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Removing Smearing-Effect Artifacts in Angle-DomainCommon-Image Gathers from Reverse Time Migration

©2015 Society of Exploration Geophysicists

Citation:

Jin, Hu, and George A. McMechan. 2015. "Removing smearing-effect artifacts in angle-domain common-image gathers from reverse time migration." Geophysics 80(3), doi:10.1190/GEO2014-0210.1

Removing smearing-effect artifacts in angle-domain common-imagegathers from reverse time migration

Hu Jin1 and George A. McMechan2

ABSTRACT

Local plane-wave decomposition (LPWD) and local shift im-aging condition (LSIC) methods for extracting angle-domaincommon-image gathers (ADCIGs) from prestack reverse timemigration are based on the local plane-wave assumption, andboth suffer from a trade-off in choosing the local window size.Small windows produce clean ADCIGs, but with low angle res-olution, whereas large windows produce noisy ADCIGs, whichinclude smearing-effect artifacts, but with high angle resolution.The cause of the smearing-effect artifacts in LPWD is the cross-correlation of plane waves obtained by decomposition of thesource and receiver wavefronts, at points that do not lie on

the source wavefront excitation time trajectory. The cause of thesmearing-effect artifacts in LSIC is the decomposition of curvedevents of offset-domain common-image gathers (ODCIGs) atincorrect depth points at zero offset. These artifacts can occureven if the migration velocity model is correct. Two methodswere proposed to remove the artifacts. In the LPWD method,the smearing-effect artifacts were removed by decomposing andcrosscorrelating the resulting source and receiver plane wavesonly at image points and excitation (image) times. In the LSICmethod, the artifacts were removed by decomposing curvedevents in ODCIGs into planar events only at zero-offset targetimage points. Numerical tests with synthetic data revealed thesuccess of the proposed methods.

INTRODUCTION

Angle-domain common-image gathers (ADCIGs) from prestackreverse time migration (RTM) are a key component of wave-equa-tion migration velocity analysis. Reflection events in ADCIGs areflat when a correct velocity model is used for migration, curvedup when the migration velocity model is slower, or curved downwhen the migration velocity model is faster than the correct model(Figure 1). Velocity updates are related to the curvature of theevents, as defined by picking residual depth moveouts (Xie andYang, 2008b; Zhang and McMechan, 2013), or by differential sem-blance optimization (Shen and Symes, 2008).Several methods are proposed to extract the ADCIGs from RTM.

They fall into three main categories (Jin et al., 2014). The first cat-egory is direction-vector-based (DVB) methods, which are based onthe first spatial and time derivatives of the wavefield amplitudes (thePoynting-vector-based method or the polarization-vector-basedmethod) (Yoon and Marfurt, 2006; Dickens and Winbow, 2011;

Zhang and McMechan, 2011a, 2011b), or of the instantaneousphase of the wavefield (Zhang and McMechan, 2011b; Jin et al.,2014). These methods are fast, but they are not stable for compli-cated wavefields that contain overlapping events because they cal-culate only one propagation direction per image point per time step(Vyas et al., 2011).The second category is local plane-wave decomposition

(LPWD), which decomposes source and receiver wavefronts usinglocal spatial windows and calculating propagation angles of the lo-cal wavefields by Fourier transforms (FTs) (Xie and Wu, 2002; Xuet al., 2011) or slant-stack transforms (SSs) (Xie and Yang, 2008a;Yan and Xie, 2011, 2012) based on an assumption that the wave-fields are locally planar. The third category is the local-shift imagingcondition (LSIC), which first produces subsurface-offset-domaincommon-image gathers (ODCIGs) by crosscorrelating the wave-fields that are shifted horizontally and/or vertically with respect toeach other and then transforming the ODCIGs into ADCIGs by FTs(Sava and Fomel, 2003, 2005a, 2005b) or by SSs (Sava and Fomel,

Manuscript received by the Editor 5 May 2014; revised manuscript received 2 November 2014; published online 17 March 2015.1Formerly The University of Texas at Dallas, Center for Lithospheric Studies, Richardson, Texas, USA; presently BPAmerica, Inc., Houston, Texas, USA. E-

mail: hu.jin@bp.com.2The University of Texas at Dallas, Center for Lithospheric Studies, Richardson, Texas, USA. E-mail: mcmec@utdallas.edu.© 2015 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 80, NO. 3 (MAY-JUNE 2015); P. U13–U24, 13 FIGS.10.1190/GEO2014-0210.1

2003; Biondi and Symes, 2004). This is also based on an assump-tion that the events in the ODCIGs are locally planar.The LPWD and LSIC methods assume local plane waves, and so

they produce very similar results if they are implemented with thesame window size (Jin et al., 2014). When the windows are large (orwithout using local windows, by transforming each individual OD-CIG in the LSIC [Biondi and Symes, 2004] or transforming wholesource and receiver wavefields in the LPWD [Jin et al., 2014]), thelocal wavefields (in the LPWD) and the events in the ODCIGs (inthe LSIC) are less likely to be straight, so smearing-effect artifacts(Jin et al., 2014) are generated. Figure 2 shows ADCIGs that con-tain smearing-effect artifacts (most of the nonflat events). The AD-CIGs in Figure 2a–2d, and in the bottom panel of Figure 2e areobtained using the LSIC method; those in the upper panel ofFigure 2e are obtained using the LPWD method. All are syntheticexamples and are migrated using a smoothed version of correctvelocity models, so the events in the ADCIGs are expected to beflat. However, some events are curved up or down, especially atsalt boundaries, which look like (but are not) the velocity-dependent

moveouts illustrated in Figure 1. Consequently, current migrationvelocity analysis algorithms (Shen and Symes, 2008; Xie and Yang,2008b; Zhang and Biondi, 2013) or automatic picking of moveouts(Siliqi et al., 2007; Siliqi and Talaalout, 2009; Liu and Han, 2010;Panizzardi et al., 2011) face challenges because of the smearing-effect artifacts. Stolk and Symes (2004) note that different datapartitions (e.g., common-source, common-receiver, plane-wave,or offset) produce angle gathers with different moveouts. Differentextrapolators (e.g., Kirchhoff, two-way wave equations, or one-waywave equations) (Stolk and de Hoop, 2002; Stolk and Symes, 2002;Stolk et al., 2009; Fomel, 2011) also generate different character-istic artifacts. Neglect of multipathing also produces artifacts asso-ciated with the imaging of such waves in the wrong place. Theartifacts that we consider below are produced by crosscorrelationof plane waves in the source and receiver wavefield decompositions(in the LPWD algorithm) and those produced by the plane-wavedecompositions of ODCIGs at zero offset (in the LSIC algorithm).On the other hand, when the windows are small (with a size of a

few wavelengths at the dominant frequency [Xie and Wu, 2002; Xuet al., 2011]), the ADCIGs from the LPWD andthe LSIC are cleaner because a small window ismore likely to satisfy the local plane-wave as-sumption, but the angle resolution is low (Jinet al., 2014). The low angle resolution does notprovide reliable moveouts at far angles (as willbe shown below). In the LSIC method, smallsubsurface offsets can be considered as small lo-cal windows.To achieve high angle resolution in ADCIGs,

large windows are needed in the LPWD andLSIC methods. Thus, it is important to solve theproblem of the smearing-effect artifacts that arealso a consequence of large windows. The trade-off between angle resolution and artifacts as afunction of the window size has been an issue formore than a decade (e.g., Xie and Wu, 2002; Yanand Xie, 2012; Jin et al., 2014), and it remainsunsolved; we provide some insights below.We first demonstrate the importance of high

angle resolution and the causes of the smear-ing-effect artifacts in the LPWD and LSIC. Then,we propose separate solutions for the LPWDand LSIC to remove the smearing-effect artifactsusing large windows, to generate ADCIGs thathave high angle resolution and are free of thesmearing-effect artifacts.

ANGLE RESOLUTION

Jin et al. (2014) illustrate the fact that angleresolution in ADCIGs depends on the range ofwavenumbers, rather than the wavenumber sam-pling (Xu et al., 2011). Below, we show that therange of wavenumbers is directly quantifiable interms of the Fresnel zone; the latter provides amore useful physical explanation. Zero paddingor wavenumber-domain oversampling (Xu et al.,2010) increases the wavenumber sampling, but itdoes not increase the angle resolution (Jin et al.,2014); the angle resolution (via the width of the

Reflection angles

Dep

th

Reflection angles

Dep

th

Reflection angles

Dep

th

b)a) c)

Figure 1. ADCIGs migrated with (a) a correct velocity model, (b) a velocity model thatis higher than the correct velocity model, and (c) a velocity model that is lower than thecorrect velocity model. Compare with the artifacts in Figure 2.

a) b) c) d) e)

(Dickens and Winbow, 2011)

SEAM model

(Yoon et al., 2011)

2D BP TTI data set

(Xu et al., 2011)

2D Sigsbee data set

(Jin et al., 2014)

Circular target

(Zhang and Sun, 2009)

2004 BP 2D data set

Figure 2. These parts of RTM ADCIGs are from five recent papers. The ADCIGs con-tain the smearing-effect artifacts, even though they are migrated with correct velocitymodels and use synthetic data. The transparent red polygons represent salt bodies.

U14 Jin and McMechan

Fresnel zone) depends only on the window size before zero pad-ding; large windows produce high angle-resolution ADCIGs, andsmall windows produce low angle-resolution ADCIGs (Jin et al.,2014).Although large windows produce high angle-resolution ADCIGs,

they are not always preferred because the large windowimplementation increases the computation time of the LPWD (Xuet al., 2011), and it also increases the smearing-effect artifactsbecause wavefronts in large windows are less likely to satisfy theplane-wave assumption (Jin et al., 2014). Then why do we need thelarge windows in the LPWD and LSIC methods? Also, why is theangle resolution important in ADCIGs? To answer these questions,we need to compare the ADCIGs obtained with different windowsizes (and hence, different angle resolutions).The first of the following three subsections defines angle reso-

lution and its distinction from the smearing artifacts. The secondsubsection describes the relation between the Fresnel zone, angleresolution, and smearing artifacts in the content of the window size,in data from a single source. The third subsection illustrates the im-portance of angle resolution by showing the ADCIGs using multi-ple sources, as a function of window size.

Definition of angle resolution

The two panels in Figure 3 show source wavefronts with the samefrequency and wavelength at depth D, but with different radii ofcurvature. The propagation angle resolution at depth D for thehigh-curvature wavefront in Figure 3a is α1, and it is α2 in the low-curvature wavefront in Figure 3b. These angle apertures are definedby the two plane waves from the plane-wave decomposition (theblue lines) that are tangent to the wavefronts and that pass throughthe black dots that are one-half cycle away (along the plane waves)from the target point (at depth D). The angle resolution is inverselyproportional to the wavefront curvature and to the wavelength. Forexample, a plane wave has an infinitely large radius of curvature, sothe angle resolution is the highest. For a constant curvature, the an-gle resolution will be higher for smaller wavelengths.The horizontal dashed lines labeled F in Figure 3 denote the

diameter of the corresponding Fresnel zones. At depths other thanD, such as A in Figure 3a, there will be plane-wave contributions(i.e., the red solid line in Figure 3a) which, when crosscorrelatedwith the plane waves of the receiver wavefields, produce the smear-ing artifacts which are trajectories in the reflection angle and depthplane as described in the following sections. So, the angle resolutionin ADCIGs corresponds to the image extension over reflection an-gles at a single depth, and the smearing-effect artifacts correspondto the image extension over incident/reflection angles at the incor-rect depths.

Relation between angle resolution, smearing-effectartifacts, and the effect of window size

To illustrate the relation between the angle resolution, smearing-effect artifacts, and the effect of window size, we generate syntheticdata from a simple velocity model (Figure 4), with a 2D scalarfinite-difference extrapolator (eighth order in space and second or-der in time). The grid spacing in the vertical and the horizontal di-mensions is 10 m, and the time sample increment is 1 ms. The 100sources are located from the horizontal position 0.0 to 3.0 km at thetop of the model, with 30 m spacing. The 300 receivers are located

along the entire surface of the model, with 20 m spacing. The re-cording geometry is a fixed spread. The source is a Ricker waveletwith 20-Hz dominant frequency, so the dominant wavelength at theaverage velocity of 2.0 km∕s is ∼0.1 km (10 grid points). For mi-gration, source and receiver wavefields are extrapolated through asmoothed version of the correct velocity model in Figure 4.

In Figure 5, the ADCIGs for a single source, obtained from theLSIC (the upper panels) and the LPWD (the lower panels) are al-most the same if the same window size is used, except that the angleresolution of the LPWD is a little lower because of convolution overthe source and receiver wavenumbers. Because the ADCIGs arefrom a single source, they are expected to contain a single event(with a small spread in angles, related to the local Fresnel zone)for each image point. Events of high angle resolution have smallerspreads in angles than those of low angle resolution. The angle res-olution for both methods increases with the window size before zeropadding (from 30 × 30 to 150 × 150 grid points) or wavenumbersampling. However, for the LSIC and LPWD, the wavenumbersampling increment (using the same the window size after zero pad-ding to ∼600 × 600 grid points) and the corresponding angle binsampling increment are the same (1°) (see Appendix A).The angle resolution does not continuously increase with increas-

ing window size beyond the point at which it is equal to, or larger

a) b)

D

A

F F

Figure 3. The two wavefronts in panels (a and b) have the samewavelength at the same depths D. The wavefront in panel (a) hashigher curvature, and the corresponding angle resolution is lower(α1 > α2). The two square windows (a and b) represent a local spa-tial window at one time step. Point A and the blue and red lines aredescribed in the text. The lengths of the dashed lines labeled F arethe diameters of the Fresnel zones.

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2.1 km/s

2.2 km/s

2.4 km/s

2.6 km/s

Figure 4. A simple velocity model contains four flat layers. Theblack dot, at horizontal position 2.1 km, represents a source posi-tion. The vertical dashed line, at horizontal position 3.0 km, repre-sents a CIP position.

Smearing-effect artifacts in ADCIGs U15

than, the Fresnel zone. For example, for the first-layer reflection(which has a Fresnel zone width of ∼54 grid points), no increaseof angle resolution is seen for the window sizes >60 × 60 gridpoints in Figure 5a and 5b. For the second- and third-layer reflec-tions (which have Fresnel zone widths of ∼70 grid points and ∼84grid points, respectively), no increase is seen in Figure 5a and 5b forwindow sizes >90 × 90 grid points. Each frequency has a differentFresnel zone width, so there is a corresponding range of windowsizes.A larger Fresnel zone width does not necessarily mean higher

angle resolution. The Fresnel zone width just indicates the maxi-mum angle resolution, which can be achieved when the width isthe same as the window size. The angle resolution will be high forsmall wavelengths and less curved wavefronts. Similarly, the am-plitudes of the images in the ADCIGs increase with the window sizewhen the window size is smaller than the Fresnel zone, but they donot change when the window size is larger than the Fresnel zone(Figure 5).On the other hand, as the local window size increases, the wave-

fronts in the local windows are less likely to satisfy the local plane-wave assumption and consequently, the smearing-effect artifacts in-crease in terms of their amplitudes and the ranges of slopes of thefour “legs” (marked by the black arrows in Figure 5b). The artifactsin the shallowest reflection event are stronger because the wavefrontat the shallowest reflector has a smaller radius of curvature than

those at the deeper reflectors. The artifacts are more obvious (whenthe window sizes are large) for more complicated velocity models,especially at salt boundaries, which locally generate wavefrontswith small radii of curvature; e.g., Jin et al. (2014) show smearing-effect artifacts in small and large window implementations. Thecurvatures (up or down) and the amplitudes of the smearing-effectartifacts (in ADCIGs for all sources) are model and data dependent;they depend on the combined effects of the local window size, re-ceiver, and source geometries, the wavefields in the local windows,and the velocity errors.

Importance of angle resolution

When the ADCIGs are stacked over all 100 sources, those thatare migrated with the correct velocity are flat and look the same forall window sizes (see the upper panels in Figure 6) because the lowangle-resolution events spread horizontally. However, when the mi-gration velocity model is not correct, e.g., if it is slower than thecorrect model, the events will be curved up, and different windowsizes (and thus, the different angle resolutions) produce differentresults (see the lower panels in Figure 6). A low angle resolutionprovides reliable angle-dependent residual moveouts only at thesmall reflection angles, but not at the large reflection angles, so thevelocity cannot be updated effectively because the residual move-outs at large angles are more sensitive to velocity errors. This isbecause a small subsurface offset window will mute the larger sub-

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30 × 30 60 × 60 120 × 120 90 × 90 150 × 150

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D

Figure 5. ADCIGs for a single source at a CIP position marked by the dashed line in Figure 4, The upper panels are from the LSIC (FT)method, and the lower panels are from the LPWD method. The numbers below the lower panels are the horizontal and vertical window size ingrid points. The arrows indicate the smearing-effect artifacts.

U16 Jin and McMechan

surface offsets that tend to include energy from large offsets (orscattering angles) when the velocity is inaccurate. This simple testdemonstrates the importance of high angle resolution in ADCIGs,for which a large window size of at least six to nine wavelengths isrequired at the dominant frequency. The smearing-effect artifactsare not evident (Figure 6) because the artifacts at each image pointadd destructively in the ADCIGs for all sources, in this simplevelocity model.

SMEARING-EFFECT ARTIFACTS IN LOCALPLANE-WAVE DECOMPOSITION

Source and receiver wavefields that are decomposed into planewaves using large windows are more likely to be curved. In theLPWD implemented with FTs, a curved wavefront is decomposedinto a series of plane waves, each with a slope and an amplitude(Figure 7a). The LPWD (FT) method (Xu et al., 2011) is imple-mented by convolution of the source and receiver wavefields inthe frequency/wavenumber domain, which are obtained by apply-ing 3D FTs (for 2D wavefields within local windows) in the time/space domain. Within the loops over the frequencies and across lo-cal windows, the wavenumber-domain convolution multiplies theamplitude of each point in the source wavenumber-domain wave-field with the amplitudes of all points in the receiver wavenumber-domain wavefield. Angle estimations are made from each source

and receiver wavenumber pair. Thus, in the corresponding time/space domain (Figure 7a), within the loops over the image time andacross local windows, a decomposed source plane wave (the thinblue straight lines) is multiplied (point-by-point) with all the decom-posed receiver plane waves (the thin green straight lines) in the localspatial window. The smearing-effect artifacts result from the cross-correlation of the decomposed source and receiver plane waves atdepths of nonphysical reflectors, such as at representative points Aand B in Figure 7a and 7b, which do not lie on the undecomposedsource wavefront (the source excitation image time trajectory), sopoints A and B are not at image depth C, but are spread over in-correct image depths as the incident/reflection angles get fartherfrom the imaged specular reflection at C. Points A and B are con-tributions to the ADCIG from one source at one time. The completeartifact trajectory in Figure 7b is the sum over all times. The com-plete ADCIG is a second sum over all sources. The smearing-effectartifacts in the ADCIGs appear over continuously varying depths,reflection angles, and amplitudes of the plane waves that result fromthe decompositions of the source and receiver wavefronts.In the LPWD implemented with SSs (Yan and Xie, 2009, 2010,

2012), the curved wavefield is decomposed into a series of planewaves by SS. The method is (mathematically) equivalent to the FTimplementation (Jin et al., 2014), and so it also suffers from thesmearing-effect artifacts when the window size is large, for the samereason as in the LPWD (with FTs) method does. Having determined

30 × 30 90 × 9060 × 60 120 × 120 150 ×1 50

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Figure 6. ADCIGs with contributions from 100 sources at one CIP position marked by the dashed line in Figure 4. The upper panels aremigrated with the correct velocity model, and the lower panels are migrated with an incorrect (low) velocity model. The numbers below thelower panels are the (horizontal × vertical) window size, in grid points.

Smearing-effect artifacts in ADCIGs U17

the causes of the smearing-effect artifacts, we can now propose asolution to remove them by constraining the decomposition of thesource and receiver plane waves to be only at the points on thesource excitation time trajectory (Figure 7c).The migration is implemented as the variant of prestack RTM, of

a common-source gather, that uses the excitation amplitude imagingcondition (Deng and McMechan, 2007; Nguyen and McMechan,2009, 2013, 2015), as follows. During the source wavefield extrapo-lation, saving all the source time snapshots (as in the conventionalcrosscorrelation implementation) is replaced with a single-valuedtrajectory of amplitudes at the points that satisfy the imaging con-dition at that time, and these are saved in a single array that has thesame dimensions as the migrated image. The image time at eachpoint is defined as the time at which the maximum amplitude occursin the source wavefield as it passes through that point. Then, at eachtime step during the reverse time propagation of the receiver wave-field, the amplitudes in the receiver wavefield are multiplied, point-by-point by the amplitudes in the precomputed source array at thepoints that are imaged at that time; this is a crosscorrelation imagingcondition that minimizes storage and computational requirements.The main modification required for the present context is that the

propagation angles also need to be computed at the points that areimaged at each time step. Multipathing is included by saving morethan one excitation time and amplitude at each point during thesource extrapolation, and applying them independently to thepropagating receiver wavefield, as shown by Jin et al. (2015) andby Nguyen and McMechan (2015).We find multiple excitation (image) times in the source wavefield

at each grid point by picking multiple highest amplitude arrivals,which can be local peaks. Two events picked at one grid pointshould have a minimum time difference of the time duration of awavelet (Yoon et al., 2011) to make each a separate reflection event,rather than local peaks within a single event (a wavelet). If themodel is complicated, there may be multipaths, and so more arrivalswill needed to be picked at each grid point to include them. The Npexcitation times at all grid points TsMAX

ðNx;Ny; Nz;NpÞ, and thecorresponding Np highest amplitudes and propagation angles aresaved into disk storage. Here, Nx, Ny, and Nz are the numbersof grid points along x-, y-, and z-directions, respectively. This partis the first step of the multiple-excitation imaging condition(MEXIC) developed by Jin et al. (2015).At each time step t, the source wavefield is decomposed into

plane waves by SSs in the time/space domain (Yan and Xie,2009, 2010), using

Usðθs; x; tÞ ¼Xx 0

Us½θs; x 0; tþ ps · ðx 0 − xÞ�: (1)

Similarly, at each time step during the receiver backward extrapo-lation in time, the receiver wavefield is decomposed using

Urðθr; x; tÞ ¼Xx 0

Ur½θr; x 0; tþ pr · ðx 0 − xÞ�: (2)

In equations 1 and 2, Us and Ur are the source and receiver wave-field amplitudes; respectively, at the spatial location x where thewavefield is decomposed. Here, x 0 represents every grid point ina local space window, and ps and pr are the source and receiverslowness vectors, associated with the propagation angles θs andθr, respectively. In this method, we do not decompose the sourceand receiver wavefields at every grid point and every time (Yan andXie, 2009, 2010), but only at the points along the excitation (image)time trajectory. So, x in equations 1 and 2 represents only the gridpoints, where one excitation time TsMAX

ðNx;Ny;Nz;NpÞ is equalto the time t (i.e., the points along the blue curved wavefield in thewindow at that time in Figure 7c). The maximum amplitude con-tribution (at C in Figure 7b) occurs at the image time when the twothick curved blue and green (source and receiver) wavefronts (inFigure 7a and 7c) coincide at the common-image point (CIP) posi-tion; this also corresponds to the coincidence of the two thick (blueand green) plane waves that are tangent to their correspondingwavefronts at the same point. The coincidence of plane-wave posi-tions at other times (such as that in Figure 7a and 7b) contribute tothe smearing artifacts at points, such as at A and B. The angle-domain image Iðθ; xÞ in Figure 7b is obtained by crosscorrelatingthe decomposed source and the receiver wavefields (Yan and Xie,2009, 2010), but again, we apply it only along the image pointtrajectory, where the time t is equal to the excitation timeTsMAX

ðNx;Ny;Nz;NpÞ

CIP positiona)

c)

°)0° 60°

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A

B

b)

C

Figure 7. (a) Crosscorrelation of the source (the thick curved blueline) and receiver (the thick curved green line) spherical wavefrontsand decomposed (thin straight) plane waves in the LPWD method,(b) the ADCIG contains the smearing-effect artifacts. Representa-tive contributions at A and B are explained in the text. Panel(c) shows the source and receiver wavefield decomposition con-strained to the points along the source wavefront excitation timetrajectory (the blue curved line) in the LPWD-MEXmethod. In pan-els (a and c), the thick straight blue and green plane waves (tangentto the source and receiver wavefronts) correspond to the dominantcontribution at the CIP position.

U18 Jin and McMechan

Iðθ; xÞ ¼Xt

Xθs

Xθr

Usðθs; x; tÞUrðθr; x; tÞ; (3)

where θ ¼ ðθs − θrÞ∕2 is the reflection angle. We call this newmethod LPWD with multiple-excitation imaging conditions(LPWD-MEX). The MEXIC multiplies the amplitudes of thereceiver wavefields with the amplitudes along the trajectory of thesource spikes that are extracted at each time step during the sourceforward extrapolation. Thus, the wavelengths of the reflectionevents in the images are approximately half of those obtained bycrosscorrelation of the source and receiver wavefields (Chattopad-hyay and McMechan, 2008).This method is as expensive as the LPWD (FT) method because

the wavefield decomposition and convolution are applied in loopsof θs (from 0° to 360°), θr (from 0° to 360°) and t (see equations 1, 2,and 3). To speed up the computation, we use the Poynting vector-based method (Yoon and Marfurt, 2006; Dickens and Winbow,2011) to calculate approximate reference source and receiver propa-gation angles, which are then used to constrain the scanning rangesof the angles in equations 1 and 2, instead of using all angles in therange from 0° to 360°. If the model is complicated, the Poyntingvector-based method is less reliable (Vyas et al., 2011; Jin et al.,2014), so the scanning ranges should be larger than those for a sim-ple velocity model.

SMEARING-EFFECT ARTIFACTS IN THE LOCALSHIFT IMAGING CONDITION

The LSIC method involves two steps (Sava and Fomel, 2003,2005a, 2005b; Biondi and Symes, 2004; Fomel, 2004, 2011). Thefirst step generates horizontal-subsurface offset-domain common-image gathers (HOCIGs)

Iðx; z; hxÞ ¼Xs

Xω

Usðx − hx; z;ωÞU�rðxþ hx; z;ωÞ; (4)

where hx is the subsurface horizontal shift, ω is the angular fre-quency, and * represents the conjugate of a complex value. In thesecond step, each HOCIG Iðx; z; hxÞ is transformed to the wave-number-domain HOCIG Iðx; kz; khxÞ by applying 2D FTs, fromwhich reflection angles are calculated from each pair of subsurfacehorizontal-offset wavenumber khx and depth wavenumber kz (Savaand Fomel, 2003, 2005a, 2005b) using

tan θ ¼ khxkz

: (5)

Then, the wavenumber-domain images Iðx; kz; khx ; θÞ are trans-formed back to space-domain images Iðx; z; θÞ by inverse 2DFTs. Alternatively, the reflection angles can be calculated by apply-ing SSs (Schultz and Claerbout, 1978) to each HOCIG Iðx; z; hxÞ todirectly calculate the reflection angles from slopes of plane waves(Sava and Fomel, 2003; Biondi and Symes, 2004). The SSs alter thewaveforms with phase shifts (Luo et al., 2010; Jin et al., 2014), sohere, we use FTs instead of SSs when transforming from HOCIGsto ADCIGs.The local plane-wave assumption occurs in the first step. If the

source and the receiver wavefronts for a single source in a localwindow are perfectly planar (Figure 8a), the relationship betweenthe horizontal shifts and the image depths at a CIP position is linear,

and the event in the HOCIG is straight (Figure 8b). The slope of thestraight event is the reflection angle, which can be calculated by anFT or a SS. However, when the wavefronts are curved (or even ifonly one wavefront is curved; Figure 8c), the event in the HOCIGwill not be straight (Figure 8d). This is very common for largewindows.In the second step, when transforming the HOCIG to the ADCIG,

a straight event (Figure 8b) will be transformed to a single imageevent in the ADCIGs (e.g., the point A in Figure 9b), and a curvedevent is decomposed into a series of the planar events (Figure 9a)by an FT or a SS; the smearing-effect artifacts result from the de-composed planar events at zero offset, such as at points B, C, and Din Figure 9a and 9b. They usually appear at continuously varyingdepths, reflection angles, and amplitudes of the plane waves (Fig-ure 9b) that result from decomposition of the HOCIGs (Figure 9a).The solution to remove the smearing-effect artifacts in the LSIC

is similar to that in the LPWD, but this is now done in the postmi-grated (HOCIG) domain. We constrain the decomposition of thecurved event (Figure 9a) to be only at the depth point in thecorresponding image (Figure 9c), at zero offset in each HOCIGfor each source. Locating the target image events is automatedby searching along the depth vector at zero offset (Figure 9a). Lo-cating the events in the HOCIGs and transforming them to ADCIGsare done for each source independently because the image depthsfor each location, migrated from different sources, are differentwhen the migration velocity model is incorrect. We call this new

Dep

thD

epth

a)

c)

b)

d)

CIP position

Figure 8. (a) The crosscorrelation of the horizontally shifted sourceand receiver plane waves for one source, (b) the corresponding HO-CIG at the CIP position marked by the solid vertical line, (c) thecrosscorrelation of the horizontally shifted source spherical andreceiver plane waves, and (d) the corresponding HOCIG at the CIPposition. The blue lines represent the source wavefields, and thegreen lines represent the receiver wavefield.

Smearing-effect artifacts in ADCIGs U19

method target-oriented LSIC (or T-LSIC). The implementation ofthe T-LSIC is the same as for the LSIC (FT) method (Sava and Fo-mel, 2005a, 2005b), except that z in Iðx; z; θÞ represents every im-age depth sample in the LISC (FT) method, whereas in the proposedT-LSIC method, z in Iðx; z; θÞ represents only the image depths ofthe most prominent events at zero offset in the HOCIGs for eachsource (Figure 8d), and the plane-wave decomposition (Figure 9c)is done only for those target image points; Iðx; z; θÞ is assigned thevalue of zero at depths at which there are no image events.

NUMERICAL EXAMPLE

The two new methods, LPWD-MEX and T-LSIC are tested onsynthetic data for the velocity model in Figure 10 to compare theresults with those previously obtained by the LPWD and LSICmethods, for the same model by Jin et al. (2014). A 2D scalar fi-nite-difference extrapolator that is eighth order in space and secondorder in time is used with grid spacing of 10 m in the vertical andhorizontal directions, with a time sample increment of 1 ms. The250 sources used are located from horizontal positions 2.5 to

7.5 km at the top of the model with spacing of 20 m; the 500 receiv-ers used are located over the entire surface of the model with spac-ing of 20 m. Each source is recorded on all receivers. The source is aRicker wavelet with 20-Hz dominant frequency, so the dominantwavelength is ∼0.1 km (10 grid points), and the length of Fresnelzone at the dipping reflector is ∼70 grid points, and ∼60–85 gridpoints at the circular velocity anomaly. The green box markedin Figure 10 is the area in which the ADCIGs are displayed inFigure 11. For migration, the source and receiver wavefields areextrapolated through a smoothed version of the correct velocitymodel in Figure 10.Figure 11 shows representative ADCIGs (left panels) and the cor-

responding stacked images (right panels) obtained from the twonew methods (LPWD-MEX and T-LSIC) as well as three previousmethods (the Poynting-vector-based method and the LSIC with FTand SS). The results from the LPWD (FT) method are not shownhere because they are very similar to those from the LSIC (FT)method, which is shown in Figure 5, and by Jin et al. (2014).

All the ADCIGs in Figure 11b–11e, are obtained with the samelarge window size, which has 64 × 64 grid points (∼6 wavelengths

at the dominant frequency), so the correspondingangle resolutions are all high. The two new meth-ods, the LPWD-MEX (Figure 11d) and the T-LSIC (Figure 11e) remove the smearing-effectartifacts that are previously seen in the LSICmethods implemented by FTs (Figure 11b) andSSs (Figure 11c).In the LSIC (SS) method (Figure 11c) and

the LPWD-MEX (Figure 11d), the phase of thewaveforms is shifted by π∕2 because of the SSimplementation (Phinney et al., 1981; Durraniand Bisset, 1984; Tatham, 1984); the formergives inaccurate phases because the SS used inthe LSIC method is a radial trace transform,which stacks amplitudes only at the points alonga search line, whereas the LPWD-MEX stacksamplitudes at all the points in a local window(equations 1 and 2), and so it is more accurate(Figure 11d). The ADCIGs obtained from thePoynting-vector-based method (Figure 11a) lookthe best, but the reflection angles and the ampli-tudes of the images migrated from overlappingevents are not reliable (Jin et al., 2014).

DISCUSSION

The LPWD-MEX and T-LSIC methods not only remove smear-ing-effect artifacts, but also reduce computation time. The com-putational performance of the LPWD-MEX is improved by decom-posing and crosscorrelating only at the points along the sourceexcitation time trajectory, and also by incorporating initial angleestimates from Poynting vectors to limit the angle-scan range whendecomposing the wavefield into plane waves. The actual computa-tion time depends on the window size, the model size, structural andvelocity complexity, as well as scan range of angles, and so it is datadependent and not easy to quantify. The computational performanceof the T-LSIC is improved by decomposing the events in the HO-CIGs only at the target image depths; this may lose some small-am-plitude events, depending on the amplitude threshold in the picking

-offset 0 +offset +offset

Dep

th

-offset 0

Dep

th

b)Reflection angle (°)

0° 60°

ABC

D

c)a)

Figure 9. (a) The curved event (the green line) in the HOCIG for one source is decom-posed into the planar events (the black lines) in the LSIC method and (b) the ADCIGcontains the smearing-effect artifacts. In panels (a and b), A is the correct image point,and B, C, and D are the smearing-effect artifacts. In panel (c), the decomposition of thecurved event in the HOCIG (a) is constrained to be at the image point in the target-oriented LSIC (T-LSIC) method.

0.0

1.0

2.0

3.0

4.0

0.0 2.0 4.0 6.0 8.0 10.0

Dep

th (

km)

Horizontal position (km)

2.2 km/s

2.0 km/s

3.4 km/s

Figure 10. The velocity model used to generate the synthetic data.The ADCIGs in later figures are displayed for the area in the greenbox. The black dot, at the horizontal position 5.0 km, is a represen-tative source position.

U20 Jin and McMechan

0.0

1.0

2.0

3.0

4.0

Dep

th (

km)

a)

b)

c)

d)

e)

'tmpa.bin' binary array=(2940,400) format="%float"

0.0

1.0

2.0

3.0

4.0

Dep

th (

km)

'tmpb.bin' binary array=(2940,400) format="%float"

0.0

1.0

2.0

3.0

4.0

Dep

th (

km)

'tmpc.bin' binary array=(2940,400) format="%float"

0.0

1.0

2.0

3.0

4.0

Dep

th (

km)

'tmpd.bin' binary array=(2940,400) format="%float"

0.0

1.0

2.0

3.0

4.0

Dep

th (

km)

'tmpe.bin' binary array=(2940,400) format="%float"

0° 60°

DV

B

ADCIGs RTM stacked images

LSIC

(F

T)

T-L

SIC

LSIC

(S

S)

LPW

D-M

EX

(S

S)

Figure 11. The ADCIGs (the left panels) are obtained from the (a) Poynting-vector-based method (a DVB method), (b) LSIC (FT) method,(c) LSIC (SS) method, (d) LPWD-MEX method, and (e) T-LSIC method. The right panels show the corresponding images, by stacking theADCIGs.

Smearing-effect artifacts in ADCIGs U21

algorithm, but it has a negligible effect on migration velocity analy-sis, which uses only the dominant reflectors.The T-LSIC method cannot remove the parts of smearing-effect

artifacts that appear at the same depths as the target image points,such as the artifacts in the black oval in Figure 12b, which are at thesame depth as the upper image point in Figure 12c. The smearing-effect artifacts in the black oval are contributions from plane waves(e.g., the representative plane wave marked by the black line in Fig-ure 12c) that are decomposed from the lower green curved event(Figure 12c). These artifacts cannot be removed because this planewave also passes through the upper image point (Figure 12c) at zerooffset. So, in the areas where there are a sequence of thin sediments,the T-LSIC method may not be applicable; e.g., the smearing-effectartifacts in the thin sediments above the top of the salt in Figure 2dcannot be removed with the T-LSIC method. In these areas, theLPWD-MEX will be more successful.The extensions of the LPWD-MEX and the T-LSIC methods to

3D are similar to those of the LPWD (Xu et al., 2011) and the LSIC(Fomel, 2004), and the computation time of the LPWD-MEX andthe T-LSIC methods in 3D will be very expensive. Amplitudes inthe ADCIGs are not explicitly, quantitatively considered here; onlydepth residuals are needed for velocity updates, but amplitudeswould also be needed for subsequent application of AVA analysis,together with true-amplitude RTM (Deng and McMechan, 2007;Zhang and Sun, 2009; Qin and McGarry, 2013), by source normali-zation of the imaging conditions and by compensating the attenu-ation and transmission loss (Deng and McMechan, 2007). These arebeyond the scope of the present paper.An elastic LPWD is developed (in the frequency domain) by Yan

and Xie (2012), which decomposes receiver P- and S-waves intoplane waves during elastic extrapolations and separation by diver-gence and curl. Elastic LPWD-MEX may follow the procedures ofthe elastic LPWD, requiring additional picking of source P-waveexcitation (image) times. Elastic T-LSIC may follow the proceduresof the elastic LSIC (FT) (Yan and Sava, 2008), where target imagepoints should be picked at zero offset of HOCIGs migrated by thecrosscorrelation of pure wave modes. The LPWD-MEX and T-LSIC methods calculate phase propagation angles, which are per-

pendicular to the local plane waves. So, both can be directly appliedto anisotropic media if ADCIGs are binned by phase angles.In the LPWD-MEX method, the picking of a few highest ampli-

tude arrivals is done in the source wavefields, so this step is notaffected by the presence of noise in the recorded data. In the T-LSICmethod, picking of target image points is done in HOCIGs for asingle source, which may be affected by the presence of coherent,incoherent, or random noise. Angle-domain decomposition of thereceiver wavefields (equation 2) in the LPWD-MEX and transform-ing HOCIGs to ADCIGs in the T-LSIC method may be affected bycoherent noise, but not by incoherent or random noise because suchnoise will interfere destructively when stacking in SS or FT imple-mentations.

CONCLUSIONS

We have investigated and illustrated the causes of the smearing-effect artifacts and of the angle resolution in the ADCIGs obtainedfrom the LPWD and the LSIC methods, which are based on localplane-wave assumptions. The smearing-effect artifacts in LPWDresult from crosscorrelating decomposed source and receiver planewaves at points that do not lie on the source wavefront excitationtime trajectory. The smearing-effect artifacts in LSIC result fromdecomposition of curved HOCIG events at incorrect depth pointsat zero offset. From this understanding of the causes, two methodsare developed to remove the artifacts.The LPWD-MEX method generates ADCIGs that have high an-

gle resolution and are free of the smearing-effect artifact by decom-posing the source and receiver wavefields only at the points alongthe source excitation time trajectory in large spatial windows. Thecomputational efficiency in the LPWD-MEX method is improvedby incorporating the Poynting-vector-based method to provideapproximate reference angles to reduce the scanning ranges of theangles in the source and receiver wavefield decomposition. The T-LSIC method generates ADCIGs that have high angle resolutionand are free of the smearing-effect artifacts by decomposing theevents in HOCIGs only at specific target image depths that corre-spond to the main reflectors, at zero offset.

ACKNOWLEDGMENTS

The research leading to this paper was sup-ported by the sponsors of the University of Texasat Dallas (UT-Dallas) Geophysical Consortium.We thank the Texas Advanced Computing Cen-ter for providing high-performance computingtime. Constructive reviews by associate editorT. Alkhalifah, by F. Perrone, and by an anony-mous reviewer are appreciated. This paper iscontribution no. 1267 from the Geosciences De-partment at the UT-Dallas.

APPENDIX A

ANGLE SAMPLING IS DIFFERENTFROM ANGLE RESOLUTION

There is some potential confusion on the roleof wavenumber sampling in the estimation ofangle resolution. Angle resolution is inversely

1.0

2.0

3.0

4.0

0 20 40 60

Dep

th (

km)

Angle (°)0 20 40 60

Angle (°)a) b) c)

-offset +offset0

Figure 12. The ADCIGs at a CIP position (at horizontal position of 1.81 km) for thesource position indicated by the black dot in Figure 10, obtained from the (a) LSIC (FT)and (b) T-LSIC methods, respectively. Panel (c) illustrates an HOCIG containing twoimage points (the black dots) at zero offset. The two curved green lines in panel (c) arethe HOCIG events. The black line in panel (c) is one of decomposed plane waves thatproduce the smearing-effect artifacts marked by oval in panel (b).

U22 Jin and McMechan

proportional to curvature and wavelength of the wavefront in thechosen local window, which increases with the window size beforezero padding or wavenumber oversampling, until the window size isequal to the Fresnel zone. So, angle resolution is independent ofwavenumber sampling. Extending a spatial window by zero-pad-ding, or by increasing the wavenumber sampling, cannot increasethe angle resolution, but it does increase angle sampling.To illustrate angle sampling in detail, consider an example of the

uppermost reflector in the velocity model in Figure 4. The velocityof this layer is 2.1 km∕s, and the dominant frequency is 20 Hz, sothe wavelength is 0.105 km. The depth of the shallowest reflector is1.0 km, and the offset from the source to the CIP is 0.9 km, so theincident/reflection angle at the reflection point of the shallowest re-flector is 42°, and the diameter of the Fresnel zone is 0.54 km, whichis 54 grid points at a 10-m grid point increment. The maximumwavenumber is

kmax ¼1

space increment¼ 1

10 m¼ 100 km−1; (A-1)

and the wavenumber increment is

dk ¼ kmax

nk; (A-2)

where nk is the number of wavenumbers after zero padding orwavenumber oversampling. The value of wavenumber k at the sam-ple index j is

kj ¼ jdk; (A-3)

and

kj ¼ λ−1j ; (A-4)

so

j ¼ kjdk

¼ kj nk

kmax

¼ nkkmaxλj

; (A-5)

where λj is the wavelength corresponding to kj. The angle samplingincrement for the window size of 30 × 30 grid points (Figure A-1) is

angle increment ¼ arctan

�dkkj

�

¼ arctan

�100 × 0.105

30

�≈ 19°; (A-6)

which from equation A-5 corresponds to j ¼ 3. To reduce this anglesampling increment to ∼1°, the window needs to be extended byzero padding from 30 × 30 to ∼600 × 600, so that

angle increment ¼ arctan

�dkkj

�

¼ arctan

�100 × 0.105

600

�≈ 1°; (A-7)

which from equation A-5 corresponds to j ¼ 57. So, the angle sam-pling increment can be quantified, from the space increment, thefrequency, and the window size after zero-padding. If the windowsize (before zero padding or wavenumber oversampling) is less thanthe Fresnel zone, then the angle resolution is correspondingly re-duced. Angle sampling and angle resolution are not the same.

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