Application of selective filtering non-staggered finite difference method to seismic wave simulation
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摘要: 引入计算空气声学领域的选择性滤波同位网格有限差分算法（SFFD法）用于二维地震波数值模拟．SFFD法使用经过优化的11点DRP同位网格差分格式，对空间一阶导数进行离散近似，同时采用选择性滤波方法来消除同位网格差分所产生的格点高频振荡，它既提高了数值模拟的精度， 又保证了求解过程的稳定性．数值实验结果表明，SFFD法能够达到O(Delta;x8, Delta;t4)阶交错网格算法同样的精度，同时该方法还具有很强的适应性，能够应用于存在着强泊松比差异的介质模型中，完整地模拟地震波传播过程中各类型的波场，并且对复杂非均匀介质的适应能力也很好．此外，由于避免了交错网格算法在曲线坐标系和一般各向异性介质的数值模拟时所需进行的复杂的插值运算， SFFD法在这些问题上也有着很好的应用前景．Abstract: In this study a selective filtering non-staggered finite difference method, called SFFD, is introduced to simulate seismic wave propagation in 2D media. SFFD utilizes optimized 11-point DRP (dispersion relation preserving) non-staggered finite difference scheme to discretize first-order spatial derivatives. In addition, selective filtering is applied to removing grid-to-grid oscillation caused by non-staggered algorithm. The selective filtering enhances the numerical accuracy and makes the simulation stable to implement. Test result demonstrates that SFFD can achieve the same accuracy as O(Delta;x8, Delta;t4) order staggered finite difference scheme. Moreover, the proposed algorithm is able to handle the media with high Poissonrsquo;s ratio. The media with strong inhomogeneity can also be treated by SFFD. As a non-staggered method, SFFD has potential application to seismic wave simulation in curvilinear coordinate system and general anisotropic media, in which complex interpolation must be performed for staggered scheme.
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董良国，马在田，曹景忠，王华忠，耿建华，雷兵，许世勇. 2000. 一阶弹性波方程交错网格高阶差分解法［J］. 地球物理学报，43(3)： 411——419.
祝贺君，张伟，陈晓非. 2009. 二维各向异性介质中地震波场的高阶同位网格有限差分模拟［J］. 地球物理学报，52(6)：1536——1546.
郑海山，张中杰. 2005. 横向各向同性（VTI）介质中非线性地震波场模拟［J］. 地球物理学报，48(3)：660——671.
Aki K, Richards P G. 2002. Quantitative Seismology［M］. 2nd ed. Sausalito, California: University Science Books：218——244.
Berland J, Bogey C, Marsden O, Bailly C. 2007. High——order, low dispersive and low dissipative explicit schemes for multiple——scale and boundary problems［J］. J Comput Phys, 224(2): 637——662.
Bogey C, Bailly C. 2004. A family of low dispersive and low dissipative explicit schemes for flow and noise computations［J］. J Comput Phys, 194(1): 194——214.
Bohlen T, Saenger E H. 2006. Accuracy of heterogeneous staggered——grid finite——difference modeling of Rayleigh waves［J］. Geophysics, 71(4): T109——T115.
Cerveny V, Firbas P. 1984. Numerical modeling and inversion of travel——time of seismic body waves in inhomogeneous anisotropic media［J］. Geophys J R astr Soc, 76(1): 41——51.
Dablain M A. 1986. The application of high——order differencing to the scalar wave equation［J］. Geophysics, 51(1): 54——66.
Graves R W. 1996. Simulating seismic wave propagation in 3D elastic media using staggered——grid finite differences［J］. Bull Seism Soc Amer, 86(4): 1091——1106.
Komatitsch D, Coutel F, Mora P. 1996. Tensorial formulation of the wave equation for modeling curved interfaces［J］. Geophys J Int, 127(1): 156——168.
Kosloff D D， Baysal E. 1982. Forward modeling by a Fourier method［J］. Geophysics，47(10): 1402——1412.
Lan H Q, Zhang Z J. 2011a. Three——dimensional wave——field simulation in heterogeneous transversely isotropic medium with irregular free surface［J］. Bull Seism Soc Amer, 101(4): 1354——1370.
Lan H Q, Zhang Z J. 2011b. Comparative study of the free——surface boundary condition in two——dimensional finite——difference elastic wave field simulation［J］. J Geophys Eng, 8(2): 275——286.
Marfurt K J. 1984. Accuracy of finite——difference and finite——element modeling of the scalar and elastic wave equation［J］. Geophysics, 49(5): 533——549.
Muijres A J, Herman G C, Bussink P G. 1998. Acoustic wave propagation in two——dimensional media containing small——scale heterogeneities［J］. Wave Motion, 27(2): 137——154.
Tsingas C, Vafidis A, Kanasewich E R. 1990. Elastic wave propagation in transversely isotropic media using finite——differences［J］. Geophys Prospect, 38(8): 933——949.
Virieux J. 1986. P——SV wave propagation in heterogeneous media: velocity——stress finite difference method［J］. Geophysics, 51(4): 889——901.
Yang D H, Liu E R, Zhang Z J, Teng J W. 2002. Finite——difference modelling in two——dimensional anisotropic media using a flux——corrected transport technique［J］. Geophys J Int, 148(2): 320——328.
Yang D H, Song G J, Chen S, Hou B Y. 2007. An improved nearly analytical discrete method: An efficient tool to simulate the seismic response of 2——D porous structures［J］. J Geophys Eng, 4(1): 40——52.
Zhang W, Chen X F. 2006. Traction image method for irregular free surface boundaries in finite——difference seismic wave simulation［J］. Geophys J Int, 167(1): 337——353.
Zhang Z, Wang G J, Harris J M. 1999. Multi——component wavefield simulation in viscous extensively dilatancy anisotropic media［J］. Phys Earth Planet Inter, 114(1——2): 25——38.
Zheng H S, Zhang Z J, Liu E R. 2006. Non——linear seismic wave propagation in anisotropic media using the flux——corrected transport technique［J］. Geophys J Int, 165(3): 943——956.
董良国，马在田，曹景忠，王华忠，耿建华，雷兵，许世勇. 2000. 一阶弹性波方程交错网格高阶差分解法［J］. 地球物理学报，43(3)： 411——419.
祝贺君，张伟，陈晓非. 2009. 二维各向异性介质中地震波场的高阶同位网格有限差分模拟［J］. 地球物理学报，52(6)：1536——1546.
郑海山，张中杰. 2005. 横向各向同性（VTI）介质中非线性地震波场模拟［J］. 地球物理学报，48(3)：660——671.
Aki K, Richards P G. 2002. Quantitative Seismology［M］. 2nd ed. Sausalito, California: University Science Books：218——244.
Berland J, Bogey C, Marsden O, Bailly C. 2007. High——order, low dispersive and low dissipative explicit schemes for multiple——scale and boundary problems［J］. J Comput Phys, 224(2): 637——662.
Bogey C, Bailly C. 2004. A family of low dispersive and low dissipative explicit schemes for flow and noise computations［J］. J Comput Phys, 194(1): 194——214.
Bohlen T, Saenger E H. 2006. Accuracy of heterogeneous staggered——grid finite——difference modeling of Rayleigh waves［J］. Geophysics, 71(4): T109——T115.
Cerveny V, Firbas P. 1984. Numerical modeling and inversion of travel——time of seismic body waves in inhomogeneous anisotropic media［J］. Geophys J R astr Soc, 76(1): 41——51.
Dablain M A. 1986. The application of high——order differencing to the scalar wave equation［J］. Geophysics, 51(1): 54——66.
Graves R W. 1996. Simulating seismic wave propagation in 3D elastic media using staggered——grid finite differences［J］. Bull Seism Soc Amer, 86(4): 1091——1106.
Komatitsch D, Coutel F, Mora P. 1996. Tensorial formulation of the wave equation for modeling curved interfaces［J］. Geophys J Int, 127(1): 156——168.
Kosloff D D， Baysal E. 1982. Forward modeling by a Fourier method［J］. Geophysics，47(10): 1402——1412.
Lan H Q, Zhang Z J. 2011a. Three——dimensional wave——field simulation in heterogeneous transversely isotropic medium with irregular free surface［J］. Bull Seism Soc Amer, 101(4): 1354——1370.
Lan H Q, Zhang Z J. 2011b. Comparative study of the free——surface boundary condition in two——dimensional finite——difference elastic wave field simulation［J］. J Geophys Eng, 8(2): 275——286.
Marfurt K J. 1984. Accuracy of finite——difference and finite——element modeling of the scalar and elastic wave equation［J］. Geophysics, 49(5): 533——549.
Muijres A J, Herman G C, Bussink P G. 1998. Acoustic wave propagation in two——dimensional media containing small——scale heterogeneities［J］. Wave Motion, 27(2): 137——154.
Tsingas C, Vafidis A, Kanasewich E R. 1990. Elastic wave propagation in transversely isotropic media using finite——differences［J］. Geophys Prospect, 38(8): 933——949.
Virieux J. 1986. P——SV wave propagation in heterogeneous media: velocity——stress finite difference method［J］. Geophysics, 51(4): 889——901.
Yang D H, Liu E R, Zhang Z J, Teng J W. 2002. Finite——difference modelling in two——dimensional anisotropic media using a flux——corrected transport technique［J］. Geophys J Int, 148(2): 320——328.
Yang D H, Song G J, Chen S, Hou B Y. 2007. An improved nearly analytical discrete method: An efficient tool to simulate the seismic response of 2——D porous structures［J］. J Geophys Eng, 4(1): 40——52.
Zhang W, Chen X F. 2006. Traction image method for irregular free surface boundaries in finite——difference seismic wave simulation［J］. Geophys J Int, 167(1): 337——353.
Zhang Z, Wang G J, Harris J M. 1999. Multi——component wavefield simulation in viscous extensively dilatancy anisotropic media［J］. Phys Earth Planet Inter, 114(1——2): 25——38.
Zheng H S, Zhang Z J, Liu E R. 2006. Non——linear seismic wave propagation in anisotropic media using the flux——corrected transport technique［J］. Geophys J Int, 165(3): 943——956.
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