起伏地表条件下有限差分地震波数值模拟——基于广义正交曲线坐标系
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摘要
为了解决常规有限差分法在处理起伏地表自由边界条件时存在的需进行复杂坐标旋转和插值运算,以及对起伏地表进行阶梯状离散近似所产生的虚假绕射波等问题,把待求解的物理域离散成贴体正交曲线网格,将该网格映射到计算域的矩形规则网格上,在该计算域内求解一阶弹性波速度-应力方程及相应的自由边界条件,实现了广义正交曲线坐标系下的起伏地表地震波数值模拟问题.该算法简化了自由边界条件的实施,避免了因起伏地表阶梯化网格离散近似所产生的虚假波.数值模拟结果表明,该方法是可行的,可以用于起伏地表条件下地震波传播规律的研究.
A staggered-grid finite-difference approach based on generally orthogonal curvilinear coordinate system was performed to simulate seismic waves including topographic surface in order to solve the problems when dealing with free surface boundary conditions: complex coordinate rotations and interpolations in conventional finite-difference methods and artifacts caused by staircase approximation to irregular free surface.The physical domain under study was discretized by orthogonal boundary-conforming curvilinear grids,which were then mapped onto regularly rectangular grids in computational domain.The first order elastic velocity-stress equations and free surface boundary conditions were solved in computational domain.The algorithm can not only avoid artifacts caused by staircase approximation to irregular free surface,but also make free surface boundary conditions simple and direct to implement.Numerical results show that the method is feasible and can be applied to the study of seismic wave propagation with topographic surface.
A staggered-grid finite-difference approach based on generally orthogonal curvilinear coordinate system was performed to simulate seismic waves including topographic surface in order to solve the problems when dealing with free surface boundary conditions: complex coordinate rotations and interpolations in conventional finite-difference methods and artifacts caused by staircase approximation to irregular free surface.The physical domain under study was discretized by orthogonal boundary-conforming curvilinear grids,which were then mapped onto regularly rectangular grids in computational domain.The first order elastic velocity-stress equations and free surface boundary conditions were solved in computational domain.The algorithm can not only avoid artifacts caused by staircase approximation to irregular free surface,but also make free surface boundary conditions simple and direct to implement.Numerical results show that the method is feasible and can be applied to the study of seismic wave propagation with topographic surface.
引文
[1]阎世信,刘怀山,姚雪根.山地地球物理勘探技术[M].北京:石油工业出版社,2000:1-85.
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[8]FU L.Numerical study of generalized Lipmann-Schwinger integral equation including surface topogra-phy[J].Geophysics,2003,68(2):665-671.
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[10]孙章庆,孙建国,韩复兴,等.波前快速推进法起伏地表地震波走时计算[J].勘探地球物理进展,2007,30(5):392-395.SUN Zhang-qing,SUN Jian-guo,HAN Fu-xing,et al.Traveltime computation using fast marching methodfrom rugged topography[J].Progress in ExplorationGeophysics,2007,30(5):392-395.
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[13]SUN J,SUN Z,HAN F.A finite difference scheme forsolving the eikonal equation including surface topogra-phy[J].Geophysics,2011,76(4):T53-T63.
[14]RICHTER G R.An explicit finite element method forthe wave equation[J].Applied Numerical Mathemat-ics,1994,16(1/2):65-80.
[15]KOMATITSCH D,VILOTTE J P.The spectral ele-ment method:an efficient tool to simulate the seismicresponse of 2Dand 3Dgeological structures[J].Bulle-tin of the Seismological Society of America,1998,88(2):368-392.
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[19]HAYASHI K,BURNS D,TOKSOZ M.Discontinu-ous-grid finite-difference seismic modeling includingsurface topography[J].Bulletin of the SeismologicalSociety of America,2001,91(6):1750-1764.
[20]WANG Y,XU J,SCHUSTER G.Viscoelastic wavesimulation in basin by a variable-grid finite-differencemethod[J].Bulletin of the Seismological Society ofAmerica,2001,91(6):1741-1749.
[21]HESTHOLM S,RUUD B.3Dfinite-difference elasticwave modeling including surface topography[J].Geo-physics,1998,63(2):613-622.
[22]RUUD B,HESTHOLM S.2Dsurface topographyboundary conditions in seismic wave modeling[J].Geophysical Prospecting,2001,49(4):445-460.
[23]HESTHOLM S,RUUD B.3Dfree-boundary condi-tions for coordinate-transform finite-difference seismicmodeling[J].Geophysical Prospecting,2002,50(5):463-474.
[24]TARRASS I,GIRAUD L,THORE P.New curvilinearscheme for elastic wave propagation in presence ofcurved topography[J].Geophysical Prospecting,2011,59(5):889-906.
[25]ZHANG W,CHEN X F.Traction image method forirregular free surface boundaries in finite differenceseismic wave simulation[J].Geophysical Journal Inter-national,2006,167(1):337-353.
[26]APPELO D,PETERSSON N.A stable finite differencemethod for the elastic wave equation on complex geom-etries with free surfaces[J].Communications in Com-putational Physics,2009,5(1):84-107.
[27]LAN H,ZHONG Z.Three-dimensional wave-fieldsimulation in heterogeneous transversely isotropic me-dium with irregular free surface[J].Bulletin of theSeismological Society of America,2011,101(3):1354-1370.
[28]THOMPSON J F,WARSI Z U A,MASTIN C W.Numerical grid generation foundations and applications[M].New York:North Hollad Publishing Company,1985:188-263.
[29]蒋丽丽,孙建国.基于Poisson方程的曲网格生成技术[J].世界地质,2008,27(3):298-305.JIANG Li-li,SUN Jian-guo.Source terms of ellipticsystem in grid generation[J].Global Geology,2008,27(3):298-305.
[30]孙建国,蒋丽丽.用于起伏地表条件下地球物理场数值模拟的正交曲网格生成技术[J].石油地球物理勘探,2009,44(4):494-500.SUN Jian-guo,JIANG Li-li.Orthogonal curvilinear gridgeneration technique used for numeric simulation of geo-physical fields in undulating surface condition[J].OilGeophysical Prospecting,2009,44(4):494-500.
[31]THOMPSON J,SONI B,WEATHERILL N.Hand-book of grid generation[M].New York:CRCPress,1999.
[32]AKCELIK V,JARAMAZ B,GHATTAS O.Nearlyorthogonal two-dimensional grid generation with aspectratio control[J].Journal of Computational Physics,2001,171(2):805-821.
[33]ZHANG Y,JIA Y,WANG S.2Dnearly orthogonalmesh generation[J].International Journal for Numeri-cal Methods in Fluids,2004,46(7):685-707.
[34]THOMAS P,MIDDLECOFF J.Direct control of thegrid point distribution in meshes generated by ellipticequations[J].AIAA Journal,1980,18(6):652-656.
[35]SORENSON R,STEGER J.Automatic mesh-pointclustering near a boundary in grid generation[J].Journal of Computational Physics,1979,33(3):405-410.
[36]AKI K,RICHARDS P G.Quantitative seismology[M].2nd ed.Sausalito:University Science Books,2002:30-34.
[37]SADD M.Elasticity theory,applications,and numerics[M].2nd ed.Burlington:Academic Press,2009:55-76.
[38]XIE Z,CHAN C,ZHANG B.An explicit fourth-orderorthogonal curvilinear staggered-grid FDTD method forMaxwell’s equations[J].Journal of ComputationalPhysics,2002,175(2):739-763.
[39]MARTIN R,KOMATITSCH D,GEDNEY S,et al.A high-order time and space formulation of the unsplitperfectly matched layer for the seismic wave equationusing auxiliary differential equation(ADE-PML)[J].Computer Modeling in Engineering and Sciences,2010,56(1):17-41.
[40]HU F,HUSSAINI M,MANTHEY J.Low-dissipa-tion and low-dispersive Runge-Kutta schemes for com-putational acoustic[J].Journal of Computational Phys-ics,1996,124(1):177-191.
[41]马啸,杨顶辉,张锦华.求解声波方程的辛可分Runge-Kut-ta方法[J].地球物理学报,2010,53(8):1993-2003.MA Xiao,YANG Ding-hui,ZHANG Jin-hua.Sym-plectic partitioned Runge-Kutta method for solving theacoustic wave equation[J].Chinese Journal of Geo-physics-Chinese Edition,2010,53(8):1993-2003.
[42]ALLAMPALLI V,HIXON R,NALLASAMY M,etal.High-accuracy large-step explicit Runge-Kutta(HALE-RK)schemes for computational aeroacoustics[J].Journal of Computational Physics,1996,228(10):3837-3850.
[43]KOMATITSCH M,COUTEL F,MORA P.Tensorialformulation of the wave equation for modeling curvedinterfaces[J].Geophysical Journal International,1996,127(1):156-168.
[44]CUNHA C A.Elastic modeling in discontinuous media[J].Geophysics,1993,58(12):1840-1851.
[2]邓志文.复杂山地地震勘探[M].北京:石油工业出版社,2006:1-30.
[3]郑鸿明,吕焕通,娄兵,等.地震勘探近地表异常校正[M].北京:石油工业出版社,2009.
[4]孙建国.复杂地表条件下地球物理场数值模拟方法评述[J].世界地质,2007,26(3):345-362.SUN Jian-guo.Methods for numerical modeling of geo-physical fields under complex topographical conditions:a critical review[J].Global Geology,2007,26(3):345-362.
[5]牟永光,裴正林.三维复杂介质地震波数值模拟[M].北京:石油工业出版社,2005:1-13.
[6]张永刚.复杂介质地震波场模拟分析与应用[M].北京:石油工业出版社,2007:153-168.
[7]BOUCHON M,SCHULTZ C,TOKSOZ M.Effect ofthree-dimensional topography on seismic motion[J].Journal of Geophysical Research,1996,101(B3):5835-5846.
[8]FU L.Numerical study of generalized Lipmann-Schwinger integral equation including surface topogra-phy[J].Geophysics,2003,68(2):665-671.
[9]CARCIONE J,HERMAN G,KROODE A.Seismicmodeling[J].Geophysics,2002,67(4):1304-1325.
[10]孙章庆,孙建国,韩复兴,等.波前快速推进法起伏地表地震波走时计算[J].勘探地球物理进展,2007,30(5):392-395.SUN Zhang-qing,SUN Jian-guo,HAN Fu-xing,et al.Traveltime computation using fast marching methodfrom rugged topography[J].Progress in ExplorationGeophysics,2007,30(5):392-395.
[11]孙章庆,孙建国,韩复兴.复杂地表条件下快速推进法地震波走时计算[J].计算物理,2010,27(2):281-286.SUN Zhang-qing,SUN Jian-guo,HAN Fu-xing.Traveltime computation using fast marching methodunder complex topographical conditions[J].ChineseJournal of Computational Physics,2010,27(2):281-286.
[12]孙章庆,孙建国,韩复兴.复杂地表条件下基于线性插值和窄带技术的地震波走时计算[J].地球物理学报,2009,52(11):2846-2853.SUN Zhang-qing,SUN Jian-guo,HAN Fu-xing.Trav-eltime computation using linear interpolation and nar-row band technique under complex topographical condi-tions[J].Chinese Journal of Geophysics,2009,52(11):2846-2853.
[13]SUN J,SUN Z,HAN F.A finite difference scheme forsolving the eikonal equation including surface topogra-phy[J].Geophysics,2011,76(4):T53-T63.
[14]RICHTER G R.An explicit finite element method forthe wave equation[J].Applied Numerical Mathemat-ics,1994,16(1/2):65-80.
[15]KOMATITSCH D,VILOTTE J P.The spectral ele-ment method:an efficient tool to simulate the seismicresponse of 2Dand 3Dgeological structures[J].Bulle-tin of the Seismological Society of America,1998,88(2):368-392.
[16]GRAVES R W.Simulating seismic wave propagationin 3Delastic media using staggered-grid finite differ-ences[J].Bulletin of the Seismological Society of Amer-ica,1996,86(4):1091-1106.
[17]ROBERTSSON J O A.A numerical free-surface condi-tion for elastic/viscoelastic finite-difference modeling inthe presence of topography[J].Geophysics,1996,61(6):1921-1934.
[18]OHMINATO T,CHOUET B A.A free-surfaceboundary condition for including 3Dtopography in thefinite-difference method[J].Bulletin of the Seismolog-ical Society of America,1997,87(2):494-515.
[19]HAYASHI K,BURNS D,TOKSOZ M.Discontinu-ous-grid finite-difference seismic modeling includingsurface topography[J].Bulletin of the SeismologicalSociety of America,2001,91(6):1750-1764.
[20]WANG Y,XU J,SCHUSTER G.Viscoelastic wavesimulation in basin by a variable-grid finite-differencemethod[J].Bulletin of the Seismological Society ofAmerica,2001,91(6):1741-1749.
[21]HESTHOLM S,RUUD B.3Dfinite-difference elasticwave modeling including surface topography[J].Geo-physics,1998,63(2):613-622.
[22]RUUD B,HESTHOLM S.2Dsurface topographyboundary conditions in seismic wave modeling[J].Geophysical Prospecting,2001,49(4):445-460.
[23]HESTHOLM S,RUUD B.3Dfree-boundary condi-tions for coordinate-transform finite-difference seismicmodeling[J].Geophysical Prospecting,2002,50(5):463-474.
[24]TARRASS I,GIRAUD L,THORE P.New curvilinearscheme for elastic wave propagation in presence ofcurved topography[J].Geophysical Prospecting,2011,59(5):889-906.
[25]ZHANG W,CHEN X F.Traction image method forirregular free surface boundaries in finite differenceseismic wave simulation[J].Geophysical Journal Inter-national,2006,167(1):337-353.
[26]APPELO D,PETERSSON N.A stable finite differencemethod for the elastic wave equation on complex geom-etries with free surfaces[J].Communications in Com-putational Physics,2009,5(1):84-107.
[27]LAN H,ZHONG Z.Three-dimensional wave-fieldsimulation in heterogeneous transversely isotropic me-dium with irregular free surface[J].Bulletin of theSeismological Society of America,2011,101(3):1354-1370.
[28]THOMPSON J F,WARSI Z U A,MASTIN C W.Numerical grid generation foundations and applications[M].New York:North Hollad Publishing Company,1985:188-263.
[29]蒋丽丽,孙建国.基于Poisson方程的曲网格生成技术[J].世界地质,2008,27(3):298-305.JIANG Li-li,SUN Jian-guo.Source terms of ellipticsystem in grid generation[J].Global Geology,2008,27(3):298-305.
[30]孙建国,蒋丽丽.用于起伏地表条件下地球物理场数值模拟的正交曲网格生成技术[J].石油地球物理勘探,2009,44(4):494-500.SUN Jian-guo,JIANG Li-li.Orthogonal curvilinear gridgeneration technique used for numeric simulation of geo-physical fields in undulating surface condition[J].OilGeophysical Prospecting,2009,44(4):494-500.
[31]THOMPSON J,SONI B,WEATHERILL N.Hand-book of grid generation[M].New York:CRCPress,1999.
[32]AKCELIK V,JARAMAZ B,GHATTAS O.Nearlyorthogonal two-dimensional grid generation with aspectratio control[J].Journal of Computational Physics,2001,171(2):805-821.
[33]ZHANG Y,JIA Y,WANG S.2Dnearly orthogonalmesh generation[J].International Journal for Numeri-cal Methods in Fluids,2004,46(7):685-707.
[34]THOMAS P,MIDDLECOFF J.Direct control of thegrid point distribution in meshes generated by ellipticequations[J].AIAA Journal,1980,18(6):652-656.
[35]SORENSON R,STEGER J.Automatic mesh-pointclustering near a boundary in grid generation[J].Journal of Computational Physics,1979,33(3):405-410.
[36]AKI K,RICHARDS P G.Quantitative seismology[M].2nd ed.Sausalito:University Science Books,2002:30-34.
[37]SADD M.Elasticity theory,applications,and numerics[M].2nd ed.Burlington:Academic Press,2009:55-76.
[38]XIE Z,CHAN C,ZHANG B.An explicit fourth-orderorthogonal curvilinear staggered-grid FDTD method forMaxwell’s equations[J].Journal of ComputationalPhysics,2002,175(2):739-763.
[39]MARTIN R,KOMATITSCH D,GEDNEY S,et al.A high-order time and space formulation of the unsplitperfectly matched layer for the seismic wave equationusing auxiliary differential equation(ADE-PML)[J].Computer Modeling in Engineering and Sciences,2010,56(1):17-41.
[40]HU F,HUSSAINI M,MANTHEY J.Low-dissipa-tion and low-dispersive Runge-Kutta schemes for com-putational acoustic[J].Journal of Computational Phys-ics,1996,124(1):177-191.
[41]马啸,杨顶辉,张锦华.求解声波方程的辛可分Runge-Kut-ta方法[J].地球物理学报,2010,53(8):1993-2003.MA Xiao,YANG Ding-hui,ZHANG Jin-hua.Sym-plectic partitioned Runge-Kutta method for solving theacoustic wave equation[J].Chinese Journal of Geo-physics-Chinese Edition,2010,53(8):1993-2003.
[42]ALLAMPALLI V,HIXON R,NALLASAMY M,etal.High-accuracy large-step explicit Runge-Kutta(HALE-RK)schemes for computational aeroacoustics[J].Journal of Computational Physics,1996,228(10):3837-3850.
[43]KOMATITSCH M,COUTEL F,MORA P.Tensorialformulation of the wave equation for modeling curvedinterfaces[J].Geophysical Journal International,1996,127(1):156-168.
[44]CUNHA C A.Elastic modeling in discontinuous media[J].Geophysics,1993,58(12):1840-1851.
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