in

abia

Image gatherOffset domainWave equation

le tatelffsehhoifcingy a

image. The offset domain CIGs are then achieved after separately migrating data with different offsets. Wegenerate offset domain CIGs on both the Marmousi synthetic data and 2D Gulf of Mexico real data using thisapproach.

Published by Elsevier B.V.

lored tin indue of inr (COG

ed by placing a sourcesolution at the surface.se solutions by simple. The COG is migrated

Journal of Applied Geophysics 103 (2014) 99103

Contents lists available at ScienceDirect

Journal of Applie

j ourna l homepage: www.e lsRTM traditionally migrates a shot prole with traces from differentoffsets at a time, it may be impossible to separate these offsets afterthe fact. Therefore, with the existing RTM algorithm, it is difcult to

to different depth levels using the precalculated RTM operators.Etgen (2012) revisited the derivation of Ehinger et al. (1996) and

described its application to 3D migration cases. He showed that it isimaging through complex geology. Although it is computationallyexpensive, it can deal with complex wave phenomena such as multiplearrivals and multiply-scattered arrivals than Kirchhoff migration. But itis not exible in the same ways that Kirchhoff migration is. Because

operators for different depth levels are computat these depths, and nding the nite-differenceThe RTM operators are then computed from theconvolutions of surface trace with its neighborsit has difculty in complex velocity models which precludes theaccurate use of ray tracing and the modeling of multiply-scatteredarrivals.

Reverse-time migration (RTM) is a sophisticated wave equationdepth migration method that has now become the routine tool for

combines the superior imaging properties of the wave-equationmethod (i.e., accuracy) with the superior capabilities of common-offsetmigration (i.e., outputting offset gathers).

Schuster (2002) described a different implementation of RTM,which provides a means for migrating COGs. In his method, exact RTM Corresponding author at: King Abdullah Universi(KAUST), Earth Sciences and Engineering, Thuwal 23955-02 808 0295.

E-mail address: ge.zhan@kaust.edu.sa (G. Zhan).

0926-9851/$ see front matter. Published by Elsevier B.Vhttp://dx.doi.org/10.1016/j.jappgeo.2014.01.005)), but also about theexpensive, but subjecttracing. Consequently,

method based on the one-way waveeld extrapolation method. Intheir method, depth migration operators are explicitly constructed justlike Kirchhoff migration via the use of Green's functions. Therefore, itoutput it produces. It is computationally into the high-frequency approximation of ray1. Introduction

Kirchhoff migration has been expdecade. And it has been a major toolIt is very exible not only to any typgather (CSG), common-offset gathehoroughly over the laststry for depth imaging.put (i.e., common-shot

QC the velocity model and post-process the image in the offset domainas Kirchhoff migration.

To partly remediate this deciency of RTM, people havebeen lookingfor common-offset solutions for the wave equation migration methods.Ehinger et al. (1996) proposed a common-offset depth migrationCommon-image gathers in the offset domareverse-time migration

Ge Zhan a,, Minyu Zhang b

a King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arb University of Houston, Houston, TX 77004, United States

a b s t r a c ta r t i c l e i n f o

Article history:Received 5 September 2013Accepted 16 January 2014Available online 24 January 2014

Keywords:RTM

Kirchhoff migration is exibwith different offsets separamplitude-variation-with-ocomplex geology than Kircwave equation. But, it has ddevelop a method for obtainan offset gather, followed bty of Science and Technology6900, Saudi Arabia. Tel.: +966

.from

o output common-image gathers (CIGs) in the offset domain by imaging datay. These CIGs supply important information for velocity model updates andt (AVO) analysis. Reverse-time migration (RTM) offers more insights intoff migration by accurately describing wave propagation using the two-wayulty to produce offset domain CIGs like Kirchhoff migration. In this paper, weoffset domain CIGs from RTM. The method rst computes the RTM operator ofdot product of the operator and the offset data to form a common-offset RTM

d Geophysics

ev ie r .com/ locate / j appgeopossible to retain the ability to sort migrated output by surface offsetusing the one-way depth-extrapolation method.

In this paper, we extend Ehinger et al. (1996)'s approach to RTM in astraightforward way. That is, we reformulate the RTM equation so thatit can be reinterpreted as a wave-equation Kirchhoff migrationalgorithm. Thus, RTM of COGs becomes feasible with the proposed

100 G. Zhan, M. Zhang / Journal of Applied Geophysics 103 (2014) 99103a)

c)method, and RTM is now able to produce offset domain common-imagegathers (CIGs).

2. Method

Aprestackmigration image of a CSG can be computed using the Bornformula (Stolt and Benson, 1986)

mmig x; s g x; tjs;0 g x;tjr;0 d r; tjs;0

h idrdt; 1

where mmig(x,s) is the migration image at the trial image point x withthe shot at s. The term d(r, t|s, 0) represents the bandlimited data inthe time domain for a source at s and receivers at r. g(x, t|s, 0) andg(x, t|r, 0) are the Green's functions in the space-time domain withreceivers at x and sources at s and r, respectively. The symbol * standsfor convolution, and the double dot symbol represents the tracedifferentiated twice in time. The dr integration is over all receivercoordinates along the surface, and the dt integration is over the durationtime of the trace.

The term g x;tjr;0 d r; tjs;0 represents the convolution of thetime reversed Green's function traces with the time-differentiatedtraces. This operation backpropagates the trace energy at r to the

Fig. 1. Green's function implementationb)

d)subsurface at x. In contrast, the Green's function g(x, t|s, 0) forwardpropagates the energy at the source point s to the subsurface point x,and the migration image at x is formed by taking the zero-lag temporalcorrelation of g(x, t|s, 0) with the backpropagated trace at x. TraditionalRTM simulates backpropagation by a nite-difference solution to theacoustic wave equation, where the point sources are at the traceslocated on the surface and the traces act as the time histories forbackpropagating seismic waveelds at the receiver locations.

A different implementation of shot-domain RTM can be obtained byleft shifting the square brackets in Eq. (1) to get (Schuster, 2002)

mmig x; s g x; tjs;0 g x; tjr;0 d r; tjs;0 drdt: 2

Here, the bracketed term

G r; s; x; t g x; tjs;0 g x; tjr;0 ; 3

is the migration operator that refocuses reection energy recorded at r(for a source at s) back to the scatterer at x. It is obtained by convolvingthe Green's function g(x, t|s, 0) at the source side with the receiver sideGreen's function g(x, t|r, 0). In practice, the Green's functions arecomputed by solving the two-way wave equation with a nite-difference method. And the angular frequency of the source wavelet

of common-shot RTM using Eq. (2).

101G. Zhan, M. Zhang / Journal of Applied Geophysics 103 (2014) 99103a)

c)associated with the Green's function is set to be

pto generate the

migration operator with the frequency of .Eq. (3) says that the migration image at x is computed by taking the

dot product between the time-differentiated shot gatherd r; tjs;0 and

themigration operatorG r; s;x; t . This is similar to the interpretation ofKirchhoff migration, except that only primary events are accounted forin conventional Kirchhoff migration, while RTM takes into accountboth primaries and multiples.

A COG along a line is a function of midpoint xm = (s + r)/2 andhalf-offset h = (s r)/2 coordinates, and so the COG for a xed xmis d(xm h, t|xm + h, 0), which can be extracted from the CSGdata. Similarly, the source side and receiver side Green's functions inthe midpoint-offset coordinate are respectively given by g(x, t|xm +h, 0) and g(x, t|xm h, 0), which can be used to construct themigrationoperator in the common-offset domain. Therefore, we get the common-offset RTM formula

mmig x;h g x; tjxm h;0 g x; tjxmh;0 d xmh; tjxm h;0 dxmdt; 4

and the new migration operator in the common-offset domain isexpressed as

Go xm;h; x; t g x; tjxm h;0 g x; tjxmh;0 : 5

Fig. 2. Green's function implementation ob)

d)Notice that, Eqs. (2) and (4) have identical computational steps(i.e., computing the migration operator by convolution, followedby a dot product with data), differing only in the ordering of convo-lution and summation. Therefore, the resultingmigration images areidentical for common-shot (Eq. (2)) and common-offset (Eq. (4))migration.

3. Numerical examples

3.1. Marmousi example

The Marmousi data set (Bourgeois et al., 1991) is generally consid-ered as a reference test for migration algorithms. A typical CSG fromthe Marmousi data set is displayed in Fig. 1a. To migrate it, we rstconstruct themigration operator using Eq. (3) with the correct velocity.And the corresponding migration operator for this CSG at the depth of1.0 km is shown in Fig. 1b. The RTM image at this depth is just the dotproduct (using Eq. (2)) between Figs. 1b and a. A summation of thedot product results along all depth locations gives the RTM image(Fig. 1c) of the CSG. Fig. 1d presents the shot-domain CIG at the sourcelocation as indicated by the blue line on Fig. 1c.

The four panels in Fig. 2 demonstrate the migration process similarto that in Fig. 1, but they are now computed in the common-offsetdomain using Eqs. (4) and (5). It is obvious that, given the correctmigration velocity, the geologic structure is clearly interpretable over

f common-offset RTM using Eq. (4).

a) a)102 G. Zhan, M. Zhang / Journal of Applied Geophysics 103 (2014) 99103b)almost the entire survey (except the two end regions) in Fig. 2c com-pared to Fig. 1c, which is limited by a nite aperture in model space.Moreover, the events in the offset domain CIG of Fig. 2d are atterthan those in the shot-domain CIG of Fig. 1d.

Fig. 2 shows the RTM image computed from an intermediate offsetsection. By doing common-offset RTM for different offsets separatelyfollowed by a stack, Fig. 3 is obtained. Fig. 3a displays the RTM image,and Fig. 3b presents the offset domain CIGs which are sorted by theirCDP locations and surface offsets.

The computational costs for both common-shot RTM and common-offset RTM of the Marmousi data set running on a 12-core Intel Xeoncomputing node are listed in Table 1. For migrating one gather, thecommon-offset RTM is slower than the common-shot RTM by a factorof 15 in this case. However, due to less input gathers for migration aswell as the reusability of the Green's functions in migrating differentoffsets, the total runtime of common-offset RTM is greatly reduced.And it only runs around 5 times slower than conventional RTM butwith the supply of surface-offset CIGs.

3.2. Gulf of Mexico example

The Mississippi Canyon data set (Dragoset, 1999) from the Gulf ofMexico is used here to test the common-offset RTM algorithm. 1001

Fig. 3. Common-offset RTM image of the Marmousi data set and the associated offsetdomain CIGs. The offset range of each CDP is from 0.2 km to 2.575 km.

Table 1Runtime comparison: common-shot RTM versus common-offset RTM.

Number of input gathers Migration aperture (km)

Common-shot RTM 240 4.0Common-offset RTM 96 7.0b)shots are acquired and each one is recorded by 180 hydrophones withan off-end spread. All 1001 CSGs are rst sorted into 180 COGs. Then,time-space Green's functions are modeled by solving the acoustictwo-way wave equation with an accurate migration velocity model,where numerical sources are at the surface air-gun/hydrophone loca-tions, and the Green's functions are recorded at all subsurface imagepoints. The offset domain migration operator for an input COG trace iscomputed by convolving the two Green's functions associated withthe air-gun and hydrophone locations of this input trace. A dot productbetween the migration operator and the COG trace contributes to theimage of this trial image point. A similar processing for all image pointsleads to a completemigration image of this COG. The two panels in Fig. 4display the raw RTM image of this Gulf of Mexico data set and the asso-ciated offset domain CIGs. And Fig. 5 shows the image after post-migration processing, where migration artifacts are greatly suppressed.

4. Conclusions

Wedescribe a common-offset RTM algorithm that uses the convolu-tion of two computedGreen's functions. The common-offset RTM imageat a trial image point is obtained by taking the dot product of the appro-priate offset domain migration operator with the offset data. The keybenet of this method is the output of offset domain CIGs. Such gathers

Runtime of migrating 1 gather (min) Runtime of migrating all gathers (min)

0.5 120.07.4 607.5

Fig. 4. Results of common-offset RTM applied to the Gulf of Mexico data set. The offsetrange of each CDP is from 0.1 km to 4.875 km.

are easily understood by processors used to thinking in updating thesubsurface velocity from residual moveouts, and inversion for rockproperties from amplitude-variation-with-offset. Another advantageof this method is that we do not need to output every CIG duringmigration, instead we can focus on local target areas. The maindrawback of this algorithm is the memory and computation expenseof creating and saving Green's function tables, and the subsequentmigration operator calculation through the convolution of traces.

However, a wavelet compression strategy (Zhan et al., 2010), Green'sfunction skeletonization (Zhan and Schuster, 2010), and data/geometryregularization (Etgen, 2012)might allow this method to be practical fortarget-oriented 3D RTM.

Acknowledgments

The authors wish to thank the Center for Subsurface Imaging andFluid Modeling (CSIM) sponsors for their nancial support. We alsoappreciate the anonymous reviewers who made a number of helpfulsuggestions that improved the quality of our manuscript.

References

Bourgeois, A., Bourget, M., Lailly, P., Poulet, M., Ricarte, P., Versteeg, R., 1991. Marmousi,model and data. Proceedings of 1990 EAGE workshop on practical aspects of seismicdata inversion.

Dragoset, B., 1999. A practical approach to surface multiple attenuation. Lead. Edge 18,104108.

Ehinger, A., Lailly, P., Marfurt, K., 1996. Green's function implementation of common-offset, wave-equation migration. Geophysics 61, 18131821.

Etgen, J.T., 2012. 3Dwave equation Kirchhoffmigration. SEG Technical Program ExpandedAbstracts, 31, pp. 15.

Schuster, G.T., 2002. Reverse-time migration=generalized diffraction stack migration.SEG Technical Program Expanded Abstracts, 21, pp. 12801283.

Stolt, R.H., Benson, A.K., 1986. Seismic Migration: Theory and Practice. Handbook ofGeophysical Exploration, vol. 5. Geophysical Press, London.

Zhan, G., Schuster, G.T., 2010. Skeletonized least squares wave equation migration. SEGTechnical Program Expanded Abstracts, 29, pp. 33803384.

Zhan, G., Luo, Y., Schuster, G.T., 2010. Modied form of reverse time migration tuned tomultiple. 72nd Annual Conference and Exhibition, EAGE, Extended Abstracts (P594).

Fig. 5. The RTM image of the Gulf of Mexico data after post-migration processing.

103G. Zhan, M. Zhang / Journal of Applied Geophysics 103 (2014) 99103

Common-image gathers in the offset domain from reverse-time migration1. Introduction2. Method3. Numerical examples3.1. Marmousi example3.2. Gulf of Mexico example

4. ConclusionsAcknowledgmentsReferences