广义Stein无偏风险估计与地球物理反问题正则化参数求取
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摘要
地球物理反演是获取地球信息的重要手段,其求解具有严重的不适定性.为获得稳定的反问题结果,通常需要在目标泛函中加入正则化约束项.正确地估计正则化参数一直是地球物理反问题中的难点.目前存在的选取方法需要根据大量的试验来确定正则化参数,工作量十分巨大,并且存在很大的经验性,很难得到最优的正则化参数.针对这个问题,本文提出了一种基于广义Stein无偏风险估计的正则化参数求取方法.该方法的具体思路是通过求解模型参数均方误差的广义Stein无偏风险估计函数,在反问题求解过程中自动求取正则化参数.本文模型测试结果表明,相比于目前常用的方法,通过该方法得到的正则化参数是最优的.
Inversion in geophysics is an important way to obtain earth′s information.However,it is usually an ill-posed problem.To obtain a stable inversion result,one needs to add a regularization constraint term into the objective function.The accurate estimation of this regularization parameter is a difficult task in geophysical inversion problems all the time.The existing methods determine this parameter based on trails which are very work-consuming.In addition,these methods are very empirical and hard to find the best one.To solve this problem,we propose a new method to select the regularization parameter based on estimation of the generalized Stein′s unbiased risk.Its specific idea is to solve the generalized Stein′s unbiased risk estimation function of the model′s mean-squared error and automatically to estimate the regularization parameter in the process of the geophysical inversion problem.Numerical tests on models indicate that the proposed method can estimatethe optimal regularization parameter compared to the other existing methods.
Inversion in geophysics is an important way to obtain earth′s information.However,it is usually an ill-posed problem.To obtain a stable inversion result,one needs to add a regularization constraint term into the objective function.The accurate estimation of this regularization parameter is a difficult task in geophysical inversion problems all the time.The existing methods determine this parameter based on trails which are very work-consuming.In addition,these methods are very empirical and hard to find the best one.To solve this problem,we propose a new method to select the regularization parameter based on estimation of the generalized Stein′s unbiased risk.Its specific idea is to solve the generalized Stein′s unbiased risk estimation function of the model′s mean-squared error and automatically to estimate the regularization parameter in the process of the geophysical inversion problem.Numerical tests on models indicate that the proposed method can estimatethe optimal regularization parameter compared to the other existing methods.
引文
Aki K,Richards P G.2002.Quantitative Seismology.2nd ed.Sausalito:University Science Books.
Antoniou A,Lu W S.2007.Practical Optimization:Algorithms and Engineering Applications.New York:Springer.
Aster R C,Borchers B,Thurber C H.2013.Parameter Estimation and Inverse Problems.2nd ed.Waltham:Elsevier Academic Press.
Beck A,Teboulle M.2009.A fast iterative shrinkage-thresholding algorithm for linear inverse problems.SIAM Journal of Imaging Sciences,2(1):183-202.
Claerbout J F,Muir F.1973.Robust modeling with erratic data.Geophysics,38(5):826-844.
Craven P,Wahba G.1979.Smoothing noisy data with spline functions:Estimating the correct degree of smoothing by the method of generalized cross-validation.Numerische Mathematik,31(4):377-403.
Cui Y,Wang Y F.2015.Tikhonov regularization and gradient descent algorithms for tomography using first-arrival seismic traveltimes.Chinese Journal of Geophysics(in Chinese),58(4):1367-1377,doi:10.6038/cjg20150423.
Dai Q W,Jiang F B,Dong L.2014.RBFNN inversion for electrical resistivity tomography based on Hannan-Quinn criterion.Chinese Journal of Geophysics(in Chinese),57(4):1335-1344,doi:10.6038/cjg20140430.
Dai R H,Zhang F C,Liu H Q.2016.Seismic inversion based on proximal objective function optimization algorithm.Geophysics,81(5):R237-R246.
Dong L G,Chi B X,Tao J X,et al.2013.Objective function behavior in acoustic full-waveform inversion.Chinese Journal of Geophysics(in Chinese),56(10):3445-3460,doi:10.6038/cjg20131020.
Donoho D L,Johnstone I M.1995.Adapting to unknown smoothness via wavelet shrinkage.Journal of the American Statistical Association,90(432):1200-1224.
Eldar Y C.2009.Generalized SURE for exponential families:applications to regularization.IEEE Transactions on Signal Processing,57(2):471-481.
Gholami A.2015.Nonlinear multichannel impedance inversion by total-variation regularization.Geophysics,80(5):R217-R224.
Golub G H,Von Matt U.1997.Generalized cross-validation for large-scale problems.Journal of Computational and Graphical Statistics,6(1):1-34.
Hansen P C.1992.Analysis of discrete ill-posed problems by means of the L-curve.SIAM Review,34(4):561-580.
Kay S M.1993.Fundamentals of Statistical Signal Processing,Volume I:Estimation Theory.New Jersey:Prentice-Hall.
Li C,Zhang F.2017.Amplitude-versus-angle inversion based on the L1-norm-based likelihood function and the total variation regularization constraint.Geophysics,82(3):R173-R182.
Robinson E A,Treitel S.1980.Geophysical Signal Analysis.Englewood Cliffs:Prentice-Hall.
Sacchi M D.1997.Reweighting strategies in seismic deconvolution.Geophysical Journal International,129(3):651-656.
Sen M K,Stoffa P L.1991.Nonlinear one-dimensional seismic waveform inversion using simulated annealing.Geophysics,56(10):1624-1638.
Sen M K,Biswas R.2015.Choice of regularization weight in basis pursuit reflectivity inversion.Journal of Geophysics and Engineering,12(1):70-79.
Sirgue L,Pratt R G.2004.Efficient waveform inversion and imaging:Astrategy for selecting temporal frequencies.Geophysics,69(1):231-248.
Stein C M.1981.Estimation of the mean of a multivariate normal distribution.Annals of Statistics,9(6):1135-1151.
Tarantola A.2005.Inverse Problem Theory and Methods for Model Parameter Estimation.Philadelphia:Society for Industrial and Applied Mathematics.
Tikhonov A N,Arsenin V Y.1977.Solutions of Ill-Posed Problems.Washington DC:Winston and Sons.
Wahba G.1990.Spline Models for Observational Data.Philadelphia:Society for Applied Mathematics.
Wang J Y.1998.Inverse Theory in Geophysics(in Chinese).Wuhan:China University of Geosciences Press.
Xu S L,Ma E N.2013.Commonly Used Algorithm Assemblies(in Chinese).5th ed.Beijing:Tsinghua University Press.
Yang W C.1997.Theory and Methods of Geophysical Inversion(in Chinese).Beijing:Geological publishing house.
Yao Y.2002.Basic Theory and Application Methods of Geophysical Inversion(in Chinese).Wuhan:China University of Geosciences Press.
Yilmaz O.2001.Seismic data analysis:processing,inversion,and interpretation of seismic data.Tulsa:Society of Exploration Geophysicists.
Zhang F C,Dai R H,Liu H Q.2014.Seismic inversion based on L1-norm misfit function and total variation regularization.Journal of Applied Geophysics,109:111-118.
Zhang F C,Dai R H.2016.Nonlinear inversion of pre-stack seismic data using variable metric method.Journal of Applied Geophysics,129:111-125.
Zhang F X,Wu Q J,Li Y H.2014.A traveltime tomography study by teleseismic S wave data in the Northeast China area.Chinese Journal of Geophysics(in Chinese),57(1):88-101,doi:10.6038/cjg20130818.
Zhang R,Castagna J.2011.Seismic sparse-layer reflectivity inversion using basis pursuit decomposition.Geophysics,76(6):R147-R158.
Zhu X S,Wu R S,Chen X F.2014.Generalized diffraction tomography.Chinese Journal of Geophysics(in Chinese),57(1):241-260,doi:10.6038/cjg20140120.
崔岩,王彦飞.2015.基于初至波走时层析成像的Tikhonov正则化与梯度优化算法.地球物理学报,58(4):1367-1377,doi:10.6038/cjg20150423.
戴前伟,江沸菠,董莉.2014.基于汉南-奎因信息准则的电阻率层析成像径向基神经网络反演.地球物理学报,57(4):1335-1344,doi:10.6038/cjg20140430.
董良国,迟本鑫,陶纪霞等.2013.声波全波形反演目标函数性态.地球物理学报,56(10):3445-3460,doi:10.6038/cjg20131020.
王家映.1998.地球物理反演理论.武汉:中国地质大学出版社.
徐士良,马尔妮.2013.常用算法程序集.5版.北京:清华大学出版社.
杨文采.1997.地球物理反演的理论与方法.北京:地质出版社.
姚姚.2002.地球物理反演基本理论与应用方法.武汉:中国地质大学出版社.
张风雪,吴庆举,李永华.2014.中国东北地区远震S波走时层析成像研究.地球物理学报,57(1):88-101,doi:10.6038/cjg20130818.
朱小三,吴如山,陈晓非.2014.广义散射层析成像反演.地球物理学报,57(1):241-260,doi:10.6038/cjg20140120.
Antoniou A,Lu W S.2007.Practical Optimization:Algorithms and Engineering Applications.New York:Springer.
Aster R C,Borchers B,Thurber C H.2013.Parameter Estimation and Inverse Problems.2nd ed.Waltham:Elsevier Academic Press.
Beck A,Teboulle M.2009.A fast iterative shrinkage-thresholding algorithm for linear inverse problems.SIAM Journal of Imaging Sciences,2(1):183-202.
Claerbout J F,Muir F.1973.Robust modeling with erratic data.Geophysics,38(5):826-844.
Craven P,Wahba G.1979.Smoothing noisy data with spline functions:Estimating the correct degree of smoothing by the method of generalized cross-validation.Numerische Mathematik,31(4):377-403.
Cui Y,Wang Y F.2015.Tikhonov regularization and gradient descent algorithms for tomography using first-arrival seismic traveltimes.Chinese Journal of Geophysics(in Chinese),58(4):1367-1377,doi:10.6038/cjg20150423.
Dai Q W,Jiang F B,Dong L.2014.RBFNN inversion for electrical resistivity tomography based on Hannan-Quinn criterion.Chinese Journal of Geophysics(in Chinese),57(4):1335-1344,doi:10.6038/cjg20140430.
Dai R H,Zhang F C,Liu H Q.2016.Seismic inversion based on proximal objective function optimization algorithm.Geophysics,81(5):R237-R246.
Dong L G,Chi B X,Tao J X,et al.2013.Objective function behavior in acoustic full-waveform inversion.Chinese Journal of Geophysics(in Chinese),56(10):3445-3460,doi:10.6038/cjg20131020.
Donoho D L,Johnstone I M.1995.Adapting to unknown smoothness via wavelet shrinkage.Journal of the American Statistical Association,90(432):1200-1224.
Eldar Y C.2009.Generalized SURE for exponential families:applications to regularization.IEEE Transactions on Signal Processing,57(2):471-481.
Gholami A.2015.Nonlinear multichannel impedance inversion by total-variation regularization.Geophysics,80(5):R217-R224.
Golub G H,Von Matt U.1997.Generalized cross-validation for large-scale problems.Journal of Computational and Graphical Statistics,6(1):1-34.
Hansen P C.1992.Analysis of discrete ill-posed problems by means of the L-curve.SIAM Review,34(4):561-580.
Kay S M.1993.Fundamentals of Statistical Signal Processing,Volume I:Estimation Theory.New Jersey:Prentice-Hall.
Li C,Zhang F.2017.Amplitude-versus-angle inversion based on the L1-norm-based likelihood function and the total variation regularization constraint.Geophysics,82(3):R173-R182.
Robinson E A,Treitel S.1980.Geophysical Signal Analysis.Englewood Cliffs:Prentice-Hall.
Sacchi M D.1997.Reweighting strategies in seismic deconvolution.Geophysical Journal International,129(3):651-656.
Sen M K,Stoffa P L.1991.Nonlinear one-dimensional seismic waveform inversion using simulated annealing.Geophysics,56(10):1624-1638.
Sen M K,Biswas R.2015.Choice of regularization weight in basis pursuit reflectivity inversion.Journal of Geophysics and Engineering,12(1):70-79.
Sirgue L,Pratt R G.2004.Efficient waveform inversion and imaging:Astrategy for selecting temporal frequencies.Geophysics,69(1):231-248.
Stein C M.1981.Estimation of the mean of a multivariate normal distribution.Annals of Statistics,9(6):1135-1151.
Tarantola A.2005.Inverse Problem Theory and Methods for Model Parameter Estimation.Philadelphia:Society for Industrial and Applied Mathematics.
Tikhonov A N,Arsenin V Y.1977.Solutions of Ill-Posed Problems.Washington DC:Winston and Sons.
Wahba G.1990.Spline Models for Observational Data.Philadelphia:Society for Applied Mathematics.
Wang J Y.1998.Inverse Theory in Geophysics(in Chinese).Wuhan:China University of Geosciences Press.
Xu S L,Ma E N.2013.Commonly Used Algorithm Assemblies(in Chinese).5th ed.Beijing:Tsinghua University Press.
Yang W C.1997.Theory and Methods of Geophysical Inversion(in Chinese).Beijing:Geological publishing house.
Yao Y.2002.Basic Theory and Application Methods of Geophysical Inversion(in Chinese).Wuhan:China University of Geosciences Press.
Yilmaz O.2001.Seismic data analysis:processing,inversion,and interpretation of seismic data.Tulsa:Society of Exploration Geophysicists.
Zhang F C,Dai R H,Liu H Q.2014.Seismic inversion based on L1-norm misfit function and total variation regularization.Journal of Applied Geophysics,109:111-118.
Zhang F C,Dai R H.2016.Nonlinear inversion of pre-stack seismic data using variable metric method.Journal of Applied Geophysics,129:111-125.
Zhang F X,Wu Q J,Li Y H.2014.A traveltime tomography study by teleseismic S wave data in the Northeast China area.Chinese Journal of Geophysics(in Chinese),57(1):88-101,doi:10.6038/cjg20130818.
Zhang R,Castagna J.2011.Seismic sparse-layer reflectivity inversion using basis pursuit decomposition.Geophysics,76(6):R147-R158.
Zhu X S,Wu R S,Chen X F.2014.Generalized diffraction tomography.Chinese Journal of Geophysics(in Chinese),57(1):241-260,doi:10.6038/cjg20140120.
崔岩,王彦飞.2015.基于初至波走时层析成像的Tikhonov正则化与梯度优化算法.地球物理学报,58(4):1367-1377,doi:10.6038/cjg20150423.
戴前伟,江沸菠,董莉.2014.基于汉南-奎因信息准则的电阻率层析成像径向基神经网络反演.地球物理学报,57(4):1335-1344,doi:10.6038/cjg20140430.
董良国,迟本鑫,陶纪霞等.2013.声波全波形反演目标函数性态.地球物理学报,56(10):3445-3460,doi:10.6038/cjg20131020.
王家映.1998.地球物理反演理论.武汉:中国地质大学出版社.
徐士良,马尔妮.2013.常用算法程序集.5版.北京:清华大学出版社.
杨文采.1997.地球物理反演的理论与方法.北京:地质出版社.
姚姚.2002.地球物理反演基本理论与应用方法.武汉:中国地质大学出版社.
张风雪,吴庆举,李永华.2014.中国东北地区远震S波走时层析成像研究.地球物理学报,57(1):88-101,doi:10.6038/cjg20130818.
朱小三,吴如山,陈晓非.2014.广义散射层析成像反演.地球物理学报,57(1):241-260,doi:10.6038/cjg20140120.
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