence of G −1 in equation 2.5 (p. 48 of (Scales et al., 2001)). . 16
2.2 Review of possible cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Possible model parameterisations. . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1 Frequency discretisation sequences for the near and far offset surveys. The
last frequency 8 Hz is chosen arbitrarily so that both experiments use identical
frequency ranges within the spectrum of the source signature. . . . . . . . . . . 70
4.1 Number of nonzero singular values with frequency and offset range. . . . . . . 95
4.3 Strategy of preconditioning for the waveform inversion for the 2-D extended
Marmousi experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.1 Synthetic data characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
xix

xx
LIST OF TABLES

Chapter 1
Introduction
Seismic exploration techniques are used to determine the geological structure of the subsurface
as a routine component in the search of hydrocarbon reservoirs. The seismic acquisition experiment
is triggered by a controlled source which in turns, initiates a propagating wave within
the earth that is ultimately recorded at the surface by receivers. The recorded signal is then
processed in order to determine a model representing the geological structure of the subsurface.
This model is most often characterised by the propagation velocities and is in practice, most
often decomposed by spatial scales: the low wavenumbers define the large scales of the model
while the high wavenumbers describe the fine details of the structure.
This decomposition by scale is an approach that is justified by the information content of
the experiment. Jannane et al. (1989) showed that typical near offset, seismic reflection data are
sensitive only to the very low and the very high wavenumbers; the intermediate wavenumbers
are invisible to the data. The standard approach in exploration imaging has, as a result, evolved
into a recovery of the low wavenumbers of the velocity field (macro-model) followed by the
reconstruction of the very high wavenumbers components. During the last decades, this decomposition
has been successfully applied to many areas of seismic exploration presenting simple
geological structure. More recently, seismic exploration targets have included geographical areas
in which the subsurface contains more complex features. Examples include areas with significant
salt flows (Ward et al., 1994), or significant amounts of volcanics (Planke et al., 1999),
both of which cause severe imaging problems. The strong heterogeneity as well as the 2D/3D
nature of the media significantly diminish the efficacy of standard imaging techniques. The
most widely used methods for imaging complex media are prestack depth migration. These approaches
however, are currently still applied within the standard approach, in which no attempt
1

2 CHAPTER 1. INTRODUCTION
is made to recover the intermediate wavenumbers. The accuracy of prestack imaging has been
a popular topic of investigation and synthetic seismic data generated in very complex velocity
models have been available to the research community. Some examples are the SEG/EAGE salt
and overthrust models, and the Marmousi model (Versteeg, 1994). These models are representative
of the current challenges of imaging and are widely used as benchmarks for testing the
accuracy of current methods.
One of the main difficulties in imaging is that the accuracy of the reconstruction of the high
wavenumbers by migration strongly depends on the accuracy of the macro-model. Versteeg
(1993) used the Marmousi model to show that prestack depth migration may be significantly
improved when the macro-model contains some of the intermediate wavenumbers. On the other
hand, none of the techniques routinely employed for the determination of the macro-model are
capable of recovering these wavenumbers.
The aims of this thesis is to apply waveform inversion of wide-angle seismic data, to reconstruct
a continuous range of wavenumbers of a complex structure. The goal is to produce a
velocity model which contains both the low and intermediate wavenumbers of the velocity field
so that the reconstruction of the high wavenumbers may be significantly improved. I will use
two synthetic case studies: the first is an extension of the Marmousi model, in which additional,
wide-angle data are used to supplement the original survey geometry. The second synthetic
example is used to illustrate the issues involved in imaging through high velocity subsurface
basaltic deposits.
1.1 Determination of the macro-model
The determination of the very low wavenumber components, referred to as the macro-model or
velocity model, aims to solve the kinematic aspect of the problem. The methods employed for
the determination of the macro-model thus utilise the traveltime information of the data. The
most frequently used technique is based on the derivation of the velocity model by stacking
velocity analysis in which the velocities are defined from the normal moveout (NMO) of the
reflection hyperbola (Yilmaz, 1987). The stacking velocities may then be converted to interval
velocities using the Dix’s formula (Dix, 1955). However, such a velocity model is defined
in time and must be converted to depth; moreover, the NMO correction also implies that the
medium is tabular and therefore 1-D. Because the macro-model is often associated with a complex
structure, the 2/3-D nature of the low wavenumbers must be properly taken into account.

1.1. DETERMINATION OF THE MACRO-MODEL 3
More advanced techniques such as traveltime tomography have therefore been developed
and are able to handle laterally varying media (Bishop et al., 1985; Farra and Madariaga, 1988).
The traveltime tomography method consists of the resolution of an inverse problem that seeks
to minimise the mismatch between observed and calculated traveltime. The observed traveltime
are picked in the data volume and are identified by coherent events such as reflection hyperbolas
or diving/refracted arrivals. The calculated data are computed using ray tracing techniques
based on the high frequency asymptotic approximation of the wave equation ( ˘Cervený et al.,
1977; Chapman, 1985). Such modelling methods are accurate if the model varies slowly with
respect to the wavelength of the propagating wave and are therefore not adapted to simulation of
propagation within highly heterogeneous media (Chapman, 1985). For this reason, the velocity
model estimated from ray based tomography is required to be smooth; the blocky nature of
the model may however be accounted for by including velocity discontinuities (interfaces).
A widely used 2-D traveltime inversion technique is the program rayinvr developed by Zelt
and Smith (1992) that allows the simultaneous inversion of various types of ray propagation:
reflections, refractions and diving waves.The implementation of this application requires the
traveltime picking of events in time that must be identified with features in the model: for
example reflections as well as refractions must be associated with reflectors (interface). Such a
method therefore demands a strong a priori knowledge of the velocity structure as the number
of layers and lateral discontinuities must be known which may be difficult to determine in the
presence of a complex structure.
Traveltime tomography relying on the use of the first arrival traveltimes alone may also be
used (Zelt and Barton, 1998). The calculated traveltimes are computed using a finite difference
solver of the eikonal equation (Vidale, 1990; Podvin and Lecomte, 1991); diving rays are traced
by following the normal of the wavefront connecting the receiver to the source. This method
presents a great advantage: the first arrival picked traveltimes are not required to be associated
with particular horizons of the model. However, large offset data are necessary to assure the
proper illumination of the deep part of the model. Also, the exploitation of the first arrival
information is not very efficient in the determination of low velocity zones which tend to be
avoided by first arrival diving rays.
Alternative methods have emerged that aims to avoid the “hand” picking and interpretation
of coherent events in the data. Billette and Lambaré (1998) proposed the stereotomography approach
that relies on the picking of locally coherent events that may be performed automatically.
This picking does not require the association of traveltime picks with continuous reflections in

4 CHAPTER 1. INTRODUCTION
the data volume or the introduction of interfaces in the model.
In order to avoid the task of picking the traveltime prerequisite to tomographic inversion, migration
velocity analysis methods have been proposed (Al-Yahya, 1989; Symes and Carazzone,
1991; Docherty et al., 1997; Chauris et al., 2002). These approaches rely on the optimisation of
trace coherence in the migrated domain. An inverse problem is posed that seeks to update the
velocity model in a way that optimises the flatness/semblance of common image gathers (CIG).
The determination of the macro-model by migration velocity analysis is often considered the
most desirable method by the seismic industry since it optimises the process of migration which
is the most widely used techniques for the recovery of the high wavenumbers.
1.2 Determination of the high wavenumbers by migration
The recovery of the very high wavenumbers is the complementary step to the determination of
the macro-model. The reflectivity field provides information on the fine structure of the model
thereby allowing the localisation of reflectors in depth. Migration is the most commonly used
technique for the determination of the high wavenumber components. It was applied to zero
offset data after NMO correction (Claerbout and Doherty, 1972) and relies on the concept of
exploding reflectors (Claerbout, 1985). The principle of summation is the basis of Kirchoff
migration: the magnitude of a diffraction point is determined by Kirchhoff summation (Schneider,
1978), which sums the amplitude of the data along the diffraction hyperbola. Alternative
poststack migrations however exist such, as f-k migration (Stolt, 1978; Gazdag, 1978), reverse
time migration (Baysal et al., 1983) and finite difference migration (Claerbout, 1985). Because
poststack migration is applied after NMO correction of the data, this approach is not accurate
for imaging dipping reflectors. Dip-moveout correction (DMO), or prestack partial migration
(PSPM) have emerged in an attempt to correct the effect of dips on the data thus improving
poststack imaging methods (Yilmaz and Claerbout, 1980; Hale, 1984).
Prestack migration is now commonly used as it is the most efficient method for the imaging
of complex media (Schultz and Sherwood, 1980; Wiggins, 1984). The migration of prestack
data often uses the asymptotic approximation of the wave equation for Kirchoff migration
(Carter and Frazer, 1984; Beylkin, 1985; Miller et al., 1987; Keho and Beydoun, 1988). Some
limitations of the prestack migration have nevertheless been pointed out as ray methods exploit
the first arrivals which may not be the most energetic arrival (Geoltrain and Brac, 1993; Moser,
1991; Gray and May, 1994). Alternative approaches based on the one-way approximation of

1.3. QUANTITATIVE MIGRATION AND LINEARISED INVERSION 5
the wave equation have shown to be more robust since they account for later arrivals (Ehinger
et al., 1996). However, even when the macro-model explains correctly the kinematic aspect of
the imaging, an accurate migration was not originally expected to provide a quantitative estimation
of the reflectivity field. Therefore, the result of migration cannot be linked to physical
parameters that may more rigorously describe the model.
1.3 Quantitative migration and linearised inversion
Many techniques have been proposed to provide quantitative estimates of the model parameters.
Most of these techniques rely on: 1- the ray theory and 2- the Born approximation (Born, 1923)
or the Kirchhoff approximation (Schneider, 1978). They can all be defined within the context
of linearised inverse problem theory (Lailly, 1983; Tarantola, 1984b), where the inverse of the
linear forward operator must be estimated. In an early work, Cohen and Bleistein (1979) defined
a direct inverse operator for zero offset data. Thereafter, inversion procedures for prestack
data were proposed (Clayton and Stolt, 1981; Beylkin, 1985; Ikelle et al., 1986; Miller et al.,
1987). These developments initiated the concept of migration/inversion using kinematic Kirchhoff
migration where amplitude are estimated using the Born or the Kirchoff approximations
(Bleistein, 1987; Bleistein et al., 1987; Beydoun and Mendes, 1989). These types of migration
methods are commonly used because of their computational efficiency for manipulating very
large data sets (Thierry et al., 1999). As an alternative to the direct inverse method, iterative
methods are often considered more robust (Lambaré et al., 1992; Jin et al., 1992; Nemeth et al.,
1999). Iterative methods rely on the explicit minimisation of least-square misfit function and are
more efficient in handling incomplete data sets. As discussed in the previous section, one of the
difficulty of ray-based migration is its difficulty in accounting for multipathing. More advanced
ray tracing methods have therefore been proposed with the concept of waveform construction
(Vinje et al., 1993; Lambaré et al., 1996) and have shown to improve the quantitative estimation
of migration/inversion (Operto et al., 2000).
However, the accuracy of the estimation of the model parameter using linearised inversion
is limited by the validity of the Born or the Kirchoff approximation. For example, the
Born approximation only accounts for first order scattering and is only valid in the presence of
small velocity perturbation (Keller, 1969; Beylkin and Ostroglio, 1985; Beydoun and Tarantola,
1988). On the other hand, the Kirchoff approximation accounts for the reflectivity which cannot
directly be related to velocity. Amplitude versus offset (AVO) analysis are then required to

6 CHAPTER 1. INTRODUCTION
quantify the model in terms of velocity (see for example Yilmaz (1987)).
1.4 Migration/Inversion and non-linearity
It is well known that the determination of model parameters from seismic data is a non-linear
inverse problem. The issue of non-linearity is partially addressed in imaging by decomposing
the process into a determination of the macro-model followed by (quantitative) migration. It
is commonly accepted that migration will fail, at least kinematically, if the low wavenumber
components of the model are inaccurate. Therefore, the non-linearity of the inverse problem is
expected to be addressed by the determination of the macro-model. In addition, the linearised
inversion approach assumes that the quantification of the model parameters is a linear process
once the kinematic aspects of the data is resolved. The non-linear inverse problem is therefore
somewhat solved by decomposing the problem into a recovery of the low wavenumbers, followed
by the reconstruction of the high wavenumbers. However, as illustrated by Stork (1992a)
in the case of flattening of CIGs, the problem of determination of the macro-model is nonunique
and can easily provide a wrong answer. As a result, migration may locate the reflectors
at a wrong depth and linearised inversion may provide an inaccurate evaluation of the model
parameters.
1.5 Non-linear waveform inversion
As an alternative to imaging techniques, the non-linear waveform inversion approach has been
proposed. The objective of waveform inversion is to estimate a quantitative model of the subsurface
in a way that minimises the differences (residuals) between observed and calculated data,
by modelling the physics of the measurement. This minimisation is achieved by localising the
global minimum of the misfit function. The calculated data must therefore be generated using
a fully non-linear modelling algorithm, usually a finite difference solver of the wave equation
(Kelly et al., 1976). Some of the complexity of the wave propagation in complex media is thus
taken into account and both phase and amplitude data misfit must be minimised.
Among non-linear waveform inversion methods, the global minimum may most accurately
be found by performing a global search of the misfit function (Cary and Chapman, 1988; Sen
and Stoffa, 1991). These methods carry out a systematic exploration of the multidimensional

1.5. NON-LINEAR WAVEFORM INVERSION 7
misfit function using for example, Monte-carlo, genetic or simulated annealing algorithms. Although
these techniques are capable of handling non-linear behaviour by inverting for both low
and high wavenumbers, they require the computation of the same order of forward modellings
as there are model parameters involved. Another approach is to decouple the low and high
wavenumber information, inverting for each parameter set alternately (Snieder et al., 1989; Cao
et al., 1990; Hicks and Pratt, 2001). This latter approach achieves the required decoupling by
a re-parametrisation in time of the velocity model, thus ensuring the zero offset traveltimes
remain fixed during the reflectivity reconstruction. This method assumes a near-vertical propagation
of the near offset events, and is thus limited to situations in which the 1-D approximation
can be applied locally.
Because of the high computational cost of calculating synthetic seismic data, non-linear
waveform inversion is usually formulated as an iterative “descent” method, in which the minimisation
of residuals is achieved through the repeated calculation of a local gradient. The
gradient at each iteration provides the direction of minimisation of the “objective” functional,
usually the L 2 norm of the data residuals possibly combined with some form of regularising
constrains. Since 1983, it has been recognised that the calculation of the gradient is of the same
computational order as the forward modelling task, and that the gradient is closely related to
seismic migration (Lailly, 1983; Tarantola, 1984a). Success depends upon the topography of
the misfit function at the location of an initial guess of the velocity model (starting model). In
the presence of strong non-linearities, the inversion may fail and convergence into local minimum
may occur (Gauthier et al., 1986). The starting model for such scheme must therefore be
located in the neighbourhood of the global minimum, i.e., a descent path leading to the global
minimum must exist if the method is to succeed.
Iterative non-linear waveform inversion was first applied to near offset reflection data to
recover the high wavenumber components of the model (Tarantola, 1986; Mora, 1987; Pica et
al., 1990; Crase et al., 1990). The advantage of the non-linear approach over the linearised
approach is that it will locate more efficiently the global minimum since the misfit function is
not required to be locally quadratic. In other words, the recovery of the high wavenumbers is
not limited by the validity of the Born approximation, since the forward problem used does
take into account multiple scattering. However, the determination of the macro-model remains
a critical aspect of the non-linear inversion of near offset data as it will cause convergence into
a local minimum if the starting macro-model velocities are not sufficiently accurate. Non-linear
inversion of near offset data therefore suffers from the same limitation than migration/inversion

8 CHAPTER 1. INTRODUCTION
techniques and depends strongly on the accuracy of the macro-model.
1.6 Non-linear inversion of wide-angle data
Mora (1988) recognised that the inversion of transmission data may allow the recovery of some
of the low wavenumbers of the velocity model. Non-linear waveform inversion may therefore
be utilised for the recovery of both the low and the high wavenumbers as in Mora’s (1989)
expression
inversion = migration + tomography.
The standard concept of imaging that decomposes the model of subsurface into the very low
and the very high wavenumbers may be redefined: waveform inversion may be used to describe
the velocity model over a continuous range of wavenumbers so that the “hole” of wavenumber
information present in classical methods no longer exists. The determination of the intermediate
wavenumbers is particularly important for the recovery of the high wavenumbers. This was
shown by Versteeg (1993) who demonstrated that prestack depth migration in the Marmousi
model may be significantly improved when spatial wavelengths up to 200 m are present in the
macro-model.
In order to exploit the information contained in transmission data such as diving and refracted
waves, waveform inversion has been applied on wide-angle (large offset) seismic data
(Mora, 1988, 1989; Sun and McMechan, 1992; Pratt et al., 1996; Shipp and Singh, 2002). The
introduction of large offset data in the misfit function however contributes to an increase of
the non-linearity of the inverse problem. Since the traveltime error is the result of integration
of slowness errors over the ray path, errors contained in the starting velocity model are
expected to have a greater effect on the large offset data which correspond to the longest ray
paths. Convergence into a local minimum will yield a wrong update of the low and intermediate
wavenumbers of the velocity model and thus will have a dramatic effect on the reconstruction
of the high wavenumbers. Therefore, local minima must be very carefully avoided during the
recovery of the low and intermediate wavenumbers to insure an accurate reconstruction of the
high wavenumbers.
The linearity of the waveform inverse problem also varies with temporal frequency of the
data as low frequencies are more linear than high frequencies. The inversion of the lowest
frequencies in the data thus appears to be desirable in order to increase the chance of conver-

1.7. AIM OF THE THESIS 9
gence into the global minimum (Bunks et al., 1995). The typical seismic data bandwidth is
nevertheless limited by acquisition practice.
Another important aspect of the waveform inversion is the determination of the starting
model where the initial descent direction of the misfit function is computed. The starting model
must be obtained using the standard methods discussed previously for the determination of
the macro-model (stacking velocity analysis, traveltime inversion...). The requirements on the
starting model are often very vague as it is considered that it must be in the neighbourhood of the
global minimum. This begs the important question of the compatibility between the waveform
inversion of the lowest frequencies available in the data and the limitations of classical methods
in terms of accuracy of the macro-model. It is primarily this question that I seek to investigate
in this thesis.
1.7 Aim of the thesis
The objective of this thesis is to apply local, non-linear waveform inversion methods on wideangle
seismic data for the recovery of a continuous range of wavenumbers of a complex velocity
model. Two velocity models will be used to investigate this question: the 2D Marmousi model
and a 1-D velocity model representing the sub-basalt imaging problem.
The 2D Marmousi numerical experiment explores the application of waveform inversion
using a realistic starting velocity model that was obtained from a standard method such as
traveltime tomography. The 1D sub-basalt experiment analyses the requirements of the starting
model that ensure the success of waveform inversion.
The method of waveform inversion in the frequency domain will be used (Pratt and Worthington,
1990), as it presents significant advantages in term of computational efficiency. This
work will examine the potential for applying waveform inversion in the frequency domain. The
difficulties caused by the non-linearity of the waveform inverse problem will also be investigated,
particularly with respect to the minimum frequency available in the data. This thesis also
seeks to propose a methodology to incorporate a range of preconditioning tools in the inversion
process (Tarantola, 1987; Mora, 1987), that may improve the linearity of the inverse problem.

10 CHAPTER 1. INTRODUCTION
1.8 Summary of the thesis
This thesis begins by reviewing the basics of inverse theory in Chapter 2. The resolution of
linear inverse problem using the singular value decomposition (SVD) method is first described.
This is followed by the definition of local methods using Gauss-Newton and gradient techniques.
A useful link is then established between the SVD and local methods. The resolution
of non-linear inverse problem using the local methods is then detailed. Numerical examples of
resolution of a linear and non-linear problem using local methods are shown. This review of
inverse theory provides the elements for the introduction of the frequency domain waveform
inversion.
In Chapter 3, I aim to further understand the relation between temporal frequencies, offset
and wavenumber reconstruction of the model. A strategy for selecting frequencies is derived
that assures a continuous reconstruction of the wavenumber spectrum of the model. This strategy
makes use of the redundancy of information present when a range of offset is available and
that allows to discretise temporal frequencies with a much larger sampling interval than that dictated
by the sampling theorem. It is shown that the larger the maximum effective offset is, the
fewer frequencies are required. The accuracy of the strategy, exact in a 1-D earth, is validated
on a 1-D numerical experiment. The strategy is also useful in more general complex media
as it is successfully applied on the 2-D Marmousi model using a starting model that was obtained
from traveltime tomography. In this chapter, all waveform inversion however start from
unrealistically low frequencies. A minimum frequency of 5 Hz is used in the 2-D Marmousi
experiment. Such a frequency is considered too low to be present in real seismic data.
In Chapter 4, I therefore investigate waveform inversion starting from a more realistic frequency:
7Hz. It is shown that at this frequency, waveform inversion can easily converge into
a local minimum. The aim of this chapter is therefore to further understand the mechanism
governing the inversion process in order to design a strategy that improves the linearity of the
inverse problem. A SVD is applied on simple 1-D models and indicates that the gradient vector
is dominated by the high wavenumbers. A linearity study is also carried out showing that the
low wavenumbers are the more non-linear components of the model. As a result, the gradient
vector must be smoothed in order to assure the proper reconstruction of the low wavenumber
components. In order to mitigate the risk of convergence into a local minimum during the recovery
of the low wavenumbers, adequate preconditioning of the data residuals may be applied
that focus on the inversion of the early arrivals.

1.8. SUMMARY OF THE THESIS 11
In Chapter 5, I explore the feasibility of an application of waveform inversion to the subbasalt
imaging problem. A non-linearity study is carried out that aims to define the requirements
of the starting model that ensure the success of waveform inversion. This study is performed on
a 1-D velocity model and emphasises the importance of the determination of the starting model
in the waveform inversion process. It is shown that the requirements on the starting model
depends on the minimum frequency available in the data: the higher this frequency is, the more
accurate the starting model needs to be.

12 CHAPTER 1. INTRODUCTION

Chapter 2
Inverse theory
2.1 Introduction
The laws of physics allow us to simulate the action and interaction of physical parameters within
a given system called the model. This model is a representation of a natural system (the earth,
the subsurface,... the atom...) and is described by a set of model parameters (velocity, density,
conductivity....). This model, once excited, yields a set of measurable quantities: the data. The
system of equations relating the data d to the model m is called the forward problem and is
noted
g (m) = d (2.1)
where g(m) is a set of mathematical equations dependent on m.
The reverse action consists in finding the model m that explains a set of observed data d obs
and is called the inverse problem. The complexity of the inverse problem is closely related to
the complexity of the forward problem. The latter must be known in order to determine the
inverse solution. Often, this answer is an estimate and differs from the true model because
the forward modeling provides with incomplete information on the model. Furthermore, the
forward problem is an approximation and the model is a simplified representation of the true
system. In addition, the observed data are recorded by instruments and thus contain noise.
In waveform inversion, we aim to recover a quantitative representation of a model of the
subsurface. The model is the most often characterized by the P-wave velocity varying with depth
although other quantities may be accounted for (S-wave, quality factor....). The information
used as data are the seismic traces recorded at the earth’s surface by receivers. These receivers
13

14 CHAPTER 2. INVERSE THEORY
Data
Model
Inverse Problem
Theory
Depth
Time
Forward Problem
Figure 2.1: The Inverse and the Forward Problem in waveform inversion. The inverse problem
seeks to estimate a depth velocity model of the subsurface from seismic data recorded on the
field.
record in time the signal propagated through the sub-surface from a source. The concept of
forward and inverse problem in the context of waveform inversion is illustrated Figure 2.1.
A straight forward solution to the inverse problem is to define the inverse operator g −1 such
as the estimated solution is
m † = g −1 (d) , (2.2)
these methods will be refered to as direct methods.
In order to avoid the estimation of the inverse operator, local methods may be used which
aim to estimate the model update ∆m of an priori model m o such as
m † = m o + ∆m. (2.3)
The model update is found by minimizing the misfit function, measuring the mismatch between
the observed and forward modelled data. These methods are local in the sense that the path
leading to the inverse solution depends on the initial guess of the model m o .
Direct and local methods are widely used in seismic exploration and I will focus in this
chapter on the resolution of the inverse problem in the case where:
• The forward problem is linear
• The forward problem is non-linear

2.2. INVERSION OF LINEAR PROBLEMS 15
The resolution of the waveform inversion problem, implemented in the frequency domain will
be introduced in the final section.
2.2 Inversion of linear problems
For linear system, the forward problem is a linear combination of model parameters and can be
expressed in discretised form as
Gm = d (2.4)
where m is the model vector of dimension (n m × 1), and d is the data vector of dimension
(n d × 1). G is the (n d × n m ) forward operator matrix of coefficients independent of m, that
projects a model vector into the data space. Finding the solution to equation (2.4) consists in
finding the vector m † that explains the observed data d obs .
The direct method consists in finding the inverse matrix G −1 of the matrix G such that the
inverse solution is
m † = G −1 d. (2.5)
The resolution of the inverse problem using direct methods thus implies that the matrix inverse
can be found or at least estimated. The inverse matrix G −1 is defined as
and/or
G −1 G = I nm for n m ≤ n d
GG −1 = I nd for n m ≥ n d
(2.6)
where I n is the identity matrix of dimension (n × n).
The inverse matrix G −1 will not exist if the matrix G is not invertible in which case a
solution of the type given by equation (2.5) cannot be obtained. Furthermore, the existence of
the inverse matrix does not assure the solution to be unique as there may be none, one or an
infinite number of solutions as shown Table 2.1 (p. 48 of Scales et al. (2001)). If G is invertible
and of dimension (n m × n d ), direct solver such as LU decomposition may be used (Press et al.,
1992) in the case of a well posed inverse problem.
In the next section, I will introduce the technique of Singular Value Decomposition (SVD)
which allows one to obtain an estimate of the inverse matrix called the generalized inverse
G † . The SVD is a powerful method since it also diagnoses the problem by providing useful
information on how well the problem is posed. It also yields an estimate of the solution m † ,
even in the case where G is not invertible.

16 CHAPTER 2. INVERSE THEORY
Unknowns/Equations G −1 exists if G has Solution
n m < n d linearly independent columns At most one
n m = n d linearly independent columns Only one
n m > n d n d linearly independent columns More than one
Table 2.1: Condition of existence of G −1 in equation 2.5 (p. 48 of Scales et al. (2001)).
2.2.1 Singular value decomposition
The Singular Value Decomposition (SVD) method was introduced by Lanczos (1961). I briefly
review in this section the principle results of the SVD. Further details on the method may be
found in Aki and Richards (1980); Menke (1989); Scales et al. (2001).
The singular value decomposition of any matrix G of dimension (n d × n m ) is expressed as
G = UΛV t (2.7)
where the columns of U (dimension (n d × n d )) are the data space eigenvectors of GG t , and
the column of V (dimension (n m × n m )) are the model space eigenvectors of G t G. The matrix
Λ of dimension (n d × n m ) is diagonal where the diagonal elements are called singular values.
There are min(n d , n m ) number of singular values which are defined as the square roots of the
first min(n d , n m ) eigenvalues of
G t Gv i = λ 2 i v i i = 1, n m
GG t u i = λ 2 i u i i = 1, n d
(2.8)
Each data eigenvector u i and model eigenvector v i are columns of U and V respectively such
as U and V form a basis of orthonormal vectors. Any vector of the data or model space can
thus be expressed as a linear combination of eigenvectors and we have
U t U = UU t = I nd (2.9)
V t V = VV t = I nm (2.10)
Note that the eigenvalues λ 2 i defined in equation (2.8) may be positive or zero real numbers so
that the diagonal terms of Λ may contain zeros. If n m ≠ n d , the number of eigenvectors of
the data and model space are different. If p ≤ min(n d , n m ) is the number of non-zero singular
values, the data and model space have p eigenvectors with associated non-zero eigenvalues.

£
2.2. INVERSION OF LINEAR PROBLEMS 17
¤¦¥ §©¨
¡ ¢
Figure 2.2: Illustration of the Singular Value Decomposition of the forward operator G. The
SVD identifies the model and data null spaces V o and U o . G can be reduced to a projection
operator from V p to U p .
Therefore, there will be (n m − p) space eigenvectors and (n d − p) data eigenvectors with the
eigenvalue zero. The projection of eigenvectors from the model to the data space is
⎧
⎨
and from the data to the model space is
⎩
⎧
⎨
⎩
Gv i = λ i u i for i = 1, p
Gv i = 0 for i = p, n, m
(2.11)
G t u i = λ i v i for i = 1, p
G t u i = 0 for i = p, n d
. (2.12)
Equation (2.11) and (2.12) implies that, for the p non-zero eigenvalues, an eigenvector of the
model space is coupled with an eigenvector of the data space which have the same eigenvalue.
The projection of an eigenvector with a zero eigenvalue has no component in the corresponding
space.
Understanding the forward problem
The matrix U and V can be decomposed into two sub-spaces represented by the eigenvectors
with non-zero and zero eigenvalues (as shown Figure 2.2)
U = (U p , U 0 )
V = (V p , U 0 )
(2.13)

18 CHAPTER 2. INVERSE THEORY
where the columns U p (n d × p) and V p (n m × r) are the eigenvectors associated with non zero
eigenvalues and U 0 (n d × (n d − p)) and V 0 (n m × (n m − p)) are the eigenvectors associated
with zero eigenvalues. U 0 and V 0 are respectively representing the data and model null spaces.
Although the equations (2.9) and (2.10) are always verified, it may not be true for the subspaces
U p and V p if either a model or data null space exist. Since the eigenvectors are normalized,
we will always have
U t pU p = VpV t p = I p (2.14)
but as the eigenvectors with non-zero eigenvalues may not span the entire space we have
U p U t p ≠ I nd if U o exists
V p V t p ≠ I nm if V o exists
(2.15)
The determination of the eigenvectors and eigenvalues belonging to each sub-space provides
us with very useful information. It allows us to identify the portions of the model and data spaces
effectively involved in the forward problem since:
• U p is sensitive to V p
• U p is insensitive to V o
• U o is insensitive to V p , V o
Within U p , the most sensitive portion of the data space are the eigenvectors with the highest
eigenvalues.
By defining the diagonal square matrix Λ p of dimension (p × p), a sub-matrix of Λ which
as non-zero value along its diagonal, we have
Λ =
⎛
⎝ Λ p 0
0 0
We can express the forward operator of equation (2.7) as
G = (U p , U 0 )
⎛
⎝ Λ p 0
0 0
The forward problem can then be reduced to the expression
⎞
⎠ . (2.16)
⎞
⎠ (V p , V o ) t . (2.17)
G = U p Λ p V t p. (2.18)

2.2. INVERSION OF LINEAR PROBLEMS 19
Equation (2.18) shows that the SVD allows the reduction of the forward operator G to its
efficient state of information. The forward problem may use the portion of the model space
constrained to V p to model the data contained in U p only. The other portion of the data U o
cannot be modeled.
Solution estimate of the inverse problem
We have seen that the SVD allows the diagnosis of the forward problem by identifying the
portion of the data and model spaces U p , V p . The SVD technique may also be used to estimate
a solution of the inverse problem. By reformulating the state of information with the perspective
of inversion, the sub-space
• V p is constrained by U p only
• V 0 is undetermined
• U 0 is useless
Since V p can only be determined using the information contained in U p only, we define the
generalised inverse G † as
G † = V p Λ −1
p Ut p . (2.19)
The generalised inverse G † is an estimate of the inverse matrix G −1 (which may not exist, see
table 2.1). The estimate of the inverse problem solution m † is then defined as
m † = G † d. (2.20)
In order to evaluate how well the model is resolved, equation (2.4) may be included into equation
(2.20) yielding
m † = G † Gm
= R m m (2.21)
where R m = G † G is the model resolution matrix. We can also measure how well the estimated
solution explained the data by defining d † , the data predicted by the generalised inverse solution
d † = Gm †
= GG † d
= R d d (2.22)

20 CHAPTER 2. INVERSE THEORY
Where R d = GG † , is the data resolution matrix.
If the model and data resolution are equal to the identity matrix, the model is perfectly
resolved and the data perfectly explained. This is possible only if p = n m = n d . A review of
possibles cases are shown Table 2.2. The interesting point is that even though G −1 may not
exist for p < (n d , n m ), the generalised inverse will still provide an estimate of the solution but
neither the data nor the model will perfectly resolved (R m , R d ≠ I).
Case V o U o R m = I R d = I G † = G −1
p = n m < n d No Yes Yes No Yes
p = n m = n d No No Yes Yes Yes
p = n d < n m Yes No No Yes Yes
p < (n d , n m ) Yes Yes No No No
Table 2.2: Review of possible cases.
2.2.2 Local methods
The local methods rely on the optimisation of the fit between observed and calculated data. The
data are calculated in a reference model m o using the forward problem and the optimisation
process provides the model update direction ∆m that minimises the data residuals vector ∆d
defined as
∆d = d calc − d obs , (2.23)
where d calc and d obs are respectively, the calculated and observed data vectors. Therefore,
an initial guess of the model m o (or a priori model, starting model) is required to initiate the
inversion process. In some cases, we will see that iterations are necessary as the first model
update will not complete the goal of optimisation. The criteria used for the optimisation process
is the minimisation of the misfit function (or objective function, cost function or least squares
function). This misfit function E is the most often defined as the L 2 norm of the data residuals
(Tarantola, 1987)
E (m) = 1 2 (d calc (m) − d obs ) t (d calc (m) − d obs )
= 1 2 ∆dt ∆d (2.24)

2.2. INVERSION OF LINEAR PROBLEMS 21
The minimisation of the function E (m) is achieved by finding the model parameter m † such
that E ( m †) is minimum. The minimisation is local: the quantities involved in the optimisation
of the misfit function depend on the starting model m o . All local methods require the calculation
of the local gradient of the misfit function ∇ m E, indicating the directions of increase of the
misfit function : the steepest ascent. The gradient vector contains the first derivatives of the
misfit function E (m) with respect to the model parameters m and is defined as
∇ m E = ∂E
∂m . (2.25)
By deriving the misfit function expressed in equation (2.24), the gradient is given by
∇ m E (m) = F t ∆d (2.26)
where F is the Fréchet derivative matrix: the derivative of the data with respect to the model
F = ∂d
∂m . (2.27)
For linear problem, the Fréchet derivative matrix is constant with respect to m and expressed as
F = ∂d
∂m = ∂ (Gm) = G, (2.28)
∂m
is simply the forward operator G and is therefore independent of m.
2.2.2.1 The Gauss-Newton method
Since the forward problem is linear, the misfit function is exactly quadratic and can be expressed
as a Taylor series of the second order
E ( m †) = E (m o + ∆m) = E (m o ) + ∇ m E t ∆m + 1 2 ∆mt H∆m (2.29)
where H is the Hessian, the second derivative matrix defined as
H = ∂ (∇E)
∂m
= ∂Ft (∆d · · · ∆d)
∂m } {{ }
+F t F. (2.30)
n m
For linear problem, as the Fréchet derivative matrix is independent of m, the Hessian is given
by
H = F t F. (2.31)

22 CHAPTER 2. INVERSE THEORY
The minimum of the misfit function will be reached at m † if the gradient at this location is zero
such as ∇ m E ( m †) = 0. Using equation (2.29) to express the derivative of E in terms of m †
we have
∇ m E ( m †) = ∇ m E (m o ) + H∆m † . (2.32)
and the model update that will locate the minimum of the misfit function is thus
∆m † = −H −1 ∇ m E (m o ) . (2.33)
The starting model is then updated, yielding the Gauss-Newton solution
m † = m o − H −1 ∇ m E. (2.34)
Each step involved in the calculation of the Gauss-Newton solution is illustrated Figure 2.3.
The minimum of the misfit function is expected to be found at the first model update. The gradient
vector is easy to calculate as it is simply the transpose of the forward modelling operator
multiplied by the data residuals. However, the determination of the inverse of the Hessian may
require significant additional steps.
Starting Model
Forward Modeling
#"$&%('*),+.-0/ !
1243527698;:2@!ACB
Residuals
Gradient
Hessian
JLKNM
D
EGFIH
Inverse Hessian
Model Update
OQPSRTOQUSVXW*Y[Z]\^_
Figure 2.3: Scheme of the Gauss-Newton method applied to linear problem.

2.2. INVERSION OF LINEAR PROBLEMS 23
2.2.2.2 The gradient method
The gradient method (also called steepest descent method) avoids the determination of the inverse
of the Hessian by updating the model in the steepest descent direction. Since the gradient
vector points towards the steepest ascent direction, the model may be updated according to
m (1) = m o − γ∇ m E (2.35)
where γ is the step length, a scalar real number. The update of the model in the steepest descent
direction assures the decrease of the misfit function so that E ( m (1)) < E (m o ). The expression
of the step length in the linear case is straightforward as it is easy to solve the inverse problem
where γ is the only parameter to resolve since the update direction is set by the gradient. Finding
the step length γ, consist of estimating γ such that the misfit function E is minimum along the
gradient direction. The solution of the step length is strictly equivalent to a Gauss-Newton
method where the misfit function is expressed as
E (m o − γ∇ m E) = E (m o ) − γ (F ∇ m E) t ∆d + 1 2 γ2 (F ∇ m E) t (F ∇ m E) (2.36)
where the steplength γ is the unknown. The minimum of E is obtained if
which yields a value of γ given by
γ =
∇ γ E = ∂E
∂γ = 0 (2.37)
(F∇ m E) t ∆d
(F∇ m E) t (F∇ m E) . (2.38)
The minimum of the misfit function is unlikely to be reached at the first model update and
m (1) ≠ m † . Several model updates may be necessary in order to complete the optimisation
process. The required steps to determine the model update are part of an iteration as shown
Figure 2.4. Several iterations are often needed and the minimum of the misfit function will be
reached in a finite number of iterations until some convergence criteria are met.
2.2.2.3 Interpretation of the Gradient and the Hessian using the SVD
In this section, I will use the Singular Value Decomposition as a tool to analyse the role of the
gradient ∇ m E and the Hessian H.
We can evaluate the gradient in term of the Singular Value Decomposition of G described
in section 2.2.1 as we have F = G and ∇ m E = F t ∆d. The eigenvectors of the Hessian are
simply the model eigenvectors since H = F t F.

‰
24 CHAPTER 2. INVERSE THEORY
Starting Model
d calc = g (m)
Forward Modeling
`ba
Residuals
Iteration k+1
~X4€579‚uƒ„k…>7†!‡,ˆ
c
dfehg
Misfit Function
Gradient
Steplength
Model Update
{}| ikjl

2.2. INVERSION OF LINEAR PROBLEMS 25
where Λ 2 is a (n m × n m ) diagonal matrix with the eigenvalues values of H on the diagonal
(which may contain some zeros if a model null space exists), and zero elsewhere. For a better
understanding of equation (2.42), it can be written as a sum of the model eigenvectors such as
∇ m E =
n m ∑
i=1
(
λ
2
i v i v t i ∆m true
)
. (2.43)
The product v i vi t is an operator in the model space of dimension (n m × n m ), that projects any
model vector on the eigenvector v i (Scales et al. (2001), section 4.9). As the eigenvectors form
an orthonormal basis, the sum of the projection of the vector ∆m true on each eigenvector v i is
simply the vector itself as we have (see equation (2.10))
∆m true =
n m ∑
i=1
(
vi v t i∆m true
)
(2.44)
where each of v i vi∆m t true is the orthogonal projection of the vector ∆m true along the direction
given by the eigenvector v i .
From equation (2.43), it is clear that the gradient vector is the sum of the projected component
of ∆m true along the eigenvectors v i , multiplied by the eigenvalue of v i . Therefore, the
components of the gradient vector on the eigenvector basis are the component of the true perturbation,
stretched by the eigenvalues values of H as shown Figure 2.5. The gradient direction
is the direction of the true model perturbation in a stretched model space.
Remarks on ∇ m E and the eigenvalues of H
• If ∆m true can be expressed as a unique component along one eigenvector (with a nonzero
eigenvalue), the gradient method will converge in one iteration.
• If the eigenvectors have all the same eigenvalues, the gradient method will converge in
one iteration.
• If ∆m true has components in the null space, the gradient will only contain information
about the component of ∆m true along eigenvectors with non-zero eigenvalues.
• The solution ∆m † is the projection of ∆m true on the model subspace V p .

œS œ *ž œ7Ÿ• ›&œ
žu¡[¢¦£
¤*¥S¤*¦¥7§©¨ ¦uª«@¬
26 CHAPTER 2. INVERSE THEORY
­©®©¯
Žu[‘¦’
°*±
“•” –u—˜@š
ÇÈ
½Q¾½*¿¾wÀ©Á ¿uÂÃ@Ä
Figure 2.5: Illustration of the construction of the gradient vector from eigenvectors and eigenvalues
for a 2 dimensional model space. The gradient vector is the component of the true model
perturbation stretched by the eigenvalues of H.
ÅQÆ
²&³´µ ´ µ*·ẃ¸•¹ ·uº[»¦¼
The Hessian
We have seen that for the Gauss-Newton method, the gradient direction is multiplied by the
inverse of the Hessian so that inverse solution can be written as
∆m † = H −1 H∆m true .
The inverse Hessian undo the stretch of the component of ∆m true present in the gradient vector.
If a model null space exists, H is not invertible and a Gauss-Newton solution of the type
given by equation (2.34) can not be obtained. Therefore, H must be constrained within V p to
be invertible and the inverse Hessian becomes
H −1
p
= V p Λ −2
p Vt p . (2.45)
The Fréchet derivative matrix F may also be restricted to U p , V p and the Gauss-Newton solution
is then defined as
m † = m o + H −1
p F t p∆d
The Gauss-Newton solution therefore yields the same solution as the gradient method: the
portion of ∆m true constrained within V p . If the starting model m o = 0, the solution given by

2.2. INVERSION OF LINEAR PROBLEMS 27
local methods will be identical to the generalised inverse solution. This solution nevertheless
requires the dual implementation of the SVD and Gauss-Newton in order to discriminate the
model null-space. An alternative is to damp the Hessian by adding a constant along its diagonal
such as
H d = H + κI (2.46)
where κ is a real, positive number. Because H d does not have eigenvectors with zero eigenvalues,
it is always invertible. Applying a damping term to the Hessian is equivalent to finding the
Gauss-Newton solution that minimises a new weighted misfit function defined as
E = 1 (
∆d t ∆d + κ∆m t ∆m ) (2.47)
2
which minimises both the data residuals and the model length update in proportion to κ.
Gauss-Newton solution and generalised inverse
It can easily be demonstrated that the matrix H −1
p Ft p is in fact the generalised inverse obtained
from the SVD since we have
H −1
p F t p = V p Λ −1
p
= G †
The Gauss-Newton solution is in fact equivalent to the solution given by the generalised inverse
U t p
when there is no a priori knowledge of the model (m o = 0).
2.2.2.4 Preconditioned gradient methods
We have seen that the component of the gradient along the space eigenvectors are the component
of the true model perturbation stretched by the eigenvalues of H. In order to mitigate this effect
without recourse of the inverse Hessian, it is possible to precondition the gradient direction by :
1. Adding a vector to the gradient
2. Weight the data residuals
3. Multiply the gradient by a matrix
The preconditioned gradient direction can always be associated with an equivalent misfit function
E, in which the preconditioned gradient is the steepest ascent. Therefore, an equivalence
exist between modifying the misfit function or the gradient direction.

28 CHAPTER 2. INVERSE THEORY
Adding a vector to the gradient
If the misfit function is defined as
E (m) = 1 2 (Gm − d obs) t (Gm − d obs ) + m t p. (2.48)
where p is a (n m × 1) vector, the steepest ascent becomes
∇ m E = ∇ m E + p. (2.49)
Adding a vector p to the gradient vector is equivalent to minimising the normal misfit function
with an additional term m t p.
Data residuals weighting versus gradient multiplication
The data residuals may be weighted by a matrix W of dimension(n d × n d ) such as the transformed
data residuals ∆¯d are
∆¯d = W∆d (2.50)
The misfit function minimising the L 2 norm of the data misfit ∆¯d is
E = 1 2 ∆¯d t ∆¯d
= 1 2 ∆dt W t W∆d
= 1 2 (Gm − d obs) t W t W (Gm − d obs ) . (2.51)
The gradient of E is
∇ m E = G t W t W (Gm − d obs ) (2.52)
The gradient of E is the preconditioned gradient of ∇ m E with the matrix P of dimension
(n m × n m ) such as
∇E = P∇ m E
where P is
P = G t W t W ( G t) −1
Therefore, weighting the data residuals is equivalent to preconditioning the gradient vector by
multiplication of a matrix. In other words, any action of multiplying the gradient by a matrix
will modify the data residuals involved in the inversion process.
It is often considered that the ideal preconditioning operator P is simply the inverse Hessian
H −1 so that the steepest descent of E points in the same direction than ∆m † .

2.2. INVERSION OF LINEAR PROBLEMS 29
2.2.2.5 The conjugate gradient method
While the steepest descent method assures that the convergence will be reached in a finite number
of iterations, conjugate gradient method improve the convergence rate at little additional
cost. The conjugate gradient optimises the model space search by avoiding to update the model
in a direction that has already been searched. The current update direction ∇ m E is then the
conjugate of the previous update direction. The inversion process is thus assured to converge in
n m iterations, where n m is the number of model parameter. The conjugate gradient is defined
as (Polak and Ribière, 1969)
(
∇m E (k) − ∇
∇ m E (k) = ∇ m E (k) m E (k−1)) ∇ m E (k)
+
∇
∇ m E (k−1)t ∇ m E (k−1) m E (k−1) . (2.53)
From equation (2.53), we can see that conjugate gradient belongs to the family of gradient
preconditioning that add a vector to the steepest ascent direction as described in the previous
section.
2.2.2.6 Numerical examples
In order to illustrate the resolution of linear system using local techniques, I chose as an example,
the linear system of 2 equations and 2 unknowns defined as
⎧
⎨
⎩
m 1 + m 2 = 2
−2m 1 + 3m 2 = 1 . (2.54)
The solution of (2.54) is equivalent to finding the intersection of the 2 straight lines as shown
Figure 2.6a). The solution is evident as the straight lines intersect in m true
forward problem can be expressed in algebraic form using equation (2.4) with
⎛
G = ⎝ 1 1
⎞
⎛
⎠ , m = ⎝ m ⎞
⎛
1
⎠ , d obs = ⎝ 2 ⎞
⎠ .
−2 3
m 2 1
The corresponding misfit function expressed with respect to the component of m is
E = 5m 2 1 + 10m2 2 − 10m 1m 2 − 10m 2 + 5
= (1, 1). The
A SVD of G was carried out using the subroutine svdcmp (Press et al., 1992), yielding the
eigenvectors and singular values
⎛
⎞ ⎛
0.996 0.09
U ≃ ⎝ ⎠ , Λ ≃ ⎝ 1.382 0
⎞
⎠ ,
−0.09 0.996
0 3.618
⎛
⎞
0.851 −0.526
V ≃ ⎝ ⎠ .
0.526 0.851

30 CHAPTER 2. INVERSE THEORY
a) b)
3
7
6
2
5
4
3
1
2
ÉËÊ$Ì]Í#Î
ÒÓ
ÜÝ
1
0
0 1 2 3
0 1 2 3 4 5 6 7
ÔÖÕ
×ÙØ
Figure 2.6: Example of a 2-D, linear inverse problem where a) the solution is the intersection
point m = (1, 1), between two straight lines, b) the corresponding misfit function with the two
eigenvectors v 1 , v 2 are shown centred on the minimum.
ÏÑÐ
ÚÑÛ
Since there are no zero singular values, we have p = n m = n d . There is neither a model nor a
data null space, the solution of the inverse problem yields the true solution: m † = m true (see
Table 2.2). The misfit function is shown Figure 2.6b), with the model eigenvectors centred on
the minimum of the misfit function.
Figure 2.7 shows the minimisation of the misfit function using (a) Gauss-Newton , (b) gradient
, (c) preconditioned gradient and (d) conjugated gradient methods. As expected, the Gauss-
Newton method locates the minimum in one iteration whereas the gradient methods converges
to the minimum in several iterations. The convergence rate is improved by preconditioning
the gradient direction with the diagonal term of the inverse Hessian and the conjugated gradient
convergences in 2 iterations. The evolution of the misfit function with iterations is shown Figure
2.8 for each method .

2.2. INVERSION OF LINEAR PROBLEMS 31
a) b)
7
7
6
6
5
5
4
3
−H −1 ∇E
4
3
&('
)+*-,/.¡0£1¥23,/.¡0
"#
2
2
1
1
$%
7
0 1 2 3 4 5 6 7
c) d)
7
0 1 2 3 4 5 6 7
!
6
6
5
5
öç÷
4
4
åçæ
úüû@ýÿþ¡ £¢¥¤ ýþ¡
3
3
áâ
2
2
èêéìëîí]ï9ðñ@ëîí]ï
ôõ
1
1
¦¨§©¡©
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
ã&ä
ø&ù
Figure 2.7: Local methods applied to a 2-D, linear inverse problem for a) Gauss-Newton , b)
Gradient, c) Preconditioned gradient and d) Conjugated gradient methods. The gradient in c)
was preconditioned with the diagonal term of the inverse Hessian.
Þàß
òàó

32 CHAPTER 2. INVERSE THEORY
Misfit Function
80
70
60
50
40
30
20
10
0 1 2 3 4 5 6 7 8 9 10
Iterations
Figure 2.8: Evolution of the misfit function with iterations for the Gauss-Newton (dotted
black), the gradient (solid black), the preconditioned gradient (solid grey) and the conjugated
gradient (dotted grey).
2.3 Inversion of non-linear problems
In the non-linear case, the forward problem is represented by equation (2.1) where the function
g(m) is a non-linear function of the model parameter m. The misfit function is thus
E = 1 2 (g (m) − d obs) t (g (m) − d obs ) (2.55)
and the Fréchet derivative matrix is
F (m o ) =
∂g (m)
∂m
(2.56)
∣
mo
and can not be directly related to the forward modelling operator and will thus require extra
computational steps.
As for linear problems, non-linear local methods make use of the local gradient of the misfit
function. The non-linear gradient remains given by equation (2.26) but the Fréchet derivative
matrix F is model dependent. A dramatic effect of the non-linearity is that, in order to obtain
the gradient vector, F must be re-defined for any new reference model since
∇ m E (m o ) = F t m o
∆d (m o ) . (2.57)
We will see that the importance of the reference model becomes fundamental for strong nonlinear
problems.

2.3. INVERSION OF NON-LINEAR PROBLEMS 33
2.3.1 Newton method
The Newton method relies on the assumption that the misfit function is approximately quadratic
in the neighbourhood of m o and may thus be expressed as a Taylor series of the second order
E (m o + ∆m) = E (m o ) + ∆m t ∇ m E (m o ) + 1 2 ∆mt H∆m + O (∣ ∣ ∣
∣ ∣∣m 3 ∣ ∣ ∣
∣ ∣∣
)
≃ E (m o ) + ∆m t ∇ m E (m o ) + 1 2 ∆mt H∆m. (2.58)
We can express the Hessian H as the sum of two matrices
H = H s + H a (2.59)
where the non-linear Hessian H s and the approximate Hessian H a are defined as
H s = ∂Ft (∆d · · · ∆d)
∂m } {{ }
, H a = F t F. (2.60)
n d
For the non-linear problem, the matrix H s is non-zero since the Fréchet matrix F is dependent
on m. The demonstration is identical to the Gauss-Newton solution for linear problems. However,
if the misfit function is not exactly quadratic, the Newton method will not converge in one
iteration. An iterative process may be implemented to account for remaining non-linearities and
the model update is given by
m (k+1) = m (k) − H −1 (k) ∇ m E (k)
Remark : The misfit function is locally quadratic if and only if the forward problem is locally
linear. Nevertheless, the Newton method relies on a discrepancy as it is assumed that the misfit
function is quadratic but its curvature is determined accounting for the non linearity of the
problem.
2.3.2 Linearised inversion
The linearised methods relies on the assumption that the forward problem is linear in the neighbourhood
of m o so that the data residuals ∆d have a linear relation with the model perturbation
∆m given by
∆d = F∆m. (2.61)
The data in the perturbed model can thus be written as
d (m o + ∆m) = d (m o ) + F∆m (2.62)

34 CHAPTER 2. INVERSE THEORY
Gauss-Newton method
- Demonstration 1
While Newton method takes into account the full Hessian, the Gauss-Newton method approximates
the Hessian by
H ≃ H a
and thus neglect the non-linear Hessian term H s . The model update then becomes
- Demonstration 2
∆m = H −1
a ∇ mE.
Equation (2.61) may be considered as a linear forward problem where the data is ∆d and the
model is ∆m (not a perturbation !). A solution can hence be found based on the resolution of
linear system shown in section 2.2.2.1. A new optimisation process is set seeking to minimise
the difference between observed data ∆d obs and calculated data ∆d calc . The new misfit function
is defined as
E (∆m) = (∆d calc − ∆d obs ) t (∆d calc − ∆d obs )
where the observed data and calculated data are
= (F∆m − ∆d obs ) t (F∆m − ∆d obs ) (2.63)
∆d obs = d obs − d (m o ) , ∆d calc = F∆m. (2.64)
A reference model m o must be chosen so that the linear assumption holds reasonably and the
data in the reference model d (m o ) are expressed using the non-linear forward problem. The
misfit function E is minimum in ∆m if the gradient of E at this point is null. The gradient with
respect to the model perturbation is
∂E
∂∆m = Ft (F∆m − ∆d obs ) .
The starting model is chosen to be the reference model so that the initial perturbation is zero:
∆m o = 0. The Gauss-Newton solution yields the model update
∆m † = H −1
a F t ∆d obs (2.65)

2.3. INVERSION OF NON-LINEAR PROBLEMS 35
For strong non-linear problems, the quadratic approximation is inaccurate and it may then
be preferable to use a damped Hessian such as
H a = H a + κI. (2.66)
The damped Hessian is not here only motivated by the existence of a model null space as in
the linear case (see section 2.2.2.3). In order to counter the effect of non-linearity and the
irrelevance of the the quadratic approximation, the damped Gauss-Newton model update is
influenced by the steepest descent direction which assures the decrease of the misfit function.
For large κ, the damped Gauss-Newton is closer to simple gradient methods and a step length
should be defined.
Linearised gradient method
An iterative linearised gradient method may be used as an alternative to Gauss-Newton method
in order to avoid the determination of the inverse Hessian. The model perturbation ∆m † is
determined after convergence of a linear, iterative gradient scheme such as
∆m (k+1) = ∆m (k) − γ (k) ∇ ∆m E
= ∆m (k) − γ (k) F t (F∆m − ∆d obs ) .
It is important to keep in mind that for linearised gradient method, the Fréchet derivatives remain
constant as they are not re-defined at each linear iteration. Therefore, after the first iteration, the
gradient does not exactly correspond to the steepest ascent direction.
Compensate for non-linearities
If the misfit function is not exactly quadratic, a non-linear iterative scheme for Gauss-Newton,
as for linearised gradient methods may be implemented as shown Figure 2.9. These non-linear
iteration are compensating for the linearization. A new non-linear forward modelling as well as
the Fréchet matrix are thus re-computed in the updated reference model.
2.3.3 Non-linear gradient method
In non-linear gradient methods, the model is updated at each iteration in the direction of the
steepest descent given by
m (k+1) = m (k) − γ k ∇ m E (k)

O
P
36 CHAPTER 2. INVERSE THEORY
Starting Model
d calc = g
(m (k))
Forward Modeling
Non−linear Iteration k+1
Gauss−
Newton
Linear
Gradient
Linear Iteration
Linear Inversion
687:9@A687:9CBEDF687:9
Linear Model Update
GIHKJML-N
4 5
Convergence
Figure 2.9: Scheme of linearised inversion. For each reference model, a linear update is found
using either Gauss-Newton or iterative gradient method. To compensate for non-linearities, the
reference model can be updated and a new linear inversion carried out until convergence is met.

2.3. INVERSION OF NON-LINEAR PROBLEMS 37
where γ is the step length. Since the Fréchet derivative is re-defined at each iteration, no assumption
of linearity is made and the gradient is the true steepest ascent direction. Finding the
step length γ such that the misfit function is a minimum for E ( m (k+1)) may be achieved by
performing a line search along the gradient direction using parabolic fit (Press et al., 1992). This
method requires the test of several values of E along the gradient direction and an additional
forward modelling is thus needed for each test. In order to save the extra cost of several forward
modellings, the linear estimate of the step length γ can be determined using equation (2.38)
and often provides an acceptable result. It is important to notice that at he first iteration, the
linearised and non-linear gradient are identical.
As for the linear case, a precondition operator may be applied on the gradient. The (damped)
approximate Hessian is often consider to be the optimal preconditioning operator. In order to
speed the convergence rate, conjugate gradient may be used.
2.3.4 Importance of the starting model
Depending on the degree of non-linearity of the forward problem, the misfit function may
present some local minima. A local minimum is a location in the misfit function where the
gradient vanishes but which does not correspond to the lowest possible minimum (the global
minimum). The success of local methods will depend upon the accuracy of the starting model.
If this starting model is located close to a local minimum, local methods will converge into this
minimum and will thus yield a wrong solution. Therefore, the starting model must be located
in the neighbourhood of the global minimum to assure success of convergence.
For Newton and linearised methods, the misfit function must be reasonably quadratic and
the model error must have a quasi linear relation with the data residuals. The remaining nonlinearities
can be compensated for by carrying out non-linear iterations.
The non-linear gradient methods do not rely on the assumption of linearity and tend to
be more robust than methods relying on the quadratic approximation. If the starting model is
located within the global minimum, gradient methods will converge accurately.

38 CHAPTER 2. INVERSE THEORY
a)
YZ
10
9
8
7
6
5
4
3
b)
QR
10
9
8
7
6
5
4
3
Local Minima
2
2
1
VXW
0
0 1 2 3 4 5 6 7 8 9 10
1
Global Minimum
0 1 2 3 4 5 6 7 8 9 10
SUT
Figure 2.10: A 2-D non-linear inverse problem. a) the solution is the intersection point between
a cosine function and a straight line at m = (5, 4) . b) the misfit function shows the global
minimum surrounded by two local minimum.
2.3.5 Numerical examples
A two dimensional non-linear forward operator g(m) and a set of observed data is defined
g (m) =
⎛
⎝ −2 cos ( π
m ⎞
2 1)
+ m2
⎠ , d obs =
−m 1 + m 2
⎛
⎝
4 −1
⎞
⎠ .
The solution of the non-linear inverse problem is the intersection point between the cosine and
the straight line shown Figure 2.10a. This inverse problem has a unique solution in m true =
(5, 4) but the direct inverse operator g −1 can not easily be obtained. The misfit function in
Figure 2.10b shows, as expected, the global minimum in m = (5, 4) but surrounded by two
local minima.
Since the analytical solution of the forward problem is known, the Fréchet derivative matrix
can easily be found and is given by
⎛
F = ⎝
π
π
)
2 2 1 1
−1 1
⎞
⎠

2.3. INVERSION OF NON-LINEAR PROBLEMS 39
and the non-linear and approximate Hessian matrices are
H s =
H a =
⎛
⎝
⎛
⎝
) 2 (
cos
π m ) ⎞
2 1 ∆d1 0
⎠ ,
0 0
) 2 ( sin
2 π
m (
2 1)
+ 1 −π sin
π m ) ⎞
2 1 + 1
−π sin ( π
m ) ⎠ .
2 1 + 1 2
( π
2
( π
2
In order to show the importance of the starting model as well as the performance of the different
local methods, I applied a Newton, Gauss-Newton, non-linear gradient and conjugated gradient
to the resolution of our non-linear inverse problem. Figure 2.11 shows the results of the
inversion for a starting model located in m o = (4.5, 7). For this starting model, the quadratic
approximation is accurate and Newton and Gauss-Newton methods converge into the global
minimum. The non-linear gradients converge as well towards the global minimum. Figure 2.12
shows the same experiment but with a starting model located at m o
= (4, 6). The Newton
method fails to find the global minimum as the quadratic approximation is not adequate (Figure
2.12a). The Gauss-Newton method would therefore also fail and Figure 2.12b shows that
instead, it is preferable to perform a Gauss-Newton methods with a damped Hessian according
to equation (2.66) with κ = 0.5. Although the quadratic approximation is not accurate, both
non-linear gradient and conjugate gradient methods converge into the global minimum.
Figure 2.13 shows the results for a starting model in m o = (7, 9), located in the neighbourhood
of a local minimum. All methods converge into the local minimum and the inversions
fails to provide with the correct answer.
These results demonstrate the importance of the starting model in applying local methods to
non-linear inverse problems. Non-linear gradient methods are more robust as they do not rely
on the quadratic approximation, but require the recomputation of the Fréchet derivatives at each
iteration.

40 CHAPTER 2. INVERSE THEORY
a) b)
10
10
9
8
7
6
5
4
9
8
7
6
5
4
hi
[\
3
2
1
3
2
1
c)
10
9
8
7
6
5
0 1 2 3 4 5 6 7 8 9 10
]_^
10
9
8
7
6
5
j_k
0 1 2 3 4 5 6 7 8 9 10
d)
de
`a
4
3
2
1
4
3
2
1
f_g
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Figure 2.11: Inversion with a starting model at m o = (4.5, 7) showing a) the Newton , b)
the Gauss-Newton, c) the gradient and d) the conjugate gradient methods. For this starting
model, the quadratic approximation is accurate and all methods converge rapidly to the global
minimum.
b_c

2.3. INVERSION OF NON-LINEAR PROBLEMS 41
10
a) b)
10
9
8
7
6
5
9
8
7
6
5
tu
4
3
4
3
xy
2
1
2
1
10
0 1 2 3 4 5 6 7 8 9 10
c) d)
v_w
10
0 1 2 3 4 5 6 7 8 9 10
z_{
9
9
8
7
6
5
8
7
6
5
pq
lm
4
3
2
1
4
3
2
1
r_s
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Figure 2.12: Inversion with a starting model at m o = (4, 6) showing a) the Newton , b) the
Damped Gauss-Newton, c) the gradient and d) the conjugate gradient methods. For this starting
model, the quadratic approximation is not accurate and Newton fails to locate the global
minimum. The damped Gauss-Newton and gradient methods are successful.
n_o

€
42 CHAPTER 2. INVERSE THEORY
10
a) b)
10
9
8
7
6
5
4
9
8
7
6
5
4
ˆ‰
|}
3
2
1
3
2
1
10
0 1 2 3 4 5 6 7 8 9 10
c) d)
~_
10
Š_‹
0 1 2 3 4 5 6 7 8 9 10
9
8
7
6
5
4
9
8
7
6
5
4
„…
3
2
1
3
2
1
†_‡
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Figure 2.13: Inversion with a starting model at m o = (7, 9) showing a) the Newton , b) the
Damped Gauss-Newton, c) the gradient and d) the conjugate gradient methods. This starting
model is not adequate as all local methods converge into a local minimum.
‚_ƒ

2.4. FREQUENCY DOMAIN WAVEFORM INVERSION 43
2.4 Frequency domain waveform inversion
The frequency domain waveform inversion method requires, as in any inverse problem, the
definition of the forward problem.
2.4.1 The forward problem
In order to simulate the propagation of seismic waves, a forward problem must be chosen.
The forward problem is always a simplified representation and takes only into account a limited
aspect of the phenomena involved. The most complete representation (so far) of the propagation
is the 3D, visco-elastic and anisotropic case. Accounting accurately for all these effects is
however highly computationally expensive and the 2D, acoustic, isotropic approximation is
often used.
We thus define the 2D, constant density, acoustic wave equation in the frequency domain as
(
∇ 2 +
)
ω2
c (x) 2 Ψ (x, s, ω) = −S (s, ω) (2.67)
where x is the position vector, c(x) is the velocity, Ψ(x, s, ω) is the complex wavefield and ω is
the circular frequency. A point source S(s, ω) at the location s is given by
S(s, ω) = δ(x − s)A(s, ω)e iϕ(s,ω) (2.68)
where A is the amplitude term and ϕ is the phase of the source wavelet.
The main difficulty of the wave equation (2.67) is that it is not in the form of a forward problem
(equation (2.1)). The data are indeed, not expressed as a combination of model parameters.
In order to establish the forward problem, we need to solve an independent inverse problem.
In this inverse problem, the wavefield Ψ (x, s, ω) plays the role of the model parameter and the
source term S (s, ω), the role of the data parameter. Defining the operator
equation (2.67) can be expressed as
ω2
g (c (x) , ω) = ∇ 2 +
c (x) 2 , (2.69)
g (c(x), ω) Ψ (x, s, ω) = S (s, ω) . (2.70)

44 CHAPTER 2. INVERSE THEORY
The solution of the forward problem requires that the inverse of the operator g must be found
such that
Ψ (x, s, ω) = g −1 (c (x) , ω) S (s, ω)
= G (c(x), x, s, ω) S(s, ω) (2.71)
where G(x, s) is the point source Green’s function for a source located at s. The Green’s
function is the wavefield at the location x for a point source of zero-phase and unit amplitude.
For a given angular frequency ω, it is non-linearly dependent on the velocities of the model. For
homogeneous media (constant velocity c o ), G(x, s) can be expressed analytically and the 2-D
and 3-D free space Green’s function are (Morse and Feshbach, 1953)
where H (1)
o
G 2D (x, s) = i 4 H(1) o (ω ||x − s|| /c o ) , G 3D (x, s) = exp (iω ||x − s|| /c o)
4π ||x − s||
is the zero order Hankel function of the first kind.
In heterogeneous media, the Green’s function cannot be solved analytically and the forward
problem must be solved numerically. The use of ray theory is often very useful as it is a computationally
efficient method to evaluate the Green’s functions ( ˘Cervený et al., 1977; Chapman,
1985). Ray methods nevertheless rely on the high-frequency asymptotic approximation of the
wave-equation and are only accurate in smoothed velocity models, i.e. when the velocity model
varies slowly compared to the wavelength. In order to simulate the propagation of waves in
realistic media (possibly discontinuous), the wave-equation can be discretized (Marfurt, 1984)
and the use of finite difference solvers may be used (Jo et al., 1996; Stekl and Pratt, 1998). The
forward problem is then a solution of a linear inverse problem in which the wavefield Ψ (x, s)
is the unknown.
2.4.2 The inverse problem
The misfit function of waveform inversion is defined as the summation over source-receiver
pairs, of the square of the data residuals:
E = 1 ∑ ∑
(Ψ calc (r, s) − Ψ obs (r, s)) ∗ (Ψ calc (r, s) − Ψ obs (r, s))
2 s r
= 1 ∑ ∑
∆Ψ ∗ (r, s) ∆Ψ (r, s) (2.72)
2
s
r

2.4. FREQUENCY DOMAIN WAVEFORM INVERSION 45
where the superscript ∗ denotes complex conjugation. If the model is represented by m (x), that
is, as a function of spatial position, x, the steepest descent direction is given by
∇ m E (x) = ∂E
∂m (x)
{ ∑ ∑
= Re
s r
∂Ψ ∗ (r, s)
∂m
(Ψ calc (r, s) − Ψ obs (r, s))
}
. (2.73)
The gradient in equation (2.73) is expressed as the sum of the individual source-receiver pair
gradients. For a given source-receiver pair, the Fréchet derivative can be derived from equation
(2.71) and can be written
∂Ψ (r, s) ∂G (r, s)
= S (ω) . (2.74)
∂m ∂m
The main difficulty of computing the Fréchet derivative is that the Green’s function can not be
expressed analytically so that equation (2.74) cannot be used analytically.
Explicit numerical computation of the Fréchet derivative
The Fréchet derivative can be computed explicitly by measuring the effect on the data, of the
perturbation of a single model parameter. The model parameter m (x) is thus perturbed of a
factor β, chosen so that the model perturbation remains small. The explicit Fréchet derivative
is then
( )
∂Ψ (r, s) ∆Ψ
∂m
= lim
Ψ (m + βm) − Ψ (m)
≃ (2.75)
∆m→0 ∆m
βm
Such a computation of the Fréchet derivatives for many model parameters is however very expensive.
This method requires the resolution of a forward problem for each model parameter
to compute the term Ψ (m + βm). For computational efficiency, this approach cannot be envisaged
for the resolution of large problem in which a significant number of model parameters are
involved.
Efficient calculation of the Fréchet derivative using the Born approximation
A more efficient approach is to use the Born approximation to compute the partial derivatives.
The Born approximation linearises the perturbed wavefield according to the relation
Ψ ′ (m + ∆m, r, s) ≃ Ψ (m, r, s) + ∂Ψ ∆m. (2.76)
∂m
The perturbed wavefield Ψ (m + ∆m) is the solution of the wave equation
g(m + ∆m)Ψ ′ (r, s) = S (ω) . (2.77)

46 CHAPTER 2. INVERSE THEORY
For a small model perturbation, the operator g may be linearised so that we have
By introducing equation (2.78) into (2.77) we have
g (m + ∆m) ≃ g (m) + ∂g ∆m. (2.78)
∂m
g(m)Ψ ′ (r, s) + ∂g
∂m ∆mΨ′ (r, s) = S(ω). (2.79)
Using equation (2.76), the perturbed wavefield in (2.79) can be expressed in its linearised form
so that the perturbed wave equation may be written
g(m)Ψ(r, s) + g(m) ∂Ψ ∂g
∂g ∂Ψ
∆m + ∆mΨ(r, s) +
∂m ∂m ∂m ∂m ∆m2 = S(ω). (2.80)
In equation (2.80), the unperturbed wavefield Ψ(r, s) is solution of the wave equation in (2.70).
As the second order term (∆m 2 ) is neglected since we lie within th Born approximation, the
partial derivative can be written
∂Ψ (r, s)
g(m)
∂m
= − ∂g Ψ(r, s). (2.81)
∂m
Equation (2.81) is simply the wave equation in (2.67) where the Fréchet derivative is the partial
derivative wavefield, and the right hand side term is the virtual source (Pratt et al., 1998). The
resolution of equation (2.81) thus requires the determination of a new forward problem: the
Green’s for an excitation at the receiver location and we have
∂Ψ (r, s)
∂m
∂g
(x) = −G (x, r) Ψ (r, s)
∂m
= −G (x, r) ∂g G (x, s) S (ω) . (2.82)
∂m
Equation (2.82) expresses the Fréchet derivative in a way that can be computationally more
efficient than its explicit formula (equation 2.75) as it involves the resolution of two forward
problems: the Green’s functions with an excitation at the source location and at the receiver location.
The calculation of the complete Fréchet matrix for all source-receiver pairs thus requires
n s × n r Green’s function where n s and n r are the number of source and receivers respectively.
2.4.2.1 The linearised inversion and the Born approximation
Linearised inversion methods rely on the assumption that the forward problem is linear in the
neighbourhood of the reference model. Therefore, the Born approximation (equation (2.76)) is

2.4. FREQUENCY DOMAIN WAVEFORM INVERSION 47
used to establish a linear relation between the model perturbation and the data residuals. The
data in the reference model Ψ (m o ) are most often computed using an approximate Green’s
function in smooth or homogeneous media using ray theory. The estimate of the direct inverse
operator may be obtained for stacked data (Cohen and Bleistein, 1979) and for unstacked data
(Clayton and Stolt, 1981; Miller et al., 1987). The result of diffraction tomography yields a
practical analytical solution in the (ω, k) domain (Devaney, 1984; Wu and Toksöz, 1987; Pratt
and Worthington, 1988) assuming the reference model is homogeneous. Often, the formulation
of the inverse problem based on the explicit minimisation of the data misfit as described in
section 2.3.2 are considered more robust (Tarantola, 1984b; Ikelle et al., 1988; Lambaré et al.,
1992).
Linearised inversion based on the ray theory does however suffer from a severe limitation:
the asymptotic ray approximation will break down if the updated reference model contains
strong heterogeneities. This would therefore prevent non-linear iteration from being carried
out. Therefore, if the starting model is not adequate and contains large errors, the Born approximation
will not be valid and the solution of the linearised problem inaccurate.
2.4.2.2 The non-linear gradient methods
As described in section 2.3.3, non-linear gradient methods require the computation of the
Fréchet derivative at each iteration involving the computation of the Green’s function from the
sources and receivers location. Since the updated model may contain strong heterogeneities,
the non-linear gradient methods must be used with an accurate wave propagation solver such as
finite difference solver of the wave equation.
The gradient computation
The gradient direction can be obtained by including equation (2.82) into equation (2.73) so that
we have
{ ( ) ∑ ∑
∗ }
∂g
∇ m E (x) = −Re
s r ∂m (x) Ψ (x, r) G (x, r) ∆Ψ (r, s)
{ }
∂g ∑ ∑
= −Re
G ∗ (x, s) S ∗ (ω)G (x, r) ∗ ∆Ψ (r, s) . (2.83)
∂m
s
r

48 CHAPTER 2. INVERSE THEORY
The gradient in equation (2.83) can be decomposed into the multiplication of two wavefields:
the forward propagated wavefield P f (x, s) of the source term, defined as
P f (x, s) = G (x, s) S (ω) (2.84)
and the back-propagated wavefield P b (x, r, s) of the data residuals from the receiver location is
given by
P b (x, r, s) = G (x, r) ∗ ∆Ψ (r, s) . (2.85)
The gradient then becomes
∇ m E (x) = −Re
{ ∂g
∂m
}
∑ ∑
Pf ∗ (x, s) P b (x, r, s)
s r
(2.86)
(see Pratt et al. (1996), equation 12)). In equation (2.86), the gradient is the result of the multiplication
of complex quantities as both forward and back-propagated wavefield are complex.
Attenuation may be accounted for through the introduction of imaginary velocity in the wave
equation, in which case the imaginary velocities are updated according to the imaginary part
of the gradient (Song et al., 1995; Hicks and Pratt, 2001). Only (real) velocity model is here
considered and is updated at iteration (k + 1) according
m (k+1) (x) = m (k) (x) − γ (k) ∇ m E (x) (2.87)
Model parameterisation
The estimate of the velocity model may be achieved by parameterising the inverse problem with
respect to velocity, slowness or square of slowness. The choice of parameterisation influences
the gradient direction through the term ∂g (Table 2.3). Since the parameterisation does not
∂m
modify the shape of the misfit function, it cannot be classified as a precondition gradient as it is
explicitly part of definition of the gradient.
Velocity Slowness Slowness Squared
m c 1/c 1/c 2
∂g
∂m −2ω2 /c 3 2ω 2 /c ω 2
Table 2.3: Possible model parameterisations.

2.4. FREQUENCY DOMAIN WAVEFORM INVERSION 49
Efficient computation of the gradient
Equation (2.86) showed that the gradient involves for each source-receiver pair, the multiplication
of two wavefields, from the source and the receiver locations. As the forward propagated
wavefield from the source, P f (x, s) does not depend on the receiver location, equation (2.86)
can be written
∇ m E (x) = −Re
{ ∂g
∂m
∑
Pf ∗ (x, s) ∑
s
r
P b (x, r, s)
}
. (2.88)
This trivial manipulation shows that for a given source gather, the forward propagated field can
be multiplied by the sum of the back-propagated wavefield from the corresponding receivers.
Equation (2.88) offers a critical improvement as the back-propagation from all receivers can
in fact be carried out simultaneously at no additional computational cost. Therefore, the cost
of back-propagation of the data residuals for all receivers together is the same than for a single
receiver. The gradient calculation thus requires n s ×2 forward modelling instead of n s ×n r if all
the data residuals were back-propagated independently. This computational gain creates a major
inconvenience: the Fréchet derivative matrix is not known as it would require the computation
of the individual source-receiver pair gradients according to equation (2.86) . The efficient
computation of the gradient thus implies that the Fréchet matrix is not computed explicitly.
Effective computation of the steplength
The linear estimate of the step length γ for the waveform inverse problem, is often considered
sufficient although line search techniques are possible. However, since the Fréchet derivative
matrix is not known, the vector F∇ m E of equation (2.38) must be evaluated by perturbing
the model in the gradient direction. An additional forward modelling must be performed to
compute the data in the perturbed model d (m o − β∇ m E). The Fréchet derivative matrix is
thus given by
F∇ m E = d (m o − β∇ m E) − d (m o )
. (2.89)
−β
The term F∇E is then used to compute the linear estimate of the step length γ .
Why the approximate Hessian cannot be efficiently computed
The approximate Hessian given by equation (2.60) requires the knowledge of the Fréchet derivative
matrix. Since the efficient computation of gradient is done without building the Fréchet
derivative matrix, the Hessian cannot be easily obtained. However, some waveform inversions

50 CHAPTER 2. INVERSE THEORY
using Gauss-Newton methods have been carried out (Pratt et al., 1998; Hicks, 1999; Hicks and
Pratt, 2001). This requires the explicit computation of the Fréchet derivative matrix. Such an
approach thus involves a forward modelling per model parameter and can only be applied in
cases where the number of model parameters is limited.
Non-linear gradient methods and the Born approximation
It is very important to keep in mind that the non-linear, iterative gradient approach do not rely
on the linear assumption. The confusion is often made that the success of non-linear inversion
methods depends upon the validity of the Born approximation, i.e. that the data residuals have a
linear relation with the true model perturbation. This is a wrong assumption as the non-linearity
is attacked by recomputing the Fréchet derivative at each iteration. The Born approximation
is only used for the computation of the Fréchet derivatives. This should allow compensation
of some of the errors contained in the starting model. The success of non-linear waveform
inversion only depends on the fact that a descent path leading to the global minimum exists. Of
course, since the waveform inverse problem is highly non-linear, the misfit function will present
many local minima and the determination of the starting model will therefore be a critical step
in the waveform inversion process.
2.5 Conclusion
The inverse problem may be solved using two main different approaches. The direct methods
aims to determine the direct inverse operator while the local methods are based on the minimisation
of the misfit between observed and calculated data.
In the linear case, the accuracy of the inverse solution will depend on the existence of a
model null space. If no model null space exists, the direct and local methods yield the exact
solution and the generalised inverse and the Gauss-Newton solver are equivalent.
In the case where a model null space exists, the solution of the generalised inverse has
no components in V o . The gradient method will update the starting model with components
contained in V p only so that the components in V o remain unchanged. Since the Hessian is
not invertible, the Gauss-Newton solution can be obtained if the Hessian is restrained to V p in
which case it remains identical to the generalised inverse method. A more widely used approach
for handling the null-space is to minimise a misfit function with an additional damping term.

2.5. CONCLUSION 51
The resolution of non-linear inverse problems introduces additional difficulties as the Fréchet
derivative matrix is model dependant and the misfit function is not quadratic and may present local
minima. Newton and linearised methods assumes that the misfit function is locally quadratic.
If this assumption holds reasonably, these approaches will be successful but non-linear iterations
may be required to compensate for non-linearities. For strong non-linear problems, non-linear
gradient methods appear to be more robust as they do not rely on the quadratic approximation
and define the Fréchet matrix at each iteration. In order to improve the convergence rate,
the preconditioning of the gradient could be optimally performed by the inverse of a damped
approximate Hessian.
The waveform inverse problem is non-linear and the efficiency of the local methods will
depend on the accuracy of the starting model. If linearised inversion are to be used, the starting
model must be good enough so that the minimum of the quadratic function corresponds to
the global minimum. Because linearised inversion is the most often implemented using ray
theory, non-linear iteration can not be carried out. The non-linear gradient methods appears to
be more robust as there are not bounded by the Born approximation. The convergence rate can
be improved by preconditioning of the gradient. The damped approximate Hessian can not be
easily used as a precondition operator as the Fréchet matrix is not explicitly computed. The
success of the non-linear gradient will depend on the location of the starting model with respect
to the global minimum. If the starting model is not adequate, the inversion will converge into a
local minimum and will thus yields a wrong answer.

52 CHAPTER 2. INVERSE THEORY

Chapter 3
A strategy for selecting temporal
frequencies
3.1 Introduction
1 Waveform inversion may be implemented either in the time domain (Tarantola, 1986; Mora,
1987; Bunks et al., 1995; Shipp and Singh, 2002), or in the frequency domain (Pratt and Worthington,
1990; Liao and McMechan, 1996); implementations for both acoustic and elastic wave
equations exist. The frequency domain approach is equivalent to the time domain approach
when all frequencies are inverted simultaneously (Pratt et al., 1998). Reciprocally, the inversion
of a single sinusoidal component of time domain data is equivalent to the inversion of a single
component of frequency domain data.
The choice of a domain for inversion allows one to apply specific methodologies to precondition
either the data residuals or the gradient in ways that may improve the convergence and/or
the linearity of the inverse problem. For example, time windowing the residuals is often useful
(Shipp and Singh, 2002); this requires a time domain representation of the data residuals. It
is nevertheless possible to devise strategies for time domain conditioning of the data by making
use of complex-valued frequencies, as we will see in Chapter 4. Conversely, when using a
frequency domain approach it is straightforward to carry out the inversion adopting a strategy
that proceeds sequentially from the low to the high frequencies (at no additional computational
cost). Since low frequency data are more linear with respect to the model perturbations than
1 This chapter is an extended version of an article accepted for publication in Geophysics (co-author G. Pratt).
53

54 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
high frequency data, such a strategy dramatically improves the chance for the inversion to locate
the global minimum of the full, wide band inverse problem. Such a methodology may of course
also be applied in the time domain by low pass filtering of the data (Bunks et al., 1995), however
the computational efficiency of the frequency domain approach is far superior when this strategy
is to be employed (Marfurt, 1984). Furthermore, the frequency domain is also well suited
to compensate for variations in the source signature, which will cause certain frequencies to be
under-represented in the image. Although this effect may be compensated for by preconditioning
of the time domain gradient (Causse et al., 1999), it is easier to apply the preconditioning in
the frequency domain as we process each mono-frequency gradient individually (Hicks, 1999).
One of the most important advantages of frequency domain inversion is the ability to provide
a unaliased image using a very limited number of frequencies, as observed by Pratt and
Worthington (1988); Liao and McMechan (1996) and Forgues et al. (1998). This observation
is consistent with the results of Wu and Toksöz (1987), who demonstrated that even a single
frequency of a given seismic survey geometry yields information on a finite portion of the
model in the wavenumber domain. Since the computational cost of frequency domain inversion
is directly proportional to the number of frequencies used, this is a significant advantage for
waveform inversion of real, multi-shot seismic data.
The claim that a limited number of frequencies would suffice when inverting reflection seismic
data was investigated by Freudenreich and Singh (2000), who pointed out that no strategy
for selecting the appropriate frequencies had yet been developed. They concluded that the frequency
domain inversion fails in the presence of a limited offset range.
In this chapter, it is shown that frequency domain inversion of reflection data using only a
few frequencies (properly selected) can yield a result that is comparable to full time domain
inversion, for any available range of offsets. I develop a strategy that adapts the selection of
imaging frequencies to the available offset range. The main idea is that the larger the offset,
the fewer frequencies are required. This strategy takes advantage of the effect of an “image
stretch”, which increases with increasing offset. This effect is commonly observed in prestack
depth migration (Gardner et al., 1974)- in a 1-D earth, it is equivalent to NMO stretch. Stretch is
often considered to have a negative impact on seismic imaging, although Haldorsen and Farmer
(1989) showed that the stretch effect can be utilised to compensate for the lack of low frequencies
of the source and hence to improve the spectral content of stacked data. In the strategy
defined, each successive frequency is selected by evaluating the image stretch at the maximum
offset for the previous frequency. Although the strategy is developed using a 1-D assumption, I

3.2. GRADIENT VECTOR AND WAVENUMBER ILLUMINATION 55
will show that the same approach can also be applied effectively in 2-D, heterogeneous media.
Because of the kinematic equivalence between the first iteration gradient vector and prestack
depth imaging for reflection data (Tarantola, 1986), the strategy also has important implications
for frequency domain implementations of prestack depth migration (such as that advocated by
Schleicher et al. (1993)).
This chapter begins by deriving a specific formula for the gradient using a plane wave approximation
in homogeneous background media that predicts analytically the wavenumber illumination
of a given target. This allows us to define a strategy for choosing frequencies for
continuous wavenumber illumination of a 1-D target. The key idea is to take advantage of the
maximum offset data in the acquisition. This strategy is then validated on the 1-D imaging
experiment of Freudenreich and Singh (2000). I go on in the chapter to show that the strategy
can be successfully applied to data generated in the 2-D heterogeneous Marmousi model.
3.2 Gradient vector and wavenumber illumination
In chapter 2, the formulation of the gradient is applicable to arbitrary reference media: equation
(2.83) may be used in cases where the background media is inhomogeneous, provided the
Green’s functions are calculated appropriately. In order to proceed, the following assumptions
are now introduced:
1. We assume we may ignore amplitude effects.
2. We assume the reference medium is homogeneous, with a velocity c o .
3. We assume we are in the far field, so that we may replace the Green’s functions with plane
wave approximations.
Note that these are introduced only in order to proceed with the analysis that will lead to a
frequency selection strategy; none of these are required for the actual inversions. Under these
three assumptions both Green’s functions, G o (x, s) and G o (x, r) may be approximated by
incident and scattered plane waves:
G o (x, s) ≈ exp (ik o ŝ · x)
G o (x, r) ≈ exp (ik oˆr · x)
(3.1)

56 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
where k o = ω/c o is the wavenumber of the incident and scattered waves in the homogeneous
reference medium, and ŝ and ˆr are respectively unit vectors in the incident propagation direction
(source to scatterer) and the inverse scattering direction (receiver to scatterer) direction.
Inserting equation (3.1) into equation (2.83) yields
∇E (x) = −ω 2 ∑ ∑
Re {exp (−ik o ŝ · x) exp (−ik oˆr · x) ∆Ψ (r, s)}
s r
= −ω 2 ∑ ∑
Re {exp (−ik o (ŝ + ˆr) · x) ∆Ψ (r, s)} . (3.2)
s r
The only space-dependent term in equation (3.2) is the complex exponential, which, for a given
source-receiver pair, oscillates at a single wavenumber, given by the vector k o (ŝ + ˆr). Equation
(3.2) is therefore an inverse Fourier summation in which the weight function for each
source/receiver pair plane wave combination is determined by the data residuals ∆Ψ(r, s).
A single frequency, single source-receiver pair in the imaging algorithm described by equation
(3.2) would therefore yield only a single wavenumber in the image, ∇E(x) (see Figure
3.1). In order to reconstruct useful images, we need to provide additional wavenumbers. There
are two ways to recover a range of wavenumbers: (1) we can use a wider band of frequencies (as
we do if we use a time-domain algorithm), or (2) we can use a range of different source-receiver
pairs that sample the same image point at different directions, ˆr and ŝ as shown Figure 3.2. A
combination of both approaches seems sensible. Almost all seismic imaging strategies take advantage
of the first approach: the wavenumber bandwidth of the image is provided by temporal
bandwidth in the experiment. My suggestion is that some of the required image bandwidth can
in fact be provided by making optimal use of different offsets in the reflection experiment to
create a variety of plane wave imaging directions. Temporal frequencies may then be selected
to avoid redundancy of the wavenumber information.
3.2.1 Analysis of the gradient within the Born approximation
The gradient image of equation (3.2) is now examined by estimating the data residuals ∆Ψ(r, s)
in the context of the Born approximation, through which we may write
∆Ψ(r, s) ≈ −ω 2 ∫
dx G o (r, x)G o (x, s) δm(x) (3.3)
where δm(x) is the perturbation in the parameter m(x) = 1/c 2 (x) (see for example Miller et
al. (1987), equation (8)). Equation (3.3) approximates the data residuals by establishing a linear
relation with the model perturbation δm. Although in the inversion scheme, the data residuals

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3.2. GRADIENT VECTOR AND WAVENUMBER ILLUMINATION 57
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58 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
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3.2. GRADIENT VECTOR AND WAVENUMBER ILLUMINATION 59
Substituting the Born estimate of the data residuals, equation (3.5) into the plane wave gradient
image equation (3.2) we obtain
g(x) = ω 4 ∑ s
∑
Re { exp (−ik o (ŝ + ˆr) · x) ˜M (k o (ŝ + ˆr)) } , (3.6)
r
which, as stated above in reference to equation (3.2) is an inverse Fourier summation. We have
now written this summation in a form that makes it clear that the weights in the summation
are given by the Fourier components of the model. Thus the summation will yield an image of
the true model, distorted however due to the limited availability of scattering directions. I may
emphasise this fundamental observation:
One source-receiver pair with an incidence direction ŝ and an inverse scattering direction ˆr
yields information on only a single wavenumber in the spectrum of the model, given by k o (ŝ+ˆr).
Multiple source-receiver pairs yield information on a finite region in wavenumber space, even
if only a single frequency component is used. This basic relationship is illustrated in Figure
3.1.This result is similar to demonstrations made in the context of diffraction tomography (Devaney,
1981; Wu and Toksöz, 1987) and linearised inversion in the (ω, k) domain (for example
(Clayton and Stolt, 1981; Ikelle et al., 1986)).
A comment on the gradient in the linear case
Equation (3.6) is in the form of an inverse Fourier summation: if we have a complete and uniform
sampling in the source and receiver scattering directions (an impossibility for most geophysical
scenarios), then this inverse Fourier summation will undo the forward Fourier transform
in ˜M(k), i.e.
g(x) → ω 4 δm(x), (3.7)
in which case we see that the gradient will recover a scaled image of the original model. Equation
(3.7) illustrates the limitation of the gradient as an imaging operator: the factor ω 2 in the
forward problem is not cancelled by the gradient operator, instead it is re-used. The gradient
thus provides a filtered image due to the distortion of the true perturbation as described in section
2.2.2.3, using the singular value decomposition. The model space eigenvectors and singular
values of the multi-frequency inverse problem (time-domain) are therefore frequency dependant
and the high frequencies are associated with the highest singular values. A correct inverse modelling
operator would cancel this term. In diffraction tomography and linearised inversion, the
inverse operator contains a filter term inversely proportional to ω 2 , thus ensuring the fidelity of

60 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
the reconstruction (Wu and Toksöz, 1987). In descent schemes, an iterative approach should
ensure that the distortions in the gradient are eventually removed.
In multi-frequency, or time-domain inversions the factor ω 4 implies a double application of
the second time derivative operator. For this reason, time domain schemes are not very good
at recovering the very low wavenumbers, which are suppressed by the derivative operators.
Iterations may eventually recover some of the low wavenumbers, but without preconditioning
in some form the convergence rate will be slow. Furthermore, since the low frequencies have a
more linear relation with the low wavenumber components of the model, the chance of locating
the global minimum will be diminished if this information is not used at an early stage of the
inversion. This effect can be compensated for by low-pass filtering of the data in time (Bunks
et al., 1995), and progressively including higher frequencies at a later stage. However, if the
inversion only involves a few frequencies, it is always more efficient to carry this out in the
frequency domain (Marfurt, 1984; Jo et al., 1996). In single frequency inversions the factor ω 4
is of no consequence, as the step length used in equation (2.38) will compensate for this factor
correctly by rescaling the gradient.
3.3 The 1-D case
Having reviewed the basic relationship between the wavenumber spectrum of the image and
the incident and scattered wave directions, we now wish to use this relationship to develop
a strategy for the optimal selection of frequencies in frequency domain imaging of seismic
reflection data. The fundamental assumption here will be that the strategy may be developed
by examining the illumination of a 1-D model (i.e., one in which the velocity varies only as a
function of depth), for which we need only recover the vertical wavenumbers from a range of
source-receiver offsets. At a later stage, the strategy is tested on a model with significant 2-D
structure in order to evaluate the generality of the approach.
3.3.1 Wavenumber illumination for the 1-D case
As shown previously using a plane wave approximation, the contribution to the gradient image
of a single source-receiver pair has only a single wavenumber component, given by k o (ŝ + ˆr).
If the model is 1-D, then all reflection points in the subsurface are midpoint reflection points.
In other words, we may consider a common shot gather to be equivalent to a common midpoint

E
7
3.3. THE 1-D CASE 61
;=<
S
CMP
R
65
43
BDC
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>
:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:
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:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:9:
898989898989898989898989898989898989898989898989898
Figure 3.3: The 1-D basic scattering experiment. The incident plane wave is scattered from a
1-D thin layer at the midpoint between source and receiver. For the 1-D case the incident and
scattered angles, θ and −θ are equal.
(CMP) gather. The basic configuration is illustrated in Figure 3.3: each source-receiver pair
records data from a given midpoint with a particular combination of symmetric plane waves
(the equivalent wavenumber diagram for this configuration appears in Figure 3.1). For a 1-D
earth, the incident and scattering angles are symmetric, so that
k o ŝ = (k o sin θ, k o cos θ) and k oˆr = (−k o sin θ, k o cos θ). (3.8)
In equation (3.8) the angles θ and −θ are the propagation directions of the source and receiver
wave vectors, defined with respect to the vertical axis. From Figure 3.3 we observe that
cos θ =
z √
h 2 +z 2
sin θ =
h √
h 2 +z 2 ,
(3.9)
where h is the half offset and z is the depth of the scattering layer.
(3.9) into equation (3.8) we find that the components of the vector k o (ŝ + ˆr) are
Substituting expression
k x = 0
k z = 2k o α,
(3.10)
with
α = cos θ

62 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
=
=
z
√
h2 + z 2
1
√
1 + R
2 , (3.11)
where R = h/z is the half offset-to-depth ratio.
Equation (3.10) defines the wavenumber illumination of a given 1-D layer for a given
source-receiver pair. The illumination varies with the wavenumber k o = ω/c o and with the
factor α, which in turn depends on the half offset-to-depth ratio, R of the source-receiver pair
relative to the target. The horizontal wavenumber of the gradient is always zero, as we expect
for a 1-D model. Each source-receiver pair contributes a different vertical wavenumber to the
gradient image; the larger the offset-to-depth ratio, the smaller is the vertical wavenumber of
the contribution. Thus, for a given frequency and offset, the vertical wavenumber recovered
will decrease with depth.
3.4 Strategy for choosing frequencies for 1-D imaging
In the preceding sections, the wavenumber spectral coverage, for a given source-receiver pair
at a single frequency over a 1-D earth, was clarified. Using equations (3.10) and (3.11), but
now for a range of offsets, we find that for a given surface seismic acquisition characterised by
an offset range [0, x max ], the vertical wavenumber coverage k z of a 1-D thin layer for a given
frequency is limited to the range [k zmin , k zmax ], where
k zmin = 2k o α min
k zmax = 2k o
(3.12)
with
α min =
1
√
1 + R
2
max
(3.13)
where R max
= h max /z is the half offset-to-depth ratio obtained at the maximum half offset
h max , and z is the depth of the target layer. The minimum and maximum wavenumbers, k zmin
and k zmax are produced by the furthest and nearest offsets, respectively. Expressing equation
(3.12) in terms of frequency we have
k zmin = 4πfα min /c o
k zmax = 4πf/c o
(3.14)

3.4. STRATEGY FOR CHOOSING FREQUENCIES FOR 1-D IMAGING 63
where f is the frequency expressed in Hz and c o is the velocity in the background medium.
From relation (3.14), we may define the wavenumber coverage of a multi-offset acquisition as
∆k z ≡ |k zmax − k zmin |
= 4π (1 − α min ) f/c o , (3.15)
while the bandwidth
k zmax
= 1 √
= 1 + Rmax k zmin α 2 . (3.16)
min
The wavenumber coverage increases linearly with frequency and is therefore greater for high
than for low frequencies (Wu and Toksöz, 1987), while the wavenumber bandwidth is a function
only of the offset-to-depth ratio.
The strategy for selecting frequencies is determined as follows: each frequency has a limited,
finite band contribution to the image spectrum. In order to recover the target accurately
over a broad range of wavenumbers, the continuity of the coverage of the object in the wavenumber
domain must be preserved as the imaging frequencies are selected. I choose
k zmin (f n+1 ) = k zmax (f n ) (3.17)
where f n+1 is the next frequency to be chosen following the frequency f n . The principle (illustrated
in Figure 3.4) is that the maximum wavenumber of the smaller frequency must equal the
minimum wavenumber of the larger frequency. This strategy thus relies on using the full range
of offsets available.
Using the condition defined in equation (3.17) and substituting equation (3.14) we arrive at
the relation
f n+1 =
f n
. (3.18)
α min
This lead us to a discretisation law in which the frequency increment ∆f n+1 is given by
∆f n+1 = f n+1 − f n
( ) 1 − αmin
=
α min
f n
= (1 − α min ) f n+1 (3.19)
Equation (3.19) shows that the optimum frequency increment is not constant and increases
linearly with frequency. This is an interesting result as the commonly used frequency domain
sampling theorem sets the frequency increment to be constant and equal to
∆f st = 1
T max
, (3.20)

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64 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
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Figure 3.4: Illustration of the frequency discretisation strategy. A single discrete frequency
yields a range of vertical wavenumbers in the image (governed by the maximum and minimum
offsets in the survey). Each successive frequency is selected in such a way as to obtain a
continuous coverage in vertical wavenumbers.
where T max is the maximum recorded time. Equation (3.20) prevents “time-aliasing” (i.e.,
wrap-around in the time domain due to insufficient frequency sampling) . Since ∆f n in equation
(3.19) is typically bigger than ∆f st in equation (3.20), the strategy we are proposing is a
discretisation that would produce time-aliasing of a single seismic trace in time, but will yield
an unaliased image of the target in depth from a set of seismic traces with a range of offsets.
3.5 Numerical test I: 1-D models
The 1-D experiment used in this section (see Figure 3.5) was originally used by Freudenreich
and Singh (2000), who applied a time domain waveform inversion method (Shipp et al., 1997)
to evaluate and compare the time and frequency domain inversion approaches. (Freudenreich
and Singh simulated a frequency domain approach by inverting a single sinusoidal component
of the time domain data.) They observed that when far offsets were included (out to 10 km),
their frequency domain approach using 3 discrete frequencies did indeed provide an accurate
imaging result equivalent to their time domain inversion. However, when attempting waveform

3.5. NUMERICAL TEST I: 1-D MODELS 65
inversion for the same 3 frequencies, but using only a near offset range (limited to 3 km),
their inversion result suffered from oscillatory artifacts due to the limited set of input frequency
components. They concluded that when only near offsets were available, the frequency domain
approach was less robust.
Freudenreich and Singh raised the important question of exactly how the inversion frequencies
should be chosen. In this section, I use the strategy on the same experiment and thus
provide an answer to this question.
3.5.1 The synthetic data
The data for the 1-D experiment (Figure 3.5) consist of a single shot gather comprising 201
receivers, with a receiver interval of 50 meters. The maximum offset range available is [0, 10]
km. The target of the imaging experiment is the thin layer located at 2 km depth. The velocity
model is exactly 1-D, and has thus only wavenumber components along the vertical k z axis. For
all inversion results I present below, the starting model was identical to this model, but without
the thin layer. All the modelling and inversion tests were performed with absorbing boundary
at the top of the model, and using a point source with a Ricker wavelet centred on 4 Hz. The
time domain seismograms shown in Figure 3.5 are synthesised from the frequency domain
finite difference results, using 50 frequencies within a range [0.2 − 10] Hz (thus representing a
maximum recording time of 5 s).
3.5.2 Validation of the plane wave predictions
In the previous section, I quantified the wavenumber coverage k z of a single source/receiver
pair within the plane wave assumption, for a 1-D model (equation (3.10)). In this section,
the gradient (descent direction) is generated using equation (2.86), with point source Green’s
functions and the results are tested against equation (3.10).
Figure 3.6 depicts the first stage in the numerical computation of the gradient at 5 Hz using
equation (2.86) for 2 individual source-receiver pairs (at 3 km and 10 km offsets). The 5 Hz
Green’s functions were calculated by the method of frequency-domain finite differences (Pratt
and Worthington, 1990; Jo et al., 1996). The background model is as described above, i.e.,
as in Figure 3.5a), without the thin layer. The figure shows the real parts of the forward and
back propagated wavefields and the gradient, the product Re { Pf ∗(x, s)P b(x, r, s) } (see equation
(2.86)). Note that to compute the back-propagated wavefield P b (x, r, s), the data residuals

66 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
a)
Velocity (km/s)
2.3 2.4 2.5 2.6 2.7 2.8
0
0.5
1.0
b) c)
S
10 km
R R
1 2
R R
... ...
R
...
R
201
0
1
Offset (km)
0 1 2 3 4 5 6 7 8 9 10
Depth (km)
1.5
2.0
2.5
3.0
3.5
4.0
°=°=°=°=°=°=°=°=°=°=°=°=°=°=°=°=°=°
±=±=±=±=±=±=±=±=±=±=±=±=±=±=±=±=±
Time (s)
2
3
4
5
Figure 3.5: Velocity model after Freudenreich and Singh, 2000. a) Velocity profile of the 1-D
model, b) acquisition geometry of the 1-D basic imaging experiment: a single source and 201
receivers were used, c) synthetic, time-domain data from the model, using a Ricker wavelet
centred on 4 Hz and with 400 ms of time delay.
, ∆Ψ(r, s) were first computed for the experiment by subtraction of the data generated with and
without the thin layer.

ÄÅ
ÆÇ
ÈÊÉ ËÍÌ¿Î
Ï
ÑÐ
Ò
Ô
º
¸¹
s·
»½¼¿¾
À ¾
Á
ÃÂ
Ó
Õ
µ
³H´ ²
Depth (km)
Depth (km)
0
1
2
3
4
0
1
2
3
4
Distance (km)
0 1 2 3 4 5 6 7 8 9 10
Distance (km)
0 1 2 3 4 5 6 7 8 9 10
Depth (km)
Depth (km)
0
1
2
3
4
0
1
2
3
4
Distance (km)
0 1 2 3 4 5 6 7 8 9 10
Distance (km)
0 1 2 3 4 5 6 7 8 9 10
Depth (km)
Depth (km)
0
1
2
3
4
Distance (km)
0 1 2 3 4 5 6 7 8 9 10
Distance (km)
0 1 2 3 4 5 6 7 8 9 10
0
Figure 3.6: Illustration of the computation of the 5 Hz gradient ( equation (2.86)) for the model in Figure 3.5, using two sourcereceiver
pairs, at offsets of 3 km (top row) and 10 km (bottom row). The midpoint for each offset pair is marked.
1
2
3
4
CMP
CMP
3.5. NUMERICAL TEST I: 1-D MODELS 67

68 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
The fundamental properties of the image in 1-D media (noted previously) may also be noted
on Figure 3.6 by examining the wavepaths at the midpoint locations:
1. At the midpoint location the gradient image contains only vertical wavenumbers.
2. Large offsets generate low vertical wavenumbers and near offsets generate high wavenumbers.
3. The wavenumber of the reconstruction increases with decreasing offset-to-depth ratio
(i.e., the deeper in the model we go, the higher the wavenumber for a given offset).
A full reconstruction is provided by combining gradient images such as those in Figure 3.6
for all source-receiver pairs (i.e., all offsets). The reconstructed images for 2 Hz and for 5 Hz
are illustrated in Figure 3.7a). Because of the equivalence between shot gathers and midpoint
gathers for 1-D models, the images may be considered to be equivalent to a set of individual
1-D images (or common image point gathers), each obtained from a midpoint gather with a
different half-offset, h. Each result is a 2-D image which manifests the expected decrease in
vertical wavenumber as the offset increases.
Figure 3.7b) shows the Fourier transform of the gradient images in depth, with an overlay
representing the predicted wavenumber coverage from equation (3.10). The comparison confirms
that the plane wave prediction is an adequate representation of the wavenumber coverage
in the far field and with a homogeneous background velocity model. This shows that the analytic
equations developed within the plane wave assumption may be safely used in the development
of the frequency selection strategy.
3.5.3 A remark on “image stretch”
Equation (3.10) predicts a decrease in vertical wavenumber content of the gradient image with
offset; this effect is evident on Figure 3.7. This shift towards low wavenumbers with offset is
referred to as “image stretch”. The notion that the seismic image from large offsets is lower
in resolution has long been observed (Buchholtz, 1972; Dunkin and Levin, 1973): this effect
is usually termed “Normal Moveout (NMO) stretch” (see for example Yilmaz (1987)). As it is
shown in the appendix A, the stretch factor 1/α in equation (3.10) is identical to that caused by
NMO stretch. If the processes of NMO correction, stacking and depth conversion are applied,
NMO stretch will, inevitably, add low wavenumber content to the image in a manner identical

3.5. NUMERICAL TEST I: 1-D MODELS 69
a) b)
Depth (km)
Half Offset (km)
0 1 2 3 4 5
0
1
2
3
4
ãØäÚåŒæÝçåßèSéê½ë¬ç
Wavenumber (1/km)
0
1
2
3
4
5
Half Offset (km)
0 1 2 3 4 5
Depth (km)
0
1
2
3
4
Half Offset (km)
0 1 2 3 4 5
ÖØ×ÚÙŒÛÝÜÙßÞSàá½â¬Ü
Wavenumber (1/km)
0
1
2
3
4
5
Half Offset (km)
0 1 2 3 4 5
Figure 3.7: a) A set of 1-D amplitude normalised gradient images at 2 Hz (top) and 5 Hz
(bottom) as a function of half-offset, h for a mid-point gather. The 1-D model used in this test is
shown in Figure 3.5a); the location of the expected anomaly is at 2 km depth. The 1-D gradient
image at a single frequency, for a single source-receiver pair contains only a single vertical
wavenumber, and hence is oscillatory. b) the vertical Fourier transform of the images in a) .
The white line is the vertical wavenumber predicted from plane wave theory in equation (3.10).
The vertical wavenumber decreases with increasing half offset in a manner exactly equivalent
to NMO stretch.

70 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
to that observed in Figure 3.7. If these processes are replaced with prestack depth migration,
the effect remains (Gardner et al., 1974; Levin, 1998). Usually this is considered detrimental,
and data with a large offset-to-depth ratio are muted in standard data processing. An exception
to this is the work of Haldorsen and Farmer (1989), who showed that NMO stretch may be used
to compensate for a lack of low frequency content in the data spectrum in order to improve the
spectral contain of a stack section.
3.5.4 1-D waveform inversion
In this section, I describe the application of the frequency discretisation strategy developed
above to the 1-D basic imaging experiment in Figure 3.5, using two survey geometries: (1) A
near offset survey with a maximum offset of 3 km, and (2) a far offset survey with a maximum
offset of 10 km. Given the depth of the bottom of the scatterer z = 2.2 km, equation (3.18)
gives the following sequence of frequencies for the two offset ranges:
Near offset survey [0, 3] km
Far offset survey [0, 10] km
f = 2, 2.4, 2.9, 3.5, 4.3, 5.2, 6.3, 7.6, 8 Hz
f = 2, 5, 8 Hz
Table 3.1: Frequency discretisation sequences for the near and far offset surveys. The last
frequency 8 Hz is chosen arbitrarily so that both experiments use identical frequency ranges
within the spectrum of the source signature.
The two discretisation sequences are illustrated in Figure 3.8, in which the continuity of the
wavenumber coverage is evident. The near offset survey requires 9 frequencies, whereas the far
offset survey only 3 frequencies. It is implicit in the strategy that the larger the offset range is,
the fewer frequencies are required.
The inversion results for each of the surveys are shown Figure 3.9. In order to complete
the imaging of the midpoints, the full set of common image point gradient images (shown
in Figure 3.7) were stacked to form a single, 1-D image at an equivalent midpoint location
for each frequency. In selecting frequencies, I followed the sequence given in Table 3.1, one
frequency at a time (i.e., “sequentially”), iterating 20 times at each discrete frequency. Figure
3.9 shows both the final results (top row) and the individual contributions for each frequency
to the image (bottom row). This selection of frequencies, although discretized far below the
frequency domain sampling theorem, yields a continuous wavenumber coverage of the target
without aliasing (see the wavenumber domain representations in Figure 3.9). Each imaging

3.5. NUMERICAL TEST I: 1-D MODELS 71
Wavenumber (1/km)
a)
10
9
8
7
6
5
4
3
2
1
ö÷Qømù¢ú
ûüQý`þgÿ
0
0
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Frequency (Hz)
Frequency (Hz)
Wavenumber (1/km)
b)
10
9
8
7
6
5
4
3
2
1
ìíQî`ïgð
ñòQómô¢õ
Figure 3.8: Frequency sequence generated by equation (3.18), for a) the near offset survey (0–
3 km) and b) the far offset survey (0–10 km). The near offset survey requires 9 frequencies,
whereas the far offset survey only 3 frequencies.
frequency for the far offset survey clearly yields a wider band of vertical wavenumbers than
does the equivalent frequency for the near offset survey.

a)
Depth (Km)
Depth (Km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Velocity perturbation (m/s)
-200 -100 0 100 200
Near Offset Range
Final Result
b)
FT
FT
Contribution of each frequency
Wavenumber (1/km)
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Amplitude
0 100 200 300 400 500 600 700 800 900
e)
Depth (Km)
Depth (Km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Velocity perturbation (m/s)
-200 -100 0 100 200
Far Offset Range
Final Result
f)
c) d)
g)
h)
FT
FT
Contribution of each frequency
Wavenumber (1/km)
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Amplitude
0 100 200 300 400 500 600 700 800 900
Figure 3.9: Sequential frequency domain inversion results for the near offset survey (a-d), and the far offset survey (e-h). The
true perturbation is shown as a dotted line. The top row shows the final results in both the depth domain (a,e) and the wavenumber
domain (b,h) while the bottom row shows the contributions at each discrete frequency, both in the depth domain (c, g) and in the
wavenumber domain (d, h).
72 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.

3.5. NUMERICAL TEST I: 1-D MODELS 73
The far offset survey is also more successful in recovering the magnitude of the velocity
perturbation. This is due to the fact that the inversion of the first frequency (2 Hz) from the far
offset survey yields information on much lower wavenumbers than does the 2 Hz result from
the near offset survey. This is fundamental, and it is due to the image stretching remarked on
above. The low wavenumbers are equally important in recovering the true velocity perturbation.
Time domain waveform inversion can be simulated using the frequency domain approach by
inverting all frequency components within the data simultaneously, rather than sequentially. By
“All” frequencies, I mean with a complete sampling according to the sampling theorem (3.20).
The gradient for this simultaneous, “time-like” inversion procedure is the sum of the singlefrequency
gradients at each iteration. Figure 3.10 shows the results of the inversion using the
same offset ranges using time-like inversion (i.e., inverting for 50 frequencies simultaneously).
The far offset survey, once more, does a better job of recovering the magnitude of the velocity
perturbation (as in frequency domain inversion, and for the same reasons). Comparing the
sequential, frequency domain inversions and the time-like inversions, we note that performing
the waveform inversion with a limited number of frequencies according to the strategy yields
an acceptable result at a much smaller computational cost, i.e., a factor of 5 fewer frequencies
for the near offset range, and a factor of 16 fewer frequencies for the far offset range.
3.5.5 Sensitivity to noise
The strategy developed suppresses the redundancy in the wavenumber coverage of the 1-D target
by reducing the number of frequencies given by the sampling theorem. However, some
of this redundancy is most desired when noise is present in the seismic data to assure the destructive
summation of incoherent signal. In order to investigate the sensitivity to noise of our
frequency selection strategy, the 1-D inversion is carried out after having added white noise to
the data using the program suaddnoise of the Seismic Un*x package (Stockwell, 1997). The
reflection hyperbola of the 1-D target with noise corresponding to a very high level of noise
(signal to noise ratio of 1) is shown Figure 3.11b and can be compared with the original reflection
on Figure 3.11a. The 1-D sequential inversion of the near offset data (0-3 km) using
the 9 frequencies dictated by the strategy is shown Figure 3.11c. Since this experiment was
carried out by inverting a single frequency at a time, the reconstruction is highly sensitive to
noise and yields a poor result. Although the level of noise is here very high, this begs the important
question of the efficiency of single frequency waveform inversion in the presence of

74 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
Near Offset Range
a) b)
0
Velocity perturbation (m/s)
-200 -100 0 100 200
0
Amplitude
0 100 200 300 400 500 600 700 800 900
0.5
1
1.0
2
3
Depth (Km)
1.5
2.0
2.5
FT
Wavenumber (1/km)
4
5
6
3.0
7
8
3.5
9
c)
4.0
0
0.5
1.0
Velocity perturbation (m/s)
-200 -100 0 100 200
Far Offset Range
d)
0
1
2
3
10
Amplitude
0 100 200 300 400 500 600 700 800 900
Depth (Km)
1.5
2.0
2.5
FT
Wavenumber (1/km)
4
5
6
3.0
7
8
3.5
9
4.0
10
Figure 3.10: Time-like waveform inversion of the near offset range (a,b) and the far offset range
(c,d). The true perturbation shown as a dotted line. 50 frequencies were inverted simultaneously
in order to simulate time domain waveform inversion. Results are shown in both depth domain
(a,c) and wavenumber domain (b,d).

3.5. NUMERICAL TEST I: 1-D MODELS 75
a)
Offset (km)
0 1 2 3 4 5 6 7 8 9 10
b)
Offset (km)
0 1 2 3 4 5 6 7 8 9 10
2
2
Time (s)
3
4
Time (s)
3
4
0
5
c) d) e) f)
Velocity perturbation (m/s)
-200 -100 0 100 200
0
Velocity perturbation (m/s)
-200 -100 0 100 200
0
5
Velocity perturbation (m/s)
-200 -100 0 100 200
0
Velocity perturbation (m/s)
-200 -100 0 100 200
0.5
0.5
0.5
0.5
1.0
1.0
1.0
1.0
1.5
1.5
1.5
1.5
Depth (Km)
2.0
Depth (Km)
2.0
Depth (Km)
2.0
Depth (Km)
2.0
2.5
2.5
2.5
2.5
3.0
3.0
3.0
3.0
3.5
3.5
3.5
3.5
4.0
4.0
4.0
4.0
Figure 3.11: Impact of noise on the reconstruction of the 1-D target for an inversion of the near
offset data (0-3 km). a) Original reflection hyperbola on the 1-D target. b) Reflection hyperbola
with white noise (signal to noise ratio of 1). c) Sequential inversion using 9 frequencies
according to the selection strategy as in Figure 3.9. d) Simultaneous inversion using the same 9
frequencies. e) Simultaneous using 16 frequencies obtained with an effective maximum offset
of 2 km. f) Time-like (simultaneous) inversion using 50 frequencies.
noise. Figure 3.11c shows that inverting the same 9 frequencies simultaneously, the reconstruction
is much improved. The reconstructed target contains more noise than the equivalent time
domain inversion using 50 frequencies (Figure 3.11e). The noise can nevertheless be attenuated
by increasing some of the redundancy using 16 frequencies in the inversion as shown Figure
3.11d. The sequence of 16 frequencies was generated using a maximum offset of 2 km in the
calculation of α min (equation 3.13).
Despite the high level of noise present in the data, the simultaneous inversion using 9 frequencies
does yield a good image. The frequency decimation suppresses some of the redundancy
but the stack of each offset contribution to the image does nevertheless allow destructive
summation of some of the random noise.

76 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
3.6 Numerical test II: 2-D models
This strategy explicitly ensures the continuity of the wavenumber coverage only along the k z
axis, and only for a 1-D target in a homogeneous medium. The applicability of the strategy to
2-D, heterogeneous velocity models is therefore unknown. In the 2-D case, other directions in
the wavenumber spectrum of the model must be recovered. In this section, the strategy is tested
using a more demanding, 2-D application.
3.6.1 Marmousi velocity model
In order to evaluate the applicability of the strategy to 2-D models, the approach is tested with
synthetic seismic data from the 2-D Marmousi velocity model (Versteeg, 1994). Waveform
inversion has already been applied to data from the Marmousi model using both a time domain
approach (Bunks et al., 1995) and a frequency domain approach (Forgues et al., 1998). Forgues
et al. (1998) performed frequency domain waveform inversion with a starting velocity model
which was a smoothed version of the true model. They also used an arbitrary selection of
imaging frequencies. These two over-simplifications are avoided in the test described below.
3.6.1.1 Synthetic data
A key conclusion of this analysis is that large offsets are important if we want to recover the
lowest wavenumbers in the model (which, in turn allow us to recover the true magnitude of
the velocity anomalies). The original Marmousi experiment was limited in offset to only 3
km. The seismic survey in the Marmousi model was therefore modified and remodelled to
create new, wide-angle data. The velocity model was restricted to 2 km in depth to assure the
efficient illumination of the model at large offset-to-depth ratios by the diving and refracted
waves (Figure 3.12). These arrivals will be needed for the determination of the starting model
of the waveform inversion (see below). The new synthetic survey comprises 96 sources (every
96 meters) and 384 receivers (every 24 meters), with a maximum offset of 9.2 km (simulating
the geometry of an Ocean Bottom Cable (OBC)). The synthetic data shown in Figure 3.12
were created using frequency domain finite differences; time domain data were extracted from
122 frequencies in the range [0.12 − 15] Hz by Fourier synthesis. A Ricker source wavelet
with a spectrum centred on 5.5 Hz was used in the simulation. The forward modelling and
the inversions were all performed with a free surface boundary at the top of the model; the

3.6. NUMERICAL TEST II: 2-D MODELS 77
a)
V (m/s)
b)
3500
3000
2500
2000
1500
Depth (km)
Offset (km)
0 1 2 3 4 5 6 7 8 9
0
0
0.5
1.0
1.5
2.0
Distance (km)
0 1 2 3 4 5 6 7 8 9
0
Offset (km)
-4 -3 -2 -1 0 1 2 3 4
0
Offset (km)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
1
1
1
2
2
2
Time (s)
3
4
Time (s)
3
4
Time (s)
3
4
5
5
5
6
6
6
7
7
Shot 1 at 48 m Shot 48 at 4560 m Shot 96 at 9168 m
7
Figure 3.12: Marmousi wide angle synthetic data. a) The velocity model, unchanged from the
original Marmousi model but truncated at 2 km depth, b) three representative wide angle shot
gathers. The acquisition was expanded to wide offsets by recording data for all shots at all 384
receiver points along the surface (simulating an OBC recording geometry).
simulations therefore include the effect of free surface multiples.
3.6.2 Determination of the starting model by traveltime inversion
In order to apply waveform inversion to the wide angle Marmousi data set, a starting velocity
model is required that must be located in the neighbourhood of the global minimum of the
objective function for the starting frequency. A suitable starting model may be obtained by
combining waveform inversion with ray-based traveltime inversion (Pratt and Goulty, 1991). I
used traveltime inversion of the hand-picked first arrivals, using the program “FAST” (Zelt and
Barton, 1998).

78 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
Depth (km)
0
0.5
1.0
1.5
2.0
Distance (km)
0 1 2 3 4 5 6 7 8 9
0.5
Eikonal
1 1.5 2 2.5 3 3.5
4
Shot 1
Depth (km)
0
0.5
1.0
1.5
2.0
Ray Tracing
Distance (km)
0 1 2 3 4 5 6 7 8 9
Depth (km)
0
0.5
1.0
1.5
Distance (km)
0 1 2 3 4 5 6 7 8 9
0.5
2
Shot 48
Depth (km)
0
0.5
1.0
1.5
Distance (km)
0 1 2 3 4 5 6 7 8 9
2.0
2
1.5 1 1
1.5
2.0
Depth (km)
Distance (km)
0 1 2 3 4 5 6 7 8 9
0
0.5
1.0
1.5
4
0.5
Shot 96
Depth (km)
0
0.5
1.0
1.5
Distance (km)
0 1 2 3 4 5 6 7 8 9
2.0
3.5
3
2.5
2
1.5
1
2.0
Figure 3.13: Example of the finite difference solver of the eikonal in a smoothed Marmousi
model for the 3 shot points (1,48 and 96). The ray tracing uses the negative time gradient to
define the rays connecting the receivers to the source location.
The forward problem
The FAST software uses a finite difference solver of the eikonal equation to forward model
the first arrival times (Vidale, 1990; Podvin and Lecomte, 1991). Figure 3.13 shows the first
arrivals forward modelling in a smoothed Marmousi model. The finite difference method yields
the first arrival at each node of the model thus allowing the determination of isochrones. Each
source-receiver ray is defined from the receiver location and is propagated to the source location
using the path given by the negative time gradient (normal to the isochrones).
What information should we expect from traveltime inversion
The eikonal equation is based on the high-frequency asymptotic approximation of the wave
equation. This approximation is valid if the medium varies slowly with respect to propagated
wave length. In order to investigate the type of velocity model that should be expected from
the traveltime inversion, I carried out a traveltime finite difference modelling of the eikonal in
a increasingly smoothed version of the true Marmousi model. The smoothing was carried out

3.6. NUMERICAL TEST II: 2-D MODELS 79
in the vertical and horizontal directions using the program smooth2 of the Seismic for Unix
(SU) package. For each smoothed model, the traveltime are compared with the hand picked
first arrivals of the waveform data. The evolution of the traveltime misfit function with respect
to the degree of smoothness of the true Marmousi model is shown Figure 3.14. The true model
(not smoothed) corresponds to a smoothness factor of 0. The misfit function decreases with
smoothness until it reaches a minimum for a smoothing factor of 36 and then re-increases.
This is a critical observation as the true model is not the model for which the misfit function is
minimum. This leads us to a conclusion that is consistent with the asymptotic approximation:
the global minimum of the traveltime misfit function does not correspond to the true model but
rather a smoothed version of it.
Traveltime inversion
The traveltime inversion was first carried out to estimate the 1-D medium that best fit the data
followed by solving for a strongly smoothed 2-D model, and then progressively less smoothed
models. The final traveltime result is a relatively smooth, 2-D velocity structure that fits the
picked time with an RMS residual of 27 ms (Figure 3.15a-b)) — a reasonable level considering
the 200 ms period of the central frequency of the source. Finite difference modelling in this result
produces seismic traces that closely match the first arrival waveforms, but contain virtually
no reflected waves (Figure 3.15d)).
3.6.3 Waveform inversion
The 2-D, smooth traveltime inversion velocity model in Figure 3.15a) was used as a starting
model for the subsequent waveform inversion. The frequency sequence in the inversion was
calculated using equation (3.18), with a target depth of z = 2 km (i.e, the maximum depth in
the model), and a maximum offset x max = 4 km. Although the maximum offset is in reality 9.2
km, the sequence was calculated with this “effective” offset range because the far offsets are
under represented in the OBC shooting geometry, and hence these will carry a disproportionally
small weight in the inversion. Furthermore, the edges of the model are not illuminated by the
widest offsets, especially at depth. Using these parameters, the frequency selection strategy
yields three frequencies within the bandwidth of the data: 5,7 and 10 Hz. 20 iterations were
carried out at each imaging frequency sequentially.
Figure 3.16 shows the contribution of each frequency and the inversion results are shown

80 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
a)
200
150
RMS residuals
100
50
V (m/s)
b)
3500
3000
2500
2000
1500
Depth (km)
0
0.5
1.0
1.5
2.0
0
0 10 20 30 40 50 60 70 80
Smoothing Factor
Distance (km)
0 1 2 3 4 5 6 7 8
Figure 3.14: a) Evolution of the traveltime misfit function with respect to increasing degree of
smoothness of the true Marmousi model. b) The smoothed model for which the misfit function
is minimum (smoothing factor of 36).

0
0
0
3.6. NUMERICAL TEST II: 2-D MODELS 81
a)
Initial residuals
Final residuals
b)
V (m/s)
3500
3000
2500
2000
1500
Depth (km)
0
0.5
1.0
1.5
Velocity Model
Distance (km)
0 1 2 3 4 5 6 7 8 9
c)
Ray Tracing
2.0
2.0
2.0
Depth (km)
1.5 1.0 0.5
Depth (km)
1.5 1.0 0.5
Depth (km)
1.5 1.0 0.5
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
d)
Offset (km)
0 1 2 3 4 5 6 7 8 9
0
0
Waveform Modeling
Offset (km)
-4 -3 -2 -1 0 1 2 3 4
0
Offset (km)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
1
1
1
2
2
2
Time (s)
3
4
Time (s)
3
4
Time (s)
3
4
5
5
5
6
6
6
7
7
7
Figure 3.15: First arrival traveltime inversion showing a) the traveltime residuals versus offset,
b) the final traveltime inversion velocity model, and c) examples of ray tracing in the final model
for shots 1,48 and 96. This velocity model fits the first arrival picks to an RMS level of 27 ms.
d) Finite difference seismic traces (normalised) in this velocity model are also shown, for the 3
shot positions.

82 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
V (m/s)
V (m/s)
V (m/s)
250
0
-250
250
0
-250
250
0
-250
Depth (km)
Depth (km)
Depth (km)
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
Distance (km)
0 1 2 3 4 5 6 7 8 9
At 5 Hz
Distance (km)
0 1 2 3 4 5 6 7 8 9
At 7 Hz
Distance (km)
0 1 2 3 4 5 6 7 8 9
At 10 Hz
Figure 3.16: Waveform inversion in Marmousi: velocity perturbation introduced following the
completion of each frequency.
Figure 3.17. As expected, the resolution of the image improves as higher frequencies are used
in the inversion. The final velocity model (following the use of 10 Hz data) is very close to the
true model. Except at the very edges of the model where the coverage is incomplete, the full
complexity of the 2-D structure is recovered. Figure 3.18 shows three representative velocity
profiles within the model, and compares these to the true model and the starting model from
traveltime inversion, from which it may be concluded that not only are the structures correctly
imaged, but also that the magnitudes of the velocity perturbations are accurately estimated. In
spite of the oscillation of the individual contributions to the image at intermediate frequencies
(Figure 3.17), there is no evidence of any oscillatory artifacts (“ringing”) in these results.

3.6. NUMERICAL TEST II: 2-D MODELS 83
V (m/s)
V (m/s)
V (m/s)
V (m/s)
3500
3000
2500
2000
1500
3500
3000
2500
2000
1500
3500
3000
2500
2000
1500
3500
3000
2500
2000
1500
Depth (km)
Depth (km)
Depth (km)
Depth (km)
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
Distance (km)
0 1 2 3 4 5 6 7 8 9
After 5 Hz
Distance (km)
0 1 2 3 4 5 6 7 8 9
After 5 and 7 Hz
Distance (km)
0 1 2 3 4 5 6 7 8 9
After 5,7 and 10 Hz
Distance (km)
0 1 2 3 4 5 6 7 8 9
True Model
Figure 3.17: Waveform inversion results in the Marmousi model at progressively higher frequencies.

84 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
Depth (m)
Depth (m)
Depth (m)
500
1000
1500
2000
500
1000
1500
2000
500
1000
1500
2000
Velocity (m/s)
1000 1500 2000 2500 3000 3500 4000
0
At 2.3 km
Velocity (m/s)
1000 1500 2000 2500 3000 3500 4000
0
At 4.5 km
Velocity (m/s)
1000 1500 2000 2500 3000 3500 4000
0
At 6.9 km
Figure 3.18: Velocity profiles at three locations in the Marmousi model (see Figure 3.17), showing
the true model (grey), the starting model (dotted) and the result of the waveform inversion
(solid).

3.7. DISCUSSION 85
An essential aspect of any inversion is an illustration of the degree to which the result actually
predicts the observations. Figure 3.19 shows the data residuals for the same shot gathers
showed in Figure 3.12 and 3.15 computed in the starting model and in the final velocity model
(after 10 Hz). Although only 3 frequencies were used for the waveform inversion, the waveforms
in the time domain (created using all 122 frequencies) show an excellent fit with the true
waveforms. Some misfit remains at the edge of the mode (near offset of shot 1 and 96) due to
the lack of coverage of the edge of the model.
3.7 Discussion
3.7.1 Efficiency of the strategy for 2-D heterogeneous media
I have shown in this chapter that the frequency selection strategy defined yields very accurate
results in both 1-D and 2-D heterogeneous media, despite the fact that the strategy was developed
using the simplest case of a 1-D target embedded in an homogeneous model. The strategy
assumes 1-D illumination by straight rays only, in which case the incident and scattering angles
yield information only on the vertical component of the target. In more realistic 2-D, heterogeneous
medium, other wavenumber directions must be recovered. The success of the strategy
when applied to more complex models (such as the Marmousi model in the previous section)
may be explained by the nature of the background velocity model. A more complex model will
produce additional coverage due to two distinct effects:
1. Non symmetric illumination: In complex, 2-D models the incident and scattering angles
are different. Energy reaches the receivers from non-horizontal structures (containing
non-vertical wavenumber components). Hence the inversion will recover additional
wavenumbers away from the vertical axis.
2. Ray bending: In models with an increase in velocity with depth, the incident and scattering
angle are wider than those predicted in a homogeneous model due to the progressive
curvature of ray paths toward the horizontal axis with depth. Wider scattering angles
provide lower wavenumber information than that predicted in a homogeneous medium;
at very wide angle diving waves exist, which are scattered from the very low wavenumber
components of the velocity model (i.e., they tell us about the background velocities).

86 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
Initial Residuals
Final Residuals
Offset (km)
0 1 2 3 4 5 6 7 8 9
0
Shot 1
Offset (km)
0 1 2 3 4 5 6 7 8 9
0
1
1
2
2
Time (s)
3
4
Time (s)
3
4
5
5
6
6
7
0
1
2
Offset (km)
-4 -3 -2 -1 0 1 2 3 4
7
Shot 48
0
1
2
Offset (km)
-4 -3 -2 -1 0 1 2 3 4
Time (s)
3
4
Time (s)
3
4
5
5
6
6
7
Offset (km)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
0
7
Shot 96
Offset (km)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
0
1
1
2
2
Time (s)
3
4
Time (s)
3
4
5
5
6
6
7
7
Figure 3.19: Data residuals in the Marmousi model , in the starting model (left column) and
in the final model of the waveform inversion (right column) for shots 1,48 and 96 . Despite
using only 3 frequencies in the waveform inversion, the modelling using 122 frequencies shows
a good fit between the real and estimated data. This is a true amplitude display, the colour scale
of initial and final residuals are identical.

3.7. DISCUSSION 87
The application of the frequency selection strategy will thus in fact produce some redundancy
in wavenumber coverage, since the wavenumber coverage of a single frequency is
actually wider than the one predicted.
These effects together seem to permit an effective 2-D coverage of the model observed with
the Marmousi experiment, even with a limited number of frequencies selected according to a
strategy based on 1-D, homogeneous models.
3.7.2 Practical implementation of the frequency selection strategy
My strategy seeks to suppress redundancy of information by taking advantage of the wavenumber
coverage inherent in multi-offset surveys. Some redundancy remains when the strategy is
applied in heterogeneous media; in cases one may also wish to further increase this redundancy.
For example, in the presence of non-coherent noise, redundancy will allow the destructive summation
of unwanted signal. Furthermore, since the far offsets are more sensitive to errors in the
background velocity model and therefore more non-linear (due to longer propagation distances),
it may be preferable to decrease the dependence of the imaging on the far offset information.
This can be easily done by decreasing the effective offset to depth ratio R max used in equation
(3.13) by using a half offset h max that is smaller than the true maximum value (as was done
in the Marmousi model). This will likely improve the quality of the imaging, particularly in
the case of significant dip on the reflectors (in which case the scattering angles are somewhat
smaller than for horizontal reflectors).
3.7.3 The equivalence between gradient images and migration
Many authors have identified the kinematic equivalence between the first iteration gradient image
and prestack migration (e.g., Tarantola (1986); Mora (1989)). Migration maps the data, after
muting of the first arrivals, to “isochrones” in the model space (Miller et al., 1987), whereas the
gradient maps the data residuals (which may contain transmitted as well as reflected arrivals)
to the wavepath (see Figure 3.6). Transmitted events map within the first Fresnel zones of the
wavepath, while reflected events map to the higher order Fresnel zones, which are elliptical
zones corresponding to the isochrones used in migration (Woodward, 1992).
In the first iteration of a waveform inversion scheme the starting model is normally a highly
smoothed model, as in the numerical test of the Marmousi model above. Such a model will gen-

88 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.
erate accurate transmitted arrivals but will generate no reflected energy (as in Figure 3.15). As
a result, the first iteration data residuals will be dominated by reflections. In the numerical test
using the 1-D model above, the initial data residuals consisted only of reflected energy from the
target at 2 km depth. Under these conditions the first iteration image is kinematically equivalent
to a migration of the data. The gradient is not a particularly well-designed migration operator,
since there are no focusing terms or amplitude terms to act as preconditioners, nevertheless
the correspondence is significant and frequency domain migration algorithms could potentially
make use of the strategy developed in this chapter.
This correspondence between gradient images and migration operators implies that the spectral
properties of the gradient image noted in this chapter also apply to migration. Multi-offset
migration also contains an “image stretch” effect (Gardner et al., 1974; Tygel et al., 1994) that
is normally considered detrimental (Brown, 1994; Levin, 1998). The meaning commonly given
to migration is the recovery of discontinuities in the subsurface, so that the suppression of low
wavenumbers may even be considered desirable. It was shown in this chapter that the image
stretch, when properly handled, will in fact contribute the required low wavenumbers to the
imaged velocity structure.
However, the successful application of the strategy to prestack migration would depend on
the accurate combination of multi-offset images. This implies that the contribution of each
offset to the image must carry true amplitude information. The result of a simple stack of the
multi-offset images will fail to recover the true velocity perturbations if the images contain
inaccurate relative amplitudes. This is also true for the simple gradient in waveform inversion,
but the iterative implementation of the descent methods corrects for the inaccuracies in the stack.
Migration is usually formulated as a one-step process, without iteration, in which case effects
such as spherical divergence, and the number of input traces contributing to each offset image
must be explicitly compensated for. In order to use this approach to frequency selection for
prestack depth migration, the required migration operators need to at least preserve amplitudes
during the combination of multiple offset images (e.g. Schleicher et al. (1993)).
As an alternative, since the implementation of the strategy implies a significant reduction
of the computational cost of the imaging, iterative migration/inversion techniques may become
more feasible.

3.8. CONCLUSION 89
3.8 Conclusion
I have defined a strategy for selecting temporal frequencies for efficient waveform inversion.
This selection scheme defines a much sparser set of frequencies than does the frequency domain
sampling theorem. This is possible because this approach makes use of the multi-offset
coverage of each reflection point and optimises the use of information contributed by each
source-receiver pair. By taking advantage of the image stretch occurring with increasing offset,
the strategy adapts the selection of frequencies to the maximum offset present in the data. The
idea is simple: the further this maximum offset is, the fewer frequencies are needed. In deriving
the strategy, I made the approximations of a 1-D target in an homogeneous background media,
illuminated by plane waves. I nevertheless show that this strategy is also very efficient when
applied to data from 2-D heterogeneous media.
Waveform inversion was used in order to illustrate numerically the accuracy of the method.
I showed that, for a 1-D target in a homogeneous medium, using only the frequencies defined
by our strategy produces a result comparable with equivalent, wide band, time domain inversion
(using the full set of frequencies as defined by the sampling theorem).
I also used the 2-D Marmousi velocity model to test the strategy in a more realistic, heterogeneous
medium. This experiment yielded very good results, in which the full range of
wavenumbers of the velocity model was recovered, demonstrating the imaging potential of parameterising
the full wavefield using only a limited set of frequencies. A traveltime tomographic
inversion of the first arrivals was used in order to create a starting model for our waveform inversion
that fell within the neighbourhood of the global minimum of the misfit function at the
minimum frequency used. This minimum frequency (5 Hz) was chosen so that the resolution
limit imposed by traveltime tomography was compatible with the linearity requirement of the
waveform inversion. This frequency is unrealistically low for real data and I will investigate in
the following Chapter, waveform inversion starting from a higher frequency.
Because of the kinematic equivalence between gradient images in waveform inversion and
prestack depth migration, the strategy can be easily applied in practice. Large offsets in the
acquisition are not required in order to take advantage of the effect of stretch. Implementation
of the strategy to standard reflection profiles with offsets limited to less than 3 km still allow
a significant decimation of the number of required frequencies. However, in order to make
use of the information arising from all offsets, amplitude preserved migration operators will be
required to ensure that the relative amplitudes of the common offset images are accurate.

90 CHAPTER 3. STRATEGY FOR SELECTING TEMPORAL FREQUENCIES.

Chapter 4
Waveform inversion starting from realistic
frequencies
4.1 Introduction
The previous chapter demonstrated that waveform inversion may be carried out in the frequency
domain using only a very limited number of temporal frequencies. The underlying empirical
assumption is that if the inversion of the first frequency (f min ) yields successful convergence
to the global minimum, the inversion of higher frequencies should then be also successful.
The 2-D Marmousi numerical experiment of Chapter 3 explored the possibility that a suitable,
smooth starting model may be determined from first arrival traveltime tomography. The fundamental
challenge of waveform inversion is therefore to assure that the starting velocity model
is located in the neighbourhood of the global minimum of the misfit function when using the
smallest frequency available in the data. The Marmousi experiment showed that a smooth starting
model from traveltime tomography is clearly adequate for a waveform inversion starting
at f min = 5 Hz. Unfortunately, such a frequency is considered unrealistically low since real
exploration seismic data contain higher minimum frequencies (> 7 Hz). Because non-linearity
rapidly increases with increasing frequency, waveform inversion at higher frequencies present
more dramatic non-linearity effects which are more likely to cause convergence into a local
minimum.
Assuming that the starting model is smooth, the goals of waveform inversion are to: 1-
correct velocity errors, and 2- “fill in” the missing wavenumber information. It is often consid-
91

92 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
ered that the effect of non-linearity is caused by velocity errors contained in the macro model.
This argument is however a simplification as the lack of intermediate and high wavenumber
information may also be a source of non-linearity despite the fact that the low wavenumber
components are accurate. The low and intermediate wavenumbers need to be recovered using
the minimum frequency available. At the same time, local minimum must be avoided. Jannane
et al. (1989) showed that for near offset surface seismic acquisition, seismic data with a typical
bandwidth were not sensitive to the intermediate wavenumbers. The imaging potential of
near offset data is thus limited to a determination of the low wavenumbers (referred to as the
macro model), using traveltime methods (tomography, stacking velocities analysis, etc....) and
the high wavenumbers (reflectivity), using prestack depth migration. The use of wide-angle
(large offset) seismic data in waveform inversion has hence been considered (Mora, 1989; Sun
and McMechan, 1992; Pratt et al., 1996; Shipp and Singh, 2002), as wide-angle data contain
information from the lower wavenumbers of the model (see chapter 3).
In this chapter, I will focus on the application of waveform inversion at 7 Hz for the recovery
of the continuous wavenumber information of a large offset seismic survey. The frequency 7
Hz presents a great increase of the non-linearity of the waveform inverse problem compared
to the frequency 5 Hz. The issue of local minima is therefore expected to be more critical,
and standard gradient methods are likely to be ineffective. Therefore, a strategy is required
that improves the chance of convergence to the global minimum. The strategy proposed in this
chapter relies on the preconditioning of both the gradient vector and the data residuals.
As the derivation of such a methodology requires better understanding of the behaviour of
waveform inversion, I will first carry out a singular value decomposition (SVD) of the single
frequency, Fréchet derivative matrix for simple 1-D models. This study will allow us to identify
the portion of the model space carrying the highest singular values and which therefore have the
strongest influence on the gradient direction in the first iterations of the inversion process. The
strong non-linearity of the forward problem will then be clarified by identifying the components
of the model that have the most non-linear relation with the data. The results of both the SVD
and non-linearity studies will lead to the definition of a strategy for improving the convergence
of the inversion. The strategy relies on the preconditioning of both the gradient vector and the
data residuals. This approach, applied to an inversion at 7 Hz of a 1-D heterogeneous velocity
model, will show that the global minimum may be found more effectively. Finally, the strategy
will be applied to an extended version of the 2-D Marmousi velocity model.

4.2. SVD OF THE FRÉCHET DERIVATIVE MATRIX 93
4.2 SVD of the Fréchet derivative matrix
The SVD technique has been used for geophysical problems in the context of tomography
(White, 1989; Pratt and Chapman, 1992; Stork, 1992b; Michelena, 1993) and waveform inversion
(Woodward, 1989; Lebrun et al., 2001). I propose in this section to apply SVD to the
Fréchet derivative matrix at a single frequency. The identification of the model eigenvectors
and their associated singular values will allow us to determine the nature of the model space
that will first be updated during the inversion. As shown in chapter 2, the gradient is the result
of the projection of the true model perturbation on the nonzero eigenvectors, multiplied by the
square of the singular values. Therefore, at the early stage of the inversion, the gradient will be
dominated by the reconstruction of the component of the model corresponding to the highest
singular values. The eigenvectors with lower singular values will carry a smaller weight and
their convergence rate will be slower, the eigenvectors with near zero singular values will not
be recovered.
The SVD will first be applied to the 1-D (quasi-homogeneous) velocity model of Chapter
3. A vertical, constant velocity gradient will then be introduced in order to apply the decomposition
to more realistic velocity model. The SVD will be carried out for a single frequency at a
time, for the near offset (0-3 km) and the far offset range (0-10 km).
4.2.1 1-D medium in homogeneous background
The SVD applied to the thin layer model used in section 3.5 (Figure 3.5b)) implies the numerical
determination of the Fréchet derivative matrix F according to equation (2.75). A single
frequency, finite difference forward modelling is therefore required for the computation of the
perturbed wavefield for each of the model parameter. The parameterisation is defined with respect
to velocity (m = c in equation 2.75). The acquisition geometry is as described in chapter
3, section 3.5, but with a free surface on the top of the model in order to take into account the
influence of the ghost effect on the amplitude of the data. The n m = 75 vertical velocity model
parameters were perturbed by 1 % of the velocity at each node of the finite difference grid.
Because the background model is quasi-homogeneous, the perturbations of the velocities only
produce reflection hyperbolas and no diving waves are present in the data perturbation. The
resulting Fréchet matrix F is complex valued, and the SVD was performed using the subroutine
csvd of the LINPACK package (Dongarra et al., 1979). This subroutine yields the model space

94 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
eigenvectors and the singular values of the Fréchet matrix. The SVD was carried out for the
near offset range (0-3 km) and the far offset range (0-10 km). The approximate Hessian was
also determined from the Fréchet matrix according to
H a = Re { F t∗ F } . (4.1)
Figure 4.1,4.2 and 4.3 show the singular value decomposition of the Fréchet matrix at 3,
5 and 10 Hz for the near (a-c) and far offset (e-g) ranges. For each offset range, the model
eigenvectors with the p nonzero singular values are shown in the depth domain (a,e) and in
the wavenumber domain (amplitude spectrum) (b,f). In the wavenumber domain, the spectral
coverage [k min , k max ], of the 1-D target (at 2 km depth) according to equation (3.14) is shown
as two horizontal straight lines. The corresponding nonzero singular values are displayed in
(c,g) in decibels according to the relation
λ i (dB) = 10 log
( )
λi
× 100 . (4.2)
λ 1
The nonzero singular values are defined as the p first singular values satisfying
λ i (dB) > 0 , i = 1, p. (4.3)
This corresponds to a threshold of 1 % of the highest singular value. The Hessian computed
using equation (4.1) is shown in (d,h).
The singular values
When comparing the number of singular values for each frequency with respect to the offset
range (c and g), we observe that the number of nonzero singular values is larger for the far
offset range than for the near offset range. Also, for the same offset range, the number of
nonzero singular values increases with frequency. The number p of nonzero singular values
with respect to the offset ranges is summarised Table 4.1.
The eigenvectors
The eigenvectors with the highest singular values correspond to the highest wavenumbers (see
for example Figure 4.2a,b, eigenvectors 1 to 5 ). We observe a shift of the wavenumber spectrum
towards the low wavenumbers as the singular values decrease.

4.2. SVD OF THE FRÉCHET DERIVATIVE MATRIX 95
Frequency / Offset range Near Offset Far Offset
3 Hz 8 11
5 Hz 10 15
10 Hz 15 26
Table 4.1: Number of nonzero singular values with frequency and offset range.
For the near offset range, the eigenvectors corresponding to the low wavenumber (for example,
eigenvector number 5 of Figure 4.2a) have energy in the shallow part of the model. This
effect is mitigated by the introduction of the far offsets.
The Hessian
As discussed in Chapter 2 (section 2.2.2.3) for the linear case, the gradient vector is the true
model perturbation filtered by the Hessian. The gradient vector for the model parameter m i is
the result of the multiplication of the i th line of the Hessian with the true perturbation and is
therefore a linear combination of all model parameters. The off diagonal terms of the Hessian
contribute to the strong filtering effect whereas the diagonal elements are weighting terms. The
distribution of amplitudes of the elements of each row away from the main diagonal tells us the
extend to which a given element of the gradient is corrupted. The display of the Hessian will
tell us about the imaging potential of a single frequency inversion which is directly linked to the
wavenumber illumination detailed in Chapter 3, for homogeneous media.
Although the Hessian matrix shown in (d) and (h) is dominated by high amplitudes along
its main diagonal (see for example Figure 4.1d) there is a strong oscillatory nature of the single
frequency Hessian which. The oscillatory effect in the Hessian is less important with increasing
frequency and increasing offset. This can be explained by the fact that the wavenumber coverage
∆k z , as defined in equation (3.14), is larger with increasing frequency and/or increasing offset
range. The ringing effect of the single frequency gradient will therefore decrease as larger
offsets or higher frequencies are used.
Furthermore, the Hessian presents an amplitude decay along the main diagonal. This shows
that the Hessian increasingly damps the gradient parameters the deeper they are in the model.
This damping is however less important for the far offset range than for the near offset range,
as large offsets allow better illumination of the deepest part of the model.

96 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
Implications for waveform inversion
The 1D inversion experiment that was carried out in Chapter 3, section 3.5 aims to recover
the velocity perturbation located at 2 km depth in a quasi-homogeneous background model.
It was shown that the single frequency inversion contributes to the reconstruction of a finite
portion of the wavenumber spectrum of this perturbation that may be expressed analytically
by the wavenumber coverage [k min , k max ] (see equation 3.14). This wavenumber coverage is
shown as horizontal straight lines in the wavenumber spectrum of the eigenvectors on Figures
4.1 to 4.3 (b and f). The results of the SVD indicate that the gradient vector is dominated by the
reconstruction of the high wavenumber part of the wavenumber coverage. As a result, the low
wavenumber components carry a smaller weight in the gradient direction, and their convergence
rate is slower.

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5
c) d)
Singular Value (dB)
20
15
10
5
0
5
Singular Value Index
Near Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Eigenvectors
5 10
g) h)
5 10
Singular Value Index
Far Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10
Figure 4.1: SVD at 3 Hz, of the thin layer velocity model as shown 3.5: a) the model eigenvectors in depth domain and b) in
wavenumber domain. The wavenumber coverage of the thin layer target after equation (3.14) is shown as horizontal lines. c) The
nonzero singular values. d) The approximate Hessian.
4.2. SVD OF THE FRÉCHET DERIVATIVE MATRIX 97

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5 10
c) d)
Singular Value (dB)
20
15
10
5
0
Near Offset Range
5 10
Singular Value Index
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Far Offset Range
Eigenvectors
5 10 15
g) h)
5 10 15
Singular Value Index
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15
Figure 4.2: SVD at 5 Hz: a) The model eigenvectors in depth domain and b) in wavenumber domain. he wavenumber coverage
of the thin layer target after equation (3.14) is shown as horizontal lines. c) The nonzero singular values. d) The approximate
Hessian.
98 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5 10 15
c) d)
Singular Value (dB)
20
15
10
5
0
Near Offset Range
5 10 15
Singular Value Index
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Eigenvectors
5 10 15 20 25
g) h)
5 10 15 20 25
Singular Value Index
Far Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15 20 25
Figure 4.3: SVD at 10 Hz: a) the model eigenvectors in depth domain and b) in wavenumber domain. The wavenumber coverage
of the thin layer target after equation (3.14) is shown as horizontal lines. c) The nonzero singular values. d) The approximate
Hessian.
4.2. SVD OF THE FRÉCHET DERIVATIVE MATRIX 99

100 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
a) b)
0
Velocity (km/s)
2.5 3.0 3.5 4.0
0
Offset (km)
0 1 2 3 4 5 6 7 8 9 10
Depth (km)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Time (s)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
4.0
Figure 4.4: a) 1-D velocity model with a constant vertical velocity gradient. b) Waveform
modelling using a Ricker wavelet with a peak frequency at 4 Hz.
4.2.2 1-D medium with a constant vertical velocity gradient
I now wish to carry out the SVD of a more realistic model containing a vertical velocity gradient
as shown Figure 4.4a. The time representation of the waveform modelling in this model is
shown Figure 4.4b.
The SVD results for this new velocity model at 3,5 and 10 Hz, for the near and far offset
ranges, are shown Figure 4.5,4.6 and 4.7. The introduction of a vertical velocity gradient
produces a strong variation of the amplitude of the oscillations of the eigenvectors with depth
(a,e) as the wavenumber spectra shift towards the low wavenumbers (b,f). This results in a very
strong amplitude decay occurring along the diagonal of the Hessian. This amplitude decay is
more important for the far offset range.
When comparing the wavenumber spectrum of the eigenvectors with the ones for the homogeneous
medium, we observe that the eigenvectors in the model with a vertical velocity gradient
contains lower wavenumbers (compare for example Figure 4.2b and Figure 4.6b). This observation
is consistent with the discussion in Chapter 3, section 3.7 that argues that a model that
contains an increase of velocity with depth produces ray bending and therefore provide access
to a wider range of wavenumber.

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5
c) d)
Singular Value (dB)
20
15
10
5
0
5
Singular Value Index
Near Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Eigenvectors
5 10
g) h)
5 10
Singular Value Index
Far Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10
Figure 4.5: SVD at 3 Hz of the 1-D velocity model with a constant vertical velocity gradient as shown 4.4: a) the model
eigenvectors in depth domain and b) in wavenumber domain. c) The nonzero singular values. d) The approximate Hessian.
4.2. SVD OF THE FRÉCHET DERIVATIVE MATRIX 101

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5 10
c) d)
Singular Value (dB)
20
15
10
5
0
Near Offset Range
5 10
Singular Value Index
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Eigenvectors
5 10 15
g) h)
5 10 15
Singular Value Index
Far Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15
Figure 4.6: SVD at 5 Hz: a) the model eigenvectors in depth domain and b) in wavenumber domain. c) The nonzero singular
values. d) The approximate Hessian.
102 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5 10 15
c) d)
Singular Value (dB)
20
15
10
5
0
5 10 15
Singular Value Index
Near Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Eigenvectors
5 10 15 20 25 30
g) h)
5 10 15 20 25 30
Singular Value Index
Far Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15 20 25 30
Figure 4.7: SVD at 10 Hz: a) the model eigenvectors in depth domain and b) in wavenumber domain. c) The nonzero singular
values. d) The approximate Hessian.
4.2. SVD OF THE FRÉCHET DERIVATIVE MATRIX 103

104 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
Slowness parameterisation
Until now, the SVD was carried out by defining the Fréchet matrix with a velocity parameterisation
(m = c). The SVD may also be performed with respect to a slowness parameterisation
(m = 1/c) defining the Fréchet matrix by perturbing the model with respect to 1 % of the
slowness at each node of the finite difference grid. The SVD of this new Fréchet matrix in the
model with the vertical velocity gradient, for the frequency 3,5 and 10 Hz are shown Figure
4.8,4.9 and 4.10. The main consequence of the slowness parameterisation is that the eigenvectors
are not as depth dependent as for the velocity parameterisation. The amplitude decay along
the main diagonal of the Hessian is therefore much less important in the case of the slowness
parameterisation.
A waveform inversion should therefore be carried out with a parameterisation in slowness
to insure proper reconstruction of the components of the model located in depth where the
background velocities are typically the highest. This observation is consistent with the role
of the derivative term ∂g/∂m used for the computation of the Fréchet derivative defined in
equation (2.82) (see Table 2.3). In the case of a velocity parameterisation, the term ω 2 /c 3
will damp the components of the gradient corresponding to the high velocities of the reference
model. The slowness parameterisation mitigates this effect as the derivative term ω 2 /c is used.
Following the same argument, the parameterisation with respect to the square of slowness (m =
1/c 2 ) should suppress any weight of the Fréchet derivative with respect to increasing velocity
as only the term ω 2 is used.

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5
c) d)
Singular Value (dB)
20
15
10
5
0
5
Singular Value Index
Near Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Eigenvectors
5 10
g) h)
5 10
Singular Value Index
Far Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10
Figure 4.8: Slowness parameterisation. SVD at 3 Hz a) the model eigenvectors in depth domain and b) in wavenumber domain.
c) The nonzero singular values. d) The approximate Hessian.
4.2. SVD OF THE FRÉCHET DERIVATIVE MATRIX 105

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5 10
c) d)
Singular Value (dB)
20
15
10
5
0
Near Offset Range
5 10
Singular Value Index
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Eigenvectors
5 10 15
g) h)
5 10 15
Singular Value Index
Far Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15
Figure 4.9: Slowness parameterisation. SVD at 5 Hz: a) the model eigenvectors in depth domain and b) in wavenumber domain.
c) The nonzero singular values. d) The approximate Hessian.
106 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES

a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Eigenvectors
5 10 15
c) d)
Singular Value (dB)
20
15
10
5
0
5 10 15
Singular Value Index
Near Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15
e) f)
Depth (km)
Singular Value (dB)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
20
15
10
5
0
Eigenvectors
5 10 15 20 25 30
g) h)
5 10 15 20 25 30
Singular Value Index
Far Offset Range
Wavenumber (1/km)
0
1
2
3
4
5
6
7
8
9
10
Eigenvectors
5 10 15 20 25 30
Figure 4.10: Slowness parameterisation. SVD at 10 Hz: a) the model eigenvectors in depth domain and b) in wavenumber
domain. c) The nonzero singular values. d) The approximate Hessian.
4.2. SVD OF THE FRÉCHET DERIVATIVE MATRIX 107

108 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
4.3 Non-linearity of the waveform inverse problem
It is well known that the waveform inversion is a non-linear problem and that as a result, the true
model perturbation cannot be expressed as a linear combination of the data residuals. Furthermore,
the degree of non-linearity varies and depends on the different components of the model
and the data considered. This section clarifies the relations of non-linearity between the model
and the data.
4.3.1 The effect of cycle skipping
The issue of non-linearity and local minima may be observed in the data domain by the phenomenon
of cycle skipping (Figure 4.11). Cycle skipping occurs when there exists a traveltime
mismatch between observed and calculated data greater than half a period of the frequency considered.
In this case, the steepest descent direction will yield an inaccurate model update and
will attempt to fit the calculated data to the wrong cycle of the observed data. The issue of
convergence to local minima is hence closely related to the effect of cycle skipping. A good
starting model should therefore explain the observed data to within half a cycle of the starting
frequency used in the inversion to avoid convergence into a local minimum.
4.3.2 Analytic linearity study of the 3D homogeneous Green’s function
Since the non-linearity of the waveform inverse problem is defined by the forward problem, I
propose in this section to study the linearity of the 3D homogeneous, free space Green’s function
G 3D
G 3D (R, f, c o ) = exp (i2πfR/c o)
, (4.4)
4πR
(Morse and Feshbach, 1953), where R is the propagation distance, f is the frequency and c o
is the constant velocity . The degree of non-linearity may be measured by defining the linearization
error as the difference between the true analytic Green’s function and its linearised
formulation. The linearization of equation (4.4) is expressed as
˜G 3D (c o + ∆c, m) = G 3D (c o ) + ∂G 3D
∆m (4.5)
∂m
where m is the model parameter used for the linearization. We then may define the linearization
error as the L 2 norm of the difference between the linearised and the true wavefield given by
E = ∣∣ ˜G 3D (c o + ∆c) − G 3D (c o + ∆c) ∣∣ 2 . (4.6)

4.3. NON-LINEARITY OF THE WAVEFORM INVERSE PROBLEM 109
a)
0
0
∆Τ
b)
0
0
∆Τ
Time
5
Time
5
10
10
Figure 4.11: Illustration of the cycle skipping problem showing a sine (single frequency) in
time for the observed (solid) and calculated (dotted) data: the traveltime mismatch is a) inferior
to half a cycle and b) superior to half a cycle. The gradient methods will minimise the misfit
between observed and calculated data in a way which is consistent with the minimisation of
traveltime error in a). If cycle skipping occurs as in b), the inversion will fit the calculated data
to the wrong cycle thus increasing the travel-time error.

110 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
Four parameters are involved in the determination of the Green’s function: the constant velocity
c o , the velocity perturbation ∆c, the frequency f and the propagation distance R. Figure 4.6
shows 4 plots of the linearisation error defined in equation (4.6) with respect to the variation
of one of the four parameters, the other three remaining constant (see Table Figure 4.12). The
linearization was performed with respect to a velocity (black) and a slowness (grey) parameterisation.
The evolution of the linearization error with respect to frequency (Figure 4.12a ) shows
that the non-linearity of the Green’s function increases with frequency (a), propagation distance
(b) and velocity perturbation (d). On the other hand, the Green’s function is more linear as
velocity increases. Also, the linearization with respect to slowness (grey) yields less error than
the linearization with respect to velocity (black) showing that a parameterisation in slowness
improves the linearity. This result is consistent with the conclusions of Beydoun and Tarantola
(1988) who studied the accuracy of the Born approximation for various parameterisations. The
oscillations of the misfit function (a,b,d) are due to the effect of cycle skipping: a local minima
can be identified at values of the misfit function for which ∂E/∂m = 0. Although the Green’s
function is more linear with respect to slowness, the effect of cycle skipping is not avoided.
4.3.3 Non-linearity and wavenumber domain
In the previous section, the model was assumed to be homogeneous and thus had only a single
wavenumber component at k = 0 km −1 . Heterogeneous models must be represented with
a range of wavenumbers which do not play the same role in the non-linearity of the inverse
problem. This characteristic of the wavenumbers may be illustrated by carrying out a linearity
study on the 1-D, heterogeneous velocity model of Figure 4.12a. This model was obtained
using the central velocity trace of the 2D Marmousi model and will be referred to as the 1-D
Marmousi model (Figure 4.13). The misfit function is computed by perturbation of the amplitude
of a single wavenumber component of the model in a similar fashion to that of Jannane
et al. (1989). Offsets up to 10 km were included in the misfit function. This wavenumber
perturbation is equivalent to adding a sinusoidal component to the depth model with the appropriate
phase. A taper is however applied to the sine function so that the very shallow part of
the model remains unchanged, thus excluding the direct arrival from the misfit function. Figure
4.14 shows the evolution of the misfit function as one of the following six wavenumbers
k = 0, 0.5, 1, 1.5, 2, 2.5 km −1 are perturbed for 5 Hz (black) and 10 Hz (grey). The global minimum
of the misfit function is much narrower as the lowest wavenumbers are perturbed. The

4.3. NON-LINEARITY OF THE WAVEFORM INVERSE PROBLEM 111
a)
0.3
b)
0.10
0.2
Misfit Function
0.1
Misfit Function
0.05
c)
0
0 5 10 15 20
Frequency (Hz)
0.15
d)
0
0 1 2 3 4 5 6 7 8 9 10
Distance (km)
0.4
0.3
0.10
Misfit Function
0.05
Misfit Function
0.2
0.1
0
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Velocity (km/s)
0
0 5 10 15 20 25
Velocity perturbation (%)
a d c d
f ↗ 10 Hz 10 Hz 10 Hz
R 2 km ↗ 2 km 2 km
c o 2 km.s −1 2 km.s −1 ↗ 2 km.s −1
∆c/c o 10 % 10 % 10% ↗
Figure 4.12: Misfit function of the linearization of the 3D homogeneous Green’s function with
variation of: a) The frequency f. b) The propagation distance R. c) The velocity c o . d) The
velocity perturbation. The constants used for each plot are shown in the table above.

112 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
a)
Velocity (m/s)
1500 2000 2500 3000 3500
0
b)
Offset (km)
0 1 2 3 4 5 6 7 8 9
0
1
Depth (km)
0.5
1.0
1.5
Time (s)
2
3
4
5
6
7
2.0
Figure 4.13: The 1-D Marmousi velocity model. a) 1-D Velocity model of the middle trace of
the Marmousi model. b) Waveform modelling in time.
low wavenumbers are therefore more non linear than the high wavenumbers. The comparison
of the misfit function of 5 Hz and 10 Hz confirms that the low frequencies are more linear (the
width of the global minimum is greater for the frequency 5 Hz).
4.3.4 Non-linearity and offset
The analytic study of the linearity of the 3D Green’s function showed that the non-linearity
increased with the propagation distance. For surface seismic data, the far offsets are the most
non-linear part of the data as they correspond to the largest propagation distance. Figure 4.15
shows the misfit function as the wavenumber k = 0.5 km -1 is perturbed for the near (0-3 km)
and far (3-10 km) offsets. The global minimum of the misfit function of the near offset is wider
than the one of the far offset showing that the far offset are more non-linear.

4.3. NON-LINEARITY OF THE WAVEFORM INVERSE PROBLEM 113
1.0
k = 0 ¢¡
−1
a) b)
0
Velocity (m/s)
-10 0 10
1.0
k = 0.5 £¢¤
−1
0
Velocity (m/s)
-10 0 10
Misfit Function
0.5
Depth (km)
0.5
1.0
Misfit Function
0.5
Depth (km)
0.5
1.0
1.5
1.5
-300 -200 -100 0 100 200 300
Velocity Perturbation (m/s)
2.0
-300 -200 -100 0 100 200 300
Velocity Perturbation (m/s)
2.0
1.0
k = 1 ¥¢¦
−1
0
Velocity (m/s)
-10 0 10
1.0
k = 1.5 §¢¨
−1
0
Velocity (m/s)
-10 0 10
0.5
0.5
Misfit Function
0.5
Depth (km)
1.0
Misfit Function
0.5
Depth (km)
1.0
1.5
1.5
-300 -200 -100 0 100 200 300
Velocity Perturbation (m/s)
2.0
-300 -200 -100 0 100 200 300
Velocity Perturbation (m/s)
2.0
k = 2 ¢
−1
k = 2.5 ©¢
−1
1.0
0
Velocity (m/s)
-10 0 10
1.0
0
Velocity (m/s)
-10 0 10
Misfit Function
0.5
Depth (km)
0.5
1.0
Misfit Function
0.5
Depth (km)
0.5
1.0
1.5
1.5
-300 -200 -100 0 100 200 300
Velocity Perturbation (m/s)
2.0
-300 -200 -100 0 100 200 300
Velocity Perturbation (m/s)
2.0
Figure 4.14: Non-linearity and wavenumbers. a) Evolution of the misfit function as a single
wavenumber component of the model of Figure 4.13 is perturbed in amplitude. The misfit
functions of the frequencies 5 Hz (black) and 10 Hz (grey) are shown. b) A taper is applied to
prevent the perturbation of the very shallow part of the model hence removing the direct arrival
from the misfit function. The misfit functions are normalised with respect to their maximum
value.

114 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
1.0
Misfit Function
0.5
-300 -200 -100 0 100 200 300
Velocity Perturbation (m/s)
Figure 4.15: Non-linearity and offset. Misfit function for a wavenumber perturbation of k =
0.5 km −1 for the near offset (0-3 km) (black) and the far offset (3-10 km) (grey). The far offsets
are more non-linear than the near offset. The misfit functions are normalised with respect to
their maximum value.
4.4 Tools for the mitigation of non-linearities
I now wish to attempt to improve the convergence accuracy of the waveform inversion using the
results of the SVD and the basic relations of non-linearity described in the previous sections.
The approach used in the previous chapter which consist in inverting from the low to the high
frequencies of the source spectrum presents a great advantage, as the low frequencies have a
more linear relation with the model. This strategy may nevertheless not be sufficient as the lowest
frequency available in the data spectrum may still present some local minima. The challenge
of waveform inversion is to assure that the inversion of the minimum frequency available in the
data converges successfully into the global minimum. In this section, I will introduce some
of the techniques that help in achieving this goal by using appropriate preconditioning of the
gradient vector and the data residuals.
4.4.1 Gradient preconditioning by wavenumber filtering
The singular value decomposition showed that the low wavenumbers of the model correspond
to the small singular values of the inverse problem. The gradient of the misfit function will
hence be dominated by the high wavenumbers. This behaviour can potentially be the cause of

4.4. TOOLS FOR THE MITIGATION OF NON-LINEARITIES 115
convergence into local minima, as the determination of the high wavenumbers depends in turn
on the accuracy of the low wavenumbers. This notion is commonly accepted in the migration
community: it is well known that the quality of the migration highly depends on the accuracy
of the macro model. Waveform inversion of large offset data allow both tomographic-like and
migration-like reconstructions (Mora, 1989; Pratt et al., 1996). From the arguments above, it
is desirable that the tomographic regime should occur before the migration. It hence appears
necessary to enhance the update of the low wavenumbers to facilitate the full convergence of
the low wavenumbers.
In order to achieve this goal, a smoothing operator may be applied to the gradient direction
removing the high wavenumber components of the gradient vector. This approach is similar
to the multi-grid technique proposed by Bunks et al. (1995) in the context of time domain
waveform inversion. The multi-grid method projects the gradient vector onto a coarse grid by
smoothing and sub-sampling the finite difference modelling grid (see for example the appendix
of Pratt et al. (1998)). Bunks et al. (1995) argued that such an approach improves the chance
of convergence into the global minimum. Unfortunately they failed to validate their claim on
numerical experiments as they used instead, a strategy that inverts band-pass filtered data, from
the very low (0-7 Hz !) to the high frequencies. The non-linearities were therefore avoided by
inverting unrealistically low frequencies.
The strategy that is proposed here relies on the preconditioning of the gradient vector by
smoothing which is performed using a 2-D low pass wavenumber filter of the gradient vector.
After 2-D spatial Fourier transform of the gradient using the subroutine rlft3 of Numerical
Recipe package (Press et al., 1992), a low-pass 2-D elliptic filter is applied to the Fourier components
of the gradient as shown Figure 4.16. A taper zone is applied to prevent artifacts of
the filter. A different filter of the horizontal and vertical directions may be applied in which
case, the filter is elliptic. I will however use a circular filter characterised by (k max , ∆k) where
k max defines the maximum preserved wavenumber in all directions, and ∆k defines the width
of the taper zone. After filtering of the gradient in the wavenumber domain, an inverse Fourier
transform is performed to recover the filtered gradient in the space domain.
An example of an inversion of a simple 1-D model with a standard and preconditioned gradient
inversion is shown Figure 4.17 at the frequency 7 Hz. The wavenumber filtered gradient
inversion (c,d) is compared to the standard inversion (a,b). A total of 30 iterations were performed
for both inversions. The preconditioned inversion was carried out with 3 sets of 10
iterations with different gradient wavenumber filtering: for the 10 first iterations the wavenum-

116 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
Figure 4.16: 2-D low-pass wavenumber filter of the gradient vector. The dashed zone represent
the preserved wavenumber and the tapering zone is shown in grey.
ber filter (k max , ∆k) = (1, 0.5 km −1 ) was applied to the gradient, the next 10 iterations were
carried out with a filter (k max , ∆k) = (2, 1 km −1 ) and for the last 10 iterations, no filter was
applied. The 2-D standard and preconditioned gradient of the first iteration are shown Figure
4.17a,c. These 2-D gradient are stacked horizontally in order to carry out the 1-D inversion
so that only the vertical component of the model is updated. These results show that the preconditioning
by low-pass wavenumber filtering of the gradient can significantly improve the
convergence accuracy of an inversion.
4.4.2 Time damping of the data residuals
The strategy of inverting from the low to the high wavenumbers components of the model may
be carried out by smoothing the gradient vector in the first steps of the waveform inversion. This
strategy may however not be sufficient as the low wavenumbers correspond to the most nonlinear
components of the model. Therefore, some components of the data wavefield contributing
to the low wavenumbers may be cycle skipped. The risk of cycle skipping is particularly critical
for the far offset data because they correspond the longest propagation distance (see section
4.3.4).
So far, for a given offset range, the full wavefield was taken into account. All events, including
reflections and refracted/diving waves where included in the data residuals. As we will

4.4. TOOLS FOR THE MITIGATION OF NON-LINEARITIES 117
Depth (km)
a)
Distance (km)
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
b)
0
Velocity (km/s)
2500 3000 3500 4000 4500 5000
Depth (km)
c)
Distance (km)
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
d)
0
Velocity (km/s)
2500 3000 3500 4000 4500 5000
0.5
0.5
1.0
1.0
Depth (km)
1.5
2.0
2.5
Depth (km)
1.5
2.0
2.5
3.0
3.0
3.5
3.5
4.0
4.0
Figure 4.17: 1-D Inversion and gradient preconditioning by wavenumber filtering. Inversion at
7 Hz with standard (a,b) and preconditioned gradient with wavenumber filtering (c,d). A total
of 30 iterations were carried out for each experiment. a) the gradient of the first iteration . b)
The result of the standard inversion. c) Filtered gradient at the first iteration. d) Result of the
precondition gradient inversion. The true model is shown in grey, the starting model in dotted
line and the result of inversion in solid line.

118 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
see, the risk of cycle skipping may be reduced by selecting the early arrivals in the data i.e., the
wavefields arriving close in time to the first arrivals. The focusing of the inversion on the fit of
the early arrivals is motivated by the fact that:
1. early arrivals are more linear than late arrivals because they optimise the tradeoff between
the propagation distance and propagation within high velocities according to the Fermat’s
principle. As shown in the linearity study of the homogeneous Green’s function in section
4.3.2, the non-linearity decreases with increasing velocity and decreasing propagation
distance.
2. early arrivals are fully compatible with the recovery of the low wavenumbers because
they arise from the first Fresnel zones
3. early arrivals contains the most trusted information if the starting model is smooth and
was obtained using first arrival traveltime tomography
I will further develop these items in the next sections. However, the selection of the early
arrivals in the frequency domain requires the definition of a preconditioning operator of the data
residuals. This operator may be carried out in the time domain by windowing the data residuals
(see for example Shipp and Singh (2002)), in which case a simultaneous inversion of several
frequencies at a time is required. Therefore, higher frequencies must be used at the expense
of an increase of non-linearity of the inverse problem. In order to preserve the advantage of
inverting for the lowest frequency available, the technique of time damping will be introduced
as a tool to damp late data arrivals of a single frequency inversion.
4.4.2.1 Time damping of the late arrivals
The use of complex frequencies for the time representation of a frequency domain modelling is
commonly used to prevent time aliasing (wrap-around) (Mallick and Frazer, 1987). The time
function of the wavefield is obtain using the inverse Fourier transform
f(t) =
∫ +∞
−∞
e −iωt Ψ (ω) dω, (4.7)
where f(t) is the wavefield in time, Ψ(ω) is the wavefield in the frequency domain. The use of
a complex angular frequency defined as
ω ′ = ω + i/τ, (4.8)

4.4. TOOLS FOR THE MITIGATION OF NON-LINEARITIES 119
is equivalent to multiplying the time domain wavefield by an exponential time decay function
since we have
f ′ (t) = e −t/τ f(t)
∫
= e −iωt Ψ (ω + i/τ) dω. (4.9)
The use of a complex ω in the waveform modelling thus allows one to damp later arrivals using
a frequency domain representation provided we calculated Ψ with the correct complex ω ′ . The
exponential decay function at t = 0 where a weighting of 1 is applied. For each trace (sourcereceiver
pair), it is possible to shift the time damping function to a new origin t o . The equivalent
time function is written as
f ′′ (t) = e −(t−to)/τ f(t)
∫
= e −iωt Ψ (ω + i/τ) e to/τ dω (4.10)
In equation (4.10), the complex wavefield is weighted by the factor e to/τ so that the equivalent
exponential time decay function is equal to 1 at the time t = t o . As for the time damping,
the time shift of the exponential decay function may be implemented in the frequency domain,
i.e. without the recourse to a time representation, by multiplying the wavefield by the term
e to/τ . Since the time representation is not required, it is possible to apply the time damping
to the single frequency wavefield Ψ(ω). Shin et al. (2002) showed that the use a very strong
time damping in the wave equation was equivalent to solve simultaneously for the eikonal and
transport equations, at a fixed frequency.
An example of modelling with complex frequency is shown Figure 4.18 in the 1-D Marmousi
model ( Figure 4.13a). The first arrivals t o , were picked on the wavefield with no time
damping (a). The wavefield in the time domain modelled with complex frequencies and amplitude
compensation according to equation (4.10) shows that different values of τ may be used to
vary the damping of the late arrivals.
4.4.2.2 Linearity of early arrivals
In order to demonstrate the linearity of the early arrivals, especially at far offsets, Figure 4.19
shows the misfit function at 7 Hz of the far offset data (3-10 km) as a single low wavenumber
component of the 1-D Marmousi model is perturbed (k = 0.5 km −1 ). The misfit functions
with no time damping (black) and with a time damping of τ = 0.25 s (grey) show that the

120 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
a)
Time (s)
c)
Time (s)
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
Offset (km)
0 1 2 3 4 5 6 7 8 9
Offset (km)
0 1 2 3 4 5 6 7 8 9
b)
Time (s)
d)
Time (s)
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
Offset (km)
0 1 2 3 4 5 6 7 8 9
Offset (km)
0 1 2 3 4 5 6 7 8 9
Figure 4.18: Finite difference modelling with different value of τ: a) No time damping, b)
τ = 0.5 s, c) τ = 0.25 s and d) τ = 0.1 s. The traveltime picks t o are shown as a white line in
a). Amplitude are normalised.

4.4. TOOLS FOR THE MITIGATION OF NON-LINEARITIES 121
1.0
Misfit Function
0.5
-100 -50 0 50 100
Velocity Perturbation (m/s)
Figure 4.19: Misfit function of the far offset at 7 Hz with no time damping (black) and τ =
0.25 s (grey). The model is perturbed at the single wavenumber k = 0.5 km -1 .
damping of late arrivals widen the global minimum. The application of time damping to the
data residuals will therefore improve the linearity of the inverse problem for the recovery of the
low wavenumbers.
4.4.2.3 Wavenumber and Fresnel zones
Chapter 3 defined the contribution of the far offsets in the determination of the low wavenumbers.
For a same source-receiver pair (same offset), the concept of a wavepath (Woodward,
1992; Pratt et al., 1996) ( see Figure 3.6) shows that arrivals contained within the first Fresnel
zone correspond to the lowest wavenumber information. Later arrivals arise from outer Fresnel
zones and are associate with higher wavenumbers. Therefore a strategy aiming to first recover
the low wavenumbers is fully compatible with the selection of the early arrivals. The smoothing
of the gradient partially achieves this goal by removing the high wavenumber components of the
gradient vector. Nevertheless, the wavenumber low-pass filtering of the gradient does not fully
prevent late arrivals from being taken into account. Furthermore, as the arrivals contained in the
first Fresnel zone may not be the most energetic arrivals, later arrivals with higher amplitudes
may dominate the inversion (as for example in Figure 4.18a). Time damping diminishes the
importance of later arrivals that may carry strong amplitudes, thus assuring that early arrivals
are taken into account early in the inversion.

122 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
4.4.3 f-k filtering of the data residuals
The f-k filtering of seismic data is routinely applied in processing to select/remove, data with
linear moveout in the t-x domain ( see for example Yilmaz (1987)). The f-k domain can thus
be used to select events with various dips such as reflection hyperbola at far offset, direct arrivals
and refracted waves. The f-k projection corresponds to a plane wave decomposition of
the wavefield where each component is associated with an emergence angle that depends on the
horizontal component of the slowness vector. Because typical velocity models present an increase
of the reference velocity with depth, it is possible to discriminate events in the wavefield
coming from the deepest part of the model as they will show different dips (more horizontal)
than shallower events.
Figure 4.20a shows an example of a velocity model containing a shallow and a deep heterogeneity
with a reference model presenting a constant vertical velocity gradient. The time
representation of the waveform modelling in this model is shown 4.20b. The application of a
f-k filter, using the program sudipfilt of the Seismic of Un*x package (Stockwell, 1997), on the
time representation of the wavefield allows to remove events with specific dips as shown Figure
4.20c. The f-k filter applied suppresses apparent velocities greater than dx/dt = 3 km.s −1 and
thus removed the reflection hyperbola of the deep heterogeneity but nevertheless also altered
the shallow reflection at near offset.
The f-k filter may be used for the single frequency waveform inversion by applying a
wavenumber filter to the data residuals in the spatial receiver Fourier domain (for a single source
gather). An inverse spatial Fourier transform is then carried out to recover the data residuals in
the source-receiver domain. No time representation is required for the application of such filter
to the data residuals.
In order to illustrate the application of the f-k filter to waveform inversion, a 1-D waveform
inversion at 7 Hz is carried out (Figure 4.21). The true velocity model is the model shown Figure
4.20a and the two heterogeneities are removed to create the starting model. The waveform
inversion using the standard data residuals (Figure 4.21a ) does reconstruct both perturbations.
If a wavenumber filter is applied to the data residuals that removes events with apparent velocities
greater than dx/dt = 3 km.s −1 , only the shallow heterogeneity is recovered. Since
the wavenumber filter of the residuals removed the near offset portion of the shallow reflection
hyperbola, the reconstruction of the shallow heterogeneity is deficient in the high wavenumbers.

4.4. TOOLS FOR THE MITIGATION OF NON-LINEARITIES 123
Depth (km)
a)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Velocity (km/s)
2.5 3.0 3.5 4.0 4.5
Time (s)
b)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Offset (km)
0 1 2 3 4 5 6 7 8 9 10
Time (s)
c)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Offset (km)
0 1 2 3 4 5 6 7 8 9 10
Figure 4.20: a) 1-D velocity model containing a shallow and a deep heterogeneity. b) Time
waveform modelling. c) f-k filter of apparent velocities greater than dx/dt = 3 km.s −1 .
a) b)
0
Velocity (km/s)
2.5 3.0 3.5 4.0 4.5
0
Velocity (km/s)
2.5 3.0 3.5 4.0 4.5
Depth (km)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Depth (km)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Figure 4.21: Preconditioning of the data residuals using an f-k filter. Waveform inversion at
7 Hz. a) standard data residuals. b) The data residuals are f-k filtered and apparent velocities
greater than 3 km/s are removed.

124 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
4.4.4 Offset windowing of the data residuals
Shipp and Singh (2002) proposed that in order to recover the low wavenumber components of
the model, the far offsets should be used in the early stages of an inversion. At a later stage,
the near offset data are inverted providing the high wavenumber information. A strategy that
windows the data residuals from the far to the near offsets may be efficient provided that no
cycle skipping occurs in the far offset data. The risk of cycle skipping may be mitigated by
applying a time damping or windowing operator but in such a case, the far offset information
will provide only information on the deepest part of the model, the low wavenumbers of the
shallow part of the model will not be constrained by the inversion. The lowest wavenumber
information of the shallow part of the model is contained in the late arrivals at far offset and are
therefore subject to a greater risk of cycle skipping (see section 4.4.2.2). The use of far offsets,
late arrivals should therefore be avoided at the early stage of the inversion.
On the other hand, the early arrivals of the near offsets can bring information on the low
wavenumber components of the shallow part of the model. This information yields higher
wavenumbers than that of the late arrivals at far offsets, but are nevertheless more linear as
they correspond to shorter propagation distance. There is therefore some justification for the
implementation of a strategy that would invert the early arrivals, from the near to the far offset
information since:
1. the early arrivals of the near offset bring low wavenumber information on the shallow part
of the model
2. the near offset data correspond to the shortest propagation distance and are less likely to
be cycle skipped
3. the inversion from the near to the far offset is consistent with the implementation of a layer
stripping strategy which proceeds by the determination of the shallowest to the deepest
part of the model and aims to reduce the risk of cycle skipping of the far offset data
4. the offset windowing selecting a range of offset assures that the far offset information that
is associated with the small singular values are properly taken into account.

4.5. DEFINITION OF A STRATEGY 125
4.4.5 Model parameterisation
The result of the SVD has shown that the parameterisation in velocity of Fréchet derivative
matrix was yielding a damping of the portion of the model associated with high velocities. In
order to compensate for this unwanted effect, a slowness or square of slowness parameterisation
should be advocated. Moreover, the linearity study of the homogeneous Green’s function
showed that the linearity is improved by a parameterisation in slowness. The various possible
parameterisations were tested (not shown here). The results demonstrated a great improvement
of the slowness over the velocity parameterisation for the determination of the magnitude of the
velocities. No improvement however was noticed when using squared slowness parameterisation.
4.5 Definition of a strategy
The previous section described some of the techniques available for the preconditioning of
the single frequency, waveform inverse problem. A strategy aiming to improve the chance of
convergence into the global minimum should hence combine the preconditioning of both the
gradient vector and the data residuals. The definition of a universal strategy is nevertheless
delicate as the non-linearity is highly dependent on the type of velocity model involved. The
practical implementation of the mitigation of the non-linearities should therefore be adapted to
the type of imaging problem involved.
In the numerical experiments shown in the next section, the strategy proposed will rely on
the use of the wavenumber filtering of the gradient vector and the time damping of the data
residuals. This strategy may be decomposed into two main stages:
1. the low wavenumbers are recovered using the early arrivals information. Time damping
and wavenumber filtering are used. The data are inverted from the near to the far offsets
thus adopting a layer stripping strategy.
2. the high wavenumbers are recovered later in the inversion by using the full, near offset
wavefield (without time damping).

126 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
Velocity (m/s)
1500 2000 2500 3000 3500
0
0.5
Depth (km)
1.0
1.5
2.0
Figure 4.22: Standard waveform inversion at 7 Hz showing the true model (grey), the starting
model (dotted line) and the result of the inversion (solid line). The inversion fails since the
starting model is not close enough to the global minimum.
4.6 Numerical experiments
4.6.1 1-D velocity model
The efficacy of the strategy of preconditioning of both the gradient vector and the data residuals
is tested on the 1-D Marmousi velocity model. The 1-D starting model used is the central trace
of the smooth, 2-D “FAST” model obtained in chapter 3, section 3.6.2. The issue of convergence
into a local minimum may be illustrated by carrying out 40 iterations of the standard
waveform inversion at 7 Hz as shown Figure 4.22. Using such starting model and inverting
for the entire wavefield (all time, all offsets), the inversion fails to provide an accurate velocity
model. The preconditioned inversion using the 2 stages approach dramatically improves the
chance of convergence to the global minimum (Figure 4.23). The first stage consist of inverting
from the near to the far offset data, with a sliding window selecting the data residuals within a
given offset range. Time damping of the data residuals and wavenumber low-pass filtering of
the gradient vector are used (the values of the parameters are detailed in the Table of Figure
4.23). The choice of the parameter were chosen empirically.

4.6. NUMERICAL EXPERIMENTS 127
a) b) c)
Velocity (m/s)
1500 2000 2500 3000 3500
0
Stage 1
Velocity (m/s)
1500 2000 2500 3000 3500
0
Velocity (m/s)
1500 2000 2500 3000 3500
0
0.5
0.5
0.5
Depth (km)
1.0
Depth (km)
1.0
Depth (km)
1.0
1.5
1.5
1.5
2.0
2.0
d)
Velocity (m/s)
1500 2000 2500 3000 3500
0
2.0
0.5
Stage 2
Depth (km)
1.0
1.5
2.0
a b c d
Offset (m) 200-3000 3000-6000 6000-10000 200-3000
Time damping: τ (s) 0.25 0.25 0.25 No
Gradient filtering: k max , ∆k (km −1) 1,2 1,2 1,2 No
Figure 4.23: 1-D Waveform inversion at 7 Hz with preconditioning of the data residuals and
gradient vector. a-c) Stage 1: Recovery of the low wavenumbers by inverting the early arrivals
using time damping and gradient wavenumber filtering. The inversion is carried out from the
far to the near offset. d) Stage 2: recovery of the high wavenumbers inverting the full wavefield
of the near offset data. The true model is shown as a grey line, the starting model as a dotted
line and the result of the inversion as a solid line.

128 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
a)
V (m/s)
3500
3000
2500
2000
1500
Depth (km)
0
0.5
1.0
1.5
2.0
Distance (km)
-4 -2 0 2 4 6 8 10 12
Figure 4.24: 2D extended Marmousi model. The original model is contained in the black
frame.
4.6.2 The 2-D extended Marmousi model
In order to test the efficiency of the preconditioned inversion on a 2-D velocity model, a dense
large offset, surface acquisition survey was carried out in an extended version of the original 2-
D Marmousi model (Figure 4.24). The original 2-D Marmousi experiment defined in Chapter 3
( section 3.6.1.1), was carried out with an OBC acquisition geometry where the very far offsets
(~ 9 km) were very poorly represented. The original model was therefore duplicated on each
side of the model to create a 18 km long model. In this new extended Marmousi model, 187
shot gathers were modelled using a finite difference solver of the acoustic wave equation. The
maximum offset present in the data is 10 km. The origin of the horizontal coordinate (x = 0 km)
is set to be at the location of the beginning of the original model.
4.6.2.1 Inversion starting from the “FAST” model
The “FAST” starting model which was obtained from first arrival travel-time tomography (see
chapter 3, section 3.6.2) was also extended in order to be used in this new wide angle inversion
experiment. Figure 4.25 shows the result of the standard waveform inversion at 7 Hz starting
from the “FAST” model. The inversion fails because the starting model is not close enough to
the global minimum of the misfit function.
In order to improve the convergence efficiency of the waveform inversion starting at 7 Hz,
the 2 stage approach of the previous section is applied. The time damping of late arrivals is
progressively decreased by gradually increasing the value of τ. At the same time, the smoothing
of the gradient is relaxed by progressively allowing the update of higher wavenumbers. The
different stages of the inversion are detailed in Table 4.3 and the results are shown Figure 4.26.
The reconstruction of the model is correct up to 1.5 km depth, beyond which the velocities are
not well recovered. The velocity profiles (Figure 4.27) confirms that the velocity are reasonably

4.6. NUMERICAL EXPERIMENTS 129
V (m/s)
3500
3000
2500
2000
1500
Depth (km)
0
0.5
1.0
1.5
2.0
Distance (km)
0 1 2 3 4 5 6 7 8 9
Figure 4.25: Standard waveform inversion at 7 Hz. Only the part of the extended model corresponding
to the original model is shown.
Offset Windowing Time Damping Gradient Smoothing
km τ (s) k max , ∆k (km −1 )
Stage 1.1 0.2-3 ⇒ 3-6 ⇒ 6-10 0.1 1,2
Stage 1.2 0.2-3 ⇒ 3-6 ⇒ 6-10 0.25 2,2
Stage 1.3 0.2-3 ⇒ 3-6 ⇒ 6-10 0.5 3,2
Stage 2 0.2-5 X X
Table 4.3: Strategy of preconditioning for the waveform inversion for the 2-D extended Marmousi
experiment. For each stage, the inversion proceeds from the near to the far offsets.
well estimated up to 1.5 km depth. The representation of the modelling in time in the true
model and the inversion result for 3 shot gathers is shown Figure 4.28. A close look at the
shot gathers shows that many traveltime mismatch are presents, especially within the far offsets
which implies that cycle skipping occurs in the residuals.
The FAST model was created in the original 2-D Marmousi model with an OBC acquisition
geometry. Because of the limited length of the model, this experiment suffers from a lack of
dense, large offset information (The maximum offset of 9.2 km was only represented twice in
the acquisition at the first and last shot of the acquisition). This causes the deeper part of the
model to be poorly constrained by the traveltime inversion. Hence the starting model did not
prevent cycle skipping of the early arrivals of the very far offset data. Since cycle skipping
produces convergence into a local minimum, the recovery of the low wavenumbers in the deep
part of the model is inaccurate. The strategy inverting from the near to far offset data did not
allow to prevent cycle skipping of the far offset. This may be explained by the fact that diving
waves travel horizontally and are thus very sensitive to velocity error at fixed depth, which

130 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
V (m/s)
V (m/s)
V (m/s)
V (m/s)
a)
3500
3000
2500
2000
1500
b)
c)
3500
3000
2500
2000
1500
3500
3000
2500
2000
1500
d)
3500
3000
2500
2000
1500
e)
V (m/s)
3500
3000
2500
2000
1500
Depth (km)
Depth (km)
Depth (km)
Depth (km)
Depth (km)
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Figure 4.26: Preconditioned Inversion at 7 Hz starting from the extended “FAST” model. a)
Near to far offset with τ = 0.1 s, F ilter(k) = 1, 2 km −1 . b) Near to far offset, τ = 0.25 s,
F ilter(k) = 2, 2 km −1 . c) Near to far offset, τ = 0.5 s, F ilter(k) = 3, 2 km −1 . d) Offset 2-5
km, no preconditioning. e) True model.

4.7. CONCLUSION 131
a) b) c)
Velocity (m/s)
1500 2000 2500 3000 3500 4000
0
Velocity (m/s)
1500 2000 2500 3000 3500 4000
0
Velocity (m/s)
1500 2000 2500 3000 3500 4000
0
500
500
500
Depth (m)
1000
Depth (m)
1000
Depth (m)
1000
1500
1500
1500
2000
2000
2000
Figure 4.27: Velocity profiles at 3 locations in the model showing the true model (grey), the
starting model (dotted) and the result of the preconditioned waveform inversion (solid). a) At
x=2.2 km. b) At x=4.6 km. b) At x=6.9 km.
cannot be fully compensated by a layer stripping approach.
4.6.2.2 Inversion starting from an improved starting model
The previous inversion experiment yielded a poor reconstruction of the deep velocities due to
the insufficient accuracy of the deeper part of the model. In order to test the inversion strategy
starting from a model that contains more accurate low wavenumbers, a waveform inversion
is carried out starting from a model that is a smoothed version of the true model. The true
extended Marmousi was thus smoothed using the program smooth2 of the SU package (Figure
4.29a). The result of the standard gradient inversion is shown Figure 4.29b. Despite the fact
that the starting model now contains accurate low wavenumbers of the true model, the standard
inversion fails to locate the global minimum. Figure 4.29 shows that when the preconditioning
tools are applied according to the strategy detailed in the previous section, the inversion result
is dramatically improved.
4.7 Conclusion
For a smooth starting model and a starting frequency of 7 Hz, the waveform inversion can easily
fail and converge into a local minimum. Some preconditioning operators must therefore be
applied to the gradient vector and data residuals in order to improve the linearity of the inverse
problem. The SVD indicates that the high wavenumbers dominate the gradient vector. Thus
when wide angle seismic data are inverted, the migration regime of the waveform inversion
carries a stronger weight than the tomographic reconstruction of the model. Since the low
wavenumbers need to be accurate to allow proper reconstruction of higher wavenumbers, the

132 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
0
1
2
True Shot
Offset (km)
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Shot at x=0 km
0
1
2
Inversion Shot
Offset (km)
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Time (s)
3
4
5
6
Time (s)
3
4
5
6
7
Offset (km)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
0
Shot at x=4.6 km
7
Offset (km)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
0
Time (s)
1
2
3
4
5
6
Time (s)
1
2
3
4
5
6
7
0
1
2
Offset (km)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Shot at x=9.2 km
7
0
1
2
Offset (km)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Time (s)
3
4
5
6
7
Time (s)
3
4
5
6
7
Figure 4.28: Shot gathers at 3 locations in the true model (left) and in the result of the preconditioned
inversion (right). Amplitude are normalised.

4.7. CONCLUSION 133
a)
V (m/s)
b)
V (m/s)
c)
V (m/s)
d)
V (m/s)
3500
3000
2500
2000
1500
3500
3000
2500
2000
1500
3500
3000
2500
2000
1500
3500
3000
2500
2000
1500
Depth (km)
Depth (km)
Depth (km)
Depth (km)
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Distance (km)
0 1 2 3 4 5 6 7 8 9
Figure 4.29: Inversion at 7 Hz starting from an improved macro model. a) Starting model. b)
Standard waveform inversion. c) Preconditioned inversion. d) True model.

134 CHAPTER 4. STARTING FROM REALISTIC FREQUENCIES
preconditioning operator should insure that the low wavenumber have fully converged before
the high wavenumbers are recovered. This can be achieved by smoothing the gradient vector.
The determination of the low wavenumbers is nevertheless delicate, as they correspond the most
non-linear components of the model. Time damping can be applied to the data residuals in order
to improve the linearity of the low wavenumber reconstruction.
The use of preconditioning tools on the 2-D extended Marmousi experiment, using dense
large offset data, has been shown to significantly improve the convergence accuracy of the waveform
inversion. However, this experiment will require to be performed with a starting model
created from a first arrival traveltime inversion to fully evaluate the efficacy of the preconditioning
strategy in realistic conditions. The “FAST” model used in this study was in fact produced
in the original Marmousi model with an OBC acquisition geometry in which the large offsets
are poorly represented.

Chapter 5
Waveform inversion and starting models:
the sub-basalt imaging problem
5.1 Introduction
The previous chapter showed that the non-linearity of the waveform inverse problem may be
mitigated by using appropriate preconditioning techniques. This preconditioning is required
because at high frequency, the topography of the misfit function in the neighbourhood of a
smooth starting model is likely to guide gradient methods into a local minimum. The linearity
may of course always be enhanced by improving the accuracy of the starting model. This begs
the important question of the definition of the requirements of the starting model that allow
convergence of the waveform inversion to the global minimum, rather than a local minimum.
Since the starting model is to be obtained from standard techniques such as stacking velocity
analysis or travel time tomography, the accuracy of the starting model is bounded by limitations
in resolution, i.e. by the maximum wavenumber that can be recovered using standard methods.
Both the requirements of waveform inversion and the limitations of standard techniques will
govern of the success of the waveform inversion.
In order to investigate the requirements of the starting model in the case of standard waveform
inversion (without preconditioning), I propose in this chapter to carry out a linearity study
by analysing the evolution of the misfit function with respect to the smoothness of the true
model. Since the inverse problem is highly dependent on the characteristics of the model to be
recovered, I will perform the study on an geophysical case that has drawn an increasing interest
135

136 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
over the past few years: the sub-basalt imaging problem.
The particularity of the sub-basalt imaging problem will first be introduced. The linearity
study will then be implemented on a 1-D velocity model. This study will provide useful indications
on the characteristic of a suitable starting model for waveform inversion. I will then
discuss the limitations of waveform inversion when applied to such an imaging problem.
5.2 The sub-basalt imaging problem
In many areas of high hydrocarbon prospectivity, extensive lava flows covered sedimentary
basins during continental breakup. The basalt flow may be thin sills intruded in the sediments
or a very thick layer of extruded lava (Planke et al., 1999). The main geographic area of interest
is the northeast Atlantic, particularly around the Faeroe islands region where the Norwegian
margin has been strongly affected by volcanism, although similar breakups have occurred along
the western Australian, the western African and the Brazilian margins (White and D., 1989).
In these areas, sub-basalt sediments may contain significant hydrocarbon reservoir that have
not yet been properly imaged because of the screening effect due to the presence of the basalt.
Many phenomena are involved in the screening of sub-basalt reflections and the full complexity
of the imaging problem is yet to be understood. The effects discussed below have however been
identified as causing standard near offset seismic data to fail to provide an accurate image of
the basalt and sub-basalt structures.
5.2.1 Transmission loss at the top basalt interface and multiples
The main feature of sub-basalt imaging is the strong velocity contrast at the top basalt interface.
Basalts are associated with much higher P-wave velocities (V p > 5 km/s) than the overburden
sediments. This contrast is responsible for very strong transmission loss of the incident wave
at the basalt interface. The issue of weak amplitude of intra-basalt and sub-basalt reflections is
exacerbated by the presence of very strong free surface and internal multiples in the data.
In recent years, large offset (wide angle) seismic acquisition has been considered in an
attempt to overcome some of the difficulties of near offset data (Jarchow et al., 1994; Samson
et al., 1995; Fruehn et al., 1998). At large offset, wide angle reflections, diving waves and
refractions may be exploited as they carry stronger amplitudes than the near offset data. These
arrivals also present the advantage of being separated in time from the free surface multiples.

5.2. THE SUB-BASALT IMAGING PROBLEM 137
Sediment
P
P
P
P
Basalt
S
P
S
S
P
Subbasalt
S
P
S
P
P
P
Figure 5.1: Converted waves in Sub-basalt imaging. The PPPPPP ray path is expected to be
weak in amplitude because of the transmission loss at the top basalt interface. The symmetric
conversions PSSSP (dotted) and PSPPSP (grey) are therefore considered (after Li et al. (1998))
The ray based traveltime inversion method of Zelt and Smith (1992) is often used and provides
information on the long wavelengths of the velocity structure (Hughes et al., 1998). The velocity
model produced by such methods may be combined with near incidence data to recover the fine
scale of the basalt structure (Fruehn et al., 2001). However, the use of traveltime techniques
require the picking of events that must be carefully associated with horizons and layers in the
model. The top of the basalt interface is easy to identify and diving waves traveltime information
may be used for the determination of the internal basalt velocity gradient. Much more delicate
is the identification of the intra-basalt, and sub-basalt reflections. However, Fliedner and White
(2001) proposed to use the amplitude versus offset (AVO) information of the diving-waves to
obtain information on the basalt thickness and the internal velocity gradient.
As an alternative to the use of refracted and diving waves information, the exploitation of
converted waves has also been investigated. The strong reflectivity of the top basalt interface
allows an efficient mode conversion beyond a certain critical angle, from an incident P-wave to
an S-wave travelling within the basalt (Purnell, 1992). Two main types of symmetric mode conversion
were considered: the PSPPSP and the PSSSSP wave (Figure 5.1 after Li et al. (1998)).
The interpretation of the converted waves is nevertheless very delicate as they are very difficult
to discriminate from internal multiples.

138 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
5.2.2 Interface scattering at the top of the basalt
In addition to a strong velocity contrast, the scattering caused by irregularities of the top basalt
interface is presumed to play a significant role in the screening of the sub-basalt reflections.
The effect of the roughness of this interface on the wave propagation may be mitigated by using
wave equation datuming techniques (Martini and Bean, 2002). This approach consist of a downward
continuation of the data to below the top basalt interface. The downward continuation is
performed using a model containing a rough interface that was determined from a stack section.
The data are then upward back-propagated to the surface using a flat top basalt interface. This
techniques as shown to improve the coherence of sub-basalt reflection.
5.2.3 Body wave scattering within the basalt
The basalt layer presents strong heterogeneities that cause the scattering of the wavefront propagating
within the basalt (Lafond et al., 1999; Martini et al., 2000). The effect of scattering is
often considered as noise and is referred to as “apparent” attenuation since it is at the origin
of some of the transmission loss of the seismic wave amplitude (Gibson and Levander, 1988).
Since the scattering affects the high frequencies more dramatically (Ziolkowski and Fokkema,
1986), seismic acquisition using with low frequencies sources (~5-7 Hz) is now being seriously
envisaged to mitigate the screening due to scattering (Ziolkowski et al., 2001).
5.2.4 Sub-basalt imaging and waveform inversion
Many additional problems are likely to be associated with the sub-basalt imaging problems
(anisotropy, 3D effects...). However, the combination of large offsets and lower frequencies is
now typically present in modern sub-basalt imaging seismic data. This offers some real potential
for the use of full waveform inversion techniques, for the reasons described in the previous
chapters: while the linearity of the waveform inverse problem is very much improved at low
frequencies, the far offsets allow wide angle illumination and provide information about the low
wavenumbers. Shipp and Singh (2002) applied time domain, elastic waveform inversion to a
sub-basalt data set. The starting model was created using ray-based traveltime inversion (Zelt
and Smith, 1992). The velocity model provided by this experiment was limited to a determination
of the overburden sediments and the basalt layer itself. The bottom basalt interface and the
sub-basalt layers were not imaged. The reconstruction however determined some of the basalt

5.3. THE 1-D SUB-BASALT NUMERICAL EXPERIMENT 139
structure and showed a good fit between the synthetic and the observed data suggesting that the
inversion converged to a model reasonably close to the global minimum.
The work presented here aims to further understand the difficulties as well as the imaging
potential that should be expected when applying waveform inversion to the sub-basalt imaging
problem. Many of the screening effects described previously will be neglected as only the
nature of the waves propagating in each layer will be taken into account.
5.3 The 1-D sub-basalt numerical experiment
The velocity model chosen to represent the sub-basalt imaging problem in this chapter is shown
Figure 5.2a. The model comprises a water layer (1 km thick), a sediment layer (1.5 km thick),
a basalt layer (2 km thick) and the sub-basalt sediments. Seismic data were generated in this
model using a finite difference solver of the acoustic wave equation. Only a single shot gather
is modelled (Figure 5.2b ); and the characteristics of the modelling are described in Table 5.1.
The free surface multiples were not modelled in this experiment.
Number of Shot 1
Number of Receivers 601
Shot/Receivers depth 9 m
Receiver spacing
25 m
Offset range
0-15 Km
Free surface on top
No
Frequency range 0.1-10 Hz
Number of frequencies modelled 99
Grid size (n x × n z ) 355x140
Grid spacing
45 m
Table 5.1: Synthetic data characteristics.

140 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
a) b)
0
Velocity (m/s)
2000 3000 4000 5000
Offset (km)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
1
1
2
WBR
2
3
Depth (km)
3
4
Time (s)
4
5
6
7
BBR
TBR
5
8
9
6
Figure 5.2: The Sub-basalt synthetic data. a) Velocity model. b) Shot point gather in the
true model with a source Ricker with a pick frequencies at 4 Hz. The water bottom reflection
(WBR), the top basalt reflection (TBR) and the bottom basalt reflection (BBR) are identified on
the right panel. The free surface multiples are not present in the data.

5.4. INVERSION STARTING FROM A “CORRECT” MACRO MODEL 141
5.4 Inversion starting from a “correct” macro model
In order to illustrate the difficulties of waveform inversion applied to sub-basalt imaging, the
results of the single frequency waveform inversion, when the starting model contains the correct
macro model (low wavenumbers) are shown Figure 5.3, for the frequencies: 3, 5 and 7 Hz. The
starting model is a smoothed version of the true model (The water bottom discontinuity is not
smoothed and its depth is therefore assumed to be known). The final model of the inversion at
3 Hz (Figure 5.3a ) is close enough to the global minimum to allow a sequential inversion of
higher frequencies (Figure 5.3d ). The inversion fails when starting at 5 or 7 Hz (Figure 5.3b
and c) which precludes any inversion that start at higher frequencies. In order to investigate the
origin of this behaviour, the next section will study the misfit function using a varying degree
of smoothness of the true model.
5.4.1 Misfit function and model smoothness
Figure 5.4b shows the misfit functions for 3,5 and 7 Hz as the entire model (sediments, basalt
and sub-basalt) is decreasingly smoothed (increasingly more accurate) as shown in Figure 5.4a.
The smoothing was carried out using the program smooth2 of the Seismic for Un*x package
(Stockwell, 1997). The degree of smoothness is defined with respect to a smoothing factor: the
model will get smoother as the smoothing factor increases. The smoothest model (smoothing
factor of 10) is the macro model used as a starting model in the previous inversion tests.
The 3 Hz misfit function shows a continuous decrease with decreasing smoothness, whereas
for 5 and 7 Hz, the function presents a local maximum. The latter strongly suggests the presence
of a local minimum in the misfit function. Any starting model located on the left side of this
maximum is therefore likely to produce cycle skipping in the data and to cause the waveform
inversion to converge within a local minimum. This behaviours explains the poor results of the
previous inversions at 5 and 7 Hz since the starting model is located on the wrong side of the
maximum for these frequencies.
5.4.2 The top basalt and bottom basalt interfaces
In order to identify the main origin of the non-linearity in the previous section, the same experiment
is carried out again, but in this case, only the sediment and top basalt interface are
decreasingly smoothed (Figure 5.5a). The misfit functions in Figure 5.5b are identical in shape

142 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
a)
0
Velocity (m/s)
2000 3000 4000 5000
b) c)
0
Velocity (m/s)
2000 3000 4000 5000
0
Velocity (m/s)
2000 3000 4000 5000
1
1
1
2
2
2
Depth (km)
3
Depth (km)
3
Depth (km)
3
4
4
4
5
5
5
6
6
6
d)
0
Velocity (m/s)
2000 3000 4000 5000
1
2
Depth (km)
3
4
5
6
Figure 5.3: Inversion starting from an accurate macro-model. Velocity models showing the
true model (grey), the starting model (dotted) and the result of inversion (solid) for a) 3 Hz, b) 5
Hz and c) 7 Hz. (d) The result in (a) is close enough to the global minimum to allow sequential
inversion of higher frequencies (5 and 7 Hz).

5.5. WAVEFIELD SEPARATION BY LAYER STRIPPING 143
a) b)
Depth (km)
0
1
2
3
4
5
6
Velocity (m/s)
2000 3000 4000 5000
1.5
1.0
0.5
Normalised Misfit Function2.0
10
8
6 4
Smoothing Factor
2
3 Hz
5 Hz
7 Hz
0
Figure 5.4: Evolution of the misfit function as the entire model is decreasingly smoothed.
a) True model (black) and the true model decreasingly smoothed (grey). b) Misfit functions
normalised with respect to the initial value for the frequency 3,5 and 7 Hz.
to the misfit functions presented when the entire model is smoothed (Figure5.4b). The intrabasalt
and sub-basalt layers appears relatively unimportant in governing the topology of the
misfit function. This leads to the conclusion that the top basalt interface is the main origin of
the failure of the waveform inversion and that the top basalt interface needs to be discontinuous
(blocky) when using frequencies of 5 and 7 Hz (or higher). Since the bottom basalt interface
corresponds to a velocity contrast of the same order as the top of the basalt, it seems reasonable
to think that this interface should therefore also be discontinuous.
5.5 Wavefield separation by layer stripping
The non-linearity of the sub-basalt waveform inverse problem is apparently caused by the top
basalt interface. This interface creates the strongest reflection in the data and is therefore the
easiest to determine. Even if the top of the basalt is known in depth and velocity, some nonlinearity
issues remain for each part of the model. In order to investigate the non-linearity
associated with each layer, a separation of the wavefield is carried out by implementing an
idealised layer stripping strategy by decomposing the model into the sediment, the basalt and
the sub-basalt layers as shown Figure 5.6. The wavefield of the basalt layer (Figure 5.6d)
was obtained by subtraction of the waveform in the sediment (b) to the waveform generated

144 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
2.5
Velocity (m/s)
2000 3000 4000
Normalised Misfit Function
2.5
2.0
1.5
1.0
0.5
3 Hz
5 Hz
7 Hz
3.0
10
8
6 4
Smoothing Factor
2
0
Figure 5.5: Evolution of the misfit function as the sediment and top basalt interface are decreasingly
smoothed. a) True model (black) and the true model decreasingly smoothed (grey).
b) Misfit function at 3,5 and 7 Hz.
in the model in (c). The wavefield in the sub-basalt in (f) was obtained by subtraction of the
total wavefield (Figure 5.1b) from the wavefield generated in (c). The sum of each of these
wavefields is the total wavefield shown Figure 5.2b.
5.5.1 The sediment layer
Figure 5.7 shows the misfit function as the sediment layer is decreasingly smoothed. The water
bottom interface is assumed to be known and remain unsmoothed. The starting point (smoothing
factor of 50) of the experiment is a quasi homogeneous layer. The 3 Hz misfit function
decreases monotonically, whereas the 5 and 7 Hz frequencies encounter a local maximum. A
satisfactory starting model for the frequency 5 or 7 Hz should therefore be located beyond the
maximum of the misfit function to assure convergence toward the global minimum.
5.5.2 The basalt layer
Figure 5.8 shows the misfit function as the intra basalt velocities are decreasingly smoothed.
Both misfit functions at 3 and 5 Hz decrease towards the true model, while the 7 Hz frequency
encounters a local maximum. This maximum at 7 Hz however occurs for a smoother model
than for the sediment layer. This can be explained by the fact that, as described in Chapter 4,

5.5. WAVEFIELD SEPARATION BY LAYER STRIPPING 145
a) c) e)
0
Velocity (m/s)
2000 3000 4000 5000
0
Velocity (m/s)
2000 3000 4000 5000
0
Velocity (m/s)
2000 3000 4000 5000
1
1
1
2
2
2
Depth (km)
3
Depth (km)
3
Depth (km)
3
4
4
4
5
5
5
6
b) d) f)
Offset (km)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
6
Offset (km)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
6
Offset (km)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Time (s)
1
2
3
4
5
6
7
8
9
Time (s)
1
2
3
4
5
6
7
8
9
Time (s)
1
2
3
4
5
6
7
8
9
Figure 5.6: Illustration of the idealised layer stripping strategy showing the part of the model
involved in grey (a,c,e) and the corresponding wavefield (b,d,f). a,b) The sediment only and the
corresponding wavefield. c,d) The basalt. e,f) The sub-basalt. The sum of the wavefields in
b,d,f is equal to the total wavefield shown Fig. 5.2. Amplitude are normalised.

146 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
a) b)
Depth (km)
0
0.5
1.0
1.5
2.0
Velocity (m/s)
1500 1750 2000 2250
Normalised Misfit Function
2.0
1.5
1.0
0.5
50
3 Hz
5 Hz
7 Hz
40
30 20
Smoothing Factor
10
0
Figure 5.7: Evolution of the misfit function as the sediment layer is decreasingly smoothed, a)
true model (black) and the true model smoothed (grey).b) Misfit function.
section 4.3.2, the non-linearity decreases with increasing velocity. As a result, the requirements
for the starting model are less demanding in the basalt than in the sediment layer.
5.5.3 The sub-basalt layer
Figure 5.9 shows the misfit functions as the sub-basalt velocities are decreasingly smoothed.
Although all misfit functions are monotonically decreasing, the data at 5 and 7 Hz show very
little sensitivity to the low and intermediate wavenumbers of the sub-basalt. This can be explained
by the presence of the basalt layer which prevents the wide angle illumination of the
sub-basalt sediments. Therefore, only narrow incidence angles propagate within the sub-basalt
sediments and these provide only information about the high wavenumbers.
5.6 Waveform inversion and starting model requirements
5.6.1 Frequency dependence of the starting model requirements
The analysis of the misfit function curves with respect to the model smoothness allows us to
specify the degree of a priori information that assures success of the waveform inversion in
the context of a layer stripping strategy. Figure 5.10 shows the minimum accuracy (maximum

5.6. WAVEFORM INVERSION AND STARTING MODEL REQUIREMENTS 147
a) b)
Depth (km)
Velocity (m/s)
2000 3000 4000 5000
2.5
3.0
3.5
4.0
1.0
0.5
Normalised Misfit Function1.5
3 Hz
5 Hz
7 Hz
4.5
50
40
30 20
Smoothing Factor
10
0
Figure 5.8: Evolution of the misfit function as the basalt layer is decreasingly smoothed. a)
True model (black) and the true model smoothed (grey). b) Misfit functions at 3,5 and 7 Hz.
a) b)
Depth (km)
Velocity (m/s)
2500 3000 3500 4000
4.5
5.0
5.5
1.5
1.0
0.5
Normalised Misfit Function2.0
3 Hz
5 Hz
7 Hz
6.0
50
40
30 20
Smoothing Factor
10
0
Figure 5.9: Evolution of the misfit function as the sub-basalt layer is decreasingly smoothed,
a) true model (black) and the true model smoothed (grey). b) Misfit functions at 3,5 and 7 Hz.

148 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
a) b) c)
0
0.5
Velocity (m/s)
1500 1750 2000 2250
3 Hz
5 Hz 2.5
7 Hz
Velocity (m/s)
3000 4000 5000
Velocity (m/s)
2500 3000 3500 4000
4.5
Depth (km)
1.0
1.5
2.0
Depth (km)
3.0
3.5
4.0
Depth (km)
5.0
5.5
2.5
4.5
6.0
Figure 5.10: Adequate velocity models for the a) sediment, b) the basalt and c) the sub-basalt
showing the true model (grey) and the velocity models for 3,5 and 7 Hz.
allowed smoothness) of the velocity model for an inversion starting at 3,5 or 7 Hz. A different
level of accuracy is required for each of the sediment, the basalt and the sub-basalt layers. The
sediment layer should contain some knowledge of the medium wavenumbers, the basalt layer
needs less accuracy as some bulk information on the vertical gradient appears to be sufficient.
The sub-basalt layer is the most demanding, as the main features of the heterogeneities seem to
be required.
5.6.2 Waveform inversion at 7 Hz
The Figure 5.11 shows a set of waveform inversions for the sediment (a), the basalt (b) and
the sub-basalt (c) layers. For each layer, two inversions were carried out: the first one from a
poor starting model, the second with a good starting model as defined from the misfit function
analysis (Figure 5.10). The sediment and the basalt poor and good starting models are on either
side of the local maxima of their respective misfit function. For the sub-basalt, the limit is
more difficult to define as no maximum occurs in the misfit functions. The waveform inversion
results appear to validate the conclusions drawn from the linearity study. When inverting with a
good starting model at 7 Hz, the sediments are recovered very accurately as the offset range is
able to reconstruct a wide range of wavenumbers. The imaging in the basalt, although accurate,
is limited by a maximum resolution, in turn controlled by the background velocities and the

5.6. WAVEFORM INVERSION AND STARTING MODEL REQUIREMENTS 149
frequency (7 Hz). For the sub-basalt layer, the starting model needs to contain much of the low
and intermediate wavenumbers as it is only illuminated by narrow incidence angles which, as
detailed in Chapter 3, provide information on the high wavenumbers. The sub-basalt wavefield
is therefore unable to contribute to the low wavenumbers. The consequence is that the inversion
will provide only a migration-like (high wavenumber) image of the sub-basalt structure.

150 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
0
0.5
Poor Starting Model
Velocity (m/s)
1500 2000
(a)
0
0.5
Good Starting Model
Velocity (m/s)
1500 2000
Depth (km)
1.0
1.5
Depth (km)
1.0
1.5
2.0
2.0
Smoothing Factor of 20
Smoothing Factor of 6
Velocity (m/s)
3000 4000 5000
2.5
(b)
Velocity (m/s)
3000 4000 5000
2.5
3.0
3.0
Depth (km)
3.5
Depth (km)
3.5
4.0
4.0
4.5
4.5
Smoothing Factor of 28
Smoothing Factor of 24
Velocity (m/s)
2500 3000 3500 4000
4.5
Velocity (m/s)
2500 3000 3500 4000
4.5
5.0
(c)
5.0
Depth (km)
5.5
Depth (km)
5.5
6.0
Smoothing Factor of 50 Smoothing Factor of 2
6.0
Figure 5.11: Waveform inversion at 7 Hz when the starting model (dotted) leads the convergences
towards a local minimum (left) and close to the global minimum (right). The true model
is shown in grey, and the result of the inversion in solid line.

5.7. DISCUSSION: STARTING MODEL AND STANDARD METHODS 151
5.7 Discussion: starting model and standard methods
The analysis of the misfit functions with respect to the model smoothness lead us to conclude
that for a realistic starting frequency of 7 Hz, the starting model should contain discontinuous
top and bottom basalt interfaces. Ray-based traveltime methods, such as the one proposed by
Zelt and Smith (1992) can easily determine the top-basalt interface. Nevertheless, the determination
of the bottom basalt interface, which must be included in the starting model, remains a
real issue as this reflection is often difficult to identify in the data due to its weak amplitude and
the presence of strong multiples.
The determination of the sediment layer velocity structure may be delicate, as the starting
model should contain some information about the medium wavenumbers. This can be explained
by the relatively low velocities in the sediments since non-linearity increases with low velocities.
Standard methods such as stacking velocity analysis may provide useful information on the
vertical velocity gradient of the sediments, but the recovery of the medium wavenumbers will
be difficult. The application of preconditioning operators for the determination of the sediment
layer is likely to be inefficient as the sediment wavefield is dominated by reflected and scattered
waves and no diving/refracted waves may be exploited. A selection of early arrivals is therefore
poorly adapted and the smoothing of the gradient will involved the far offset data which is the
most non-linear part of the reflection hyperbola. Therefore, the recovery of the sediments using
waveform inversion is restricted to a migration-like reconstruction and the result will depend
on the accuracy of the starting model. An important point is that inaccurate velocities in the
sediments may not be easily noticeable when comparing predicted and observed data. Figure
5.12 shows the wavefield of the sediment in the true model and in the wrong inversion result at
7 Hz, Figure 5.11, left hand side). The wavefield in the wrong model is difficult to differentiate
from the wavefield in the true model, especially at near offset.
The requirements for the starting model are less demanding for the basalt layer, as the wavefield
is more linear due to higher velocities. The vertical gradient of the basalt layer may be
determined in the starting model by exploiting the traveltime of the diving/refracted waves at
far offsets. During the waveform inversion process, time windowing/damping or f-k filtering of
the data residuals will be necessary in order to invert for the diving/refracted waveform of the
basalt and thus exclude shallower reflection and multiples (as advocated by Shipp and Singh
(2002).).
Some imaging limitations exist for the sub-basalt sediments. Because the presence of the

152 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL
a) b)
0
Offset (km)
0 1 2 3 4 5 6 7 8 9 10
0
Offset (km)
0 1 2 3 4 5 6 7 8 9 10
1
1
2
2
Time (s)
3
4
Time (s)
3
4
5
5
6
6
Figure 5.12: Sediment wavefield in time for a) the true sediments and b) the wrong model
recovered by the inversion at 7 Hz starting from a smooth model (see Figure 5.11a, left). The
source is a Ricker wavelet with a pick frequency at 4 Hz.
basalt prevents any wide-angle illumination of the sub-basalt target such as diving or refracted
waves. Therefore, only a migration-like reconstruction (high wavenumber) may be achieved.
5.8 Conclusion
Sub-basalt imaging is an important exploration problem as it is associated with geographical areas
that may contain important hydrocarbon reservoirs. The application of waveform inversion
is considered since sub-basalt seismic data contains lower frequencies and larger offset data
than standard seismic acquisition.
A linearity study of the waveform inverse problem was carried out on a 1-D experiment
representing the sub-basalt imaging problem. The characteristics of an adequate starting model
could then be defined. The overburden sediments must contain some of the intermediate wavenumbers
to allow an accurate reconstruction. The basalt layer is less demanding as only some information
on the vertical gradient appears to be sufficient. Only the high wavenumber components
of the sub-basalt layer may be recovered due to the lack of wide-angle information illuminating
this target.
This study was carried out in the context of an idealised layer stripping strategy that relies
on a perfect separation of the wavefield. Moreover, the free surface multiples were not taken

5.8. CONCLUSION 153
into account. The characteristics of the required starting model are likely to be more demanding
in reality. More realistic layer stripping have yet to be studied. The results of this study however
provide us with some useful indications as to how accurate the starting model needs to be. These
requirements change with the lowest frequency available in the data.
The determination of the starting model is a critical step in the waveform inversion process.
The linearity of the waveform inverse problem may be significantly enhanced by improving
the accuracy of the starting model. A good starting model should thus insure that a descent
path exists that allow gradient methods to locate the global minimum of the misfit function.
This study demonstrates the dependence of the starting model requirements on the starting
frequency: the higher the starting frequency is, the more accurate the model needs to be.

154 CHAPTER 5. WAVEFORM INVERSION AND STARTING MODEL

Chapter 6
Conclusions
In this thesis, I examined the application of waveform inversion on wide-angle surface seismic
data. Since the data frequency bandwidth is limited in term of its low frequency content,
wide-angle data are necessary to determine the medium wavenumber components of the velocity
model. The medium wavenumbers cannot be recovered by classical near incident angle
acquisition since the macro model, as defined by the standard imaging principle, is typically
deficient in the medium wavenumbers. This information is of significant interest as in complex
media, it is required for an accurate recovery of the high wavenumbers. Thus, the objective is
to use waveform inversion for the reconstruction of a continuous wavenumber spectrum of the
velocity model.
Due to the high computational cost of generating the synthetic data, the waveform inverse
problem was solved using local methods based on the calculation of the steepest descent of the
misfit function. This approach however, is very sensitive to the issue of local minima associated
with the strong non-linearity of the inversion. This non-linearity is aggravated by the introduction
of wide-angle/large offset data in the inverse problem. These data which correspond to the
longest propagation distances and are thus more likely to present cycle skipping.
To summarise, in order to determine a continuous wavenumber spectrum of the velocity
model, the following difficulties must be addressed:
1. The recovery of the high wavenumbers is a reasonably linear problem if the low and
medium wavenumbers are already present and accurate in the velocity model
2. The recovery of the medium wavenumbers requires the introduction of large offset data
which are the most non-linear part of the data
155

156 CHAPTER 6. CONCLUSIONS
The waveform inversion of large offset data must hence be considered very carefully. Success
depends on the accuracy of the low wavenumbers and the minimum frequency accessible in the
data. In order to carry out this study, the frequency domain waveform inversion approach was
adopted because of its efficiency in the implementation of a strategy that inverts from the low
to the high frequencies. This strategy is fundamental as the inverse problem is more linear for
the low frequencies.
6.1 Wavenumber, frequency and offset
In Chapter 3, I investigated the relation between frequency, offset and the wavenumber coverage
of the model. It was shown that, within the plane wave and homogeneous background
velocity approximations, the contribution of a single frequency to the reconstruction of the velocity
perturbation is limited to a range of wavenumbers that may be expressed analytically.
The wavenumber coverage increases linearly with frequency which led to a strategy for selecting
frequency that takes advantage of a redundancy of information along the offset axis. This
strategy takes advantage of the well known effect of wavelet stretch in NMO correction and
migration and allows the discretisation of frequencies without degradation of the image quality.
As the maximum offset increases, the wavenumber coverage gets wider since the largest
offset controls the minimum wavenumber that may be recovered. As a result, the larger the
maximum offset is, the fewer frequencies are needed to assure a continuous reconstruction of
the velocity wavenumber spectrum. Because of the similarity between the first iteration steepest
descent schemes and migration, this strategy also has implications for the more commonly
used migration techniques. In frequency domain, the computational cost of inversion is directly
proportional to the amount of frequencies processed. The topic is therefore of fundamental
interest.
This study helped to further illustrate the relation between frequency, offset and wavenumber
coverage. The lowest wavenumber that may be recovered from the waveform data depends
on both the minimum frequency inverted and on the maximum offset present in the seismic
acquisition. The importance of wide-angle data for the recovery of the low wavenumbers was
illustrated on a 1-D experiment, in which it was shown that the reconstruction of velocity is
much improved by the introduction of large offset data.
The same principle was illustrated using the 2-D Marmousi numerical experiment. In the
Marmousi study, frequency domain waveform inversion was very effective in recovering the

6.2. WAVEFORM INVERSION AND STARTING FREQUENCY 157
2-D structure of the true model, using only 3 frequencies. In this experiment, the waveform
inversion was initiated at 5 Hz with a starting velocity model obtained from traveltime tomography.
In the final velocity model of the waveform inversion, a continuous range of wavenumber
was recovered, i.e. both velocities and discontinuities were recovered. However frequencies as
low as 5 Hz are unlikely to be present in real exploration data. The use of realistic frequencies
in waveform inversion was therefore investigated in the Chapter 4 of the thesis.
6.2 Waveform inversion and starting frequency
The inversion of wide-angle data is particularly delicate as the introduction of large offset increases
the non-linearity of the inverse problem, due to the increased risk of cycle skipping.
While the inversion of the 2-D Marmousi data set yielded accurate velocities when initiated
at 5 Hz, the same experiment failed when starting the inversion at 7 Hz. This may be easily
explained by the fact that non-linearity increases dramatically with frequency and that standard
gradient method are less likely to converge properly at 7 Hz. In Chapter 4, I therefore
investigated techniques for improving the linearity of the inverse problem at 7 Hz.
In order to further understand the behaviour of gradient-based waveform inversion, a singular
value decomposition was applied to the mono-frequency Fréchet derivative matrix. The
results show that the high wavenumbers are associated with the highest singular values and
that the gradient will thus be dominated by the update of the high wavenumbers. On the other
hand, as mentioned previously, the low and medium wavenumbers must be present to ensure
the accurate recovery of the high wavenumbers. Therefore the update of the high wavenumber
should be penalised during the early stages of the inversion. This may be achieved by applying a
smoothing operator to the gradient vector. Unfortunately, the recovery of the low wavenumbers
involves the far offset information - the most non-linear part of the data. The linearity of the far
offset data may however be improved by applying time damping to the data residuals, which
has the effect of focusing on the inversion of early arrivals.
The application of the preconditioning of both the gradient image and the data residuals
were shown to improve the convergence accuracy of the waveform inversion using an extended
version of the 2-D Marmousi model in which a dense, wide-angle acquisition survey was modelled.

158 CHAPTER 6. CONCLUSIONS
6.3 Waveform inversion and starting model
In Chapter 5, I explored requirements on the starting model for the waveform inversion process.
Suitable starting models must be obtained using classical methods which in general provide
only a low wavenumber velocity model. The question of the characteristics of adequate starting
models must therefore be addressed. These characteristics depend on the minimum frequency
available in the data: the higher this frequency is, the more demanding are the requirements of
the starting model. The characteristic also depend on the survey geometry, and on the velocity
structure itself.
In order to investigate this question, a non-linearity study was carried out on a 1-D velocity
model representing the sub-basalt imaging problem. The requirements of the starting model are
determined by examining the evolution of the misfit function with respect to various degrees
of smoothing of the true model. The results allowed me to determine the degree of resolution
required for each part of the velocity model. The potential of the application of waveform
inversion to the sub-basalt imaging problem could then be evaluated. My primary conclusion
are that waveform inversion may be used for the determination of intra-basalt velocities, but
the reconstruction of the overburden sediments will be more difficult. Only a migration likeimage
of the sub-basalt velocity structure may be obtained due to the lack of illumination at
wide-angles below the basalt layer.
6.4 Towards the waveform inversion of real data
An adequate real seismic data set should contain large offsets and frequencies as low as 7
Hz although ideally, lower frequencies would significantly improve the linearity of the inverse
problem. The importance of the low frequencies is however largely underestimated by the
seismic industry, which often relies only on the high frequencies to improve the resolution of
the reconstructed image. Some technical difficulties exist generating seismic data with very low
frequencies (Ziolkowski et al., 2001), but the seismic community should certainly be aware that
low frequencies offer a great potential to improve the determination of the velocity model.
The starting model is a fundamental aspect of the waveform inversion I showed in Chapter
5. Any application of waveform inversion to real data must address the question of the determination
of the starting model. Furthermore, any application to real data should be preceded
by inversion tests on synthetic data in order to evaluate the difficulties that may be encoun-

6.4. TOWARDS THE WAVEFORM INVERSION OF REAL DATA 159
tered by waveform inversion. Linearity studies such as that as carried out in Chapter 5, should
contribute to a better understanding of the true potential of applying waveform inversion on a
specific imaging problem. More advanced analysis may be carried out by including the preconditioning
of the data residuals (especially through time damping), in the non-linearity study.
The starting model for my 2-D Marmousi study was obtained using first arrival traveltime
tomography, which is limited by the asymptotic ray approximation. New methods are currently
under investigation in which the first arrival travel time and amplitude are computed using a full
wave propagator with very strong time damping. Shin et al. (2002) show that the use of strong
time damping in the wave equation is equivalent to solving simultaneously for the eikonal and
the transport equations, at a single frequency. This approach could potentially offer a good
alternative to ray based tomography and would be consistent with the use of time damping later
in the waveform inversion as proposed in Chapter 4.
In this thesis, I demonstrated that the frequency domain is very efficient, since only a few
frequencies are required for the reconstruction of the velocity model. It is also particularly
appropriate for the implementation of a strategy of inverting from the lowest to the highest
frequencies. However, as pointed out in Chapter 3, the inversion of a single frequency is likely
to cause problem in the presence of random noise in which case, the more robust simultaneous
inversion of several frequencies must be considered. The question of the number of frequencies
that should be inverted, as a function of the signal to noise ratio should therefore be investigated.
All numerical experiments in this thesis were performed within the acoustic approximation
of the wave equation. Waveform inversion using a frequency domain elastic forward modelling
would benefit from the more accurate simulation of wave propagation. Such simulations are
however computationally very expensive. Elastic, time domain waveform inversions have already
been carried out (Shipp and Singh, 2002). The advantage and tradeoffs of the elastic
versus the acoustic should be further investigated.
Frequency domain and time domain waveform inversion approaches are intrinsically equivalent
as they both make use of the wave equation. The differences are primarily the types of
preconditioning that may be applied during the inversion process. The main advantage of the
time over the frequency domain is that a precise time windowing of the data residuals is possible.
It was shown in Chapter 4, that time windowing may be adequately performed in the
frequency domain using time damping, in which case the inversion of a single frequency of the
data at a time still remains possible.

160 CHAPTER 6. CONCLUSIONS

Appendix A
Image stretch and NMO stretch
In a homogeneous medium the zero offset two-way traveltime for a single horizontal reflection
is given by
τ = 2z c ,
(A.1)
where z is the depth of the reflector and c is the velocity. At finite offsets the two-way traveltime
is
t = 2 √
z2 + h
c 2
⎛ ( ) ⎞ 2
2h
= ⎝τ 2 + ⎠
c
1
2
, (A.2)
where h is the half-offset.
The NMO correction seeks to move all reflections to their zero offset times. NMO stretch
occurs because reflections are associated with a wavelet of finite (non-zero) duration. Therefore
the start of the reflection is moved to the correct zero offset time, whereas the end of the wavelet
is moved to a different zero offset time. If we characterise the width of the wavelet as ∆t and
the width of the wavelet after NMO correction as ∆τ, the NMO stretch is
∆τ
∆t ,
(A.3)
which varies as a function of depth and offset. Both these dependencies are captured by considering
the NMO stretch to be a function of two-way time, t. The instantaneous value of the
NMO stretch is therefore
( ) ∆τ
lim = dτ
∆t→0 ∆t dt .
161
(A.4)

162 APPENDIX A. IMAGE STRETCH AND NMO STRETCH
We may evaluate this derivative with the use of equation (A.2), from which we obtain
dτ
dt = √ 1 + R 2 = 1 α ,
(A.5)
where R = h/z is the offset-to-depth ratio and α = cos θ as in equation (3.10).
It can therefore be concluded that the shift towards the low wavenumbers with increasing
offset predicted by equation (3.10) is exactly the same as that caused by NMO stretch after
depth correction.

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