贴体网格各向异性对坐标变换法求解起伏地表下地震初至波走时的影响
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摘要
笛卡尔坐标系中的经典程函方程在静校正、叠前偏移、走时反演、地震定位、层析成像等很多地球物理工作中都有应用,然而用其计算起伏地表的地震波走时却比较困难.本文通过把曲线坐标系中的矩形网格映射到笛卡尔坐标系的贴体网格,推导出曲线坐标中的程函方程,而后,用Lax-Friedrichs快速扫描算法求解曲线坐标系的程函方程.研究表明本文方法能有效处理地表起伏的情况,得到准确稳定的计算结果.由于地表起伏,导致与之拟合的贴体网格在空间上的展布呈各向异性,且这种各向异性的强弱对坐标变换法求解地震初至波的走时具有重要影响.本文研究表明,随着贴体网格的各向异性增强,用坐标变换法求解地表起伏区域的走时计算误差增大,且计算效率降低,这在实际应用具有指导意义.
The classical eikonal equation is commonly used in Cartesian coordinate system for problems that involve static correction,prestack migration,earthquake location and seismic tomography,but is less effective for calculating travel times in an earth model that has an irregular surface.We have presented a topography-dependent eikonal equation in a curvilinear coordinate system that makes use of the surface-fitting grid and map a rectangular grid onto a curved grid.Then,we utilized the efficient Lax-Friedrichs sweeping scheme to approximate the viscosity solutions of the eikonal equation in the curvilinear coordinate system.In this paper,we investigate the impacts due to the anisotropic stretching of the surface-fitting grid on the traveltime computation by using the topography-dependent eikonal equation,which has a significant meaning in the direction of our method in geophysical application.
The classical eikonal equation is commonly used in Cartesian coordinate system for problems that involve static correction,prestack migration,earthquake location and seismic tomography,but is less effective for calculating travel times in an earth model that has an irregular surface.We have presented a topography-dependent eikonal equation in a curvilinear coordinate system that makes use of the surface-fitting grid and map a rectangular grid onto a curved grid.Then,we utilized the efficient Lax-Friedrichs sweeping scheme to approximate the viscosity solutions of the eikonal equation in the curvilinear coordinate system.In this paper,we investigate the impacts due to the anisotropic stretching of the surface-fitting grid on the traveltime computation by using the topography-dependent eikonal equation,which has a significant meaning in the direction of our method in geophysical application.
引文
[1]Aki K,Richards P G.Quantitative Seismology.Univ ScienceBooks,2002.
[2]Hole J.Nonlinear high-resolution three-dimensional seismictravel time tomography.Journal of Geophysical Research,1992,97(B5):6553-6562.
[3]Chen Y,Badal J,Hu J.Love and Rayleigh wave tomographyof the Qinghai-Tibet Plateau and surrounding areas.Pureand Applied Geophysics,2010,167(10):1-33.
[4]Zelt C,Smith R.Seismic traveltime inversion for 2-D crustalvelocity structure.Geophysical Journal International,1992,108(1):16-34.
[5]Zhang Z,Wang G,Teng J,et al.CDP mapping to obtain thefine structure of the crust and upper mantle from seismicsounding data:an example for the southeastern China.Physics of the Earth and Planetary Interiors,2000,122(1-2):133-146.
[6]张中杰,秦义龙,陈赟等.由宽角反射地震资料重建壳幔反射结构的相似性剖面.地球物理学报,2004,47(3):469-474.Zhang Z J,Qin Y L,Chen Y,et al.Reconstruction of thesemblance section for the crust and mantle reflection structureby wide-angle reflection seismic data.Chinese J.Geophys.(in Chinese),2004,47(3):469-474
[7]朱金明,王丽燕.地震波走时的有限差分法计算.地球物理学报,1992,35(1):86-92.Zhu Z M and Wang L Y.Finite difference calculation ofseismic travel times.Chinese J.Geophys.(in Chinese),1992,35(1):86-92.
[8]Cerveny V,Molotkov I A,Psencik I.Ray Methods inSeismology.Univ.of Karlova Press,1977.
[9]Zhang Z,Badal J,Li Y,et al.Crust-upper mantle seismicvelocity structure across Southeastern China.Tectonophysics,2005,395(1-2):137-157.
[10]Zhang Z,Li Y,Lu D,et al.Velocity and anisotropystructure of the crust in the Dabieshan orogenic belt fromwide-angle seismic data.Physics of the Earth and PlanetaryInteriors,2000,122(1-2):115-131.
[11]Zhang Z,Wang Y,Chen Y,et al.Crustal structure acrossLongmenshan fault belt from passive source seismic profiling.Geophys.Res.Lett.,2009,36.
[12]Xu T,Xu G,Gao E,et al.Block modeling and segmentallyiterative ray tracing in complex 3Dmedia.Geophysics,2006,71:T41-T51.
[13]Xu T,Zhang Z,Gao E,et al.Segmentally iterative raytracing in complex 2Dand 3Dheterogeneous block models.Bulletin of the Seismological Society of America,100(2):841-850.
[14]Vidale J E.Finite-difference calculation of travel times.Bulletin of the Seismological Society of America,1988,78(6):2062-2076.
[15]Vidale J E.Finite-difference calculation of traveltimes inthree dimensions.Geophysics,1990,55(5):521-526.
[16]Qin F,Luo Y,Olsen K B,et al.Finite-difference solution ofthe eikonal equation along expanding wavefronts.Geophysics,1992,57:478-487.
[17]Van Trier J,Symes W W.Upwind finite-differencecalculation of traveltimes.Geophysics,1991,56(6):812-821.
[18]Sethian J.Evolution,implementation,and application oflevel set and fast marching methods for advancing fronts.Journal of Computational Physics,2001,169(2):503-555.
[19]Sethian J A.Fast marching methods.SIAM review,1999,41(2):199-235.
[20]Sethian J A,Popovici A M.Three dimensional traveltimescomputation using the Fast Marching Method.Geophysics,1999,64(2):516-523.
[21]Sethian J A.A fast marching level set method formonotonically advancing fronts.Proceedings of the NationalAcademy of Sciences of the United States of America,1996,93(4):1591-1595.
[22]Alkhalifah T,Fomel S.Implementing the fast marchingeikonal solver:spherical versus Cartesian coordinates.Geophys.Prospect.,2001,49:165-178.
[23]Fomel S.A variational formulation of the fast marchingeikonal solver.SEP-95:Stanford Exploration Project,1997:127-147.
[24]Kim S.3-D eikonal solvers:First-arrival traveltimes.Geophysics,2002,67(4):1225-1231.
[25]Rawlinson N,Sambridge M.Wave front evolution instrongly heterogeneous layered media using the fast marchingmethod.Geophysical Journal International,2004,156(3):631-647.
[26]Rawlinson N,Sambridge M.Multiple reflection andtransmission phases in complex layered media using amultistage fast marching method.Geophysics,2004,69(5):1338.
[27]Rawlinson N,Sambridge M.The fast marching method:aneffective tool for tomographic imaging and tracking multiplephases in complex layered media.Exploration Geophysics,2005,36(4):341-350.
[28]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.地球物理学报,2009,52(11):2846-2853.Sun Z Q,Sun J G,Han F X.Travehimes computation usinglinear interpoiation and narrow band technique under complextopographical conditions.Chinese J.Geophys.(in Chinese),2009,52(11):2846-2853
[29]De Kool M,Rawlinson N,Sambridge M.A practical grid-based method for tracking multiple refraction and reflectionphases in three-dimensional heterogeneous media.Geophysical Journal International,2006,167(1):253-270.
[30]Zhao H.A fast sweeping method for eikonal equations.Mathematics of Computation,2005,74(250):603-628.
[31]Zhang Y T,Zhao H K,Qian J.High order fast sweepingmethods for static Hamilton-Jacobi equations.Journal ofScientific Computing,2006,29(1):25-56.
[32]兰海强,张智,徐涛等.地震波走时场模拟的快速推进法和快速扫描法比较研究.地球物理学进展,2012,待刊.Lan H Q,Zhang Z,Xu T,et al.A comparative study on thefast marching and fast sweeping methods in the calculation offirst-arrival traveltime field.Progress in Geophys.(inChinese),2012,in press.
[33]Lan H,Zhang Z.Three-dimensional wave-field simulation inheterogeneous transversely isotropic medium with irregularfree surface.Bulletin of the Seismological Society ofAmerica,2011 101(3):1354-1370.
[34]阎世信,刘怀山,姚雪根.山地地球物理勘探技术.北京:石油工业出版社,2000.Yan S X,Liu H S and Yao X G.Geophysical ExplorationTechnology in the Mountainous Area(in Chinese).Beijing:Petroleum Industry Press,2000.
[35]Kao C Y,Osher S,Qian J.Legendre-transform-based fastsweeping methods for static Hamilton-Jacobi equations ontriangulated meshes.Journal of Computational Physics,2008,227(24):10209-10225.
[36]Kimmel R,Sethian J A.Computing geodesic paths onmanifolds.Proceedings of the National Academy ofSciences of the United States of America,1998,95(15):8431-7435.
[37]Lelièvre P G,Farquharson C G,Hurich C A.Computingfirst-arrival seismic traveltimes on unstructured 3-Dtetrahedral grids using the Fast Marching Method.Geophysical Journal International,184(2):885-896.
[38]Qian J,Zhang Y T,Zhao H K.A fast sweeping method forstatic convex Hamilton-Jacobi equations.Journal ofScientific Computing,2007,31(1):237-271.
[39]Qian J,Zhang Y T,Zhao H K.Fast sweeping methods forEikonal equations on triangular meshes.SIAM Journal onNumerical Analysis,2008,45(1):83-107.
[40]Sethian J A,Vladimirsky A.Fast methods for the Eikonaland related Hamilton-Jacobi equations on unstructuredmeshes.Proceedings of the National Academy of Sciencesof the United States of America,2000,97(11):5699.
[41]Appelo D,Petersson N.A stable finite difference method forthe elastic wave equation on complex geometries with freesurfaces.Communications in Computational Physics,2009,5(1):84-107.
[42]孙章庆,孙建国,张东良.二维起伏地表条件下坐标变换法直流电场数值模拟.吉林大学学报:地球科学版,2009(3):528-534.Sun Z Q,Sun J G,Zhang D L.2DDC electric field numericalmodeling including surface topography using coordinatetransformation method.Journal of Jilin University:EarthScience Edition,2009,39(3):528-534.
[43]Hestholm S,Ruud B.2-D finite-difference elastic wavemodeling including surface topography.Geophys.Prosp.,1994,42:371-390.
[44]Lan H,Zhang Z.Seismic wavefield modeling in media withfluid-filled fractures and surface topography.AppliedGeophysics,2012,9(3):301-312.
[45]Tessmer E,Kosloff D.3-D elastic modeling with surfacetopography by a Chebychev spectral method.Geophysics,1994,59:464.
[46]兰海强,刘佳,白志明.VTI介质起伏地表地震波场模拟.地球物理学报,2011,54(8):2072-2084.Lan H Q,Liu J,Bai Z M.Wave-field simulation in VTImedia with irregular free surface.Chinese J.Geophys.(inChinese),2011,54(8):2072-2084.
[47]孙建国.复杂地表条件下地球物理场数值模拟方法评述.世界地质,2007,26(3):345-362.Sun J G.Methods for numerical modeling of geophysicalfields under complex topographicaI conditions:a criticalreview.Global Geology(in Chinese),2007,26(3):345-362.
[48]Lan H,Zhang Z.Comparative study of the free-surfaceboundary condition in two-dimensional finite-difference elasticwave field simulation.Journal of Geophysics andEngineering,2011,8:275-286.
[49]Sun J,Sun Z,Han F.A finite difference scheme for solvingthe eikonal equation including surface topography.Geophysics,2011,76(4):T53-T63.
[50]Lan H,Zhang Z.Topography-dependent eikonal equationand its solver for calculating first-arrival travel times with anirregular surface.Geophysical Journal International,2012a,in review.
[51]刘一峰,兰海强.曲线坐标系程函方程的求解方法研究.地球物理学报,2012,55(6):2014-2026.Liu Y F,Lan H Q.Study on the numerical solutions of theeikonal equation in curvilinear coordinate system.Chinese J.Geophys.(in Chinese),2012,55(6):2014-2026.
[52]Lan H,Zhang Z.A high order fast sweeping scheme forcalculating first-arrival travel times with an irregular surface.Bull.Seismol.Soc.Am.,2012b,in review.
[53]Hvid S L.Three dimensional algebraic grid generation[Ph.D.thesis].Technical University of Denmark,1994.
[54]Thompson J,Warsi Z,Mastin C.Numerical grid generation:foundations and applications.1985,North-hollandAmsterdam.
[55]Hestholm S,Ruud B.3-D finite-difference elastic wavemodeling including surface topography.Geophysics,1998,63:613-622.
[56]Tessmer E,Kosloff D,Behle A.Elastic wave propagationsimulation in the presence of surface topography.Geophys.J.Int.,1992,108(2):621-632.
[57]Alton K,Mitchell I M.Fast marching methods for stationaryHamilton-Jacobi equations with axis-aligned anisotropy.SIAM J.Numer.Anal,2008,47:363-385.
[58]Sethian J A,Vladimirsky A.Ordered upwind methods forstatic Hamilton-Jacobi equations.Proceedings of theNational Academy of Sciences,2001,98(20):11069.
[59]Tsai Y H R,Cheng L T,Osher S,et al.Fast sweepingalgorithms for a class of Hamilton-Jacobi equations.SIAMJournal on Numerical Analysis,2004:673-694.
[60]Kao C Y,Osher S,Qian J.Lax-Friedrichs sweeping schemefor static Hamilton-Jacobi equations*1.Journal ofComputational Physics,2004,196(1):367-391.
[61]Sethian J A.Level Set Methods and Fast Marching Methods:Evolving Interfaces in Computational Geometry,FluidMechanics,Computer Vision,and Materials Science.Cambridge University Press,1999.
[62]Luo S,Qian J.Factored singularities and high-order Lax-Friedrichs sweeping schemes for point-source traveltimes andamplitudes.J.Comput.Phys.,2011,230:4742-4755.
[63]Lin C T and Tadmor E.L1-stability and error estimates forapproximate Hamilton-Jacobi solutions.Numer.Math.,2001,87:701-735.
[64]Qian J,Symes W W.Finite-difference quasi-P traveltimes foranisotropic media.Geophysics,2002,67:147-155.
[2]Hole J.Nonlinear high-resolution three-dimensional seismictravel time tomography.Journal of Geophysical Research,1992,97(B5):6553-6562.
[3]Chen Y,Badal J,Hu J.Love and Rayleigh wave tomographyof the Qinghai-Tibet Plateau and surrounding areas.Pureand Applied Geophysics,2010,167(10):1-33.
[4]Zelt C,Smith R.Seismic traveltime inversion for 2-D crustalvelocity structure.Geophysical Journal International,1992,108(1):16-34.
[5]Zhang Z,Wang G,Teng J,et al.CDP mapping to obtain thefine structure of the crust and upper mantle from seismicsounding data:an example for the southeastern China.Physics of the Earth and Planetary Interiors,2000,122(1-2):133-146.
[6]张中杰,秦义龙,陈赟等.由宽角反射地震资料重建壳幔反射结构的相似性剖面.地球物理学报,2004,47(3):469-474.Zhang Z J,Qin Y L,Chen Y,et al.Reconstruction of thesemblance section for the crust and mantle reflection structureby wide-angle reflection seismic data.Chinese J.Geophys.(in Chinese),2004,47(3):469-474
[7]朱金明,王丽燕.地震波走时的有限差分法计算.地球物理学报,1992,35(1):86-92.Zhu Z M and Wang L Y.Finite difference calculation ofseismic travel times.Chinese J.Geophys.(in Chinese),1992,35(1):86-92.
[8]Cerveny V,Molotkov I A,Psencik I.Ray Methods inSeismology.Univ.of Karlova Press,1977.
[9]Zhang Z,Badal J,Li Y,et al.Crust-upper mantle seismicvelocity structure across Southeastern China.Tectonophysics,2005,395(1-2):137-157.
[10]Zhang Z,Li Y,Lu D,et al.Velocity and anisotropystructure of the crust in the Dabieshan orogenic belt fromwide-angle seismic data.Physics of the Earth and PlanetaryInteriors,2000,122(1-2):115-131.
[11]Zhang Z,Wang Y,Chen Y,et al.Crustal structure acrossLongmenshan fault belt from passive source seismic profiling.Geophys.Res.Lett.,2009,36.
[12]Xu T,Xu G,Gao E,et al.Block modeling and segmentallyiterative ray tracing in complex 3Dmedia.Geophysics,2006,71:T41-T51.
[13]Xu T,Zhang Z,Gao E,et al.Segmentally iterative raytracing in complex 2Dand 3Dheterogeneous block models.Bulletin of the Seismological Society of America,100(2):841-850.
[14]Vidale J E.Finite-difference calculation of travel times.Bulletin of the Seismological Society of America,1988,78(6):2062-2076.
[15]Vidale J E.Finite-difference calculation of traveltimes inthree dimensions.Geophysics,1990,55(5):521-526.
[16]Qin F,Luo Y,Olsen K B,et al.Finite-difference solution ofthe eikonal equation along expanding wavefronts.Geophysics,1992,57:478-487.
[17]Van Trier J,Symes W W.Upwind finite-differencecalculation of traveltimes.Geophysics,1991,56(6):812-821.
[18]Sethian J.Evolution,implementation,and application oflevel set and fast marching methods for advancing fronts.Journal of Computational Physics,2001,169(2):503-555.
[19]Sethian J A.Fast marching methods.SIAM review,1999,41(2):199-235.
[20]Sethian J A,Popovici A M.Three dimensional traveltimescomputation using the Fast Marching Method.Geophysics,1999,64(2):516-523.
[21]Sethian J A.A fast marching level set method formonotonically advancing fronts.Proceedings of the NationalAcademy of Sciences of the United States of America,1996,93(4):1591-1595.
[22]Alkhalifah T,Fomel S.Implementing the fast marchingeikonal solver:spherical versus Cartesian coordinates.Geophys.Prospect.,2001,49:165-178.
[23]Fomel S.A variational formulation of the fast marchingeikonal solver.SEP-95:Stanford Exploration Project,1997:127-147.
[24]Kim S.3-D eikonal solvers:First-arrival traveltimes.Geophysics,2002,67(4):1225-1231.
[25]Rawlinson N,Sambridge M.Wave front evolution instrongly heterogeneous layered media using the fast marchingmethod.Geophysical Journal International,2004,156(3):631-647.
[26]Rawlinson N,Sambridge M.Multiple reflection andtransmission phases in complex layered media using amultistage fast marching method.Geophysics,2004,69(5):1338.
[27]Rawlinson N,Sambridge M.The fast marching method:aneffective tool for tomographic imaging and tracking multiplephases in complex layered media.Exploration Geophysics,2005,36(4):341-350.
[28]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.地球物理学报,2009,52(11):2846-2853.Sun Z Q,Sun J G,Han F X.Travehimes computation usinglinear interpoiation and narrow band technique under complextopographical conditions.Chinese J.Geophys.(in Chinese),2009,52(11):2846-2853
[29]De Kool M,Rawlinson N,Sambridge M.A practical grid-based method for tracking multiple refraction and reflectionphases in three-dimensional heterogeneous media.Geophysical Journal International,2006,167(1):253-270.
[30]Zhao H.A fast sweeping method for eikonal equations.Mathematics of Computation,2005,74(250):603-628.
[31]Zhang Y T,Zhao H K,Qian J.High order fast sweepingmethods for static Hamilton-Jacobi equations.Journal ofScientific Computing,2006,29(1):25-56.
[32]兰海强,张智,徐涛等.地震波走时场模拟的快速推进法和快速扫描法比较研究.地球物理学进展,2012,待刊.Lan H Q,Zhang Z,Xu T,et al.A comparative study on thefast marching and fast sweeping methods in the calculation offirst-arrival traveltime field.Progress in Geophys.(inChinese),2012,in press.
[33]Lan H,Zhang Z.Three-dimensional wave-field simulation inheterogeneous transversely isotropic medium with irregularfree surface.Bulletin of the Seismological Society ofAmerica,2011 101(3):1354-1370.
[34]阎世信,刘怀山,姚雪根.山地地球物理勘探技术.北京:石油工业出版社,2000.Yan S X,Liu H S and Yao X G.Geophysical ExplorationTechnology in the Mountainous Area(in Chinese).Beijing:Petroleum Industry Press,2000.
[35]Kao C Y,Osher S,Qian J.Legendre-transform-based fastsweeping methods for static Hamilton-Jacobi equations ontriangulated meshes.Journal of Computational Physics,2008,227(24):10209-10225.
[36]Kimmel R,Sethian J A.Computing geodesic paths onmanifolds.Proceedings of the National Academy ofSciences of the United States of America,1998,95(15):8431-7435.
[37]Lelièvre P G,Farquharson C G,Hurich C A.Computingfirst-arrival seismic traveltimes on unstructured 3-Dtetrahedral grids using the Fast Marching Method.Geophysical Journal International,184(2):885-896.
[38]Qian J,Zhang Y T,Zhao H K.A fast sweeping method forstatic convex Hamilton-Jacobi equations.Journal ofScientific Computing,2007,31(1):237-271.
[39]Qian J,Zhang Y T,Zhao H K.Fast sweeping methods forEikonal equations on triangular meshes.SIAM Journal onNumerical Analysis,2008,45(1):83-107.
[40]Sethian J A,Vladimirsky A.Fast methods for the Eikonaland related Hamilton-Jacobi equations on unstructuredmeshes.Proceedings of the National Academy of Sciencesof the United States of America,2000,97(11):5699.
[41]Appelo D,Petersson N.A stable finite difference method forthe elastic wave equation on complex geometries with freesurfaces.Communications in Computational Physics,2009,5(1):84-107.
[42]孙章庆,孙建国,张东良.二维起伏地表条件下坐标变换法直流电场数值模拟.吉林大学学报:地球科学版,2009(3):528-534.Sun Z Q,Sun J G,Zhang D L.2DDC electric field numericalmodeling including surface topography using coordinatetransformation method.Journal of Jilin University:EarthScience Edition,2009,39(3):528-534.
[43]Hestholm S,Ruud B.2-D finite-difference elastic wavemodeling including surface topography.Geophys.Prosp.,1994,42:371-390.
[44]Lan H,Zhang Z.Seismic wavefield modeling in media withfluid-filled fractures and surface topography.AppliedGeophysics,2012,9(3):301-312.
[45]Tessmer E,Kosloff D.3-D elastic modeling with surfacetopography by a Chebychev spectral method.Geophysics,1994,59:464.
[46]兰海强,刘佳,白志明.VTI介质起伏地表地震波场模拟.地球物理学报,2011,54(8):2072-2084.Lan H Q,Liu J,Bai Z M.Wave-field simulation in VTImedia with irregular free surface.Chinese J.Geophys.(inChinese),2011,54(8):2072-2084.
[47]孙建国.复杂地表条件下地球物理场数值模拟方法评述.世界地质,2007,26(3):345-362.Sun J G.Methods for numerical modeling of geophysicalfields under complex topographicaI conditions:a criticalreview.Global Geology(in Chinese),2007,26(3):345-362.
[48]Lan H,Zhang Z.Comparative study of the free-surfaceboundary condition in two-dimensional finite-difference elasticwave field simulation.Journal of Geophysics andEngineering,2011,8:275-286.
[49]Sun J,Sun Z,Han F.A finite difference scheme for solvingthe eikonal equation including surface topography.Geophysics,2011,76(4):T53-T63.
[50]Lan H,Zhang Z.Topography-dependent eikonal equationand its solver for calculating first-arrival travel times with anirregular surface.Geophysical Journal International,2012a,in review.
[51]刘一峰,兰海强.曲线坐标系程函方程的求解方法研究.地球物理学报,2012,55(6):2014-2026.Liu Y F,Lan H Q.Study on the numerical solutions of theeikonal equation in curvilinear coordinate system.Chinese J.Geophys.(in Chinese),2012,55(6):2014-2026.
[52]Lan H,Zhang Z.A high order fast sweeping scheme forcalculating first-arrival travel times with an irregular surface.Bull.Seismol.Soc.Am.,2012b,in review.
[53]Hvid S L.Three dimensional algebraic grid generation[Ph.D.thesis].Technical University of Denmark,1994.
[54]Thompson J,Warsi Z,Mastin C.Numerical grid generation:foundations and applications.1985,North-hollandAmsterdam.
[55]Hestholm S,Ruud B.3-D finite-difference elastic wavemodeling including surface topography.Geophysics,1998,63:613-622.
[56]Tessmer E,Kosloff D,Behle A.Elastic wave propagationsimulation in the presence of surface topography.Geophys.J.Int.,1992,108(2):621-632.
[57]Alton K,Mitchell I M.Fast marching methods for stationaryHamilton-Jacobi equations with axis-aligned anisotropy.SIAM J.Numer.Anal,2008,47:363-385.
[58]Sethian J A,Vladimirsky A.Ordered upwind methods forstatic Hamilton-Jacobi equations.Proceedings of theNational Academy of Sciences,2001,98(20):11069.
[59]Tsai Y H R,Cheng L T,Osher S,et al.Fast sweepingalgorithms for a class of Hamilton-Jacobi equations.SIAMJournal on Numerical Analysis,2004:673-694.
[60]Kao C Y,Osher S,Qian J.Lax-Friedrichs sweeping schemefor static Hamilton-Jacobi equations*1.Journal ofComputational Physics,2004,196(1):367-391.
[61]Sethian J A.Level Set Methods and Fast Marching Methods:Evolving Interfaces in Computational Geometry,FluidMechanics,Computer Vision,and Materials Science.Cambridge University Press,1999.
[62]Luo S,Qian J.Factored singularities and high-order Lax-Friedrichs sweeping schemes for point-source traveltimes andamplitudes.J.Comput.Phys.,2011,230:4742-4755.
[63]Lin C T and Tadmor E.L1-stability and error estimates forapproximate Hamilton-Jacobi solutions.Numer.Math.,2001,87:701-735.
[64]Qian J,Symes W W.Finite-difference quasi-P traveltimes foranisotropic media.Geophysics,2002,67:147-155.
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