基于Shannon奇异核理论的褶积微分算子在地震波场模拟中的应用
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摘要
本文在前人工作的基础上,建立了一种基于shannon奇异核的交错网格褶积微分算子方法.文中不仅详细讨论了影响算子精度的各种因素,同时也着重分析了其在弹性波模拟中的频散关系和稳定性条件.通过和交错网格有限差分算子比较,发现该算子即使在高波数域也具有较高的精度.均匀介质中的数值试验也表明,该方法9点格式就基本上达到了解析解精度.而分层均匀介质和复杂介质中的地震波数值模拟也同时证实了该方法精度高,稳定性好,是一种研究复杂介质中地震波传播的有效数值方法.
This article provides a technique to model seismic wave propagation in complex media using convolutional differentiator method based on Shannon singular kernel theory. The high accuracy of the differentiator has been demonstrated through comparisons with numerical results computed by staggered-grid finite difference method. Numerical tests suggest that the differentiator allows sufficient accuracy even in high wave-number domain. Dispersion and stability condition of the method in velocity-stress equation are also thoroughly discussed. Experiment in homogeneous medium shows that the 9-point scheme of this differentiator can achieve analytical accuracy. The application of this differentiator to model seismic wave propagation in homogenous layered medium and complex medium also shows high accuracy and robust stability. These appealing characters of this improved method would make it effective to model seismic wave propagation in complex media.
This article provides a technique to model seismic wave propagation in complex media using convolutional differentiator method based on Shannon singular kernel theory. The high accuracy of the differentiator has been demonstrated through comparisons with numerical results computed by staggered-grid finite difference method. Numerical tests suggest that the differentiator allows sufficient accuracy even in high wave-number domain. Dispersion and stability condition of the method in velocity-stress equation are also thoroughly discussed. Experiment in homogeneous medium shows that the 9-point scheme of this differentiator can achieve analytical accuracy. The application of this differentiator to model seismic wave propagation in homogenous layered medium and complex medium also shows high accuracy and robust stability. These appealing characters of this improved method would make it effective to model seismic wave propagation in complex media.
引文
[1] 殷文,印兴耀,吴国忱等.高精度频率域弹性波方程有限差分方法及波场模拟.地球物理学报,2006,49(2) :561～568Yin W,Yin X Y,Wu G C,et al.The method of finite difference of high precision elastic equations in the frequency domain and wave field simulation.Chinese J.Geophys.(in Chinese),2006,49(2) :561 ～568
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[3] 张美根.各向异性弹性波正反演问题研究[博士论文].北京:中国科学院地质与地球物理研究所,2000Zhang M G.The researches of forward modeling and inversion of anisotropic elastic wave[Ph.D.thesis](in Chinese).Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2000
[4] Takenaka H,Wang Y B,Furumura T.An efficient approach of the pseudospectral method for modeling of geometrically symmetric seismic wavefield.Earth Planets Space,1999,51 (2) :73～79
[5] Fornberg B.The pseudospectral method:Accurate representation of interfaces in elastic wave calculations.Geophysics,1988,53(5) :625～637
[6] Komatitsch D.Spectral and spectral-element methods for the 2D and 3D electrodynamics equations in heterogeneous media.[Ph.D.thesis].Paris:Institute de Physique du Globe,1997[7] Roberts R A,Mullis C T.Digital Signal Processing.Addison-Wesley,Reading,Massachusetts,1987
[8] Mora P.Elastic finite difference with convolutional operators.Stanford Exploration Project Report,1986,48:277～289
[9] Holberg O.Computational aspects of the choice of operator and sampling interval for numerical differentiation in largescale simulation of wave phenomena.Geophys.Prosp.,2006,35(6) :629～655
[10] Mittet R,Holberg O.Fast finite-difference modeling of 3-D elastic wave propagation.58~(th) Ann.Int.Mtg.,Soc.Explor.Geophys.,Expanded Absttacts,1988． 1308～1311
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[13] 戴志阳,孙建国,查显杰.地震波场模拟中的褶积微分算子法.吉林大学学报,2005,35(4) :520～524Dai Z Y,Sun J G,Cha X J.Seismic wave field modeling with convolutional differentiator algorithm.Journal of Jilin University (in Chinese),2005,35(4) :520～524
[14] 程冰洁.基于广义正交多项式迭积微分算子的地震波场数值模拟研究[博士论文].北京:中国科学院地质与地球物理研究所,2007Cheng B J.Study on seismic waves modeling by convolutional differentiator based on Forsyte polynomial[Ph.D.thesis](in Chinese).Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2007
[15] Zhou Y C,Gu Y,Wei G W.Conjugated filter approach for solving Burgers' equation.J.Comput.Appl.Math.,2002,149(2) :439～456
[16] Zhou Y C,Wei G W.High resolution conjugate filters for the simulation of flows.J.Comput.Appl.Math.,2003,189(1) :159～179
[17] Sun Y H,Zhou Y C,Li S G,et al.A windowed Fourier pseudospectral method for hyperbolic conservation laws.J.Comput.Appl.Math.,2006,214(2) :466～490
[18] Wei G W.Discrete singular convolution for the solution of the Fokker-Planck equations.J.Chem.Phys.,1999,110 (18) :8930～8942
[19] Bérenger J.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,1994,114 (2) :185～200
[20] Komatitsch D,Tromp J.A perfectly matched layer absorbing boundary condition for the second order seismic wave equation.Geophys.J.Int.,2003,154(1) :146～153
[21] Alford R M,Kelly K R,Boore D M.Accuracy of finitedifference modeling of the acoustic wave equation.Geophysics,1974,39(6) :834～842
[22] 董良国,李培明.地震波传播数值模拟中的频散问题.天然气工业,2004,24(6) :53～56Dong L G,Li P M.Dispersive problem in seismic wave propagation numerical modeling.Natur.Gas Ind.(in Chinese),2004,24(6) :53～56
[23] 董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶有限差分解法.地球物理学报,2000,43(3) :411～419Dong L G,Ma Z T,Cao J Z.The staggered-grid high-order difference method of one-order elastic equation.Chinese J.Geophys.(in Chinese),2000,43(3) :411～419
[24] Lines L R,Slawinski R,Bording R P.A recipe for stability of finite-difference wave-equation computations.Geophysics,1999,64(3) :967～969
[25] Mufti I R.Large scale three dimensional seismic models and their interpretive significance.Geophysics,1990,55(9) :1166～1182
[26] Wu W,Lines L R,Lu H.Analysis of high order finite difference schemes in 3D reverse time migration.Geophysics,1996,61(3) :845～856
[27] 董良国,马在田,曹景忠等.一阶弹性波方程交错网格高阶差分解法稳定性研究.地球物理学报,2000,43(6) :856～864DongLG,MaZT,CaoJ Z,et al.A study on stability of the staggered-grid high-order difference method of first-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(6) :856～864
[2] 王红落.地震波传播与成像若干问题的研究[博士论文].北京:中国科学院地质与地球物理研究所,2005Wang H L.The study of several problems in seismic wave propagation and imaging[Ph.D.thesis](in Chinese).Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2005
[3] 张美根.各向异性弹性波正反演问题研究[博士论文].北京:中国科学院地质与地球物理研究所,2000Zhang M G.The researches of forward modeling and inversion of anisotropic elastic wave[Ph.D.thesis](in Chinese).Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2000
[4] Takenaka H,Wang Y B,Furumura T.An efficient approach of the pseudospectral method for modeling of geometrically symmetric seismic wavefield.Earth Planets Space,1999,51 (2) :73～79
[5] Fornberg B.The pseudospectral method:Accurate representation of interfaces in elastic wave calculations.Geophysics,1988,53(5) :625～637
[6] Komatitsch D.Spectral and spectral-element methods for the 2D and 3D electrodynamics equations in heterogeneous media.[Ph.D.thesis].Paris:Institute de Physique du Globe,1997[7] Roberts R A,Mullis C T.Digital Signal Processing.Addison-Wesley,Reading,Massachusetts,1987
[8] Mora P.Elastic finite difference with convolutional operators.Stanford Exploration Project Report,1986,48:277～289
[9] Holberg O.Computational aspects of the choice of operator and sampling interval for numerical differentiation in largescale simulation of wave phenomena.Geophys.Prosp.,2006,35(6) :629～655
[10] Mittet R,Holberg O.Fast finite-difference modeling of 3-D elastic wave propagation.58~(th) Ann.Int.Mtg.,Soc.Explor.Geophys.,Expanded Absttacts,1988． 1308～1311
[11] Zhou B,Greenhalgh S.Seismic scalar wave equation modeling by a convolutional differentiator.Bull.Seism.Am.,1992,82(1) :289～303
[12] 张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法.地震学报,1996,18(1) :63～69Zhang Z J,Teng J W,Yang D H.The convolutional differentiator method for numerical modeling of acoustic and elastic wave-field.Acta Seismological Sinica (in Chinese),1996,18(1) :63～69
[13] 戴志阳,孙建国,查显杰.地震波场模拟中的褶积微分算子法.吉林大学学报,2005,35(4) :520～524Dai Z Y,Sun J G,Cha X J.Seismic wave field modeling with convolutional differentiator algorithm.Journal of Jilin University (in Chinese),2005,35(4) :520～524
[14] 程冰洁.基于广义正交多项式迭积微分算子的地震波场数值模拟研究[博士论文].北京:中国科学院地质与地球物理研究所,2007Cheng B J.Study on seismic waves modeling by convolutional differentiator based on Forsyte polynomial[Ph.D.thesis](in Chinese).Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2007
[15] Zhou Y C,Gu Y,Wei G W.Conjugated filter approach for solving Burgers' equation.J.Comput.Appl.Math.,2002,149(2) :439～456
[16] Zhou Y C,Wei G W.High resolution conjugate filters for the simulation of flows.J.Comput.Appl.Math.,2003,189(1) :159～179
[17] Sun Y H,Zhou Y C,Li S G,et al.A windowed Fourier pseudospectral method for hyperbolic conservation laws.J.Comput.Appl.Math.,2006,214(2) :466～490
[18] Wei G W.Discrete singular convolution for the solution of the Fokker-Planck equations.J.Chem.Phys.,1999,110 (18) :8930～8942
[19] Bérenger J.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,1994,114 (2) :185～200
[20] Komatitsch D,Tromp J.A perfectly matched layer absorbing boundary condition for the second order seismic wave equation.Geophys.J.Int.,2003,154(1) :146～153
[21] Alford R M,Kelly K R,Boore D M.Accuracy of finitedifference modeling of the acoustic wave equation.Geophysics,1974,39(6) :834～842
[22] 董良国,李培明.地震波传播数值模拟中的频散问题.天然气工业,2004,24(6) :53～56Dong L G,Li P M.Dispersive problem in seismic wave propagation numerical modeling.Natur.Gas Ind.(in Chinese),2004,24(6) :53～56
[23] 董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶有限差分解法.地球物理学报,2000,43(3) :411～419Dong L G,Ma Z T,Cao J Z.The staggered-grid high-order difference method of one-order elastic equation.Chinese J.Geophys.(in Chinese),2000,43(3) :411～419
[24] Lines L R,Slawinski R,Bording R P.A recipe for stability of finite-difference wave-equation computations.Geophysics,1999,64(3) :967～969
[25] Mufti I R.Large scale three dimensional seismic models and their interpretive significance.Geophysics,1990,55(9) :1166～1182
[26] Wu W,Lines L R,Lu H.Analysis of high order finite difference schemes in 3D reverse time migration.Geophysics,1996,61(3) :845～856
[27] 董良国,马在田,曹景忠等.一阶弹性波方程交错网格高阶差分解法稳定性研究.地球物理学报,2000,43(6) :856～864DongLG,MaZT,CaoJ Z,et al.A study on stability of the staggered-grid high-order difference method of first-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(6) :856～864
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