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
|
摘要
本文将DRP/opt MacCormack有限差分格式用于模拟二维各向异性介质中的地震波传播.DRP/optMacCorma ck是一种同位网格下的差分格式,避免了传统的交错网格在计算各向异性问题时由于变量插值而导致的误差.而且相对于低阶同位网格差分格式,它具有低色散、低耗散的优点.此格式将中心差分算子分成前向和后向两个空间单边差分,然后在4-6步Runge-Kutta时间积分中使用单边差分组合.在具有垂直对称轴的横向各向同性(VTI)模型下,通过对比DRP/opt MacCormack有限差分和谱元方法的模拟结果,验证了前者具有很高的精度和稳定性.由于实际地质条件下TI介质的对称轴通常是倾斜的(TTI),本文在二维三分量框架下模拟TTI介质中的地震波场.结果显示横波分裂和切平面/反平面运动耦合的特征.数值实验表明DRP/opt MacCormack是一种有效的研究各向异性介质中地震波传播规律的差分格式.
In this paper,a DRP/opt MacCormack finite difference scheme is used to simulate seismic wave propagation in two-dimensional anisotropic media.DRP/opt MacCormack is a non-staggered finite difference scheme which avoids the stress/strain interpolation in traditional staggered finite difference method.Compared to low-order non-staggered finite different scheme, it has low dispersion and dissipation.This scheme divides center difference to forward and backward one-side difference operators which are combined in 4-6 steps Runge-Kutta time integration.In a transverse isotropic media with vertical axis (VTI),we validate the high accuracy and stability of DRP/opt MacCormack scheme by comparing with spectral element method.In the realistic geological situations,the symmetric axes of transverse isotropic media are tiled (TTI) ,we simulate three components seismic wavefields in two-dimensional TTI media. The results show shear wave splitting and decoupling of in-plane/anti-plane movements.The numerical simulations show that DRP/opt MacCormack scheme is an efficient tool to study the wavefields in arbitrary anisotropic media.This scheme is suitable for media with arbitrary topographic variations and can be easily implemented in three-dimensional simulations.
In this paper,a DRP/opt MacCormack finite difference scheme is used to simulate seismic wave propagation in two-dimensional anisotropic media.DRP/opt MacCormack is a non-staggered finite difference scheme which avoids the stress/strain interpolation in traditional staggered finite difference method.Compared to low-order non-staggered finite different scheme, it has low dispersion and dissipation.This scheme divides center difference to forward and backward one-side difference operators which are combined in 4-6 steps Runge-Kutta time integration.In a transverse isotropic media with vertical axis (VTI),we validate the high accuracy and stability of DRP/opt MacCormack scheme by comparing with spectral element method.In the realistic geological situations,the symmetric axes of transverse isotropic media are tiled (TTI) ,we simulate three components seismic wavefields in two-dimensional TTI media. The results show shear wave splitting and decoupling of in-plane/anti-plane movements.The numerical simulations show that DRP/opt MacCormack scheme is an efficient tool to study the wavefields in arbitrary anisotropic media.This scheme is suitable for media with arbitrary topographic variations and can be easily implemented in three-dimensional simulations.
引文
[1] Igel H,Mora P,Riollet B.Anisotropic wave propagation through finite difference grids.Geophysics,1995,60 (4):1203-1216
[2] Zhang Z J,He Q D,Teng J W.The Simulation of 3-component seismic records in 2D inhomogeneous transversely isotropic media.Can.J.Explorational Geophys.,1993,29:51～58
[3] Zhang Z J,Wang G J,Harris J M.Multi-component wavefield simulation in viscous extensively dilatancy anisotropic media.Phys.Earth Planet.Interiors.,1999,114:25-38
[4] Hokstad K.3-D elastic finite-difference modeling in tilted transversely isotropic media.SEG Expanded Abstracts,2002,21:1951
[5] Okaya D A,McEvilly T V.Elastic wave propagation in anisotropic crustal material possessing arbitrary internal tilt.Geophys.J.Int.,2003,153:344-358
[6] Yang D H,Liu E,Zhang Z J,et al.Finite-difference modeling in two-dimensional anisotropic media using a flux-corrected transport technique.Geophys.J.Int.,2002,148:320-328
[7] 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
[8] 侯安宁,何樵登.各向异性介质中弹性波动高阶差分法及其稳定性的研究.地球物理学报,1995,38(2) :243～251 Hou A N,He Q D.Study of an elastic wave higB order difference method and its stability in anisotropic media.ChineseJ.Geophys.(in Chinese),1995,38(2) :243-251
[9] 孙卫涛,杨慧珠.各向异性介质弹性波传播的三维不规则网格有限差分方法.地球物理学报,2004,47(2) :332～337 Sun W T,Yang H Z.A 3-D finite difference method using irregular grids for elastic wave propagation in anisotropic media.Chinese J.Geophys.(in Chinese),2004,47(2) :332-337
[10] Zhu J L,Dorman J.Two-dimensional,three-component wave propagation in a transversely isotropic medium with arbitrary-orientation-finite-element modeling.Geophysics,2000,65(3) :934-942
[11] Zhang J F,Verschuur D J.Elastic wave propagation in heterogeneous anisotropic media using the lumped finite-element method.Geophysics,2002,67 (2) :625-638
[12] 周辉,徐世浙,刘斌等.各向异性介质中波动方程有限元法模拟及其稳定性.地球物理学报,1997,40(6) :833～841 Zhou H,Xu S Z,Liu B,et al.Modeling of wave equation in anisotropic media using finite element method and its stability condition.Chinese J.Geophys.(Acta Geophysica Sinica)(in Chinese),1997,40(6) :833-841
[13] 杨顶辉.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,2002,45(4) :575～583 Yang D H.Finite element method of the elastic wave equation and wavefield simulation in two-phase anisotropic media.Chinese J.Geophys.(in Chinese),2002,45(4) :575-583
[14] Carcione J M,Kosloff D,Behle A,et al.A spectral scheme for wave propagation simulation in 3-D elastic-anisotropic media.Geophysics,1992,57(12) :1593-1607
[15] Carcione J M.Staggered mesh for the anisotropic and viscoelastic wave equation.Geophysics,1999,64(6) :1863-1866
[16] Hung S H,Forsyth D W.Modelling anisotropic wave propagation in oceanic inhomogenous structures using the parallel multidomain pseudo-spectral method.Geophys.J.Int.,1998,133:726-740
[17] Komatitsch D,Barnes C,Tromp J.Simulation of anisotropic wave propagation based upon a spectral element method.Geophysics,2000,65(4) :1251-1260
[18] Chen M,Tromp J.Theoretical and numerical investigations of global and regional seismic wave propagation in weakly anisotropic earth models.Geophys.J.Int.,2007,168:1130-1152
[19] Booth D C,Crampin S.The anisotropic reflectivity technique;theory.Geophys.J.Roy.Astr.Soc.,1983,72:755-766
[20] Gajewski D,Psencik L.Computation of high frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media.Geophys.J.Roy.Astr.Soc.,1987,91:383-411
[21] Yang D H,Teng J W,Zhang Z J,et al.A nearly analytic discrete method for acoustic and elastic wave in anisotropic media.Bull.Seism.Soc.Am.,2003,93(2) :882-890
[22] Yang D H,Song G J,Lu M.Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3D anisotropic media.Bull.Seism.Soc.Am.,2007,97 (5) :1557-1569
[23] Gao H W,Zhang J F.Parallel 3-D simulation of seismic wave propagation in heterogeneous anisotropic media:a grid method approach.Geophys.J.Int.,2006,165:875-888
[24] Moczo P.Introduction to modeling seismic wave propagation by the finite-difference method. Lecture notes.Kyoto University Japan.,1998． 1-161
[25] Madariaga R.Dynamics of an expanding circular fault.Bull.Seism.Soc.Am.,1976,66:639-666
[26] Graves R W.Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences.Bull.Seism.Soc.Am.,1996,86:1091-1106
[27] Tsingas C,Vafidis A,Kanasewich E R.Elastic wave-propagation in transversely isotropic media using finite differences.Geophys.Prospecting,1990,38:933-949
[28] Zhang W,Chen X F.Traction image method for irregular free surface boundaries in finite difference seismic wave simulation.Geophys.J.Int.,2006,167:337-353
[29] Pitarka A,Irikura K.Modeling 3D surface topography by finite difference method:Kobe-JMA station site,Japan,case study.Geophys.Res.Lett.,1996,23,2729-2732
[30] Olsen K B.Site amplification in the Los Angeles basin from three dimensional modeling of ground motion.Bull.Seism. Soc.Am.,2000,90:S77-S94
[31] 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.Bull.Seism.Soc.Am.,2002,92:3042-3066
[32] Igel H,Riollet B,Mora P.Accuracy of staggered 3 D finite difference grids for anisotropic wave propagation.SEG Expanded Abstracts,1992,11:1244
[33] Hixon R.On increasing the accuracy of MacCormack schenes for aeroacoustic applications.AIAA Paper.1997． 97-1586
[34] Hu F Q,Hussaini M Y,Manthey J L.Low dissipation and low-dispersion Runge-Kutta schemes for computational acoustics.Journal of Computational Physics.,1996,124:177-191
[35] Berenger J.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Geophys.,1994,114:185-200
[36] Wang Y B,Takenaka H,Furumura T.Modelling seismic wave propagation in a two-dimensional cylindrical whole earth model using the pseudospectral method.Geophys.J.Int.,2001,145:689-708
[37] Komatitsch D,Tromp J.Spectral-element simulation of global seismic wave propagationI.Validation.Geophys.J.Int.,2002,149:390-412
[38] Komatitsch D,Liu Q,Tromp J,et al.Simulations of ground motion in the Los Angeles basin upon the spectralelement method.Bull.Seism.Soc.Am.,2004,94(1) :187-206
[39] Ampuero J P.SEM2DPACK.-A Spectral Element Method tool for 2D wave propagation and earthquake source dynamics,User's Guide.Version 2． 2． 11 Swiss Federal Institute of Technology.Switzerland.2007． 1-37
[40] Clayton R,Engquist B.Absorbing boundary conditions for acoustic and elastic wave equation.Bull.Seism.Soc.Am.,1977,67:1529-1540
[41] Virieux J.P-SV-wave propagation in heterogeneous media velocity-stress finite difference method.Geophysics,1986,51:889-901
[42] Komatitsch D,Roland M.An unsplit convolution Perfectly Matched Layer improved at grazing incidence for the seismic wave equation.Geophysics,2007,72(5) :SM155-SM167
[43] Faria E L,Stoffa P L.Finite-difference modeling in transversely isotropic meida.Geophysics,1994,59(2) :282-289
[44] Zhang L B,Rector J W,Hoversten G W.Finite difference modeling of wave propagate on in acoustic tilted TI media.Geophys.Prospecting,2005,53:843-852
[45] Hixon R.Evaluation of a high accuracy MacCormack-type scheme using benchmark problems.Journal of Computational Acoustics,1998,6:291-305
[2] Zhang Z J,He Q D,Teng J W.The Simulation of 3-component seismic records in 2D inhomogeneous transversely isotropic media.Can.J.Explorational Geophys.,1993,29:51～58
[3] Zhang Z J,Wang G J,Harris J M.Multi-component wavefield simulation in viscous extensively dilatancy anisotropic media.Phys.Earth Planet.Interiors.,1999,114:25-38
[4] Hokstad K.3-D elastic finite-difference modeling in tilted transversely isotropic media.SEG Expanded Abstracts,2002,21:1951
[5] Okaya D A,McEvilly T V.Elastic wave propagation in anisotropic crustal material possessing arbitrary internal tilt.Geophys.J.Int.,2003,153:344-358
[6] Yang D H,Liu E,Zhang Z J,et al.Finite-difference modeling in two-dimensional anisotropic media using a flux-corrected transport technique.Geophys.J.Int.,2002,148:320-328
[7] 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
[8] 侯安宁,何樵登.各向异性介质中弹性波动高阶差分法及其稳定性的研究.地球物理学报,1995,38(2) :243～251 Hou A N,He Q D.Study of an elastic wave higB order difference method and its stability in anisotropic media.ChineseJ.Geophys.(in Chinese),1995,38(2) :243-251
[9] 孙卫涛,杨慧珠.各向异性介质弹性波传播的三维不规则网格有限差分方法.地球物理学报,2004,47(2) :332～337 Sun W T,Yang H Z.A 3-D finite difference method using irregular grids for elastic wave propagation in anisotropic media.Chinese J.Geophys.(in Chinese),2004,47(2) :332-337
[10] Zhu J L,Dorman J.Two-dimensional,three-component wave propagation in a transversely isotropic medium with arbitrary-orientation-finite-element modeling.Geophysics,2000,65(3) :934-942
[11] Zhang J F,Verschuur D J.Elastic wave propagation in heterogeneous anisotropic media using the lumped finite-element method.Geophysics,2002,67 (2) :625-638
[12] 周辉,徐世浙,刘斌等.各向异性介质中波动方程有限元法模拟及其稳定性.地球物理学报,1997,40(6) :833～841 Zhou H,Xu S Z,Liu B,et al.Modeling of wave equation in anisotropic media using finite element method and its stability condition.Chinese J.Geophys.(Acta Geophysica Sinica)(in Chinese),1997,40(6) :833-841
[13] 杨顶辉.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,2002,45(4) :575～583 Yang D H.Finite element method of the elastic wave equation and wavefield simulation in two-phase anisotropic media.Chinese J.Geophys.(in Chinese),2002,45(4) :575-583
[14] Carcione J M,Kosloff D,Behle A,et al.A spectral scheme for wave propagation simulation in 3-D elastic-anisotropic media.Geophysics,1992,57(12) :1593-1607
[15] Carcione J M.Staggered mesh for the anisotropic and viscoelastic wave equation.Geophysics,1999,64(6) :1863-1866
[16] Hung S H,Forsyth D W.Modelling anisotropic wave propagation in oceanic inhomogenous structures using the parallel multidomain pseudo-spectral method.Geophys.J.Int.,1998,133:726-740
[17] Komatitsch D,Barnes C,Tromp J.Simulation of anisotropic wave propagation based upon a spectral element method.Geophysics,2000,65(4) :1251-1260
[18] Chen M,Tromp J.Theoretical and numerical investigations of global and regional seismic wave propagation in weakly anisotropic earth models.Geophys.J.Int.,2007,168:1130-1152
[19] Booth D C,Crampin S.The anisotropic reflectivity technique;theory.Geophys.J.Roy.Astr.Soc.,1983,72:755-766
[20] Gajewski D,Psencik L.Computation of high frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media.Geophys.J.Roy.Astr.Soc.,1987,91:383-411
[21] Yang D H,Teng J W,Zhang Z J,et al.A nearly analytic discrete method for acoustic and elastic wave in anisotropic media.Bull.Seism.Soc.Am.,2003,93(2) :882-890
[22] Yang D H,Song G J,Lu M.Optimally accurate nearly analytic discrete scheme for wave-field simulation in 3D anisotropic media.Bull.Seism.Soc.Am.,2007,97 (5) :1557-1569
[23] Gao H W,Zhang J F.Parallel 3-D simulation of seismic wave propagation in heterogeneous anisotropic media:a grid method approach.Geophys.J.Int.,2006,165:875-888
[24] Moczo P.Introduction to modeling seismic wave propagation by the finite-difference method. Lecture notes.Kyoto University Japan.,1998． 1-161
[25] Madariaga R.Dynamics of an expanding circular fault.Bull.Seism.Soc.Am.,1976,66:639-666
[26] Graves R W.Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences.Bull.Seism.Soc.Am.,1996,86:1091-1106
[27] Tsingas C,Vafidis A,Kanasewich E R.Elastic wave-propagation in transversely isotropic media using finite differences.Geophys.Prospecting,1990,38:933-949
[28] Zhang W,Chen X F.Traction image method for irregular free surface boundaries in finite difference seismic wave simulation.Geophys.J.Int.,2006,167:337-353
[29] Pitarka A,Irikura K.Modeling 3D surface topography by finite difference method:Kobe-JMA station site,Japan,case study.Geophys.Res.Lett.,1996,23,2729-2732
[30] Olsen K B.Site amplification in the Los Angeles basin from three dimensional modeling of ground motion.Bull.Seism. Soc.Am.,2000,90:S77-S94
[31] 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.Bull.Seism.Soc.Am.,2002,92:3042-3066
[32] Igel H,Riollet B,Mora P.Accuracy of staggered 3 D finite difference grids for anisotropic wave propagation.SEG Expanded Abstracts,1992,11:1244
[33] Hixon R.On increasing the accuracy of MacCormack schenes for aeroacoustic applications.AIAA Paper.1997． 97-1586
[34] Hu F Q,Hussaini M Y,Manthey J L.Low dissipation and low-dispersion Runge-Kutta schemes for computational acoustics.Journal of Computational Physics.,1996,124:177-191
[35] Berenger J.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Geophys.,1994,114:185-200
[36] Wang Y B,Takenaka H,Furumura T.Modelling seismic wave propagation in a two-dimensional cylindrical whole earth model using the pseudospectral method.Geophys.J.Int.,2001,145:689-708
[37] Komatitsch D,Tromp J.Spectral-element simulation of global seismic wave propagationI.Validation.Geophys.J.Int.,2002,149:390-412
[38] Komatitsch D,Liu Q,Tromp J,et al.Simulations of ground motion in the Los Angeles basin upon the spectralelement method.Bull.Seism.Soc.Am.,2004,94(1) :187-206
[39] Ampuero J P.SEM2DPACK.-A Spectral Element Method tool for 2D wave propagation and earthquake source dynamics,User's Guide.Version 2． 2． 11 Swiss Federal Institute of Technology.Switzerland.2007． 1-37
[40] Clayton R,Engquist B.Absorbing boundary conditions for acoustic and elastic wave equation.Bull.Seism.Soc.Am.,1977,67:1529-1540
[41] Virieux J.P-SV-wave propagation in heterogeneous media velocity-stress finite difference method.Geophysics,1986,51:889-901
[42] Komatitsch D,Roland M.An unsplit convolution Perfectly Matched Layer improved at grazing incidence for the seismic wave equation.Geophysics,2007,72(5) :SM155-SM167
[43] Faria E L,Stoffa P L.Finite-difference modeling in transversely isotropic meida.Geophysics,1994,59(2) :282-289
[44] Zhang L B,Rector J W,Hoversten G W.Finite difference modeling of wave propagate on in acoustic tilted TI media.Geophys.Prospecting,2005,53:843-852
[45] Hixon R.Evaluation of a high accuracy MacCormack-type scheme using benchmark problems.Journal of Computational Acoustics,1998,6:291-305
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