Comparison of angle decomposition methods for wave-equation migrationNatalya Patrikeeva∗ and Paul Sava, Center for Wave Phenomena, Colorado School of Mines

SUMMARY

Angle domain common image gathers offer advantagesfor subsurface image analysis in complex media. Wecompare two angle decomposition methods using ex-tended images and wave propagation directions based onPoynting vectors of the source and receiver wavefields.We evaluate the ability of each method to produce accu-rate and efficient angle gathers for wave-equation migra-tion in the presence of multi-arrivals and in the subsaltenvironment. Both methods are characterized by fea-tures that make them attractive in different situations.

INTRODUCTION

Common image gathers (CIGs) produced by Kirchhoffmigration are often used for tomographic subsalt veloc-ity updates. However, these gathers suffer from artifactsinduced by multipathing and by poor subsalt data qual-ity. To address these problems, subsurface angle gath-ers can be constructed from non-ray based migrationssuch as reverse time migration (RTM) and wave equa-tion migration (WEM). Wave-equation migration prop-erly handles multiple propagation paths and producesaccurate images in complex environments as opposed toKirchhoff migration (Gray et al., 2001).

Migrated images partitioned based on the reflection andazimuth angles characterize the reflector illuminationin the subsurface and provide information for migra-tion velocity analysis (MVA) and amplitude variationwith angle (AVA) (de Bruin et al., 1990; Prucha et al.,1999; Sava et al., 2001; Fomel, 2004; Biondi and Symes,2004)). Subsurface angle gathers avoid some imagingartifacts and improve migration velocity updates by re-ducing the impact of multipathing associated with theoffset-domain CIGs.

Different techniques have been proposed to address theproblem of angle-domain imaging in complex geologicstructures. Generally speaking, we can separate thesetechniques into two categories: techniques based on de-composition of wavefields prior to the imaging condition,and techniques based on decomposition of migrated im-ages after the imaging condition. Although they share acommon objective, the techniques in the two categorieshave significant differences in quality and computationalcost.

In this abstract, we describe two methods capable ofconstructing subsurface angle gathers and compare themfrom the perspectives of cost, accuracy and applicabil-ity to imaging in complex media. The first consideredmethod uses decomposition of extended images – theEI method (Rickett and Sava, 2002; Sava and Fomel,

2003, 2005; Sava and Vlad, 2011a; Fomel, 2011), andthe second method uses decomposition of extrapolatedvectors using Poynting vectors – the PV method (Yoonand Marfurt, 2004; Yoon et al., 2011; Vyas et al., 2011;Dickens and Winbow, 2011; Crawley et al., 2012). Bothmethods use seismic wavefields reconstructed by reverse-time extrapolation or by downward continuation. Weassume that we compare similar outputs, i.e. angle-gathers computed at all locations in space, although asindicated by Sava and Vasconcelos (2011) this is not anefficient approach.

ANGLE DECOMPOSITION

A conventional cross-correlation imaging condition forwave-equation migration is (Claerbout, 1985)

R (x) =∑e

∑t

Wr (e,x, t)Ws (e,x, t) , (1)

where x = {x, y, z} are space coordinates, Ws and Wr

are the source and receiver wavefields, e is the experi-ment (shot) index, t is time, and R (x) is the migratedimage.

The conventional imaging condition is a special case ofthe generalized or extended imaging condition (Sava andFomel, 2005):

R (x,λ, τ) =∑e

∑t

Ws (e,x− λ, t− τ)Wr (e,x + λ, t+ τ) ,

(2)where λ = {λx, λy, λz} are the cross-correlation spacelags, τ is the cross-correlation time lag, and R (x,λ, τ)is the extended image. The conventional imaging condi-tion, equation 1, is a special case of the extended imag-ing condition, equation 2, when λ = 0 and τ = 0.

In this paper we consider a special case of the extendedimages for τ = 0, i.e. we discuss space-lag gathers. Thisextended imaging condition produces migrated imagesas a function of space x and a local subsurface offset λ.The space-lag between the wavefields can be convertedto the reflection and azimuth angles at each image pointusing a local slant-stack, or equivalently, a radial-tracetransform (Fomel, 1997; Sava and Fomel, 2000; Rickettand Sava, 2002; Sava and Fomel, 2003). This decomposi-tion method assumes a smooth velocity field near the im-age point, thus justifying the local slant-stack. This op-eration is performed in extended common-image gathers(CIGs), although a more efficient implementation takesadvantage of common-image-point gathers (CIPs) (Savaand Vlad, 2011b). Algorithm 1 summarizes the methodused to convert the extended images to angle gathers.

Yoon and Marfurt (2004) propose a method to computeangle gathers during reverse time migration (RTM) via a

Angle decomposition comparison

Poynting vector imaging condition. They use the Poynt-ing vector to compute the opening angle between thesource and receiver directions and filter the RTM back-scattered energy artifacts. Other authors follow theirmethod to output angle gathers as a product of migra-tion (Vyas et al. (2011); Yoon et al. (2011); Dickens andWinbow (2011); Crawley et al. (2012)).

Yoon and Marfurt (2004) calculate Poynting vectors asa product of the time derivative and the gradient ofthe wavefield. This method requires accurate numericalderivatives to produce correct angles and access to mul-tiple time steps during wavefield extrapolation, whichis not always practical for time extrapolation methodslike RTM. Alternatively, we can construct the Poyntingvectors for acoustic waves as a product of pressure andparticle velocity (Kaufman et al., 2002):

P (e,x, t) = W (e,x, t)V (e,x, t) , (3)

where P is the Poynting vector, W is the pressure wave-field, and V is the particle velocity.

By definition, the Poynting vector is the energy flux den-sity vector. Its magnitude defines the amount of energytransmitted through the unit area per unit time. In thehigh frequency approximation, the Poynting vector isoriented along the ray, i.e. it captures the group veloc-ity angle in general anisotropic media. Here we use thePoynting vector to define the propagation directions ofthe source and receiver wavefields. Then, we can definethe reflection and azimuth angles as

sin (2θ) =(Ps ×Pr) · k̂|Ps||Pr|

, sin (2φ) =(Pr −Ps) · k̂||Pr −Ps||

,

(4)where Ps (e,x, t) and Pr (e,x, t) are the Poynting vec-tors of the source and receiver wavefields, respectively,and k̂ is the unit vector along the z axis. The valuesof the reflection angle range from −90o to +90o. Usingthe cross product instead of the dot product of the twoPoynting vectors allows us to differentiate between neg-ative and positive values of reflection angle θ. Algorithm2 outlines the angle decomposition method using Poynt-ing vectors of the source and receiver wavefields. Afterwe calculate the angles, we map the image contributionto the appropriate angle bin.

The Poynting vectors oscillate rapidly since they arehighly sensitive to noise present in the extrapolated wave-fields. This noise could either originate in the data, or itcould be the result of interference between waves propa-gating in different directions. Therefore, reliable Poynt-ing vectors require spatial and/or temporal smoothingbefore the angle calculations in order to reduce theirrapid spatial and temporal oscillations which degradeangle calculation. For example, Yoon et al. (2011) aver-age the Poynting vector over four periods of the sourcewavelet, and Dickens and Winbow (2011) smooth thevector components over a small region in space. Weuse similar space and time smoothing in our examples.The downside of this operation is that the accuracy of

(a)

(b)

Figure 1: Poynting vectors represented on wavefields(a) in a constant medium and (b) in a medium with anegative Gaussian anomaly.

the Poynting vector direction degrades, especially in ar-eas of high wavefield complexity like sub-salt. Further-more, if different wavefield components interfere, e.g. attriplications, then the Poynting vector captures an av-erage direction which does not represent either of thebranches of the reconstructed wavefield, thus degradingangle-domain accuracy.

(a) (b)

Figure 2: Reflection angles computed on a horizontalreflector in constant velocity using the EI and the PVmethods. The overlay depicts the theoretical reflectionangles along the horizontal reflector.

Figure 1 illustrates the behavior of Poynting vectors forsimple wavefields (a) and for complex wavefields (b) dueto a negative velocity anomaly. The vectors accuratelydescribe the propagation directions in simple media butdo not achieve the same accuracy and reliability in com-plex media. Nevertheless, if the medium is fairly simple,then both methods based on extended images (EI) andPoynting vectors (PV) lead to accurate angles, as illus-

Angle decomposition comparison

(a)

(b)

(c)

Figure 3: Simple synthetic illustrating angle-domainimaging with multi-pathing. The model (panel a) causeswavefield triplications which lead to multiple illumina-tion paths in the image (panel b). The EI method cap-tures all propagation paths (panel c, left) while the PVmethod does not (panel c, right).

trated in Figure 2. The figure shows the reflection anglecalculations of both methods in a homogeneous isotropicmodel with a horizontal reflector for one shot at 5.5 km.Both methods output accurate reflection angles.

ALGORITHMS

We compare the two algorithms using the following cri-teria: wavefield reconstruction cost, imaging conditioncost, angular resolution, ability to handle multipathing,robustness for reflectors with large curvature, and theability to accurately describe anisotropic media.

1: The wavefield reconstruction step is computationallycheaper for the EI method than for the PV method be-cause we do not require additional calculation of the par-ticle velocity and PV components at each image point,at all times, and for all experiments. Algorithm 2 shows

that the PV method requires both the pressure W andthe particle velocity V fields, in contrast with the EImethod which requires just the pressure field.

(a)

(b)

(c)

Figure 4: Synthetic example of complex geology. Themodel (panel a) contains a salt body causing wavefieldtriplications. These events result in a strong amplitudeevent in the image (panel b) and in artifacts in PV anglegathers (panel c, right). The EI method produces morereliable angle gathers (panel c, left).

2: The EI method is costlier due to the computation andstorage of the additional cross-correlation lags whichgenerate a larger-dimensional image R (x,λ). Even if weonly compute the horizontal space-lags λ = {λx, λy, 0}at all locations, the output image is a 5D cube whichrequires longer access due to its increased size. TheEI angle decomposition is performed after summationover experiments at an additional, albeit small, compu-tational cost. In contrast, the PV method allows anglecalculation at each grid point on the fly, as we propagatethe wavefields in the medium. The PV method does notrequire computation of the additional cross-correlation

Angle decomposition comparison

lags nor any additional conversions post-migration, al-though this cost savings are partially offset by the needto compute and store the local Poynting vectors at allpositions and times. It appears, however, that the PVmethod requires fewer computational loops, thus poten-tially being more cost-effective for 3D angle decomposi-tion.

3: The PV algorithm produces higher angular resolu-tion because the method uses only local information atthe image point (Dickens and Winbow, 2011). However,a complicated receiver wavefield makes it difficult tocompute the receiver Poynting vector accurately (Yoonet al., 2011), which might lead to high-resolution butinaccurate angle-dependent reflectivity.

4: Another limitation of the PV method is that it as-sumes one dominant propagation direction for the sourceand receiver wavefields and therefore computes only oneopening reflection angle. However, one reflection angleis not an accurate representation of the subsurface incomplex geology where the wavefields are complicatedby multipathing. Figures 3(a)-3(c) illustrate one suchexample. The model has a negative Gaussian anomalywhich causes triplication of the wavefield, Figure 3(a).Migration of one single shot shows the effects of multi-pathing, i.e. multiple illumination angles at the reflec-tion point, Figure 3(b). The EI and PV angle gathersdisplayed in Figure 3(c) show multiple (EI, left), or sin-gle (PV, right) illumination angles.

Algorithm 1 – EI angle decomposition

1: for e do2: for t,x do3: for λ do4: R (x,λ) + = Ws (e,x + λ, t)Wr (e,x− λ, t)5: end for6: end for7: end for8: for x do9: for θ,φ do

10: R (x,λ) −→ R (x, θ, φ)11: end for12: end for

Algorithm 2 – PV angle decomposition

1: for e do2: for t,x do3: Ps (e,x, t) = Ws (e,x, t)V s (e,x, t)4: Pr (e,x, t) = Wr (e,x, t)V r (e,x, t)5: {Ps,Pr} (e,x, t) −→ {θ, φ} (e,x, t)6: {Ws,Wr} (e,x, t) −→ R (x, θ, φ)7: end for8: end for

5: The EI method degrades if the imaged reflectors arecharacterized by high curvature or by rapid lateral veloc-ity variation. In contrast, the PV method does not have

similar constraints on reflector curvature because of thelocalized angle computation, although the local spaceand/or time smoothing degrades its ability to identifyprecise angles when the wavefields are too complex. Fig-ures 4(a)-4(c) illustrate the behavior of the two angle-decomposition methods in complex geology similar tothe one derived from the Sigsbee 2A model, Figure 4(a).The migrated image for one shot, Figure 4(b), showsthe imprint of multipathing. This is also visible in theangle gathers shown in Figure 4(c). The EI methodproduces low-resolution angle gathers but captures themultipathing observed in this area. In contrast, the PVmethod produces high-resolution angle gathers but doesnot capture the wavefild multipathing and thus it doesnot correctly evaluate all incidence angles in the sub-saltregion.

6: As demonstrated by Sava and Alkhalifah (2013), theEI method is applicable to wave-equation imaging inanisotropic media. This is mainly because this methodexploits the wavefront geometry of the propagating wave-fields, i.e. the gathers are related to the phase anglesat the image points. In contrast, the PV method pro-duces gathers related to the group angles which need tobe converted to the phase angles (Dickens and Winbow(2011)).

CONCLUSIONS

We extract reflection and azimuth angles using the ex-tended images (EI) and Poynting vector (PV) methods.Each method has its own advantages and applications.In simple models, the PV method produces high an-gular resolution and is less computationally intensivethan the EI method. In areas characterized by multi-pathing, the PV method produces only one angle perimage point and provides a less accurate representationof angle-dependent reflectivity. The PV components os-cillate rapidly and are impacted by wavefield noise. Fur-thermore, in anisotropic media, PV angle decompositionis further complicated by the need to convert group tophase angles at each image point. The EI method is po-tentially costlier since it requires cross-correlation lagsat all locations where angle decomposition is needed.Otherwise, the method accurately and simultaneouslymaps all reflections into the angle domain and is readilyapplicable to anisotropic media. However, lateral ve-locity variations or high reflector curvature reduce theaccuracy of EI angle calculations.

ACKNOWLEDGMENTS

This work was supported by the sponsors of the Cen-ter for Wave Phenomena at Colorado School of Mines.The reproducible numeric examples in this paper use theMadagascar open-source software package freely avail-able from http://www.ahay.org.

Angle decomposition comparison

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