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GEOPHYSICS, VOL. 71, NO. 6 �NOVEMBER-DECEMBER 2006�; P. S209–S217, 16 FIGS.10.1190/1.2338824

ime-shift imaging condition in seismic migration

aul Sava1 and Sergey Fomel2

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ABSTRACT

Seismic imaging based on single-scattering approxima-tion is in the analysis of the match between the source and re-ceiver wavefields at every image location. Wavefields atdepth are functions of space and time and are reconstructedfrom surface data either by integral methods �Kirchhoff mi-gration� or by differential methods �reverse-time or wave-field extrapolation migration�. Different methods can be usedto analyze wavefield matching, of which crosscorrelation is apopular option. Implementation of a simple imaging condi-tion requires time crosscorrelation of source and receiverwavefields, followed by extraction of the zero time lag. Ageneralized imaging condition operates by crosscorrelationin both space and time, followed by image extraction at zerotime lag. Images at different spatial crosscorrelation lags areindicators of imaging accuracy and are also used for image-angle decomposition. In this paper, we introduce an alterna-tive prestack imaging condition in which we preserve multi-ple lags of the time crosscorrelation. Prestack images are de-scribed as functions of time shifts as opposed to space shiftsbetween source and receiver wavefields. This imaging condi-tion is applicable to migration by Kirchhoff, wavefield ex-trapolation, or reverse-time techniques. The transformationallows construction of common-image gathers presented asfunctions of either time shift or reflection angle at every loca-tion in space. Inaccurate migration velocity is revealed by an-gle-domain common-image gathers with nonflat events.Computational experiments using a synthetic data set from acomplex salt model demonstrate the main features of themethod.

INTRODUCTION

Akey challenge for imaging in complex areas is accurate determi-ation of a velocity model in the area under investigation. Migra-

Manuscript received by the Editor September 29, 2005; revised manuscrip1Colorado School of Mines, Department of Geophysics, 1500 Illinois Stree2University of Texas -Austin, Bureau of Economic Geology,Austin, Texas2006 Society of Exploration Geophysicists.All rights reserved.

S209

ion-velocity analysis is based on the principle that image-accuracyndicators are optimized when data are correctly imaged.Acommonrocedure for velocity analysis is to examine the alignment of imag-s created with multioffset data.An optimal choice of image analysisan be done in the angle domain, which is free of some complicatedrtifacts present in surface offset gathers in complex areas �Stolk andymes, 2004�.Migration-velocity analysis after migration by wavefield extrapo-

ation requires image decomposition in scattering angles relative toeflector normals. Several methods have been proposed for such de-ompositions �de Bruin et al., 1990; Prucha et al., 1999; Mosher andoster, 2000; Rickett and Sava, 2002; Xie and Wu, 2002; Sava andomel, 2003; Soubaras, 2003; Biondi and Symes, 2004; Fomel,004�. These procedures require decomposition of extrapolatedavefields in variables that are related to the reflection angle.A key component of such image decompositions is the imaging

ondition. A careful implementation of the imaging condition pre-erves all information necessary to decompose images in their angle-ependent components. The challenge is efficient and reliable con-truction of these angle-dependent images for velocity or amplitudenalysis.

In migration with wavefield extrapolation, a prestack imagingondition based on spatial shifts of the source and receiver wave-elds allows for angle composition �Rickett and Sava, 2002; Savand Fomel, 2005�. Such formed angle gathers describe reflectivity asfunction of reflection angles and are powerful tools for migration-elocity analysis or amplitude-versus-angle analysis. However, be-ause of the large expense of space-time crosscorrelations, especial-y in three dimensions, this imaging methodology is not used rou-inely in data processing.

This paper presents a different form of imaging condition. Theey idea of this new method is to use time shifts instead of spacehifts between wavefields computed from sources and receivers. Asor the space-shift imaging condition, an image is built by space-ime crosscorrelations of subsurface wavefields, and multiple lags ofhe time crosscorrelation are preserved in the image. Time shiftsave physical meaning that can be related directly to reflection ge-metry, similar to the procedure used for space shifts. Furthermore,

ed February 23, 2006; published online September 21, 2006.en, Colorado 80401. E-mail: psava@mines.edu.E-mail: sergey.fomel@beg.utexas.edu.

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ime-shift imaging is cheaper to apply than space-shift imaging, andhus it might alleviate some of the difficulties posed by costly cross-orrelations in 3D space-shift imaging condition.

The idea of a time-shift imaging condition is related to the idea ofepth-focusing analysis �Faye and Jeannot, 1986; MacKay andbma, 1992, 1993; Nemeth, 1995, 1996�. The main novelty of our

pproach is that we employ time shifting to construct angle-domainathers for prestack depth imaging.

The time-shift imaging concept is applicable to Kirchhoff migra-ion, migration by wavefield extrapolation, or reverse-time migra-ion. We present a theoretical analysis of this new imaging condition,ollowed by a physical interpretation leading to angle decomposi-ion. Finally, we illustrate the method with images of the complexigsbee 2Adata set �Paffenholz et al., 2002�.

IMAGING CONDITION INWAVE-EQUATION IMAGING

A traditional imaging condition for shot-record migration, ofteneferred to as UD* imaging condition �Claerbout, 1985�, consists ofime crosscorrelation at every image location between the sourcend receiver wavefields, followed by image extraction at zero time:

U�m,t� = Ur�m,t� � Us�m,t� , �1�

R�m� = U�m,t = 0� , �2�

here the symbol � denotes crosscorrelation in time. Here, m =mx,my,mz� is a vector describing the locations of image points.

s�m,t� and Ur�m,t� are source and receiver wavefields, respective-y; and R�m� denotes a migrated image. A final image is obtained byummation over shots.

pace-shift imaging condition

A generalized prestack imaging condition �Sava and Fomel,005� estimates image reflectivity using crosscorrelation in spacend time, followed by image extraction at zero time:

U�m,h,t� = Ur�m + h,t� � Us�m − h,t� , �3�

R�m,h� = U�m,h,t = 0� . �4�

ere, h = �hx,hy,hz� is a vector describing the space shift betweenhe source and receiver wavefields prior to imaging. Special cases ofhis imaging condition are horizontal space shift �Rickett and Sava,002� and vertical space shift �Biondi and Symes, 2004�.

For computational reasons, this imaging condition is usually im-lemented in the Fourier domain using the expression

R�m,h� = ��

Ur�m + h,��Us*�m − h,�� . �5�

he � sign represents a complex conjugate applied on the receiveravefield Us in the Fourier domain.

ime-shift imaging condition

Another possible imaging condition, advocated in this paper, in-olves shifting the source and receiver wavefields in time, as op-osed to space, followed by image extraction at zero time:

U�m,t,�� = U �m,t + �� � U �m,t − �� , �6�

r s

R�m,� � = U�m,� ,t = 0� . �7�

ere, � is a scalar describing the time shift between the source andeceiver wavefields prior to imaging. This imaging condition can bemplemented in the Fourier domain using the expression

R�m,� � = ��

Ur�m,��Us*�m,��e2i�� , �8�

hich involves a phase shift applied to the wavefields prior to sum-ation over frequency � for imaging at zero time.

pace-shift and time-shift imaging condition

To be even more general, we can formulate an imaging conditionnvolving both space shift and time shift, followed by image extrac-ion at zero time:

U�m,h,t� = Ur�m + h,t + � � � Us�m − h,t − �� , �9�

R�m,h,�� = U�m,h,�,t = 0� . �10�

owever, the cost involved in this transformation is large, so thiseneral form does not have immediate practical value. Imaging con-itions described by equations 3 and 4 and equations 6 and 7 are spe-ial cases of equations 9 and 10 for h = 0 and � = 0, respectively.

ANGLE TRANSFORMATION INWAVE-EQUATION IMAGING

Using the definitions introduced in the preceding section, we canake the standard notations for source and receiver coordinates: sm − h and r = m + h. The traveltime from a source to a receiver

s a function of all spatial coordinates of the seismic experiment tt�m,h�. Differentiating t with respect to all components of the

ectors m and h, and using the standard notations p� = ��t, where= �m,h,s,r�, we can write

pm = pr + ps, �11�

ph = pr − ps. �12�

rom equations 11 and 12, we can write

2ps = pm − ph, �13�

2pr = pm + ph. �14�

By analyzing the geometric relations of various vectors at an im-ge point �Figure 1�, we can write the following trigonometric ex-ressions:

�ph�2 = �ps�2 + �pr�2 − 2�ps��pr�cos�2�� , �15�

�pm�2 = �ps�2 + �pr�2 + 2�ps��pr�cos�2�� . �16�

quations 15 and 16 relate wavefield quantities ph and pm to a geo-etric quantity, reflection angle �. Analysis of these expressions

rovides sufficient information for complete decompositions of mi-rated images in components for different reflection angles.

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pace-shift imaging condition

Defining km and kh as location and offset wavenumber vectorsnd assuming �ps� = �pr� = s, where s�m� is the slowness at imageocations, we can replace �pm� = �km�/� and �ph� = �kh�/� in equa-ions 15 and 16:

�kh�2 = 2��s�2�1 − cos 2�� , �17�

�km�2 = 2��s�2�1 + cos 2�� . �18�

sing the trigonometric identity

cos�2�� =1 − tan2 �

1 + tan2 �, �19�

e can eliminate from equations 17 and 18 the dependence on fre-uency and slowness and obtain an angle decomposition formula-ion after imaging by expressing tan � as a function of position andffset wavenumbers �km,kh�:

tan � =�kh��km�

. �20�

We can construct angle-domain common-image gathers by trans-orming prestack migrated images using equation 20:

R�m,h� Þ R�m,�� . �21�

n two dimensions, this transformation is equivalent with a slanttack on migrated offset gathers. For 3D, this transformation is de-cribed in more detail by Fomel �2004� or Sava and Fomel �2005�.

ime-shift imaging condition

Using the same definitions as those introduced in the precedingubsection, we can rewrite equation 18 as

�pm�2 = 4s2 cos2 � , �22�

rom which we can derive an expression for angle transformation af-er time-shift prestack imaging:

cos � =�pm�2s

. �23�

elation 23 can be interpreted using ray-parameter vectors at imageocations �Figure 2�. Angle-domain common-image gathers cane obtained by transforming prestack-migrated images using equa-ion 23:

igure 1. Geometric relations between ray vectors at a reflectionoint.

R�m,� � Þ R�m,�� . �24�

quation 23 can be written as

cos2 � =��m2� �2

4s2�m�=

� x2 + � y

2 + � z2

s2�x,y,z�, �25�

here � x,� y,� z are partial derivatives of � relative to x,y,z. We canewrite equation 25 as

cos2 � =� z

2

s2�x,y,z��1 + zx

2 + zy2� , �26�

here zx and zy denote the partial derivative of coordinate z relativeo coordinates x and y, respectively. Equation 26 describes an algo-ithm in two steps for angle decomposition after time-shift imaging:ompute cos � through a slant stack in z − � panels �find a change

n � with respect to z�; then apply a correction using the migrationlowness s and a function of the structural dips 1 + zx

2 + zy2.

MOVEOUT ANALYSIS

We can use the Kirchhoff formulation to analyze the moveout be-avior of the time-shift imaging condition in the simplest case of aat reflector in a constant-velocity medium �Figures 3–5�.The synthetic data are imaged using shot-record wavefield extrap-

lation migration. Figure 4 shows offset common-image gathers forhree different migration slownesses s, one of which is equal to the

odeling slowness s0. Figure 4a corresponds to the space-shift im-ging condition and Figure 4b to the time-shift imaging condition.

For the space-shift common-image gathers constructed with cor-ect slowness �Figure 4a�, the energy is focused at zero offset, but itpreads in a region of offsets when the slowness is wrong. Slanttacking produces the images in Figure 5a.

igure 2. Interpretation of angle decomposition based on equation3 for time-shift gathers.

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For the time-shift common-image gathers constructed with cor-ect slowness �Figure 4b�, the energy is distributed along a line withslope equal to the local velocity at the reflector position, but it

preads around this region when the slowness is wrong. Slant stack-ng produces the images in Figure 5b.

Let s0 and z0 represent the true slowness and reflector depth, and snd z stand for the corresponding quantities used in migration. Anmage is formed when the Kirchoff stacking curve t� h� =

sz2 + h2 + 2� touches the true reflection response t0� h� =

s0z0

2 + h2 �Figure 3�. Solving for h from the envelope condition

�� h� = t0�� h� yields two solutions:

igure 3. An image is formed when the Kirchoff stacking curvedashed line� touches the true reflection response. �a� The case of un-ermigration. �b� Overmigration.

igure 4. Common-image gathers for �a� space-shift imaging and �b�ime-shift imaging.

h = 0 �27�

nd

h = s02z2 − s2z0

2

s2 − s02 . �28�

ubstituting solutions 27 and 28 in the condition t� h� = t0� h� pro-uces two images in the �z,� � space. The first image is a straight line,

z�� � =z0s0 − �

s, �29�

nd the second image is a segment of the second-order curve,

z�� � = z02 +

� 2

s2 − s02 . �30�

pplying a slant-stack transformation with z = z1 − �� turns line 29nto a point �z0s0/s,1/s� in �z1,�� space, while curve 30 turns into theurve

z1��� = z01 + �2�s0

2 − s2� . �31�

he curvature of the z1��� curve at � = 0 is a clear indicator of theigration-velocity errors.By contrast, the moveout shape z�h� appearing in wave-equationigration with the lateral-shift imaging condition is

igure 5. Common-image gathers after slant stack for �a� space-shiftmaging and �b� time-shift imaging. The vertical line indicates the

igration velocity.

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z�h� = s0z02

s2 +h2

s2 − s02 . �32�

fter the slant transformation z = z1 + h tan �, the moveout curve2 turns into the curve

z1��� =z0

ss0

2 + tan2 ��s02 − s2� , �33�

hich is applicable for velocity analysis. A formal connection be-ween �-parameterization in equation 31 and �-parameterization inquation 33 is given by

tan2 � = s2�2 − 1 �34�

r

cos � =1

�s=

� z

s, �35�

here �z = ���/�z�. Equation 35 is a special case of equation 23 forat reflectors. Curves of shapes 31 and 33 are plotted on top of thexperimental moveouts in Figure 5.

TIME-SHIFT IMAGING INKIRCHHOFF MIGRATION

The imaging condition described in the preceding section has anquivalent formulation in Kirchhoff imaging. Traditional construc-ion of common-image gathers using Kirchhoff migration is repre-ented by the expression

�m,h� = �m

U�m,h,ts�m,m − h� + tr�m,m + h�� , �36�

here U�m,h,t� is the recorded wavefield at the surface as a func-ion of surface midpoint m and offset h �Figure 6�. Terms ts and tr

tand for traveltimes from sources and receivers at coordinates mh and m + h to points in the subsurface at coordinates m. For sim-

licity, the amplitude- and phase-correction term A�m,m,h�� /�t ismitted in equation 36. The time-shift imaging condition can be im-lemented in Kirchhoff imaging using a modification of equation 36hat is equivalent to equations 6 and 8:

R�m,� � = �m

�h

U�m,h,ts�m,m − h�

+ tr�m,m + h� + 2� � . �37�

Images obtained by Kirchhoff migration as discussed in equation7 differ from images constructed with equation 36. Relation 37 in-olves a double summation over surface midpoint m and offset h toroduce an image at location m. Therefore, the entire input data con-ributes potentially to every image location. This is advantageousecause migrating using relation 36 different offsets h independent-y may lead to imaging artifacts as discussed by Stolk and Symes2004�. After Kirchhoff migration using relation 37, images can beonverted to the angle domain using equation 23.

EXAMPLES

We demonstrate the imaging condition introduced in this paperith the Sigsbee 2Asynthetic model �Paffenholz et al., 2002�. Figure

shows the correct migration velocity and the image created byhot-record migration with wavefield extrapolation using the time-hift imaging condition. The image in Figure 7a is extracted at �

0.Figure 8a shows common-image gathers at locations x =

7,9,11,13,15,17� km obtained by time-shift imaging conditions.s in the preceding synthetic example, we can observe events with

inear trends at slopes corresponding to local migration velocity.ince the migration velocity is correct, the strongest events in com-on-image gathers correspond to � = 0. For comparison, Figure 8b

hows common-image gathers at the same locations obtained bypace-shift imaging condition. In the latter case, the strongest events

igure 6. Notations for Kirchhoff imaging. S is a source and R is a re-eiver.

igure 7. Sigsbee 2A model. �a� Correct velocity and �b� migratedmage obtained by shot-record wavefield extrapolation migrationith time-shift imaging condition.

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S214 Sava and Fomel

ccur at h = 0. The zero-offset images �� = 0 and h = 0� are identi-al.

Figure 9 shows the angle decomposition for the common-imageather at location x = 7 km. From left to right, the panels depict theigrated image, a common-image gather resulting from migration

igure 8. Imaging-gathers at positions x = �7,9,11,13,15,17� km. �ng condition and �b� space-shift imaging condition.

igure 9. Time-shift imaging condition gather at x = 7 km. The pan-ls depict �a� the image, �b� the time-shift gather, �c� the slant-tacked time-shift gather, and �d� the angle gather.

y wavefield extrapolation with time-shift imaging, the common-mage gather after slant stacking in the z−� plane, and an angle gath-r derived from the slant-stacked panel using equation 23.

For comparison, Figure 10 depicts a similar process for a com-on-image gather at the same location obtained by space-shift im-

ging. Despite the fact that the offset-gathers are completely differ-ent, the angle gathers are comparable, showingsimilar trends of angle-dependent reflectivity.

Figure 11a shows angle-domain common-im-age gathers for time-shift imaging at locations x= �7,9,11,13,15,17� km. Since the migration ve-locity is correct, all events are mostly flat, indicat-ing correct imaging. For comparison, Figure 11bshows angle-domain common-image gathers forspace-shift imaging condition at the same loca-tions in the image.

Finally, we illustrate the behavior of time-shiftimaging with incorrect velocity. Figure 12ashows an incorrect velocity model used to imagethe Sigsbee 2Adata, and Figure 12b shows the re-sulting image. The incorrect velocity is a smoothversion of the correct interval velocity, scaled by10% from a depth z = 5 km downward. The un-collapsed diffractors at depth z = 7 km clearly in-dicate velocity inaccuracy.

Figures 13 and 14 show image gathers and thederived angle gathers for time-shift and space-shift imaging at the same location, x = 7 km. Be-cause of incorrect velocity, focusing does not oc-cur at � = 0 or h = 0 as in the preceding case.Likewise, the reflections in angle gathers are non-flat, indicating velocity inaccuracies. CompareFigures 9 and 13 and Figures 10 and 14. Thosemoveouts can be exploited for migration-velocityanalysis �Biondi and Sava, 1999; Clapp et al.,2004; Sava and Biondi, 2004a, b�.

e-shift imag-

igure 10. Space-shift imaging condition gather at x = 7 km. Theanels depict �a� the image, �b� the space-shift gather, �c� the slant-tacked space-shift gather, and �d� the angle gather.

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Time-shift imaging S215

DISCUSSION

As discussed in a preceding section, time-shiftgathers consist of linear events with slopes corre-sponding to the local migration velocity. In con-trast, space-shift gathers consist of events fo-cused at h = 0. Those events can be mapped tothe angle domain using transformations 20 and23, respectively.

To understand the angle-domain mapping, weconsider a simple synthetic in which we modelcommon-image gathers corresponding to inci-dence at a particular angle. The experiment is de-picted in Figure 15 for time shift imaging and inFigure 16 for space-shift imaging. For this exper-iment, the sampling parameters are �z = 0.01km, �h = 0.02 km, and �� = 0.01 s.

Areflection event at a single angle of incidencemaps in common-image gathers as a line of a giv-en slope. The left panels in Figures 15 and 16show three cases corresponding to angles of 0°,20°, and 40°. Since we want to analyze how suchevents map to angle, we subsample each line tofive selected samples lining up at the correctslope.

The middle panels in Figures 15 and 16 showthe data in the left panels after slant stacking inz−� or z−h panels, respectively. Each individualsample from the common-image gathers maps ina line of a different slope intersecting in a point.For example, normal incidence in a time-shiftgather maps at the migration velocity v = 2 km/s

Figure 15, top row, middle panel�, and normal incidence in a space-hift gather maps at slant-stack parameter tan � = 0.

Figures 15f and 16f show the data from the middle panels afterapping to angle using equations 23 and 20, respectively. All lines

rom the slant-stack panels map into curves that intersect at the anglef incidence.

shift imaging

igure 13. Time-shift imaging condition gather at x = 7 km. Theanels depict �a� the image, �b� the offset gather, �c� the slant-stackedather, and �d� the angle gather. Compare with Figure 9.

igure 11. Angle gathers at positions x = �7,9,11,13,15,17� km. �a� Time-

igure 12. Sigsbee 2A model imaging shows �a� incorrect velocitynd �b� migrated image obtained by shot-record wavefield extrapo-ation migration with time-shift imaging condition. Compare with

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We note that all curves for the time-shift angle gathers have zerourvature at normal incidence. Therefore, the resolution of the time-hift mapping around normal incidence is lower than the corre-ponding space-shift resolution. However, the storage and computa-ional cost of time-shift imaging is smaller than the cost of equiva-ent space-shift imaging. The choice of the appropriate imaging con-ition depends on the imaging objective and on the trade-offetween the cost and the desired resolution.

igure 14. Space-shift imaging condition gather at x = 7 km. Theanels depict �a� the image, �b� the offset gather, �c� the slant-stackather, and �d� the angle gather. Compare with Figure 10.

igure 15. Image-gather formation using time-shift imaging. Hori-ontal rows depict events at �a� 0°, �b� 20°, and �c� 40°. Vertical col-mns correspond to subsampled �d� time-shift gathers, �e� slant-tack gathers, and �f� angle gathers.

CONCLUSIONS

We develop a new imaging condition based on time shifts be-ween source and receiver wavefields. This method is applicable toirchhoff, reverse-time, and wave-equation migrations and produc-

s common-image gathers indicative of velocity errors. In wave-quation migration, time-shift imaging is more efficient than space-hift imaging because it only involves a simple phase shift prior tohe application of the usual imaging crosscorrelation. Disk storage islso reduced, because the output volume depends on only one pa-ameter �time shift � � instead of three parameters �space shift h�. Wehow how this imaging condition can be used to construct angleathers from time-shift gathers. More research is needed on how totilize this new imaging condition for velocity and amplitudenalysis.

ACKNOWLEDGMENT

We would like to acknowledge ExxonMobil for partial financialupport of this research.

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