Residual migration in VTI media using anisotropy continuation

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1 Stanford Exploration Project, Report SERGEY, Noember 9, 2000, pages 671?? Residual migration in VTI media using anisotropy continuation Tariq Alkhalifah Sergey Fomel 1 ABSTRACT We introduce anisotropy continuation as a process which relates changes in seismic images to perturbations in the anisotropic medium parameters. This process is constrained by two kinematic equations, one for perturbations in the normal-moeout NMO) elocity the other for perturbations in the dimensionless anisotropy parameter η. We consider separately the case of post-stack migration show that the kinematic equations in this case can be soled explicitly by conerting them to ordinary differential equations by the method of characteristics. Comparing the results of kinematic computations with synthetic numerical experiments confirms the theoretical accuracy of the method. INTRODUCTION A well-known paradox in seismic imaging is that the detailed information about the subsurface elocity is required before a reliable image can be obtained. In practice, this paradox leads to an iteratie approach to building the image. It looks attractie to relate small changes in elocity parameters to inexpensie operators perturbing the image. This approach has been long known as residual migration. A classic result is the theory of residual post-stack migration Rothman et al., 1985), extended to the prestack case by Etgen 1990). In a recent paper, Fomel 1996) introduced the concept of elocity continuation as the continuous model of the residual migration process. All these results were based on the assumption of the isotropic elocity model. Recently, emphasis has been put on the importance of considering anisotropy its influence on data. Alkhalifah Tsankin 1995) demonstrated that, for TI media with ertical symmetry axis VTI media), just two parameters are sufficient for performing all time-related processing, such as normal moeout NMO) correction including non-hyperbolic moeout correction, if necessary), dip-moeout DMO) correction, prestack poststack time migration. One of these two parameters, the short-spread NMO elocity for a horizontal reflector, is gien by nmo 0) = 1 + 2δ, 1) where is the ertical P-wae elocity, δ is one of Thomsen s anisotropy parameters 1 tariq@sep.stanford.edu,sergey@sep.stanford.edu 671

2 672 Alkhalifah Fomel SEP 94 Thomsen, 1986). Taking h to be the P-wae elocity in the horizontal direction, the other parameter, η, is gien by η h ɛ δ nmo 2 1) = 0) 1 + 2δ, 2) where ɛ is another of Thomsen s parameters. In addition, Alkhalifah 1997) has showed that the dependency on just two parameters becomes exact when the ertical shear wae elocity V S0 ) is set to zero. Setting V S0 = 0 leads to remarkably accurate kinematic representations. It also results in much simpler equations that describe P-wae propagation in VTI media. Throughout this paper, we use these simplified, yet accurate, equations, based on setting V S0 = 0, to derie the continuation equations. Because we are only considering time sections, for the sake of simplicity, we denote nmo by. Thus, time processing in VTI media, depends on two parameters η), whereas in isotropic media only counts. In this paper, we generalize the elocity continuation concept to hle VTI media. We define anisotropy continuation as the process of seismic image perturbation when either or η change as migration parameters. This approach is especially attractie, when the initial image is obtained with isotropic migration that is with η = 0). In this case, anisotropy continuation is equialent to introducing anisotropy in the model without the need of repeating the migration step. For the sake of simplicity, we start from the post-stack case purely kinematic description. We define howeer the guidelines for moing to the more complicated interesting cases of prestack migration dynamic equations. The results are preliminary, but they open promising opportunities for seismic data processing in presence of anisotropy. THE GENERAL THEORY In the case of zero-offset reflection, the ray trael distance, l, from the source to the reflection point is related to the two-way zero-offset time, t, by the simple equation l = g t, 3) where g is the half of the group elocity, best expressed in terms of its components, as follows: g = 2 gx gτ. Here gx denotes the horizontal component of group elocity, is the ertical P-wae elocity, gτ is the -normalized ertical component of the group elocity. Under the assumption of zero shear-wae elocity in VTI media, these components hae the following analytic expressions: gx = 2 p x 1 2η + 2η pτ 2 ) η) p x 2 + p τ 2, 4)

3 SEP 94 Anisotropy Continuation 673 gτ = η p x 2 ) p τ η) p x 2 + p τ 2, 5) where p x is the horizontal component of slowness, p τ is the normalized again by the ertical P-wae elocity ) ertical component of slowness. The two components of the slowness ector are related by the following eikonal-type equation Alkhalifah, 1997): p τ = 1 2 p x η p x 2. 6) Equation 6) corresponds to a normalized ersion of the dispersion relation in VTI media. If we consider η as imaging parameters migration elocity migration anisotropy coefficient), the ray length l can be taken as an imaging inariant. This implies that the partial deriaties of l with respect to the imaging parameters are zero. Therefore, l = g t + g l η = g η t + g t = 0, 7) t = 0. 8) η Applying the simple chain rule to equations 7) 8), we obtain t = t, t η = t η, 9) where t = p τ, the two-way ertical traeltime is gien by τ = gτ t. Combining equations 7-9) eliminates the two-way zero-offset time t, which leads to the equations = g τ p τ gτ g, 10) η = g η τ p τ gτ g. 11) After some tedious algebraic manipulation, we can transform equations 5) 6) to the general form = τ F p x,,η), 12)

4 674 Alkhalifah Fomel SEP 94 η = τ F η p x,,η). 13) Since the residual migration is applied to migrated data, with the time axis gien by τ the reflection slope gien by x, instead of t p x, respectiely, we need to eliminate p x from equations 12) 13). This task can be achieed with the help of the following explicit relation, deried in Appendix A, p 2 x = 2τ x η) τ x 2 + S, 14) where τ x = x, S = 8 2 η τ x η) τ x 2 ) 2. Inserting equation 14) into equations 12) 13) yields exact, yet complicated equations, describing the continuation process for η. In summary, these equations hae the form ) = τ f x,,η 15) ) η = τ f η x,,η. 16) Equations of the form 15) 16) contain all the necessary information about the kinematic laws of anisotropy continuation in the domain of zero-offset migration. Linearization A useful approximation of equations 15) 16) can be obtained by simply setting η equal to zero in the right-h side of the equations. Under this approximation, equation 15) leads to the kinematic elocity-continuation equation for elliptically anisotropic media, which has the following relatiely simple form: = τ ) τ x τ x 2 ) τ x 2. 17) It is interesting to note that setting =, yields Fomel s expression for isotropic media Fomel, 1996) gien by = τ τ x 2. 18)

5 SEP 94 Anisotropy Continuation 675 Alkhalifah Tsankin 1995) hae shown that time-domain processing algorithms for elliptically anisotropic media should be the same as those for isotropic media. Howeer, in anisotropic continuation, elliptical anisotropy isotropy differ by a ertical scaling factor that is related to the difference between the ertical NMO elocities. In isotropic media, when elocity is continued, both the ertical NMO elocities which are the same) are continued together, whereas in anisotropic media including elliptically anisotropic) the NMO-elocity continuation is separated from the ertical-elocity one, equation 19) corresponds to continuation only in the NMO elocity. This also implies that equation 19) is more flexible than equation 18), in that we can isolate the ertical-elocity continuation a parameter that is usually ambiguous in surface processing) from the rest of the continuation process. Using τ = z, where z is depth, we immediately obtain the equation = τ, which represents the ertical-elocity continuation. Setting η = 0 = in equation 16) leads to the following kinematic equation for η-continuation: η = τ4 4 τ x τ 2. 19) x We include more discussion about different aspects of linearization in Appendix B. The next section presents the analytic solution of equation 19). Later in this paper, we compare the analytic solution with a numerical synthetic example. ORDINARY DIFFERENTIAL EQUATION REPRESENTATION: ANISOTROPY RAYS According to the classic rules of mathematical physics, the solution of the kinematic equations 15) 16) can be obtained by soling the following system of ordinary differential equations: dx dm = τ f m x, dτ m dm = τ f m m + τ m f m, dτ dm = ττ f m x + τ m, x dτ x dm = τ x f m. 20) Here m sts for either or η, τ x = x, f = m. To trace the η rays, we must first identify the initial alues x 0, τ 0, τ x0, τ m0 from the boundary conditions. The ariables x 0 τ 0 describe the initial position of a reflector in a time-migrated section, τ x0 describes its migrated slope, τ m0 is simply f m 0,τ 0,τ x0 ). Using the exact kinematic expressions for f results in rather complicated representations of the ordinary differential equations. The linearized expressions, on the other h, are simple allow for a straightforward analytical solution.

6 676 Alkhalifah Fomel SEP 94 From kinematics to dynamics The kinematic η-continuation equation 19) corresponds to the following linear fourth-order dynamic equation 4 P t 3 η + 4 P 2 x 2 t η + t4 4 P x 4 = 0, 21) where the t coordinate refers to the ertical traeltime τ, Pt, x,η) is the migrated image, parameterized in the anisotropy parameter η. To find the correspondence between equations 19) 21), it is sufficient to apply a ray-theoretical model of the image Pt, x,η) = Ax,η) f t τx,η)) 22) as a trial solution to 21). Here the surface t = τx,η) is the anisotropy continuation waefront - the image of a reflector for the corresponding alue of η, the function A is the amplitude. Substituting the trial solution into the partial differential equation 21) considering only the terms with the highest asymptotic order those containing the fourth-order deriatie of the waelet f ), we arrie at the kinematic equation 19). The next asymptotic order the third-order deriaties of f ) gies us the linear partial differential equation of the amplitude transport, as follows: τx 2 ) A η + 22 τ x τη 2 2 ττx 2 ) A x + 2 A 2τ x τ xη + τ η τ xx 6 2 ττx 2 τ ) xx = 0. 23) We can see that when the reflector is flat τ x = 0 τ xx = 0), equation 23) reduces to the equality A η = 0, the amplitude remains unchanged for different η. This is of course a reasonable behaior in the case of a flat reflector. It doesn t guarantee though that the amplitudes, defined by 23), behae equally well for dipping cured reflectors. The amplitude behaior may be altered by adding low-order terms to equation 21). According to the ray theory, such terms can influence the amplitude behaior, but do not change the kinematics of the wae propagation. An appropriate initial-alue condition for equation 21) is the result of isotropic migration that corresponds to the η = 0 section in the t, x,η) domain. In practice, the initial-alue problem can be soled by a finite-difference technique. SYNTHETIC TEST Residual post-stack migration operators can be obtained by generating synthetic data for a model consisting of diffractors for gien medium parameters then migrating the same data with different medium parameters. For example, we can generate diffractions for isotropic media migrate those diffractions using an anisotropic migration. The resultant operator

7 SEP 94 Anisotropy Continuation 677 Distance km) Distance km) Time s) 1.0 Time s) Figure 1: Residual post-stack migration operators calculated by soling equation 19), oerlaid aboe synthetic operators. The synthetic operators are obtained by applying TI poststack migration with η = 0.1 left) η = 0.2 right) to three diffractions generated considering isotropic media. The NMO elocity for the modeling migration is 2.0 km/s. anico-impres [NR]

8 678 Alkhalifah Fomel SEP 94 describes the correction needed to transform an isotropically migrated section to an anisotropic one, that is the anisotropic residual migration operator. Figure 1 shows such synthetic operators oerlaid by kinematically calculated operators that were computed with the help of equation 19) the continuation equations for the case of small η). Despite the inherent accuracy of the synthetic operators, they suffer from the lack of aperture in modeling the diffractions, therefore, beyond a certain angle the operators anish. The agreement between the synthetic calculated operators for small angles, especially for the η = 0.1 case, promises reasonable results in future dynamic implementations. CONCLUSIONS We hae extended the concept of elocity continuation in isotropic media to continuations in both the NMO elocity the anisotropy parameter η for VTI media. Despite the fact that we hae considered the simple case of post-stack migration separately, the exact kinematic equations describing the continuation process are anything but simple. Howeer, useful insights into this problem are deduced from linearized approximations of the continuation equations. These insights include the following conclusions: The leading order behaior of the elocity continuation is proportional to τ 2 x, which corresponds to small or moderate dips. The leading order behaior of the η continuation is proportional to τx 4, which corresponds to moderate or steep dips. Both leading terms are independent of the strength of anisotropy η). In practical applications, the initial migrated section is obtained by isotropic migration,, therefore, the residual process is used to correct for anisotropy. Setting η = 0 in the continuation equations for this type of an application is a reasonable approximation, gien that η=0 is the starting point we consider only weak to moderate degrees of anisotropy η 0.1). Numerical experiments with synthetically generated operators confirm this conclusion. Future directions of research will focus on a practical implementation of anisotropy continuation as well as on extending the theory to the case of residual prestack migration. REFERENCES Alkhalifah, T., Tsankin, I., 1995, Velocity analysis for transersely isotropic media: Geophysics, 60, Alkhalifah, T., 1997, Acoustic approximations for processing in transersely isotropic media: submitted to Geophysics. Etgen, J., 1990, Residual prestack migration interal elocity estimation: Ph.D. thesis, Stanford Uniersity.

9 SEP 94 Anisotropy Continuation 679 Fomel, S., 1996, Migration elocity analysis by elocity continuation: SEP 92, Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., 1992, Numerical recipes, the art of scientific computing: Cambridge Uniersity Press, New York. Rothman, D. H., Lein, S. A., Rocca, F., 1985, Residual migration applications limitations: Geophysics, 50, no. 1, Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, no. 10, APPENDIX A RELATING THE ZERO-OFFSET AND MIGRATION SLOPES The chain rule of differentiation leads to the equality p x = t x = p τ x, where p τ = t. It is conenient to transform equality A-1) to the form x = p x p τ. A-1) A-2) Using the expression for p τ from the main text, we can write equation A-2) as a quadratic polynomial in px 2 as follows apx 4 + bp2 x + c = 0, A-3) where a = 2 2 η, b = x ) η) + 1, c = x )2. Since η can be small as small as zero for isotropic media), we use the following form of solution to the quadratic equation px 2 = 2c b ± b 2 4ac Press et al., 1992). This form does not go to infinity as η approaches 0. We choose the solution with the negatie sign in front of the square root, because this solution complies with the isotropic result when η is equal to zero.

10 680 Alkhalifah Fomel SEP 94 APPENDIX B LINEARIZED APPROXIMATIONS Although the exact expressions might be sufficiently constructie for actual residual migration applications, linearized forms are still useful, because they gie us aluable insights into the problem. The degree of parameter dependency for different reflector dips is one of the most obious insights in the anisotropy continuation problem. Perturbation of a small parameter proides a general mechanism to simplify functions by recasting them into power-series expansion oer a parameter that has small alues. Two ariables can satisfy the small perturbation criterion in this problem: The anisotropy parameter η η << 1) the reflection dip τ x τ x << 1 or p x << 1). Setting η = 0 yields equation 19) for the elocity continuation in elliptical anisotropic media η = 4 4 τ τ x τ 2 x τ )) x τ ) 2 x τ ). B-1) 2 x which represents the case when we initially introduce anisotropy into our model. Because p x the zero-offset slope) is typically lower than τ x the migrated slope), we perform initial expansions in terms of y = p x. Applying the Taylor series expansion of equations 12) 13) in terms of y dropping all terms beyond the fourth power in y, we obtain = τ p 2 x ) 3 4 τ p x ) η) ), B-2) 2 4 η = 4 4 τ p x ). B-3) 2 Although both equations are equal to zero for p x =0, the leading term in the elocity continuation is proportional to px 2, whereas the the leading term in the η continuation is proportional to px 4. As a result the elocity continuation has greater influence at lower angles than the η continuation. It is also interesting to note that both leading terms are independent of the size of anisotropy η). Despite the typically lower alues of p x, expansions in terms of τ x are more important, but less accurate. For small τ x, p x τ x,, therefore, the leading-term behaior of τ x expansions is the same as that of p x As a result, we arrie at the equation = τ ) 2 τ x + 4 τ ) + τ ) + 12η τ ) ) τ x, B-4)

11 SEP 94 Anisotropy Continuation 681 η = 4 τ ) 4 τ x. B-5) 2 Most of the terms in equations B-4) B-5) are functions of the difference between the ertical NMO elocities. Therefore, for simplicity without a loss of generality, we set = keep only the terms up to the eighth power in τ x. The resultant expressions take the form = τ τ x η τ τ x η 4 25η) τ τ x η 5 83η + 144η 2) τ τ x 8 η = 4 τ τ x η) τ τ x η + 156η 2) τ τ x 8. B-6) B-7) Curiously enough, the second term of the η continuation heaily depends on the size of anisotropy 20η). The first term of equation B-6) τx 2 ) is the isotropic term; all other terms in equations B-6) B-7) are induced by the anisotropy.

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