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)

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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