地震波数值模拟中的最优Shannon奇异核褶积微分算子
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摘要
如何有效协调地震波数值模拟过程中的计算精度和计算效率,一直是许多地球物理工作者密切关注且努力寻求解决的问题.这一问题求解的好坏,将直接影响到人们正确认识地下构造的地球物理响应.本文从离散Shannon奇异核理论出发,推导了基于Shannon奇异核的交错网格褶积微分算子,分析并求出了影响算子精度窗函数的最优化参数.通过将最优褶积微分算子与不同长度的交错网格有限差分算子进行比较,发现该算子8点格式的精度相当于16阶交错网格有限差分.对于同长度的算子,交错网格褶积微分算子可以在一个波长内具有较少的采样点数.将本文算子运用于Corner-Edge固液介质模型,其数值试验结果同样表明,该方法计算精度高,稳定性好,可作为研究复杂介质中地震波传播的又一有效数值方法.
One of many problems that geophysicists concern with and endeavor to solve for a long time has been lacking of new methods that can well balance the incompatible relation between accuracy and computation efficiency in seismic modeling. In order to solve this problem and correctly recognize the geophysical response of the subsurface structures, we derived a kind of numerical method named staggered-grid convolutional differentiator to solve the one-order velocity-stress equation in elastic media based on discrete Shannon singular kernel theory, whose optimal coefficients are obtained with nonlinear optimization method. Accuracy comparison between optimal staggered-grid convolutional differentiator and staggered-grid finite differentiators with different lengths indicates that the accuracy of the 8th-order scheme differentiator proposed in this paper is comparable with 16th-order staggered=grid finite difference, which, in other words, means that optimal staggered-grid convolutional differentiator has fewer points per wavelength when the two differentiators have the same length. Numerical modeling in Corner-Edge model also demonstrates that the optimal staggered-grid convolutional difference method is of high efficiency and robustness, and this method can be chosen as an alternative approach to compute the wave field in complex media.
One of many problems that geophysicists concern with and endeavor to solve for a long time has been lacking of new methods that can well balance the incompatible relation between accuracy and computation efficiency in seismic modeling. In order to solve this problem and correctly recognize the geophysical response of the subsurface structures, we derived a kind of numerical method named staggered-grid convolutional differentiator to solve the one-order velocity-stress equation in elastic media based on discrete Shannon singular kernel theory, whose optimal coefficients are obtained with nonlinear optimization method. Accuracy comparison between optimal staggered-grid convolutional differentiator and staggered-grid finite differentiators with different lengths indicates that the accuracy of the 8th-order scheme differentiator proposed in this paper is comparable with 16th-order staggered=grid finite difference, which, in other words, means that optimal staggered-grid convolutional differentiator has fewer points per wavelength when the two differentiators have the same length. Numerical modeling in Corner-Edge model also demonstrates that the optimal staggered-grid convolutional difference method is of high efficiency and robustness, and this method can be chosen as an alternative approach to compute the wave field in complex media.
引文
陈浩,王秀明,赵海波.2006.旋转交错网格有限差分及其完全匹配层吸收边界条件[J].科学通报,51(17):1985-1994.
程冰洁.2007.基于广义正交多项式跌积微分算子的地震波场数值模拟研究[D].北京:中国科学院地质与地球物理研究所:1-87.
戴志阳,孙建国,查显杰.2005.地震波场模拟中的褶积微分算子法[J].吉林大学学报,35(4):520-524.
董良国,马在田,曹景忠,王华忠,耿建华,雷兵,许世勇.2000a.一阶弹性波方程交错网格高阶有限差分解法[J].地球物理学报,43(3):411-419.
董良国,马在田,曹景忠.2000b.一阶弹性波方程交错网格高阶有限差分解法稳定性研究[J].地球物理学报,43(6):856-864.
王红落.2005.地震波传播与成像若干问题的研究[D].北京:中国科学院地质与地球物理研究所：1-107.
王秀明,张海澜,王东.2003.利用高阶交错网格有限差分算法模拟地震波在非均匀孔隙介质中的传播[J].地球物理学报,46(6):842-849.
张中杰,滕吉文,杨顶辉.1996.声波与弹性波场数值模拟中的褶积微分算子法[J].地震学报,18(1):63-69.
朱生旺,魏修成.2006.波动方程数值模拟的隐式差分法[J].石油物探,45(2):151-156.
Bale R A.2002.StaggeredGridsfor3 D Pseudospectral Modeling in Anisotropic Elastic Media[R].CREWES Re-search Report,Calgary:1-14.
Baltensperger R,Trummer M.2003.Spectral differencing with a twist[J].SIAMJ Sci Comp,24(5):1465-1487.
Bérenger,J.1994.A perfectly matched layer for the absorption of electromagnetic waves[J].J Comput Phys,114(2):185-200.
Carcione J M,Helle H B.1999.Numerical solution of the poroviscoelastic wave equation on a staggered mesh[J].J Comput Phys,154(2):520-527.
Diaz J,Ezziani A.2010.Analytical solution for wave propagation in stratified poroelastic medium.PartΙ:the 2Dcase[J].Comm Comput Phys,7(1):171-174.
Fornberg B.1987.The pseudospectral method:Comparisons with finite differences for the elastic wave equation[J].Geophysics,52(4):483-501.
Fornberg B.1988.The pseudospectral method:Accurate representation of interfaces in elastic wave calculations[J].Geophysics,53(5):625-637.
Graves R W.1996.Simulating seismic wave propagation in 3Delastic media using staggered-grid finite differences[J].Bull Seism Soc Amer,86(4):1091-1106.
Holberg O.1987.Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena[J].Geophys Prospect,35(6):629-655.
Levander A.1988.Fourth-order finite-difference P-SV seismograms[J].Geophysics,53(11):1425-1436.
Mora P.1986.Elastic Finite Difference with Convolutional Operators[R].Stanford Exploration Project Report,Cali-fornia:277-290.
Sun Y H,Zhou Y C,Li S G,Wei G W.2005.A windowed Fourier pseudospectral method for hyperbolic conservation laws[J].J Comput Phys,214(2):466-490.
Takenaka H,Wang Y B,Furumura T.1999.An efficient approach of the pseudospectral method for modeling of geomet-rically symmetric seismic wavefield[J].Earth Planes Space,51(2):73-79.
Virieux J.1986.P-SV wave propagation in heyerogeneous media:velocity-stress finite difference method[J].Geophysics,51(40):889-901.
Virieux J.1988.SH-wave propagation in heterogeneous media:velocity-stress finite difference method[J].Geophysics,53(11):1425-1436.
Weideman J,Reddy S.2000.A matlab differencing with a twist[J].ACM Transactions on Mathematical Software,26(4):465-519.
Zhou B,Greenhalgh S.1992.Seismic scalar wave equation modeling by a convolutional differentiator[J].Bull Seism Soc Amer,82(1):289-303.
Zhou Y C,Gu Y,Wei G W.2002.Conjugated filter approach for solving Burgers’equation[J].J Comput Appl Math,149(2):439-456.
Zhou Y C,Wei G W.2003.High resolution conjugate filters for the simulation of flows[J].J Comput Appl Math,189(1):159-179.
程冰洁.2007.基于广义正交多项式跌积微分算子的地震波场数值模拟研究[D].北京:中国科学院地质与地球物理研究所:1-87.
戴志阳,孙建国,查显杰.2005.地震波场模拟中的褶积微分算子法[J].吉林大学学报,35(4):520-524.
董良国,马在田,曹景忠,王华忠,耿建华,雷兵,许世勇.2000a.一阶弹性波方程交错网格高阶有限差分解法[J].地球物理学报,43(3):411-419.
董良国,马在田,曹景忠.2000b.一阶弹性波方程交错网格高阶有限差分解法稳定性研究[J].地球物理学报,43(6):856-864.
王红落.2005.地震波传播与成像若干问题的研究[D].北京:中国科学院地质与地球物理研究所：1-107.
王秀明,张海澜,王东.2003.利用高阶交错网格有限差分算法模拟地震波在非均匀孔隙介质中的传播[J].地球物理学报,46(6):842-849.
张中杰,滕吉文,杨顶辉.1996.声波与弹性波场数值模拟中的褶积微分算子法[J].地震学报,18(1):63-69.
朱生旺,魏修成.2006.波动方程数值模拟的隐式差分法[J].石油物探,45(2):151-156.
Bale R A.2002.StaggeredGridsfor3 D Pseudospectral Modeling in Anisotropic Elastic Media[R].CREWES Re-search Report,Calgary:1-14.
Baltensperger R,Trummer M.2003.Spectral differencing with a twist[J].SIAMJ Sci Comp,24(5):1465-1487.
Bérenger,J.1994.A perfectly matched layer for the absorption of electromagnetic waves[J].J Comput Phys,114(2):185-200.
Carcione J M,Helle H B.1999.Numerical solution of the poroviscoelastic wave equation on a staggered mesh[J].J Comput Phys,154(2):520-527.
Diaz J,Ezziani A.2010.Analytical solution for wave propagation in stratified poroelastic medium.PartΙ:the 2Dcase[J].Comm Comput Phys,7(1):171-174.
Fornberg B.1987.The pseudospectral method:Comparisons with finite differences for the elastic wave equation[J].Geophysics,52(4):483-501.
Fornberg B.1988.The pseudospectral method:Accurate representation of interfaces in elastic wave calculations[J].Geophysics,53(5):625-637.
Graves R W.1996.Simulating seismic wave propagation in 3Delastic media using staggered-grid finite differences[J].Bull Seism Soc Amer,86(4):1091-1106.
Holberg O.1987.Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena[J].Geophys Prospect,35(6):629-655.
Levander A.1988.Fourth-order finite-difference P-SV seismograms[J].Geophysics,53(11):1425-1436.
Mora P.1986.Elastic Finite Difference with Convolutional Operators[R].Stanford Exploration Project Report,Cali-fornia:277-290.
Sun Y H,Zhou Y C,Li S G,Wei G W.2005.A windowed Fourier pseudospectral method for hyperbolic conservation laws[J].J Comput Phys,214(2):466-490.
Takenaka H,Wang Y B,Furumura T.1999.An efficient approach of the pseudospectral method for modeling of geomet-rically symmetric seismic wavefield[J].Earth Planes Space,51(2):73-79.
Virieux J.1986.P-SV wave propagation in heyerogeneous media:velocity-stress finite difference method[J].Geophysics,51(40):889-901.
Virieux J.1988.SH-wave propagation in heterogeneous media:velocity-stress finite difference method[J].Geophysics,53(11):1425-1436.
Weideman J,Reddy S.2000.A matlab differencing with a twist[J].ACM Transactions on Mathematical Software,26(4):465-519.
Zhou B,Greenhalgh S.1992.Seismic scalar wave equation modeling by a convolutional differentiator[J].Bull Seism Soc Amer,82(1):289-303.
Zhou Y C,Gu Y,Wei G W.2002.Conjugated filter approach for solving Burgers’equation[J].J Comput Appl Math,149(2):439-456.
Zhou Y C,Wei G W.2003.High resolution conjugate filters for the simulation of flows[J].J Comput Appl Math,189(1):159-179.
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