VTI介质起伏地表地震波场模拟
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摘要
起伏地表下地震波场模拟有助于解释主动源和被动源地震探测中穿过山脉和盆地的测线所获得的资料.然而传统的有限差分法处理起伏的自由边界比较困难,为了克服这一困难,我们将笛卡尔坐标系的各向异性介质弹性波方程和自由边界条件变换到曲线坐标系中,采用一种稳定的、显式的二阶精度有限差分方法离散(曲线坐标系)VTI介质中的弹性波方程;对地表自由边界条件处理时采用了一种修饰的差分算子来计算弹性波方程中的混合导数项在自由边界上的法向导数.兰姆问题的解析解与本文的数值解对比结果表明该方法可以有效地处理自由地表边界条件.模拟实例表明:起伏地表对地震波场有重要影响,各向异性导致弹性波波前形状复杂且具有明显的方向性.
Modeling of seismic wave propagation in anisotropic media with irregular topography is a powerful tool that may help to interpret seismic data acquired by active and passive source seismology conducted in areas of interest like mountain ranges and basins.The major challenge in this context is the numerical implementation of the free-surface boundary condition.To implement the free-surface boundary condition,we use the boundary-conforming grid and transform a rectangular grid onto a curved grid.We use a stable and explicit second-order finite difference scheme to discretize the elastic wave equations(in a curvilinear coordinate system)in heterogeneous anisotropic medium.The free-surface boundary conditions are numerically implemented by introducing a discretization that uses boundary-modified difference operators for the mixed derivatives in the governing equations.The accuracy of the proposed method is checked by comparing the numerical results obtained by the trial algorithm with the analytical solution of the Lamb's problem,for a transversely isotropic medium with a vertical symmetry axis. Efficiency tests performed by different numerical experiments illustrate clearly the influence of an irregular(non-flat)free surface on seismic wave propagation.
Modeling of seismic wave propagation in anisotropic media with irregular topography is a powerful tool that may help to interpret seismic data acquired by active and passive source seismology conducted in areas of interest like mountain ranges and basins.The major challenge in this context is the numerical implementation of the free-surface boundary condition.To implement the free-surface boundary condition,we use the boundary-conforming grid and transform a rectangular grid onto a curved grid.We use a stable and explicit second-order finite difference scheme to discretize the elastic wave equations(in a curvilinear coordinate system)in heterogeneous anisotropic medium.The free-surface boundary conditions are numerically implemented by introducing a discretization that uses boundary-modified difference operators for the mixed derivatives in the governing equations.The accuracy of the proposed method is checked by comparing the numerical results obtained by the trial algorithm with the analytical solution of the Lamb's problem,for a transversely isotropic medium with a vertical symmetry axis. Efficiency tests performed by different numerical experiments illustrate clearly the influence of an irregular(non-flat)free surface on seismic wave propagation.
引文
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[6] 张永刚.复杂介质地震波场模拟分析与应用.北京:石油工业出版社,2007 Zhang Y G.Seismic Wave Field Simulation Analysis and Application in Complex Media (in Chinese).Beijing:Petroleum Industry Press,2007
[7] 郑鸿明,吕焕通,娄兵等.地震勘探近地表异常校正.北京:石油工业出版社,2009 Zheng H M,Lv H T,Lou B,et al.Near-Surface Anomaly Correction in Seismic Exploration (in Chinese).Beijing:Petroleum Industry Press,2009
[8] 裴正林,何光明,谢芳.复杂地表复杂构造模型的弹性波方程正演模拟.石油地球物理勘探,2010,45(6) :807～818 Pei Z L,He H M,Xie F.Elastic wave equation forward modeling for complex surface and complex structure model.Oil Geophysical Prospecting (in Chinese),2010,45(6) :807～818
[9] Huang Z P,Zhang M,Wu W Q,et al.A domain decomposition method for numerical simulation of the elastic wave propagation.Chinese].Geophys.,2004,47(6) :1094~1100
[10] Rial J A,Saltzman N G,Ling H.Earthquake-induced resonance in sedimentary basins.American Scientist,1992,80(6) :566-578
[11] Toshinawa T,Ohmachi T.Love-wave propagation in a three-dimensional sedimentary basin.Bull.Seism.Soc.Am.,1992,82(4) :1661-1667
[12] Komatitsch D,Tromp].Introduction to the spectral element method for three-dimensional seismic wave propagation.Geophysical Journal International,1999,139(3) :806～822
[13] Komatitsch D,Vilotte J.The spectral element method;An efficient tool to simulate the seismic response of 2D and 3D geological structures.Bulletin of the Seismological Society of America,1998,88(2) .368～392
[14] Nielsen P,If F,Berg P,et al.Using the pseudospectral technique on curved grids for 2D acoustic forward modelling.Geophysical Prospecting ,1994,42(4) :321～342
[15] Tessmer E,Kosloff D.3-D elastic modeling with surface topography by a Chebychev spectral method.Geophysics,1994,59(3) :464～473
[16] Tessmer E,Kosloff D,Behle A.Elastic wave propagation simulation in the presence of surface topography.Geophys.J.Int.,1992,108(2) :621～632
[17] Bouchon M,Campillo M,Gaffet S.A boundary integral equation-discrete wavenumber representation method to study wave propagation in multilayered media having irregular interfaces.Geophysics,1989,54(9) :1134～1140
[18] Liu E R,Zhang Z J,Yue J H,et al.Boundary integral modelling of elastic wave propagation in multi-layered 2D medium with irregular interfaces.Commun.Comput.Phys.,2008,3(1) :52-62
[19] 符力耘,牟永光.弹性波边界元法正演模拟.地球物理学报, 1994,37(4) :521-529 Fu LY,Mou Y G.Boundary element method for elastic wave forward modeling.Chinese J.Geophys.C Acta Geophysica Sinica) (in Chinese),1994,37(4) ,521～529
[20] Gao H W,Zhang J F.Parallel 3-D simulation of seismic wave propagation in heterogeneous anisotropic media:a grid method approach.Geophysical Journal International,2006,165(3) :875-888
[21] Hestholm S,Ruud B.2D finite-difference elastic wave modelling including surface topography.Geophys.Prosp,1994,42(5) :371-390
[22] Jih R S,McLaughlin K L,Der Z A.Free-boundary conditions of arbitrary polygonal topography in a two-dimensional explicit elastic finite-difference scheme.Geophysics,1988,53(8) :1045～1055
[23] Lombard B,Piraux J,Gelis C,et al.Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves.Geophysical Journal International,2008,172(1) :252-261
[24] Robertsson J O A.A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography.Geophysics,1996,61(6) :1921～1934
[25] Zhang W,Chen X F.Traction image method for irregular free surface boundaries in finite difference seismic wave simulation.Geophysical Journal International,2006,167(1) :337～353
[26] 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(S1) :357～366 Zhang J F.Non-orthogonal grid finite difference method for numerical simulation of elastic wave propagation.Chinese J.Geophys.(in Chinese),1998,41(S1) :357～366
[27] 张金海,王卫民,赵连锋等.傅里叶有限差分法三维波动方程正演模拟.地球物理学报,2007,50(6) :1854～1862 Zhang J H,Wang W M,Zhao L F,et al.Modeling 3-D scalar waves using the Fourier finite-difference method.ChineseJ.Geophys.(in Chinese),2007,50(6) :1854～1862
[28] Komatitsch D,Barnes C,Tromp J.Simulation of anisotropic wave propagation based upon a spectral element method.Geophysics,2000,65(4) :1251～1260
[29] Bouchon M,Schultz C A,Nafi Toks? z M.A fast implementation of boundary integral equation methods to calculate the propagation of seismic waves in laterally varying layered media.Bulletin of the Seismological Society of America,1995,85(6) :1679～1687
[30] Alterman Z S,Karal F C Jr.Propagation of elastic waves in layered media by finite difference methods.Bulletin of the Seismological Society of America ,1968,58(1) :367～398
[31] Alterman Z S,Rotenberg A.Seismic waves in a quarter plane.Bulletin of the Seismological Society of America ,1969,59(1) :347~368
[32] Yang D H,Liu E,Zhang Z J,et al.Finite-difference modelling in two-dimensional anisotropic media using a flux- corrected transport technique.Geophys.J.Int.,2002,148 (2) :320～328
[33] Zhang Z J,He Q D,Teng J W.Forward modeling of 3-component seismic records in 2-D transversely isotropic media with finite difference method.Can.J.Exp.Geophys.,1993,29(1) :51v58
[34] Zhang Z J,Wang G J,Harris J M.Multi-component wavefield simulation in viscous extensively dilatancy anisotropic media.Physics of the Earth and Planetary Interiors ,1999,114(1-2) :25～38
[35] Ilan A,Loewenthal D.Instability of finite difference schemes due to boundary conditions in Elastic Media.Geophysical Prospecting,1976,24(3) :431～453
[36] Lan H Q,Zhang Z J.Comparative study of the free-surface boundary condition in two-dimensional finite-difference elastic wave field simulation.Journal of Geophysics and Engineering,2011,8(2) :275～286
[37] Ilan A.Stability of finite difference schemes for the problem of elastic wave propagation in a quarter plane.Journal of Computational Physics ,1978,29(3) :389～403
[38] Vidale J E,Clayton R W.A stable free-surface boundary condition for two-dimensional elastic finite-difference wave simulation.Geophysics,1986,51(12) :2247
[39] Levander A R.Fourth-order finite-difference P-SV seismograms.Geophysics,1988,53(11) :1425
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