基于LO范数最小化的地球物理数据稀疏重构(英文)
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摘要
在实际的地球物理数据采集工作中,会因为多方面的客观原因导致数据缺失,对缺失数据进行插值重构是地球物理数据处理和解释的基础问题。基于地球物理数据自身或在变换域内的稀疏性,将地球物理数据的重构转化为稀疏优化问题可提高数据重构的精确度与稳定性。本文建立了LO范数最小化的地球物理数据稀疏重构模型,针对不同规模、不同特征的地球物理数据引入了两种不同类型的LO范数最小优化问题的近似求解算法,即基于LO范数最小化的迭代再加权最小二乘算法与具有快速收敛性的快速迭代硬阈值法。理论分析与数值试验表明,将迭代再加权最小二乘算法应用到位场数据重构中可发挥其收敛速度快,计算时间短,精度高的优势,而快速迭代硬阈值法更适合处理地震数据,相对于传统的迭代硬阈值法计算效率有了很大的提高。
Missing data are a problem in geophysical surveys,and interpolation and reconstruction of missing data is part of the data processing and interpretation.Based on the sparseness of the geophysical data or the transform domain,we can improve the accuracy and stability of the reconstruction by transforming it to a sparse optimization problem.In this paper,we propose a mathematical model for the sparse reconstruction of data based on the L0-norm minimization.Furthermore,we discuss two types of the approximation algorithm for the L0-norm minimization according to the size and characteristics of the geophysical data:namely,the iteratively reweighted least-squares algorithm and the fast iterative hard thresholding algorithm.Theoretical and numerical analysis showed that applying the iteratively reweighted least-squares algorithm to the reconstruction of potential field data exploits its fast convergence rate,short calculation time,and high precision,whereas the fast iterative hard thresholding algorithm is more suitable for processing seismic data,moreover,its computational efficiency is better than that of the traditional iterative hard thresholding algorithm.
Missing data are a problem in geophysical surveys,and interpolation and reconstruction of missing data is part of the data processing and interpretation.Based on the sparseness of the geophysical data or the transform domain,we can improve the accuracy and stability of the reconstruction by transforming it to a sparse optimization problem.In this paper,we propose a mathematical model for the sparse reconstruction of data based on the L0-norm minimization.Furthermore,we discuss two types of the approximation algorithm for the L0-norm minimization according to the size and characteristics of the geophysical data:namely,the iteratively reweighted least-squares algorithm and the fast iterative hard thresholding algorithm.Theoretical and numerical analysis showed that applying the iteratively reweighted least-squares algorithm to the reconstruction of potential field data exploits its fast convergence rate,short calculation time,and high precision,whereas the fast iterative hard thresholding algorithm is more suitable for processing seismic data,moreover,its computational efficiency is better than that of the traditional iterative hard thresholding algorithm.
引文
Beck,A.,and Teboulle,M.,2009,A fast iterative shrinkage-thresholding algorithm for linear inverse problems:Siam J.Imaging Sciences,2(1),183-202.
Bioucas-Dias,J.M.,and Figueiredo,M.A.T.,2007,Anew twist: two-step iterative shrinkage/thresholding algorithms for image restoration: IEEE Transactions OnImage Processing,16(12),2992-3004.
Blumensath,T.,and Davies,M.E.,2010,Normalizediterative hard thresholding: guaranteed stability andperformance: IEEE Journal of Selected Topics in SignalProcessing,4(2),298-309.
Blumensath,T.,and Davies,M.,2009,Iterative hardthresholding for compressed sensing: Applied andComputational Harmonic Analysis,27(3),265-274.
Blumensath,T.,2012,Accelerated iterative hardthresholding: Signal Processing,92,752-756.
Candès,E.,and Donoho,D.,2000,Curvelets—Asurprisingly active nonadaptive representation for objectswith edges: Vanderbilt University Press,Nashville,105-120.
Candès,E.,and Wakin,M.,2008,An introductionto compressive sampling: IEEE Signal ProcessingMagazine,25(2),21-30.
Cao,J.J.,Wang,Y.F.,and Yang,C.C.,2012,Seismicdata restoration based on compressive sensing usingthe regularization and zero-norm sparse optimization:Chinese J.Geophys.(in Chinese) 55(2),596-607.
Donoho,D.L.,and Tsaig,Y.,2006,Extensions ofcompressed sensing: Signal Processing,86(3),533-548.
Donoho,D.L.,2006,Compressed sensing: IEEETransactions on Information Theory,52(4),1289-1306.
Duijndam,A.J.W.,Schonewille,M.A.,and Hindriks,C.O.H.,1999,Reconstruction of band-limited signals,irregularly sampled along one spatial direction :Geophysics,2,524-538.
Dunbar,D.,and Humphreys,G.,2006,A spatial datastructure for poisson-disk sample generation: ACMTransactions on Graphics,25(3),503-508.
Guo,L.H.,Meng,X.H.,and Guo,Z.H.,2005,Griddingmethods of geophysical irregular data in space domain:Geophysical & Geochemical Exploration.(in Chinese),29(5),438-442.
Hennenfen,G.,and Herrmann,F.J.,2008,Simply denoise:wavefield reconstruction via jittered undersampling:Geophysics,73(3),19-28.
Herrmann,J.,and Hennenfent,G.,2008,Nonparametricseismic data recovery with curvelet frames: GeophysicalJournal International,173(1),1-5.
Jiao,L.C.,Yang,S.Y.,and Liu,F.,2011,Developmentand Prospect of Compressive Sensing: Acta ElectronicaSinica,39(7),1651-1662.
Levy,S.,and Fullagar,P.k.,1981,Reconstruction of a sparsespike train from a portion of its spectrum and applicationto high-resolution deconvolution: Geophysics,46(9),1235-1243.
Li,X.,Aravkin,A.Y.,and Leeuwen,T.V.,2012,Fast-randomized full-waveform inversion with compressivesensing: Geophysics,77(3),13-17.
Liu,X.W.,Liu,H.,and Liu,B.,2004,A study on algorithmfor reconstruction of de-alias uneven seismic data:Chinese J.Geophys.(in Chinese),47(2),299-305.
Mallat,S.,and Zhang,Z.,1993,Matching pursuit withtime-frequency dictionaries: IEEE Trans.on SignalProcessing,41(12),3397-3415.
Meng,X.H.,Hou,J.Q.,and Liang,H.Y.,2002,Thefast realization of discrete smooth interpolation inthe interpolation of potential data: Geophysical &Geochemical Exploration (in Chinese),26(4),302-306.
Pei,Y.L.,2009,The method research of sparse constraintdeconvolution and wave impedance inversion: MasterThesis Beijing,China University of Geosciences(Beijing).
Qiu,K.,and Dogandzic,A.,2012,Sparse signalreconstruction via ECME Hard Thresholding: IEEETransactions on Signal Processing,60(9),4551-4569.
Tang,G.,Ma,J.W.,and Yang,H.Z.,2012,Seismicdata denoising overcomplete based on learning-typeovercomplete dictionaries: Applied Geophysics,9(1),27-32.
Tang,G.,and Yang,H.Z.,2010,Seismic data compressionand reconstruction based on poisson disk sampling:Chinese J Geophys (in Chinese),53(9),2181-2188.
Tikhonov,A.N.,and Arsenin,V.Y.,1977,Solutions of illposed problems: John Wiley and Sons,New York.
Trad,D.,Ulrych,T.,and Sacchi,M.,2003,Latest view ofsparse radon transform: Geophysics,68(1),386-399.
Tropp,J.A.,and Gilbert,A.C.,2007,Signal recovery fromrandom measurements via orthogonal matching pursuit:IEEE Transactions on Information Theory,53(12),4655-4666.
Wang,W.Y.,and Qiu,Z.Y.,2011,The research to a stableminimum curvature gridding method in potential dataprocessing: Progress in Geophys.(in Chinese),26(6),2003-2010.
Bioucas-Dias,J.M.,and Figueiredo,M.A.T.,2007,Anew twist: two-step iterative shrinkage/thresholding algorithms for image restoration: IEEE Transactions OnImage Processing,16(12),2992-3004.
Blumensath,T.,and Davies,M.E.,2010,Normalizediterative hard thresholding: guaranteed stability andperformance: IEEE Journal of Selected Topics in SignalProcessing,4(2),298-309.
Blumensath,T.,and Davies,M.,2009,Iterative hardthresholding for compressed sensing: Applied andComputational Harmonic Analysis,27(3),265-274.
Blumensath,T.,2012,Accelerated iterative hardthresholding: Signal Processing,92,752-756.
Candès,E.,and Donoho,D.,2000,Curvelets—Asurprisingly active nonadaptive representation for objectswith edges: Vanderbilt University Press,Nashville,105-120.
Candès,E.,and Wakin,M.,2008,An introductionto compressive sampling: IEEE Signal ProcessingMagazine,25(2),21-30.
Cao,J.J.,Wang,Y.F.,and Yang,C.C.,2012,Seismicdata restoration based on compressive sensing usingthe regularization and zero-norm sparse optimization:Chinese J.Geophys.(in Chinese) 55(2),596-607.
Donoho,D.L.,and Tsaig,Y.,2006,Extensions ofcompressed sensing: Signal Processing,86(3),533-548.
Donoho,D.L.,2006,Compressed sensing: IEEETransactions on Information Theory,52(4),1289-1306.
Duijndam,A.J.W.,Schonewille,M.A.,and Hindriks,C.O.H.,1999,Reconstruction of band-limited signals,irregularly sampled along one spatial direction :Geophysics,2,524-538.
Dunbar,D.,and Humphreys,G.,2006,A spatial datastructure for poisson-disk sample generation: ACMTransactions on Graphics,25(3),503-508.
Guo,L.H.,Meng,X.H.,and Guo,Z.H.,2005,Griddingmethods of geophysical irregular data in space domain:Geophysical & Geochemical Exploration.(in Chinese),29(5),438-442.
Hennenfen,G.,and Herrmann,F.J.,2008,Simply denoise:wavefield reconstruction via jittered undersampling:Geophysics,73(3),19-28.
Herrmann,J.,and Hennenfent,G.,2008,Nonparametricseismic data recovery with curvelet frames: GeophysicalJournal International,173(1),1-5.
Jiao,L.C.,Yang,S.Y.,and Liu,F.,2011,Developmentand Prospect of Compressive Sensing: Acta ElectronicaSinica,39(7),1651-1662.
Levy,S.,and Fullagar,P.k.,1981,Reconstruction of a sparsespike train from a portion of its spectrum and applicationto high-resolution deconvolution: Geophysics,46(9),1235-1243.
Li,X.,Aravkin,A.Y.,and Leeuwen,T.V.,2012,Fast-randomized full-waveform inversion with compressivesensing: Geophysics,77(3),13-17.
Liu,X.W.,Liu,H.,and Liu,B.,2004,A study on algorithmfor reconstruction of de-alias uneven seismic data:Chinese J.Geophys.(in Chinese),47(2),299-305.
Mallat,S.,and Zhang,Z.,1993,Matching pursuit withtime-frequency dictionaries: IEEE Trans.on SignalProcessing,41(12),3397-3415.
Meng,X.H.,Hou,J.Q.,and Liang,H.Y.,2002,Thefast realization of discrete smooth interpolation inthe interpolation of potential data: Geophysical &Geochemical Exploration (in Chinese),26(4),302-306.
Pei,Y.L.,2009,The method research of sparse constraintdeconvolution and wave impedance inversion: MasterThesis Beijing,China University of Geosciences(Beijing).
Qiu,K.,and Dogandzic,A.,2012,Sparse signalreconstruction via ECME Hard Thresholding: IEEETransactions on Signal Processing,60(9),4551-4569.
Tang,G.,Ma,J.W.,and Yang,H.Z.,2012,Seismicdata denoising overcomplete based on learning-typeovercomplete dictionaries: Applied Geophysics,9(1),27-32.
Tang,G.,and Yang,H.Z.,2010,Seismic data compressionand reconstruction based on poisson disk sampling:Chinese J Geophys (in Chinese),53(9),2181-2188.
Tikhonov,A.N.,and Arsenin,V.Y.,1977,Solutions of illposed problems: John Wiley and Sons,New York.
Trad,D.,Ulrych,T.,and Sacchi,M.,2003,Latest view ofsparse radon transform: Geophysics,68(1),386-399.
Tropp,J.A.,and Gilbert,A.C.,2007,Signal recovery fromrandom measurements via orthogonal matching pursuit:IEEE Transactions on Information Theory,53(12),4655-4666.
Wang,W.Y.,and Qiu,Z.Y.,2011,The research to a stableminimum curvature gridding method in potential dataprocessing: Progress in Geophys.(in Chinese),26(6),2003-2010.
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