错格傅里叶伪谱微分算子在波场模拟中的应用
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摘要
本文在传统傅里叶微分矩阵的基础上,对原始微分算子进行改进,引入了错格微分算子的傅里叶伪谱方法.尽管该方法增加了一些计算量,但却极大地提高了计算精度和稳定性.而且,该方法将微分计算过程由传统的傅里叶变换转换为一般的矩阵矢量乘积,大大降低了微分求解过程的复杂程度.在均匀介质中,将错格伪谱微分算子计算的结果和解析解进行比较,结果表明本文算子几乎达到了解析解的精度.而在分层均匀介质中的实验结果同时显示,该方法精度高、稳定性好,是一种研究层状介质中地震波传播的有效数值方法.
Based on traditional Fourier differentiation matrix and some modifications of the original differentiation operator,this paper refers to a new kind of method——staggered-grid Fourier pseudospectral differentiation operator.The staggered-grid strategy proposed in this paper,though a little more time-consuming than the traditional method,greatly improves the accuracy and stability of the computing process.And the thought of converting the traditional fast Fourier transform into matrix vector multiplication,at the same time,reduces the complexity of the difference,which then makes them more easily to be treated.A comparison with the analytic result for vertical displacement in homogeneous medium demonstrates that the staggered-grid differentiation operator almost achieves as the same accuracy as the analytic method.The application of this method in homogeneous layered media also shows high accuracy and robust stability.These attractive properties of this method make it an effective modeling method for seismic wave propagation in homogeneous layered media.
Based on traditional Fourier differentiation matrix and some modifications of the original differentiation operator,this paper refers to a new kind of method——staggered-grid Fourier pseudospectral differentiation operator.The staggered-grid strategy proposed in this paper,though a little more time-consuming than the traditional method,greatly improves the accuracy and stability of the computing process.And the thought of converting the traditional fast Fourier transform into matrix vector multiplication,at the same time,reduces the complexity of the difference,which then makes them more easily to be treated.A comparison with the analytic result for vertical displacement in homogeneous medium demonstrates that the staggered-grid differentiation operator almost achieves as the same accuracy as the analytic method.The application of this method in homogeneous layered media also shows high accuracy and robust stability.These attractive properties of this method make it an effective modeling method for seismic wave propagation in homogeneous layered media.
引文
[1]王红落.地震波传播与成像若干问题的研究[博士论文].北京:中国科学院地质与地球物理研究所,2005Wang H L.The study of several problems in seismic wave propagation and imaging[Ph.D.thesis](in Chinese).Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2005
[2]董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶有限差分解法.地球物理学报,2000,43(3):411～419Dong L G,Ma Z T,Cao J Z.A staggered-grid high-order difference method of one-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(3):411～419
[3]Virieux J.P-SV wave propagation in heterogeneous media:velocity-stress finite difference method.Geophysics,1986,51(4):889～901
[4]ˇStekl I,Pratt R G.Accurate viscoelastic modeling by frequency-domain finite differences using rotated operators.Geophysics,1998,63(5):1779～1794
[5]Saenger E H,Gold N,Shapiro S A.Modeling the propagation of elastic waves using a modified finite-difference grid.Wave Motion,2000,31:77～92
[6]张剑锋.弹性波数值模拟的非规则网格差分法.地球物理学报,1998,41(增刊):357～365Zhang J F.Irregular grid finite-difference method for numerical simulation of elastic wave propagation.Chinese J.Geophys.(in Chinese),1998,41(Suppl.):357～365
[7]郭宗汾,陈昆华,龚钟等.在空间域和时间域解弹性动力学问题的有限元方法.石油物探,1980,1:26～33Guo Z F,Chen K H,Gong Z,et al.The finite element method in space andtime for solving elastodynamic problems.Geophysical Prospecting for Petroleum(in Chinese),1980,1:26～33
[8]牟永光.有限元法弹性波偏移.地球物理学报,1984,27(3):268～278Mu Y G.Finite element elastic wave migration.Chinese J.Geophys.(in Chinese),1984,27(3):268～278
[9]邓玉琼,张之立.弹性波的三维有限元模拟.地球物理学报,1990,33(1):44～51Teng Y C,Zhang Z L.Three-dimensional finite element modeling of elastic wave.Chinese J.Geophys.(in Chinese),1990,33(1):44～51
[10]底青云,王妙月.弹性波有限元逆时偏移技术研究.地球物理学报,1997,40(4):570～579Di Q Y,Wang MY.The study of finite element reverse-time migrationfor elastic wave.Chinese J.Geophys.(in Chinese),1997,40(4):570～579
[11]Kreiss H O,Oliger J.Comparison of accurate methods for the integration hyperbolic equations.Tellus,24:199～215
[12]Fornberg B.The pseudospectral method:accurate representation of interfaces in elastic wave calculations.Geophysics,1988,53(5):625～637
[13]Dault C R,Brail L W,Nowack R L,et al.Acomparison of finite-difference and Fourier method calculations of synthetic seismograms.Bull.Seism.Am.,1989,79(4):1210～1230
[14]Baltensperger R,Trummer MR.Spectral differencing with a Twist.J.Sci.Comp.,2003,24:1465～1487
[15]Renaut R A,Fr hlich J H.A pseudospectral Chebyshev method for the2D wave equation with domain stretching and absorbing boundary conditions.J.Comput.Phys.,1996,124(2):324～336
[16]Canuto C,Hussaini M Y,Quarteroni A,et al.Spectral Methods in Fluid Dynamics.New York:Springer Verlag,1988
[17]Weideman J A,Reddy S C.A MATLAB differentiationmatrix suite.ACMTransactions on Mathematical Soft ware,2000,26(4):465～519
[18]Corrêa GJ,Spiegel man M,Carbotle S,et al.Centered and staggered Fourier derivatives and Hilbert transforms.Geophysics,2002,67(5):1558～1563
[19]Bale R A.Modeling3D anisotropic elastic data using the pseudospectral approach.CREWES Research Report,Calgary,2002
[20]Bale R A.Staggered grids for3D pseudospectral modeling in anisotropic elastic media.CREWES Research Report,Calgary,2002
[21]zdenvar T,Mc Mechan G A.Cause and reduction of numerical artifacts in pseudo-spectral wavefield extrapolation.Geophy.J.Int.,1996,126(3):819～828
[22]Graves R W.Simulating seismic wave propagation in3D elastic media using staggered-grid finite differences.Bull.Seism.Am.,1996,86(4):1091～1106
[23]Pitarka A.3D elastic finite-difference modeling of seismic motion using staggered grids with nonuniformspacing.Bull.Seism.Am.,1999,89(1):54～68
[24]Bord J P.Chebyshev and Fourier Spectral Methods.New York:Springer Verlag,1974
[25]Welfert B D.A remark on pseudospectral differentiation matrix suite.Report,Depart ment of Mathematics,Corvallis:Oregon State University,1999
[26]武晔,李小凡,赵玉莲等.2.5维各向异性介质中地震波场数值模拟.物探化探计算技术,2006,28(4):289～293Wu Y,Li X F,Zhao Y L,et al.2.5-D numerical simulation of seismic wave propagation in heterogeneous media by velocity-stress Pseudo-spectral method.Computing Techniques for Geophysical and Geochemical Exploration(in Chinese),2006,28(4):289～293
[27]牟永光,裴正林.三维复杂介质地震波数值模拟.北京:石油工业出版社,2004Mou Y G,Pei Z L.Numerical Modeling of Seismic Wave Propagation in3D Complex Media(in Chinese).Beijing:PetroleumIndustry Press,2004
[28]Cerjan C,Kosloff D,Kosloff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equation.Geophysics,1985,50(4):705～708
[29]赵志新,徐纪人,堀内茂木.错格实数傅里叶变换微分算子及其在非均匀介质波动传播研究中的应用.地球物理学报,2003,46(2):234～240Zhao Z X,Xu J R,Horiuchi S.Staggered grid real value FFT differentiation operator and its application on wave propagation simulationinthe heterogeneous medium.Chinese J.Geophys.(in Chinese),2003,46(2):234～240
[30]Aki K,Richards P G.Quantitative seismology theory and methods.San Francisco:W.H.Freeman and Company,1980
[2]董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶有限差分解法.地球物理学报,2000,43(3):411～419Dong L G,Ma Z T,Cao J Z.A staggered-grid high-order difference method of one-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(3):411～419
[3]Virieux J.P-SV wave propagation in heterogeneous media:velocity-stress finite difference method.Geophysics,1986,51(4):889～901
[4]ˇStekl I,Pratt R G.Accurate viscoelastic modeling by frequency-domain finite differences using rotated operators.Geophysics,1998,63(5):1779～1794
[5]Saenger E H,Gold N,Shapiro S A.Modeling the propagation of elastic waves using a modified finite-difference grid.Wave Motion,2000,31:77～92
[6]张剑锋.弹性波数值模拟的非规则网格差分法.地球物理学报,1998,41(增刊):357～365Zhang J F.Irregular grid finite-difference method for numerical simulation of elastic wave propagation.Chinese J.Geophys.(in Chinese),1998,41(Suppl.):357～365
[7]郭宗汾,陈昆华,龚钟等.在空间域和时间域解弹性动力学问题的有限元方法.石油物探,1980,1:26～33Guo Z F,Chen K H,Gong Z,et al.The finite element method in space andtime for solving elastodynamic problems.Geophysical Prospecting for Petroleum(in Chinese),1980,1:26～33
[8]牟永光.有限元法弹性波偏移.地球物理学报,1984,27(3):268～278Mu Y G.Finite element elastic wave migration.Chinese J.Geophys.(in Chinese),1984,27(3):268～278
[9]邓玉琼,张之立.弹性波的三维有限元模拟.地球物理学报,1990,33(1):44～51Teng Y C,Zhang Z L.Three-dimensional finite element modeling of elastic wave.Chinese J.Geophys.(in Chinese),1990,33(1):44～51
[10]底青云,王妙月.弹性波有限元逆时偏移技术研究.地球物理学报,1997,40(4):570～579Di Q Y,Wang MY.The study of finite element reverse-time migrationfor elastic wave.Chinese J.Geophys.(in Chinese),1997,40(4):570～579
[11]Kreiss H O,Oliger J.Comparison of accurate methods for the integration hyperbolic equations.Tellus,24:199～215
[12]Fornberg B.The pseudospectral method:accurate representation of interfaces in elastic wave calculations.Geophysics,1988,53(5):625～637
[13]Dault C R,Brail L W,Nowack R L,et al.Acomparison of finite-difference and Fourier method calculations of synthetic seismograms.Bull.Seism.Am.,1989,79(4):1210～1230
[14]Baltensperger R,Trummer MR.Spectral differencing with a Twist.J.Sci.Comp.,2003,24:1465～1487
[15]Renaut R A,Fr hlich J H.A pseudospectral Chebyshev method for the2D wave equation with domain stretching and absorbing boundary conditions.J.Comput.Phys.,1996,124(2):324～336
[16]Canuto C,Hussaini M Y,Quarteroni A,et al.Spectral Methods in Fluid Dynamics.New York:Springer Verlag,1988
[17]Weideman J A,Reddy S C.A MATLAB differentiationmatrix suite.ACMTransactions on Mathematical Soft ware,2000,26(4):465～519
[18]Corrêa GJ,Spiegel man M,Carbotle S,et al.Centered and staggered Fourier derivatives and Hilbert transforms.Geophysics,2002,67(5):1558～1563
[19]Bale R A.Modeling3D anisotropic elastic data using the pseudospectral approach.CREWES Research Report,Calgary,2002
[20]Bale R A.Staggered grids for3D pseudospectral modeling in anisotropic elastic media.CREWES Research Report,Calgary,2002
[21]zdenvar T,Mc Mechan G A.Cause and reduction of numerical artifacts in pseudo-spectral wavefield extrapolation.Geophy.J.Int.,1996,126(3):819～828
[22]Graves R W.Simulating seismic wave propagation in3D elastic media using staggered-grid finite differences.Bull.Seism.Am.,1996,86(4):1091～1106
[23]Pitarka A.3D elastic finite-difference modeling of seismic motion using staggered grids with nonuniformspacing.Bull.Seism.Am.,1999,89(1):54～68
[24]Bord J P.Chebyshev and Fourier Spectral Methods.New York:Springer Verlag,1974
[25]Welfert B D.A remark on pseudospectral differentiation matrix suite.Report,Depart ment of Mathematics,Corvallis:Oregon State University,1999
[26]武晔,李小凡,赵玉莲等.2.5维各向异性介质中地震波场数值模拟.物探化探计算技术,2006,28(4):289～293Wu Y,Li X F,Zhao Y L,et al.2.5-D numerical simulation of seismic wave propagation in heterogeneous media by velocity-stress Pseudo-spectral method.Computing Techniques for Geophysical and Geochemical Exploration(in Chinese),2006,28(4):289～293
[27]牟永光,裴正林.三维复杂介质地震波数值模拟.北京:石油工业出版社,2004Mou Y G,Pei Z L.Numerical Modeling of Seismic Wave Propagation in3D Complex Media(in Chinese).Beijing:PetroleumIndustry Press,2004
[28]Cerjan C,Kosloff D,Kosloff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equation.Geophysics,1985,50(4):705～708
[29]赵志新,徐纪人,堀内茂木.错格实数傅里叶变换微分算子及其在非均匀介质波动传播研究中的应用.地球物理学报,2003,46(2):234～240Zhao Z X,Xu J R,Horiuchi S.Staggered grid real value FFT differentiation operator and its application on wave propagation simulationinthe heterogeneous medium.Chinese J.Geophys.(in Chinese),2003,46(2):234～240
[30]Aki K,Richards P G.Quantitative seismology theory and methods.San Francisco:W.H.Freeman and Company,1980
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