地球物理资料非线性反演方法讲座(七) 同伦反演法
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摘要
同伦是代数拓扑学中的概念,同伦算法已应用于一些工程实际问题的求解。同伦方法的基本思想是:通过引入同伦参数,将原来复杂的非线性映射零点求解问题转化为跟踪同伦零点曲线问题,该同伦零点曲线的一端为已知值(给定的初始模型),另一端即为所求原映射的解。本讲座概要地介绍了同伦方法的基本原理,跟踪同伦曲线的具体算法,并给出了同伦方法在地震勘探波阻抗反演中的应用实例,最后归纳总结了同伦方法的特点及目前存在的不足。
Homotopyis a concept of algebra topology,and homotopy algorithmhas been applied to solving the practical problems.The main spirit of the homotopyis that the original complicated problemis translatedintothe one of tracking homotopy zero curves which start with the knowninitial solutions and end withthe solutions of the object functions by adding homotopy parameter.This paper briefly introduces the homotopy theory and algorithm of tracking zero curves.At the same ti me,the paper gives the example of the i mpedance inversionin the seismic exploration.In the end,the advantages and disadvantages of the homotopy method are pointed out in this paper.
Homotopyis a concept of algebra topology,and homotopy algorithmhas been applied to solving the practical problems.The main spirit of the homotopyis that the original complicated problemis translatedintothe one of tracking homotopy zero curves which start with the knowninitial solutions and end withthe solutions of the object functions by adding homotopy parameter.This paper briefly introduces the homotopy theory and algorithm of tracking zero curves.At the same ti me,the paper gives the example of the i mpedance inversionin the seismic exploration.In the end,the advantages and disadvantages of the homotopy method are pointed out in this paper.
引文
[1]蒲保明,蒋继光,胡淑礼.拓扑学[M].北京:高等教育出版社,1985.
[2]Keller HB,Perozzi DJ.Fast seismic ray tracing:SI-AM[J].J.Appl.Math.,1983,43:981~992.
[3]Vasco,D W.Singularity and branching:A path-following formalismfor geophysical inverse problems[J].Geophys J.Internat.,1994,119:809~830.
[4]Vasco D W.Regularization and trade-off associatedwith nonlinear geophysical inverse problems:penaltyhomotopies[J].Inverse Problem,1998,14:1033~1052.
[5]Bube K P,and Langan R T,On a continuation ap-proach to regularization for crosswell tomography[A].69th Ann.Internat.Mtg.,Soc.Expl.Geo-phys.,Expanded Abstracts,1990,bf2.
[6]Everett M E.Homotopy,polynomial equations,gross boundary data,and small Hel mholtz systems[J].J.Comput.Phys.,1996,124:431~441.
[7]Jegen M D,Everett ME,and Schultz A.Using ho-motopy to invert geophysical data[J],Geophysics,2001,66:1749~1760.
[8]王则柯,高堂安.同伦方法引论[M]重庆:重庆出版社,1990.
[9]B Han,H Y Yin,J Q Liu.Regularizing-numerical-continuation method for solving the resistivities ofthe earth[J].Mathematics and Computers in Si mu-lation,1995,39:109~114.
[10]Liu Jiaqi,Qian Jianliang,Han Bo,A method ofwide convergence region for solving inverse prob-lems of wave equations[J].Journal of Harbin In-stitute of Technology,1995,E 2(4):
[11]B Han,ML Zhang,J Q Liu.A Widely ConvergentGeneralized Pulse-Spectrum Technique for the Co-efficient Inverse Problem of Differential Equations[J].Applied Mathematics and Computation,1997,81:97~112.
[12]冯国峰,韩波,刘家琦.二维波动方程约束反演的大范围收敛广义脉冲谱方法[J].地球物理学报,2003(2):265~270.
[13]B Han,G F Feng,J Q Liu.A Widely ConvergentGeneralized Pulse-Spectrum Technique for the in-version of two-di mensional acoustic wave equation[J],Applied Mathematics and Computation,2005(1):1~15.
[14]HS Fu,B Han.A homotopy method for nonlinearinverse problems[J],Applied Mathematics andComputation,2006(1):1~10.
[15]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].北京:科学出版社,1997.
[16]黄光远.地震勘探数据处理的一种算法——分布参数系统最优理论的应用[J].地球物理学报,1985,28(1):74~83.
[17]张丽琴,王家映,严德天.一维波动方程波阻抗反演的同伦方法[J].地球物理学报,2004(6):1111~1117.
[2]Keller HB,Perozzi DJ.Fast seismic ray tracing:SI-AM[J].J.Appl.Math.,1983,43:981~992.
[3]Vasco,D W.Singularity and branching:A path-following formalismfor geophysical inverse problems[J].Geophys J.Internat.,1994,119:809~830.
[4]Vasco D W.Regularization and trade-off associatedwith nonlinear geophysical inverse problems:penaltyhomotopies[J].Inverse Problem,1998,14:1033~1052.
[5]Bube K P,and Langan R T,On a continuation ap-proach to regularization for crosswell tomography[A].69th Ann.Internat.Mtg.,Soc.Expl.Geo-phys.,Expanded Abstracts,1990,bf2.
[6]Everett M E.Homotopy,polynomial equations,gross boundary data,and small Hel mholtz systems[J].J.Comput.Phys.,1996,124:431~441.
[7]Jegen M D,Everett ME,and Schultz A.Using ho-motopy to invert geophysical data[J],Geophysics,2001,66:1749~1760.
[8]王则柯,高堂安.同伦方法引论[M]重庆:重庆出版社,1990.
[9]B Han,H Y Yin,J Q Liu.Regularizing-numerical-continuation method for solving the resistivities ofthe earth[J].Mathematics and Computers in Si mu-lation,1995,39:109~114.
[10]Liu Jiaqi,Qian Jianliang,Han Bo,A method ofwide convergence region for solving inverse prob-lems of wave equations[J].Journal of Harbin In-stitute of Technology,1995,E 2(4):
[11]B Han,ML Zhang,J Q Liu.A Widely ConvergentGeneralized Pulse-Spectrum Technique for the Co-efficient Inverse Problem of Differential Equations[J].Applied Mathematics and Computation,1997,81:97~112.
[12]冯国峰,韩波,刘家琦.二维波动方程约束反演的大范围收敛广义脉冲谱方法[J].地球物理学报,2003(2):265~270.
[13]B Han,G F Feng,J Q Liu.A Widely ConvergentGeneralized Pulse-Spectrum Technique for the in-version of two-di mensional acoustic wave equation[J],Applied Mathematics and Computation,2005(1):1~15.
[14]HS Fu,B Han.A homotopy method for nonlinearinverse problems[J],Applied Mathematics andComputation,2006(1):1~10.
[15]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].北京:科学出版社,1997.
[16]黄光远.地震勘探数据处理的一种算法——分布参数系统最优理论的应用[J].地球物理学报,1985,28(1):74~83.
[17]张丽琴,王家映,严德天.一维波动方程波阻抗反演的同伦方法[J].地球物理学报,2004(6):1111~1117.
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