不分裂卷积完全匹配层与旋转交错网格有限差分在孔隙弹性介质模拟中的应用
|
摘要
完全匹配层吸收边界在地震波模拟中已广泛使用,但常用的场分裂格式完全匹配层吸收边界(SPML)和传统的不分裂完全匹配层吸收边界(NPML)对极低频入射波或大角度入射波的边界吸收效果不好.一种无需分裂和显式卷积计算的完全匹配层吸收边界(CPML)不仅能够解决常规PML吸收边界的不足,而且具有存储量小、计算效率高、易于编程实现的特点.本文将这种完全匹配层(CPML)吸收边界引入到孔隙弹性介质速度应力格式的旋转交错网格有限差分算法中,对完全匹配层吸收边界参数进行数值分析,得到一组优化的参数.孔隙弹性介质数值模拟结果表明这种不分裂卷积完全匹配层的吸收效果优于常规完全匹配层.
Perfectly matched layer (PML) absorbing boundary has been widely used in seismic simulation.At grazing incidence and low frequency, the classical split PML (SPML) method and the traditional nonsplit PML (NPML) method suffer from large spurious reflections.An unsplit convolutional PML (CPML) not only fixes the disadvantages of conventional PML, but also needs less memory, computes efficiently and is easy to program.In this paper, this CPML absorbing boundary is imported to the rotated grid finite-difference method, in velocity-stress formulation, of poroelastic media.The numerical analysis of parameters in the PML absorbing boundary gives a set of optimum parameters.Numerical test shows that the CPML absorbing boundary gives a better result than the conventional PML absorbing boundary.
Perfectly matched layer (PML) absorbing boundary has been widely used in seismic simulation.At grazing incidence and low frequency, the classical split PML (SPML) method and the traditional nonsplit PML (NPML) method suffer from large spurious reflections.An unsplit convolutional PML (CPML) not only fixes the disadvantages of conventional PML, but also needs less memory, computes efficiently and is easy to program.In this paper, this CPML absorbing boundary is imported to the rotated grid finite-difference method, in velocity-stress formulation, of poroelastic media.The numerical analysis of parameters in the PML absorbing boundary gives a set of optimum parameters.Numerical test shows that the CPML absorbing boundary gives a better result than the conventional PML absorbing boundary.
引文
[1] Carcione J M.Wave fields in real media:wave propagation in anisotropic,anelastic,porous and electromagnetic media.Elsevier,2007
[2] 牟永光,裴正林.三维复杂介质地震数值模拟.北京:石油工业出版社,2005 Mou Y G,Pei Z L.Seismic Numerical Modeling for 3 D Complex Media(in Chinese).Beijing:Petroleum Industry Press,2005
[3] Biot M A.Theory of propagation of elastic waves in a fluidsaturated porous solid.I.Low-frequency range.The Journal of the Acoustical Society of America,1956,28(2) :168～178
[4] Biot M A.Theory of propagation of elastic waves in a fluidsaturated porous solid.Ⅱ.Higher frequency range.The Journal of the Acoustical Society of America,1956,28(2) :179～191
[5] Biot M A.Mechanics of deformation and acoustic propagation in porous media.Journal of Applied Physics,1962,33(4) :1482～1498
[6] Mavko G,Mukerji T,Dvorkin J.The rock physics handbook.Cambridge University Press,1998
[7] Seron F J,Sanz F J,Kindelan M,et al.Finite-element method for elastic wave propagation.Communications in Applied Numerical Methods,1990,6(5) :359～368
[8] 符力耘,牟永光.弹性波边界元法正演模拟.地球物理学报,1994,37(4) :521～529 Fu L Y,Mou Y G.Boundary element method for elastic wave forward modeling.Chinese J.Geophys.(in Chinese),1994,37(4) :521～529
[9] 胡善政,符力耘,裴正林.流体饱和多孔隙介质弹性波方程边界元解法研究.地球物理学报,2009,52(9) :2364～2369 Hu S Z,Fu L Y,Pei Z L.A boundary element method for the 2-D wave equation in fluid-saturated porous media.Chinese J.Geophys.(in Chinese),2009,52(9) :2364～2369
[10] Gazdag J.Modeling of the acoustic wave equation with transform methods.Geophysics,1981,46(6) :854～859
[11] Kosloff D,Baysal E.Forward modeling by a Fourier method.Geophysics,1982,47(10) :1402～1412
[12] Faccioli E,Maggio F,Paolucci R,et al.2d and 3D elastic wave propagation by a pseudo-spectral domain decomposition method.Journal of Seismology,1997,1(3) :237～251
[13] Mikhailenko B G,Mikhailov a A,Reshetova G V.Numerical Modeling of Transient Seismic Fields in Viscoelastic Media Based on the Laguerre Spectral Method.Pure and Applied Geophysics,2003,160(7) :1207～1224
[14] Marfurt K J.Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations.Geophysics,1984,49(5) :533～549
[15] Jo C H,Shin C S,Suh J H.An optimal 9 point,finite- difference,frequency-space,2-D scalar wave extrapolator.Geophysics,1996,61(2) :529～537
[16] Wenzlau F,M(u|¨)ller T M.Finite difference modeling of wave propagation and diffusion in poroelastic media.Geophysics,2009,74(4) :T55～T66
[17] Wu C,Harris J M,Nihei K T,et al.Two-dimensional finite-difference seismic modeling of an open fluid-filled fracture:Comparison of thin-layer and linear slip models.Geophysics,2005,70(4) :T57～T62
[18] Yao Z,Margrave G F.Fourth-order finite-difference scheme for P and SV wave propagation in 2D transversely isotropic media.CREWES Research Report,1999,11
[19] Virieux J.P-SV wave propagation in heterogeneous media:velocity stress finite-difference method.Geophysics,1986,51(4) :889～901
[20] Levander a R.Fourth order finite difference P-SV seismograms.Geophysics,1988,53(11) :1425～1436
[21] Sheen D H,Tuncay K,Baag C E,et al.Parallel implementation of a velocity stress staggered grid finitedifference method for 2-D poroelastic wave propagation.Computers and Geosciences,2006,32(8) :1182～1191
[22] Moczo P,Kristek J,Vavrycuk V,et al.3D Heterogeneous Staggered-Grid Finite-Difference Modeling of Seismic Motion with Volume Harmonic and Arithmetic Averaging of Elastic Moduli and Densities.Bulletin of the Seismological Society of America,2002,92(8) :3042～3066
[23] 陈浩,王秀明,赵海波.旋转交错网格有限差分及其完全匹配层吸收边界条件.科学通报,2006,51(17) :1985～1994 Chen H,Wang X M,Zhao H B.A rotated staggered grid finite difference with the absorbing boundary condition of a perfectly matched layer.Chinese Science Bulletin(in Chinese),2006,51(17) :1985～1994
[24] Saenger E H,Gold N,Shapiro S A.Modeling the propagation of elastic waves using a modified finite-difference grid.Wave Motion,2000,31(1) :77～92
[25] Saenger E H,Shapiro S A.Effective velocities in fractured media:a numerical study using the rotated staggered finite difference grid.Geophysical Prospecting,2002,50(2) :183～194
[26] Saenger E H,Bohlen T.Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid.Geophysics,2004,69(2) :583～591
[27] Saenger E H,Ciz R,Kr(u|¨)ger O S,et al.Finite-difference modeling of wave propagation on microscale:a snapshot of the work in progress.Geophysics,2007,72(5) :SM293～ SM300
[28] Wang X M,Dodds K,Zhao H B.An improved high-order rotated staggered finite-difference algorithm for simulating elastic waves in heterogeneous viscoelastic/anisotropic media.Exploration Geophysics,2006,37(2) :160～174
[29] Cerian C,Kosloff D,Kosloff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equations.Geophysics,1985,50(4) :705～708
[30] Keys R G.Absorbing boundary conditions for acoustic media.Geophysics,1985,50(6) :892～902
[31] Berenger J P.A perfectly matched layer for the absorption of electromagnetic waves.Journal of Computational Physics,1994,114(2) :185～200
[32] Collino F,Monk P B.Optimizing the perfectly matched layer.Computer Methods in Applied Mechanics and Engineering,1998,164(1-2) :157～171
[33] 刘顺坤,陈向跃,聂鑫.有耗介质空间完全匹配层吸收边界条件及其应用.强激光与粒子束,2009,21(11) :1701～1704 Liu S K,Chen X Y,Nie X.Perfectly matched layer absorbing boundary condition and its application in truncation of lossy media space.High Power Laser and Particle Beams(in Chinese),2009,21(11) :1701～1704
[34] Zeng Y Q,Liu Q H.A staggered grid finite-difference method with perfectly matched layers for poroelastic wave equations.The Journal of the Acoustical Society of America,2001,109(6) :2571～2580
[35] Komatitsch D,Tromp J.A perfectly matched layer absorbing boundary condition for the second order seismic wave equation.Geophysical Journal International,2003,154(1) :146～153
[36] Marcinkovich C,Olsen K.On the implementation of perfectly matched layers in a three-dimensional fourth order velocity-stress finite difference scheme.Journal of Geophysical Research-Solid Earth,2003,108(B5) :2276～2291
[37] 赵海波,王秀明,王东等.完全匹配层吸收边界在孔隙介质弹性波模拟中的应用.地球物理学报,2007,50(2) :581～591 Zhao H B,Wang X M,Wang D,et al.Application of the boundary absorption using a perfectly matched layer for elastic wave simulation in poroelastic media.Chinese J.Geophys.(in Chinese),2007,50(2) :581～591
[38] 陈可洋.声波完全匹配层吸收边界条件的改进算法.石油物探,2009,48(1) :76～79 Chen K Y.The improved algorithm of the perfectly matched layer absorbing boundary for acoustic wave.Geophysical Prospecting for Petroleum.(in Chinese),2009,48(1) :76～79
[39] Kristek J,Moczo P,Galis M.A brief summary of some PML formulations and discretizations for the velocity-stress equation of seismic motion.Studia Geophysica et Geodaetica,2009,53(4) :459～474
[40] Roden J A,Gedney S D.Convolutional PML(CPML):An efficient FDTD implementation of the CFS-PML for arbitrary media.Microwave and Optical Technology Letters,2000,27(5) :334～338
[41] 姜永金,毛钧杰,柴舜连.CFS-PML边界条件在PSTD算法 中的实现与性能分析.微波学报,2004,20(4) :36～39 Jiang Y J,Mao J J,Chai S L.Implementation and Analysis of the Perfectly Matched Layer Media with CFS for the PSTD Method.Journal of Microwaves(in Chinese),2004,20(4) :36～39
[42] Komatitsch D,Martin R.An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation.Geophysics,2007,72(5) :SM155～SM167
[43] Martin R,Komatitsch D,Ezziani A.An unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media.Geophysics,2008,73(4) :T51～T61
[44] Martin R,Komatitsch D.An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation.Geophysical Journal International,2009,179(1) :333～344
[45] Qin Z,Lu M H,Zhang X D,et al.The implementation of an improved NPML absorbing boundary condition in elastic wave modeling.Applied Geophysics,2009,6(2) :113～121
[46] Carcione J,Quiroga-Goode G.Some aspects of the physics and numerical modeling of Biot compressional waves.Journal of Computational Acoustics,1995,3(4) :261～280
[47] Zhao H B,Wang X M,Chen H.A method of solving the stiffness problem in Biot' s poroelastic equations using a staggered high-order finite-difference.Chinese Physics,2006,15(12) :2819～2827
[48] Masson Y J,Pride S R,Nihei K T.Finite difference modeling of Biot s poroelastic equations at seismic frequencies.Journal of Geophysical Research(Solid Earth),2006,111(B10) :10305
[49] Zeng Y,Liu Q.Acoustic detection of buried objects in 3-D fluid saturated porous media:numerical modeling.IEEE Transactions on Geoscience and Remote Sensing,2001,39(6) :1165～1173
[50] Zeng Y Q,He J Q,Liu Q H.The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media.Geophysics,2001,66(4) :1258～1266
[51] Masson Y J,Pride S R.Poroelastic finite difference modeling of seismic attenuation and dispersion due to mesoscopic-scale heterogeneity.Journal of Geophysical Research-Solid Earth,2007,112(B3) :B03204
[52] Wenzlau F,M(u|¨)ller T M.Finite-difference modeling of wave propagation and diffusion in poroelastic media.Geophysics,2009,74(4) :T55～T66
[53] Chew W C,Weedon W H.A 3-D perfectly matched medium from modified Maxwell ' s equations with stretched coordinates.Microwave and Optical Technology Letters,1994,7(13) :599～604
[54] Collino F,Tsogka C.Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media.Geophysics,2001,66(1) :294～307
[55] Song R L,Ma J,Wang K X.The application of the nonsplitting perfectly matched layer in numerical modeling of wave propagation in poroelastic media.Applied Geophysics,2005,2(4) .:216～222
[56] Wang T,Tang X M.Finite-difference modeling of elastic wave propagation:A nonsplitting perfectly matched layer approach.Geophysics,2003,68(5) :1749～1755
[57] Drossaert F H,Giannopoulos A.Complex frequency shifted convolution PML for FDTD modelling of elastic waves.Wave Motion,2007,44(7-8) :593～604
[58] Luebbers R J,Hunsberger F.FDTD for N th-order dispersive media.IEEE transactions on Antennas and Propagation,1992,40(11) :1297～1301
[59] Kindelan M,Kamel A,Sguazzero P.On the construction and efficiency of staggered numerical differentiators for the wave equation.Geophysics,1990,55(1) :107～110
[2] 牟永光,裴正林.三维复杂介质地震数值模拟.北京:石油工业出版社,2005 Mou Y G,Pei Z L.Seismic Numerical Modeling for 3 D Complex Media(in Chinese).Beijing:Petroleum Industry Press,2005
[3] Biot M A.Theory of propagation of elastic waves in a fluidsaturated porous solid.I.Low-frequency range.The Journal of the Acoustical Society of America,1956,28(2) :168～178
[4] Biot M A.Theory of propagation of elastic waves in a fluidsaturated porous solid.Ⅱ.Higher frequency range.The Journal of the Acoustical Society of America,1956,28(2) :179～191
[5] Biot M A.Mechanics of deformation and acoustic propagation in porous media.Journal of Applied Physics,1962,33(4) :1482～1498
[6] Mavko G,Mukerji T,Dvorkin J.The rock physics handbook.Cambridge University Press,1998
[7] Seron F J,Sanz F J,Kindelan M,et al.Finite-element method for elastic wave propagation.Communications in Applied Numerical Methods,1990,6(5) :359～368
[8] 符力耘,牟永光.弹性波边界元法正演模拟.地球物理学报,1994,37(4) :521～529 Fu L Y,Mou Y G.Boundary element method for elastic wave forward modeling.Chinese J.Geophys.(in Chinese),1994,37(4) :521～529
[9] 胡善政,符力耘,裴正林.流体饱和多孔隙介质弹性波方程边界元解法研究.地球物理学报,2009,52(9) :2364～2369 Hu S Z,Fu L Y,Pei Z L.A boundary element method for the 2-D wave equation in fluid-saturated porous media.Chinese J.Geophys.(in Chinese),2009,52(9) :2364～2369
[10] Gazdag J.Modeling of the acoustic wave equation with transform methods.Geophysics,1981,46(6) :854～859
[11] Kosloff D,Baysal E.Forward modeling by a Fourier method.Geophysics,1982,47(10) :1402～1412
[12] Faccioli E,Maggio F,Paolucci R,et al.2d and 3D elastic wave propagation by a pseudo-spectral domain decomposition method.Journal of Seismology,1997,1(3) :237～251
[13] Mikhailenko B G,Mikhailov a A,Reshetova G V.Numerical Modeling of Transient Seismic Fields in Viscoelastic Media Based on the Laguerre Spectral Method.Pure and Applied Geophysics,2003,160(7) :1207～1224
[14] Marfurt K J.Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations.Geophysics,1984,49(5) :533～549
[15] Jo C H,Shin C S,Suh J H.An optimal 9 point,finite- difference,frequency-space,2-D scalar wave extrapolator.Geophysics,1996,61(2) :529～537
[16] Wenzlau F,M(u|¨)ller T M.Finite difference modeling of wave propagation and diffusion in poroelastic media.Geophysics,2009,74(4) :T55～T66
[17] Wu C,Harris J M,Nihei K T,et al.Two-dimensional finite-difference seismic modeling of an open fluid-filled fracture:Comparison of thin-layer and linear slip models.Geophysics,2005,70(4) :T57～T62
[18] Yao Z,Margrave G F.Fourth-order finite-difference scheme for P and SV wave propagation in 2D transversely isotropic media.CREWES Research Report,1999,11
[19] Virieux J.P-SV wave propagation in heterogeneous media:velocity stress finite-difference method.Geophysics,1986,51(4) :889～901
[20] Levander a R.Fourth order finite difference P-SV seismograms.Geophysics,1988,53(11) :1425～1436
[21] Sheen D H,Tuncay K,Baag C E,et al.Parallel implementation of a velocity stress staggered grid finitedifference method for 2-D poroelastic wave propagation.Computers and Geosciences,2006,32(8) :1182～1191
[22] Moczo P,Kristek J,Vavrycuk V,et al.3D Heterogeneous Staggered-Grid Finite-Difference Modeling of Seismic Motion with Volume Harmonic and Arithmetic Averaging of Elastic Moduli and Densities.Bulletin of the Seismological Society of America,2002,92(8) :3042～3066
[23] 陈浩,王秀明,赵海波.旋转交错网格有限差分及其完全匹配层吸收边界条件.科学通报,2006,51(17) :1985～1994 Chen H,Wang X M,Zhao H B.A rotated staggered grid finite difference with the absorbing boundary condition of a perfectly matched layer.Chinese Science Bulletin(in Chinese),2006,51(17) :1985～1994
[24] Saenger E H,Gold N,Shapiro S A.Modeling the propagation of elastic waves using a modified finite-difference grid.Wave Motion,2000,31(1) :77～92
[25] Saenger E H,Shapiro S A.Effective velocities in fractured media:a numerical study using the rotated staggered finite difference grid.Geophysical Prospecting,2002,50(2) :183～194
[26] Saenger E H,Bohlen T.Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid.Geophysics,2004,69(2) :583～591
[27] Saenger E H,Ciz R,Kr(u|¨)ger O S,et al.Finite-difference modeling of wave propagation on microscale:a snapshot of the work in progress.Geophysics,2007,72(5) :SM293～ SM300
[28] Wang X M,Dodds K,Zhao H B.An improved high-order rotated staggered finite-difference algorithm for simulating elastic waves in heterogeneous viscoelastic/anisotropic media.Exploration Geophysics,2006,37(2) :160～174
[29] Cerian C,Kosloff D,Kosloff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equations.Geophysics,1985,50(4) :705～708
[30] Keys R G.Absorbing boundary conditions for acoustic media.Geophysics,1985,50(6) :892～902
[31] Berenger J P.A perfectly matched layer for the absorption of electromagnetic waves.Journal of Computational Physics,1994,114(2) :185～200
[32] Collino F,Monk P B.Optimizing the perfectly matched layer.Computer Methods in Applied Mechanics and Engineering,1998,164(1-2) :157～171
[33] 刘顺坤,陈向跃,聂鑫.有耗介质空间完全匹配层吸收边界条件及其应用.强激光与粒子束,2009,21(11) :1701～1704 Liu S K,Chen X Y,Nie X.Perfectly matched layer absorbing boundary condition and its application in truncation of lossy media space.High Power Laser and Particle Beams(in Chinese),2009,21(11) :1701～1704
[34] Zeng Y Q,Liu Q H.A staggered grid finite-difference method with perfectly matched layers for poroelastic wave equations.The Journal of the Acoustical Society of America,2001,109(6) :2571～2580
[35] Komatitsch D,Tromp J.A perfectly matched layer absorbing boundary condition for the second order seismic wave equation.Geophysical Journal International,2003,154(1) :146～153
[36] Marcinkovich C,Olsen K.On the implementation of perfectly matched layers in a three-dimensional fourth order velocity-stress finite difference scheme.Journal of Geophysical Research-Solid Earth,2003,108(B5) :2276～2291
[37] 赵海波,王秀明,王东等.完全匹配层吸收边界在孔隙介质弹性波模拟中的应用.地球物理学报,2007,50(2) :581～591 Zhao H B,Wang X M,Wang D,et al.Application of the boundary absorption using a perfectly matched layer for elastic wave simulation in poroelastic media.Chinese J.Geophys.(in Chinese),2007,50(2) :581～591
[38] 陈可洋.声波完全匹配层吸收边界条件的改进算法.石油物探,2009,48(1) :76～79 Chen K Y.The improved algorithm of the perfectly matched layer absorbing boundary for acoustic wave.Geophysical Prospecting for Petroleum.(in Chinese),2009,48(1) :76～79
[39] Kristek J,Moczo P,Galis M.A brief summary of some PML formulations and discretizations for the velocity-stress equation of seismic motion.Studia Geophysica et Geodaetica,2009,53(4) :459～474
[40] Roden J A,Gedney S D.Convolutional PML(CPML):An efficient FDTD implementation of the CFS-PML for arbitrary media.Microwave and Optical Technology Letters,2000,27(5) :334～338
[41] 姜永金,毛钧杰,柴舜连.CFS-PML边界条件在PSTD算法 中的实现与性能分析.微波学报,2004,20(4) :36～39 Jiang Y J,Mao J J,Chai S L.Implementation and Analysis of the Perfectly Matched Layer Media with CFS for the PSTD Method.Journal of Microwaves(in Chinese),2004,20(4) :36～39
[42] Komatitsch D,Martin R.An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation.Geophysics,2007,72(5) :SM155～SM167
[43] Martin R,Komatitsch D,Ezziani A.An unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media.Geophysics,2008,73(4) :T51～T61
[44] Martin R,Komatitsch D.An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation.Geophysical Journal International,2009,179(1) :333～344
[45] Qin Z,Lu M H,Zhang X D,et al.The implementation of an improved NPML absorbing boundary condition in elastic wave modeling.Applied Geophysics,2009,6(2) :113～121
[46] Carcione J,Quiroga-Goode G.Some aspects of the physics and numerical modeling of Biot compressional waves.Journal of Computational Acoustics,1995,3(4) :261～280
[47] Zhao H B,Wang X M,Chen H.A method of solving the stiffness problem in Biot' s poroelastic equations using a staggered high-order finite-difference.Chinese Physics,2006,15(12) :2819～2827
[48] Masson Y J,Pride S R,Nihei K T.Finite difference modeling of Biot s poroelastic equations at seismic frequencies.Journal of Geophysical Research(Solid Earth),2006,111(B10) :10305
[49] Zeng Y,Liu Q.Acoustic detection of buried objects in 3-D fluid saturated porous media:numerical modeling.IEEE Transactions on Geoscience and Remote Sensing,2001,39(6) :1165～1173
[50] Zeng Y Q,He J Q,Liu Q H.The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media.Geophysics,2001,66(4) :1258～1266
[51] Masson Y J,Pride S R.Poroelastic finite difference modeling of seismic attenuation and dispersion due to mesoscopic-scale heterogeneity.Journal of Geophysical Research-Solid Earth,2007,112(B3) :B03204
[52] Wenzlau F,M(u|¨)ller T M.Finite-difference modeling of wave propagation and diffusion in poroelastic media.Geophysics,2009,74(4) :T55～T66
[53] Chew W C,Weedon W H.A 3-D perfectly matched medium from modified Maxwell ' s equations with stretched coordinates.Microwave and Optical Technology Letters,1994,7(13) :599～604
[54] Collino F,Tsogka C.Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media.Geophysics,2001,66(1) :294～307
[55] Song R L,Ma J,Wang K X.The application of the nonsplitting perfectly matched layer in numerical modeling of wave propagation in poroelastic media.Applied Geophysics,2005,2(4) .:216～222
[56] Wang T,Tang X M.Finite-difference modeling of elastic wave propagation:A nonsplitting perfectly matched layer approach.Geophysics,2003,68(5) :1749～1755
[57] Drossaert F H,Giannopoulos A.Complex frequency shifted convolution PML for FDTD modelling of elastic waves.Wave Motion,2007,44(7-8) :593～604
[58] Luebbers R J,Hunsberger F.FDTD for N th-order dispersive media.IEEE transactions on Antennas and Propagation,1992,40(11) :1297～1301
[59] Kindelan M,Kamel A,Sguazzero P.On the construction and efficiency of staggered numerical differentiators for the wave equation.Geophysics,1990,55(1) :107～110
版权所有：© 2023 中国地质图书馆 中国地质调查局地学文献中心