修正辛格式有限元法的地震波场模拟
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摘要
三角网格有限元法具有网格剖分的灵活性,能有效模拟地震波在复杂介质中的传播.但传统有限元法用于地震波场模拟时计算效率较低,消耗较大计算资源.本文采用改进的核矩阵存储(IKMS)策略以提高有限元法的计算效率,该方法不用组合总体刚度矩阵,且相比于常规有限元法节省成倍的内存.对于时间离散,将有限元离散后的地震波运动方程变换至Hamilton体系,在显式二阶辛Runge-Kutta-Nystr9m(RKN)格式的基础之上加入额外空间离散算子构造修正辛差分格式,通过Taylor展开式得到具有四阶时间精度时间格式,且辛系数全为正数.本文从理论上分析了时空改进方法相比传统辛-有限元方法在频散压制、稳定性提升等方面的优势.数值算例进一步证实本方法具有内存消耗少、稳定性强和数值频散弱等优点.
Finite element method(FEM)with triangular mesh has the properties of flexibility and adaptability in complex medium.However,the traditional finite element method is inefficient in seismic wavefield modeling because this method commonly requires a large amount of computation resources.Here we propose a so-called improved kernel matrix storage(IKMS)strategy to improve the computation efficiency of FEM.The improved FEM does not need to assemble global stiffness matrix and the memory requirement is many times less than that of the conventional FEM.In terms of temporal discretization,the elastic wave equation after FEM discretization is first transformed into a Hamilton system.Then,an additional spatial discretization term is addedto the second-order explicit Runge-Kutta-Nystr9 m(RKN)scheme.Based on Taylor series expansion,we obtain a fourth-order symplectic scheme with all positive symplectic coefficients.Theoretical analysis shows that the temporal-spatial modified numerical scheme is superior over the traditional symplectic FEM in suppressing numerical dispersion and increasing numerical stability.Furthermore,numerical results verify that this method requires less computer resource and achieves higher numerical accuracy.
Finite element method(FEM)with triangular mesh has the properties of flexibility and adaptability in complex medium.However,the traditional finite element method is inefficient in seismic wavefield modeling because this method commonly requires a large amount of computation resources.Here we propose a so-called improved kernel matrix storage(IKMS)strategy to improve the computation efficiency of FEM.The improved FEM does not need to assemble global stiffness matrix and the memory requirement is many times less than that of the conventional FEM.In terms of temporal discretization,the elastic wave equation after FEM discretization is first transformed into a Hamilton system.Then,an additional spatial discretization term is addedto the second-order explicit Runge-Kutta-Nystr9 m(RKN)scheme.Based on Taylor series expansion,we obtain a fourth-order symplectic scheme with all positive symplectic coefficients.Theoretical analysis shows that the temporal-spatial modified numerical scheme is superior over the traditional symplectic FEM in suppressing numerical dispersion and increasing numerical stability.Furthermore,numerical results verify that this method requires less computer resource and achieves higher numerical accuracy.
引文
Alford R M,Kelly K R,Boore D M.1974.Accuracy of finitedifference modeling of the acoustic wave equation.Geophysics,39(6):834-842.
Amaru M L.2007.Global travel time tomography with 3-D reference models[Ph.D.thesis].Netherlands:Utrecht University.
Berenger J P.1994.A perfectly matched layer for the absorption of electromagnetic waves.Journal of Computational Physics,114(2):185-200.
Butcher J C.1996.A history of Runge-Kutta methods.Applied Numerical Mathematics,20(3):247-260.
Cai W J,Zhang H,Wang Y S.2017.Dissipation-preserving spectral element method for damped seismic wave equations.Journal of Computational Physics,350:260-279.
Chen J B.2007.High-order time discretizations in seismic modeling.Geophysics,72(5):SM115-SM122.
Cheng B,Zhao D P,Zhang G W.2011.Seismic tomography and anisotropy in the source area of the 2008Iwate-Miyagi earthquake(M7.2).Physics of the Earth and Planetary Interiors,184(3-4):172-185.
Cohen G C.2002.Higher-Order Numerical Methods for Transient Wave Equations.Berlin:Springer-Verlag.
Dablain M A.1986.The application of high-order differencing to the scalar wave equation.Geophysics,51(1):54-66.
De Hoop A T.1960.A modification of Cagniard′s method for solving seismic pulse problems.Applied Scientific Research,Section B,8(1):349-356.
De Basabe J D,Sen M K.2007.Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations.Geophysics,72(6):T81-T95.
De Basabe J D.2009.High-order finite element methods for seismic wave propagation[Ph.D.thesis].Texas:University of Texas at Austin.
Feng K,Qin M Z.2010.Symplectic Geometric Algorithms for Hamiltonian Systems.Berlin:Springer.
Fornberg B.1988.The pseudospectral method;Accurate repre sentation of interfaces in elastic wave calculations.Geophysics,53(5):625-637.
Fornberg B.1990.High-order finite differences and the pseudospectral method on staggered grids.SIAM Journal on Numerical Analysis,27(4):904-918.
Gao Y J,Song H J,Zhang J H,et al.2015.Comparison of artificial absorbing boundaries for acoustic wave equation modelling.Exploration Geophysics,48(1):76-93.
Gao Y J,Zhang J H,Yao Z X.2016.Third-order symplectic integration method with inverse time dispersion transform for long-term simulation.Journal of Computational Physics,314:436-449.
Kane C,Marsden J E,Ortiz M,et al.2015.Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems.International Journal for Numerical Methods in Engineering,49(10):1295-1325.
Komatitsch D,Vilotte J P.1998.The spectral element method:An efficient tool to simulate the seismic response of 2Dand 3Dgeological structures.Bulletin of the Seismological Society of America,88(2):368-392.
Komatitsch D,Ritsema J,Tromp J.2002.The spectral-element method,Beowulf computing,and global seismology.Science,298(5599):1737-1742.
Lan H Q,Zhang Z J.2011.Comparative study of the free-surface boundary condition in two-dimensional finite-difference elastic wave field simulation.Journal of Geophysics and Engineering,8(2):275-286.
Lax P D,Wendroff B.1964.Difference schemes for hyperbolic equations with high order of accuracy.Communications on Pure and Applied Mathematics,17(3):381-398.
Li J S,Yang D H,Liu F Q.2013.An efficient reverse time migration method using local nearly analytic discrete operator.Geophysics,78(1):S15-S23.
Li X F,Li Y Q,Zhang M G,et al.2011.Scalar seismic-wave equation modeling by a multisymplectic discrete singular convolution differentiator method.Bulletin of the Seismological Society of America,101(4):1710-1718.
Li X F,Wang W S,Lu M W,et al.2012.Structure-preserving modelling of elastic waves:A symplectic discrete singular convolution differentiator method.Geophysical Journal International,188(3):1382-1392.
Liu S L,Li X F,Liu Y S,et al.2014.Dispersion analysis of trianglebased finite element method for acoustic and elastic wave simulations.Chinese Journal of Geophysics(in Chinese),57(8):2620-2630,doi:10.6038/cjg20140821.
Liu S L,Li X F,Wang W S,et al.2014.A mixed-grid finite element method with PML absorbing boundary conditions for seismic wave modelling.Journal of Geophysics and Engineering,11(5):055009.
Liu S L,Li X F,Wang W S,et al.2015.A symplectic RKNscheme for solving elastic wave equations.Chinese Journal of Geophysics(in Chinese),58(4):1355-1366,doi:10.6038/cjg20150422.
Liu S L,Li X F,Wang W S,et al.2015.A modified symplectic scheme for seismic wave modeling.Journal of Applied Geophysics,116:110-120.
Liu S L,Yang D H,Dong X P,et al.2017.Element-by-element parallel spectral-element methods for 3-D teleseismic wave modeling.Solid Earth,8(5):969-986.
Liu S L,Yang D H,Lang C,et al.2017a.Modified symplectic schemes with nearly-analytic discrete operators for acoustic wave simulations.Computer Physics Communications,213:52-63.
Liu S L,Yang D H,Ma J.2017b.A modified symplectic PRKscheme for seismic wave modeling.Computers&Geosciences,99:28-36.
Liu Y K,Chang X.1998.The implicit multigrid method for the secondary hydrocarbon migration drived by water.Acta Geophysics Sinica(in Chinese),41(3):342-348.
Liu Y S,Teng J W,Xu T,et al.2014.Numerical modeling of seismic wavefield with the SEM based on Triangles.Progress in Geophysics(in Chinese),29(4):1715-1726,doi:10.6038/pg20140430.
Liu Y S.2015.Seismic wavefield modeling using the spectral element method and migration/inversion[Ph.D.thesis].Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences.
Marfurt K J.1984.Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations.Geophysics,49(5):533-549.
Mazzieri I,Rapetti F.2012.Dispersion analysis of triangle-based spectral element methods for elastic wave propagation.Numerical Algorithms,60(4):631-650.
McLachlan R I,Atela P.1992.The accuracy of symplectic integrators.Nonlinearity,5(2):541-562.
Meng W J,Fu L Y.2017.Seismic wavefield simulation by a modified finite element method with a perfectly matched layer absorbing boundary.Journal of Geophysics and Engineering,14(4):852-864.
Moczo P,Kristek J,Halada L.2000.3Dfourth-order staggeredgrid finite-difference schemes:Stability and grid dispersion.Bulletin of the Seismological Society of America,90(3):587-603.
Mullen R,Belytschko T.1982.Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation.International Journal for Numerical Methods in Engineering,18(1):11-29.
Nilsson S,Petersson N A,Sj9green B,et al.2007.Stable difference approximations for the elastic wave equation in second order formulation.SIAM Journal on Numerical Analysis,45(5):1902-1936.
Padovani E,Priolo E,Seriani G.1994.Low and high order finite element method:Experience in seismic modeling.Journal of Computational Acoustics,2(4):371-422.
Pratt R G,Shin C,Hick G J.1998.Gauss-newton and full newton methods in frequency-space seismic waveform inversion.Geophysical Journal International,133(2):341-362.
Saad Y.2001.Parallel Iterative Methods for Sparse Linear Systems.Studies in Computational Mathematics,8:423-440.
Shragge J.2014.Reverse time migration from topography.Geophysics,79(4):S141-S152.
Smith W D.1975.The application of finite element analysis to body wave propagation problems.Geophysical Journal International,42(2):747-768.
Sonneveld P.1989.CGS,a fast Lanczos-type solver for nonsymmetric linear systems.SIAM Journal on Scientific and Statistical Computing,10(1):36-52.
Sun W J,Fu L Y.2013.Two effective approaches to reduce data storage in reverse time migration.Computers&Geosciences,56:69-75.
Sun W J,Zhou B Z,Fu L Y.2015.A staggered-grid convolutional differentiator for elastic wave modelling.Journal of Computational Physics,301:59-76.
Tape C,Liu Q Y,Maggi A,et al.2009.Adjoint tomography of the southern California crust.Science,325(5943):988-992.
Tape C,Liu Q Y,Maggi A,et al.2010.Seismic tomography of the southern California crust based on spectral-element and adjoint methods.Geophysical Journal of the Royal Astronomical Society,180(1):433-462.
Tarantola A.1984.Inversion of seismic reflection data in the acoustic approximation.Geophysics,49(8):1259-1266.
Tong P,Yang D H,Hua B L,et al.2013.A high-order stereomodeling method for solving wave equations.Bulletin of the Seismological Society of America,103(2A):811-833.
Virieux J.1984.SH-wave propagation in heterogeneous media;Velocitystress finite-difference method.Geophysics,49(11):1933-1942.
Virieux J.1986.P-SV wave propagation in heterogeneous media:Velocity-stress finite-difference method.Geophysics,51(4):889-901.
Virieux J,Operto S.2009.An overview of full-waveform inversion in exploration geophysics.Geophysics,74(6):WCC1-WCC26.
Wang Y B,Takenaka H,Furumura T.2010.Modelling seismic wave propagation in a two-dimensional cylindrical whole-earth model using the pseudospectral method.Geophysical Journal of the Royal Astronomical Society,145(3):689-708.
Wang Y F,Liang W Q,Nashed Z,et al.2016.Determination of finite difference coefficients for the acoustic wave equation using regularized least-squares inversion.Journal of Inverse and Illposed Problems,24(6):743-760.
Wang Z W,Zhao D P,Liu X,et al.2017.Pand S wave attenuation tomography of the Japan subduction zone.Geochemistry,Geophysics,Geosystems,18(4):1688-1710.
Wu S R.2006.Lumped mass matrix in explicit finite element method for transient dynamics of elasticity.Computer Methods in Applied Mechanics and Engineering,195(44-47):5983-5994.
Yan Z Z,Zhang H,Yang C C,et al.2009.Spectral element analysis on the characteristics of seismic wave propagation
triggered by Wenchuan MS8.0earthquake.Science in China Series D:Earth Sciences,52(6):764-773.
Yang D H.2002.Finite element method of the elastic wave equation and wavefield simulation in two-phase anisotropic media.Chinese Journal of Geophysics(in Chinese),45(4):575-583.
Yang D H,Teng J W,Zhang Z J,et al.2003.A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media.Bulletin of the Seismological Society of America,93(2):882-890.
Yang D H,Peng J M,Lu M,et al.2006.Optimal nearly analytic discrete approximation to the scalar wave equation.Bulletin of the Seismological Society of America,96(3):1114-1130.
Yang D H,Song G J,Lu M.2007.Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3danisotropic media.Bulletin of the Seismological Society of America,97(5):1557-1569.
Yang Y P,Shen H N.2010.The rationality and limitation on the substitution of the concentrated mass matrix for the consistent mass matrix.Journal of Qinghai University(Natural Science)(in Chinese),28(1):35-39.
Zhang J H,Yao Z X.2013.Optimized explicit finite-difference schemes for spatial derivatives using maximum norm.Journal of Computational Physics,250:511-526.
Zhao D P,Yamamoto Y,Yanada T.2013.Global mantle heterogeneity and its influence on teleseismic regional tomography.Gondwana Research,23(2):595-616.
Zhebel E,Minisini S,Kononov A,et al.2014.A comparison of continuous mass-lumped finite elements with finite differences for 3-D wave propagation.Geophysical Prospecting,62(5):1111-1125.
刘少林,李小凡,刘有山等.2014.三角网格有限元法声波与弹性波模拟频散分析.地球物理学报,57(8):2620-2630,doi:10.6038/cjg20140821.
刘少林,李小凡,汪文帅等.2015.求解弹性波方程的辛RKN格式.地球物理学报,58(4):1355-1366,doi:10.6038/cjg20150422.
刘伊克,常旭,1998.盆地模拟水动力油气二次运移隐式多重网格法.地球物理学报,41(3):342-348.
刘有山,滕吉文,徐涛等.2014.三角网格谱元法地震波场数值模拟.地球物理学进展,29(4):1715-1726,doi:10.6038/pg20140430.
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杨顶辉.2002.双相各向异性介质中弹性波方程的有限元解法及波场模拟.地球物理学报,45(4):575-583.
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Amaru M L.2007.Global travel time tomography with 3-D reference models[Ph.D.thesis].Netherlands:Utrecht University.
Berenger J P.1994.A perfectly matched layer for the absorption of electromagnetic waves.Journal of Computational Physics,114(2):185-200.
Butcher J C.1996.A history of Runge-Kutta methods.Applied Numerical Mathematics,20(3):247-260.
Cai W J,Zhang H,Wang Y S.2017.Dissipation-preserving spectral element method for damped seismic wave equations.Journal of Computational Physics,350:260-279.
Chen J B.2007.High-order time discretizations in seismic modeling.Geophysics,72(5):SM115-SM122.
Cheng B,Zhao D P,Zhang G W.2011.Seismic tomography and anisotropy in the source area of the 2008Iwate-Miyagi earthquake(M7.2).Physics of the Earth and Planetary Interiors,184(3-4):172-185.
Cohen G C.2002.Higher-Order Numerical Methods for Transient Wave Equations.Berlin:Springer-Verlag.
Dablain M A.1986.The application of high-order differencing to the scalar wave equation.Geophysics,51(1):54-66.
De Hoop A T.1960.A modification of Cagniard′s method for solving seismic pulse problems.Applied Scientific Research,Section B,8(1):349-356.
De Basabe J D,Sen M K.2007.Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations.Geophysics,72(6):T81-T95.
De Basabe J D.2009.High-order finite element methods for seismic wave propagation[Ph.D.thesis].Texas:University of Texas at Austin.
Feng K,Qin M Z.2010.Symplectic Geometric Algorithms for Hamiltonian Systems.Berlin:Springer.
Fornberg B.1988.The pseudospectral method;Accurate repre sentation of interfaces in elastic wave calculations.Geophysics,53(5):625-637.
Fornberg B.1990.High-order finite differences and the pseudospectral method on staggered grids.SIAM Journal on Numerical Analysis,27(4):904-918.
Gao Y J,Song H J,Zhang J H,et al.2015.Comparison of artificial absorbing boundaries for acoustic wave equation modelling.Exploration Geophysics,48(1):76-93.
Gao Y J,Zhang J H,Yao Z X.2016.Third-order symplectic integration method with inverse time dispersion transform for long-term simulation.Journal of Computational Physics,314:436-449.
Kane C,Marsden J E,Ortiz M,et al.2015.Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems.International Journal for Numerical Methods in Engineering,49(10):1295-1325.
Komatitsch D,Vilotte J P.1998.The spectral element method:An efficient tool to simulate the seismic response of 2Dand 3Dgeological structures.Bulletin of the Seismological Society of America,88(2):368-392.
Komatitsch D,Ritsema J,Tromp J.2002.The spectral-element method,Beowulf computing,and global seismology.Science,298(5599):1737-1742.
Lan H Q,Zhang Z J.2011.Comparative study of the free-surface boundary condition in two-dimensional finite-difference elastic wave field simulation.Journal of Geophysics and Engineering,8(2):275-286.
Lax P D,Wendroff B.1964.Difference schemes for hyperbolic equations with high order of accuracy.Communications on Pure and Applied Mathematics,17(3):381-398.
Li J S,Yang D H,Liu F Q.2013.An efficient reverse time migration method using local nearly analytic discrete operator.Geophysics,78(1):S15-S23.
Li X F,Li Y Q,Zhang M G,et al.2011.Scalar seismic-wave equation modeling by a multisymplectic discrete singular convolution differentiator method.Bulletin of the Seismological Society of America,101(4):1710-1718.
Li X F,Wang W S,Lu M W,et al.2012.Structure-preserving modelling of elastic waves:A symplectic discrete singular convolution differentiator method.Geophysical Journal International,188(3):1382-1392.
Liu S L,Li X F,Liu Y S,et al.2014.Dispersion analysis of trianglebased finite element method for acoustic and elastic wave simulations.Chinese Journal of Geophysics(in Chinese),57(8):2620-2630,doi:10.6038/cjg20140821.
Liu S L,Li X F,Wang W S,et al.2014.A mixed-grid finite element method with PML absorbing boundary conditions for seismic wave modelling.Journal of Geophysics and Engineering,11(5):055009.
Liu S L,Li X F,Wang W S,et al.2015.A symplectic RKNscheme for solving elastic wave equations.Chinese Journal of Geophysics(in Chinese),58(4):1355-1366,doi:10.6038/cjg20150422.
Liu S L,Li X F,Wang W S,et al.2015.A modified symplectic scheme for seismic wave modeling.Journal of Applied Geophysics,116:110-120.
Liu S L,Yang D H,Dong X P,et al.2017.Element-by-element parallel spectral-element methods for 3-D teleseismic wave modeling.Solid Earth,8(5):969-986.
Liu S L,Yang D H,Lang C,et al.2017a.Modified symplectic schemes with nearly-analytic discrete operators for acoustic wave simulations.Computer Physics Communications,213:52-63.
Liu S L,Yang D H,Ma J.2017b.A modified symplectic PRKscheme for seismic wave modeling.Computers&Geosciences,99:28-36.
Liu Y K,Chang X.1998.The implicit multigrid method for the secondary hydrocarbon migration drived by water.Acta Geophysics Sinica(in Chinese),41(3):342-348.
Liu Y S,Teng J W,Xu T,et al.2014.Numerical modeling of seismic wavefield with the SEM based on Triangles.Progress in Geophysics(in Chinese),29(4):1715-1726,doi:10.6038/pg20140430.
Liu Y S.2015.Seismic wavefield modeling using the spectral element method and migration/inversion[Ph.D.thesis].Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences.
Marfurt K J.1984.Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations.Geophysics,49(5):533-549.
Mazzieri I,Rapetti F.2012.Dispersion analysis of triangle-based spectral element methods for elastic wave propagation.Numerical Algorithms,60(4):631-650.
McLachlan R I,Atela P.1992.The accuracy of symplectic integrators.Nonlinearity,5(2):541-562.
Meng W J,Fu L Y.2017.Seismic wavefield simulation by a modified finite element method with a perfectly matched layer absorbing boundary.Journal of Geophysics and Engineering,14(4):852-864.
Moczo P,Kristek J,Halada L.2000.3Dfourth-order staggeredgrid finite-difference schemes:Stability and grid dispersion.Bulletin of the Seismological Society of America,90(3):587-603.
Mullen R,Belytschko T.1982.Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation.International Journal for Numerical Methods in Engineering,18(1):11-29.
Nilsson S,Petersson N A,Sj9green B,et al.2007.Stable difference approximations for the elastic wave equation in second order formulation.SIAM Journal on Numerical Analysis,45(5):1902-1936.
Padovani E,Priolo E,Seriani G.1994.Low and high order finite element method:Experience in seismic modeling.Journal of Computational Acoustics,2(4):371-422.
Pratt R G,Shin C,Hick G J.1998.Gauss-newton and full newton methods in frequency-space seismic waveform inversion.Geophysical Journal International,133(2):341-362.
Saad Y.2001.Parallel Iterative Methods for Sparse Linear Systems.Studies in Computational Mathematics,8:423-440.
Shragge J.2014.Reverse time migration from topography.Geophysics,79(4):S141-S152.
Smith W D.1975.The application of finite element analysis to body wave propagation problems.Geophysical Journal International,42(2):747-768.
Sonneveld P.1989.CGS,a fast Lanczos-type solver for nonsymmetric linear systems.SIAM Journal on Scientific and Statistical Computing,10(1):36-52.
Sun W J,Fu L Y.2013.Two effective approaches to reduce data storage in reverse time migration.Computers&Geosciences,56:69-75.
Sun W J,Zhou B Z,Fu L Y.2015.A staggered-grid convolutional differentiator for elastic wave modelling.Journal of Computational Physics,301:59-76.
Tape C,Liu Q Y,Maggi A,et al.2009.Adjoint tomography of the southern California crust.Science,325(5943):988-992.
Tape C,Liu Q Y,Maggi A,et al.2010.Seismic tomography of the southern California crust based on spectral-element and adjoint methods.Geophysical Journal of the Royal Astronomical Society,180(1):433-462.
Tarantola A.1984.Inversion of seismic reflection data in the acoustic approximation.Geophysics,49(8):1259-1266.
Tong P,Yang D H,Hua B L,et al.2013.A high-order stereomodeling method for solving wave equations.Bulletin of the Seismological Society of America,103(2A):811-833.
Virieux J.1984.SH-wave propagation in heterogeneous media;Velocitystress finite-difference method.Geophysics,49(11):1933-1942.
Virieux J.1986.P-SV wave propagation in heterogeneous media:Velocity-stress finite-difference method.Geophysics,51(4):889-901.
Virieux J,Operto S.2009.An overview of full-waveform inversion in exploration geophysics.Geophysics,74(6):WCC1-WCC26.
Wang Y B,Takenaka H,Furumura T.2010.Modelling seismic wave propagation in a two-dimensional cylindrical whole-earth model using the pseudospectral method.Geophysical Journal of the Royal Astronomical Society,145(3):689-708.
Wang Y F,Liang W Q,Nashed Z,et al.2016.Determination of finite difference coefficients for the acoustic wave equation using regularized least-squares inversion.Journal of Inverse and Illposed Problems,24(6):743-760.
Wang Z W,Zhao D P,Liu X,et al.2017.Pand S wave attenuation tomography of the Japan subduction zone.Geochemistry,Geophysics,Geosystems,18(4):1688-1710.
Wu S R.2006.Lumped mass matrix in explicit finite element method for transient dynamics of elasticity.Computer Methods in Applied Mechanics and Engineering,195(44-47):5983-5994.
Yan Z Z,Zhang H,Yang C C,et al.2009.Spectral element analysis on the characteristics of seismic wave propagation
triggered by Wenchuan MS8.0earthquake.Science in China Series D:Earth Sciences,52(6):764-773.
Yang D H.2002.Finite element method of the elastic wave equation and wavefield simulation in two-phase anisotropic media.Chinese Journal of Geophysics(in Chinese),45(4):575-583.
Yang D H,Teng J W,Zhang Z J,et al.2003.A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media.Bulletin of the Seismological Society of America,93(2):882-890.
Yang D H,Peng J M,Lu M,et al.2006.Optimal nearly analytic discrete approximation to the scalar wave equation.Bulletin of the Seismological Society of America,96(3):1114-1130.
Yang D H,Song G J,Lu M.2007.Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3danisotropic media.Bulletin of the Seismological Society of America,97(5):1557-1569.
Yang Y P,Shen H N.2010.The rationality and limitation on the substitution of the concentrated mass matrix for the consistent mass matrix.Journal of Qinghai University(Natural Science)(in Chinese),28(1):35-39.
Zhang J H,Yao Z X.2013.Optimized explicit finite-difference schemes for spatial derivatives using maximum norm.Journal of Computational Physics,250:511-526.
Zhao D P,Yamamoto Y,Yanada T.2013.Global mantle heterogeneity and its influence on teleseismic regional tomography.Gondwana Research,23(2):595-616.
Zhebel E,Minisini S,Kononov A,et al.2014.A comparison of continuous mass-lumped finite elements with finite differences for 3-D wave propagation.Geophysical Prospecting,62(5):1111-1125.
刘少林,李小凡,刘有山等.2014.三角网格有限元法声波与弹性波模拟频散分析.地球物理学报,57(8):2620-2630,doi:10.6038/cjg20140821.
刘少林,李小凡,汪文帅等.2015.求解弹性波方程的辛RKN格式.地球物理学报,58(4):1355-1366,doi:10.6038/cjg20150422.
刘伊克,常旭,1998.盆地模拟水动力油气二次运移隐式多重网格法.地球物理学报,41(3):342-348.
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