基于WNAD方法的非一致网格算法及其弹性波场模拟
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摘要
加权近似解析离散化(WNAD)方法是近年发展的一种在粗网格步长条件下能有效压制数值频散的数值模拟技术.在地震勘探的实际应用中,不是所有情况都适合使用空间大网格步长.为适应波场模拟的实际需要,本文给出了求解波动方程的非一致网格上的WNAD算法.这种方法在低速区、介质复杂区域使用细网格,在其他区域采用粗网格计算.在网格过渡区域,根据近似解析离散化方法的特点,采用了新的插值公式,使用较少的网格点得到较高的插值精度.数值算例表明,非一致网格上的WNAD方法能够有效压制数值频散,显著减少计算内存需求量和计算时间,进一步提高了地震波场的数值模拟效率.
Weighted nearly-analytical discrete (WNAD) method is a new numerical technology developed in recent years,which can effectively suppress the numerical dispersion when a coarse grid is used.However,using a large spatial step is not always suitable for any seismic exploration cases.In order to meet the actual requirement of seismic wave-field simulations,we suggest a non-uniform grid algorithm based on the WNAD method to solve the wave equations in this paper.This algorithm uses a fine spatial grid in special computational domains such as the low-velocity areas and complicated media,and adopts a coarse spatial grid in the rest computational domains.Based on the characteristics of the WNAD method,this non-uniform grid algorithm uses a new interpolation formula to connect the fine grids and the coarse grids in the transition zone,thus obtains higher interpolation accuracy through using fewer grids.Numerical results show that the non-uniform grid algorithm can suppress effectively the numerical dispersion,and reduce the storage spaces and computational costs,resulting in further increasing the computational efficiency of seismic wave-field simulations.
Weighted nearly-analytical discrete (WNAD) method is a new numerical technology developed in recent years,which can effectively suppress the numerical dispersion when a coarse grid is used.However,using a large spatial step is not always suitable for any seismic exploration cases.In order to meet the actual requirement of seismic wave-field simulations,we suggest a non-uniform grid algorithm based on the WNAD method to solve the wave equations in this paper.This algorithm uses a fine spatial grid in special computational domains such as the low-velocity areas and complicated media,and adopts a coarse spatial grid in the rest computational domains.Based on the characteristics of the WNAD method,this non-uniform grid algorithm uses a new interpolation formula to connect the fine grids and the coarse grids in the transition zone,thus obtains higher interpolation accuracy through using fewer grids.Numerical results show that the non-uniform grid algorithm can suppress effectively the numerical dispersion,and reduce the storage spaces and computational costs,resulting in further increasing the computational efficiency of seismic wave-field simulations.
引文
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[8] Dablain M A.The application of high-order differencing to the scale wave equation Laser.Geophysics,1986,51(1) :54～66
[9] Virieux J.SH-wave propagation in heterogeneous media:Velocity-stress finite-difference method.Geophysics,1984,49(11) :1933～1957
[10] Yang D H,Teng J,Zhang Z J,et al.A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media.Bull.Seis.Soc.Am.,2003,93(2) :882～890
[11] Yang D H,Lu M,Wu R S,et al.An optimal nearly analytic discrete method for 2D acoustic and elastic wave equations,Bull.Seis.Soc.Am.,2004,94(5) :1982～1991
[12] Yang D H,Song G J,Chen S,et al.An improved nearly analytical discrete method:an efficient tool to simulate the seismic response of 2-D porous structures.J.Geophys.Eng,2007,4:40～52
[13] 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
[14] Demmel J W.Applied Numerical Linear Algebra.Philadelphia:SIAM,1997
[15] 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.Seis.Soc.Am.,2003,93(6) :2389～2401
[2] Moczo P.Finite-difference technique for SH waves in 2-D media,using irregular grids:application to the seismic response problem.Geophys.J.Int.,1989,99:321～329
[3] Jastram C,Tessemer E.Elastic modeling on a grid of vertically varying spacing.Geophys.Prosp.,1994,42:357～370
[4] 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 3D finite difference method using irregular grids for elastic wave propagation in anisotropic media.Chinese J.Geophys.(in Chinese),2004,47(2) :332～337
[5] 李胜军,孙成禹,倪长宽等.声波方程有限差分数值模拟的变网格步长算法.工程地球物理学报,2007,4(3) :207～212 Li S J,Sun C Y,Ni C W,et al.Acoustic equation numerical modeling on a grid of varying spacing.Chinese J.Eng.Geophys.(in Chinese),2007,4(3) :207～212
[6] 朱生旺,曲寿利,魏修成等.变网格有限差分弹性波方程数值模拟方法.石油地球物理勘探,2007,42(6) :634～639 Zhu S W,Qu S L,Wei X C,et al.Numeric simulation by grid-various finite-difference elastic wave equation.Oil Geophys.Prosp.(in Chinese),2007,42(6) :634～639
[7] 黄超,董良国.可变网格和局部时间步长的高阶差分地震波数值模拟.地球物理学报,2009,52(1) :176～186 Huang C,Dong L G.High-order finite-difference method in seismic wave simulation with variable grids and local time-steps.Chinese J.Geophys.(in Chinese),2009,52(1) :176～186
[8] Dablain M A.The application of high-order differencing to the scale wave equation Laser.Geophysics,1986,51(1) :54～66
[9] Virieux J.SH-wave propagation in heterogeneous media:Velocity-stress finite-difference method.Geophysics,1984,49(11) :1933～1957
[10] Yang D H,Teng J,Zhang Z J,et al.A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media.Bull.Seis.Soc.Am.,2003,93(2) :882～890
[11] Yang D H,Lu M,Wu R S,et al.An optimal nearly analytic discrete method for 2D acoustic and elastic wave equations,Bull.Seis.Soc.Am.,2004,94(5) :1982～1991
[12] Yang D H,Song G J,Chen S,et al.An improved nearly analytical discrete method:an efficient tool to simulate the seismic response of 2-D porous structures.J.Geophys.Eng,2007,4:40～52
[13] 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
[14] Demmel J W.Applied Numerical Linear Algebra.Philadelphia:SIAM,1997
[15] 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.Seis.Soc.Am.,2003,93(6) :2389～2401
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