四阶龙格-库塔方法的一种改进算法及地震波场模拟
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摘要
提出了求解波动方程的四阶龙格-库塔方法的一种改进算法.首先将原四阶龙格-库塔方法合并为两级格式,然后在第一级中引入加权参数以获得加权算法.针对这种改进方法,研究了它的稳定性条件;对一维问题导出了频散关系,给出了数值频散结果,并与四阶的Lax-Wendroff(LWC)方法和位移-应力交错网格方法进行了对比;对二维问题,使用我们的改进方法、四阶LWC和交错网格三种方法进行了声波波场模拟,并进行了计算效率分析和不同方法计算结果的比较;最后选取两个层状介质模型进行了声波和弹性波波场模拟.数值结果表明,本文的改进方法具有非常弱的数值频散和高的计算效率,是一种在地震勘探领域具有巨大应用潜力的数值方法.
In this article,we present an improved algorithm of the fourth-order Runge-Kutta (RK) method to solve the wave equations.We first change the original fourth-order Runge-Kutta method into a 2-stage scheme,and then introduce a weighting parameter in the first stage to obtain a weighted scheme.To study this new improved method,first of all,we analyze its stability condition for 1D and 2D cases.Secondly,we derive the dispersion relation for 1D problem and give the numerical dispersion results,and compare the method against the fourth-order Lax-Wendroff correction (LWC) and the displacement-stress staggered-grid methods. Thirdly,for 2D case we use the improved RK,LWC and staggered-grid methods to simulate the acoustic wave fields,and present some comparisons of the computational efficiency and numerical results for different methods.Finally,two layered-medium models are further selected to investigate the computational validity of the acoustic and elastic wave-field simulations.These numerical results show that the improved method has weak numerical dispersion,high computational efficiency,and great potentiality of application in seismic exploration.
In this article,we present an improved algorithm of the fourth-order Runge-Kutta (RK) method to solve the wave equations.We first change the original fourth-order Runge-Kutta method into a 2-stage scheme,and then introduce a weighting parameter in the first stage to obtain a weighted scheme.To study this new improved method,first of all,we analyze its stability condition for 1D and 2D cases.Secondly,we derive the dispersion relation for 1D problem and give the numerical dispersion results,and compare the method against the fourth-order Lax-Wendroff correction (LWC) and the displacement-stress staggered-grid methods. Thirdly,for 2D case we use the improved RK,LWC and staggered-grid methods to simulate the acoustic wave fields,and present some comparisons of the computational efficiency and numerical results for different methods.Finally,two layered-medium models are further selected to investigate the computational validity of the acoustic and elastic wave-field simulations.These numerical results show that the improved method has weak numerical dispersion,high computational efficiency,and great potentiality of application in seismic exploration.
引文
[1] Chen K H.Propagating numerical model of elastic wave in anisotropic in homogeneous media-finite element method.Symposium of 54~(th) SEG,1984,S4:631 ～ 632
[2] Seron F J,Sanz F J,Kindelan M,et al.Finite-element method for elastic wave propagation.Comm.Appl.Numerical Methods ,1990,6(2) :359 ～368
[3] 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.ChineseJ.Geophys.(in Chinese),2002,45(4) :575 ～ 583
[4] Alterman Z,Karal F C.Propagation of elastic waves in layered media by finite-difference methods.Bull.Seism.Soc.Am.,1968,58:367 ～ 398
[5] Kelly K,Ward R,Treitel S,et al.Synthetic seismograms:a finite-difference approach.Geophysics,1976 ,41:2 ～27
[6] Dablain M A.The application of high-order differencing to the scalar wave equation.Geophysics,1986,51:54 ～ 66
[7] Virieux J.P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method.Geophysics,1986,41:889 ～ 901
[8] 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.地球物理学报,2000,43(3) :411～419 Dong L G,Ma Z T,Gao J Z,et al.A staggered-grid highorder difference method of one-order elastic wave equation.ChineseJ.Geophys.(in Chinese),2000,43(3) :411 ～ 419
[9] Kosloff D,Baysal E.Forward modeling by a Fourier method.Geophysics,1982,47:1402 ～ 1412
[10] Huang B S.A program for two-dimensional seismic wave propagation by the pseudo-spectrum method.Comput.Geosci.,1992,18:289 ～ 307
[11] Booth D C,Crampin S.The anisotropic reflectivity technique:anomalous arrives from an anisotropic upper mantle.Geophys.J.R.Astr.Soc.,1983,72:767 ～ 782
[12] Chen X F.A Systematic and efficient method of computing normal modes for multi-layered half-space.Geophys.J.Int.,1993,115:391 ～ 409
[13] Zhang H M,Chen X F,Chang S H.An efficient numerical method for computing synthetic seismograms for a layered half-space with sources and receivers at close or same depths.Pure Appl.Geophys.,1993,160(3/4) :467 ～ 486
[14] Chapman C H,Shearer P M.Ray tracing in azimuthally anisotropic media-II:quasi-shear-wave coupling.Geophys.J.,1989,96:65 ～ 83
[15] Yang D H,Peng J M,Lu M,et al.Optimal nearly analytic discrete approximation to the scalar wave equation.Bull.Seism.Soc.Am.,2006,96:1114 ～ 1130
[16] Igel J,Weber M.SH-wave propagation in the whole mantle using high-order finite differences.Geophys.Res.Lett.,1995,22:731 ～ 734
[17] 王书强,杨顶辉,杨宽德.弹性波方程的紧致差分方法.清华大学学报(自然科学版),2002,42(8) :1128～1131 Wang S Q,Yang D H,Yang K D.Compact finite difference scheme for elastic equations.J.Tsinghua Univ.(Sci.& Tech.) (in Chinese) ,2002,42(8) :1128-1131
[18] Levander A R.Fourth-order finite-difference P-SV seismograms.Geophysics,1988,53:1425 ～ 1436
[19] Graves R W.Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences.Bull.Seism.Soc.Am.,1996,86:1091 ～ 1106
[20] 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
[21] Yang D H,Chen S ,Li J Z.A Runge-Kutta method using high-order interpolation approximation for solving 2D acoustic and elastic wave equations.Journal of Seismic Exploration , 2007,16:331 ～ 353
[22] Moczo P,Kristek J,Halada L,3D fourth-order staggered-grid finite-difference schemes:stability and grid dispersion.Bull.Seism.Soc.Am.,2000,90(3) :587～603
[23] Jiang G S,Shu C W.Efficient implementation of Weighted ENO schemes.J.Comput.Phys.,1996,126:202～228
[24] 王磊,杨顶辉,邓小英.非均匀介质中地震波应力场的WNAD方法及其数值模拟.地球物理学报,2009,52(6) :1526～1535 Wang L,Yang D H,Deng X Y.A WNAD method for seismic stress-field modeling in heterogeneous media.Chinese J.Geophys.(in Chinese) ,2009,52(6) :1526～1535
[25] Richetmyer R D,Morton K W.Difference Methods for Initial Value Problems.New York:Interscience,2002
[26] Sei A,Symes W.Dispersion analysis of numerical wave propagation and its computational consequences.J.Sci.Comput.,1994,10:1 ～ 27
[27] Yang D H,Wang S Q,Zhang Z J,et al.N-times absorbing boundary conditions for compact finite-difference modeling of acoustic and elastic wave propagation in the 2D TI medium.Bull.Seism.Soc.Am.,2003,93:2389 ～ 2401
[28] Yang D H,Teng J W,Zhang Z J,et al.A nearly-analytic discrete method for acoustic and elastic wave equations in anisotropic media.Bull.Seism.Soc.Am.,2003 ,93 ( 2 ):882 ～ 890
[29] Yang D H,Song G J,ChenS,et al.An improved nearly analytical discrete method:An efficient tool to simulate the seismic response of 2-D porous structures.Journal of Geophysics and Engineering ,2007,4(1) :40 ～ 52．
[2] Seron F J,Sanz F J,Kindelan M,et al.Finite-element method for elastic wave propagation.Comm.Appl.Numerical Methods ,1990,6(2) :359 ～368
[3] 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.ChineseJ.Geophys.(in Chinese),2002,45(4) :575 ～ 583
[4] Alterman Z,Karal F C.Propagation of elastic waves in layered media by finite-difference methods.Bull.Seism.Soc.Am.,1968,58:367 ～ 398
[5] Kelly K,Ward R,Treitel S,et al.Synthetic seismograms:a finite-difference approach.Geophysics,1976 ,41:2 ～27
[6] Dablain M A.The application of high-order differencing to the scalar wave equation.Geophysics,1986,51:54 ～ 66
[7] Virieux J.P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method.Geophysics,1986,41:889 ～ 901
[8] 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.地球物理学报,2000,43(3) :411～419 Dong L G,Ma Z T,Gao J Z,et al.A staggered-grid highorder difference method of one-order elastic wave equation.ChineseJ.Geophys.(in Chinese),2000,43(3) :411 ～ 419
[9] Kosloff D,Baysal E.Forward modeling by a Fourier method.Geophysics,1982,47:1402 ～ 1412
[10] Huang B S.A program for two-dimensional seismic wave propagation by the pseudo-spectrum method.Comput.Geosci.,1992,18:289 ～ 307
[11] Booth D C,Crampin S.The anisotropic reflectivity technique:anomalous arrives from an anisotropic upper mantle.Geophys.J.R.Astr.Soc.,1983,72:767 ～ 782
[12] Chen X F.A Systematic and efficient method of computing normal modes for multi-layered half-space.Geophys.J.Int.,1993,115:391 ～ 409
[13] Zhang H M,Chen X F,Chang S H.An efficient numerical method for computing synthetic seismograms for a layered half-space with sources and receivers at close or same depths.Pure Appl.Geophys.,1993,160(3/4) :467 ～ 486
[14] Chapman C H,Shearer P M.Ray tracing in azimuthally anisotropic media-II:quasi-shear-wave coupling.Geophys.J.,1989,96:65 ～ 83
[15] Yang D H,Peng J M,Lu M,et al.Optimal nearly analytic discrete approximation to the scalar wave equation.Bull.Seism.Soc.Am.,2006,96:1114 ～ 1130
[16] Igel J,Weber M.SH-wave propagation in the whole mantle using high-order finite differences.Geophys.Res.Lett.,1995,22:731 ～ 734
[17] 王书强,杨顶辉,杨宽德.弹性波方程的紧致差分方法.清华大学学报(自然科学版),2002,42(8) :1128～1131 Wang S Q,Yang D H,Yang K D.Compact finite difference scheme for elastic equations.J.Tsinghua Univ.(Sci.& Tech.) (in Chinese) ,2002,42(8) :1128-1131
[18] Levander A R.Fourth-order finite-difference P-SV seismograms.Geophysics,1988,53:1425 ～ 1436
[19] Graves R W.Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences.Bull.Seism.Soc.Am.,1996,86:1091 ～ 1106
[20] 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
[21] Yang D H,Chen S ,Li J Z.A Runge-Kutta method using high-order interpolation approximation for solving 2D acoustic and elastic wave equations.Journal of Seismic Exploration , 2007,16:331 ～ 353
[22] Moczo P,Kristek J,Halada L,3D fourth-order staggered-grid finite-difference schemes:stability and grid dispersion.Bull.Seism.Soc.Am.,2000,90(3) :587～603
[23] Jiang G S,Shu C W.Efficient implementation of Weighted ENO schemes.J.Comput.Phys.,1996,126:202～228
[24] 王磊,杨顶辉,邓小英.非均匀介质中地震波应力场的WNAD方法及其数值模拟.地球物理学报,2009,52(6) :1526～1535 Wang L,Yang D H,Deng X Y.A WNAD method for seismic stress-field modeling in heterogeneous media.Chinese J.Geophys.(in Chinese) ,2009,52(6) :1526～1535
[25] Richetmyer R D,Morton K W.Difference Methods for Initial Value Problems.New York:Interscience,2002
[26] Sei A,Symes W.Dispersion analysis of numerical wave propagation and its computational consequences.J.Sci.Comput.,1994,10:1 ～ 27
[27] Yang D H,Wang S Q,Zhang Z J,et al.N-times absorbing boundary conditions for compact finite-difference modeling of acoustic and elastic wave propagation in the 2D TI medium.Bull.Seism.Soc.Am.,2003,93:2389 ～ 2401
[28] Yang D H,Teng J W,Zhang Z J,et al.A nearly-analytic discrete method for acoustic and elastic wave equations in anisotropic media.Bull.Seism.Soc.Am.,2003 ,93 ( 2 ):882 ～ 890
[29] Yang D H,Song G J,ChenS,et al.An improved nearly analytical discrete method:An efficient tool to simulate the seismic response of 2-D porous structures.Journal of Geophysics and Engineering ,2007,4(1) :40 ～ 52．
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