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An overview of full-waveform inversion in exploration geophysics
J. Virieux1and S. Operto2
ABSTRACT
Full-waveform inversion 共FWI兲is a challenging data-fitting
procedure based on full-wavefield modeling to extract quantita-
tive information from seismograms. High-resolution imaging at
half the propagated wavelength is expected. Recent advances in
high-performance computing and multifold/multicomponent
wide-aperture and wide-azimuth acquisitions make 3D acoustic
FWI feasible today. Key ingredients of FWI are an efficient for-
ward-modeling engine and a local differential approach, in
which the gradient and the Hessian operators are efficiently esti-
mated. Local optimization does not, however, prevent conver-
gence of the misfit function toward local minima because of the
limited accuracy of the starting model, the lack of low frequen-
cies, the presence of noise, and the approximate modeling of the
wave-physics complexity. Different hierarchical multiscale
strategies are designed to mitigate the nonlinearity and ill-posed-
ness of FWI by incorporating progressively shorter wavelengths
in the parameter space. Synthetic and real-data case studies ad-
dress reconstructing various parameters, from VPand VSveloci-
ties to density, anisotropy, and attenuation. This review attempts
to illuminate the state of the art of FWI. Crucial jumps, however,
remain necessary to make it as popular as migration techniques.
The challenges can be categorized as 共1兲building accurate start-
ing models with automatic procedures and/or recording low fre-
quencies, 共2兲defining new minimization criteria to mitigate the
sensitivity of FWI to amplitude errors and increasing the robust-
ness of FWI when multiple parameter classes are estimated, and
共3兲improving computational efficiency by data-compression
techniques to make 3D elastic FWI feasible.
INTRODUCTION
Seismic waves bring to the surface information gathered on the
physical properties of the earth. Since the discovery of modern seis-
mology at the end of the 19th century, the main discoveries have aris-
en from using traveltime information 共Oldham, 1906;Gutenberg,
1914;Lehmann, 1936兲. Then there was a hiatus until the 1980s for
amplitude interpretation, when global seismic networks could pro-
vide enough calibrated seismograms to compute accurate synthetic
seismograms using normal-mode summation. Differential seismo-
grams estimated through the Born approximation have been used
as perturbations for matching long-period seismograms, which can
provide high-resolution upper-mantle tomography 共Gilbert and Dz-
iewonski, 1975;Woodhouseand Dziewonski, 1984兲. The sensitivity
or Fréchet derivative matrix, i.e., the partial derivative of seismic
data with respect to the model parameters, is explicitly estimated be-
fore proceeding to inversion of the linearized system. The normal-
mode description allows a limited number of parameters to be in-
verted 共a few hundred parameters兲, which makes the optimization
procedure feasible through explicit sensitivity matrix estimation in
spite of the high number of seismograms.
Meanwhile, exploration seismology has taken up the challenge of
high-resolution imaging of the subsurface by designing dense, mul-
tifold acquisition systems. Construction of the sensitivity matrix is
too prohibitive because the number of parameters exceed 10,000. In-
stead, another road has been taken to perform high-resolution imag-
ing. Using the exploding-reflector concept, and after some kinemat-
ic corrections, amplitude summation has provided detailed images
of the subsurface for reservoir determination and characterization
共Claerbout, 1971,1976兲. The sum of the traveltimes from a specific
point of the interface toward the source and the receiver should coin-
cide with the time of large amplitudes in the seismogram. The reflec-
tivity as an amplitude attribute of related seismic traces at the select-
ed point of the reflector provides the migrated image needed for seis-
mic stratigraphic interpretation. Although migration is more a con-
cept for converting seismic data recorded in the time-space domain
Manuscript received by the Editor 7 June 2009; revised manuscript received 8 July 2009; published online 3 December 2009.
1Université Joseph Fourier, Laboratoire de Géophysique Interne et Tectonophysique, CNRS, IRD, Grenoble, France. E-mail: Jean.Virieux@obs.ujf-
grenoble.fr.
2Université Nice-Sophia Antipolis, Géoazur, CNRS, IRD, Observatoire de la Côte d’Azur, Villefranche-sur-mer,France. E-mail: operto@geoazur.obs-vlfr.fr.
© 2009 Society of Exploration Geophysicists. All rights reserved.
GEOPHYSICS, VOL. 74, NO. 6 共NOVEMBER-DECEMBER 2009兲; P. WCC127–WCC152, 15 FIGS., 1TABLE.
10.1190/1.3238367
WCC127
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into images of physical properties, we often refer to it as the geomet-
ric description of the short wavelengths of the subsurface. Avelocity
macromodel or background model provides the kinematic informa-
tion required to focus waves inside the medium.
The limited offsets recorded by seismic reflection surveys and the
limited-frequency bandwidth of seismic sources make seismic im-
aging poorly sensitive to intermediate wavelengths 共Jannane et al.,
1989兲. This is the motivation behind a two-step workflow: construct
the macromodel using kinematic information, and then the ampli-
tude projection using different types of migrations 共Claerbout and
Doherty, 1972;Gazdag, 1978;Stolt, 1978;Baysal et al., 1983;Yil-
maz, 2001;Biondi and Symes, 2004兲. This procedure is efficient for
relatively simple geologic targets in shallow-water environments,
although more limited performances have been achieved for imag-
ing complex structures such as salt domes, subbasalt targets, thrust
belts, and foothills. In complex geologic environments, building an
accurate velocity background model for migration is challenging.
Various approaches for iterative updating of the macromodel recon-
struction have been proposed 共Snieder et al., 1989;Docherty et al.,
2003兲, but they remain limited by the poor sensitivity of the reflec-
tion seismic data to the large and intermediate wavelengths of the
subsurface.
Simultaneous with the global seismology inversion scheme,
Lailly 共1983兲and Tarantola 共1984兲recast the migration imaging
principle of Claerbout 共1971,1976兲as a local optimization problem,
the aim of which is least-squares minimization of the misfit between
recorded and modeled data. They show that the gradient of the misfit
function along which the perturbation model is searched can be built
by crosscorrelating the incident wavefield emitted from the source
and the back-propagated residual wavefields. The perturbation mod-
el obtained after the first iteration of the local optimization looks like
a migrated image obtained by reverse-time migration. One differ-
ence is that the seismic wavefield recorded at the receiver is back
propagated in reverse time migration, whereas the data misfit is back
propagated in the waveform inversion of Lailly 共1983兲and Tarantola
共1984兲. When added to the initial velocity, the velocity perturbations
lead to an updated velocity model, which is used as a starting model
for the next iteration of minimizing the misfit function. The impres-
sive amount of data included in seismograms 共each sample of a time
series must be considered兲is involved in gradient estimation. This
estimation is performed by summation over sources, receivers, and
time.
Waveform-fitting imaging was quite computer demanding at that
time, even for 2D geometries 共Gauthier et al., 1986兲. However, it has
been applied successfully in various studies using forward-model-
ing techniques such as reflectivity techniques in layered media 共Kor-
mendi and Dietrich, 1991兲, finite-difference techniques 共Kolb et al.,
1986;Ikelle et al., 1988;Crase et al., 1990;Pica et al., 1990;Djik-
péssé and Tarantola, 1999兲, finite-element methods 共Choi et al.,
2008兲, and extended ray theory 共Cary and Chapman, 1988;Koren et
al., 1991;Sambridge and Drijkoningen, 1992兲. A less computation-
ally intensive approach is achieved by Jin et al. 共1992兲and Lambaré
et al. 共1992兲, who establish the theoretical connection between ray-
based generalized Radon reconstruction techniques 共Beylkin, 1985;
Bleistein, 1987;Beylkin and Burridge, 1990兲and least-squares opti-
mization 共Tarantola, 1987兲. By defining a specific norm in the data
space, which varies from one focusing point to the next, they were
able to recast the asymptotic Radon transform as an iterative least-
squares optimization after diagonalizing the Hessian operator. Ap-
plications on 2D synthetic data and real data are provided 共Thierry et
al., 1999b;Operto et al., 2000兲and 3D extension is possible 共Thierry
et al., 1999a;Operto et al., 2003兲because of efficient asymptotic for-
ward modeling 共Lucio et al., 1996兲. Because the Green’s functions
are computed in smoothed media with the ray theory, the forward
problem is linearized with the Born approximation, and the optimi-
zation is iterated linearly, which means the background model re-
mains the same over the iterations. These imaging methods are gen-
erally called migration/inversion or true-amplitude prestack depth
migration 共PSDM兲. The main difference with the waveform-inver-
sion methods we describe is that the smooth background model does
not change over iterations and only the single scattered wavefield is
modeled by linearizing the forward problem.
Alternatively, the full information content in the seismogram can
be considered in the optimization. This leads us to full-waveform in-
version 共FWI兲, where full-wave equation modeling is performed at
each iteration of the optimization in the final model of the previous
iteration. All types of waves are involved in the optimization, includ-
ing diving waves, supercritical reflections, and multiscattered waves
such as multiples. The techniques used for the forward modeling
vary and include volumetric methods such as finite-element meth-
ods 共Marfurt, 1984;Min et al., 2003兲, finite-difference methods
共Virieux, 1986兲, finite-volume methods 共Brossier et al., 2008兲, and
pseudospectral methods 共Danecek and Seriani, 2008兲; boundary in-
tegral methods such as reflectivity methods 共Kennett, 1983兲; gener-
alized screen methods 共Wu, 2003兲; discrete wavenumber methods
共Bouchon et al., 1989兲; generalized ray methods such as WKBJ and
Maslov seismograms 共Chapman, 1985兲; full-wave theory 共de Hoop,
1960兲; and diffraction theory 共Klem-Musatov and Aizenberg, 1985兲.
FWI has not been recognized as an efficient seismic imaging tech-
nique because pioneering applications were restricted to seismic re-
flection data. For short-offset acquisition, the seismic wavefield is
rather insensitive to intermediate wavelengths; therefore, the opti-
mization cannot adequately reconstruct the true velocity structure
through iterative updates. Only when a sufficiently accurate initial
model is provided can waveform-fitting converge to the velocity
structure through such updates. For sampling the initial model, so-
phisticated investigations with global and semiglobal techniques
共Koren et al., 1991;Jin and Madariaga, 1993,1994;Mosegaard and
Tarantola, 1995;Sambridge and Mosegaard, 2002兲have been per-
formed. The rather poor performance of these investigations that
arises from insensitivity to intermediate wavelengths has led many
researchers to believe that this optimization technique is not particu-
larly efficient.
Only with the benefit of long-offset and transmission data to re-
construct the large and intermediate wavelengths of the structure has
FWI reached its maturity as highlighted by Mora 共1987,1988兲,Pratt
and Worthington 共1990兲,Pratt et al. 共1996兲, and Pratt 共1999兲. FWI at-
tempts to characterize a broad and continuous wavenumber spec-
trum at each point of the model, reunifying macromodel building
and migration tasks into a single procedure. Historical crosshole and
wide-angle surface data examples illustrate the capacity of simulta-
neous reconstruction of the entire spatial spectrum 共e.g., Pratt, 1999;
Ravaut et al., 2004;Brenders and Pratt, 2007a兲. However, robust ap-
plication of FWI to long-offset data remains challenging because of
increasing nonlinearities introduced by wavefields propagated over
several tens of wavelengths and various incidence angles 共Sirgue,
2006兲.
Here, we consider the main aspects of FWI. First, we review the
forward-modeling problem that underlies FWI. Efficient numerical
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modeling of the full seismic wavefield is a central issue in FWI, es-
pecially for 3D problems.
In the second part, we review the main theoretical aspects of FWI
based on a least-squares local optimization approach. We follow the
compact matrix formalism for its simplicity 共Pratt et al., 1998;Pratt,
1999兲, which leads to a clear interpretation of the gradient and the
Hessian of the objective function. Once the gradient is estimated, we
review different optimization algorithms used to compute the pertur-
bation model. We conclude the methodology section by the source
estimation problem in FWI.
In the third part, we review some key features of FWI. First, we
highlight the relationships between the experimental setup 共source
bandwidth, acquisition geometry兲and the spatial resolution of FWI.
The resolution analysis provides the necessary guidelines to design
the multiscale FWI algorithms required to mitigate the nonlinearity
of FWI. We discuss the pros and cons of the time and frequency do-
mains for efficient multiscale algorithms. We provide a few words
concerning the parallel implementation of FWI techniques because
these are computer demanding. Then we review some alternatives to
the least-squares criterion and the Born linearization. Akey issue of
FWI is the initial model from which the local optimization is started.
We also discuss several tomographic approaches to building a start-
ing model.
In the fourth part, we review the main case studies of FWI subdi-
vided into three categories of case studies: acoustic, multiparameter,
and three dimensional. Finally, we discuss the future challenges
raised by the revival of interest in FWI that has been shown by the
exploration and the earthquake-seismology communities.
THE FORWARD PROBLEM
Let us first introduce the notations for the forward problem, name-
ly, modeling the full seismic wavefield. The reader is referred to
Robertsson et al. 共2007兲for an up-to-date series of publications on
modern seismic-modeling methods.
We use matrix notations to denote the partial-differential opera-
tors of the wave equation 共Marfurt, 1984;Carcione et al., 2002兲. The
most popular direct method to discretize the wave equation in the
time and frequency domains is the finite-difference method
共Virieux, 1986;Levander, 1988;Graves, 1996;Operto et al., 2007兲,
although more sophisticated finite-element or finite-volume ap-
proaches can be considered. This is especially true when accurate
boundary conditions through unstructured meshes must be imple-
mented 共e.g., Komatitsch and Vilotte, 1998;Dumbser and Kaser,
2006兲.
In the time domain, we have
M共x兲d2u共x,t兲
dt2⳱A共x兲u共x,t兲Ⳮs共x,t兲,共1兲
where Mand Aare the mass and the stiffness matrices, respectively
共Marfurt, 1984兲. The source term is denoted by sand the seismic
wavefield by u. In the acoustic approximation, ugenerally repre-
sents pressure, although in the elastic case ugenerally represents
horizontal and vertical particle velocities. The time is denoted by t
and the spatial coordinates by x. Equation 1generally is solved with
an explicit time-marching algorithm: The value of the wavefield at a
time step 共nⳭ1兲at a spatial position is inferred from the value of the
wavefields at previous time steps. Implicit time-marching algo-
rithms are avoided because they require solving a linear system
共Marfurt, 1984兲. If both velocity and stress wavefields are helpful,
the system of second-order equations can be recast as a first-order
hyperbolic velocity-stress system by incorporating the necessary
auxiliary variables 共Virieux, 1986兲.
In the frequency domain, the wave equation reduces to a system of
linear equations; the right-hand side is the source and the solution is
the seismic wavefield. This system can be written compactly as
B共x,
兲u共x,
兲⳱s共x,
兲,共2兲
where Bis the impedance matrix 共Marfurt, 1984兲. The sparse com-
plex-valued matrix Bhas a symmetric pattern, although it is not sym-
metric because of absorbing boundary conditions 共Hustedt et al.,
2004;Operto et al., 2007兲.
Equation 2can be solved by a decomposition of Bsuch as lower
and upper 共LU兲triangular decomposition, leading to direct-solver
techniques. The advantage of the direct-solver approach is that once
the decomposition is performed, equation 2is efficiently solved for
multiple sources using forward and backward substitutions 共Mar-
furt, 1984兲. The direct-solver approach is efficient for 2D forward
problems 共Jo et al., 1996;Stekl and Pratt, 1998;Hustedt et al., 2004兲.
However, the time and memory complexities of LU factorization
and its limited scalability on large-scale distributed memory plat-
forms prevent use of the approach for large-scale 3D problems 共i.e.,
problems involving more than 10 million unknowns; Operto et al.,
2007兲.
Iterative solvers provide an alternative approach for solving the
time-harmonic wave equation 共Riyanti et al., 2006,2007;Plessix,
2007;Erlangga and Herrmann, 2008兲. Iterative solvers currently are
implemented with Krylov subspace methods 共Saad, 2003兲that are
preconditioned by solving the dampened time-harmonic wave equa-
tion. The solution of the dampened wave equation is computed with
one cycle of a multigrid. The main advantage of the iterative ap-
proach is the low memory requirement, although the main drawback
results from a difficulty to design an efficient preconditioner because
the impedance matrix is indefinite. To our knowledge, the extension
to elastic wave equations still needs to be investigated. As for the
time-domain approach, the time complexity of the iterative ap-
proach increases linearly with the number of sources in contrast to
the direct-solver approach.
An intermediate approach between the direct and iterative meth-
ods consists of a hybrid direct-iterative approach based on a domain
decomposition method and the Schur complement system 共Saad,
2003;Sourbier et al., 2008兲. The iterative solver is used to solve the
reduced Schur complement system, the solution of which is the
wavefield at interface nodes between subdomains. The direct solver
is used to factorize local impedance matrices that are assembled on
each subdomain. Briefly, the hybrid approach provides a compro-
mise in terms of memory savings and multisource-simulation effi-
ciency between the direct and the iterative approaches.
The last possible approach to compute monochromatic wave-
fields is to perform the modeling in the time domain and extract the
frequency-domain solution either by discrete Fourier transform in
the loop over the time steps 共Sirgue et al., 2008兲or by phase-sensitiv-
ity detection once the steady-state regime is reached 共Nihei and Li,
2007兲. One advantage of the approach based on the discrete Fourier
transform is that an arbitrary number of frequencies can be extracted
within the loop over time steps at minimal extra cost. Second, time
windowing can be easily applied, which is not the case when the
modeling is performed in the frequency domain. Time windowing
allows the extraction of specific arrivals for FWI 共early arrivals, re-
flections, PS converted waves兲, which is often useful to mitigate the
Full-waveform inversion WCC129
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nonlinearity of the inversion by judicious data preconditioning
共Sears et al., 2008;Brossier et al., 2009a兲.
Among all of these possible approaches, the iterative-solver ap-
proach theoretically has the best time complexity 共here, “complexi-
ty” denotes how the computational cost of an algorithm grows with
the size of the computational domain兲if the number of iterations is
independent of the frequency 共Erlangga and Herrmann, 2008兲.In
practice, the number of iterations generally increases linearly with
frequency. In this case, the time complexities of the time-domain and
the iterative-solver approach are equivalent 共Plessix, 2007兲.
The reader is referred to Plessix 共2007,2009兲and Virieux et al.
共2009兲for more detailed complexity analyses of seismic modeling
based on different numerical approaches. A discussion on the pros
and cons of time-domain versus frequency-domain seismic model-
ing with application to FWI is also provided in Vigh and Starr
共2008b兲and Warner et al. 共2008兲.
Source implementation is an important issue in FWI. The spatial
reciprocity of Green’s functions can be exploited in FWI to mitigate
the number of forward problems if the number of receivers is signifi-
cantly smaller than the number of sources 共Aki and Richards, 1980兲.
The reciprocity of Green’s functions also allows matching data emit-
ted by explosions and recorded by directional sensors, with pressure
synthetics computed for directional forces 共Operto et al., 2006兲.Of
note, the spatial reciprocity is satisfied theoretically for the unidirec-
tional sensor and the unidirectional impulse source. However, the
spatial reciprocity of the Green’s functions can also be used for ex-
plosive sources by virtue of the superposition principle. Indeed, ex-
plosions can be represented by double dipoles or, in other words, by
four unidirectional impulse sources.
A final comment concerns the relationship between the discretiza-
tion required to solve the forward problem and that required to re-
construct the physical parameters. Often during FWI, these two dis-
cretizations are identical, although it is recommended that the fin-
gerprint of the forward problem be kept minimal in FWI.
The properties of the subsurface that we want to quantify are em-
bedded in the coefficients of matrices M,A,orBof equations 1and
2. The relationship between the seismic wavefield and the parame-
ters is nonlinear and can be written compactly through the operator
G, defined as
u⳱G共m兲共3兲
in the time domain or in the frequency domain.
FWI AS A LEAST-SQUARES LOCAL
OPTIMIZATION
We follow the simplest view of FWI based on the so-called length
method 共Menke, 1984兲. For information on probabilistic maximum
likelihood or generalized inverse formulations, the reader is referred
to Menke 共1984兲,Tarantola 共1987兲,Scales and Smith 共1994兲, and
Sen and Stoffa 共1995兲.
We define the misfit vector ⌬d⳱dobsⳮdcal共m兲of dimension N
by the differences at the receiver positions between the recorded
seismic data dobs and the modeled seismic data dcal共m兲for each
source-receiver pair of the seismic survey. Here, dcal can be related to
the modeled seismic wavefield uby a detection operator R, which
extracts the values of the wavefields computed in the full computa-
tional domain at the receiver positions for each source: dcal ⳱Ru.
The model mrepresents some physical parameters of the subsurface
discretized over the computational domain.
In the simplest case corresponding to the monoparameter acoustic
approximation, the model parameters are the P-wave velocities de-
fined at each node of the numerical mesh used to discretize the in-
verse problem. In the extreme case, the model parameters corre-
spond to the 21 elastic moduli that characterize linear triclinic elastic
media, the density, and some memory variables that characterize the
anelastic behavior of the subsurface 共Toksöz and Johnston, 1981兲.
The most common discretization consists of projection of the con-
tinuous model of the subsurface on a multidimensional Dirac comb,
although a more complex basis can be considered 共see Appendix A
in Pratt et al. 关1998兴for a discussion on alternative parameteriza-
tions兲. We define a norm C共m兲of this misfit vector ⌬d, which is re-
ferred to as the misfit function or the objective function. We focus be-
low on the least-squares norm, which is easier to manipulate mathe-
matically 共Tarantola, 1987兲. Other norms are discussed later.
The Born approximation and the linearization of the
inverse problem
The least-squares norm is given by
C共m兲⳱1
2⌬d†⌬d,共4兲
where † denotes the adjoint operator 共transpose conjugate兲.
In the time domain, the implicit summation in equation 4is per-
formed over the number of source-channel pairs and the number of
time samples in the seismograms, where a channel is one component
of a multicomponent sensor. In the frequency domain, the summa-
tion over frequencies replaces that over time. In the time domain, the
misfit vector is real valued; in the frequency domain, it is complex
valued.
The minimum of the misfit function C共m兲is sought in the vicinity
of the starting model m0. The FWI is essentially a local optimization.
In the framework of the Born approximation, we assume that the up-
dated model mof dimension Mcan be written as the sum of the start-
ing model m0plus a perturbation model ⌬m:m⳱m0Ⳮ⌬m.Inthe
following, we assume that mis real valued.
A second-order Taylor-Lagrange development of the misfit func-
tion in the vicinity of m0gives the expression
C共m0Ⳮ⌬m兲⳱C共m0兲Ⳮ兺
j⳱1
M
C共m0兲
mj
⌬mj
Ⳮ1
2兺
j⳱1
M
兺
k⳱1
M
2C共m0兲
mj
mk
⌬mj⌬mkⳭO共m3兲.
共5兲
Taking the derivative with respect to the model parameter mlresults
in
C共m兲
ml
⳱
C共m0兲
ml
Ⳮ兺
j⳱1
M
2C共m0兲
mj
ml
⌬mj.共6兲
The minimum of the misfit function in the vicinity of point m0is
reached when the first derivative of the misfit function vanishes. This
gives the perturbation model vector:
⌬m⳱ⳮ
冋
2C共m0兲
m2
册
ⳮ1
C共m0兲
m.共7兲
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The perturbation model is searched in the opposite direction of the
steepest ascent 共i.e., the gradient兲of the misfit function at point m0.
The second derivative of the misfit function is the Hessian; it defines
the curvature of the misfit function at m0. Of note, the error term
O共m3兲in equation 5is zero when the misfit function is a quadratic
function of m. This is the case for linear forward problems such as u
⫽G.m. In this case, the expression of the perturbation model of
equation 7gives the minimum of the misfit function in one iteration.
In FWI, the relationship between the data and the model is nonlinear
and the inversion needs to be iterated several times to converge to-
ward the minimum of the misfit function.
Normal equations: The Newton, Gauss-Newton, and
steepest-descent methods
Basic equations
The derivative of C共m兲with respect to the model parameter ml
gives
C共m兲
ml
⳱ⳮ
1
2兺
i⳱1
N
冋
冉
dcali
ml
冊
共dobsi
ⳮdcali兲*
Ⳮ共dobsi
ⳮdcali兲
dcali
*
ml
册
⳱ⳮ兺
i⳱1
N
R
冋
冉
dcali
ml
冊
*
共dobsi
ⳮdcali兲
册
,共8兲
where the real part and the conjugate of a complex number are denot-
ed by Rand *, respectively. In matrix form, equation 8translates to
ⵜCm⳱
C共m兲
m
⳱ⳮR
冋
冉
dcal共m兲
m
冊
†
共dobsⳮdcal共m兲兲
册
⳱ⳮR关J†⌬d兴,共9兲
where Jis the sensitivity or the Fréchet derivative matrix. In equa-
tion 9,ⵜCmis a vector of dimension M. Takingm⳱m0in equation 9
provides the descent direction along which the perturbation model is
searched in equation 7.
Differentiation of the gradient expression 8, with respect to the
model parameters gives the following expression in matrix form for
the Hessian 共see Pratt et al. 关1998兴for details兲:
2C共m0兲
m2⳱R关J0
†J0兴ⳭR
冋
J0
t
mt共⌬d0
*...⌬d0
*兲
册
.共10兲
Inserting the expression of the gradient 共equation 9兲and the Hessian
共equation 10兲into equation 7gives the following for the perturbation
model:
⌬m⳱ⳮ
再
R
冋
J0
†J0Ⳮ
J0
t
mt共⌬d0
*...⌬d0
*兲
册
冎
ⳮ1
R关J0
†⌬d0兴.
共11兲
The method solving the normal equations, e.g., equation 11, general-
ly is referred to as the Newton method, which is locally quadratically
convergent.
For linear problems 共d⫽G.m兲, the second term in the Hessian is
zero because the second-order derivative of the data with respect to
model parameters is zero. Most of the time, this second-order term is
neglected for nonlinear inverse problems. In the following, the re-
maining term in the Hessian, i.e., Ha⳱J0
†J0, is referred to as the ap-
proximate Hessian. The method which solves equation 11 when only
Hais estimated is referred to as the Gauss-Newton method.
Alternatively, the inverse of the Hessian in equation 11 can be re-
placed by a scalar
␣
, the so-called step length, leading to the gradient
or steepest-descent method. The step length can be estimated by a
line-search method, for which a linear approximation of the forward
problem is used 共Gauthier et al., 1986;Tarantola, 1987兲. In the linear
approximation framework, the second-order Taylor-Lagrange de-
velopment of the misfit function gives
C共mⳮ
␣
ⵜC共m0兲兲⳱C共m兲ⳮ
␣
具ⵜC共m兲兩ⵜC共m0兲典
Ⳮ1
2
␣
2Ha共m兲具ⵜC共m0兲兩ⵜC共m0兲典,
共12兲
where we assume a model perturbation of the form ⌬m
⳱
␣
ⵜC共m0兲. In equation 12, we replace the second-order deriva-
tive of the misfit function by the approximate Hessian in the second
term on the right-hand side. Inserting the expression of the approxi-
mate Hessian Hainto the previous expression, zeroing the partial de-
rivative of the misfit function with respect to
␣
, and using m⳱m0
gives
␣
⳱具ⵜC共m0兲兩ⵜC共m0兲典
具共Jt共m0兲ⵜC共m0兲兩Jt共m0兲ⵜC共m0兲兲典 .共13兲
The term Jt共m0兲ⵜC共m0兲is computed conventionally using a first-
order-accurate finite-difference approximation of the partial deriva-
tive of G,
G共m0兲
mⵜC共m0兲⳱1
共G共m0ⳭⵜC共m0兲兲ⳮG共m0兲兲,
共14兲
with a small parameter . Estimation of
␣
requires solving an extra
forward problem per shot for the perturbed model m0ⳭⵜC共m0兲.
This line-search technique is extended to multiple-parameter classes
by Sambridge et al. 共1991兲using a subspace approach. In this case,
one forward problem must be solved per parameter class, which can
be computationally expensive. Alternatively, the step length can be
estimated by parabolic interpolation through three points, 共
␣
,C共m0
Ⳮ
␣
ⵜC共m0兲兲兲. The minimum of the parabola provides the desired
␣
. In this case, two extra forward problems per shot must be solved
because we already have a third point corresponding to 共0,C共m0兲兲
共see Figure 1in Vigh et al. 关2009兴for an illustration兲.
Pratt et al. 共1998兲illustrate how quality and rate of convergence of
the inversion depend significantly on the Newton, Gauss-Newton, or
gradient method used. Importantly, they show how the gradient
method can fail to converge toward an acceptable model, however
many iterations, unlike the Newton and Gauss-Newton methods.
They interpret this failure as the result of the difficulty of estimating
a reliable step length. However, gradient methods can be significant-
ly improved by scaling 共i.e., dividing兲the gradient by the diagonal
terms of Haor of the pseudo-Hessian 共Shin et al., 2001a兲.
Full-waveform inversion WCC131
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Numerical algorithms: The conjugate-gradient method
Over the last decade, the most popular local optimization algo-
rithm for solving FWI problems was based on the conjugate-gradi-
ent method 共Mora, 1987;Tarantola, 1987;Crase et al., 1990兲. Here,
the model is updated at the iteration nin the direction of p共n兲, which is
a linear combination of the gradient at iteration n,ⵜC共n兲, and the di-
rection p共nⳮ1兲:
p共n兲⳱ⵜCnⳭ

共n兲p共nⳮ1兲.共15兲
The scalar

共n兲is designed to guarantee that p共n兲and p共nⳮ1兲are con-
jugate. Among the different variants of the conjugate-gradient meth-
od to derive the expression of

共n兲, the Polak-Ribière formula 共Polak
and Ribière, 1969兲is generally used for FWI:

共n兲⳱具ⵜC共n兲ⳮⵜC共nⳮ1兲兩ⵜC共n兲典
储ⵜC共n兲储2.共16兲
In FWI, the preconditioned gradient Wm
ⳮ1ⵜC共n兲is used for p共n兲, where
Wmis a weighting operator that is introduced in the next section
共Mora, 1987兲. Only three vectors of dimension M, i.e., ⵜC共n兲,
ⵜC共nⳮ1兲, and p共nⳮ1兲, are required to implement the conjugate-gradi-
ent method.
Numerical algorithms: Quasi-Newton algorithms
Finite approximations of the Hessian and its inverse can be com-
puted using quasi-Newton methods such as the BFGS algorithm
共named after its discoverers Broyden, Fletcher, Goldfarb, and Sh-
anno; see Nocedal 关1980兴for a review兲. The main idea is to update
the approximation of the Hessian or its inverse H共n兲at each iteration
of the inversion, taking into account the additional knowledge pro-
vided by ⵜC共n兲at iteration n. In these approaches, the approximation
of the Hessian or its inverse is explicitly formed.
For large-scale problems such as FWI in which the cost of storing
and working with the approximation of the Hessian matrix is prohib-
itive, a limited-memory variant of the quasi-Newton BFGS method
known as the L-BFGS algorithm allows computing in a recursive
manner H共n兲ⵜC共n兲without explicitly forming H共n兲. Only a few gradi-
ents of the previous nonlinear iterations 共typically, 3–20 iterations兲
need to be stored in L-BFGS, which represents a negligible storage
and computational cost compared to the conjugate-gradient algo-
rithm 共see Nocedal, 1980; p. 177–180兲. The L-BFGS algorithm re-
quires an initial guess H共0兲, which can be provided by the inverse of
the diagonal Hessian 共Brossier et al., 2009a兲. For multiparameter
FWI, the L-BFGS algorithm provides a suitable scaling of the gradi-
ents computed for each parameter class and hence provides a com-
putationally efficient alternative to the subspace method of Sam-
bridge et al. 共1991兲. A comparison between the conjugate-gradient
method and the L-BFGS method for a realistic onshore application
of multiparameter elastic FWI is shown in Brossier et al. 共2009a兲.
Newton and Gauss-Newton algorithms
The more accurate, although more computationally intensive,
Gauss-Newton and Newton algorithms are described in Akcelik
共2002兲,Askan et al. 共2007兲,Askan and Bielak 共2008兲, and Epano-
meritakis et al. 共2008兲, with an application to a 2D synthetic model
of the San Fernando Valley using the SH-wave equation. At each
nonlinear FWI iteration, a matrix-free conjugate-gradient method is
used to solve the reduced Karush-Kuhn-Tucker 共KKT兲optimal sys-
tem, which turns out to be similar to the normal equation system
共equation 11兲. Neither the full Hessian nor the sensitivity matrix is
formed explicitly; only the application of the Hessian to a vector
needs to be performed at each iteration of the conjugate-gradient al-
gorithm.
Application of the Hessian to a vector requires performing two
forward problems per shot for the incident and the adjoint wavefields
共Akcelik, 2002兲. Because these two simulations are performed at
each iteration of the conjugate-gradient algorithm, an efficient pre-
conditioner must be used to mitigate the number of iterations of the
conjugate-gradient algorithm. Epanomeritakis et al. 共2008兲use a
variant of the L-BFGS method for the preconditioner of the conju-
gate gradient, in which the curvature of the objective function is up-
dated at each iteration of the conjugate gradient using the Hessian-
vector products collected over the iterations.
Regularization and preconditioning of inversion
As widely stressed, FWI is an ill-posed problem, meaning that an
infinite number of models matches the data. Some regularizations
are conventionally applied to the inversion to make it better posed
共Menke, 1984;Tarantola, 1987;Scales et al., 1990兲.The misfit func-
tion can be augmented as follows:
C共m兲⳱1
2⌬d†Wd⌬dⳭ1
2共mⳮmprior兲†Wm共mⳮmprior兲,
共17兲
where Wd⳱Sd
tSdand Wm⳱Sm
tSm. Weighting operators are Wdand
Wm, the inverse of the data and model covariance operators in the
frame of the Bayesian formulation of FWI 共Tarantola, 1987兲. The
operator Sdcan be implemented as a diagonal weighting operator
that controls the respective weight of each element of the data-misfit
vector. For example, Operto et al. 共2006兲use Sdas a power of the
source-receiver offset to strengthen the contribution of large-offset
data for crustal-scale imaging. In geophysical applications where the
smoothest model that fits the data is often sought, the aim of the
least-squares regularization term in the augmented misfit function
共equation 17兲is to penalize the roughness of the model m, hence de-
fining the so-called Tikhonov regularization 共see Hansen 关1998兴for
a review on regularization methods兲. The operator Smis generally a
roughness operator, such as the first- or second-difference matrices
共Press et al., 1986, 1007兲.
For linear problems 共assuming the second term of the Hessian is
neglected兲, the minimization of the weighted misfit function gives
the perturbation model:
⌬m⳱ⳮ兵R共J0
†WdJ0兲ⳭWm其ⳮ1R关J0
†Wd⌬d0兴,共18兲
where we use mprior ⳱m0. Of note, equation 18 is equivalent to
Tarantola 共1987,p.70兲and Menke 共1984,p.55兲:
⌬m⳱ⳮWm
ⳮ1兵R共J0Wm
ⳮ1J0
†兲ⳭWd
ⳮ1其ⳮ1R关J0
†⌬d0兴.
共19兲
Equation 19 can be more tractable from a computational viewpoint
when N⬍M. Because Wmis a roughness operator, Wm
ⳮ1is a
smoothing operator. It can be implemented, for example, with a mul-
tidimensional adaptive Gaussian smoother 共Ravaut et al., 2004;Op-
erto et al., 2006兲or with a low-pass filter in the wavenumber domain
共Sirgue, 2003兲.
For the steepest-descent algorithm, the regularized solution for
the perturbation model is given by
WCC132 Virieux and Operto
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⌬m⳱ⳮ
␣
Wm
ⳮ1R关J0
†Wd⌬d0兴,共20兲
where the scaling performed by the diagonal terms of the approxi-
mate Hessian can be embedded in the operator Wm
ⳮ1in addition to the
smoothing operator.
A more complete and rigorous mathematical derivation of these
equations is presented in Tarantola 共1987兲.
Alternative regularizations based on minimizing the total varia-
tion of the model have been developed mainly by the image-process-
ing and electromagnetic communities. The aim of the total variation
or edge-preserving regularization is to preserve both the blocky and
the smooth characteristics of the model 共Vogel and Oman, 1996;Vo-
gel, 2002兲. Total variation 共TV兲regularization is conventionally im-
plemented by minimizing the L1-norm of the model-misfit function
RTV ⳱共⌬mWm⌬m兲1/2. Alternatively, van den Berg and Abubakar
共2001兲implement TV regularization as a multiplicative constraint in
the original misfit function. In this framework, the original misfit
function can be seen as the weighting factor of the regularization
term, which is automatically updated by the optimization process
without the need for heuristic tuning. TV regularization is applied to
FWI in Askan and Bielak 共2008兲. The weighted L2-norm regulariza-
tion applied to frequency-domain FWI is shown in Hu et al. 共2009兲
and Abubakar et al. 共2009兲.
The gradient and Hessian in FWI: Interpretation and
computation
A clear interpretation of the gradient and Hessian is given by Pratt
et al. 共1998兲using the compact matrix formalism of frequency-do-
main FWI. Areview is given here. Let us consider the forward-prob-
lem equation given by equation 2for one source and one frequency.
In the following, we assume that the model is discretized in a finite-
difference sense using a uniform grid of nodes.
Differentiation of equation 2with respect to the model parameter
mlgives the expression of the partial derivative wavefield
u/
mᐉ
by solving the following system:
B
u
mᐉ
⳱f共ᐉ兲,共21兲
where
f共ᐉ兲⳱ⳮ
B
mᐉ
u.共22兲
An analogy between the forward-problem equation 2and equa-
tion 21 shows that the partial-derivative wavefield can be computed
by solving one forward problem, the source of which is given by f共ᐉ兲.
This so-called virtual secondary source is formed by the product of
B/
mᐉand the incident wavefield u. The matrix
B/
mᐉis built by
differentiating each coefficient of the forward-problem operator B
with respect to mᐉ. Because the discretized differential operators in B
generally have local support, the matrix
B/
mᐉis extremely
sparse.
The spatial support of the virtual secondary source is centered on
the position of mᐉ, whereas the temporal support of f共ᐉ兲is centered
around the arrival time of the incident wavefield at the position of mᐉ.
Therefore, the partial-derivative wavefield with respect to the model
parameter mᐉcan be interpreted as the wavefield emitted by the seis-
mic source sand scattered by a point diffractor located at mᐉ. The ra-
diation pattern of the virtual secondary source is controlled by the
operator
B/
mᐉ. Analysis of this radiation pattern for different pa-
rameter classes allows us to assess to what extent parameters of dif-
ferent natures are uncoupled in the tomographic reconstruction as a
function of the diffraction angle and to what extent they can be reli-
ably reconstructed during FWI. Radiation patterns for the isotropic
acoustic, elastic, and viscoelastic wave equations are shown in Wu
and Aki 共1985兲,Tarantola 共1986兲,Ribodetti and Virieux 共1996兲, and
Forgues and Lambaré 共1997兲.
Because the gradient is formed by the zero-lag correlation be-
tween the partial-derivative wavefield and the data residual, these
have the same meaning: They represent perturbation wavefields
scattered by the missing heterogeneities in the starting model m0
共Tarantola, 1984;Pratt et al., 1998兲.This interpretation draws some
connections between FWI and diffraction tomography; the perturba-
tion model can be represented by a series of closely spaced diffrac-
tors. By virtue of Huygens’ principle, the image of the model pertur-
bations is built by the superposition of the elementary image of each
diffractor, and the seismic wavefield perturbation is built by super-
position of the wavefields scattered by each point diffractor 共Mc-
Mechan and Fuis, 1987兲.
The approximate Hessian is formed by the zero-lag correlation
between the partial-derivative wavefields, e.g., equation 10. The di-
agonal terms of the approximate Hessian contain the zero-lag auto-
correlation and therefore represent the square of the amplitude of the
partial-derivative wavefield. Scaling the gradient by these diagonal
terms removes from the gradient the geometric amplitude of the par-
tial-derivative wavefields and the residuals. In the framework of sur-
face seismic experiments, the effects of the scaling performed by the
diagonal Hessian provide a good balance between shallow and deep
perturbations. A diagonal Hessian is shown in Ravaut et al. 共2004,
their Figure 12兲. The off-diagonal terms of the Hessian are computed
by correlation between partial-derivative wavefields associated with
different model parameters. For 1D media, the approximate Hessian
is a band-diagonal matrix, and the numerical bandwidth decreases as
the frequency increases. The off-diagonal elements of the approxi-
mate Hessian account for the limited-bandwidth effects that result
from the experimental setup. Applying its inverse to the gradient can
be interpreted as a deconvolution of the gradient from these limited-
bandwidth effects.
An illustration of the scaling and deconvolution effects performed
by the diagonal Hessian on one hand and the approximate Hessian
on the other hand is provided in Figure 1. Asingle inclusion in a ho-
mogeneous background model 共Figure 1a兲is reconstructed by one
iteration of FWI using a gradient method preconditioned by the diag-
onal terms of the approximate Hessian 共Figure 1b兲and by a Gauss-
Newton method 共Figure 1c兲. The image of the inclusion is sharper
when the Gauss-Newton algorithm is used. The corresponding ap-
proximate Hessian and its diagonal elements are shown in Figure 2.
An interpretation of the second term of the Hessian 共equation 10兲is
given in Pratt et al. 共1998兲. This term accounts for multiscattering
events such as multiples in the reconstruction procedure. Through it-
erations, we might correct effects caused by this missing term as
long as convergence is achieved.
Although equation 21 gives some clear insight into the physical
sense of the gradient of the misfit function, it is impractical from a
computer-implementation point of view; with the computer explicit-
ly forming the sensitivity matrix with equation 21, it would require
performing as many forward problems as the number of model pa-
rameters mᐉ共ᐉ⳱1,M兲for each source of the survey. To mitigate this
computational burden, the spatial reciprocity of Green’s functions
can be exploited as shown below.
Full-waveform inversion WCC133
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Inserting the expression of the partial derivative of the wavefield
共equation 21兲in the expression of the gradient of equation 9gives the
following expression of the gradient:
ⵜCᐉ⳱R
冋
ut
冋
B
mᐉ
册
t
Bⳮ1t共P⌬d兲*
册
,共23兲
where Pdenotes an operator that augments the residual data vector
with zeroes in the full computational domain so that the dimension
of the augmented vector matches the dimension of the matrix Bⳮ1t
共Pratt et al., 1998兲. The column of Bⳮ1corresponds to the Green’s
functions for unit impulse sources located at each node of the model.
By virtue of the spatial reciprocity of the Green’s functions, Bⳮ1is
symmetric. Therefore, Bⳮ1tcan be substituted by Bⳮ1in equation 23,
which gives
ⵜCᐉ⳱R
冋
ut
冋
B
mᐉ
册
t
Bⳮ1共P⌬d*兲
册
⳱R
冋
ut
冋
B
mᐉ
册
t
rb
册
.
共24兲
The wavefield rbcorresponds to the back-propagated residual wave-
field. All of the residuals associated with one seismic source are as-
sembled to form one residual source. The back propagation in time is
indicated by the conjugate operator in the frequency domain. The
number of forward seismic problems for computing the gradient is
reduced to two: one to compute the incident wavefield uand one to
back propagate the corresponding residuals. The underlying imag-
ing principle is reverse-time migration, which relies on the corre-
spondence of the arrival times of the incident wavefield and the
0123
0
1
2
3
Distance (km)
Depth (km)
0123
0
1
2
3
Depth (km)
0123
0
1
2
3
Depth (km)
Distance (km)
Distance (km)
4.0 4.1 4.2 4.3
0
1
2
3
4.0 4.1 4.2 4.3
0
1
2
3
VP(km/s)
VP(km/s)
a)
b)
c)
d)
e)
Figure 1. Reconstruction of an inclusion by frequency-domain FWI.
共a兲True model and FWI models built 共b兲by a preconditioned gradi-
ent method and 共c兲by a Gauss-Newton method. Four frequencies 共4,
5, 7, and 10 Hz兲were inverted. One iteration per frequency was com-
puted. Fourteen shots were deployed along the top and left edges of
the model. Shots along the top edge were recorded by 14 receivers
along the bottom edge; shots along the left edges were recorded by
14 receivers along the right edge. The P-wave velocities in the back-
ground and in the inclusion are 4.0 and 4.2 km/s, respectively. Verti-
cal velocity profiles are extracted from the true model 共gray line兲and
the FWI models 共black line兲for 共d兲the gradient and 共e兲the Gauss-
Newton inversions.
0
200
400
600
800
Row num
b
er
Depth (km)
Column number
0 200 400 600 800
0
5
10
15
20
25
30
Distance (km)
0 5 10 15 20 25 30
a)
b)
Figure 2. Hessian operator. 共a兲Approximate Hessian corresponding
to the 31 ⫻31 model of Figure 1for a frequency of 4 Hz. Aclose-up
of the area delineated by the yellow square highlights the band-diag-
onal structure of the Hessian. 共b兲Corresponding diagonal terms of
the Hessian plotted in the distance-depth domain. The high-ampli-
tude coefficients indicate source and receiver positions. Scaling the
gradient by this map removes the geometric amplitude effects from
the wavefields.
WCC134 Virieux and Operto
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back-propagated wavefield at the position of heterogeneity 共Claer-
bout, 1971;Lailly, 1983;Tarantola, 1984兲.
The approach that consists of computing the gradient of the misfit
function without explicitly building the sensitivity matrix is often re-
ferred to as the adjoint-wavefield approach by the geophysical com-
munity. The underlying mathematical theory is the adjoint-state
method of the optimization theory 共Lions, 1972;Chavent, 1974兲. In-
teresting links exist between optimization techniques used in FWI
and assimilation methods, widely used in fluid mechanics 共Tala-
grand and Courtier, 1987兲. A detailed review of the adjoint-state
method with illustrations from several seismic problems is given in
Tromp et al. 共2005兲,Askan 共2006兲,Plessix 共2006兲, and Epanomer-
itakis et al. 共2008兲. The expression of the gradient of the frequency-
domain FWI misfit function 共equation 24兲is derived from the ad-
joint-state method and the method of the Lagrange multiplier in Ap-
pendix A.
For multiple sources and multiple frequencies, the gradient is
formed by the summation over these sources and frequencies:
ⵜCᐉ⳱兺
i⳱1
N
兺
s⳱1
Ns
R
冋
关Bi
ⳮ1ss兴t
冋
Bi
mᐉ
册
t
关Bi
ⳮ1共P⌬di,s
*兲兴
册
.
共25兲
We also need to note that matrices Bi
ⳮ1共i⳱1,N
兲do not depend on
shots; therefore, any speedup toward resolving systems that involve
these matrices with multiple sources should be considered 共Marfurt,
1984;Jo et al., 1996;Stekl and Pratt, 1998兲.
By comparing the expressions of the gradient in equations 9and
24, we can conclude that one element of the sensitivity matrix is giv-
en by
Jk共s,r兲,ᐉ⳱us
t
冋
Bt
mᐉ
册
Bⳮ1
␦
r,共26兲
where k共s,r兲denotes a source-receiver couple of the acquisition ge-
ometry, with sand rdenoting a shot and a receiver position, respec-
tively. An impulse source
␦
ris located at receiver position r.Ifthe
sensitivity matrix must be built, one forward problem for the inci-
dent wavefield and one forward problem per receiver position must
be computed. Therefore, the number of simulations to build the sen-
sitivity matrix can be higher than that required by gradient estima-
tion if the number of nonredundant receiver positions significantly
exceeds the number of nonredundant shots, or vice versa. Comput-
ing each term of the sensitivity matrix is also required to compute the
diagonal terms of the approximate Hessian Ha共Shin et al., 2001b兲.
To mitigate the resulting computational burden for coarse OBS sur-
veys, Operto et al. 共2006兲suggest computing the diagonal terms of
Hafor a decimated shot acquisition. Alternatively, Shin et al.
共2001a兲propose using an approximation of the diagonal Hessian,
which can be computed at the same cost as the gradient.
Although the matrix-free adjoint approach is widely used in ex-
ploration seismology, the earthquake-seismology community tends
to favor the scattering-integral method, which is based on the explic-
it building of the sensitivity matrix 共Chen et al., 2007兲. The linear
system relating the model perturbation to the data perturbation is
formed and solved with a conjugate-gradient algorithm such as
LSQR 共Paige and Saunders, 1982a兲. A comparative complexity
analysis of the adjoint approach and the scattering-integral approach
is presented in Chen et al. 共2007兲, who conclude that the scattering-
integral approach outperforms the adjoint approach for a regional to-
mographic problem. Indeed, the superiority of one approach over the
other is highly dependent on the acquisition geometry 共the relative
number of sources and receivers兲and the number of model parame-
ters.
The formalism in equation 25 has been kept as general as possible
and can relate to the acoustic or the elastic wave equation. In the
acoustic case, the wavefield is the pressure scalar wavefield; in the
elastic case, the wavefield ideally is formed by the components of the
particle velocity and the pressure if the sensors have four compo-
nents. Equation 25 can be translated in the time domain using Parse-
val’s relation. The expression of the gradient in equation 25 can be
developed equivalently using a functional analysis 共Tarantola,
1984兲. The partial derivatives of the wavefield with respect to the
model parameters are provided by the kernel of the Born integral that
relates the model perturbations to the wavefield perturbations. Mul-
tiplying the transpose of the resulting operator by the conjugate of
the data residuals provides the expression of the gradient. The two
formalisms 共matrix and functional兲give the same expression, pro-
vided the discretization of the partial differential operators are per-
formed consistently in the two approaches. The derivation in the fre-
quency domain of the gradient of the misfit function using the two
formalisms is explicitly illustrated by Gelis et al. 共2007兲.
Source estimation
Source excitation is generally unknown and must be considered
as an unknown of the problem 共Pratt, 1999兲. The source wavelet can
be estimated by solving a linear inverse problem because the rela-
tionship between the seismic wavefield and the source is linear
共equation 2兲. The solution for the source is given by the expression
s⳱具gcal共m0兲兩dobs典
具gcal共m0兲兩gcal共m0兲典,共27兲
where gcal共m0兲denotes the Green’s functions computed in the start-
ing model m0. The source function can be estimated directly in the
FWI algorithm once the incident wavefields have been modeled.
The source and the medium are updated alternatively over iterations
of the FWI. Note that it is possible to take advantage of source esti-
mation to design alternative misfit functions based on the differential
semblance optimization 共Pratt and Symes, 2002兲or to define more
heuristic criteria to stop the iteration of the inversion 共Jaiswal et al.,
2009兲.
Alternatively, new misfit functions have been designed so the in-
version becomes independent of the source function 共Lee and Kim,
2003;Zhou and Greenhalgh, 2003兲. The governing idea of the meth-
od is to normalize each seismogram of a shot gather by the sum of all
of the seismograms. This removes the dependency of the normalized
data with respect to the source and modifies the misfit function. The
drawback is that this approach requires an explicit estimate of the
sensitivity matrix; the normalized residuals cannot be back propa-
gated because they do not satisfy the wave equation.
SOME KEY FEATURES OF FWI
Resolution power of FWI and relationship to the
experimental setup
The interpretation of the partial-derivative wavefield as the wave-
field scattered by the missing heterogeneities provides some connec-
tions between FWI and generalized diffraction tomography 共Dev-
aney and Zhang, 1991;Gelius et al., 1991兲. Diffraction tomography
Full-waveform inversion WCC135
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recasts the imaging as an inverse Fourier transform 共Devaney, 1982;
Wu and Toksoz, 1987;Sirgue and Pratt, 2004;Lecomte et al., 2005兲.
Let us consider a homogeneous background model of velocity c0,an
incident monochromatic plane wave propagating in the direction s
ˆ
and a scattered monochromatic plane wave in the far-field approxi-
mation propagating in the direction r
ˆ共Figure 3兲. If amplitude effects
are not taken into account, the incident and scattered Green’s func-
tions can be written compactly as
G0共x,s兲⳱exp共ik0s
ˆ.x兲,
G0共x,r兲⳱exp共ik0r
ˆ.x兲,共28兲
with the relation k0⳱
/c0. Inserting the expression of the incident
and scattered plane waves into the gradient of the misfit function of
equation 24 gives the expression 共Sirgue and Pratt, 2004兲
ⵜC共m兲⳱ⳮ
2兺
兺
s兺
r
R兵exp共ⳮik0共s
ˆⳭr
ˆ兲.x兲⌬d其.
共29兲
Equation 29 has the form of a truncated Fourier series where the
integration variable is the scattering wavenumber vector given by k
⳱k0共s
ˆⳭr
ˆ兲. The coefficients of the series are the data residuals. The
summation is performed over sources, receivers, and frequencies
that control the truncation and sampling of the Fourier series.
We can express the scattering wavenumber vector k0共s
ˆⳭr
ˆ兲as a
function of frequency, diffraction angle, or aperture to highlight the
relationship between the experimental setup and the spatial resolu-
tion of the reconstruction 共Figure 4兲:
k⳱2f
c0
cos
冉
2
冊
n,共30兲
where nis a unit vector in the direction of the slowness vector 共s
ˆ
Ⳮr
ˆ兲. Equation 30 was also derived in the framework of the ray ⫹
Born migration/inversion, recast as the inverse of a generalized Ra-
don transform 共Miller et al., 1987兲or as a least-squares inverse prob-
lem 共Lambaré et al., 2003兲.
Several key conclusions can be derived from equation 30. First,
one frequency and one aperture in the data space map one wavenum-
ber in the model space. Therefore, frequency and aperture have re-
dundant control of the wavenumber coverage. This redundancy in-
creases with aperture bandwidth. Pratt and Worthington 共1990兲,Sir-
gue and Pratt 共2004兲,and Brenders and Pratt 共2007a兲propose deci-
mating this wavenumber-coverage redundancy in frequency-do-
main FWI by limiting the inversion to a few discrete frequencies.
This data reduction leads to computationally efficient frequency-do-
main FWI and allows managing a compact volume of data, two clear
advantages with respect to time-domain FWI. The guideline for se-
lecting the frequencies to be used in the FWI is that the maximum
wavenumber imaged by a frequency matches the minimum vertical
wavenumber imaged by the next frequency 共Sirgue and Pratt, 2004,
their Figure 3兲. According to this guideline, the frequency interval
increases with the frequency.
Second, the low frequencies of the data and the wide apertures
help resolve the intermediate and large wavelengths of the medium.
At the other end of the spectrum, the maximum wavenumber, con-
strained by
⫽0 and the highest frequency, leads to a maximum res-
olution of half a wavelength if normal-incidence reflections are re-
corded. Third, for surface acquisitions, long offsets are helpful for
sampling the small horizontal wavenumbers of dipping structures
such as flanks of salt domes.
A frequency-domain sensitivity kernel for point sources, referred
to as the wavepath by Woodward 共1992兲, is shown in Figure 5. The
interference picture shows zones of equiphase over which the resid-
uals are back projected during FWI. The central zone of elliptical
shape is the first Fresnel zone of width 冑
osr, where osr is the source-
receiver offset. Residuals that match the first arrival with an error
lower than half a period will be back projected constructively over
the first Fresnel zone, updating the large wavelengths of the struc-
ture. The outer fringes are isochrones on which residuals associated
with later-arriving reflection phases will be back projected, provid-
ing an update of the shorter wavelengths of the medium, just like
PSDM 共Lecomte, 2008兲. The width of the isochrones, which gives
some insight into the vertical resolution in Figure 4, is given by the
modulus of the wavenumber of equation 30.
To illustrate the relationship between FWI resolution and the ex-
perimental setup, we show the FWI reconstruction of an inclusion in
a homogeneous background for three acquisition geometries 共Figure
6兲. In the crosshole experiment 共Figure 6a兲, FWI has reconstructed a
low-pass-filtered 共smoothed兲version of the vertical section of the in-
clusion and a band-pass-filtered version of the horizontal section of
the inclusion. This anisotropy of the imaging results from the trans-
mission-like reconstruction of the vertical wavenumbers and the re-
flection-like reconstruction of the horizontal wavenumbers of the in-
clusion. In the case of the double crosshole experiment 共Figure 6b兲,
the vertical and horizontal wavenumber spectra of the inclusion have
4
0
1
2
3
01234
Distance (km)
Dept
h
(
k
m)
0
1
2
3
01234
Distance (km)
0
1
2
3
0123
4
Distance (km)
44
s
^
r
^
k
s
^r
^
a) b) c
)
Figure 3. Resolution analysis of FWI. 共a兲Incident monochromatic
plane wave 共real part兲.共b兲Scattered monochromatic plane wave
共real part兲.共c兲Gradient of FWI describing one wavenumber compo-
nent 共real part兲built from the plane waves shown in 共a兲and 共b兲.
S
R
ps
pr
x
q
θ
k=fq
λ
=c/f
q=ps+p
r
ps=pr=1/c
Figure 4. Illustration of the main parameters in diffraction tomogra-
phy and their relationships. Key:
, wavelength;
, diffraction or ap-
erture angle; c, P-wave velocity; f, frequency; pS,pR,q, slowness
vectors; k, wavenumber vector; x, diffractor point; Sand R, source
and receiver positions.
WCC136 Virieux and Operto
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been partly filled because of the combined use of transmission and
reflection wavepaths. Of note, the vertical section shows a lack of
low wavenumbers, whereas the horizontal section exhibits a deficit
of low wavenumbers because the maximum horizontal source-re-
ceiver offset is two times higher than the vertical one 共Figure 6b兲.
Therefore, the aperture illuminations of the horizontal and vertical
wavenumbers differ. For the surface acquisition 共Figure 6c兲, the ver-
tical section exhibits a strong deficit of low wavenumbers because of
the lack of large-aperture illumination. Of note, the pick-to-pick am-
plitude of the perturbation is fully recovered in Figure 6c. The hori-
zontal section of the inclusion is poorly recovered because of the
poor illumination of the horizontal wavenumbers from the surface.
The ability of the wide apertures to resolve the large wavelengths
of the medium has prompted some studies to consider long-offset ac-
quisitions as a promising approach to design well-posed FWI prob-
lems 共Pratt et al., 1996;Ravaut et al., 2004兲. For example, equation
30 can suggest that the long wavelengths of the medium can be re-
solved whatever the source bandwidth, provided that wide-aperture
data are recorded by the acquisition geometry. However, all of the
conclusions derived so far rely on the Born approximation. The Born
approximation requires that the starting model allows matching the
observed traveltimes with an error less than half the period 共Bey-
doun and Tarantola, 1988兲. If not, the so-called cycle-skipping arti-
facts will lead to convergence toward a local minimum 共Figure 7兲.
Pratt et al. 共2008兲translates this condition in terms of relative time
error ⌬t/TLas a function of the number of propagated wavelengths
N
, expressed as
⌬t
TL
⬍1
N
,共31兲
where TLdenotes the duration of the simulation. Condition 31 shows
that the traveltime error must be less than 1% for an offset involving
50 propagated wavelengths, a condition unlikely to be satisfied if
FWI is applied without data preconditioning. Therefore, some stud-
ies consider that recording low frequencies 共⬍1Hz兲is the best strat-
egy to design well-posed FWI 共Sirgue, 2006兲. Unfortunately, such
low frequencies cannot be recorded during controlled-source exper-
iments. As an alternative to low frequencies, multiscale layer-strip-
ping approaches where longer offsets, shorter apertures, and longer
recording times are progressively introduced in the inversion, have
been designed to mitigate the nonlinearity of the inversion.
Multiscale FWI: Time domain versus frequency domain
FWI can be implemented in the time domain or in the frequency
domain. FWI was originally developed in the time domain 共Taran-
X
X
X
θ
010 20 30 40 50 60 70 80 90 100
0
5
10
15
20
Distance (
k
m)
010 20 30 40 50 60 70 80 90 100
0
5
10
15
20
Depth (km)Depth (km)
sr
a)
b)
Figure 5. Wavepath. 共a兲Monochromatic Green’s function for a point
source. 共b兲Wavepath for a receiver located at a horizontal distance
of 70 km from the source. The frequency is 5 Hz and the velocity in
the homogeneous background is 6 km/s. The dashed red lines delin-
eate the first Fresnel zone and an isochrone surface. The yellow line
is a vertical section across the wavepath. The blue lines represent dif-
fraction paths within the first Fresnel zone and from the isochrone.
Depth (km)
Distance (km)
012345678
0
1
2
3
4
V(km/s)
P
4.1
4.0
Distance (km)
012345678
0
1
2
3
4
4.2
4.1
4.0
VP(km/s)
Distance (km)
012345678
0
1
2
3
4
Depth (km)
4.0
3.9
VP(km/s)
Distance (km)
012345678
0
1
2
3
4
Depth (km)
a)
b)
c)
Figure 6. Imaging an inclusion by FWI. 共a兲Crosshole experiment.
Source and receiver lines are in red and blue, respectively. The con-
tour of the inclusion with a diameter of 400 m is delineated by the
blue circle. The true velocity in the inclusion is 4.2 km/s, whereas the
velocity in the background is 4 km/s. Six frequencies 共4, 7, 9, 11, 12,
and 15 Hz兲were inverted successively, and 20 iterations per frequen-
cy were computed. The black and gray curves along the right and
bottom sides of the model are velocity profiles across the center of
the inclusion extracted from the exact model and the reconstructed
model, respectively. 共b兲Same as 共a兲for a vertical and horizontal
crosshole experiment 共the shots along the red dashed line are record-
ed by only the receivers along the vertical blue dashed line兲.共c兲
Same as 共a兲for a surface experiment.
Full-waveform inversion WCC137
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tola, 1984;Gauthier et al., 1986;Mora, 1987;Crase et al., 1990兲
whereas the frequency-domain approach was proposed mainly in
the 1990s by Pratt and collaborators 共Pratt, 1990;Pratt and Wor-
thington, 1990;Pratt and Goulty, 1991兲, first with application to
crosshole data and later with application to wide-aperture surface
seismic data 共Pratt et al., 1996兲.
The nonlinearity of FWI has prompted many studies to develop
some hierarchical multiscale strategies to mitigate this nonlinearity.
Apart from computational efficiency, the flexibility offered by the
time domain or the frequency domain to implement efficient multi-
scale strategies is one of the main criteria that favors one domain
rather than the other. The multiscale strategy successively processes
data subsets of increasing resolution power to incorporate smaller
wavenumbers in the tomographic models. In the time domain,
Bunks et al. 共1995兲propose successive inversion of subdata sets of
increasing high-frequency content because low frequencies are less
sensitive to cycle-skipping artifacts. The frequency domain provides
a more natural framework for this multiscale approach by perform-
ing successive inversions of increasing frequencies. In the frequency
domain, single or multiple frequencies 共i.e., frequency group兲can be
inverted at a time.
Although a few discrete frequencies theoretically are sufficient to
fill the wavenumber spectrum for wide-aperture acquisitions, simul-
taneous inversion of multiple frequencies improves the signal-to-
noise ratio and the robustness of FWI when complex wave phenom-
ena are observed 共i.e., guide waves, surface waves, dispersive
waves兲. Therefore, a trade-off between computational efficiency and
quality of imaging must be found. When simultaneous multifre-
quency inversion is performed, the bandwidth of the frequency
group ideally must be as large as possible to mitigate the nonlinearity
of FWI in terms of the nonunicity of the solution, whereas the maxi-
mum frequency of the group must be sufficiently low to prevent cy-
cle-skipping artifacts. An illustration of this tuning of FWI is given
in Brossier et al. 共2009a兲in the framework of elastic seismic imaging
of complex onshore models from the joint inversion of surface
waves and body waves.
The regularization effects introduced by hierarchical inversion of
increasing frequencies might not be sufficient to provide reliable
FWI results for realistic frequencies and realistic starting models in
the case of complex structures. This has prompted some studies to
design additional regularization levels in FWI. One of these is to se-
lect a subset of arrivals as a function of time. An aim of this time win-
dowing is to remove arrivals that are not predicted by the physics of
the wave equation implemented in FWI 共for example, PS-converted
waves in the frame of acoustic FWI兲. A second aim is to perform a
heuristic selection of aperture angles in the data. Considering a nar-
row time window centered on the first arrival leads to so-called ear-
ly-arrival waveform tomography 共Sheng et al., 2006兲. Time win-
dowing the data around the first arrivals is equivalent to selecting the
large-aperture components of the data to image the large and inter-
mediate wavelengths of the medium. Alternatively, time windowing
can be applied to isolate later-arriving reflections or PS-converted
phases to focus on imaging a specific reflector or a specific parame-
ter class, such as the S-wave velocity 共Shipp and Singh, 2002;Sears
et al., 2008;Brossier et al., 2009a兲.
The frequency domain is the most appropriate to select one or a
few frequencies for FWI, but the time domain is the most appropriate
to select one type of arrival for FWI. Indeed, time windowing cannot
be applied in frequency-domain modeling, in which only one or few
frequencies are modeled at a time. Alast resort is the use of complex-
valued frequencies, which is equivalent to the exponential damping
of a signal p共t兲in time from an arbitrary traveltime t0共Sirgue, 2003;
Brenders and Pratt, 2007b兲:
P
冉
Ⳮi
冊
et0/
⳱
冕
ⳮ⬁
Ⳮ⬁
p共t兲eⳮ共tⳮt0兲/
ei
tdt,共32兲
where P共
兲denotes the Fourier transform of p共t兲and
is the damp-
ing factor.
A last regularization level can be implemented by layer stripping,
in which the imaging proceeds hierarchically from the shallow part
to the deep part. Layer stripping in FWI can be applied by combined
offset and temporal windowing 共Shipp and Singh, 2002;Wang and
Rao, 2009兲.
These three levels of regularization — frequency dependent, time
dependent, and offset dependent — can be combined in one integrat-
ed multiloop FWI workflow.An example is provided in Shin and Ha
共2009兲and Brossier et al. 共2009a兲, in which the frequency- and time-
dependent regularizations are implemented into two nested loops
over frequency groups and time-damping factors. In this approach,
the frequencies increase in the outer loop and the damping factors
decrease in the inner loop. In Figure 8, the VPand VSmodels of the
overthrust model are inferred from the successive inversion of two
groups of five frequencies 共Brossier et al., 2009a兲. The frequencies
of the first group range from 1.7 to 3.5 Hz, whereas those of the sec-
ond group range from 3.5 to 7.2 Hz. Five damping factors of
be-
tween 0.67 and 30.0 s were applied hierarchically for data precondi-
tioning during the inversion of each frequency group. Without these
two regularization levels associated with frequency and aperture se-
lections, FWI fails to converge toward acceptable models.
In summary, the implementation of FWI in the frequency domain
allows the easy implementation of multiscale FWI based on the hier-
archical inversion of groups of frequencies of arbitrary bandwidth
and sampling intervals. Time-domain modeling provides the most
flexible framework to apply time windowing of arbitrary geometry.
n n +1
n–1
n
Time
(
s
)
n+1
nn –1
T/2 T/2
n+1
n–1
Figure 7. Schematic of cycle-skipping artifacts in FWI. The solid
black line represents a monochromatic seismogram of period Tas a
function of time. The upper dashed line represents the modeled
monochromatic seismograms with a time delay greater than T/2. In
this case, FWI will update the model such that the nⳭ1th cycle of
the modeled seismograms will match the nth cycle of the observed
seismogram, leading to an erroneous model. In the bottom example,
FWI will update the model such that the modeled and recorded nth
cycle are in-phase because the time delay is less than T/2.
WCC138 Virieux and Operto
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This makes frequency-domain FWI based on time-domain model-
ing an attractive strategy to design robust FWI algorithms. This is es-
pecially true for 3D problems, for which time-domain modeling has
several advantages with respect to frequency-domain modeling 共Sir-
gue et al., 2008兲.
On the parallel implementation of FWI
FWI algorithms must be implemented in parallel to address large-
scale 3D problems. Depending on the numerical technique for solv-
ing the forward problem, different parallel strategies can be consid-
ered for FWI. If the forward problem is based on numerical methods
such as time-domain modeling or iterative solvers, which are not de-
manding in terms of memory, a coarse-grained parallelism that con-
sists of distributing sources over processors is generally used and the
forward problem is performed sequentially on each processor for
each source 共Plessix, 2009兲. If the number of processors significant-
ly exceeds the number of shots, which can be the case if source en-
coding techniques are used 共Krebs et al., 2009兲, a second level of
parallelism can be viewed by domain decomposition of the physical
computational domain. A comprehensive review of different algo-
rithms to efficiently compute the forward and the adjoint wavefields
in time-domain FWI is presented by Akcelik 共2002兲.
In contract, if the forward problem is based on a method that em-
beds a memory-expensive preprocessing step, such as LU factoriza-
tion in the frequency-domain direct-solver approach, parallelism
must be based on a domain decomposition of the computational do-
main. Each processor computes a subdomain of the wavefields for
all sources. Examples of such algorithms are described in Ben Hadj
Alietal.共2008a兲,Brossier et al. 共2009a兲, and Sourbier et al. 共2009a,
2009b兲. A contrast source inversion 共CSI兲method is described by
Abubakar et al. 共2009兲, which allows a decrease in the number of LU
factorization in frequency-domain FWI at the expense of the number
of iterations.
Variants of classic least-squares and
Born-approximation FWI
Although the most popular approach of FWI is based on minimiz-
ing the least-squares norm of the data misfit on the one hand and on
the Born approximation for estimating partial-derivative wavefields
on the other, several variants of FWI have been proposed over the
last decade. These variants relate to the definition of the minimiza-
tion criteria, the representation of the data 共amplitude, phase, loga-
rithm of the complex-valued data, envelope兲in the misfit, or the lin-
earization procedure of the inverse problem.
The choice of the minimization criterion
The least-squares norm approach assumes a Gaussian distribution
of the misfit 共Tarantola, 1987兲. Poor results can be obtained when
this assumption is violated, for example, when large-amplitude out-
liers affect the data. Therefore, careful quality control of the data
must be carried out before least-squares inversion. Crase et al.
共1990兲investigate several norms such as the least-squares norm L2,
the least-absolute-values norm L1, the Cauchy criterion norm, and
the hyperbolic secant 共sech兲criterion in FWI 共Figure 9兲. The
L1-norm specifically ignores the amplitude of the residuals during
back propagation of the residuals when gradient building, making
this criterion less sensitive to large errors in the data. The Cauchy and
sechcriteria can be considered a combination of the L1- and the
L2-norms with different transitions between the norms. Crase et al.
共1990兲conclude that the most reliable FWI results have been ob-
tained with the Cauchy and the sechcriteria.
The L2and Cauchy criteria are also compared by Amundsen
共1991兲in the framework of frequency-wavenumber-domain FWI
for stratified media described by velocity, density, and layer thick-
nesses 共Amundsen and Ursin, 1991兲. They consider random noise
and weather noise and conclude in both cases that the Cauchy criteri-
on leads to the more robust results.
The Huber norm also combines the L1- and the L2-norms; it is
combined with quasi-Newton L-BFGS by Guitton and Symes
共2003兲and Bube and Nemeth 共2007兲. The Huber norm is also used in
the framework of frequency-domain FWI by Ha et al. 共2009兲and
shows an overall more robust behavior than the L2-norm.
The choice of the linearization method
The sensitivity matrix is generally computed with the Born ap-
proximation, which assumes a linear tangent relationship between
the model and wavefield perturbations 共Woodward, 1992兲. This lin-
ear relationship between the perturbations can be inferred from the
assumption that the wavefield computed in the updated model is the
wavefield computed in the starting model plus the perturbation
wavefield.
VP(km/s)
VS(km/s
)
04812
16 20
0
1
2
3
4
04812
16 20
0
1
2
3
4
048
12 16
0
2
4
6
8
10
12
14
Depth (km)
048
12 16
0
2
4
6
8
10
12
14
Depth (km)
048
12 16
0
2
4
6
8
10
12
14
Depth (km)
048
12 16
0
2
4
6
8
10
12
14
Depth (km)
Depth
(
km
)
Depth (km)
0
1
2
3
4
Depth (km)
0
1
2
3
4
Depth (km)
346
23
Offset (km) Offset (km)
Offset (km) Offset (km)
VS(km/s)
VP(km/s)
x= 7.5 km
x= 7.5 km
Distance (km)
a
)
b) c)
Figure 8. Multiscale strategy for elastic FWI with application to the
overthrust model. 共a兲FWI velocity models VP共top兲and VS共bottom兲.
共b兲Comparison between the logs from the true model 共black兲, the
starting model 共dashed gray兲, and the final FWI model 共solid gray兲.
共c兲Synthetic seismograms computed in the final FWI models for the
horizontal 共left兲and vertical 共right兲components of particle velocity.
The bottom panels are the final residuals between seismograms
computed in the true and in the final FWI models 共image courtesy R.
Brossier兲.
Full-waveform inversion WCC139
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The Rytov approach considers the generalized phase as the wave-
field 共Woodward, 1992兲.The Rytov approximation provides a linear
relationship between the complex-phase perturbation and the model
perturbation by assuming that the wavefield computed in the updat-
ed model is related to the wavefield computed in the starting model
through u共x,
兲⳱u0共x,
兲exp共⌬
共x,
兲兲, where ∆
共x,
兲denotes
the complex-phase perturbation. The sensitivity of the Rytov kernel
is zero on the Fermat raypath because the traveltime is stationary
along this path. A linear relationship between the model perturba-
tions and the logarithm of the amplitude ratio Ln关A共
兲/A0共
兲兴 is
also provided by the Rytov approximation by taking the real part of
the sensitivity kernel of the Rytov integral instead of the imaginary
part that provides the phase perturbation.
The Born approximation is valid in the case of weak and small
perturbations, but the Rytov approximation is supposed to be valid
for large-aperture angles and a small amount of scattering per wave-
length, i.e., smooth perturbations or smooth variation in the phase-
perturbation gradient 共Beydoun and Tarantola, 1988兲.Although sev-
eral analyses of the Rytov approximation have been carried out, it re-
mains unclear to what extent its domain of validity differs signifi-
cantly from that of the Born approximation. Acomparison between
the Born approximation and the Rytov approximation in the frame-
work of elastic frequency-domain FWI is presented in Gelis et al.
共2007兲. The main advantage of the Rytov approximation might be to
provide a natural separation between phase and amplitude 共e.g.,
Woodward, 1992兲. This separation allows the implementation of
phase and amplitude inversions 共Bednar et al., 2007;Pyun et al.,
2007兲from a frequency-domain FWI code using a logarithmic norm
共Shin and Min, 2006;Shin et al., 2007兲, the use of which leads to the
Rytov approximation in the framework of local optimization
problems.
Another approach in the time-frequency domain is developed by
Fichtner et al. 共2008兲for continental- and global-scale FWI, in
which the misfit of the phase and the misfit of the envelopes are min-
imized in a least-squares sense. The expected benefit from this ap-
proach is to mitigate the nonlinearity of FWI by separating the phase
and amplitude in the inversion and by inverting the envelope instead
of the amplitudes, the former being more linearly related to the data.
Building starting models for FWI
The ultimate goal in seismic imaging is to be able to apply FWI re-
liably from scratch, i.e., without the need for sophisticated a priori
information. Unfortunately, because multidimensional FWI at
present can only be attacked through local optimization approaches,
building an accurate starting model for FWI remains one of the most
topical issues because very low frequencies 共⬍1Hz兲still cannot be
recorded in the framework of controlled-source experiments.
A starting model for FWI can be built by reflection tomography
and migration-based velocity analysis such as those used in oil and
gas exploration. A review of the tomographic workflow is given in
Woodward et al. 共2008兲. Other possible approaches for building ac-
curate starting models, which should tend toward a more automatic
procedure and might be more closely related to FWI, are first-arrival
traveltime tomography 共FATT兲, stereotomography, and Laplace-do-
main inversion.
FATT performs nonlinear inversions of first-arrival traveltimes to
produce smooth models of the subsurface 共e.g., Nolet, 1987;Hole,
1992;Zelt and Barton, 1998兲. Traveltime residuals are back project-
ed along the raypaths to compute the sensitivity matrix. The tomog-
raphic system, augmented with smoothing regularization, generally
is solved with a conjugate-gradient algorithm such as LSQR 共Paige
and Saunders, 1982b兲. Alternatively, the adjoint-state method can be
applied to FATT, which avoids the explicit building of the sensitivity
matrix, just as in FWI 共Taillandier et al., 2009兲. The spatial resolu-
tion of FATT is estimated to be the width of the first Fresnel zone
共Williamson, 1991; Figure 5兲.
Examples of applications of FWI to real data using a starting mod-
el built by FATT are shown, for example, in Ravaut et al. 共2004兲,Op-
erto et al. 共2006兲,Jaiswal et al. 共2008,2009兲, and Malinowsky and
Operto 共2008兲for surface acquisitions; in Dessa and Pascal 共2003兲in
the framework of ultrasonic experimental data; in Pratt and Goulty
共1991兲for crosshole data; and in Gaoetal.共2006b兲for VSP data.
Several blind tests that correspond to surface acquisitions have
been tackled by joint FATT and FWI. Results at the oil-exploration
scale and at the lithospheric scale are presented in Brenders and Pratt
共2007a,2007b,2007c兲and suggest that very low frequencies and
very large offsets are required to obtain reliable FWI results when
the starting model is built by FATT. For example, only the upper part
of the BP benchmark model was imaged successfully by Brenders
and Pratt 共2007c兲using a starting frequency as small as 0.5 Hz and a
maximum offset of 16 km. Another drawback of FATT is that the
method is not suitable when low-velocity zones exist because these
low-velocity zones create shadow zones.
Reliable picking of first-arrival times is also a difficult issue when
low-velocity zones exist. Fitting first-arrival traveltimes does not
guarantee that computed traveltimes of later-arriving phases, such as
0
1
2
3
4
5
6
7
8
–4–3–2–1 0 1 2 3 4
L2functional
L1functional
Huber norm functional
Hybrid norm functional
Cost function
–1
–2
–3
–4
0
1
2
3
4
–4–3–2–1 0 1 2 3
4
Residual source amplitude
L2residual source
L1residual source
Huber norm functional
Hybrid norm functional
D
ata
r
es
i
dua
l
a
)
b)
Figure 9. Different functionals for FWI. 共a兲Value of four functionals
as a function of real-valued data residual. The L1,L2, Huber, and
mixed L1–L2functionals are plotted as indicated. 共b兲Amplitude of
the residual source used to compute the back-propagated adjoint
wavefield for the different functionals shown in 共a兲. Note that the
back-propagated source in the L1-norm is not sensitive to the residu-
al amplitude and therefore is less sensitive to large-amplitude residu-
als than the L2-norm 共image courtesy R. Brossier兲.
WCC140 Virieux and Operto
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reflections, will match the true reflection traveltimes with an error
that does not exceed half a period, especially if anisotropy affects the
wavefield. We should stress that FATT can be recast as a phase inver-
sion of the first arrival using a frequency-domain waveform-inver-
sion algorithm within which complex-valued frequencies are imple-
mented 共Min and Shin, 2006;Ellefsen, 2009兲. Compared to FATT
based on the high-frequency approximation, this approach helps ac-
count for the finite-frequency effect of the data in the sensitivity ker-
nel of the tomography. Judicious selection of the real and imaginary
parts of the frequency allows extraction of the phase of the first arriv-
al. The principles and some applications of the method are presented
in Min and Shin 共2006兲and Ellefsen 共2009兲for near-surface applica-
tions. This is strongly related to finite-frequency tomography 共Mon-
telli et al., 2004兲.
Traveltime tomography methods that can manage refraction and
reflection traveltimes should provide more consistent starting mod-
els for FWI. Among these methods, stereotomography is probably
one of the most promising approaches because it exploits the slope
of locally coherent events and a reliable semiautomatic picking pro-
cedure has been developed 共Lambaré, 2008兲. Applications of ste-
reotomography to synthetic and real data sets are presented in Bil-
lette and Lambaré 共1998兲,Alerini et al. 共2002兲,Billette et al. 共2003兲,
Lambaré and Alérini 共2005兲, and Dummong et al. 共2008兲.
To illustrate the sensitivity of FWI to the accuracy of different
starting models, Figure 10 shows FWI reconstructions of the syn-
thetic Valhall model when the starting model is built by FATTand re-
flection stereotomography 共Prieux et al., 2009兲. In the case of ste-
reotomography, the maximum offset is 9 km and only the reflection
traveltimes are used 共Lambaré and Alérini, 2005兲, whereas the maxi-
mum offset is 32 km for FATT 共Prieux et al., 2009兲. Stereotomogra-
phy successfully reconstructs the large wavelength within the gas
cloud down to a maximum depth of 2.5 km; FATT fails to reconstruct
the large wavelengths of the low-velocity zone associated with the
gas cloud. However, the FWI model inferred from the FATT starting
model shows an accurate reconstruction of the shallow part of the
model. These results suggest that joint inversion of refraction and re-
flection traveltimes by stereotomography can provide a promising
framework to build starting models for FWI.
A third approach to building a starting model for FWI can be pro-
vided by Laplace-domain and Laplace-Fourier-domain inversions
共Shin and Cha, 2008,2009;Shin and Ha, 2008兲. The Laplace-do-
main inversion can be viewed as a frequency-domain waveform in-
version using complex-valued frequencies 共see equation 32兲, the
real part of which is zero and the imaginary part of which controls the
time damping of the seismic wavefield. In other words, the principle
is the inversion of the DC component of damped seismograms where
the Laplace variable scorresponds to 1 /
in equation 32. The DC of
the undamped data is zero, but the DC of the damped data is not and
is exploited in Laplace-domain waveform inversion. The informa-
tion contained in the data can be similar to the amplitude of the wave-
field 共Shin and Cha, 2009兲. Laplace-domain waveform inversion
provides a smooth reconstruction of the velocity model, which can
be used as a starting model for Laplace-Fourier and classical fre-
quency-domain waveform inversions.
The Laplace-Fourier domain is equivalent to performing an inver-
sion of seismograms damped in time. The results shown in Shin and
Cha 共2009兲suggest that this method can be applied to frequencies
smaller than the minimum frequency of the source bandwidth. The
ability of the method to use frequencies smaller than the frequencies
effectively propagated by the seismic source is called a mirage resur-
rection of the low frequencies by Shin and Cha 共2009兲. An applica-
tion to real data from the Gulf of Mexico is shown in Shin and Cha
共2009兲. For the real-data application, frequencies between 0 and 2
Hz in combination with nine Laplace damping constants are used for
the Laplace-Fourier-domain inversion, the final model of which is
used as the starting model for standard frequency-domain FWI.
Joint application of Laplace-domain, Laplace-Fourier-domain
and Fourier-domain FWI to the BP benchmark model is illustrated in
Figure 11 共Shin and Cha, 2009兲. The starting model is a simple ve-
locity-gradient model 共Figure 11b兲. A first velocity model of the
large wavelengths is obtained by Laplace-domain inversion 共Figure
11c兲, which is subsequently used as a starting model for inversion in
the Laplace-Fourier-domain inversion, the final model of which is
shown in Figure 11d. During this stage, the starting frequency used
in the inversion of the damped data is as low as 0.01 Hz. The final
model obtained after frequency-domain FWI is shown in Figure 11e.
All of the structures were successfully imaged, beginning with a
very crude starting model.
4
3
2
1
0
4567891011
4
3
2
1
0
456789101
1
Depth (km)
Distance (km) Distance (km)
4
3
2
1
0
4567891011
4
3
2
1
0
4567891011
Depth (km)
1.6 2.0 2.4 2.8 3.2
km/s
1.2 2.2 3.2
0
1
2
3
4
Depth (km)
VP(km/s)
1.2 2.2 3.2
0
1
2
3
4
Depth (km)
VP(km/s)
1.2 2.2 3.2
0
1
2
3
4
Depth (km)
VP(km/s)
a)
b)
c)
d)
e) f) g)
Figure 10. 共a兲Close-up of the synthetic Valhall velocity model cen-
tered on the gas layer. 共b兲FWI model built from a starting model ob-
tained by smoothing the true model with a Gaussian filter with hori-
zontal and vertical correlation lengths of 500 m. 共c兲FWI model from
a starting model built by FATT 共Prieux et al., 2009兲.共d兲FWI model
from a starting model built by stereotomography 共Lambaré and
Alérini, 2005兲.共e兲Velocity profiles at a distance of 7.5 km extracted
from the true model 共black line兲, from the starting model built by
smoothing the true model 共blue line兲, and from the FWI model of 共b兲
共red line兲.共f兲Same as 共e兲for the starting model built by FATT and 共c兲
the corresponding FWI model. 共g兲Same as 共e兲for the starting model
built by stereotomography and 共d兲the corresponding FWI model.
The frequencies used in the inversion are between 4 and 15 Hz.
Full-waveform inversion WCC141
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CASE STUDIES
Applications of FWI have been applied essentially to synthetic
examples for which high-resolution images have been constructed.
Because FWI is an attractive approach, the number of real-data ap-
plications has increased quite rapidly, from monoparameter recon-
structions of the VPparameter to multiparameter ones.
Monoparameter acoustic FWI
Most of the recent real-data case studies of FWI at different scales
and for different acquisition geometries have been performed in the
acoustic isotropic approximation, considering only VPas the model
parameter 共Dessa and Pascal, 2003;Ravaut et al., 2004;Chironi et
al., 2006;Gao et al., 2006a,2006b;Operto et al., 2006;Bleibinhaus
et al., 2007;Ernst et al., 2007;Jaiswal et al., 2008,2009;Shin and
Cha, 2009兲. Although the acoustic approximation can be questioned
in the framework of FWI because of unreliable amplitudes, one ad-
vantage of acoustic FWI is dealing with less computationally expen-
sive forward modeling than in the elastic case. Moreover, acoustic
FWI is better posed than elastic FWI because only the dominant pa-
rameter VPcan be involved in the inversion.
Specific waveform-inversion data processing generally is de-
signed to account for the amplitude errors introduced by acoustic
modeling 共Pratt, 1999;Ravaut et al., 2004;Brenders and Pratt,
2007b兲. The amplitude discrepancies in the P-wavefield result from
incorrectly modeling the amplitude-variation-with-offset 共AVO兲ef-
fects and incorrectly modeling the directivity of the source and re-
ceiver 共Brenders and Pratt, 2007b兲. Acoustic-wave modeling gener-
ally is based on resolving the acoustic-wave equation in pressure;
therefore, the particle-velocity synthetic wavefields might not be
computed 共Hustedt et al., 2004兲. If the receivers are geophones, the
physical measurements collected in the field 共particle velocities兲are
not the same as those computed by the seismic modeling engine
共pressure兲. Amatch between the vertical geophone data and the pres-
sure synthetics can, however, be performed by exploiting the reci-
procity of the Green’s functions if the sources are explosions 共Operto
et al., 2006兲.
In contrast, if the sources and receivers are directional, the pres-
sure wavefield cannot account for the directivity of the sources and
receivers, and heuristic amplitude corrections must be applied be-
fore inversion. Brenders and Pratt 共2007b兲propose optimizing an
empirical correction law for the decay of the rms amplitudes with
offset. Applying this correction law to the modeled data matches the
main AVO trend of the observed data before FWI. Using this data
preprocessing, Brenders and Pratt 共2007b兲successfully image the
onshore lithospheric model of the CCSS blind test 共Zelt et al., 2005兲
by acoustic FWI of synthetic elastic data. This strategy is also used
successfully by Jaiswal et al. 共2008,2009兲in the framework of
acoustic FWI of real onshore data in the Naga thrust and fold belt in
India.
Successful application of acoustic FWI to synthetic elastic data
computed in the marine Valhall model from an OBC acquisition is
presented by Brossier et al. 共2009b兲. An application of acoustic FWI
to real onshore long-offset data recorded in the southern Apennines
thrust belt is illustrated in Figure 12 共Ravaut et al., 2004兲. The veloc-
ity model is validated locally by comparison with a VSP log.Appli-
cation of PSDM using the final FWI model as a starting model con-
tributes to the validation of the relevance of the velocity structure re-
constructed by FWI 共Operto et al., 2004兲. Some guidelines based on
numerical examples of the domain of validity of acoustic FWI ap-
plied to elastic data are also provided in Barnes and Charara 共2008兲.
Multiparameter FWI
Because FWI accounts for the full wavefield, the seismic model-
ing embedded in the FWI algorithm theoretically should honor as far
as possible all of the physics of wave propagation. This is especially
required by FWI of wide-aperture data, in which significant AVO
and azimuthal anisotropic effects should be observed in the data. The
requirement of realistic seismic modeling has prompted some stud-
ies to extend monoparameter acoustic FWI to account for parameter
classes other than the P-wave velocity, such as density, attenuation,
Distance
(
km
)
D
ept
h(k
m
)D
ept
h(k
m
)D
ept
h(k
m
)D
ept
h(k
m
)D
ept
h(k
m
)
Velocit
y(
km/s
)
Velocity (km/s)
Velocit
y(
km/s
)
Velocit
y(
km/s
)
Velocit
y(
km/s
)
0 102030405060
0
4.5
24.0
43.5
63.0
2.5
8
2.0
10
1.5
0 102030405060
0
4.5
24.0
43.5
63.0
2.5
8
2.0
10
1.5
0 102030405060
0
4.5
24.0
43.5
63.0
2.5
8
2.0
10
1.5
0 102030405060
0
4.5
24.0
43.5
63.0
2.5
8
2.0
10
1.5
0 102030405060
0
4.5
24.0
43.5
63.0
2.5
8
2.0
10
1.5
12
a
)
b
)
c
)
d
)
e
)
Figure 11. Laplace-Fourier-domain waveform inversion. 共a兲BP
benchmark model. 共b兲Starting model. 共c兲Velocity model after
Laplace-domain inversion. 共d兲Velocity model after Laplace-Fouri-
er-domain inversion. 共e兲Velocity model after frequency-domain
FWI 共image courtesy C. Shin and Y. H. Cha兲.
WCC142 Virieux and Operto
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shear-wave velocity or related parameters, and anisotropy. The fact
that additional parameter classes are taken into account in FWI in-
creases in the ill-posedness of the inverse problem because more de-
grees of freedom are considered in the parameterization and because
the sensitivity of the inversion can change significantly from one pa-
rameter class to the next.
Different parameter classes can be more or less coupled as a func-
tion of the aperture angle. This coupling can be assessed by plotting
the radiation pattern of each parameter class using some asymptotic
analyses. Diffraction patterns of the different combination of param-
eters for the acoustic, elastic, and viscoelastic wave equation are
shown in Wu and Aki 共1985兲,Tarantola 共1986兲,Ribodetti and
Virieux 共1996兲, and Forgues and Lambaré 共1997兲.An alternative is
to plot the sensitivity kernel, i.e., that obtained by summing all of the
rows of the sensitivity matrix for the full acquisition and for different
combinations of parameters and to qualitatively assess which com-
bination provides the best image 共Luo et al., 2009兲. Hierarchical
strategies that successively operate on different parameter classes
should be designed to mitigate the ill-posedness of FWI 共Tarantola,
1986;Kamei and Pratt, 2008;Sears et al., 2008;Brossier et al.,
2009b兲.
Density
Density is difficult to reconstruct 共Forgues and Lambaré, 1997兲.
As an illustration, acoustic radiation patterns are shown in Figure 13
for different combinations of parameters 共IP,
兲,共IP,VP兲, and 共VP,
兲,
where IPdenotes P-wave impedance. The radiation pattern of VPis
isotropic because the operator
B/
mlreduces to a scalar for VPand
therefore represents an explosion. On the other hand, the density has
the same radiation pattern as VPat short apertures but does not scatter
energy at wide apertures because the secondary source fᐉcorre-
sponds to a vertical force for the density. Because VPand
have the
same radiation pattern at short apertures, these two parameters are
difficult to reconstruct from short-offset data. For such data, the
P-wave impedance can be considered a reliable parameter for FWI.
If wide-aperture data are available, VPand IPmight provide the most
judicious parameterization because they scatter energy for different
aperture bands 共wide and short apertures, respectively; Figure 13b兲.
A successful reconstruction of the density parameter in the case of
the Marmousi case study is presented by Choi et al. 共2008兲. Howev-
er, the use of an unrealistically low frequency 共0.125 Hz兲brings into
question the practical implication of these results.
Attenuation
The attenuation reconstruction can be implemented in frequency-
domain seismic-wave modeling using complex velocities 共Toksöz
and Johnston, 1981兲. The most commonly applied attenuation/dis-
persion model is referred to as the Kolsky-Futterman model 共Kol-
sky, 1956;Futterman, 1962兲. This model has linear frequency de-
pendence of the attenuation coefficient, whereas the deviation from
24
6
VP(km/s)
1
0
–1
–2
Elevation (km)
1
0
–1
–2
Elevation (km)
1
0
–1
–2
E
l
evation (
k
m)
1
0
–1
–2
0510
15
Elevation (km)
Distance (
k
m)
a)
b)
c)
d)
Figure 12. Two-dimensional seismic imaging of a thrust belt in the
southern Apennines, Italy, from long-offset data by frequency-do-
main FWI. Thirteen frequencies from 6 to 20 Hz were inverted suc-
cessively. 共a兲Starting model for FWI developed by FATT 共Improta
et al., 2002兲.共b–d兲FWI model after inversion of 共b兲the starting
6-Hz, 共c兲10-Hz, and 共d兲20-Hz frequencies 共Ravaut et al., 2004兲.
45°
90°
135°
180°
225°
270°
315°
45°
90°
135°
180°
225°
270°
315°
θ
=0°
45°
90°
135°
180°
225°
270°
315°
ρ
ρ
θ
=0°
θ
=0°
I
P
V
P
I
PV
P
a)
b)
c)
Figure 13. Radiation pattern of different parameter classes in acous-
tic FWI. 共a兲IP-
;共b兲IP-VP;共c兲VP-
.
Full-waveform inversion WCC143
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constant-phase velocity is accounted for through a term that varies as
the logarithm of the frequency. This model is obtained by imposing
causality on the wave pulse and assuming the absorption coefficient
is strictly proportional to the frequency over a restricted range of fre-
quencies.
Using the Kolsky-Futterman model, the complex velocity c
¯
is giv-
en by
c
¯
⳱c
冋
冉
1Ⳮ1
Q兩log共
/
r兲兩
冊
Ⳮisgn共
兲
2Q
册
ⳮ1
,共33兲
where Qis the attenuation factor and
ris a reference frequency.
In frequency-domain FWI, the real and imaginary parts of the ve-
locity can be processed as two independent real-model parameters
without any particular implementation difficulty. However, the re-
construction of attenuation is an ill-posed problem. Ribodetti et al.
共2000兲show that in the framework of the ray ⫹Born inversion,
P-wave velocity and Qare uncoupled 共i.e., the Hessian is nonsingu-
lar兲only if a reflector is illuminated from both sides 共with circular
acquisition as in medical imaging; see Figure 5b of Ribodetti et al.,
2000兲. In contrast, the Hessian becomes singular for a surface acqui-
sition. Mulder and Hak 共2009兲also conclude that VPand Qcannot be
imaged simultaneously from short-aperture data because the Hilbert
transform with respect to depth of the complex-valued perturbation
model for VPand Qproduces the same wavefield perturbation in the
framework of the Born approximation as the original perturbation
model. Despite this theoretical limitation, preconditioning of the
Hessian is investigated by Hak and Mulder 共2008兲to improve the
convergence of the joint inversion for VPand Q.
Assessment and application of viscoacoustic frequency-domain
FWI is presented at various scales by Liao and McMechan 共1995,
1996兲,Song et al. 共1995兲,Hicks and Pratt 共2001兲,Pratt et al. 共2005兲,
Malinowsky et al. 共2007兲, and Smithyman et al. 共2008兲.Kamei and
Pratt 共2008兲recommend inversion for VPonly in a first step and then
joint inversion for VPand Qin a second step because the reliability of
the attenuation reconstruction strongly depends on the accuracy of
the starting VPmodel. Indeed, accurate VPmodels are required be-
fore reconstructing Qso that the inversion can discriminate the in-
trinsic attenuation from the extrinsic attenuation.
Elastic parameters
A limited number of applications of elastic FWI have been pro-
posed. Because VPis the dominant parameter in elastic FWI, Taran-
tola 共1986兲recommends inversion first for VPand IP, second for VS
and IS, and finally for density. This strategy might be suitable if the
footprint of the S-wave velocity structure on the seismic wavefield is
sufficiently small. This hierarchical strategy over parameter classes
is illustrated by Sears et al. 共2008兲, who assess time-domain FWI of
multicomponent OBC data with synthetic examples. They highlight
how the behavior of FWI becomes ill-posed for S-wave velocity re-
construction when the S-wave velocity contrast at the sea bottom is
small. In this case, the S-wave velocity structure has a minor foot-
print on the seismic wavefield because the amount of PS conversion
is small at the sea bottom. In this configuration, they recommend in-
version first for VP, using only the vertical component; second for VP
and VSfrom the vertical component; and finally for VS, using both
components. The aim of the second stage is to reconstruct the inter-
mediate wavelengths of the S-wave velocity structure by exploiting
the AVO behavior of the P-waves.
In contrast, Brossier et al. 共2009a兲conclude that joint inversion
for VPand VSwith judicious hierarchical data preconditioning by
time damping is necessary for inversion of land data involving both
body waves and surface waves. The strong sensitivity of the high-
amplitude surface waves to the near-surface S-wave velocity struc-
ture requires inversion for VSduring the early stages of the inversion.
This makes the elastic inversion of onshore data highly nonlinear
when the surface waves are preserved in the data.
A recent application of elastic FWI to a gas field in China is pre-
sented by Shi et al. 共2007兲. They invert for the Lamé parameters and
unambiguously image Poisson’s ratio anomalies associated with the
presence of gas. They accelerate the convergence of the inversion by
computing an efficient step length using an adaptive controller based
on the theory of model-reference nonlinear control. Several logs
available along the profile confirm the reliability of this gas-layer
reconstruction.
Anisotropy
Reconstruction of anisotropic parameters by FWI is probably one
of the most undeveloped and challenging fields of investigation. Ver-
tically transverse isotropy 共VTI兲or tilted transversely isotropic
共TTI兲media are generally considered a realistic representation of
geologic media in oil and gas exploration, although fractured media
require an orthorhombic description 共Tsvankin, 2001兲. The normal-
moveout 共NMO兲P-wave velocity in VTI media depends on only two
parameters: the NMO velocity for a horizontal reflector VNMO共0兲
⳱VP0冑1Ⳮ2
␦
and the
⳱共ⳮ
␦
兲/共1Ⳮ2
␦
兲parameter 共Alkhali-
fah and Tsvankin, 1995兲, which is a combination of Thomsen’s pa-
rameters and
␦
共Thomsen, 1986兲. The dependency of NMO veloc-
ity in VTI media on a limited subset of anisotropic parameters sug-
gests that defining the parameter classes to be involved in FWI will
be a key task. Another issue will be to assess to what extent FWI can
be performed in the acoustic approximation knowing that acoustic
media are by definition isotropic 共Grechka et al., 2004兲. The kine-
matic and dynamic accuracy of an acoustic TTI wave equation for
FWI is discussed in Operto et al. 共2009兲.
A feasibility study of FWI in VTI media for crosswell acquisitions
is presented in Barnes et al. 共2008兲. They invert for five parameter
classes — VP,VS, density,
␦
, and — and show reliable reconstruc-
tion of the classes, even with noisy data. Pratt et al. 共2001,2008兲ap-
ply anisotropic FWI to crosshole real data. The results of Pratt et al.
共2001兲highlight the difficulty in discriminating layer-induced an-
isotropy from intrinsic anisotropy in FWI.
Further investigations of anisotropic FWI in the case of surface
seismic data are required. In particular, the benefit of wide apertures
in resolving as many anisotropic parameters as possible needs to be
investigated 共Jones et al., 1999兲.
Three-dimensional FWI
Because of the continuous increase in computational power and
the evolution of acquisition systems toward wide-aperture and wide-
azimuth acquisition, 3D acoustic FWI is feasible today. In three di-
mensions, the computational burden of multisource seismic model-
ing is one of the main issues. The pros and cons of time-domain ver-
sus frequency-domain seismic modeling for FWI have been dis-
cussed. Another issue is assessing the impact of azimuth coverage on
FWI. Sirgue et al. 共2007兲show the footprint of the azimuth coverage
in 3D surveys on the velocity model reconstructed by FWI. Their im-
aging confirms the importance of wide-azimuth surveys for FWI of
coarse acquisitions such as node surveys.
WCC144 Virieux and Operto
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Most 3D FWI applications have been limited to low frequencies
共⬍7Hz兲. At these frequencies, FWI can be seen as a tool for veloci-
ty-model-building rather than a self-contained seismic-imaging tool
that continuously proceeds from the velocity-model-building task to
the migration task through the continuous sampling of wavenum-
bers 共Ben Hadj Ali et al., 2008b兲.
Ben Hadj Ali et al. 共2008a兲apply a frequency-domain FWI algo-
rithm to a series of synthetic data sets. The forward problem is solved
in the frequency domain with a massively parallel direct solver.
Plessix 共2009兲presents an application to real ocean-bottom-seis-
mometer 共OBS兲data. Seismic modeling is performed in the frequen-
cy domain with an iterative solver. The inverted frequencies range
between 2 and 5 Hz. The inversion converges to a similar velocity
model down to the top of a salt body using two different starting ve-
locity models, with one a simple velocity-gradient model. An appli-
cation to a real 3D streamer data set is presented by Warner et al.
共2008兲for the imaging of a shallow channel. They perform seismic
modeling in the frequency domain using an iterative solver 共Warner
et al., 2007兲.
Three-dimensional time-domain FWI is developed by Vigh and
Starr 共2008a兲, in which the input data in FWI are plane-wave gathers
rather than shot gathers. The main motivation behind the use of
plane-wave shot gathers is to mitigate the computational burden by
decimating the volume of data. The computational cost is reduced by
one order of magnitude for 2D applications and by a factor 3 for 3D
applications when the plane-wave-based approach is used instead of
the shot-based approach.
Sirgue et al. 共2009兲apply 3D frequency-domain FWI to the hy-
drophone component of OBC data from the shallow-water Valhall
field. Frequencies between 3.5 and 7 Hz are inverted successively,
using a starting model built by ray-based reflection tomography.
They successfully image a complex network of shallow channels at
150 m depth and a gas cloud between 1000 and 2500 m depth. Al-
though the spacing between the cables is as high as 300 m, a limited
footprint of the acquisition is visible in the reconstructed models.
Comparisons of depth-migrated sections computed from the reflec-
tion tomography model and the FWI velocity model show the im-
provements provided by FWI, both in the shallow structure and at
the reservoir level below the gas cloud. The step-change improve-
ment in the quality of the depth-migrated image results from the
high-resolution nature of the velocity model from FWI and the ac-
counting of the intrabed multiples by the two-way wave-equation
modeling engine. Comparisons between depth slices across the
channels and the gas cloud extracted from the reflection tomography
and the FWI models highlight the significant resolution improve-
ment provided by FWI 共Figure 14兲.
Solving large-scale 3D elastic problems is probably beyond our
current tools because of the computational burden of 3D elastic
modeling for many sources. This has prompted several studies to de-
velop strategies to mitigate the number of forward simulations re-
quired during migration or FWI of large data sets. One of these ap-
proaches stacks the seismic sources before modeling 共Capdeville et
al., 2005兲. Because the relationship between the seismic wavefield
and the source is linear, stacking sources is equivalent to emitting
each source simultaneously instead of independently. This assem-
3
4
5
6
7
8
9
10
11
3456789101112131415161718
1800
2000
2200
2400
3
4
5
6
7
8
9
10
11
3456789101112131415161718
1700
1800
1900
Dip
l
ine (
k
m)
Crossline (km)
Crossline (km)
V
P(km/s)
Dipline (km)
3
4
5
6
7
8
9
10
11
3456789101112131415161718
1
700
1
800
1
900
Crossline (km)
V
P(km/s)
V
P(km/s)
3
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11
3456789101112131415161718
1800
2000
2200
2400
Crossline (km)
V
P(km/s)
a)
b)
c)
d)
Figure 14. Imaging of the Valhall field by 3D FWI. Depth slices extracted from the velocity models. 共a,c兲Built by ray-based reflection tomogra-
phy. 共b,d兲Built by 3D FWI. 共c,d兲The shallow slice at 150 m depth shows a complex network of channels in the FWI model, although the deeper
slice at 1050 m depth shows a much higher resolution of the top of the gas cloud in the FWI model 共image courtesy L. Sirgue and O. I. Barkved,
BP兲.
Full-waveform inversion WCC145
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blage generates artifacts in the imaging that arise from the undesired
correlation between each independent source wavefield with the
back-propagated residual wavefields associated with the other
sources of the stack. Applying some random-phase encoding to each
source before assemblage mitigates these artifacts. The method orig-
inally was applied to migration algorithms 共Romero et al., 2000兲.
Promising applications to time- and frequency-domain FWI are pre-
sented in Krebs et al. 共2009兲and Ben Hadj Ali et al. 共2009a,2009b兲.
Alternatively, Herrmann et al. 共2009兲propose to recover the source-
separated wavefields from the simultaneous simulation before FWI.
An illustration of the source-encoding technique is provided in
Figure 15, in which a dip section of the overthrust model is built
three ways: by conventional frequency-domain FWI 共i.e., without
source assemblage; Figure 15b兲, by FWI with source assemblage but
without phase encoding 共Figure 15c兲, and by FWI with source as-
semblage and phase encoding 共Figure 15d兲. The models were ob-
tained by successive inversions of four groups of two frequencies
ranging between 3.5 and 20 Hz. The number of shots was 200, and
no noise was added to the data. The number of iterations per frequen-
cy group to obtain the final FWI models without and with the source
assemblage was 15 and 200, respectively. The time to compute the
model of Figure 15d was seven times less than the time to build the
model of Figure 15b. More details can be found in Ben Hadj Ali et al.
共2009兲. If the phase-encoding technique is seen as sufficiently ro-
bust, especially in the presence of noise, it is likely that 3D elastic
FWI can be viewed in the near future using sophisticated modeling
engines.
DISCUSSION
Most of the FWI methods presented and assessed in the literature
are based on the local least-squares optimization formulation, in
which the misfit between the observed seismograms and the mod-
eled ones are minimized in the time domain or in the frequency do-
main. Without very low frequencies 共⬍1Hz兲, it remains very diffi-
cult to obtain reliable results from these approaches when consider-
ing real data, especially at high frequencies. Clearly, new formula-
tions of FWI are needed to proceed further.
Recent improvements relate to Hessian estimation, which has
been shown to be quite important for better convergence toward the
solution. A systematic strategy since the beginning of FWI investi-
gations has been the progressive introduction of the entire content of
seismograms through multiscale approaches to partially prevent the
convergence toward secondary minima. Transformingthe data at the
Table 1. Nomenclature, listed as introduced in the text.
Symbol Description
xSpatial coordinates 共m兲
t,f,
⳱2
/fTime 共s兲, frequency 共Hz兲, circular frequency
共rad/s兲
k,k⳱c/fWavenumber vector, wavenumber 共1/m兲,
where cis wavespeed
⳱1/kWavelength 共m兲
M,A,BMass, stiffness, impedance matrices in
the wave equation
u共x,t/
兲,s共x,t/
兲Wavefield solution of the wave equation and
seismic source
VP,VS,
P-wave and S-wave velocities 共m/s兲, density
共kg/m3兲
QAttenuation factor for P-waves
␦
,Anisotropic Thomsen’s parameters
m,m0,∆mUpdated and starting FWI models,
perturbation model
dobs,dcal共m兲Recorded and modeled data
⌬d⳱dobs ⳮdcal共m兲Data-misfit vector
C共m兲,ⵜCmMisfit function, gradient of the misfit
function
J⫽关∂d/∂m兴Sensitivity or Fréchet derivative matrix
␣
Step length in gradient methods
nIteration number in FWI
Ha⳱J†JApproximate Hessian
p共n兲,

共n兲Descent direction and Polak and Ribière
coefficient in conjugate
gradients
Wd⳱Sd
tSdWeighting operators in the data space
Wm⳱Sm
tSmWeighting operators in the model space
Damping in damped least-squares FWI
f共ᐉ兲Virtual source in FWI for diffractor mᐉ
s,rSource and receiver indices
Diffraction or aperture angle
Damping coefficient for time damping of
seismograms
02468101214161820
06000
15000
24000
33000
42000
Distance (
k
m)
Depth (km)
V
P
(m/s)
02468101214161820
06000
15000
24000
33000
42000
02468101214161820
06000
15000
24000
33000
42000
02468101214161820
06000
15000
24000
33000
42
000
V
P
(m/s)V
P
(m/s)V
P
(m/s)
Depth (km)Depth (km)Depth (km)
a)
b)
c)
d)
Figure 15. Application of source encoding in FWI; 200 sources and
receivers were on the surface. 共a兲Dip section of the overthrust
model. 共b兲FWI model obtained without source assemblage and
phase encoding; 15 iterations per frequency were computed. 共c兲FWI
model after assemblage of all the sources in one super shot. No phase
encoding was applied. 共d兲FWI model obtained with source assem-
blage and phase encoding; 200 iterations per frequency were com-
puted 共image courtesy of H. Ben Hadj Ali兲.
WCC146 Virieux and Operto
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different stages of the local inversion procedure 共particle displace-
ment, particle velocity, logarithms of these quantities, divergence兲
can also provide benefits for global convergence of the optimization
procedure.
Do we need to mitigate the nonlinearity of the inverse problem
through more sophisticated search strategies such as semiglobal ap-
proaches or even global sampling of the model space? Because this
search is quite computationally intensive, should we reduce the di-
mensionality of the parameter models? To perform such global
searches, should we speed up the forward model at the expense of its
precision? If so, can we consider these procedures as intermediate
strategies for approaching the global minimum?
A pragmatic workflow, from low to high spatial-frequency con-
tent, should be expected to move hierarchically from imaging the
large medium wavelengths to the short medium wavelengths.A step-
by-step procedure for extracting the information, starting from trav-
eltimes and proceeding to true amplitudes, might include many in-
termediate steps for interpreting new observables 共polarization, ap-
parent velocity, envelope兲.
Another aspect is the quality control of a reconstruction. An ob-
jective analysis for uncertainty estimation is important and might
rely on semiglobal investigations once the global minimum has been
found. Various strategies can be considered that would be manage-
able as statistical procedures 共bootstrapping or jackknifing tech-
niques兲or as local nondifferential approaches 共simplex, simulation
annealing, genetic algorithms兲.
Finally, solving large-scale 3D elastic problems remains beyond
our present tools, although one must be aware that these aids are
right around the corner. Because massive data acquisition for 3D re-
construction has been achieved, we indeed expect an improvement
in our data-crunching for high-resolution imaging.
An appealing reconstruction is 4D imaging, which is based on
time-lapse evolution of targets inside the earth. Differential data are
available, providing us with new information for tracking the evolu-
tion of the subsurface parameters. Thus, fluid tracking and variations
in solid-rock matrices are possible challenges for FWI in the near fu-
ture.
CONCLUSION
FWI is the last-course procedure for extracting information from
seismograms. We have shown the conceptual efforts that have been
carried out over the last 30 years to provide FWI as a possible tool for
high-resolution imaging. These efforts have been focused on devel-
opment of large-scale numerical optimization techniques, efficient
resolution of the two-way wave equation, judicious model parame-
terization for multiparameter reconstruction, multiscale strategies to
mitigate the ill-posedness of FWI, and specific waveform-inversion
data preprocessing.
FWI is mature enough today for prototype application to 3D real
data sets. Although applications to 3D real data have shown promis-
ing results at low frequencies 共⬍7Hz兲, it is still unclear to what ex-
tent FWI can be applied efficiently at higher frequencies. To answer
this question, a more quantitative understanding of FWI sensitivity
to the accuracy of the starting model, to the noise, and to the ampli-
tude accuracies is probably required.
If FWI remains limited to low frequencies, it will remain a tool to
build background models for migration. In the opposite case, FWI
will tend toward a self-contained processing workflow that can re-
unify macromodel building and migration tasks.
The present is exciting because realistic applications are becom-
ing possible right now. However, new strategies must be found to
make this technique as attractive as the scientific issues require.
Fields of investigation should address the need to speed up the for-
ward problem by means of providing new hardware 共GPUs兲and
software 共compressive sensing兲, defining new minimization criteria
in the model and data spaces, and incorporating more sophisticated
wave phenomena 共attenuation, elasticity, anisotropy兲in modeling
and inversion.
ACKNOWLEDGMENTS
We thank L. Sirgue and O. I. Barkved from BP for providing Fig-
ure 14 and for permission to present it. L. Sirgue is acknowledged for
editing part of the manuscript. We thank C. Shin 共Seoul National
University兲and Y. H. Cha 共Seoul National University兲for providing
Figure 11. We thank A. Ribodetti 共Géosciences Azur-IRD兲for dis-
cussion on attenuation reconstruction. We are grateful to R. Brossier
共GéosciencesAzur-UNSA兲and H. Ben HadjAli 共Géosciences Azur-
UNSA兲for providing Figures 8,9, and 15 and for fruitful discus-
sions on many aspects of FWI discussed in this study, in particular
multiparameter FWI, norms, Rytov approximation, and source en-
coding. We would like to thank R. E. Plessix 共Shell兲for his explana-
tions of the adjoint-state method. J. Virieux thanks E. Canet 共LGIT兲
and A. Sieminski 共LGIT兲for an interesting discussion on the inverse
problem and its relationship with assimilation techniques. We would
like to thank S. Buske 共University of Berlin兲, T. Nemeth 共Chevron兲,
I. Lecomte 共NORSAR兲, and V. Sallares 共CMIMA-CSIC Barcelona兲
for their encouragement during the preparation of the manuscript.
Finally, we thank I. Lecomte and T. Nemeth for the time they dedi-
cated to the review of the manuscript. We thank the sponsors of the
SEISCOPE consortium 共BP, CGGVeritas, ExxonMobil, Shell, and
Total兲for their support.
APPENDIX A
APPLICATION OF THE ADJOINT-STATE
METHOD TO FWI
In this appendix, we provide some guidelines for the derivation
of the gradient of the misfit function 共equation 24兲with the adjoint-
state method and Lagrange multipliers. The reader is referred to No-
cedal and Wright 共1999兲for a review of constrained optimization
and to Plessix 共2006兲for a review of the adjoint-state method and its
application to FWI.
First, we introduce the Lagrangian function Lcorresponding to
the misfit function Caugmented with equality constraints:
L共u
¯
,m,
兲⳱1
2具P共dobsⳮRu
¯
兲兩P共dobsⳮRu
¯
兲典
ⳮ具
兩B共m兲u
¯
ⳮs典共A-1兲
The equality constraints correspond to the forward-problem equa-
tion, namely, the state equation, which must be satisfied by the seis-
mic wavefield. A realization uof the state equation is the so-called
state variable. In equation A-1, we introduce the variable u
¯
to distin-
guish any element of the state variable space from a realization of the
state equation 共Plessix, 2006兲.
The vector
, the dimension of which is that of the wavefield u,is
the Lagrange multiplier; it corresponds to the adjoint-state variables.
Full-waveform inversion WCC147
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In the framework of the theory of constrained optimization, the first-
order necessary optimality conditions known as the Karush-Kuhn-
Tucker 共KKT兲conditions state that the solution of the optimization
problem is obtained at the stationary points of L.
The first condition, 关
L/
兴u
¯
⳱cste,m⳱cste ⳱0, leads to the for-
ward-problem equation: B共m兲u⫽s. Resolving the state equation for
m⳱m0provides the incident wavefield for FWI.
The second condition, 关
L/
u
¯
兴
⳱cste,m⳱cste ⳱0, leads to the so-
called adjoint-state equation, expressed as
L共u
¯
,m,
兲
u
¯
⳱P共dobsⳮRu
¯
兲ⳮB†共m兲
⳱0.共A-2兲
For the derivation of equation A-2, we use the fact that 具
,B共m兲u
¯
典
⳱具B†共m兲
,u
¯
典and that the source does not depend on u
¯
. Choosing
m⳱m0and u
¯
⳱u共m0兲in equation A-2 leads to
B†共m0兲
⳱P共⌬d0兲,共A-3兲
which can be rewritten equivalently as
*⳱Bⳮ1共m0兲P共⌬d0
*兲,共A-4兲
where we exploit the fact that Bⳮ1is symmetric by virtue of the spa-
tial reciprocity of Green’s functions. The adjoint-state variables are
computed by solving a forward problem for a composite source
formed by the conjugate of the residuals, which is equivalent to back
propagation of the residuals in the model.
The third condition, 关
L/
m兴u
¯
⳱cste,
⳱cste ⳱0, defines the mini-
mum of Lin a comparable way as for the unconstrained minimiza-
tion of the misfit function C. We have
L共u
¯
,m,
兲
m
⳱
冓
兩
B共m兲
mu
¯
冔
.共A-5兲
For any realization of the forward problem u,L共u,m,
兲⳱C共m兲.
Therefore, equation A-5 gives the expression of the desired gradient
of Cas a function of the state variable and adjoint-state variable
when u
¯
⳱u:
C
m
⳱
冓
兩
B共m兲
mu
冔
.共A-6兲
Inserting the expression of
共equation A-4兲into equation A-3 and
choosing m⳱m0gives the expression of the gradient of Cat the
point m0in the opposite direction of which a minimum of Cis sought
for in FWI:
C共m0兲
m
⳱ut共m0兲
Bt共m0兲
mBⳮ1共m0兲P共⌬d0
*兲.共A-7兲
Equation A-7 is equivalent to equation 24 in the main text.
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