地震数据压缩重构的正则化与零范数稀疏最优化方法
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摘要
地震数据重构问题是一个病态的反演问题.本文基于地震数据在curvelet域的稀疏性,将地震数据重构变为一个稀疏优化问题,构造0范数的逼近函数作为目标函数,提出了一种投影梯度求解算法.本文还运用最近提出的分段随机采样方式进行采样,该采样方式能够有效地控制采样间隔并且保持采样的随机性.地震数值模拟表明,基于0范数逼近的投影梯度法计算效率有明显的提高;分段随机采样方式比随机欠采样有更加稳定的重构结果.
Seismic data restoration is an ill-posed inverse problem.Based on the sparseness of seismic data in the curvelet domain,this problem can be transformed into a sparse optimization problem.This paper proposes to use the approximation of zero-norm as the objective function and develop a projected gradient method to solve the corresponding minimization problem.We also employ a recently proposed piecewise random sampling method which can both control the sampling gap and keep the randomness of sampling.Numerical results show that the projected gradient method can reduce the amount of computation greatly,and the restoration based on the piecewise random sampling are better than that of random sub-sampling.
Seismic data restoration is an ill-posed inverse problem.Based on the sparseness of seismic data in the curvelet domain,this problem can be transformed into a sparse optimization problem.This paper proposes to use the approximation of zero-norm as the objective function and develop a projected gradient method to solve the corresponding minimization problem.We also employ a recently proposed piecewise random sampling method which can both control the sampling gap and keep the randomness of sampling.Numerical results show that the projected gradient method can reduce the amount of computation greatly,and the restoration based on the piecewise random sampling are better than that of random sub-sampling.
引文
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[4]Duijndam A J W,Schonewille M A,Hindriks C O H.Reconstruction of band-limited signals,irregularly sampledalong one spatial direction.Geophysics,1999,64(2):524-538.
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[8]Gülünary N.Seismic trace interpolation in the Fouriertransform domain.Geophysics,2003,68(1):355-369.
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[11]Ronen J.Wave-equation trace interpolation.Geophysics,1987,52(7):973-984.
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[13]Hennenfent G,Herrmann F J.Random sampling:Newinsights into the reconstruction of coarsely-sampled wavefields.SEG,Expanded Abstracts,2007,26:2575-2579.
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[20]Wang Y F,Fan S F,Feng X.Retrieval of the aerosol particlesize distribution function by incorporating a prioriinformation.Journal of Aerosol Science,2007,38(8):885-901.
[21]Wang Y F.Seismic impedance inversion using l1-normregularization and gradient descent methods.J.Inv.Ill-Posed Problems,2011,18(7):823-838.
[22]Wang Y F,Ma S Q,Yang H,et al.On the effectiveinversion by imposing apriori information for retrieval of landsurface parameters.Science in China D,2009,52(4):540-549.
[23]Wang Y F,Cao J J,Yuan Y X,et al.Regularizing active setmethod for nonnegatively constrained ill-posed multichannelimage restoration problem.Applied Optics,2009,48(7):1389-1401.
[24]Wang Y F,Cao J J,Yang C C.Recovery of seismicwavefields based on compressive sensing by an l1-normconstrained trust region method and the piecewise randomsubsampling.Geophysical Journal International,2011,187(1):199-213.
[25]Mohimani H,Babaie-Zadeh M,Jutten C.A fast approach forovercomplete sparse decomposition based on smoothed l0norm.IEEE Transactions on Signal Processing,2009,57(1):289-301.
[26]Zwartjes P M,Sacchi M D.Fourier reconstruction ofnonuniformly sampled,aliased seismic data.Geophysics,2007,72(1):V21-V32.
[27]van den Berg E,Friedlander M P.Probing the pareto frontierfor basis pursuit solutions.SIAM J.Sci.Comput.,2008,31(2):890-912.
[28]Wang Y F,Yang C C.Accelerating migration deconvolutionusing a nonmonotone gradient method.Geophysics,2010,75(4):S131-S137.
[29]Wang Y F,Xiao T Y.Fast realization algorithms fordetermining regularization parameters in linear inverseproblems.Inverse Problems,2001,17(2):281-291.
[30]Hennenfent G,van den Berg E,Friedlander M P,et al.Newinsights into one-norm solvers from the Pareto curve.Geophysics,2008,73(4):A23-A26.
[31]Lustig M,Donoho D L,Pauly J M.Sparse MRI:theapplication of compressed sensing for rapid MR imaging.Magnetic Resonance in Medicine,2007,58(6):1182-1195.
[32]Hennenfent G,Herrmann F J.Simply denoise:wavefieldreconstruction via jittered undersampling.Geophysics,2008,73(3):V19-V28.
[33]Taylor H L,Banks S C,McCoy J F.Deconvolution with thel1norm.Geophysics,1979,44(1):39-52.
[2]Cao J J,Wang Y F,Zhao J T,et al.A review on restorationof seismic wavefields based on regularization and compressivesensing.Inverse Problems in Science and Engineering,2011,19(5):679-704.
[3]Sacchi M D,Ulrych T J.Estimation of the discrete Fouriertransform,a linear inversion approach.Geophysics,1996,61(4):1128-1136.
[4]Duijndam A J W,Schonewille M A,Hindriks C O H.Reconstruction of band-limited signals,irregularly sampledalong one spatial direction.Geophysics,1999,64(2):524-538.
[5]Liu B.Multi-dimensional reconstruction of seismic data[D].Alberta:University of Alberta,2004.
[6]Sacchi M D,Verschuur D J,Zwartjes P M.Datareconstruction by generalized deconvolution.74th AnnualInternational Meeting,SEG,Expanded Abstracts,2004:1989-1992.
[7]Spitz S.Seismic trace interpolation in the F-X domain.Geophysics,1991,56(6):785-794.
[8]Gülünary N.Seismic trace interpolation in the Fouriertransform domain.Geophysics,2003,68(1):355-369.
[9]Claerbout J F.Earth Soundings Analysis:Processing VersusInversion.Blackwell Science,1992.
[10]Crawley S.Seismic trace interpolation with nonstationaryprediction-error filters[D].Stanford:Stanford University,2000.
[11]Ronen J.Wave-equation trace interpolation.Geophysics,1987,52(7):973-984.
[12]Stolt R H.Seismic data mapping and reconstruction.Geophysics,2002,67(3):890-908.
[13]Hennenfent G,Herrmann F J.Random sampling:Newinsights into the reconstruction of coarsely-sampled wavefields.SEG,Expanded Abstracts,2007,26:2575-2579.
[14]王彦飞.反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 andTheir Applications(in Chinese).Beijing:Higher EducationPress,2007.
[15]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 theSolution of Inverse Problems(in Chinese).Beijing:SciencePress,2003.
[16]袁亚湘,孙文瑜.最优化理论与方法.北京:科学出版社,1997.Yuan Y X,Sun W Y.Theory and Methods for Optimization(in Chinese).Beijing:Science Press,1997.
[17]Candes E J.Compressive sampling.∥Proceedings ofInternational Congress of Mathematicians.Madrid:EuropeanMathematical Society Publishing House,2006:33-52.
[18]Candes E J,Donoho D L.Curvelets:A surprisingly effectivenonadaptive representation of objects with edges.∥Cohen A,Rabut C,Schumaker L L,eds.Curves and Surfaces Fitting:Saint-Malo 1999.Nashville:Vanderbilt University Press,2000:105-120.
[19]Herrity K K,Gilbert A C,Troop J A.Sparse approximationvia iterative thresholding.∥Proceedings of the IEEE InternationalConference on Acoustics,Speech and Signal Proceeding,2006:624-627.
[20]Wang Y F,Fan S F,Feng X.Retrieval of the aerosol particlesize distribution function by incorporating a prioriinformation.Journal of Aerosol Science,2007,38(8):885-901.
[21]Wang Y F.Seismic impedance inversion using l1-normregularization and gradient descent methods.J.Inv.Ill-Posed Problems,2011,18(7):823-838.
[22]Wang Y F,Ma S Q,Yang H,et al.On the effectiveinversion by imposing apriori information for retrieval of landsurface parameters.Science in China D,2009,52(4):540-549.
[23]Wang Y F,Cao J J,Yuan Y X,et al.Regularizing active setmethod for nonnegatively constrained ill-posed multichannelimage restoration problem.Applied Optics,2009,48(7):1389-1401.
[24]Wang Y F,Cao J J,Yang C C.Recovery of seismicwavefields based on compressive sensing by an l1-normconstrained trust region method and the piecewise randomsubsampling.Geophysical Journal International,2011,187(1):199-213.
[25]Mohimani H,Babaie-Zadeh M,Jutten C.A fast approach forovercomplete sparse decomposition based on smoothed l0norm.IEEE Transactions on Signal Processing,2009,57(1):289-301.
[26]Zwartjes P M,Sacchi M D.Fourier reconstruction ofnonuniformly sampled,aliased seismic data.Geophysics,2007,72(1):V21-V32.
[27]van den Berg E,Friedlander M P.Probing the pareto frontierfor basis pursuit solutions.SIAM J.Sci.Comput.,2008,31(2):890-912.
[28]Wang Y F,Yang C C.Accelerating migration deconvolutionusing a nonmonotone gradient method.Geophysics,2010,75(4):S131-S137.
[29]Wang Y F,Xiao T Y.Fast realization algorithms fordetermining regularization parameters in linear inverseproblems.Inverse Problems,2001,17(2):281-291.
[30]Hennenfent G,van den Berg E,Friedlander M P,et al.Newinsights into one-norm solvers from the Pareto curve.Geophysics,2008,73(4):A23-A26.
[31]Lustig M,Donoho D L,Pauly J M.Sparse MRI:theapplication of compressed sensing for rapid MR imaging.Magnetic Resonance in Medicine,2007,58(6):1182-1195.
[32]Hennenfent G,Herrmann F J.Simply denoise:wavefieldreconstruction via jittered undersampling.Geophysics,2008,73(3):V19-V28.
[33]Taylor H L,Banks S C,McCoy J F.Deconvolution with thel1norm.Geophysics,1979,44(1):39-52.
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