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1 Downloaded 0/7/6 to Redistribution subject to SEG license or copright; see Terms of Use at 3D Beamlet migration in VTI medium Jian Mao *, Ru-Shan Wu, Universit of California at Santa Cruz; Biaolong Hua, Paul Williamson, TOTAL E&P INC at Houston Summar Beamlet migration is a wave equation based migration approach. The use of local background velocit and local perturbations results in a two-scale decomposition of beamlet propagators: a background propagator for largescale structures and a local phase screen correction for small-scale local perturbations. The beamlet propagator can handle strong lateral velocit variations with improved accurac. Local cosine basis beamlet propagator was proposed and developed recentl. It has been successfull applied to poststack and prestack depth migration for media with large lateral velocit variations. Here we extend the beamlet propagator into transversel isotropic medium with vertical smmetric axis (VTI) medium. We tested impulse responses, a snthetic dataset and a field dataset. The results demonstrated the validit of the VTI beamlet propagator. Introduction In most seismic processing and interpretation cases, we usuall assume that a medium is isotrop. In isotropic medium, velocit is the same regardless the direction of measurement. However, anisotrop exists in man subsurface media (Thomsen, 986), and the velocit depends on direction. If we use conventional methods to deal with the wave propagation in anisotropic media, it ma result in large errors, which can lead to lower resolution and misplaced images of subsurface targets. Therefore, we have to consider anisotrop if we want to achieve higherresolution and provide the details required for productiongeophsics. In the recent ears, a lot of efforts have been made to develop migration algorithms for anisotropic media. A number of approaches have been proposed to solve the anisotropic problem, including Gaussian beam methods and wave equation based methods (Alkhalifah, 995; Ball, 995; Alkhalifah, 998; Rousseau and Hoop, 00). Among these methods, wave equation based methods (onewa methods or full-wave based reverse time migration) can give much more accurate results than Kirchhoff migration. Zhang et al. (Zhang et al., 005) propose a prestack depth migration method based on a one-wa wave equation. The extend the Fourier finite-difference method (FFD) to VTI media. B introducing a reference velocit model at each depth during wave field extrapolation, the method can handle the strong lateral velocit variation. Beamlet migration approach is based on local reference velocities and local perturbation (Wu et al., 000). Migration methods based on this theor have been developed using the Gabor-Daubechies frame (GDF) (Chen et al., 00, 006) and local cosine bases (LCB) (Wang and Wu, 00; Luo and Wu, 003; Wang et al., 003; Luo et al., 004; Wu et al., 008). In these methods, the wavefields are localized spatiall with local windows and directionall with local wavenumbers. The wavefield at each depth is propagated with beamlet propagators (sparse propagator matrices), followed b local perturbation corrections. These methods can provide good imaging results when compared to traditional wave-equation based methods. The non-orthogonal Gabor-Daubechies frame decomposition needs at least a redundanc ratio of 4 in the wavefield representation, which results in large computational cost. Since the local cosine transform is orthogonal and has a fast algorithm, the wavefield decomposition and extrapolation using LCB beamlet have good computation efficienc. In this work, we first derived the 3D beamlet propagator for VTI medium b using the dispersion relationship in VTI medium. Then we show some numerical examples for both snthetic data and field data. The numerical results demonstrated the capabilit of VTI beamlet propagators. Theor and Methods The frequenc-space-domain wavefield can be decomposed into beamlets, which provides us the localized information in both space and direction simultaneousl. Beamlet transform uses translated window for the spatial localization and harmonic modulations for the directional localization. Both orthogonal bases (e.g., local cosine bases (LCB)) (Wu et al., 008) and tight frames (e.g., Gabor- Daubechies frame) (Chen et al., 006; Wu and Chen, 006) were introduced into the beamlet decomposition. For the consideration of computational efficienc, we prefer orthogonal bases, which have no redundanc and have efficient decomposition and reconstruction. The LCB beamlet migration has been successfull tested through 3D models (Luo and Wu, 003; Wang et al., 003). Beamlet propagator can handle models with large velocit variations (e.g. salt models) because of the local perturbation approximation (Wu et al., 000), which results in better image qualit. On the other hand, the efficienc of beamlet propagator is comparable with other wave-equation based one-wa propagators (e.g. Generalized screen propagator (de Hoop et al., 000)). As a result, we extend 3D beamlet propagator to VTI case here..two dimensional local cosine bases 0 SEG DOI SEG Las Vegas 0 Annual Meeting Page

2 3D Beamlet migration in VTI medium Downloaded 0/7/6 to Redistribution subject to SEG license or copright; see Terms of Use at A local cosine basis element can be specified b its spatial position x n, interval (the nominal length of the window) xn xn, and the wavenumber index m ( m0,, M, M denotes the sample point number of each interval; n0,, N, N denotes the total interval number) as follows where b ( c) mn ( c) mn n m n cos b x B x x x, () x denote the local cosine bases. The local m horizontal wavenumber is defined b m. The expression Bn x denotes a bell function (see Appendix A for detailed definition), which is smooth and supported in the compact interval [ xn, xn ] for x x, with, as the left and right n n overlapping radius, respectivel. For the 3D case, the two dimensional local cosine bases function is composed b two one dimensional local cosine functions as follow: b ( x, ) mm, nn mn mn b x b, cos cos Bn, n x m xx n m x n L n () where m, x n and m, n are the local horizontal wavenumber and the local window position along x-axis and -axis respectivel., are the interval length along x-axis and -axis. The expression Bn, n x, is the two dimensional bell function. Then the decomposition coefficients of a D function f x, b the two dimensional local cosine transform can be calculated b the inner product as follow fˆ f x,, b x,. (3) jl, mn mm, nn Similar with the D case, we use fast local cosine transform instead of calculating the inner product of equation (3) directl. In practice, we do local cosine transform along x- axis first and then along -axis. In the Appendix, we give more details about local cosine transform and its fast implementation..beamlet propagation in the 3D case The wave field at depth z can be decomposed into beamlets with windows along the x-axis and -axis uxz (,, ) u( x, ) n m n m n m n m z ub, ( x, ) b ( x, ) mm, nn mm, nn uˆ ( x,,, ) b ( x, ) z n n m m mm, nn where bmm, nn ( x, ) are the two dimensional beamlet atoms, uˆ z( xn,,, ) n m m are the coefficients of the decomposition locate at space locus xn, n and wavenumber locus m, m. The wavefield at depth z z can be extrapolated b the () background propagator P and perturbation operator P as follow uz z( x, ) () b ( x, ) P ( x, ). (5) jj, ll ll, nn l j l j n n P ˆ jjll, mmnnu (,,, ) z xn n m m m m The background propagator 3.Beamlet propagator in VTI medium P can be calculated as For three dimensional acoustic wave equation, the dispersion relation for isotropic medium can be expressed as follow, (6) k 0 x Where k0 ( is circular frequenc and V 0 is V0 velocit), x and are horizontal wavenumber along x- axis and -axis respectivel, is vertical wavenumber. In VTI medium, the dispersion relation is formulated as follow k Where k0, V x and V z QVx x Q Vx x 0 (4), (7) V z are horizontal and vertical phase velocit respectivel, with Vx Vz. Q and, where and are Thomsen parameter (Thomsen, 986). In isotropic medium, 0, 0, 0, Q and Vx Vz. Equation (7) can also be expressed as follow 0 SEG DOI SEG Las Vegas 0 Annual Meeting Page

3 3D Beamlet migration in VTI medium Downloaded 0/7/6 to Redistribution subject to SEG license or copright; see Terms of Use at k x x 0 k 0. (8) With this new dispersion relation, we can derive the beamlet propagator for VTI case similarl. The VTI beamlet propagator still can be spitted into two parts: the background propagator P and perturbation operator () () P. The perturbation operator P stas the same as that in isotropic case. The background propagator P can be expressed as Pjjll, mmnn P ( xl, l, j, j ; x n, n, m, m ) d xd 4 ( ) ij L / / n ˆ ij L n e b ˆ 0( x j ) e b 0( x j ). (9) im L / / n ˆ im L n e b ˆ 0( x m ) e b 0( x m ) i j L / n e bˆ i 0 ( j L / n ˆ j ) e b 0( ) j im L / / n ˆ im L n e b ˆ 0( m ) e b 0( m ) ix( xl x ) ( ) n i l n i n, nz e e e Where n n k xn, n x x, 0 local vertical wavenumber. Numerical examples. Impulse response k 0 is the Figure Impulse response using FFD propagator in VTI medium Figure Impulse response using LCB propagator in VTI medium First, we calculated the impulse response for a homogeneous medium, in which the vertical phase velocit is 000 m/s, and the Thomsen parameters 0.3, 0.. Figure and Figure are the impulse responses generated b FFD propagator and LCB propagator. We see that the FFD result has big errors along the diagonal directions due to the split-step wide angle corrections.. Snthetic data Then we tested the snthetic VTI model with steep structures. Figure 3 shows the velocit map. The anisotrop parameter is set in different laers from the top to bottom. Figure 4 gives the poststack migration images using conventional isotropic LCB propagator and VTI LCB propagator. We can see the improvement of the migration image using VTI propagators, especiall for the position of the steep events and the structures in deep area. Figure 3 Velocit map of steep model 0 SEG DOI SEG Las Vegas 0 Annual Meeting Page 3

4 Downloaded 0/7/6 to Redistribution subject to SEG license or copright; see Terms of Use at 3D Beamlet migration in VTI medium Figure 6 Poststack migration image using VTI LCB propagator Figure 4 poststack migration image with isotropic LCB (upper) and VTI LCB (lower) propagators 3. Field data Finall, we use our VTI propagator on a real dataset. Figure 5 and Figure 6 are the poststack migration images using isotropic and VTI LCB propagator respectivel. We see the continuit for some structures are improved a lot after using VTI propagator. Conclusions We derived 3D beamlet propagator for VTI medium. As FFD propagator has the anisotropic correction in the wide angle correction part, it results in inaccurac b using the split-step implementation. The impulse response shows ver clear difference compared with FFD method. Both the snthetic data and field data tests show the validit of the VTI beamlet propagator. For the future work, we will extend the beamlet propagator to TTI medium. Acknowledgement We d like to thank TOTAL E&P USA give us the opportunit to present this work. We thank Huimin Guan and Fuchun Gao in TOTAL R&T group, Yaofeng He in Exxon Mobil, Jun Cao in Conoco Philips and Xiaobi Xie from UC Santa Cruz for helpful discussions. We also appreciate the HPC help from Terrence Liao and Saber Feki in TOTAL E&P USA. Figure 5 Poststack migration image using isotropic LCB propagator 0 SEG SEG Las Vegas 0 Annual Meeting DOI Page 4

5 EDITED REFERENCES Downloaded 0/7/6 to Redistribution subject to SEG license or copright; see Terms of Use at Note: This reference list is a cop-edited version of the reference list submitted b the author. Reference lists for the 0 Annual International Meeting, SEG, Expanded Abstracts have been cop edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Alkhalifah, T., 995, Gaussian beam depth migration for anisotropic media: Geophsics, 60, Alkhalifah, T., 998, Acoustic approximations for processing in transversel isotropic media: Geophsics, 63, Ball, G., 995, Estimation of anisotrop and anisotropic 3-d prestack depth migration, offshore Zaire: Geophsics, 60, Cao, J., and R.-S. Wu, 008, Amplitude compensation for one-wa wave propagators in inhomogeneous media and its application to seismic imaging: Communications in Computational Phsics, 3, 03. Chen, L., R.-S. Wu, and Y. Chen, 00, Target-oriented prestack beamlet migration using gabordaubechies frames: 7nd Annual International Meeting, SEG, Expanded Abstracts, Chen, L., R.-S. Wu, and Y. Chen, 006, Target-oriented beamlet migration based on Gabor-Daubechies frame decomposition: Geophsics, 7, no., S37 S5. de Hoop, M.V., J.H. Le Rousseau, and R.-S. Wu, 000, Generalization of the phase-screen approximation for the scattering of acoustic waves: Wave Motion, 3, Luo, M., and R.-S. Wu, 003, 3D beamlet prestack depth migratio n using the local cosine basis propagator: 73rd Annual International Meeting, SEG, Expanded Abstracts, Luo, M., R.-S. Wu, and X.-B. Xie, 004, Beamlet migration using local cosine basis with shifting windows: 74th Annual International Meeting, SEG, Expanded Abstracts, Mao, J., and R.-S. Wu, 007, Illumination analsis using local exponential beamlets: 77th Annual International Meeting, SEG, Expanded Abstracts, Mao, J., and R.-S. Wu, 00, Target oriented 3D acquisition aperture correction in local wavenumber domain: 80th Annual International Meeting, SEG, Expanded Abstracts, Mao, J., R.-S. Wu, and J.-H. Gao, 00, Directional illumination analsis using the local exponential frame: Geophsics, 75, no. 5, S63 S74. Rousseau, J.H.L., and M.V.d. Hoop, 00, Scalar generalized-screen algorithms in transversel isotropic media with a vertical smmetr axis: Geophsics, 66, Thomsen, L., 986, Weak elastic anisotrop: Geophsics, 5, Wang, Y., R. Cook, and R.-S. Wu, 003, 3d local cosine beamlet propagator: 73rd Annual International Meeting, SEG, Expanded Abstracts, Wang, Y., and R.-S. Wu, 00, Beamlet prestack depth migration using local cosine basis propagator: 7nd Annual International Meeting, SEG, Expanded Abstracts, SEG DOI SEG Las Vegas 0 Annual Meeting Page 5

6 Downloaded 0/7/6 to Redistribution subject to SEG license or copright; see Terms of Use at Wu, R.-S., and L. Chen, 006, Directional illumination analsis using beamlet decomposition and propagation: Geophsics, 7, no. 4, S47 S59. Wu, R.-S., M. Luo, S. Chen, and X.-B. Xie, 004, Acquisition aperture correction in angle-domain and true-amplitude imaging for wave equation migration: 74th Annual International Meeting, SEG, Expanded Abstracts, Wu, R.-S., Y. Wang, and J. Gao, 000, Beamlet migration based on local perturbation theor: 70th Annual International Meeting, SEG, Expanded Abstracts, Wu, R.-S., Y. Wang, and M. Luo, 008, Beamlet migration using local cosine basis: Geophsics, 73, no. 5, S07 S7. Xie, X.-B., S. Jin, and R.-S. Wu, 006, Wave-equation-based seismic illumination analsis: Geophsics, 7, no. 5, S69 S77. Xie, X.-B., and R.-S. Wu, 00, Extracting angle domain information from migrated wavefield: 7nd Annual International Meeting, SEG, Expanded Abstracts, Zhang, L., B. Hua, and H. Calandra, 005, 3D Fourier finite difference anisotropic depth migration: 75th Annual International Meeting, SEG, Expanded Abstracts, SEG DOI SEG Las Vegas 0 Annual Meeting Page 6