地震偏移反演成像的迭代正则化方法研究
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摘要
利用伴随算子L~*,直接的偏移方法通常导致一个低分辨率或模糊的地震成像.线性化偏移反演方法需求解一个最小二乘问题.但直接的最小二乘方法的数值不稳定,为目视解译带来困难.本文建立约束正则化数学模型,研究了地震偏移反演成像问题的迭代正则化求解方法.首先对最小二乘问题施加正则化约束,接着利用梯度迭代法求解反演成像问题,特别是提出了共轭梯度方法的混合实现技巧.为了表征该方法的可实际利用性,分别对一维,二维和三维地震模型进行了数值模拟.结果表明该正则偏移反演成像方法是有效的,对于实际的地震成像问题有着良好的应用前景.
In this paper,we consider continuous solution methods for migration deconvolution imaging in seismic inverse problems.Direct migration methods,using the adjoint operator L~* ,usually yield a lower resolution or blurred image.Linearized migration deconvolution requires solving a least-squares migration (LSM) problem.However,we notice that the direct LSM method is unstable in computation which is a severe obstacle for visual explanation.We study regularized mathematical model.We first formulate the problem by incorporating regularizing constraints,and then employ iterative gradient methods for migration deconvolution and imaging.A hybrid gradient technique for ill-posed migration in verse problem is proposed.To show the potential for application of the proposed method,we use synthetic one-,two- and three-dimensional seismograms for seismic migration inversion.Numerical performance indicates that the proposed method is very promising for practical seismic migration imaging.
In this paper,we consider continuous solution methods for migration deconvolution imaging in seismic inverse problems.Direct migration methods,using the adjoint operator L~* ,usually yield a lower resolution or blurred image.Linearized migration deconvolution requires solving a least-squares migration (LSM) problem.However,we notice that the direct LSM method is unstable in computation which is a severe obstacle for visual explanation.We study regularized mathematical model.We first formulate the problem by incorporating regularizing constraints,and then employ iterative gradient methods for migration deconvolution and imaging.A hybrid gradient technique for ill-posed migration in verse problem is proposed.To show the potential for application of the proposed method,we use synthetic one-,two- and three-dimensional seismograms for seismic migration inversion.Numerical performance indicates that the proposed method is very promising for practical seismic migration imaging.
引文
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[2] Claerbout J F.Imaging the Earth's Interior.Oxford:Blackwell,1985
[3] Sj(?)berg T,Gelius L-J,Lecomte I.2D deconvolution of seismic image blur.Expanded Abstracts,SEG 73nd Annual Meeting,Dallas,2003
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[5] Guitton A.Amplitude and kinematic corrections of migrated images for nonunitary imaging operators.Geophysics,2004,69:1017～1024
[6] Sacchi M D,Wang J,Kuehl H.Regularized migration/ inversion:new generation of imaging algorithms.CSEG Recorder,2006,31:54～59
[7] Levenberg K.A method for the solution of certain nonlinear problems in least squares.Quart.Appl.Math.,1944,2:164～166
[8] Marquardt D W.An algorithm for least-squares estimation of nonlinear inequalities.SIAM J.Appl.Math.,1963,11:431～441
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[10] Wang Y F.A restarted conjugate gradient method for ill posed problems.Acta Mathematicae Applicatae Sinica,2003,19:31～40
[11] Wang Y F,Yuan Y X.Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems.Inverse Problems,2005,21:821～838
[12] Tarantola A.Inverse Problems Theory:Methods for Data Fitting and Model Parameter Estimation.Amsterdam:Elsevier,1987
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[16] Wang Y F,Yang C C,Li X W.A regularizing kernel-based brdf model inversion method for ill-posed land surface parameter retrieval using smoothness constraint.Journal of Geophysical Research,2008,113:D13101
[17] Lanczos C.Linear Differential Operators.New York:Van Nostrand,1961
[18] Trampert J,Leveque J J.Simultaneous iterative reconstruction technique:physical interpretation based on the generalized least squares solution.Journal of Geophysical Research,1990,95:12553～12559
[19] Nolet G.Solving or resolving inadequate noisy tomographic systems.J.Comp.Phys.,1985,61:463～482
[20] Nolet G,Snieder R.Solving large linear inverse problems by projection.Geophys.J.Int.,1990,103:665～568
[21] Murio D A.The Mollification Method and the Numerical Solution of Ill-posed Problems.New York:John Wiley and Sons,1993
[22] Backus G,Gilbert J.Numerical applications of a formalism for geophysical inverse problems.Geophys.J.R.Astron.Soc.,1967,13:247～276
[23] Backus G,Gilbert J.The resolving power of gross earth data.Geophys.J.R.Astron.Soc.,1968,16:169～205
[24] Van Decar J C,Snieder R.Obtaining smooth solutions to large linear inverse problems.Geophysics,1994,59:818～829
[25] 王彦飞.反演问题的计算方法及其应用.北京:高等教育出版社,2007 Wang Y F.Computational Methods for Inverse Problems and Their Applications(in Chinese).Beijing:Higher Education Press,2007
[26] Engl H W,Hanke M,Neubauer A.Regularization of Inverse Problems.Dorcrecht:Kluwer Academic Publishers,1996
[27] Schuster G T,Hu J X.Green's function for migration:Continuous recording geometry.Geophysics,2000,65 (1) :167～175
[28] Lecomte I.Resolution and illumination analyses in PSDM:A ray based approach.The Leading Edge,2008,27:650～663
[29] Tikhonov A N,Arsenin V Y.Solutions of Ill-posed Problems.New York:John Wiley and Sons,1977
[30] 肖庭延,于慎根,王彦飞.反问题的数值解法.北京:科学出版社,2003 Xiao T Y,Yu S G,Wang Y F.Numerical Methods for the Solution of Inverse Problems(in Chinese).Beijing:Science Press,2003
[31] Nashed M Z.Perturbations and approximations for generalized inverses and linear operators equations.In:Nashed,M.Z.Ed.,Generalized Inverses and Applications,1976． 325～396
[32] Schuster G T.Green's functions for migration.Abstracts of Expanded Abstracts 67th SEG meeting,1997． 1754～1758
[33] Stolt R H,Benson A K.Seismic Migration:Theory and Practice,Vol.5,Handbook of Geophysical Exploration,Section I.Seismic Exploration.London:Geophysical Press,1986
[34] Yilmaz O.Seismic Data Processing,Investigations in Geophysics No.2． Society of Exploration Geophysicists,Tulsa,Okla,1987
[35] Hu J X,Schuster G T,Valasek P A.Poststack migration deconvolution.Geophysics,2001,66(3) :939～952
[36] Chen J,Schuster G T.Resolution limits of migrated images.Geophysics,1999,64(4) :1046～1053
[37] Yu J H,Hu J X,Schuster G T,Estill R.Prestack migration deconvolution.Geophysics,71 (2) :S53～S62,2006
[38] 袁亚湘.非线性规划数值方法.上海:上海科学与技术出版社,1993 Yuan Y X.Numerical Methods for Nonlinear Programming (in Chinese).Shanghai:Shanghai Science and Technology Press,1993
[39] Touati Ahmed D,Storey C.Efficient hybrid conjugate gradient techniques.J.Optimization Theory Appl.,1990,64:379～397
[2] Claerbout J F.Imaging the Earth's Interior.Oxford:Blackwell,1985
[3] Sj(?)berg T,Gelius L-J,Lecomte I.2D deconvolution of seismic image blur.Expanded Abstracts,SEG 73nd Annual Meeting,Dallas,2003
[4] Nemeth T,Wu C J,Schuster G T.Least-squares migration of incomplete reflection data.Geophysics,1999,64(1) :208～221
[5] Guitton A.Amplitude and kinematic corrections of migrated images for nonunitary imaging operators.Geophysics,2004,69:1017～1024
[6] Sacchi M D,Wang J,Kuehl H.Regularized migration/ inversion:new generation of imaging algorithms.CSEG Recorder,2006,31:54～59
[7] Levenberg K.A method for the solution of certain nonlinear problems in least squares.Quart.Appl.Math.,1944,2:164～166
[8] Marquardt D W.An algorithm for least-squares estimation of nonlinear inequalities.SIAM J.Appl.Math.,1963,11:431～441
[9] 曾锐,刘洪,陈世军等.柯西约束自反褶积技术在井间地震的应用.地球物理学进展,2004,19:1 66～172 Zeng R,Liu H,Chen S J,et al.Application of Cauchyconstraint blind deconvolution in cross hole seismic data processing.Progress in Geophysics (in Chinese),2004,19:166～172
[10] Wang Y F.A restarted conjugate gradient method for ill posed problems.Acta Mathematicae Applicatae Sinica,2003,19:31～40
[11] Wang Y F,Yuan Y X.Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems.Inverse Problems,2005,21:821～838
[12] Tarantola A.Inverse Problems Theory:Methods for Data Fitting and Model Parameter Estimation.Amsterdam:Elsevier,1987
[13] 杨文采.地球物理反演和地震层析成像.北京:地质出版社,1989 Yang W C.Geophysical Inversion and Seismic Tomography (in Chinese).Beijing:Geological Publishing House,1989
[14] Ulrych T J,Sacchi M D,Woodbury A.A Bayesian tour to inversion.Geophysics,2000,66:55～69
[15] 张宏兵,杨长春.正则参数控制下的波阻抗约束反演.地球物理学报,2003,46:827～834 Zhang H B,Yang C C.A constrained impedance inversion method controlled by regularized parameters.Chinese J.Geophys.(in Chinese),2003,46:827～834
[16] Wang Y F,Yang C C,Li X W.A regularizing kernel-based brdf model inversion method for ill-posed land surface parameter retrieval using smoothness constraint.Journal of Geophysical Research,2008,113:D13101
[17] Lanczos C.Linear Differential Operators.New York:Van Nostrand,1961
[18] Trampert J,Leveque J J.Simultaneous iterative reconstruction technique:physical interpretation based on the generalized least squares solution.Journal of Geophysical Research,1990,95:12553～12559
[19] Nolet G.Solving or resolving inadequate noisy tomographic systems.J.Comp.Phys.,1985,61:463～482
[20] Nolet G,Snieder R.Solving large linear inverse problems by projection.Geophys.J.Int.,1990,103:665～568
[21] Murio D A.The Mollification Method and the Numerical Solution of Ill-posed Problems.New York:John Wiley and Sons,1993
[22] Backus G,Gilbert J.Numerical applications of a formalism for geophysical inverse problems.Geophys.J.R.Astron.Soc.,1967,13:247～276
[23] Backus G,Gilbert J.The resolving power of gross earth data.Geophys.J.R.Astron.Soc.,1968,16:169～205
[24] Van Decar J C,Snieder R.Obtaining smooth solutions to large linear inverse problems.Geophysics,1994,59:818～829
[25] 王彦飞.反演问题的计算方法及其应用.北京:高等教育出版社,2007 Wang Y F.Computational Methods for Inverse Problems and Their Applications(in Chinese).Beijing:Higher Education Press,2007
[26] Engl H W,Hanke M,Neubauer A.Regularization of Inverse Problems.Dorcrecht:Kluwer Academic Publishers,1996
[27] Schuster G T,Hu J X.Green's function for migration:Continuous recording geometry.Geophysics,2000,65 (1) :167～175
[28] Lecomte I.Resolution and illumination analyses in PSDM:A ray based approach.The Leading Edge,2008,27:650～663
[29] Tikhonov A N,Arsenin V Y.Solutions of Ill-posed Problems.New York:John Wiley and Sons,1977
[30] 肖庭延,于慎根,王彦飞.反问题的数值解法.北京:科学出版社,2003 Xiao T Y,Yu S G,Wang Y F.Numerical Methods for the Solution of Inverse Problems(in Chinese).Beijing:Science Press,2003
[31] Nashed M Z.Perturbations and approximations for generalized inverses and linear operators equations.In:Nashed,M.Z.Ed.,Generalized Inverses and Applications,1976． 325～396
[32] Schuster G T.Green's functions for migration.Abstracts of Expanded Abstracts 67th SEG meeting,1997． 1754～1758
[33] Stolt R H,Benson A K.Seismic Migration:Theory and Practice,Vol.5,Handbook of Geophysical Exploration,Section I.Seismic Exploration.London:Geophysical Press,1986
[34] Yilmaz O.Seismic Data Processing,Investigations in Geophysics No.2． Society of Exploration Geophysicists,Tulsa,Okla,1987
[35] Hu J X,Schuster G T,Valasek P A.Poststack migration deconvolution.Geophysics,2001,66(3) :939～952
[36] Chen J,Schuster G T.Resolution limits of migrated images.Geophysics,1999,64(4) :1046～1053
[37] Yu J H,Hu J X,Schuster G T,Estill R.Prestack migration deconvolution.Geophysics,71 (2) :S53～S62,2006
[38] 袁亚湘.非线性规划数值方法.上海:上海科学与技术出版社,1993 Yuan Y X.Numerical Methods for Nonlinear Programming (in Chinese).Shanghai:Shanghai Science and Technology Press,1993
[39] Touati Ahmed D,Storey C.Efficient hybrid conjugate gradient techniques.J.Optimization Theory Appl.,1990,64:379～397
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