二维波动方程速度的正则化-同伦-测井约束反演
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摘要
针对二维波动方程反问题,将大范围收敛的同伦方法引入速度参数的反演过程中,并将其与求解不适定问题的Tikhonov正则化有机结合,提出了一种新的、特别适用于非线性的、不适定的、多极值的地震勘探反演问题的反演策略:正则化-同伦方法.为了充分利用测井资料和地震资料的互补特征,进一步提高反演分辨率并压制噪声,设计了正则化-同伦-测井约束联合反演方法.大量数值试验结果表明了这两种方法的有效性.
Combining the Tikhonov regularization method for ill-posed problems with the widely convergent homotopy method applied to the inversion process of operator identification,a new strategy-regularization-homotopy method is proposed for the inversion of 2-D wave equation,which is suitable for the nonlinear,ill-posed and multi-extremum seismic inverse problem.In order to restrain noises and improve the quality of inversion,a well log constraint regularization-homotopy method is further constructed.Lots of numerical simulations indicate our method's effectiveness.
Combining the Tikhonov regularization method for ill-posed problems with the widely convergent homotopy method applied to the inversion process of operator identification,a new strategy-regularization-homotopy method is proposed for the inversion of 2-D wave equation,which is suitable for the nonlinear,ill-posed and multi-extremum seismic inverse problem.In order to restrain noises and improve the quality of inversion,a well log constraint regularization-homotopy method is further constructed.Lots of numerical simulations indicate our method's effectiveness.
引文
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[2]Amundsen L,Ursin B.Frequency_wavenumber inversion of acousticdata.Geophysics,1991,56:1027~1039
[3]Zhao H,Ursin B,Amundsen L.Frequency_Wavenumber elasticinversion of marine seismic data.Geophysics,1994,59:1868~1881
[4]Chunduru R K,Sen M K,Stoffa P L.Hybrid optimization methodsfor geophysical inversion.Geophysics,1997,62:1196~1207
[5]韩波,匡正,刘家琦.反演地层电阻率的单调同伦法.地球物理学报,1991,34:97~112Han B,Kuang Z,Liu J Q.A monotonous homotopy method forsolving the resistivities of the earth.Chinese J.Geophys.(inChinese),1991,34:97~112
[6]Han B,Zhang M L,Liu J Q.A widely convergent generalized pulse_spectrum technique for the coefficient inverse problems of differentialequations.Appl.Math.Comput.,1997,81:97~112
[7]Feng G F,Han B,Liu J Q.A widely convergent generalized pulse_spectrum methods for 2_D wave equation inversion.Chinese J.Geophys.,2003,46:373~380
[8]Vasco D W.Singularity and branching:a path_following formalismfor geophysical inverse problems.Geophys.J.Int.,1994,119:809~830
[9]Vasco D W.Regularization and trade_off associated with nonlineargeophysical inverse problems:penalty homotopies.Inverse Problems,1998,14:1033~1052
[10]Bube K P,Langan R T.On a continuation approach to regularizationfor crosswell tomography.69th Ann.Internat.Mtg.Soc.Expl.Geophys.,Expanded Abstracts,1999.1295~1298
[11]Everett M E.Homotopy,polynomial equations,gross boundary data,and small Helmholtz systems.J.Comput.Phys.,1996,124:431~441
[12]Jegen M D,Everett M E,Schultz A.Using homotopy to invertgeophysical data.Geophysics,2001,66:1749~1760
[13]韩华,章梓茂,汪越胜.双相介质波动方程孔隙率反演的同伦法.力学学报,2003,35(2):235~239Han H,Zhang Z M,Wang Y S.Homotopy method for inversing theporosity of 1_D wave equation in porous media.Acta MechanicaSinica(in Chinese),2003,35(2):235~239
[14]Bao G,Liu J.Numerical solution of inverse scattering problems withmulti_experimental limited aperture data.Siam J.Sci.Comput.,2003,25:1102~1117
[15]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.地球物理学报,2004,47(6):1111~1117Zhang L Q,Wang J Y,Yan D T.A homotopy method for theimpedance inversion of 1_D wave equation.Chinese J.Geophys.(inChinese),2004,47(6):1111~1117
[16]魏培君,章梓茂.弹性动力学反问题的数值反演方法.力学进展,2001,31(2):172~180Wei P J,Zhang Z M.Numerical methods for inverse problems inelastic dynamics.Advances in Mechnics(in Chinese),2001,31(2):172~180
[17]Xie G Q,Li J H,Chen Y M.Gauss_Newton_Regularzing method forsolving coefficient inverse problem of partial differential equation andits convergence.J.Comput.Math.,1987,5:38~49
[18]宋华,刘家琦.牛顿正则化方法与一类差分方程反问题的求解.计算数学,1990,3:227~231Song H,Liu J Q.Newton regularization method and its application.Mathematics Numerica Sinica(in Chinese),1990,3:227~231
[19]Bakushinskil A B.The problem of the convergence of the iterativelyregularized Gauss_Newton method.Comput.Math.Phys.,1992,32:1353~1359
[20]Scherzer O.A convergence analysis of a method of steepest descentand a two_step algorithm for nonlinear ill_posed problems.Numer.Funct.Anal.Optim.,1996,17:197~214
[21]Scherzer O.A modified Landweber iteration for solving parameterestimation problems.Appl.Math.Optim.,1998,38:45~68
[22]Fletcher R.A modified Marquardt subroutine for nonlinear leastsquares:Rep.AERE_R 6799.Atomic energy research establishment,Harwell,U.K,1971,271~307
[23]Kaltenbacher B.Some Newton_type methods for the regularization ofnonlinear ill_posed problems.Inverse Problems,1997,13:729~753
[24]Ramlau R.A modified Landweber method for inverse problems.Numer.Funct.Anal.and Optim.,1999,20(1-2):79~98
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