形态分量分析在地震数据重建中的应用
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摘要
本文从稀疏信号恢复理论出发,采用形态分量分析(MCA)方法重建地震数据。MCA方法的核心是选取合适的字典。首先从地震数据的特点和计算复杂性出发,选取非抽样小波变换(UWT)字典和曲波变换(Curve-let)字典,UWT字典用来稀疏表示地震数据的局部奇异部分,Curvelet字典用来稀疏表示地震数据平滑和线状变化部分;其次将数据分解为形态特征不同的两个分量,采用BCR(Block Coordinate Relaxation)算法求解目标函数;最后对两个分量进行插值重建、合并得到最终的重建结果。模型测试和实际资料处理结果表明:利用MCA方法不仅可以对非均匀和大间距数据进行插值重建,而且可消除空间假频;同时该方法本身还具有去噪功能,不受数据带宽的限制。
According to the theory of sparse signal recovery,morphological component analysis(MCA) method is used to reconstruct the seismic data in this paper.The key of MCA method is to select the appropriate dictionaries.In view of the characteristics of seismic data and computational complexity,two kinds of dictionaries are selected,that is the undecimated wavelet transform(UWT) dictionary and curvelet transform dictionary.One sparsely represents for local singular part of seismic data,the other sparsely represents for smooth and linear part of seismic data.BCR(Block Coordinate Relaxation) algorithm is used to solve the objective function,and seismic data is decomposed into two morphologically different components.Then the reconstruction results are obtained by summing the components after interpolated and inpainted.Model testing and real data processing show that MCA method can be used not only for reconstruction large spacing uneven data,but also for space aliasing elimination.At the same time,the method itself has the denoising effect without bandwidth restriction.
According to the theory of sparse signal recovery,morphological component analysis(MCA) method is used to reconstruct the seismic data in this paper.The key of MCA method is to select the appropriate dictionaries.In view of the characteristics of seismic data and computational complexity,two kinds of dictionaries are selected,that is the undecimated wavelet transform(UWT) dictionary and curvelet transform dictionary.One sparsely represents for local singular part of seismic data,the other sparsely represents for smooth and linear part of seismic data.BCR(Block Coordinate Relaxation) algorithm is used to solve the objective function,and seismic data is decomposed into two morphologically different components.Then the reconstruction results are obtained by summing the components after interpolated and inpainted.Model testing and real data processing show that MCA method can be used not only for reconstruction large spacing uneven data,but also for space aliasing elimination.At the same time,the method itself has the denoising effect without bandwidth restriction.
引文
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[12]Starck J L,Murtagh F and Fadili M J.Sparse Imageand Signal Processing:Wavelets,Curvelets,Morpho-logical Diversity.Cambridge University Press,2010
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[15]Herrmann F J,Moghaddam P P and Stolk C C.Spar-sity-and continuity-promoting seismic image recoverywith curvelet frames.Applied and ComputationalHarmonic Analysis,2008,24(2):150～173
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[20]Fadili M J,Starck J L and Murtagh F.Inpainting andzooming using sparse representations.The ComputerJournal,2007,52(1):64～79
[21]Sardy S,Bruce A and Tseng P.Block coordinate re-laxation methods for nonparametric wavelet denois-ing.Journal of Computational and Graphical Sta-tistics,2000,9(2):361～379
[22]Tseng P.Convergence of a block coordinate descentmethod for nondifferentiable minimizations.Journalof Optimization Theory and Applications,2001,109(3):475～494
[23]Elad M,Starck J L,Querre P and Donoho D L.Simu-lataneous cartoon and texture image inpainting usingmorphological component analysis(MCA).Appliedand Computational Harmonic Analysis,2005,19(3):340～358
[24]Starck J L,Fadili M J and Murtagh F.The Undeci-mated wavelet decomposition and its reconstruction.IEEE Transactions on Image Processing,2007,16(2):297～309
[25]Candes E J,Demanet L,Donoho D L et al.Fast dis-crete curvelet transforms.Multiscale Modeling andSimulation,2006,5(3):861～899
[26]郑静静,印兴耀,张广智.基于Curvelet变换的多尺度分析技术.石油地球物理勘探,2009,44(5):543～547Zheng Jingjing,Yin Xingyao and Zhang Guangzhi.Multi-scale analysis technique based on curvelettransform.OGP,2009,44(5):543～547
[27]Candes E J and Donoho D L.Recovering edges in ill-posed inverse problems:Optimality of curveletframes.Ann Statist,2002,30:784～842
[27]La Rivière P J and Pan X.Fourier-based approach tointerpolation in single-slice helical computed tomo-graphy.Medical Physics,2001,28(3):381～392
[2]Spitz S.Seismic trace interpolation in the F-X do-main.Geophysics,1991,56(6):785～794
[3]孟小红,刘国峰,周建军.大间距地震数据重建方法研究.地球物理学进展,2006,21(3):687～691Meng Xiaohong,Liu Guofeng and Zhou Jianjun.Thestudy of reconstruction of large gap seismic data.Progress in Geophysics,2006,21(3):687～691
[4]刘喜武,刘洪,刘彬.反假频非均匀地震数据重建方法研究.地球物理学报,2004,47(2):299～305Liu Xiwu,Liu Hong,Liu Bin.A study on algorithmfor reconstruction of de-alias uneven seismic data.Chinese Journal of Geophysics,2004,47(2):299～305
[5]Spitz S.3-D seismic interpolation in the f-x-y do-main.SEG Technical Program Expanded Ab-stracts,1990,9:1641～1643
[6]Fomel S.Seismic reflection data interpolation withdifferential offset and shot continuation.Geophysics,2006,68(2):733～744
[7]陈双全,王尚旭,季敏.基于信号保真的地震数据插值.石油地球物理勘探,2005,40(5):515～517Chen Shuangquan,Wang Shangxu and Ji Min.Seismicdata interpolation based on signal h-ifi.OGP,2005,40(5):515～517
[8]王维红,高红伟,刘洪.道均衡抛物线Radon变换法地震道重建.石油地球物理勘探,2005,40(5):518～522Wang Weihong,Gao Hongwei and Liu Hong.Seismictrace reconstruction by trace equalization parabolicRadon transform.OGP,2005,40(5):518～522
[9]李信富,李小凡.地震数据重建方法原理及运用.物探化探计算技术,2008,30(5):357～362Li Xinfu and Li Xiaofan.Comparison and theory ofreconstruction method for seismic data.ComputingTechniques for Geophysical and Geochemical Explo-ration,2008,30(5):357～362
[10]Canning A and Gerald G H F.Regularizing 3-D data-sets with DMO.Geophysics,1996,61(4):1103～1114
[11]Ronen J.Wave equation trace interpolation.Geophy-sics,1987,52(7):973～984
[12]Starck J L,Murtagh F and Fadili M J.Sparse Imageand Signal Processing:Wavelets,Curvelets,Morpho-logical Diversity.Cambridge University Press,2010
[13]Moghaddam P P.Sparsity and continuity enhancingseismic imaging.2007 CSPG CSEG Convention,2007:404～408
[14]Ma Jianwei,Plonka G and Chauris H.A new sparserepresentation of seismic data using adaptive easy-path wavelet transform.Geoscience and Remote Sens-ing Letters,2010,7(3):540～544
[15]Herrmann F J,Moghaddam P P and Stolk C C.Spar-sity-and continuity-promoting seismic image recoverywith curvelet frames.Applied and ComputationalHarmonic Analysis,2008,24(2):150～173
[16]Felix J H,Deli W et al.Curvelet-based seismic dataprocessing:A multiscale and nonlinear approach.Geophysics,2008,73(1):A1～A5
[17]Candes E,Romberg J,Tao T.Robust uncertaintyprinciples:Exact signal reconstruction from highly in-complete frequency information.IEEE Transactionson Information Theory,2006,52(2):489～509
[18]Starck J L,Elad M and Donoho D L.Redundant mul-tiscale transform and their application for morphologi-cal component analysis.Advance in Imaging and E-lectron Physics,2004,132(82):278～348
[19]Starck J L,Elad M and Donoho D L.Image decompo-sition via the combination of sparse representation anda variational approach.IEEE Transactions on ImageProcessing,2005,14(10):1570～1582
[20]Fadili M J,Starck J L and Murtagh F.Inpainting andzooming using sparse representations.The ComputerJournal,2007,52(1):64～79
[21]Sardy S,Bruce A and Tseng P.Block coordinate re-laxation methods for nonparametric wavelet denois-ing.Journal of Computational and Graphical Sta-tistics,2000,9(2):361～379
[22]Tseng P.Convergence of a block coordinate descentmethod for nondifferentiable minimizations.Journalof Optimization Theory and Applications,2001,109(3):475～494
[23]Elad M,Starck J L,Querre P and Donoho D L.Simu-lataneous cartoon and texture image inpainting usingmorphological component analysis(MCA).Appliedand Computational Harmonic Analysis,2005,19(3):340～358
[24]Starck J L,Fadili M J and Murtagh F.The Undeci-mated wavelet decomposition and its reconstruction.IEEE Transactions on Image Processing,2007,16(2):297～309
[25]Candes E J,Demanet L,Donoho D L et al.Fast dis-crete curvelet transforms.Multiscale Modeling andSimulation,2006,5(3):861～899
[26]郑静静,印兴耀,张广智.基于Curvelet变换的多尺度分析技术.石油地球物理勘探,2009,44(5):543～547Zheng Jingjing,Yin Xingyao and Zhang Guangzhi.Multi-scale analysis technique based on curvelettransform.OGP,2009,44(5):543～547
[27]Candes E J and Donoho D L.Recovering edges in ill-posed inverse problems:Optimality of curveletframes.Ann Statist,2002,30:784～842
[27]La Rivière P J and Pan X.Fourier-based approach tointerpolation in single-slice helical computed tomo-graphy.Medical Physics,2001,28(3):381～392
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