最优化辛格式广义离散奇异核褶积微分算子地震波场模拟
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摘要
将波动方程变换至Hamilton体系,构造了一种新的保结构算法,即最优化辛格式广义褶积微分算子(OSGCD).在时间离散上,首先引入了Lie算子设计二级二阶辛格式,基于最小误差原理得到了优化的辛格式.在空间离散上,引入广义离散奇异核褶积微分算子计算空间微分,提出了一种有效方法优化GCD并得到了稳定的算子系数.针对本文发展的新方法,给出了OSGCD稳定性条件.在数值实验中,将OSGCD与多种方法比较,从精度和计算效率两方面分析了OSGCD的计算优势,计算结果也表明OSGCD长时程以及非均匀介质中地震波模拟亦具有较强能力.
In this paper,seismic wave equation is transformed into Hamiltonian system,and a new symplectic numerical scheme is developed,which is so called optimal symplectic algorithm and generalized discrete convolutional differentiator(OSGCD).For temporal discretization,OSGCD introduces Lie operators to construct second-order and two stage symplectic scheme and adopts optimal symplectic scheme based on minimum error principle.For the spatial derivative,OSGCD employs generalized discrete convolution differentiator to approximate the spatial differential operators and uses derivative approximation to obtain stable operator coefficients.We obtain the stability condition for 2D case.In numerical experiments,OSGCD is compared with different methods and it has advantages in both accuracy and efficiency.OSGCD also has the ability for modeling long-term seismic wave propagation and modeling seismic wave in heterogeneous media.
In this paper,seismic wave equation is transformed into Hamiltonian system,and a new symplectic numerical scheme is developed,which is so called optimal symplectic algorithm and generalized discrete convolutional differentiator(OSGCD).For temporal discretization,OSGCD introduces Lie operators to construct second-order and two stage symplectic scheme and adopts optimal symplectic scheme based on minimum error principle.For the spatial derivative,OSGCD employs generalized discrete convolution differentiator to approximate the spatial differential operators and uses derivative approximation to obtain stable operator coefficients.We obtain the stability condition for 2D case.In numerical experiments,OSGCD is compared with different methods and it has advantages in both accuracy and efficiency.OSGCD also has the ability for modeling long-term seismic wave propagation and modeling seismic wave in heterogeneous media.
引文
[1]陈景波,秦孟兆.辛几何算法在射线追踪中的应用.数值计算与计算机应用,2000,21(4):254-265.Chen J B,Qin M Z.Ray tracing by symplectic algorithm.Journal of Numerical Methods and Computer Applications(in Chinese),2000,21(4):254-265.
[2]ˇCerveny V.Seismic Ray Theory.Cambridge:Cambridge University Press,2001.
[3]Pao Y H,Varatharajulu V.Huygen′s principle,radiation conditions,and integral formulas for the scattering of elastic wave.J.Acoust.Soc.Am.,1976,59(6):1361-1371.
[4]Ursin B.Review of elastic and electromagnetic wave propagation in horizontally layered media.Geophysics,1983,44(8):1063-1081.
[5]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 for3-D Complex Media(in Chinese).Beijing:Petroleum Industry Press(in Chinese),2005.
[6]董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法稳定性研究.地球物理学报,2000,43(6):856-864.Dong L G,Ma Z T,Cao J Z.A study on stability of the staggered-grid high-order difference method of first-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(6):856-864.
[7]张美根,王妙月,李小凡等.各向异性弹性波场的有限元数值模拟.地球物理学进展,2002,17(3):384-389.Zhang M G,Wang M Y,Li X F,et al.Finite element forward modeling of anisotropic elastic waves.Progress in Geophysics(in Chinese),2002,17(3):384-389.
[8]Gazdag J.Modeling of the acoustic wave equation with transform methods.Geophysics,1981,46(6):854-859.
[9]Komatitsch D.Spectral and spectral-element methods for the 2Dand3Delastodynamics equations in heterogeneous media.Paris,France:Institut de Physique du Globe,1997.
[10]Carcione J M,Herman G C,ten Kroode A P E.Seismic modeling.Geophysics,2002,67(4):1304-1325.
[11]Yang D H,Song G J,Lu M.Optimally accurate nearly analytic discrete scheme for wave-field simulation in3D anisotropic media.Bull.Seism.Soc.Am.,2007,97(5):1557-1569.
[12]Bayliss A,Jordan K E,Lemesurier B J,et al.A fourth-order accurate finite-difference scheme for the computation of elastic wave.Bull.Seism.Soc.Am.,1986,76(4):1115-1132.
[13]Fornberg B.The Pseudospectral method:Comparisons with finite differences for the elastic wave equation.Geophysics,1987,52(4):483-501.
[14]Fornberg B.The Pseudospectral method:Accurate representation of interfaces in elastic wave calculations.Geophysics,1988,53(5):625-637.
[15]Li X F,Wang W S,Lu M W,et al.Structure-preserving modelling of elastic waves:a symplectic discrete singular convolution differentiator method.Geophys.J.Int.,2012,188(3):1382-1392.
[16]Mora P.Elastic Finite Differences with Convolutional Operators.California:Stanford Exploration Project Report,1986:277-290.
[17]Holberg O.Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena.Geophysical Prospecting,1987,35(6):629-655.
[18]Zhou B,Greenhalgh S.Seismic scalar wave equation modeling by a convolutional differentiator.Bull.Seism.Soc.Amer.,1992,82(1):289-303.
[19]张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法.地震学报,1996,18(1):63-69.Zhang Z J.Teng J W.Yang D H.The convolutional differentiator method for numerical modeling of acoustic and elastic wave-field.Acta Seismologica Sinica(in Chinese),1996,18(1):63-69.
[20]戴志阳,孙建国,查显杰.地震波场模拟中的褶积微分算子法.吉林大学学报(地球科学版),2005,35(4):520-524.Dai Z Y,Sun J G,Zha X J.Seismic wave field modeling with convolutional differentiator algorithm.Journal of JilinUniversity(Earth Science Edition)(in Chinese),2005,35(4):520-524.
[21]龙桂华,赵宇波,赵家福.地震波数值模拟中的最优Shannon奇异核褶积微分算子.地震学报,2011,33(5):650-662.Long G H,Zhao Y B,Zhao J F.Optimal Shannon singular kernel convolutional differentiator in seismic wave modeling.Acta Seismologica Sinica(in Chinese),2011,33(5):650-662.
[22]Wei G W.Quasi wavelets and quasi interpolating wavelets.Chemical Physics Letters,1998,296(3-4):215-220.
[23]Qian L W.On the regularized Whittaker-kotel′nikov-Shannon sampling formula.American Mathematical Society,2002,131(4):1169-1176.
[24]Li R,Wu X.Two new Fourth-order Three-stage symplectic integrators.Chin.Phys.Lett.,2011,28(7):070201,doi:10.1088/0256-307X/28/7/070201.
[25]Li R,Wu X.Application of the fourth-order three-stage symplectic integrators in chaos determination.Eur.Phys.J.Plus,2011,126(8):73-80.
[26]Casas F,Murua A.An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications.Technical report,Universitat Jaume I,2008.
[27]Ruth R D.A canonical integration technique.IEEE Transactions on Nuclear Science,1983,30(4):2669-2671.
[28]Iwatsu R.Two new solutions to the third-order symplectic integration method.Physics Letter A,2009,373(34):3056-3060.
[29]李一琼,李小凡,朱童.基于辛格式奇异核褶积微分算子的地震标量波场模拟.地球物理学报,2011,54(7):1827-1834.Li Y Q,Li X F,Zhu T.The seismic scalar wave field modeling by symplectic scheme and singular kernel convolutional differentiator.Chinese J.Geophys.(in Chinese),2011,54(7):1827-1834.
[30]李一琼,李小凡,张美根.基于辛格式离散奇异褶积微分算子的弹性波场模拟.地球物理学报,2012,55(5):1725-1731.Li Y Q,Li X F,Zhang M G.The elastic wave fields modeling by symplectic discrete singular convolution differentiator method.Chinese J.Geophys.(in Chinese),2012,55(5):1725-1731.
[31]汪文帅,李小凡.一种新的三阶非力梯度辛积分算法.武汉大学学报(理学版),2012,58(3):221-228.Wang W S,Li X F.A new solution to the third-order non-gradient symplectic integration algorithm.J.Wuhan Univ.(Nat.Sci.Ed.)(in Chinese),2012,58(3):221-228.
[32]罗明秋,刘洪,李幼铭.地震波传播的哈密顿表述及辛几何算法.地球物理学报,2001,44(1):120-128.Luo M Q,Liu H,Li Y M.Hamiltonian description and symplectic method of seismic wave propagation.Chinese J.Geophys.(in Chinese),2001,44(1):120-128.
[33]孙耿.波动方程的一类显式辛格式.计算数学,1997,(1):1-10.Sun G.A class of explicitly symplectic schemes for wave equations.Mathematica Numerica Sinica(in Chinese),1997,(1):1-10.
[34]马啸,杨顶辉,张锦华.求解声波方程的辛可分Runge-Kutta方法.地球物理学报,2010,53(8):1993-2003.Ma X,Yang D H,Zhang J H.Symplectic partitioned Runge-Kutta method for solving the acoustic wave equation.Chinese J.Geophys.(in Chinese),2010,53(8):1993-2003.
[35]Ma X,Yang D H,Liu F Q.A nearly analytic symplectically partitioned Runge-Kutta method for2-D seismic wave equations.Geophys.J.Int.,2011,187(1):480-496.
[36]Li X F,Li Y Q,Zhang M G,et al.Scalar seismic-wave equation modeling by a multisymplectic discrete singular convolution differentiator method.Bull.Seism.Soc.Am.,2011,101(4):1710-1718.
[37]冯康,秦孟兆.哈密尔顿系统的辛几何算法.杭州:浙江科学技术出版社,2003.Feng K,Qin M Z.Symplectic Geometric Algorithm for Hamiltonian Systems(in Chinese).Hangzhou:Zhejiang Science&Technology Press,2003.
[38]Suzuki M.General theory of higher-order decomposition of exponential operators and symplectic integrators.Phys.Lett.A,1992,165(5-6):387-395.
[39]Dablain M A.The application of high-order differencing to the scalar wave equation.Geophysics,1986,51(1):54-56.
[40]Fei T,Larner K.Elimination of numerical dispersion in finite-difference modeling and migration by flux-corrected transport.Geophysics,1995,60(6):1380-1842.
[41]龙桂华,李小凡,张美根.基于Shannon奇异核理论的褶积微分算子在地震波场模拟中的应用.地球物理学报,2009,52(4):1014-1024.Long G H,Li X F,Zhang M G.The application of convolutional differentiator in seismic modeling based on Shannon singular kernel theory.Chinese J.Geophys.(in Chinese),2009,52(4):1014-1024.
[42]Smith G D.Numerical Solution of Partial Differential Equations:Finite Difference Methods.Oxford:Glarendon Press,1978.
[43]李信富,李小凡.地震波传播的褶积微分算子法数值模拟.地球科学-中国地质大学学报,2008,33(6):861-866.Li X F,Li X F.Numerical simulation of seismic wave propagation using convolutional differentiator.Earth Science-Journal of China University of Geosciences(in Chinese),2008,33(6):861-866.
[44]Berenger J.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,1994,114(2):185-220.
[45]Xu J,Wu X.Several fourth-order force gradient symplectic algorithms.Research in Astronomy and Astrophysics,2010,10(2):173-188.
[46]Jo C H,Shin C,Suh J H.An optimal9-point,finite-difference,frequency-space,2-D Scalar wave extrapolator.Geophysics,1996,61(2):529-537.
[2]ˇCerveny V.Seismic Ray Theory.Cambridge:Cambridge University Press,2001.
[3]Pao Y H,Varatharajulu V.Huygen′s principle,radiation conditions,and integral formulas for the scattering of elastic wave.J.Acoust.Soc.Am.,1976,59(6):1361-1371.
[4]Ursin B.Review of elastic and electromagnetic wave propagation in horizontally layered media.Geophysics,1983,44(8):1063-1081.
[5]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 for3-D Complex Media(in Chinese).Beijing:Petroleum Industry Press(in Chinese),2005.
[6]董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法稳定性研究.地球物理学报,2000,43(6):856-864.Dong L G,Ma Z T,Cao J Z.A study on stability of the staggered-grid high-order difference method of first-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(6):856-864.
[7]张美根,王妙月,李小凡等.各向异性弹性波场的有限元数值模拟.地球物理学进展,2002,17(3):384-389.Zhang M G,Wang M Y,Li X F,et al.Finite element forward modeling of anisotropic elastic waves.Progress in Geophysics(in Chinese),2002,17(3):384-389.
[8]Gazdag J.Modeling of the acoustic wave equation with transform methods.Geophysics,1981,46(6):854-859.
[9]Komatitsch D.Spectral and spectral-element methods for the 2Dand3Delastodynamics equations in heterogeneous media.Paris,France:Institut de Physique du Globe,1997.
[10]Carcione J M,Herman G C,ten Kroode A P E.Seismic modeling.Geophysics,2002,67(4):1304-1325.
[11]Yang D H,Song G J,Lu M.Optimally accurate nearly analytic discrete scheme for wave-field simulation in3D anisotropic media.Bull.Seism.Soc.Am.,2007,97(5):1557-1569.
[12]Bayliss A,Jordan K E,Lemesurier B J,et al.A fourth-order accurate finite-difference scheme for the computation of elastic wave.Bull.Seism.Soc.Am.,1986,76(4):1115-1132.
[13]Fornberg B.The Pseudospectral method:Comparisons with finite differences for the elastic wave equation.Geophysics,1987,52(4):483-501.
[14]Fornberg B.The Pseudospectral method:Accurate representation of interfaces in elastic wave calculations.Geophysics,1988,53(5):625-637.
[15]Li X F,Wang W S,Lu M W,et al.Structure-preserving modelling of elastic waves:a symplectic discrete singular convolution differentiator method.Geophys.J.Int.,2012,188(3):1382-1392.
[16]Mora P.Elastic Finite Differences with Convolutional Operators.California:Stanford Exploration Project Report,1986:277-290.
[17]Holberg O.Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena.Geophysical Prospecting,1987,35(6):629-655.
[18]Zhou B,Greenhalgh S.Seismic scalar wave equation modeling by a convolutional differentiator.Bull.Seism.Soc.Amer.,1992,82(1):289-303.
[19]张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法.地震学报,1996,18(1):63-69.Zhang Z J.Teng J W.Yang D H.The convolutional differentiator method for numerical modeling of acoustic and elastic wave-field.Acta Seismologica Sinica(in Chinese),1996,18(1):63-69.
[20]戴志阳,孙建国,查显杰.地震波场模拟中的褶积微分算子法.吉林大学学报(地球科学版),2005,35(4):520-524.Dai Z Y,Sun J G,Zha X J.Seismic wave field modeling with convolutional differentiator algorithm.Journal of JilinUniversity(Earth Science Edition)(in Chinese),2005,35(4):520-524.
[21]龙桂华,赵宇波,赵家福.地震波数值模拟中的最优Shannon奇异核褶积微分算子.地震学报,2011,33(5):650-662.Long G H,Zhao Y B,Zhao J F.Optimal Shannon singular kernel convolutional differentiator in seismic wave modeling.Acta Seismologica Sinica(in Chinese),2011,33(5):650-662.
[22]Wei G W.Quasi wavelets and quasi interpolating wavelets.Chemical Physics Letters,1998,296(3-4):215-220.
[23]Qian L W.On the regularized Whittaker-kotel′nikov-Shannon sampling formula.American Mathematical Society,2002,131(4):1169-1176.
[24]Li R,Wu X.Two new Fourth-order Three-stage symplectic integrators.Chin.Phys.Lett.,2011,28(7):070201,doi:10.1088/0256-307X/28/7/070201.
[25]Li R,Wu X.Application of the fourth-order three-stage symplectic integrators in chaos determination.Eur.Phys.J.Plus,2011,126(8):73-80.
[26]Casas F,Murua A.An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications.Technical report,Universitat Jaume I,2008.
[27]Ruth R D.A canonical integration technique.IEEE Transactions on Nuclear Science,1983,30(4):2669-2671.
[28]Iwatsu R.Two new solutions to the third-order symplectic integration method.Physics Letter A,2009,373(34):3056-3060.
[29]李一琼,李小凡,朱童.基于辛格式奇异核褶积微分算子的地震标量波场模拟.地球物理学报,2011,54(7):1827-1834.Li Y Q,Li X F,Zhu T.The seismic scalar wave field modeling by symplectic scheme and singular kernel convolutional differentiator.Chinese J.Geophys.(in Chinese),2011,54(7):1827-1834.
[30]李一琼,李小凡,张美根.基于辛格式离散奇异褶积微分算子的弹性波场模拟.地球物理学报,2012,55(5):1725-1731.Li Y Q,Li X F,Zhang M G.The elastic wave fields modeling by symplectic discrete singular convolution differentiator method.Chinese J.Geophys.(in Chinese),2012,55(5):1725-1731.
[31]汪文帅,李小凡.一种新的三阶非力梯度辛积分算法.武汉大学学报(理学版),2012,58(3):221-228.Wang W S,Li X F.A new solution to the third-order non-gradient symplectic integration algorithm.J.Wuhan Univ.(Nat.Sci.Ed.)(in Chinese),2012,58(3):221-228.
[32]罗明秋,刘洪,李幼铭.地震波传播的哈密顿表述及辛几何算法.地球物理学报,2001,44(1):120-128.Luo M Q,Liu H,Li Y M.Hamiltonian description and symplectic method of seismic wave propagation.Chinese J.Geophys.(in Chinese),2001,44(1):120-128.
[33]孙耿.波动方程的一类显式辛格式.计算数学,1997,(1):1-10.Sun G.A class of explicitly symplectic schemes for wave equations.Mathematica Numerica Sinica(in Chinese),1997,(1):1-10.
[34]马啸,杨顶辉,张锦华.求解声波方程的辛可分Runge-Kutta方法.地球物理学报,2010,53(8):1993-2003.Ma X,Yang D H,Zhang J H.Symplectic partitioned Runge-Kutta method for solving the acoustic wave equation.Chinese J.Geophys.(in Chinese),2010,53(8):1993-2003.
[35]Ma X,Yang D H,Liu F Q.A nearly analytic symplectically partitioned Runge-Kutta method for2-D seismic wave equations.Geophys.J.Int.,2011,187(1):480-496.
[36]Li X F,Li Y Q,Zhang M G,et al.Scalar seismic-wave equation modeling by a multisymplectic discrete singular convolution differentiator method.Bull.Seism.Soc.Am.,2011,101(4):1710-1718.
[37]冯康,秦孟兆.哈密尔顿系统的辛几何算法.杭州:浙江科学技术出版社,2003.Feng K,Qin M Z.Symplectic Geometric Algorithm for Hamiltonian Systems(in Chinese).Hangzhou:Zhejiang Science&Technology Press,2003.
[38]Suzuki M.General theory of higher-order decomposition of exponential operators and symplectic integrators.Phys.Lett.A,1992,165(5-6):387-395.
[39]Dablain M A.The application of high-order differencing to the scalar wave equation.Geophysics,1986,51(1):54-56.
[40]Fei T,Larner K.Elimination of numerical dispersion in finite-difference modeling and migration by flux-corrected transport.Geophysics,1995,60(6):1380-1842.
[41]龙桂华,李小凡,张美根.基于Shannon奇异核理论的褶积微分算子在地震波场模拟中的应用.地球物理学报,2009,52(4):1014-1024.Long G H,Li X F,Zhang M G.The application of convolutional differentiator in seismic modeling based on Shannon singular kernel theory.Chinese J.Geophys.(in Chinese),2009,52(4):1014-1024.
[42]Smith G D.Numerical Solution of Partial Differential Equations:Finite Difference Methods.Oxford:Glarendon Press,1978.
[43]李信富,李小凡.地震波传播的褶积微分算子法数值模拟.地球科学-中国地质大学学报,2008,33(6):861-866.Li X F,Li X F.Numerical simulation of seismic wave propagation using convolutional differentiator.Earth Science-Journal of China University of Geosciences(in Chinese),2008,33(6):861-866.
[44]Berenger J.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,1994,114(2):185-220.
[45]Xu J,Wu X.Several fourth-order force gradient symplectic algorithms.Research in Astronomy and Astrophysics,2010,10(2):173-188.
[46]Jo C H,Shin C,Suh J H.An optimal9-point,finite-difference,frequency-space,2-D Scalar wave extrapolator.Geophysics,1996,61(2):529-537.
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