Prestack exploding reflector modelling and migration for anisotropic media

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1 Geophysical Prospecting, 015, 63, 10 doi: / Prestack exploding reflector modelling and migration for anisotropic media T. Alkhalifah Physical Sciences and Engineering, King Abdullah University of Science and Technology, Mail box # 180, Thuwal , Saudi Arabia Received November 01, revision accepted May 013 ABSTRACT The double-square-root equation is commonly used to image data by downward continuation using one-way depth extrapolation methods. A two-way time extrapolation of the double-square-root-derived phase operator allows for up and downgoing wavefields but suffers from an essential singularity for horizontally travelling waves. This singularity is also associated with an anisotropic version of the double-square-root extrapolator. Perturbation theory allows us to separate the isotropic contribution, as well as the singularity, from the anisotropic contribution to the operator. As a result, the anisotropic residual operator is free from such singularities and can be applied as a stand alone operator to correct for anisotropy. We can apply the residual anisotropy operator even if the original prestack wavefield was obtained using, for example, reverse-time migration. The residual correction is also useful for anisotropic parameter estimation. Applications to synthetic data demonstrate the accuracy of the new prestack modelling and migration approach. It also proves useful in approximately imaging the Vertical Transverse Isotropic Marmousi model. Key words: Anisotropic, Prestack, Migration. INTRODUCTION One of the most under appreciated characteristics of the double-square-root DSR) equation is its ability to relate the image to prestack data using a single operator. This characteristic is vital for velocity estimation since it is easy to evaluate and use gradients of a single operator. Wave-equation depth migration methods are commonly divided into two types: one-way for wave extrapolation in depth and two-way for wave extrapolation in time or reverse-time migration Etgen, Gray and Zhang 009, Sava 011). Conventionally, both methods are applied on individual shot gathers. With the one-way approach, it is also possible to combine all data multiple shot gathers) into one wave-extrapolation procedure with the survey-sinking or DSR double-square-root) formulation of the wave equation Claerbout 1985; Popovici 1996; de Hoop, Rousseau and tariq.alkhalifah@kaust.edu.sa Biondi 003). A limitation of the DSR formulation is the oneway nature of wave extrapolation, which limits the imaging accuracy at large structural dips. An alternative is to extend the survey-sinking approach to extrapolation in time rather than depth of the full source-andreceiver wavefield. Application of two-way extrapolators to modelling and migration follows the exploding reflector concept Loewenthal et al. 1976; Claerbout 1985) and allows for upgoing as well as downgoing wavefields. However, a major hindrance in solving the DSR equation is an inherent singularity in the extrapolation operator for horizontally travelling waves Biondi 00; Duchkov and de Hoop 009; Alkhalifah and Fomel 010). Residual implementations of the DSR based operator allows us to isolate the singularity in the extrapolation operator in the leading term, resulting in residual terms that are free from such singularities. In this paper, I derive the phase operator for the time extrapolation of prestack wavefields in transversely isotropic media with a vertical symmetry axis VTI C 014 European Association of Geoscientists & Engineers

2 Anisotropic prestack exploding reflector 3 media). The operator, like its isotropic counterpart, includes an essential singularity for horizontally travelling waves. Using perturbation theory, I recast the solution as a series expansion in the anisotropy parameter η, with the anisotropic contribution providing the residual correction. The resulting residual operators for extrapolation are free of singularities. Numerical tests show the accuracy of the new formulation. THEORY Consider a seismic survey Pt, s, r, z) as a function of time t and source and receiver locations s and r at the surface at depth z. Our goal is to extrapolate the four-dimensional six-dimensional in 3D) wavefield Pt, s, r, z) in time. Let x represent the space coordinates x ={s, r, z}. The wave extrapolation operator, valid for small t, is Wards, Margrave and Lamoureux 008): Pt + t, x) + Pt t, x) Pt, k) cos[φx, k) t] e i k x dk, 1) where k ={k s, k r, k z }. In the geometrical high-frequency) approximation, the function φx, k, t) appearing in equation 1) should satisfy the appropriate eikonal equation, which is, in the case of prestack data, the double-square-root DSR) equation Duchkov and de Hoop 009; Alkhalifah and Fomel 010). For VTI media, it has the following form Alkhalifah 000; Duchkov and de Hoop 010): p z = 1 v vr vr 1 pr 1 vr η r pr + 1 v vs 1 p s v s 1 p s v s η s, ) where p z = k z, p φ s = k s and p φ r = k r. Ignoring the velocity variation between the source and receiver, which is a good ap- φ proximation for small offsets, η = η s = η r, v = v s = v r and v v = v vs = v vr and thus, equation ), after squaring quantities under the square roots has the following polynomial form: 56vv 4 v 8 η 4 kr 4 ks 4 kz 4 18vv v 6 η 3 kr ks kz )) v 1 + η)kr ks + vv kz k r + ks φ ) + 3vv v 4 η kz v 3 + 8η)kr ks k r + ks )) + vv kz 4k r ks + kr 4 + ks 4 φ 4 16vv v ηkz v 1 + 4η) ) 4kr ks + kr 4 + ks 4 )) + 4vv kz k r + ks φ 6 + v ) 4 kr ks ) + 8v v v 1 + 8η)kz k r + ks + 16v 4 v k 4 z ) φ 8 16v v k z φ 10 = 0. 3) Equation 3) is a fifth-order polynomial in φ with analytical solutions having very complicated forms. Often we solve these type of equations numerically by starting at a point near the desired solution. An alternative approach is provided by perturbation theory. In the η = 0 case, by substituting k s = k x k h and k r = k x + k h, equation 3) has the following solution v kh + ) ) v v kz v kx + vv kz φ =±, 4) v v k z which clearly demonstrates that the k z singularity is embedded in the desired solution of equation 3). Note, that this phase operator corresponds to the elliptically anisotropic case and reduces to the isotropic form Alkhalifah and Fomel 010) if v v = v. In the zero-offset case, k h = 0 and the solution reduces to φ =± v kx + vv kz, which is appropriate for post-stack exploding-reflector modelling and migration, however, for elliptically anisotropic media. Note that in this case the singularity vanishes. It also vanishes from equation 4) if we set k x = 0, which corresponds to the zero-dip case. A general bound on k z is given by kz 1 v k 4 vv r ks. PERTURBATION THEORY Analytical solutions for equation 3) are complex and they are even more complicated by recognizing that k z = 0isan essential singularity reducing the order of the polynomial. It, specifically, has a set of five complex conjugate solutions with the desired one given by the solution that reduces to equation 4) when η = 0. Thus, for η small and independent, I consider the following trial solution: φ φ 0 + φ 1 η + φ η. 5) Inserting this trial solution into equation 3) gives n i=0 ηi F i φ i ) = 0, where n for the above case is 5. Solving F 0 = 0 yields equation 4) for φ 0 φ, which is the solution for the elliptically anisotropic case. Solving F 1 = 0 yields: φ 1 = v 4 3k h k x + k 4 h v k h + v v k z + 3k h k x + k 4 x v k x + v v k z Finally, solving F = 0 yields: 4v 6 vv kz φ = v kh + ) v v kz 3 ) v kx + vv kz 3 ) 3vv v 4 kxk hk z k h + kx k h kx) v 6 khk 4 x 4 k h kx + 3kh 4 + 3kx) 4 ). 6) C 014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 10

3 4 T. Alkhalifah + vv 4 v khk xk z 4 10k h kx + 7kh 4 + 7kx) 4 ) )) + vv 6 kz 6 k h + kx 14k h kx + kh 4 + kx 4. 7) Both φ 1 and φ are free of the k z = 0 singularity. Actually, setting k z = 0givesφ 1 = 8v k h + k x) and φ = 0, which are approximations but at least they exist. An improvement to the accuracy of this polynomial based trial solution is accessible though Shanks transform Bender and Orszag 1978), which has, using the same coefficients in equation 5), the following form: φ φ 0 + ηφ 1 φ 1 ηφ. 8) This form also allows us to separate the singular elliptically anisotropic solution, φ 0, from the perturbation terms. ACCURACY We first investigate the accuracy of our perturbation based trial solution in representing the exact solution. To do so, we solve equation 3) numerically and compare the solution with the derived approximations. As demonstrated in Fig. 1, the phase per cent error associated with using the first-order approximation in equation 5) black curve) is higher than that using the second-order dashed black curve) or Shanks transform grey curve). Surprisingly, for this test the second order approximation yields lower errors than Shanks transform. This can happen if the second-order term of the expansion is dominant. Overall, all approximations show high accuracy with errors not exceeding 0.5 per cent. This holds for η = 0.1 left) and the stronger anisotropy η = 0. and thus, phase errors are generally low. IMPLEMENTATION To allow for a separate application of the anisotropic stable solution, I consider the time step, t,tobesmallandrecastthe extrapolation problem 1, using the Taylor s series expansion, as follows: Pt + t, x) + Pt t, x) = Pt, k) cos[φx, k) t] e i k x dk Pt, k) 1 1 ) t) φ0x, k) e i k x dk Pt, k) 1 t) φ x, k)e i k x dk Pt, x) t) t) Pt, k) φ 0x, k)e i k x dk Pt, k) φ x, k)e i k x dk. 9) Here φ can equal φ 1 η for a first-order approximation, φ 1 η + φ η for a second-order approximation, or for ηφ 1 φ 1 φ η the Shanks transform approximation. Equation 9) still requires computing the background elliptically anisotropic field along with the perturbation part. The second term in the approximation in equation 9) reduces to the elliptical Laplacian, v P + v x v P, in the D space domain. To separate the perturbation part, we assume Pt, x) = P 0 t, x) + Pt, x) and z thus, Pt + t, x) + Pt t, x) Pt, x) t) P 0 t, k) + Pt, k)) φ x, k)e i k x dk, 10) where P 0 is the wavenumber domain form of the prestack background wavefield for elliptically anisotropic media, which can be computed using other more stable means, like RTM Sava and Fomel 006). Note that equation 10) is free of the potentially singular φ 0 function. It can be used to estimate the residual prestack wavefield needed to correct for η. To test the accuracy of the phase operator after the second-order Taylor s series expansion of the cosine, I repeat the test in Fig. 1 using the second-order expansion approximation but for different time sampling lengths as the phase operator calculated by taking the arc tangent of the imaginary part over the real part) now depends on t. Figure shows the errors in the phase operator for the original second-order approximation solid lines) and the Taylor s series approximation of the exponential dashed lines). For t = 01 s a), the difference is negligible, however, for = 1 s c), it becomes obvious. For a reasonable sometimes considered large) time step of 04 s b), the errors are tolerable. To use equations 1) or 9) in their current form, we resort to spectral methods for time extrapolation. This, however, initially implies a prohibitively expensive implementation considering the multi-dimensional integration required at each time step. Different approaches exist to separate the space and wavenumber components of the integral so that it reduces to an inverse Fourier transform problem. These approaches replace the full integration over wavenumber with a number of inverse fast Fourier transforms with constant C 014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 10

4 Anisotropic prestack exploding reflector 5 ø Error%) a) ø Error%) b) k z km 1 ) k z km 1 ) Figure 1 Percent errors in the phase of the extrapolator operator given by the first-order solid black curves) and second-order dashed back curves) approximations, given in equation 5), as well as, the Shanks transform expansion 8) solid grey curve), for a) η = 0.1 andb)η = 0.. The model has a vertical velocity of km/s and a NMO velocity of.1 km/s. Also, k x = k h = 5km 1. ø Error%) a) b) c) k z km 1 ) ø Error%) ø Error%) k z km 1 ) k z km 1 ) Figure Percent errors in the phase of the extrapolator operator given by the second-order approximation solid black curves, same as in Fig. 1 dashed curve) and the additional Taylor s series expansion dashed back curves), given in equation 10), for t = 01 s a), t = 04 s b) and t = 1 s c). The model has a vertical velocity of km/s, NMO velocity of.1 km/s and η = 0.. Also, k x = k h = 5km 1. medium parameters. I use the low-rank approach suggested by Fomel, Ying and Song 010) to implement this solution. In addition to the feature of equation 10) in providing a residual implementation, equation 10) also has a low rank representation compared to the exponential form, resulting in a speed up in application. EXAMPLES One of the main applications of DSR extrapolation is prestack modelling and migration based on the exploding reflector assumption. In other words, we extrapolate the prestack wavefield in time in the midpoint-offset domain. For simplicity, I initially consider the French model French 1974) with a homogeneous medium with velocity equal to 000 m/s and η = 0.. To construct an exploding reflector implementation for modelling, the image is given by the reflectivity of the French model shown in Fig. 3a) and set to zero offset and time; the imaging condition in the prestack domain. I use first the second-order phase approximation given by equation 5). This phase operator is independent of the time-step, which is 05 s. Using equation 1) to extrapolate the wavefield in Fig. 3a) forward in time we obtain snapshots of the wavefield shown in Fig. 3b-i). The snapshots display the evolution of the prestack wavefield as it starts to propagate in the nonzerooffset region. As expected, a large portion of the energy travels downward and is not recorded. The prestack wavefield at depth, z = 0, as a function of time, is shown in Fig. 4 and corresponds to the recorded prestack data. All the events expected in the conventional prestack data, including diffractions, are clearly visible. If we run an isotropic DSR extrapolator backward in time starting with the data at the surface as a boundary condition Fig. 4), we obtain the final image of the wavefield shown in Fig. 5a) at time zero. It has all the hallmarks of ignoring anisotropy including a considerable residual at non-zero offset. On the other hand, if I use approximation 9), which allows for a residual implementation and the operator is now C 014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 10

5 6 T. Alkhalifah a) b) c) d) e) f) g) h) i) Figure 3 Snapshots of the VTI wavefield in the prestack domain starting with time equal to 0 a) to time equal to 4 s i) in an exploding reflector modelling application. The extrapolation operator is based on the second-order trial solution 5). The homogeneous medium has a velocity of km/sandη = 0.. The initial wavefield is the image at zero offset and time the imaging condition). The solid lines cutting through each of the sections of the 3D represent the locations of the slices of the 3D volume. C 014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 10

6 Anisotropic prestack exploding reflector 7 Figure 4 A cube plot of the resulting prestack VTI data at the surface using the proposed extrapolation method used in Fig. 3a-i). The lines crossing each section of the cube represent the location of the corresponding slices. a) b) Figure 5 Final wavefields at time equal to zero in the prestack domain obtained by DSR migration using a) an isotropic DSR operator b) a VTI opartor based on the second-order Taylor s series expansion 9). The data shown in Fig. 4 are injected into the wavefield in a reverse fashion at depth, z, equal to zero the surface boundary condition). dependent on the time step t = 05 s), I obtain the final image of the wavefield shown in Fig. 5b) at time zero. As predicted by the numerical analysis shown in Fig., the errors are small and thus, the image looks generally fine, granted that we still have minor aperture limitation artefacts. One can easily use the time and subsurface offset dependency of the wavefield to extract angle gathers Sava and Fomel 006, Fomel 011). C 014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 10

7 8 T. Alkhalifah a) b) Figure 6 The VTI anisotropic Marmousi model given by a) the original velocity model and b) an η model. THE ANISOTROPIC MARMOUSI DATA In the spirit of the isotropic Marmousi model, Alkhalifah 1997) built an anisotropic Marmousi model using the original isotropicmarmousi Fig. 6a) and an η model that follows closely the stratigraphy of the original model as shown in Fig. 6b) He also generated acoustic VTI data based on the model using the same original acquisition setting used in the original Marmousi data set. I insert the Marmousi VTI prestack data set in reverse time at the surface given by the zero depth boundary and extrapolate the three-dimensional wavefield that includes subsurface offset, in reverse time until time equals zero the imaging condition). Snapshots of the wavefield are shown in Fig. 7a-g). The snapshots display the evolution of the prestack wavefield as it starts to propagate from the surface to the zero -offset plane the image plane). Despite the many approximations used in deriving the extrapolator operator for VTI media, the resultant image given by the zero-offset plane is acceptable, with the assumed reservoir structure imaged at the right depth. DISCUSSION One of the main features of the double-square-root DSR) implementation is that we do not need to rely on the cross-correlation imaging condition and its many flaws) to relate the source and receiver wavefields in the imaging process. Among these flaws are the low-frequency artefacts associated with sharp velocity contrasts in the velocity model used in the imaging, as well as, the inaccurate treatment of amplitudes. The process also requires an artificially modelled source wavefield to cross-correlate with the backward extrapolated recorded wavefield. The DSR operator allows for direct extrapolation of the prestack data for imaging purposes. This feature is critical for migration velocity analysis as linearizing a single operator is far more effective than dealing with a cross-correlation step in the middle. The DSR operator, however, has a fundamental weakness brought about by the inherent singularity for horizontally travelling waves embedded in the operator. We can avoid the singularity in extrapolating the wavefield by applying the operator in the spectral domain, however, it does mitigate the limitation of handling horizontally travelling and overturned waves. Luckily, the anisotropic perturbation terms of the DSR operator are free of such singularities. They can be applied as standalone operators for residual corrections for anisotropy. As such, they can provide a platform for inverting for the anisotropy parameters. The single operator nature of the DSR implementation allows for direct linearization of the imaging process with respect to velocity by linearizing this single operator. In anisotropic media, we can do this by linearizing the residual anisotropic part. CONCLUSIONS The DSR equation forms the central ingredient for prestack wavefield extrapolation in time, necessary for prestack exploding reflector modelling and migration. However, the DSR equation as a time extrapolation operator has an inherent singularity for horizontally travelling waves. For VTI media, C 014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 10

8 Anisotropic prestack exploding reflector 9 a) b) d) f) c) e) g) Figure 7 Snapshots of the imaging process of the VTI wavefield in the prestack domain starting with time equal to.9 a) to time equal to 0 s i) in an exploding reflector modelling application. The extrapolation operator is based on the second-order trial solution 5). The exact velocity and η model no smoothing) was used in the imaging. The solid lines cutting through each of the sections of the 3D represent the locations of the slices of the 3D volume. this singularity still exits in the prestack time extrapolator operator. However, by recasting our operator as a pertubation series in η, I literally isolate the singularity in the background elliptical anisotropic medium operator, which makes the residual operator for η free of singularities. Using this operator, I formulate a residual extrapolation to correct for η. The accuracy of the residual operator is high and the numerical tests support this assertion. C ACKNOWLEDGEMENTS I thank Alexey Stovas, Anton Duchkov and Pavel Golikov for many stimulating discussions on the subject of DSR. I thank KAUST and specifically the AEA round 3 project for its support. I also thank the associate editor and two reviewers for their fruitful and critical review of the manuscript. 014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 10

9 10 T. Alkhalifah Figure 8 The final image of the Marmousi model extracted from the zero-offset slice of the prestack wavefield at time equal to zero, shown in Fig. 7g). REFERENCES Alkhalifah T An anisotropic Marmousi model. SEP-95: Stanford Exploration Project, Alkhalifah T Prestack phase-shift migration of separate offsets. Geophysics 65, Alkhalifah T. and Fomel S Source-receiver two-way wave extrapolation for prestack exploding-reflector modelling and migration. SEG, Expanded Abstracts 9, Bender C.M. and Orszag S.A Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill. Biondi B. 00. Reverse time migration in midpoint-offset coordinates. SEP-Report 111, Claerbout J.F Imaging the Earth s Interior. Blackwell Scientific Publications. de Hoop M.V., Rousseau J.H.L. and Biondi B.L Symplectic structure of wave-equation imaging: A path-integral approach based on the double-square-root equation. Geophysical Journal International 153, Duchkov A. and de Hoop M.V Extended isochron rays in prestack depth migration. 79th Annual International Meeting, Society of Exploration Geophysicists, Duchkov A. and de Hoop M Extended isochron rays in prestack depth map) migration. Geophysics 75, S139 S150. Etgen J., Gray S.H. and Zhang Y An overview of depth imaging in exploration geophysics. Geophysics 74, WCA5 WCA17. Fomel S., Ying L. and Song X Seismic wave extrapolation using lowrank symbol approximation. 80th Annual International Meeting, Society of Exploration Geophysicists. Fomel, S., 011. Theory of 3-D angle gathers in wave-equation seismic imaging, Journal of Petroleum Exploration and Production Technology 1, French W.S Two-dimensional and three-dimensional migration of model-experiment reflection profiles. Geophysics 39, Loewenthal D., Lu L., Roberson R. and Sherwood J The wave equation applied to migration. Geophysical Prospecting 4, Popovici A.M Prestack migration by split-step DSR. Geophysics 61, Sava, P Micro-earthquake monitoring with sparsely sampled data, Journal of Petroleum Exploration and Production Technology 1, Sava P. and Fomel S Time-shift imaging condition in seismic migration. Geophysics 71, S09 S17. Wards B.D., Margrave G.F. and Lamoureux M.P Phaseshift timestepping for reversetime migration. SEG, Expanded Abstracts 7, C 014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63, 10