CWP-681

Wide-azimuth angle gathers foranisotropic wave-equation migration

Paul Sava1 & Tariq Alkhalifah2

1Center for Wave Phenomena, Colorado School of Mines2King Abdullah University of Science and Technology

ABSTRACTExtended common-image-point gathers (CIP) constructed by wide-azimuth TI wave-equation migration contain all the necessary information for angle decomposition asa function of the reflection and azimuth angles at selected locations in the subsurface.The aperture and azimuth angles are derived from the extended images using ana-lytic relations between the space- and time-lag extensions using information which isalready available at the time of migration, i.e. the anisotropic model parameters. CIPsare cheap to compute because they can be distributed in the image at irregular locationsaligned with the geologic structure. If information about the reflector dip is availableat the CIP locations, then only two components of the space-lag vectors are required,thus reducing computational cost and increasing the affordability of the method. Thetransformation from extended images to angle gathers amounts to a linear Radon trans-form which depends on the local medium parameters. This transformation allows usto separate all illumination directions for a given experiment, or between different ex-periments. We do not need to decompose the reconstructed wavefields or to choosethe most energetic directions for decomposition. Applications of the method includeillumination studies in areas of complex geology where ray-based methods are not sta-ble, and assuming that the subsurface illumination is sufficiently dense, the study ofamplitude variation with aperture and azimuth angles.

Key words: imaging, wave-equation, angle-domain, wide-azimuth, anisotropy

1 INTRODUCTION

In regions characterized by complex subsurface structure,wave-equation depth migration is a powerful tool for accu-rately imaging the earth’s interior. The quality of the finalimage depends on the quality of the earth model and on thetechnique used for wavefield reconstruction in the subsur-face (Gray et al., 2001). This is particularly true for imag-ing in areas characterized both by strong anisotropy, as wellas complex heterogeneous geology. Such challenges can bestbe addressed by imaging using reverse-time migration (Baysalet al., 1983; McMechan, 1983), a computationally intensivemethod, but capable of providing the most accurate represen-tation of complex geologic structure.

In addition to structural imaging, it is desirable to con-struct images of reflectivity as a function of reflection an-gles. Such images not only indicate the subsurface illumina-tion patterns, but could potentially be used for velocity ver-

ification and estimation and amplitude variation with angleanalysis which could lead to more accurate lithologic inter-pretation. Angle gathers can be produced by ray methods (Xuet al., 1998; Brandsberg-Dahl et al., 2003) or by wavefieldmethods (de Bruin et al., 1990; Xie and Wu, 2002; Sava andFomel, 2003; Biondi and Symes, 2004; Wu and Chen, 2006;Xu et al., 2010; Sava and Vlad, 2011). Gathers constructedwith these methods have similar characteristics since they sim-ply describe the reflectivity as a function of incidence angles atthe reflector. Wavefield-based angle gather can be constructedeither before or after the application of an imaging condition.The methods operating before the imaging condition decom-pose the extrapolated wavefields directly (de Bruin et al., 1990;Wu and Chen, 2006; Xu et al., 2010). This type of decompo-sition is costly since it operates on large wavefields character-ized by complex multipathing, which makes event identifica-tion challenging. In contrast, the methods operating after theimaging condition decompose the images themselves which

42 P. Sava & T. Alkhalifah

are represented as a function of space and additional param-eters, typically referred to as extensions (Rickett and Sava,2002; Sava and Fomel, 2006; Sava and Vasconcelos, 2011).Both classes of methods produce angle-dependent reflectivityrepresented by the so-called scattering matrix (Wu and Chen,2006). The main differences lie in the complexity of the de-composition and in the cost required to achieve this result.

Conventionally, angle-domain imaging uses common-image-gathers (CIGs) describing the reflectivity as a functionof reflection angles and a space axis, typically the depth axis.An alternative way of constructing angle-dependent reflectiv-ity is based on common-image-point-gathers (CIP) selectedat various positions in the subsurface. In this paper, we fo-cus on angle decomposition using extended CIPs, which of-fer significant computational savings over alternative methodsbased on CIGs. For a complete discussion of the merits of us-ing common-image-point gathers we refer to Sava and Vas-concelos (2011).

Wave-equation imaging with wide-azimuth data (Re-gone, 2006; Michell et al., 2006; Clarke et al., 2006) posesadditional challenges for angle-domain imaging, mainly aris-ing from the larger data size and the interpretation difficulty ofdata in higher dimensionality. Several techniques have beenproposed for wide-azimuth angle decomposition, includingray methods (Koren et al., 2008) and wavefield methods usingwavefield decomposition before the imaging condition (Zhuand Wu, 2010; Biondi and Tisserant, 2004; Xu et al., 2010) orafter the imaging condition (Sava and Fomel, 2005; Sava andVlad, 2011). Here, we focus on imaging with wavefield meth-ods which are more accurate and robust in complex geology.

The nature of sedimentation and layering in the Earth in-duces wave propagation characteristics that can be best de-scribed by considering anisotropic media. Specifically, sincethe layering has a general preferred direction, the trans-versely isotropic (TI) assumption is the most practical typeof anisotropy applicable to large parts of the subsurface.Anisotropy (including the TI version) defines a medium inwhich the wave speed varies with the propagation angle. Anglegathers have a prominent role in defining such angular varia-tions, and thus, in estimating anisotropy parameters. Alkhal-ifah and Fomel (2010) derive analytical relations to constructangle gathers for TI media with a vertical symmetry axis (VTI)from downward continued wavefields prior to the applicationof an imaging condition. This approach requires wavefieldsin the wavenumber domain, and thus, it is generally limitedto downward continuation methods. Similarly, Alkhalifah andSava (2010) provide a framework to construct angle gathersfor dip-constrained TI media (DTI). This model allows for ananalytical description of the angle gather mapping process ap-plied prior to the imaging condition, and it is also appropriatefor downward continuation imaging methods. In addition, thistechnique requires media that satisfy the dip constraint.

In this paper, we develop a technique for mapping theimage extensions to angle gathers in general TI media withsymmetry axis direction oriented arbitrarily in a 3D space.We focus on imaging methods using time-domain wavefieldextrapolation, i.e. reverse-time migration. Our technique uti-

lizes indirectly local plane wave decompositions of extendedimage and exploits information about the anisotropy charac-terizing the imaged medium. We show that this formulationproduces an accurate description of the specular reflection forgeneral 3D TI media, while being affordable for large-scalewide-azimuth imaging projects.

2 WAVE-EQUATION IMAGING

Conventional seismic imaging is based on the concept of sin-gle scattering. Under this assumption, waves propagate fromseismic sources, interact with discontinuities and return tothe surface as reflected seismic waves. The “source” and “re-ceiver” wavefields associated with the propagation from thesources to the reflectors and with the propagation from thereflectors to the receivers coincide kinematically on the re-flectors (Berkhout, 1982; Clærbout, 1985). Imaging is a pro-cess involving two steps: the wavefield reconstruction and theimaging condition. The key elements in this imaging proce-dure are the source and receiver wavefields,Ws andWr whichare 4-dimensional objects as a function of space x = {x, y, z}and time t (or, equivalently, frequency ω). For imaging, weneed to analyze if the wavefields match kinematically in timeand then extract the reflectivity information using an imagingcondition operating along the space and time axes.

Wavefield reconstruction in anisotropic media with atilted axis of symmetry (TTI), requires a numeric solutions toa pseudo-acoustic wave-equation. Different pseudo-acousticwave-equations are reported in the literature, their main dif-ferences being in the order of the equation and the imple-mentation used. The method discussed in this paper does notdepend on any specific implementation. Our choice is to usethe time-domain method of Fletcher et al. (2009) and Fowleret al. (2010) which consists of solving a system of second-order coupled equations:

∂2p

∂t2= vp

2xH2 [p] + vp

2zH1 [q] , (1)

∂2q

∂t2= vp

2nH2 [q] + vp

2zH1 [q] . (2)

Here, p and q are two wavefields depending on space x ={x, y, z} and time t coordinates. H1 and H2 are differentialoperators applied to the quantity in the square brackets. Thevelocities vpz , vpx, and vpn are the vertical, horizontal and“NMO” velocities used to parametrize a generic TTI medium.These expressions assume that the shear-wave velocity (vsz) iszero, which leads to well-known wavefield artifacts (Alkhali-fah, 2000) visible in our numeric examples. These artifacts areheavily distorted shear waves due to the approximation madeon their propagation velocity. An alternative implementationof equations 1-2 allow for non-zero shear-wave velocity whichreduces the strength of the artifacts. We do not emphasize thischoice here, since it does not affect our angle decompositionmethod. We concentrate on angle-domain imaging with thequasi-P wave mode which is accurately described. For details,we refer to Fletcher et al. (2009) and Fowler et al. (2010) andcitations therein.

Anisotropic wide-azimuth angle-gathers 43

If we describe the medium using the parameters intro-duced by Thomsen (2001), the velocities are related to theanisotropy parameters ε and δ by the relations

vpn = vpz√1 + 2δ , (3)

vpx = vpz√1 + 2ε . (4)

Thus, the quantities H1 and H2 are defined as

H1 = sin2 θa cos2 φa

∂2

∂x2

+ sin2 θa sin2 φa

∂2

∂y2

+ cos2 θa∂2

∂z2

+ sin2 θa sin 2φa∂2

∂x∂y

+ sin 2θa sin2 φa

∂2

∂y∂z

+ sin 2θa cosψ∂2

∂z∂x, (5)

H2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2−H1 , (6)

where θa describes the angle made by the TI symmetry axiswith the vertical and φa the azimuth angle of the plane thatcontains the tilt direction. In this paper, we assume that all pa-rameters characterizing the TI medium are known, includingthe angles θa and φa. Therefore, we can use this procedureto reconstruct wavefields with arbitrary heterogeneity, includ-ing that of the anisotropy and tilt parameters. Here, we do notmake any assumption about the orientation of the tilt medium,i.e. the tilt axis does not have to be aligned with the normal ofthe reflector at any point in the image.

A conventional imaging condition based on the recon-structed wavefields can be formulated as the zero lag of thecross-correlation between the source and receiver wavefields(Clærbout, 1985):

R (x) =∑shots

∑t

Ws (x, t)Wr (x, t) ; , (7)

where R represents the migrated image which depends onposition x. An extended imaging condition preserves in theoutput image certain acquisition (e.g. source or receiver coor-dinates) or illumination (e.g. reflection angle) parameters. Inshot-record migration, the source and receiver wavefields arereconstructed on the same computational grid at all locationsin space and all times or frequencies, therefore there is no a-priori wavefield separation that can be transferred to the outputimage. In this situation, the separation can be constructed bycorrelation of the wavefields from symmetric locations relativeto the image point, measured either in space (Rickett and Sava,2002; Sava and Fomel, 2005) or in time (Sava and Fomel,2006). This separation essentially represents lags of the localcross-correlation between the source and receiver wavefields

(Sava and Vasconcelos, 2011):

R (x,λ, τ) =∑shots

∑t

Ws (x− λ, t− τ)Wr (x+ λ, t+ τ) .

(8)Equation 7 represents a special case of equation 8 for λ = 0and τ = 0. Equivalent equations can be written for imagingin the frequency domain (Sava and Vasconcelos, 2011). Thespace- and time-lag extensions can be converted to reflectionangles (Sava and Fomel, 2003, 2006; Sava and Vlad, 2011),thus enabling analysis of amplitude variation with angle forimages constructed in complex areas using wavefield-basedimaging.

In this paper, we use extended common-image-point-gathers to extract angle-dependent reflectivity at individualpoints in the image. The method described in the followingsection is appropriate for 3D anisotropic wide-azimuth wave-equation imaging and is analogous to the procedure outlinedby Sava and Vlad (2011) for isotropic media. The problem weare solving is to decompose extended CIPs as a function of az-imuth φ and reflection θ angles at selected points in the image:

R (λ, τ) =⇒ R (φ, θ, τ) . (9)

However, we assume that all parmeters characterizing theanisotropic medium are known. Therefore, all the energy in theoutput CIPs concentrates at τ = 0, so we can concentrate onthe particular case of decomposition which does not preservethe time-lag variable in the output: R (λ, τ) =⇒ R (φ, θ).The topic of angle decomposition when the gathers are con-structed with an incorrect model remains outside the scope ofthis paper, although the general ideas used here remain valid.

In the remainder of the paper, we derive the moveoutfunction for extended images and develop of procedure for an-gle decomposition applicable to arbitrary TTI media. We illus-trate the procedure with 3D wide-azimuth synthetic examples.

3 MOVEOUT ANALYSIS

An implicit assumption made by all methods of angle decom-position is that we can describe the reflection process by lo-cally planar objects. Such methods assume that (locally) thereflector is a plane, and that the incident and reflected wave-fields are also (locally) planar. Only with these assumptionswe can define vectors in-between which we measure incidenceand reflection angles and the azimuth of the reflection plane.We also formulate our method under this assumption, althoughwe do not assume that the wavefronts of the source and re-ceiver wavefields are planar. Instead, we consider each (com-plex) wavefront as a superposition of planes with different ori-entations and discuss how each one of these planes behavesduring the extended imaging condition and angle decomposi-tion.

In general, we can distinguish two situations illustratedby Figures 1(a)-1(d).

• First, if the wavefields (source or receiver) correspond tounique propagation paths but are non-planar, we can decom-pose them in planar components. Using rays, this is analogous

44 P. Sava & T. Alkhalifah

to saying that we discuss one pair of incident and reflected raysat an image point.• Second, if the wavefields (source or receiver) correspond

to multiple propagation paths, then we need to consider all(planar) components corresponding to the various branches ofthe wavefield. Using rays, this is analogous to saying that wediscuss all pairs of incident and reflected rays at a given point.

In general, the wavefields may be characterized by a superpo-sition of the two phenomena discussed here.

In our procedure, we do not explicitly decompose thewavefields in planar components corresponding to differentbranches of the wavefield. Instead, we discuss how each oneof these planes behaves during the extended imaging and angledecomposition. We can consider the entire wavefield as a su-perposition of planes corresponding to the various propagationdirections. By linearity, each pair of source and receiver planeshas an expression in the extended images, thus allowing us todecouple different propagation directions after imaging. Forexample, for the case depicted in Figure 1(d), showing threeindependent propagation branches, we would obtain three dif-ferent overlapping but distinct events in the extended image.Thus, we can obtain a map of reflectivity as a function of an-gles by decomposition of the extended images, regardless ofthe number of shots which contribute to it or the number ofbranches that characterize each wavefield at an image point.This approach is cheap relative to procedures that decomposethe seismic wavefields explicitly, and we also do not requirecomplicated selections of specific wavefield components iden-tified by their direction of propagation.

Using the planar components of the source and receiverwavefields, we can write the conventional imaging conditionas

ns · (x− xs) = vs (t− tp) , (10)

nr · (x− xr) = vr (t− ts) , (11)

where vs (θ, φ) and vr (θ, φ) are velocities for the source andreceiver wavefields which, for anisotropic media, depend onthe reflection angles θ and φ. This system indicates that thesource and receiver wavefields are coincident in space at agiven time. In anisotropic media, the velocities for the sourceand receiver wavefronts depend on the propagation direction.For the case of PP reflections in isotropic media, or in TTImedia with an axis of symmetry parallel to the reflector nor-mal, we have vs = vr . However, for the case when the axisof symmetry is tilted relative to the reflector normal, we havethat vs 6= vr . Here we consider the general case when the twovelocities are not equal.

Similarly, we can write the extended imaging conditionas

ns · (x− xs − λ) = vs (t− tp − τ) , (12)

nr · (x− xr + λ) = vr (t− ts + τ) , (13)

where λ and τ are lags in space and time. The delay is relativeto the position and time identified by the conventional imagingcondition, equations 10-11. Substituting equation 10 in 12 and

equation 11 in equation 13, we obtain the expressions

ns · λ = vsτ , (14)

nr · λ = vrτ . (15)

We can then replace the vectors ns and nr as a function ofthe normal vector n and the vector q at the intersection of thereflection and the reflector planes

ns = q sin θs − n cos θs , (16)

nr = q sin θr + n cos θr , (17)

where θs and θr are the (phase) angles made by the source andreceiver wavefronts with the normal direction. Therefore, theexpressions 14 and 15 become

(q · λ) sin θs − (n · λ) cos θs = vsτ , (18)

(q · λ) sin θr + (n · λ) cos θr = vrτ . (19)

We can separate the quantity (q · λ) from equations 18and 19 by multiplying them with cos θr and cos θs, respec-tively, followed by summation:

(q · λ) [sin θs cos θr + sin θr cos θs] = [vs cos θr + vr cos θs] τ .(20)

Likewise, we can separate the quantity (n · λ) from equa-tions 18 and 19 by multiplying them with − sin θr and sin θs,respectively, followed by summation:

(n · λ) [sin θr cos θs + cos θr sin θs] = [−vs sin θr + vr sin θs] τ .(21)

Using Snell’s law

sin θsvs

=sin θrvr

, (22)

which is valid for any type of media, including the TTImedium considered here, we have

vr sin θs − vs sin θr = 0 . (23)

Therefore, after substitution in equations 20 and 21 we canwrite

(q · λ) sin (θs + θr) = [vs cos θr + vr cos θs] τ ,(24)

(n · λ) sin (θs + θr) = 0 . (25)

The calculations simplify and become more symmetric ifwe make the notation

2θ = θr + θs , (26)

2ψ = θr − θs , (27)

which is equivalent with

θs = θ − ψ , (28)

θr = θ + ψ . (29)

The angle 2θ represents the sum of the reflection angles on thesource and receiver sides, and the angle 2ψ represents theirdifference. If we label γ the ratio of the velocities on the sourceand receiver sides, we have

γ =vsvr

=sin θssin θr

, (30)

Anisotropic wide-azimuth angle-gathers 45

n

ns

y

x

z

(a)

ns

n

y

x

z

(b)

ns

n

y

x

z

(c)

ns

ns

n

ns

y

x

z

(d)

Figure 1. Schematic representation of three planes representing independent branches of a wavefield at their interaction with a reflector. Panels(a)-(c) show the individual wavefield branches separated by propagation direction, and panel (d) shows their superposition. The extended imagingcondition uses the wavefields as in panel (d), but for the purpose of angle decomposition we can study the individual components.

so we can write

γ =sin (θ − ψ)sin (θ + ψ)

, (31)

or

tanψ =1− γ1 + γ

tan θ . (32)

This expression gives us a direct link between the angles θand ψ. For isotropic, VTI or DTI media, we have that γ = 1,therefore tanψ = 0 regardless of the angles of incidence orreflection, θs and θr . Otherwise, the angle ψ depends on theopening angle θ and on the material properties at the reflec-tion point, i.e. the velocities vs and vr . On the other hand, thevelocities vs and vr depend on the angles θ and ψ, thus lead-ing to a circular dependency which we discuss in detail in thefollowing section.

Except for the case of normal incidence (θs = θr = θ =0 when ψ = 0), we can write the system 24-25 as

(q · λ) sin (2θ) = [vs cos (θ + ψ) + vr cos (θ − ψ)] τ ,(33)

(n · λ) = 0 . (34)

The system 33-34 characterizes the moveout of PP reflec-tions in a TTI medium with arbitrary symmetry axis direction.We can use this system for wide-azimuth angle decompositionin arbitrary TTI media with a procedure similar to the one out-lined in Sava and Vlad (2011) for isotropic media when the

system reduces to

(q · λ) sin θ = vτ , (35)

(n · λ) = 0 . (36)

However, for arbitrary TTI media, the angle decompositionprocedure is complicated by the fact that the velocities on thesource and receiver sides depend on the angles of incidence.On the other hand, the angles of incidence depend on the ve-locities, thus leading to a circular dependency. This difficultycan be solved through numeric techniques, as discussed in thefollowing section.

4 ANGLE DECOMPOSITION

The moveout function derived in the preceding section allowsus to assemble an algorithm for wide-azimuth angle decom-position applicable to anisotropic media with arbitrary tilt rel-ative to the reflector. We describe the anisotropic medium bythe vertical velocity, vpz , by the anisotropic parameters ε andδ (Thomsen, 2001), and by the tilt vector t which can take anarbitrary orientation relative to the normal vector n. This de-scription characterizes the most general case of TTI medium.The DTI case (Alkhalifah and Sava, 2010) is simply a specialcase for which t = n. The vectors t and n are normalized, i.e.|t| = |n| = 1.

46 P. Sava & T. Alkhalifah

4. 1 Isotropic media

As shown in Algorithm 1, at every CIP location we use theinformation about the known normal vector n, measured fromthe image, and the azimuth reference vector v to compute areference in the reflection plane, a and extract the velocity vfrom the model used for migration. As pointed out by Savaand Vlad (2011), the azimuth reference vector v is arbitrarybut common for all CIPs, thus allowing us to compare angle-dependent reflectivity at various points in the image. Further-more, the normal vector allows us to avoid computing onecomponent of the space lag vector, thus significantly reducingthe computational cost of the method.

Algorithm 1 Isotropic angle decomposition1: for each CIP do2: input R (λ, τ)3: input n, v4: {n, v} → a5: for φ = 0◦ . . . 360◦ do6: {n, a, φ} → q7: for θ = 0◦ . . . 90◦ do8: R (λ, τ)

q,θ,v−−−→ R (φ, θ)9: end for

10: end for11: return R (φ, θ)12: end for

Then, as we loop over all possible values of the azimuthangle φ, we construct the vector q at the intersection of thereflector and the reflection planes, Figure 2(a). This operationsimply requires a rotation of the azimuth reference vector aaround the normal vector n by angle φ

q = Q (n, φ) a , (37)

where Q is a rotation matrix defined by the axis n and theangle φ. We continue by looping over all possible values ofthe reflection angle θ and apply a slant-stack to the CIP cubeR (λ, τ) using equations 35 and 36. This operation producesthe values of the reflectivity for angles θ and φ. No wavefielddecomposition prior to the imaging condition is required, andno wavefield windowing is necessary to select specific wave-field components.

4. 2 Anisotropic media

The wide-azimuth angle decomposition for anisotropic mediafollows the same general structure of the isotropic algorithm.As shown Algorithm 2, at every CIP location, we use the in-formation about the known normal vector n, measured fromthe image, and the azimuth reference vector v to compute areference in the reflection plane, a. We also extract at every lo-cation the vertical migration velocity, as well as the anisotropyparameters ε and δ, and the tilt vector t describing the TTImedium. All these parameters are known at the time of migra-tion.

Algorithm 2 TTI angle decomposition1: for each CIP do2: input R (λ, τ)3: input n, vpz , ε, δ, t4: {n, v} → a5: for φ = 0◦ . . . 360◦ do6: {n, a, φ} → q7: for θ = 0◦ . . . 90◦ do8: {n, q, θ, vpz, ε, δ, t} → {vs, vr, ψ}9: R (λ, τ)

q,θ,vs,vr,ψ−−−−−−−→ R (φ, θ)10: end for11: end for12: return R (φ, θ)13: end for

Then, as for the isotropic algorithm, we loop over all pos-sible values of the azimuth angle φ to compute the azimuthvector q, and over the reflection angles θ. The next step isthe main difference with the isotropic algorithm. Instead ofproceeding directly with the slant-stack from the CIP to theangle gather, we first need to evaluate the angle ψ defined inthe preceding section. This calculation cannot be done analyt-ically due to the circular dependence between the velocitiesand phase angles in anisotropic media, but requires a numericsolution. A simple and effective numeric solution is to loopover all possible values of the angle ψ and select the one forwhich Snell’s law is best satisfied. This is cheap calculationwhich could be done before angle decomposition at the loca-tions where CIPs are constructed. The steps of the algorithmare the following:

1 . Evaluate the incidence and reflection angles, θs and θr .2 . Evaluate the incidence and reflection vectors, ns and

nr , by a simple rotation of the normal vector n with angles θsand θr around the vector b = q× n.

3 . Evaluate the angle αs and αr between the incidenceand reflection vectors and the tilt vector, t.

4 . Evaluate the velocities corresponding to angles αs andαr relative to the tilt vector using the expression (Tsvankin,2005)

v2 (α)

vp2z= 1 + ε sin2 α− f

2(1− s) , (38)

where

f = 1− vs2z

vp2z, (39)

and

s =

√1 +

4 sin2 α

f(2δ cos2 α− ε cos 2α) + 4ε2 sin4 α

f2.

(40)In our example, we use the acoustic approximation and setvsz = 0. Here α is the angle between the direction of propa-gation of the source or receiver wavefields, ns and nr , and thetilt axis n, i.e. we are using angles αs and αr evaluated in thepreceding step.

Anisotropic wide-azimuth angle-gathers 47

5 . Evaluate the function

S (ψ) = vr sin θs − vs sin θr , (41)

and decide if the chosen value of ψ minimizes the functionS (ψ). The value of ψ that brings the function S (ψ) closest tozero corresponds to the case when Snell’s law is observed.

This numeric procedure is summarized in Algorithm 3. Fi-nally, we apply a slant-stack to the CIP cube R (λ, τ) usingequations 33 and 34 using the velocities computed before.Thus, we obtain a measure of the reflectivity as a function ofthe reflection phase angles θ and φ.

Algorithm 3 Snell’s law in TTI media

1: Get {n, q, θ, vpz, ε, δ, t}2: Compute {n, q} → b3: S = huge4: for ψ∗ = ψmin . . . ψmax do5: θ∗s = θ − ψ∗ ; {b, n, θ∗s} → ns ; {ns, t} → αs6: Compute v∗s = v (α∗

s , vpz, ε, δ)

7: θ∗r = θ + ψ∗ ; {b, n, θ∗r} → nr ; {nr, t} → αr8: Compute v∗r = v (α∗

r , vpz, ε, δ)9: Compute {v∗s , θ∗s , v∗r , θ∗r} → S∗

10: if (S∗ < S) then11: S = S∗

12: ψ = ψ∗

13: vs = v∗s14: vr = v∗r15: end if16: end for17: Return {vs, vr, ψ}

Figures 2(a)-2(b)-2(c) illustrate the procedure describedearlier. In all examples, the left panels depict the vectors char-acterizing a particular reflection process, i.e. at a given aper-ture angle 2θ and azimuth φ = 45◦, and the right panels depictthe dependence of the angle ψ with the reflection angle θ. Fig-ure 2(a) corresponds to reflections in an isotropic material, i.e.ψ = 0 for all θ, and Figures 2(b) and 2(c) are for reflectionsin an anisotropic material characterized by vpz = 3.0 km/s,ε = 0.45 and δ = −0.29. In all cases, the normal vector isvertical, i.e. n = {0, 0, 1}, representing a horizontal reflector.The anisotropic material is VTI, Figure 2(b), or TTI tilted byθa = 35◦ at azimuth φa = 90◦ measured from the x axis,Figure 2(c). For the TTI material, the angle ψ is not constantand depends on the material properties, as well as the orienta-tion of the tilt vector and the reflection azimuth. For the specialcase of DTI materials, i.e. when the normal and anisotropy tiltvectors are aligned, the angle ψ = 0. The large black dot in theright panels corresponds to the geometry of the vectors shownin the left panels.

5 EXAMPLES

We illustrate our angle decomposition methodology usingthree simple examples. The models are homogeneous, with

a horizontal reflector, as shown in Figure 3. We considerisotropic, VTI and TTI models characterized by the parame-ters discussed in the preceding section. Panels 4(a)-4(c)-4(e)show wavefield snapshots, and panels 4(b)-4(d)-4(f) show thedata acquired on the surface for the three different cases. Asexpected, the moveout observed in the isotropic and VTI casesis azimuthally symmetric, while the moveout observed for theTTI case is not.

Figures 5(a)-5(c)-5(e) show the extended images ob-tained by reverse-time migration for one shot located on thesurface at coordinates {x = 4, y = 4, z = 0} km. The CIPanalyzed here is located at {x = 4.7, y = 4.7, z = 1} km.Figures 5(b)-5(d)-5(f) show the corresponding wide-azimuthangle gathers for the extended images in Figures 5(a)-5(c)-5(e), respectively. We use this geometry because the expectedreflection and azimuth angles for the homogeneous isotropicmedium are φ = 45◦ and θ = 45◦. Similarly, Figures 6(a)-6(i) show extended CIPs constructed with the TTI data for ninedifferent locations regularly spaced 0.7 km appart on a gridcentered on the source located at {x = 4, y = 4, z = 0} km.

The yellow dots overlain on the angle-gathers in Fig-ures 5(b)-5(d)-5(f) and Figures 6(a)-6(i) correspond to a nu-meric estimation of the reflection angles using two-point raytracing in the given TTI medium. We evaluate the reflectionangle θ using the expression (Sava and Fomel, 2006)

tan θ =|ph||pm|

, (42)

where the quantities pm and ph are evaluated from the sourceand receiver ray parameters, ps and pr, using the expressions

pm = pr + ps , (43)

ph = pr − ps . (44)

For a horizontal reflector, the angle φ is given by the azimuthof the incident or reflected phase slowness vectors. These ex-amples demonstrate the accuracy of our angle decomposition,even in media characterized by strong anisotropy.

So far, we have demonstrated angle decomposition forindependent shots and CIPs, although our technique does notneed to be applied to CIPs created independently. Our pro-cedure is capable of separating the expression of differentshots from extended CIP. Figures 7(a)-7(f) are similar to Fig-ures 5(a)-5(f), except that imaging is performed using shotslocated on a regular grid at z = 0 km with spacing of 0.4 kmin x and y. Each shots illuminates this point in the subsurfaceat different angles θ and φ, as illustrated in Figures 7(b)-7(d)-7(f). When imaging in VTI medium, the reflection angles de-crease compared to the isotropic case. When imaging in theTTI medium, the reflection illuminate predominantly in thedirection of the tilt, as expected. Each shot leaves a differentimprint in the angle-gather.

6 DISCUSSION

The exploration community is relying increasingly on reverse-time imaging of data from media characterized by TI

48 P. Sava & T. Alkhalifah

q

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0 20 40 60 80

−40

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ψ(°

)

(a)

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0 20 40 60 80

−40

−30

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10

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)

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0 20 40 60 80

−40

−30

−20

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10

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40

θ(°)

ψ(°

)

(c)

Figure 2. Reflection geometry (left panels) and dependence of the angle ψ with the aperture angle θ (right panels). From top to bottom, the figurescorrespond to (a) isotropic, (b) VTI and (c) TTI materials.

Anisotropic wide-azimuth angle-gathers 49

Figure 3. Geometry of the synthetic experiments. The horizontal reflector is located at z = 1 km and the sources are distributed on the surface ona regular grid with separation of 0.4 km in x and y.

anisotropy. This imaging method can handle abrupt changesin the symmetry axis direction, as well as heterogeneity inall medium parameters, including velocity and anisotropy.Though the efficiency of the TI reverse-time migration meth-ods leave plenty of room for improvement, our biggest chal-lenge is attaining the anisotropy information required torun these migration algorithms. Building wide-azimuth anglegathers and understanding their dependence on the anisotropyparameters are helpful toward addressing this challenge. Theangle gather extraction approach developed here works withreverse-time migration and is specifically designed for pro-cessing of wide-azimuth data.

As discussed earlier, angle domain imaging can be doneby tracking the wave propagation directions in the recon-structed wavefields at all locations in the subsurface. This isboth costly and difficult, since this approach relies on identifi-cation of wavepaths in large and complex wavefields, followedby selection of a few most energetic paths for mapping in theangle domain. Our alternative approach uses the intermediatestep of constructing the extended images at relevant positionsin the subsurface, followed by angle decomposition. All rel-evant angle-domain information is available in the extendedimages, which makes the decomposition both cheap and ro-bust, since we do not need to select any specific propagationdirection.

Figures 8-9(b) illustrate this idea. Here, we consider anisotropic model which is generally homogeneous, except fortwo vertical walls which are arranged to generate additionalpropagation paths in the center of the model, Figure 8. Thesource is located at coordinates {x = 1, y = 1, z = 0} km.Figures 9(a) and 9(b) show a wavefield snapshot and the data,respectively, and indicate that there are four different prop-

agation directions causing reflections in the subsurface. Fig-ure 10(a) show the extended CIP at coordinates {x = 1, y =1, z = 1.2} km. Different reflection paths are visible in theextended cube, but angle decomposition separates these prop-agation directions as a function of reflection angles. For exam-ple, Figure 10(b) shows the corresponding angle gather for theextended CIP shown in Figure 10(a). The four different prop-agation paths are visible in the image with their respectivesamplitude and reflection angles. We can observe a reflection atnormal incidence, two reflections at approximately 40◦ in the0◦ and 90◦ azimuths, and prismatic reflections visible along45◦ azimuth. All reflection directions are simultaneously andcorrectly mapped with no additional cost over the mapping ofone individual direction. Furthermore, this conclusion remainstrue if the extended CIPs are constructed with multiple shots,as seen in the examples presented in the preceding section, orif the wavefield extrapolation is done in TTI media with arbi-trary tilt and orientation.

7 CONCLUSIONS

Wide-azimuth angle decomposition can be performed on in-dividual common-image-point gathers constructed using theextended imaging condition in space and time. The extendedimaging condition is applicable equally well when migration isdone with downward continuation or time reversal, or whetherthe model used for wavefield reconstruction is isotropic oranisotropic.

The angle decomposition can separate the precise reflec-tion angles (aperture and azimuth), regardless of the complex-ity of wave propagation in the overburden and the heterogene-

50 P. Sava & T. Alkhalifah

(a) (b)

(c) (d)

(e) (f)

Figure 4. Wavefield snapshots (left panels) and surface data (right panels) for the experiment described in Figure 3. From top to bottom, the panelscorrespond to (a)-(b) isotropic, (c)-(d) VTI and (e)-(f) TTI materials.

Anisotropic wide-azimuth angle-gathers 51

(a) (b)

(c) (d)

(e) (f)

Figure 5. Extended CIPs (left panels) and wide-azimuth angle gathers (right panels) at coordinates {x = 4.7, y = 4.7, z = 1} km obtained frommigration of one source located at {x = 4, y = 4, z = 0} km. From top to bottom, the panels correspond to (a)-(b) isotropic, (c)-(d) VTI and(e)-(f) TTI materials.

52 P. Sava & T. Alkhalifah

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 6. Wide-azimuth angle gathers at nine locations along the reflector distributed in a regular grid with separation of 0.7 km in x and y. Thesource is at coordinates {x = 4, y = 4, z = 0} km and the model is characterized by TTI anisotropy.

Anisotropic wide-azimuth angle-gathers 53

(a) (b)

(c) (d)

(e) (f)

Figure 7. Extended CIPs (left panels) and wide-azimuth angle gathers (right panels) at coordinates {x = 4, y = 4, z = 1} km obtained frommigration of shots located on a regular grid at z = 0 km with spacing of 0.4 km in x and y. From top to bottom, the panels correspond to (a)-(b)isotropic, (c)-(d) VTI and (e)-(f) TTI materials.

54 P. Sava & T. Alkhalifah

Figure 8. The velocity model for the synthetic experiment showing reflections of triplicating wavefields.

(a) (b)

Figure 9. (a) A wavefield snapshot and (b) the data acquired on the surface for the synthetic experiment showing reflections of triplicating wavefields.

Anisotropic wide-azimuth angle-gathers 55

(a) (b)

Figure 10. (a) Extended CIP and (b) wide-azimuth angle gather at coordinates {x = 1, y = 1, z = 1.2} km for the model with reflections frommultiple directions shown in Figures 8-9(b).

ity of the model parameters. In particular, our technique doesnot require the TI anisotropic model to align with the reflectororientation, thus giving us full flexibility to handle arbitrary TImedia.

The extended common-image-point gathers encode infor-mation from all wavefield branches and all experiments usedfor imaging. Thus angle decomposition unravels all illumi-nation directions equally well and at no additional computa-tional cost over that of imaging an individual event from agiven experiment. The methodology is mainly applicable tothe study of illumination in the subsurface, and potentially foramplitude-versus-angle analysis.

8 ACKNOWLEDGMENTS

We acknowledge stimulating discussions with graduatestudents Bharath Shekar and Julio Frigerio. The repro-ducible numeric examples in this paper use the Mada-gascar open-source software package freely available fromhttp://www.reproducibility.org.

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