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Image-domain and data-domain waveform tomography: a case
study
Esteban Dı́az1, Yuting Duan1, Gerhard Pratt2, and Paul
Sava11Center for Wave Phenomena, Colorado School of Mines,
2University of Western Ontario
SUMMARY
Wavefield-based tomographic methods are idoneous forrecovering
velocity models from seismic data. The useof wavefields rather than
rays is more consistent withthe bandlimited nature of seismic data.
Image domainmethods seek to improve the focusing in extended
im-ages, thus producing better seismic images. However,image domain
methods produce low resolution modelsdue to the fact that their
objective functions are smooth,particularly in the vicinity of the
global minimum. Incontrast, data-domain methods produce high
resolutionmodels but su↵er from strong non-linearity causing cy-cle
skipping if certain conditions are not met. By com-bining the
characteristics of each method, we can obtainmodels that produce
better images and contain high res-olution features at the same
time. We demonstrate a theworkflow that combines both methods with
an applica-tion to a broadband marine 2D dataset with a
variablestreamer depth.
INTRODUCTION
Velocity analysis methods based on wavefield extrapola-tion are
commonly referred to as Wavefield Tomography(WT) (Tarantola, 1984;
Woodward, 1992; Pratt, 1999;Sava and Biondi, 2004a,b; Shen and
Symes, 2008; Biondiand Symes, 2004; Symes, 2008); such tomographic
ap-proaches can be formulated either in the image domain,where one
tries to improve image quality, or in the datadomain, where one
seeks consistency between modeledand observed data.
Image-domain wavefield tomography (iWT) can be for-mulated by
many means. A common approach aims toimprove the flatness of angle
gathers; equivalently onecan improve the focusing of space-lag
gathers (Shen andCalandra, 2005). Space-lag gathers (Rickett and
Sava,2002; Sava and Fomel, 2006) measure the spatial sim-ilarity
between source and receiver wavefields. Hence,during tomography,
one seeks to increase the similar-ity of the spatial correlation
for a collection of seismicexperiments (Shen and Symes, 2008; Yang
and Sava,2011; Weibull and Arntsen, 2013; Yang et al., 2013;
Shanand Wang, 2013). Inverse problems formulated in theimage domain
are generally better-posed than those for-mulated in the data
domain.
Data-domain wavefield tomography (dWT) is generallyformulated by
improving the consistency between mod-eled and observed data.
Originally, Tarantola (1984)introduced the data di↵erence as a
similarity estimatein the time domain. Alternatively, the problem
can beposed in the frequency domain (Pratt and Worthing-ton, 1990;
Pratt et al., 1998). In contrast to the image-domain formulation,
data-domain wavefield tomography
is highly non-linear resulting in an objective functionwith many
local minima. To overcome the non-linearity,a multi-scale
separation approach is needed (Bunks et al.,1995; Brenders and
Pratt, 2007). Within each scale(wavenumber band or frequency band),
the problem canbe more linear if the initial model is closer to the
onecorresponding to the global minimum. Another loopof multi-scale
can be added by introducing time damp-ing, a method commonly
referred to as Laplace-Fourierwaveform inversion (Sirgue and Pratt,
2004; Shin andHo Cha, 2009). The purpose of the time damping isto
fit earlier arrivals first, and then to fit later
arrivalsprogressively.
Both data-domain and image-domain tomographic meth-ods share
many parts of the process: both use the sameextrapolation engine
(the two-way wave equation), andshare similarities in building the
gradient of the ob-jective function through the Adjoint State
framework(Tarantola, 1984; Plessix, 2006; Symes, 2008).
Both tomographic methods are complementary. There-fore, in this
abstract, we combine image-domain anddata-domain wavefield
tomography approaches for op-timizing the velocity model. The idea
is to produce amodel that improves focusing with image-domain
wave-field tomography and then refine it using data-domainwavefield
tomography. We apply the cascaded work-flow to a marine 2D dataset.
The data are acquiredwith a variable depth streamer cable, which
producesa diverse notch spectrum (Soubaras and Dowle, 2010).The
increased depth produces a better low frequency re-sponse, which
can be useful in the multi-scale approachdiscussed earlier.
IMAGE-DOMAIN WT
Space-lag gathers (Rickett and Sava, 2002; Sava andVasconcelos,
2011) highlight the spatial consistency be-tween wavefields by
exploring the focusing informationin the image domain. The focusing
in the gather is sen-sitive to velocity perturbations, and hence
can be opti-mized. Space-lag gathers are defined as follows:
R(x,�) =X
e,t
u
s
(e,x� �, t)ur
(e,x+ �, t), (1)
where � is the space-lag vector, x the image location, ethe
experiment index, u
s
the source wavefield, and ur
the receiver wavefield. The source wavefield us
is pro-duced by forward extrapolation of the source
function,whereas the receiver wavefield u
r
is produced by back-ward propagation of the data at the receiver
location.In matrix notation, the process is described by
L(m, t) 0
0 L>(m, t)
� u
s
u
r
�=
f
s
f
r
�, (2)
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where m = 1/v2(x) is the medium slowness squared,f
s
is the source function, fr
is the data at the receiverlocations, L(m), and L>(m) are
forward and backwardwave propagators, respectively. Here, we use
the acous-tic wave equation as wave operator:
L(m) = m(x)⇢(x)
@
2
@t
2�r · 1
⇢(x)r. (3)
For iWT we use a constant density ⇢(x) = 1 km/m3.A well-focused
gather concentrates most of its energyaround � = 0. This can be
used as an optimizationcriterion by minimizing the energy outside �
= 0. Wecan accomplish this by defining an objective function
J =12||P (�)R(x,�)||2 , (4)
where P (�) is the penalty function, which plays a vitalrole in
the inversion. Depending on the choice of thepenalty operator P
(�), equation 4 can be either mini-mized or maximized. Here, we use
P (�) = |�| (Shenand Symes, 2008) as penalty operator. This penalty
op-erator defines a smooth objective function and correctsfor most
kinematic errors in the model.
Once we have the penalized gathers (image residuals),we compute
the adjoint sources (Shen and Symes, 2008;Weibull and Arntsen,
2013),
g
s
(x, t) =X
�
P (�)2R(x+ �,�)ur
(x+ 2�, t) (5)
for the source terms, and
g
r
(x, t) =X
�
P (�)2R(x� �,�)ur
(x� 2�, t) (6)
for the receiver terms. Once we have the adjoint sources,we
solve the adjoint equations
L>(m, t) 0
0 L(m, t)
� a
s
a
r
�=
g
s
g
s
�, (7)
and compute the gradient of equation 4 using
rJ(x) =X
e,t
ü
s
(e,x, t)as
(e,x, t)+ (8)
ü
r
(e,x, t)ar
(e,x, t).
DATA-DOMAIN WT
The construction of the tomography problem in the datadomain
begins with a measure of the error (or residual)at the receiver
locations. For dWT, we normally use thedata di↵erence
J =12||u
s
� fr
||2 = 12
�������ddiff (e,x
r
,⌦)������2, (9)
or the phase residual
J =12||arg(u
s
)� arg(fr
)||2 = 12
�������dphase(e,x
r
,⌦)������2,
(10)
where xr
are the receiver locations and ⌦ is the com-plex valued
frequency described below. First, we pro-duce synthetic data by
forward propagating the sourcefunction f
s
(xs
,⌦):
L(m,⌦)us
= fs
(e,xs
,⌦) (11)
L(m,⌦) is the acoustic wave-equation transformed tothe frequency
domain. For dWT we parametrize thedensity ⇢(x) as a function of the
velocity following Gard-ner et al. (1974). The adjoint wavefield
a
s
(e,x,⌦) iscomputed by backpropagating the data residual
L>(m,⌦)as
= �d(e,xr
,⌦). (12)
Once we obtain us
(e,x,⌦) and as
(e,x,⌦), we can pro-ceed to compute the gradient:
rJ(x) = R(X
e,t
⌦2us
(e,x,⌦)a⇤s
(e,x,⌦)
), (13)
here ⇤ is the complex conjugate and R {} is the real part.The
complex valued frequency ⌦ = ! + i/⌧ , where ⌧ isthe characteristic
time for exponential damping, can beused for a multi-scale workflow
which can help to cir-cumvent the local minima problems of equation
9 (Sir-gue and Pratt, 2004; Shin and Ho Cha, 2009; Kameiet al.,
2013). The idea is to first fit lower frequencies! and earlier
arrivals, and then repeat the inversion us-ing longer damping
constant ⌧ . In order to get con-sistent observed data with the
damped modeled data,one must also scale the observed data as f
r
(e,xr
, t) =d
obs
(e,xr
, t)e�t/⌧ before the transformation to frequencydomain.
The low frequencies in the data are sensitive to the
longwavelength (smooth) components of the earth model.However, if
the data do not contain su�ciently low fre-quencies or su�ciently
large o↵sets, dWT is unable toupdate such components. In contrast,
focusing in ex-tended images is mostly sensitive to the smooth
com-ponents of the model. By implementing a joint work-flow using
iWT for updating the smooth components ofthe model and later using
dWT for the high resolutionfeatures of the model, we can obtain a
more completespectrum of the model.
APPLICATION TO A REAL 2D DATASET
In this section we demonstrate our proposed workflowwith a real
2D marine dataset acquired with a variabledepth cable (Soubaras and
Dowle, 2010). The streamercontains increased depths as a function
of o↵set, whichenhances the frequency content of the data by
produc-ing a mixed notch response. Hence, the increasing
cabledepths improve the low frequency content at intermedi-ate and
far o↵sets which is very helpful for dWT. Theacquisition setup in
the modeling software mimics thevariable depth streamer cable and
the free surface, henceit is consistent with the observed data.
We build the initial model, Figure 1(a)(top), by per-forming
time-domain NMO analysis followed by smooth-ing, RMS (stacking)
conversion to interval velocity (Dix,
-
(a)
(b)
(c)
(d)
Figure
1:Im
agingcompositionwith:velocity
(top
),RTM
imag
e(center)
andan
gle-ga
thers(bottom)from
(a)theinitialvelocity
model,(b)thedW
Tmodel
built
from
thestartingmodel,(c)theiW
Tmodel,an
d(d)thedW
Tmodel
builtfrom
theiW
Tmodel.
-
1955), and time to depth conversion. Figure 1(a)(center)shows
the corresponding RTM image. One can observethat the image is
over-migrated (the velocities are toohigh) below 3km in depth.
Figure 1(a)(bottom) showsangle gathers extracted at sparse
locations in the model.Note that we do not use the angle gathers
for inver-sion; instead, we use these particular gathers as an
inde-pendent quality control tool. The transformation fromspace-lag
gathers R(x,�) to angle domain R(x, ✓) fol-lows the method of Sava
and Fomel (2003). The anglesvary from 0 to 45� for all the gathers
shown in this ab-stract. The moveout in the gathers confirms that
thevelocity is too fast below 3 km. Some of the events inthe
gathers, however, correspond to migrated multiplesand their moveout
is not indicative of velocity error.
For dWT, we use 7 successive frequency blocks with 5simultaneous
frequencies each. The center frequency foreach block ranges from f
= 2.6 Hz to f = 8.9 Hz. Forthe time damping constant, we use ⌧ =
1.6 s. We usethe phase di↵erence for the data-domain residual.
Thefirst step in data domain wavefield tomography
involvesestimating the source function f
s
(⌦). We use 365 shotsfor the inversion with a shot interval �s =
0.09375 km.
Figure 1(b)(top) shows the dWT model built from Fig-ure
1(a)(top). The dWT process slows the velocity inthe shallow part of
the section, close to the water bot-tom, introducing a sharp
discontinuity in the model.One can see in the gathers, Figure
1(b)(bottom), thatin the shallow part the events get flatter with
the newvelocity. However, the deeper section does not
changesignificantly. This area of the model corresponds tolonger
travel-times in the data, and these late arrivalsare down-weighted
in the inversion by the exponentialtime damping (in order to avoid
cycle skipping).
We generate the model in Figure 1(c)(top) using theiWT approach.
The idea of this tomographic step is tocorrect for the bulk of the
kinematic errors in the model.The updated velocity, in general
becomes slower, espe-cially in the deep part of the section. Figure
1(c)(center)shows the corresponding RTM image, with increased
fo-cusing around z = 3.5 km. This observation is con-firmed in
Figure 1(c)(bottom), where the gathers areflatter throughout the
section.
We then update the iWT model using dWT. Figure 1(d)(top) depicts
the updated model (compare with Fig-ure 1(b)), which changes
considerably in the intervalz = 4 km to z = 6 km. From the
corresponding RTMimage and angle gathers we can see that the final
dWTstep does not significantly vary the kinematics of
theexperiment. However, it introduces subtle structuralfeatures in
the image. We can see that the structureof the line becomes flatter
with the new model (see forinstance the event at z = 4 km and x =
18 to 24 km).
Figures 2(a)-2(d) show the data-domain residuals for themodels
presented in this section. The residual corre-sponding to the
initial model, Figure 2(a), show largeamplitude and phase mismatch.
After updating the ini-tial model using dWT, we can see in Figure
2(b) how the
residuals improve for intermediate o↵sets. Figure 2(c)depicts
the residual corresponding to the iWT model.We can see that this
residuals have smaller energy thanthose of Figure 2(a). Figure 2(d)
shows the residuals forthe dWT built from the iWT model. We can see
a bet-ter match for near and intermediate o↵sets. The energyfor
this residual is smaller than any of the previous.
(a) (b)
(c) (d)
Figure 2: Data domain residuals for shot position x =18.75 km
using: (a) the initial velocity model, (b) thedWT model built from
the initial model, (c) the iWTmodel, and (d) the dWT model built
from the iWTmodel.
CONCLUSIONS
The combination of image-domain and data-domain wave-field
tomography seeks to exploit the features of eachmethod.
Image-domain wavefield tomography methodsare sensitive to the
smooth components of the modeldue to the definition of the
objective problem. Oncewe obtain a smooth model that improves
focusing inthe extended images, we can proceed to further refinethe
model using data-domain wavefield tomography. Wedemonstrate the
cascaded workflow using a real 2D ma-rine dataset. Our image-domain
wavefield tomographymodel corrects most of the kinematic errors in
the model,whereas the data-domain wavefield tomography
modelcorrects early arrival phase errors in the data, and
in-troduces discontinuities in the model directly correlatedwith
events in the image.
ACKNOWLEDGMENTS
We thank sponsor companies of the Consortium Projecton Seismic
Inverse Methods for Complex Structures.The seismic data shown in
this abstract is proprietaryto and provided courtesy of CGG. We
thank Bruce Ver-West for the support with the dataset. The
reproduciblenumeric examples in this paper use the Madagascar
open-source software package (Fomel et al., 2013) freely avail-able
from http://www.ahay.org.
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REFERENCES
Biondi, B., and W. Symes, 2004, Angle-domaincommon-image gathers
for migration velocity anal-ysis by wavefield-continuation imaging:
Geophysics,69, 1283–1298.
Brenders, A. J., and R. G. Pratt, 2007, Full waveformtomography
for lithospheric imaging: results from ablind test in a realistic
crustal model: GeophysicalJournal International, 168, 133–151.
Bunks, C., F. Saleck, S. Zaleski, and G. Chavent,
1995,Multiscale seismic waveform inversion: Geophysics,60,
1457–1473.
Dix, C., 1955, Seismic velocities from surface measure-ments:
Geophysics, 20, 68–86.
Fomel, S., P. Sava, I. Vlad, Y. Liu, and V. Bashkardin,2013,
Madagascar: open-source software projectfor multidimensional data
analysis and reproduciblecomputational experiments: Journal of Open
Re-search Software, 1, e8.
Gardner, G., L. Gardner, and A. Gregory, 1974, Forma-tion
velocity and density—the diagnostic basics forstratigraphic traps:
Geophysics, 39, 770–780.
Kamei, R., R. G. Pratt, and T. Tsuji, 2013, On acous-tic
waveform tomography of wide-angle obs data-strategies for
pre-conditioning and inversion: Geo-physical Journal
International.
Plessix, R.-E., 2006, A review of the adjoint statemethod for
computing the gradient of a functionalwith geophysical
applications: Geophysical JournalInternational, 167, 495–503.
Pratt, R., 1999, Seismic waveform inversion in the fre-quency
domain, part 1: Theory and verification in aphysical scale model:
Geophysics, 64, 888–901.
Pratt, R. G., C. Shin, and G. J. Hick, 1998, Gauss–newton and
full newton methods in frequency–spaceseismic waveform inversion:
Geophysical Journal In-ternational, 133, 341–362.
Pratt, R. G., and M. H. Worthington, 1990, Inverse the-ory
applied to multi-source cross-hole tomography.:Geophysical
Prospecting, 38, 287–310.
Rickett, J. E., and P. C. Sava, 2002, O↵set and angle-domain
common image-point gathers for shot-profilemigration: Geophysics,
67, 883–889.
Sava, P., and B. Biondi, 2004a, Wave-equation mi-gration
velocity analysis - I: Theory: GeophysicalProspecting, 52,
593–606.
——–, 2004b, Wave-equation migration velocity anal-ysis - II:
Subsalt imaging examples: GeophysicalProspecting, 52, 607–623.
Sava, P., and S. Fomel, 2006, Time-shift imaging con-dition in
seismic migration: Geophysics, 71, S209–S217.
Sava, P., and I. Vasconcelos, 2011, Extended imagingconditions
for wave-equation migration: GeophysicalProspecting, 59, 35–55.
Sava, P. C., and S. Fomel, 2003, Angle-domain common-image
gathers by wavefield continuation methods:Geophysics, 68,
1065–1074.
Shan, G., and Y. Wang, 2013, Rtm based wave equation
migration velocity analysis: 4726–4731.Shen, P., and H.
Calandra, 2005, One-way waveform
inversion within the framework of adjoint state dif-ferential
migration: 1709–1712.
Shen, P., and W. W. Symes, 2008, Automatic velocityanalysis via
shot profile migration: Geophysics, 73,VE49–VE59.
Shin, C., and Y. Ho Cha, 2009, Waveform inversion inthe
Laplace–Fourier domain: Geophysical Journal In-ternational, 177,
1067–1079.
Sirgue, L., and R. Pratt, 2004, E�cient waveform inver-sion and
imaging: A strategy for selecting temporalfrequencies: Geophysics,
69, 231–248.
Soubaras, R., and R. Dowle, 2010, Variable-depthstreamer–a
broadband marine solution: first break,28.
Symes, W. W., 2008, Migration velocity analysis andwaveform
inversion: Geophysical Prospecting, 56,765–790.
Tarantola, A., 1984, Inversion of seismic reflection datain the
acoustic approximation: Geophysics, 49, 1259–1266.
Weibull, W., and B. Arntsen, 2013, Automatic velocityanalysis
with reverse-time migration: Geophysics, 78,S179–S192.
Woodward, M., 1992, Wave-equation tomography: Geo-physics, 57,
15–26.
Yang, T., and P. Sava, 2011, Image-domain waveformtomography
with two-way wave-equation: SEG Tech-nical Program Expanded
Abstracts 2011, 508, 2591–2596.
Yang, T., J. Shragge, and P. Sava, 2013,
Illuminationcompensation for image-domain wavefield tomogra-phy:
Geophysics, 78, U65–U76.