波动方程反演的全局优化方法研究
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摘要
复杂介质波动方程反演是地球物理研究中的重要问题,通常表述为特定目标函数最优化,难点是多参数、非线性和不适定性.局部和全局优化方法都不能实现快速全局优化.本文概述了地震波勘探反演问题的理论基础和研究进展,阐述了反演中优化问题的解决方法和面临的困难,并提出了一种确定性全局优化的新方法.通过在优化参数空间识别并划分局部优化解及其附近区域,只需有限次参数空间划分过程就能发现所有局部解(集合);基于复杂目标函数多尺度结构分析,提出多尺度参数空间分区优化方法的研究方向.该方法收敛速度快,优化结果不依赖初始解的选取,是对非线性全局优化问题的一个新探索.
The wave equation inversion in complex media is an important research in geophysics.This problem is usually described as objective function optimization,which has many difficulties such as multi-parameter,nonlinear in the parameter and ill-posedness.Both local-optimization and global-optimization methods are unable to achieve fast global optimization.This paper gives a brief introduction of theoretical foundation and research advances in seismic exploration inversion problem.The optimization solutions of inversion problem are discussed and the difficulties are analyzed.Finally,a new deterministic optimization method is presented.All the local optimization solutions(sets) can be determined after finite times of parameter space identification and partition procedures.Based on multi-scale landscape analysis of complex objective functions,a multi-scale parameter space partition method is proposed.The new method has a very fast convergence speed and the optimization solution is independent on the selection of initial solutions.This is a new research direction of nonlinear global optimization methods.
The wave equation inversion in complex media is an important research in geophysics.This problem is usually described as objective function optimization,which has many difficulties such as multi-parameter,nonlinear in the parameter and ill-posedness.Both local-optimization and global-optimization methods are unable to achieve fast global optimization.This paper gives a brief introduction of theoretical foundation and research advances in seismic exploration inversion problem.The optimization solutions of inversion problem are discussed and the difficulties are analyzed.Finally,a new deterministic optimization method is presented.All the local optimization solutions(sets) can be determined after finite times of parameter space identification and partition procedures.Based on multi-scale landscape analysis of complex objective functions,a multi-scale parameter space partition method is proposed.The new method has a very fast convergence speed and the optimization solution is independent on the selection of initial solutions.This is a new research direction of nonlinear global optimization methods.
引文
[1]Tarantola A.Inversion of Seismic Reflection Data in the A-coustic Approximation[J].Geophysics,1984,49(8):1259~1266.
[2]Backus G E,Gilbert F.The resolving power of gross earth da-ta[J].Geophysical Journal of the Royal Astronomical Socie-ty,1968.16:169~205.
[3]Cohen J K,Hagin F G,Bleistein N.Three-dimensional Borninversion with an arbitrary reference[J].Geophysics,1986.51:1552~1558.
[4]Bunks C,Saleck F M,et al.Multiscale seismic waveform in-version[J].Geophysics,1995,60(5):1457~1473.
[5]杨丽华,孟绍波.一维波动方程小波逐版本反演[J].地球物理学报,1995,38(6):815~821.
[6]Wood W T.Simutaneous deconvolution and wavelet inversionas a global optimization[J].Geophysics,1999,64(4):1108~1115.
[7]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,2003,46(2):265~270.
[8]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,2004,47(6):1111~1117.
[9]张宏兵,尚作萍,杨长春,段秋梁.波阻抗反演正则参数估计[J].地球物理学报,2005,48(1):181~188.
[10]杨光大,陈湛.地震资料波阻抗多尺度融合反演[J].地球物理学进展,2005,20(3):718~723.
[11]杨文采.非线性地球物理反演方法:回顾与展望[J].地球物理学进展,2002,17(2):255~261.
[12]Rothman D H.Nonlinear inversion statistical mechanics,andresidual statics corrections[J].Geophysics,1985,50:2784~2796.
[13]Rothman D H.Automatic estimation of large residual staticscorrections[J].Geophysics,1986,51:332~346.
[14]Kirkpatrick S C,Gelatt D,Vecchi M P.Optimization bysimulated annealing[J].Science,1983,220:671~680.
[15]Holland J H.Adaptation in Natural and Artificial Systems[M].Ann Arbor:University of Michigan Press,1975.
[16]Goldberg D E.Genetic Algorithms in Search,Optimization,and Machine Learning[M].Reading,MA:Addison-Wesley,1989.
[17]Stoffa P L,Sen M K.Nonlinear multiparameter optimizationusing genetic algorithms:Inversion of plane wave seismo-grams[J].Geophysics,1991,56:1794~1810.
[18]Sambridge M and Drijkoningen G G.Genetic algorithms inseismic waveform inversion[J].Geophys.J.Int.,1992,109:323~342.
[19]Gallagher K,Sambridge M.Genetic algorithms:a powerfultool for large-scale non-linear optimization problems[J].Comput.Geosci.,1994,20(7/8):1229~1236.
[20]Sen M and Stoffa P L.Global Optimization Methods in Geo-physical Inversion[M],Advances in Exploration Geophysics,1995,4,Elsevier,Amsterdam.
[21]Fallat,M R,Dosso,S E.Geoacoustic inversion via local,global,and hybrid algorithms[J].Acoust.Soc.Amer.1999,105:3219~3230.
[22]Liu P,Hartzell S,Stephenson W.Non-linear multiparameterinversion using a hybrid global search algorithm:applicationsin reflection seismology[J].Geophys.J.Int.1995,122:991~1000.
[23]Scales J A,Smith M L,Fischer T L.Global optimizationmethods for multimodal inverse problems[J].J.Comput.Phys.1992,103:258~268.
[24]刘鹏程,纪晨,等.改进的模拟退火-单纯形综合反演方法[J].地球物理学报,1995,38(2):199~205.
[25]纪晨,姚振兴.用于地球物理反演的均匀设计优化算法[J].地球物理学报,1996,39(2):233~242.
[26]艾印双,刘鹏程,郑天愉.自适应全局混合反演[J].中国科学(D),1998,28(2):105~110.
[27]杨立强,宋海斌,郝天珧.基于BP神经网络的波阻抗反演及应用[J].地球物理学进展,2005,20(1):34~37.
[28]魏超,朱培民,王家映.量子退火反演的原理和实现.地球物理学报,2006,49(2):577~583
[29]于鹏,王家林,吴健生,王大为.重力与地震资料的模拟退火约束联合反演.地球物理学报,2007,50(2):529~538
[30]基于BP神经网络的波阻抗反演及应用[J].地球物理学进展,2005,20(1):34~37
[31]王登刚,刘迎曦,李守巨.弹性力学非线性反演方法概述[J].力学进展,2003,33(2):166~174.
[32]Cary P W,Chapman C H.Automatic 1-D waveform inver-sion of marine seismic reflection data[J].Geophys.J.1988,93:527~546.
[33]Gerstoft P.Inversion of acoustic data using a combination ofgenetic algorithms and the Gauss-Newton approach[J].J.Acoust.Soc.Amer.1995,97:2181~2190.
[34]Hibbert D B.A hybrid genetic algorithm for the estimation ofkinetic parameters[J].Chemometrics and Intelligent labora-tory systems,1993,19:319~329.
[35]Chunduru R,Sen M K,et al.Hybrid optimization methodsfor geophysical inversion[J].Geophysics,1997,62:1196~1207.
[36]张霖斌,姚振兴.层状介质参数反演的混合最优化法[J].地球物理学进展,2000,15(1):46~53.
[37]Macias C C,Sen M K,et al.Artificial neural networks forparameter estimation in geophysics[J].Geophysical Prospec-ting,2000,48:21~47.
[38]张新兵,王家林,吴健生.混合最优化算法在地球物理学中的应用现状与前景[J].地球物理学进展,2003,18(2):218-223.
[39]崔建文.一种改进的全局优化算法及其在面波频散曲线反演中的应用[J].地球物理学报,2004,47(3):521~527.
[40]邵泽辉,李正文,许多.自适应GA-BP优化方法进行反演[J].地球物理学进展,2004,19(4):942-945.
[41]杨晓春,李小凡,张美根.地震波散射非线性反演的不动点理论研究[J].地球物理学进展,2005,20(2):496~502.
[42]李琼,李勇,李正文,等.基于GA-BP理论的储层视裂缝密度地震非线性反演方法[J].地球物理学进展,2006,21(2):465~471.
[43]赵宪生,严刚峰.波阻抗混合反演全局寻优与编码的实验研究[J].地球物理学进展,2005,20(3):688~693.
[44]Sambridge M.Geophysical inversion with a neighbourhoodalgorithm-I.Searching a parameter space[J].Geophys.J.Int.,1999,138:479~494.
[45]Sambridge M.Geophysical inversion with a neighbourhoodalgorithm-II.Appraising the ensemble[J].Geophys.J.Int.,1999,138:727~746.
[46]Barhen J,Protopopescu V,Reister D.TRUST:a determin-istic algorithm for global optimization[J].Science 1997,276:1094~1097.
[47]Basso P.Iterative methods for the localization of the globalmaximum[J].SIAM J.Numer.Anal.1982,19:781~792.
[48]Piyavskii S A.An algorithm for 0nding the absolute extre-mum of a function[J].USSR Comput.Math.Math.Phys.1972,12:57~67.
[49]Shubert B O.A sequential method seeking the global maxi-mum of a function[J].SIAM J.Numer.Anal.,1972,9:379~388.
[50]Floudas C A.Deterministic Global Optimization:Theory,Methods and Applications[M],Dordrecht:Kluwer Academ-ic,2000.
[51]Caprani O,Godthaab B,Madsen K.Use of a real-valued lo-cal minimum in parallel interval global optimization[J].Inter-val Comput.1993,2:71~82.
[52]Hansen E.Global optimization using interval analysis:theone-dimensional case[J].J.Optim.Theor.Appl.1979,29:331~344.
[53]Hansen E.Global optimization using interval analysis:themulti-dimensional case[J].Numer.Math.1980,34.
[54]Hansen E.Global Optimization Using Interval Analysis[M].New York:Marcel Dekker,1992.
[55]Ichida K,Fujii Y.An interval arithmetic method for globaloptimization[J].Computing,1979,23:85~97.
[56]Kearfott R B.Rigorous Global Search:Continuous Problems[M].Dordrecht:Kluwer Academic,1996.
[57]Ratschek H,Rokne J.New Computer Methods for GlobalOptimization[M].Chichester:Ellis Horwood,1988.
[58]Tarvainen M,Tiira T,Husebye ES.Locating regional seis-mic events with global optimization based on interval arithme-tic[J].Geophys.J.Int.1999,138:879~885.
[59]Locatelli M.On the multilevel structure of global optimiza-tion problems[J].Computation Optimization and Applica-tions,2005,30:5~22.
[60]孙卫涛.复杂介质弹性波场有限差分正演和全局优化方法研究[D].北京:清华大学,2003.
[61]Sun W T.A Scalable Parallel Algorithm for Global Optimiza-tion Based on Seed-Growth Techniques[A].Lecture Notes inComputer Science 3726(Springer-Verlag,Berlin),HPCC2005,Naples,Italy,2005:839~844
[62]Sun W T,Shu J W,Zheng W M.Deterministic Global Opti-mization with a Neighbourhood Determination AlgorithmBased on Neural Networks[A].Lecture Notes in ComputerScience 3496(Springer-Verlag,Berlin),ISNN 2005,Chongqin,China,2005:700~705
[63]Sun W T.Global Optimization with Multi-grid Seed-GrowthParameter Space Division Algorithm[A].Proceedings of2005 International Conference on Scientific Computing,LasVegas,USA,2005,32~38
[64]Mallat S.A theory for multiresolution signal decomposition:the wavelet representation[J].IEEE Pattern Anal.and Ma-chine Intell.,1989,11(7):674~693.
[65]Daubechies I.Ten lectures on wavelets[M].SIAM,1992.
[2]Backus G E,Gilbert F.The resolving power of gross earth da-ta[J].Geophysical Journal of the Royal Astronomical Socie-ty,1968.16:169~205.
[3]Cohen J K,Hagin F G,Bleistein N.Three-dimensional Borninversion with an arbitrary reference[J].Geophysics,1986.51:1552~1558.
[4]Bunks C,Saleck F M,et al.Multiscale seismic waveform in-version[J].Geophysics,1995,60(5):1457~1473.
[5]杨丽华,孟绍波.一维波动方程小波逐版本反演[J].地球物理学报,1995,38(6):815~821.
[6]Wood W T.Simutaneous deconvolution and wavelet inversionas a global optimization[J].Geophysics,1999,64(4):1108~1115.
[7]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,2003,46(2):265~270.
[8]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,2004,47(6):1111~1117.
[9]张宏兵,尚作萍,杨长春,段秋梁.波阻抗反演正则参数估计[J].地球物理学报,2005,48(1):181~188.
[10]杨光大,陈湛.地震资料波阻抗多尺度融合反演[J].地球物理学进展,2005,20(3):718~723.
[11]杨文采.非线性地球物理反演方法:回顾与展望[J].地球物理学进展,2002,17(2):255~261.
[12]Rothman D H.Nonlinear inversion statistical mechanics,andresidual statics corrections[J].Geophysics,1985,50:2784~2796.
[13]Rothman D H.Automatic estimation of large residual staticscorrections[J].Geophysics,1986,51:332~346.
[14]Kirkpatrick S C,Gelatt D,Vecchi M P.Optimization bysimulated annealing[J].Science,1983,220:671~680.
[15]Holland J H.Adaptation in Natural and Artificial Systems[M].Ann Arbor:University of Michigan Press,1975.
[16]Goldberg D E.Genetic Algorithms in Search,Optimization,and Machine Learning[M].Reading,MA:Addison-Wesley,1989.
[17]Stoffa P L,Sen M K.Nonlinear multiparameter optimizationusing genetic algorithms:Inversion of plane wave seismo-grams[J].Geophysics,1991,56:1794~1810.
[18]Sambridge M and Drijkoningen G G.Genetic algorithms inseismic waveform inversion[J].Geophys.J.Int.,1992,109:323~342.
[19]Gallagher K,Sambridge M.Genetic algorithms:a powerfultool for large-scale non-linear optimization problems[J].Comput.Geosci.,1994,20(7/8):1229~1236.
[20]Sen M and Stoffa P L.Global Optimization Methods in Geo-physical Inversion[M],Advances in Exploration Geophysics,1995,4,Elsevier,Amsterdam.
[21]Fallat,M R,Dosso,S E.Geoacoustic inversion via local,global,and hybrid algorithms[J].Acoust.Soc.Amer.1999,105:3219~3230.
[22]Liu P,Hartzell S,Stephenson W.Non-linear multiparameterinversion using a hybrid global search algorithm:applicationsin reflection seismology[J].Geophys.J.Int.1995,122:991~1000.
[23]Scales J A,Smith M L,Fischer T L.Global optimizationmethods for multimodal inverse problems[J].J.Comput.Phys.1992,103:258~268.
[24]刘鹏程,纪晨,等.改进的模拟退火-单纯形综合反演方法[J].地球物理学报,1995,38(2):199~205.
[25]纪晨,姚振兴.用于地球物理反演的均匀设计优化算法[J].地球物理学报,1996,39(2):233~242.
[26]艾印双,刘鹏程,郑天愉.自适应全局混合反演[J].中国科学(D),1998,28(2):105~110.
[27]杨立强,宋海斌,郝天珧.基于BP神经网络的波阻抗反演及应用[J].地球物理学进展,2005,20(1):34~37.
[28]魏超,朱培民,王家映.量子退火反演的原理和实现.地球物理学报,2006,49(2):577~583
[29]于鹏,王家林,吴健生,王大为.重力与地震资料的模拟退火约束联合反演.地球物理学报,2007,50(2):529~538
[30]基于BP神经网络的波阻抗反演及应用[J].地球物理学进展,2005,20(1):34~37
[31]王登刚,刘迎曦,李守巨.弹性力学非线性反演方法概述[J].力学进展,2003,33(2):166~174.
[32]Cary P W,Chapman C H.Automatic 1-D waveform inver-sion of marine seismic reflection data[J].Geophys.J.1988,93:527~546.
[33]Gerstoft P.Inversion of acoustic data using a combination ofgenetic algorithms and the Gauss-Newton approach[J].J.Acoust.Soc.Amer.1995,97:2181~2190.
[34]Hibbert D B.A hybrid genetic algorithm for the estimation ofkinetic parameters[J].Chemometrics and Intelligent labora-tory systems,1993,19:319~329.
[35]Chunduru R,Sen M K,et al.Hybrid optimization methodsfor geophysical inversion[J].Geophysics,1997,62:1196~1207.
[36]张霖斌,姚振兴.层状介质参数反演的混合最优化法[J].地球物理学进展,2000,15(1):46~53.
[37]Macias C C,Sen M K,et al.Artificial neural networks forparameter estimation in geophysics[J].Geophysical Prospec-ting,2000,48:21~47.
[38]张新兵,王家林,吴健生.混合最优化算法在地球物理学中的应用现状与前景[J].地球物理学进展,2003,18(2):218-223.
[39]崔建文.一种改进的全局优化算法及其在面波频散曲线反演中的应用[J].地球物理学报,2004,47(3):521~527.
[40]邵泽辉,李正文,许多.自适应GA-BP优化方法进行反演[J].地球物理学进展,2004,19(4):942-945.
[41]杨晓春,李小凡,张美根.地震波散射非线性反演的不动点理论研究[J].地球物理学进展,2005,20(2):496~502.
[42]李琼,李勇,李正文,等.基于GA-BP理论的储层视裂缝密度地震非线性反演方法[J].地球物理学进展,2006,21(2):465~471.
[43]赵宪生,严刚峰.波阻抗混合反演全局寻优与编码的实验研究[J].地球物理学进展,2005,20(3):688~693.
[44]Sambridge M.Geophysical inversion with a neighbourhoodalgorithm-I.Searching a parameter space[J].Geophys.J.Int.,1999,138:479~494.
[45]Sambridge M.Geophysical inversion with a neighbourhoodalgorithm-II.Appraising the ensemble[J].Geophys.J.Int.,1999,138:727~746.
[46]Barhen J,Protopopescu V,Reister D.TRUST:a determin-istic algorithm for global optimization[J].Science 1997,276:1094~1097.
[47]Basso P.Iterative methods for the localization of the globalmaximum[J].SIAM J.Numer.Anal.1982,19:781~792.
[48]Piyavskii S A.An algorithm for 0nding the absolute extre-mum of a function[J].USSR Comput.Math.Math.Phys.1972,12:57~67.
[49]Shubert B O.A sequential method seeking the global maxi-mum of a function[J].SIAM J.Numer.Anal.,1972,9:379~388.
[50]Floudas C A.Deterministic Global Optimization:Theory,Methods and Applications[M],Dordrecht:Kluwer Academ-ic,2000.
[51]Caprani O,Godthaab B,Madsen K.Use of a real-valued lo-cal minimum in parallel interval global optimization[J].Inter-val Comput.1993,2:71~82.
[52]Hansen E.Global optimization using interval analysis:theone-dimensional case[J].J.Optim.Theor.Appl.1979,29:331~344.
[53]Hansen E.Global optimization using interval analysis:themulti-dimensional case[J].Numer.Math.1980,34.
[54]Hansen E.Global Optimization Using Interval Analysis[M].New York:Marcel Dekker,1992.
[55]Ichida K,Fujii Y.An interval arithmetic method for globaloptimization[J].Computing,1979,23:85~97.
[56]Kearfott R B.Rigorous Global Search:Continuous Problems[M].Dordrecht:Kluwer Academic,1996.
[57]Ratschek H,Rokne J.New Computer Methods for GlobalOptimization[M].Chichester:Ellis Horwood,1988.
[58]Tarvainen M,Tiira T,Husebye ES.Locating regional seis-mic events with global optimization based on interval arithme-tic[J].Geophys.J.Int.1999,138:879~885.
[59]Locatelli M.On the multilevel structure of global optimiza-tion problems[J].Computation Optimization and Applica-tions,2005,30:5~22.
[60]孙卫涛.复杂介质弹性波场有限差分正演和全局优化方法研究[D].北京:清华大学,2003.
[61]Sun W T.A Scalable Parallel Algorithm for Global Optimiza-tion Based on Seed-Growth Techniques[A].Lecture Notes inComputer Science 3726(Springer-Verlag,Berlin),HPCC2005,Naples,Italy,2005:839~844
[62]Sun W T,Shu J W,Zheng W M.Deterministic Global Opti-mization with a Neighbourhood Determination AlgorithmBased on Neural Networks[A].Lecture Notes in ComputerScience 3496(Springer-Verlag,Berlin),ISNN 2005,Chongqin,China,2005:700~705
[63]Sun W T.Global Optimization with Multi-grid Seed-GrowthParameter Space Division Algorithm[A].Proceedings of2005 International Conference on Scientific Computing,LasVegas,USA,2005,32~38
[64]Mallat S.A theory for multiresolution signal decomposition:the wavelet representation[J].IEEE Pattern Anal.and Ma-chine Intell.,1989,11(7):674~693.
[65]Daubechies I.Ten lectures on wavelets[M].SIAM,1992.
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