3D angle decomposition for elastic reverse time migrationYuting Duan∗ & Paul Sava, Center for Wave Phenomena, Colorado School of Mines

SUMMARY

We propose 3D angle decomposition methods from elastic re-verse time migration using time- and space-lag common imagepoint gathers, time-lag common image gathers, and space-lagcommon image gathers computed by elastic wavefield migra-tion. We compute time-lag common image gathers at multiplecontiguous locations, instead of isolated positions as is com-monly done with common image gathers. Then, we trans-form the extended images to the angle domain using slantstacks along surfaces that connect neighboring positions, in-stead of line slant stacks for isolated common image gathers.We demonstrate our methods using 2D and 3D synthetic ex-amples and show that our techniques provide accurate openingand azimuth angles, and that they can handle steeply dippingreflectors and converted wave modes.

INTRODUCTION

Common reflection-angle gathers are useful in migration ve-locity analysis and amplitude-versus-angle analysis. Amongangle decomposition techniques, wavefield-based methods arepreferred because they accurately simulate wave propagationin complex geologic structures. Wavefield-based methods gen-erally fall into two categories: pre-migration and post-migrationalgorithms (Vyas et al., 2011). Pre-migration methods decom-pose the wavefields prior to the imaging condition, for exam-ple by f − k domain decomposition, (Xu et al., 2011), or byPoynting vector methods (Yoon and Marfurt, 2006; Yan et al.,2013). Post-migration methods, transform extended imagesto angle gathers after the imaging condition, using space-lagor time-lag common image gathers (CIG), and mixed time-and space-lags common image point (CIP) gathers (De Bruinet al., 1990; Sava and Fomel, 2003; Biondi and Symes, 2004;Sava and Vlad, 2011). Compared to pre-migration algorithms,post-migration methods are cheaper and can handle compli-cated wavefields.

Angle decomposition methods can also be applied to elasticwavefields. In contrast to acoustic wavefields, elastic wave-fields contain various wave modes (e.g. P and S waves). Yanand Xie (2012) suggest a pre-migration method for angle de-composition in elastic media, while Yan and Sava (2008) pro-pose post-migration decomposition methods involving wavemode decomposition. Duan and Sava (2015) further developa scalar imaging condition for converted waves, which gener-ates 3D PS and SP scalar images cheaply and without explicitpolarity correction.

In this paper, we present methods for opening and azimuthangle decomposition from elastic RTM images. We developthe extended scalar imaging condition which generalizes thezero-lag scalar imaging condition (Duan and Sava, 2015). Ourmethods utilize local plane wave decompositions of extended

images and exploit the relationships between the time-lag, space-lag, and image shift in space, which are measured in extendedimages. These relationships lead to opening and azimuth angledecomposition methods using either time-lag common imagegather (τCIGs), space-lag common image gathers (λCIGs), ortime- and space-lag common image point (CIP) gathers. Wedemonstrate the algorithms with 3D elastic time- and space-lag CIPs and 2D elastic time-lag extended images evaluated atmultiple contiguous positions.

ELASTIC SCALAR IMAGING CONDITION

Duan and Sava (2015) propose a scalar imaging condition us-ing geometrical relationships between the propagation direc-tions for the P and S waves and the reflector orientation. ThePS and SP images computed using this imaging condition arescalars without polarity reversals. Here, we develop a moregeneral form of extended imaging conditions as a function ofspace and crosscorrelation lags (space-lag λ and time-lag τ):

IPP (x,λ ,τ) =∑e,t

Ps (e,x−λ , t− τ)Pr (e,x+λ , t + τ) , (1)

IPS (x,λ ,τ) =∑e,t

[∇Ps (e,x−λ , t− τ)×n(x)] ·Sr (e,x+λ , t + τ) ,

(2)

ISP (x,λ ,τ) =∑e,t

[∇×Ss (e,x−λ , t− τ) ·n(x)]Pr (e,x+λ , t + τ) ,

(3)

ISS (x,λ ,τ) =∑e,t

Ss (e,x−λ , t− τ) ·Sr (e,x+λ , t + τ) .

(4)

Vector n(x) is the normal to the reflector plane. QuantitiesS(e,x, t) and P(e,x, t) represent P and S wavefields obtainedby wave-mode decomposition, as functions of experiment e,time t, and space x. Subscripts s and r indicate the wavefieldorigins at source or receivers, respectively. P and S wavefieldsare obtained from displacement field using Helmholtz decom-position (Dellinger and Etgen, 1990; Yan and Sava, 2008):

P(e,x, t) = ∇ ·u(e,x, t) , (5)

S(e,x, t) = ∇×u(e,x, t) . (6)

MOVEOUT ANALYSIS

The extended imaging condition in equations 1-4 preservesnecessary information for decomposing images in angle-dependentcomponents, which can be used for velocity analysis. In 3D,the PP, PS, SP, and SS images generated using this imagingcondition are scalars, which can be used directly to constructangle gathers. We formulate our method under the assumptionthat the reflector is locally planar, and that the incident and

Angle decomposition for elastic RTM

reflected wavefields are also locally planar. However, the de-rived algorithms are not limited to cases where the wavefrontsare planar, because waves can always be decomposed to pla-nar components, even in complex wavefields characterized bymultipathing.

We define the angle domain as the set of half-opening angle θ ,and azimuth φ , {θ ,φ}. The (planar) source wavefield propa-gates along the direction indicated by vector ns with speed vs,and the (planar) receiver wavefield propagates along the direc-tion indicated by vector nr with speed vr. Quantity xo indicatesthe intersection of the two wavefronts at time t = to:{ ns · (x−xo) = vs (t− to) , (7)

nr · (x−xo) = vr (t− to) . (8)

The time-lag τ and space-lags λ in equations 7 and 7 representmovement of the wavefronts in time and space, respectively.Then, the intersection of the two wavefronts shifts to a new po-sition x. The geometrical expressions for the extended imagingcondition using the planar source and receiver wavefields are

(ns

vs− nr

vr

)· (x−xo)−

(ns

vs+

nr

vr

)·λ =−2τ , (9)(

ns

vs+

nr

vr

)· (x−xo)−

(ns

vs− nr

vr

)·λ = 0 , (10)

where vector (x−xo) measures the spatial shift of the imagepoint, corresponding to time-lag τ and space-lag λ .

We consider three types of extended images, using both space-and time-lags, or using time-lag, or using space-lags. In thefollowing, we derive the equations that link the space and/ortime lags with half-opening θ and azimuth φ angles from equa-tions 9 and 10.

Angle decomposition using CIPsFor CIP gathers, we set the image shift x− xo = 0 in equa-tions 9 and 10, and replace vectors ns and nr with vectors q,n and angle θ . Quantity n is the local normal vector, and q isthe vector lying at the intersection of the interface and the re-flection plane, thus it is a function of the azimuth angle φ . Weobtain the system of equations that describes the relationshipsbetween image shift (x−xo) and time-lag τ:

γ sin2θ√γ2 +1+2γ cos2θ

(q ·λ ) = vsτ , (11)

n ·λ = 0 . (12)

where quantity γ is the ratio of vs and vr. For angle decomposi-tion, we loop over all possible values of the reflection angles, θ

and φ , and slant-stack the extended image I(λ ,τ) using equa-tions 11 and 12 to produce the angle-domain image I(φ ,θ).Equations 11 and 12 simplify to the acoustic case, with veloc-ity ratio γ = 1 Sava and Vlad (2011):{

sinθ (q ·λ ) = vsτ , (13)

n ·λ = 0 . (14)

We illustrate our angle decomposition methodology using asimple example of a 3D homogeneous model with a horizontalreflector at depth z = 0.2 km. The acquisition geometry con-sists of a vertical displacement source at {0.4,0.4,0.02} km

and a 2D network of receivers at z = 0.02 km. We computethe PS image (Figure 1a) and an angle gather (Figure 1b) atcoordinates {0.2,0.2,0.2} km. The dot overlain on the anglegather image is the analytical solution of the opening and az-imuth angles for this specific acquisition geometry.

(a)

(b)

Figure 1: (a) PS image computed for one shot. (b)ExtendedCIP and 3D angle gather for the PS image at coordinates{0.2,0.2,0.2} km. The dot represents the analytical estima-tion of the opening and azimuth angle using the acquisitiongeometry.

Angle decomposition using time-lag CIGsFor τCIGs, we set the space-lag λ = 0 in equations 9 and 10,and replace vector ns and nr with vector q, n and angle θ . Thesystem of equations that describes the relationships betweenimage shift (x−xo) and time-lag τ is:

√γ2+1+2γ cos2θ

2[n·(x−xo)]=vsτ , (15)

sin2θ

√γ2+1+2γ cos2θ [q·(x−xo)]=vsτ

1−γ2

γ, (16)

where equation 15 corresponds to the projection of the imageshift (x−xo) on the reflector normal n, and equation 16 corre-sponds to the projection of the image shift (x−xo) on vectorq. For angle decomposition, we loop over all possible val-ues of the reflection angles, θ and φ , and slant stack the ex-tended image I(x,τ) using equations 15 and 16 to produce theangle-domain image I(φ ,θ). Equations 15 and 16 simplify to

Angle decomposition for elastic RTM

(a) (b)

Figure 2: Predicted spatial shift for (a) dipping and (b) horizontal reflectors imaged using PP and PS data, respectively. The largered dots represents the source locations, and the line of smaller green dots represents receiver positions.

the acoustic case, with velocity ratio γ = 1 (Sava and Fomel,2006): {

cosθ [n · (x−xo)] = vsτ , (17)

q · (x−xo) = 0 . (18)

Note that in equations 17 and 18, time-lag τ is not related tovector q, i.e., the time-lag extended images do not contain az-imuthal information for the acoustic case. However, for con-verted waves (γ 6= 1), time-lag extended images contain az-imuthal information according to equations 15 and 16.

We illustrate the time-lag algorithm with 2D examples. Con-sider a model with a dipping reflector in the P-velocity. Usingequations 17 and 18, we can predict the spatial shift of thereflector as a function of time-lag τ , Figure 2a. The dashedline is the intersection of two planes (equations 17 and 18),which illustrates the spatial shift for image point at x = 0.0 km,z = 0.0 km. The solid line through the image point representsthe analytical moveout of the extended image in a vertical CIG.The dashed and solid lines do not coincide with one-another,i.e., the method that applies a slant stack to a vertical CIG ata single x location does not measure the image shift (x−xo)accurately. The angle gathers computed using this method andthe method using a slant stack along the surface are shown inthe middle and right panels of Figure 3a, respectively. The sur-face slant stack leads to more focused angle gathers, indicatingsignificantly higher accuracy.

Similarly, for converted waves, the image does not only shiftvertically, but also horizontally in the time-lag gather even fora horizontal reflector. The predicted spatial shift of the reflec-tor is shown in Figure 2b. The first panel in Figure 3b showsthe computed extended image gather at x = 0.0 km. Note thatthe shape of the extended image in this panel is curved. Theangle gathers computed using a line slant stack for this gatherand a surface slant stack are shown in the middle and right pan-els of Figure 3b, respectively. As for the preceding example,we obtain more focused angle gathers using the surface slantstack.

ANGLE DECOMPOSITION USING SPACE-LAG CIG

For λCIGs, we set τ = 0 in equations 9 and 10, and replacevector ns and nr with vector q, n and angle θ . The system ofequations that describes the relationships between image shift(x−xo) and space-lag λ is:

γ2+1+2γcos2θ

2γn·(x−xo)+sin2θq·λ =

1−γ2

2γn·λ , (19)

γ2+1+2γcos2θ

2γn·λ+sin2θq·(x−xo)=

1−γ2

2γn·(x−xo) .(20)

For angle decomposition, we loop over all possible values ofthe reflection angles, θ and φ , and slant stack the extendedimage I(x,λ ) using equations 19 and 20 to produce the angle-domain image I(φ ,θ).

EXAMPLES

Figure 4: 3D salt body based on the SEG/EAGE model, show-ing the position of the image point at {0.3,0.4,0.3} km, andthe sources on the surface.

For simple geologic models, subsurface illumination from dif-ferent opening and azimuth angles tends to be uniform. How-ever, in areas with complex geologic structures, such as salt

Angle decomposition for elastic RTM

(a) (b)

Figure 3: Extended image gathers for (a) PP and (b) PS reflections. From left to right, the panels depict the extended image, andthe angle gathers constructed using line or surface slant stacks, respectively.

bodies, severe wavefield distortions lead to irregular subsur-face illumination, even if acquisition is uniform. Angle gath-ers characterize subsalt illumination, which benefits reservoircharacterization and acquisition design.

We show an angle gather computed using the proposed elasticangle decomposition methodologies for a modified version ofthe SEG/EAGE salt model (Aminzadeh et al., 1996). We adda horizontal reflector at z = 0.3 km, and compute the openingand azimuth angles for an image point on the horizontal re-flector at coordinates x = 0.3 km, y = 0.4 km. The acquisitiongeometry contains 91 sources and a 2D network of receivers atz = 0.02 km. We generate three-component shot gathers usinga vertical displacement source with a 100 Hz peak frequencyRicker wavelet. As shown in Figure 4, due to the existence ofthe salt body, it is difficult to illuminate the considered imagepoint from certain azimuths, as seen in Figures 5a-5b. Bothangle gathers show similar illumination patterns. The PP im-age is mainly illuminated from north-west and south-east, andthe PS image is illuminated from west and south-east (0◦ in-dicates due East). Compared to the PP angle gather, the PSangle gather has higher resolution, since the S wavelength isgenerally smaller than the P wavelength.

CONCLUSIONS

Elastic 3D angle decomposition can be performed on time-lagor space-lag common image gathers, as well as on mixed time-/space-lag common image point gathers. We propose extendedimaging conditions to compute scalar PP, PS, SP, and SS im-ages, and algorithms for computing opening and azimuth an-gle gathers. Our algorithms generate accurate angle gathersif we consider the complete slant-stack surface, instead of themore popular slant stacks along lines defined in common im-age gathers at fixed surface positions. Our 2D and 3D exam-ples show that the algorithms can handle dipping reflectors andimages generated using converted waves.

ACKNOWLEDGMENTS

We thank the sponsors of the Center for Wave Phenomena,whose support made this research possible. The reproduciblenumeric examples in this paper use the Madagascar open-sourcesoftware package (Fomel et al., 2013) freely available fromhttp://www.ahay.org. The authors thank Paul Fowler for help-ful discussions and valuable suggestions.

(a)

(b)

Figure 5: Opening and azimuth angle gather computed using(a) PP and (b) PS elastic reflections.

Angle decomposition for elastic RTM

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