基于Curvelet变换的缺失地震数据插值方法
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摘要
Curvelet变换具有局部性、多尺度性和多方向性,在处理非稳态地震信号中具有较大的优势,可以利用Curvelet变换的压缩特性重建缺失的地震数据。本文首先分析了基于稀疏变换的缺失地震数据插值的基本原理,在反问题正则化理论框架下,针对L2范数约束和L1范数约束条件,分析了两种约束的差异,着重阐述了基于Curvelet变换的L1范数约束的插值方法,其优点在于对非线性同相轴不需要分窗口处理,并将凸集投影算法(POCS)引入到Curvelet变换的插值方法中,通过采用按指数规律衰减的阈值参数加快了迭代收敛的速度。理论和实际算例验证了Curvelet变换插值方法的有效性。
Curvelet transform is suitable for nonstationary seismic data processing because it is multi-scale,multidirectional,and strictly localized.The compression attribute of Curvelet transform can be used for reconstruction of missing seismic data.In the paper,we first analyze the principle of missing seismic data interpolation using sparse transform.Then we discuss the difference between the interpolation methods with-norm and-norm restrictions in the inversion framework.We expatiate in detail on the interpolation method with-norm restriction,which do not need sliding windows for processing nonstationary seismic events.We introduce the Projections Onto Convex Sets(POCS) algorithm into the interpolation using Curvelet transform.The threshold in POCS algorithm is chosen to be exponentially decreased,which is a fast convergence method in iterative inversion.Synthetic and real data examples show that the interpolation method using Curvelet transform can effectively rebuild missing seismic data.
Curvelet transform is suitable for nonstationary seismic data processing because it is multi-scale,multidirectional,and strictly localized.The compression attribute of Curvelet transform can be used for reconstruction of missing seismic data.In the paper,we first analyze the principle of missing seismic data interpolation using sparse transform.Then we discuss the difference between the interpolation methods with-norm and-norm restrictions in the inversion framework.We expatiate in detail on the interpolation method with-norm restriction,which do not need sliding windows for processing nonstationary seismic events.We introduce the Projections Onto Convex Sets(POCS) algorithm into the interpolation using Curvelet transform.The threshold in POCS algorithm is chosen to be exponentially decreased,which is a fast convergence method in iterative inversion.Synthetic and real data examples show that the interpolation method using Curvelet transform can effectively rebuild missing seismic data.
引文
[1]Fomel S.Three-Di mensional Seismic Data Regulari-zation[D].Stanford University,2001
[2]Verschuur D J et al.Adaptive surface-related multi-ple eli mination.Geophysics,1992,57(9):1166~1177
[3]Claerbout J F.Basic Earth I maging∥Stanford Explo-ration Project.http:∥sepwww.stanford.edu/sep/prof/bei1005.pdf,1995
[4]Symes W W.Reverse ti me migration with opti malcheckpointing∥Technical Report 06-18,Depart mentof Computational and Applied Mathematics.Rice U-niversity,Houston,Texas,USA,2006
[5]Liu G C,Fomel S and Chen X.Stacking angle-domaincommon-i mage gathers for normalization of illumina-tion.Geophysical Prospecting,2010(in press)
[6]Naghizadeh M.Parametric Reconstruction of Multi-di mensional Seismic Records[D].University of Al-berta,2009
[7]Sacchi M D and Ulrych T J.Esti mation of the dis-crete Fourier transform,a linear inversion approach.Geophysics,1996,61(4):1128~1136
[8]Duijndam AJ W,Schonewille M Aand Hindriks C OH.Reconstruction of band-li mited signals,irregularlysampled along one spatial direction.Geophysics,1999,64(2):524~538
[9]Xu S et al.Antileakage Fourier transformfor seismicdata regularization.Geophysics,2005,70(4):V87~V95
[10]Zwartjes P and Gisolf A.Fourier reconstruction ofmarine-streamer datainfour spatial coordinates.Geo-physics,2006,71(6):V171~V186
[11]Herrmann F J and Hennenfent G.Non-parametricseismic data recovery with curvelet frames.GeophysJ Int,2008,148(1):233~248
[12]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 Shuang-quan,Wang Shang-xu and Ji Min.Seis-mic data interpolation based on signal hi-fi.OGP,2005,40(5):515~517
[13]Ronen J.Wave-equationtraceinterpolation.Geophys-ics,1987,52(7):973~984
[14]Stolt R H.Seismic data mapping and reconstruction.Geophysics,2002,67(3):890~908
[15]Fomel S.Seismic reflection data interpolation withdifferential offset and shot continuation.Geophysics,2003,68(2):733~744
[16]Malcol m A E,Hoop M V de and LeRousseau J H.The applicability of dip moveout/azi muth moveout inthe presence of caustics.Geophysics,2005,70(1):S1~S17
[17]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):518~522,560Wang Wei-hong,Gao Hong-wei and Liu Hong.Seis-mic trace reconstruction by trace equalization parabol-ic Radon transform.OGP,40(5):518~522,560
[18]Claerbout J F.I mage esti mation by example:geo-physical soundings i mage construction∥Stanford Ex-ploration Project.http:∥sepwww.stanford.edu/sep,2008
[19]崔兴福,刘东奇,张关泉.小波变换实现地震道内插.石油地球物理勘探,2003,38(增刊):93~97
[20]Cohen L.Ti me-Frequency Analysis.Prentice Hall,Inc,1995
[21]Candès E J et al.Fast discrete curvelet transforms.Multiscale Modeling and Si mulation,2005,5(3):861~899
[22]Fomel S and Liu Y.Seislet transform and seisletframe.Geophysics,2010,75(3):V25~V38
[23]Naghizadeh Mand Sacchi D.Beyond alias hierarchicalscale curvelet interpolation of regularly and irregular-ly sampled seismic data.Geophysics,2010,75(6):WB189~WB202
[24]Fomel S and Claerbout J.Multidi mensional recursivefilter preconditioning in geophysical esti mation prob-lems.Geophysics,2003,68(2):577~588
[25]Fomel S.Applications of plane-wave destruction fil-ters.Geophysics,2002,67(6):1946~1960
[26]Crawley S,Claerbout J and Clapp R.Interpolationwith smoothly nonstationary prediction-error filters.SEG Technical Program Expanded Abstracts,1999,18:1154~1157
[27]Liu B and Sacchi M D.Mini mum weighted normin-terpolation of seismic records.Geophysics,2004,69(6):1560~1568
[28]Fomel S.Shaping regularization in geophysical-esti-mation problems.Geophysics,2007,72(2):R29~R36
[29]Hennenfent Get al.Newinsights into one-normsolv-ers fromthe Pareto curve.Geophysics,2008,73(4):A23~A26
[30]Bregman L.The method of successive projection forfinding a common point of convex sets.Soviet MathDokl,1965,6(2):688~692
[31]Candès E J and Romberg J.Practical signal recoveryfrom random projections∥Wavelet Applications inSignal and I mage ProcessingⅪ,Proc SPIE Conf,5914,2005,San Diego
[32]Abma R and Kabir N.3D interpolation of irregulardata with a POCS algorithm.Geophysics,2006,71(6):E91~E97
[33]Donoho D L.De-noising by soft-thresholding.IEEETransaction on Information Theory,1995,41(3):613~627
[34]Hennenfent G and Herrmann F.Sparseness-con-strained data continuation with frames:Applicationsto missing traces and aliased signals in 2/3-D.SEGTechnical Program Expanded Abstracts,2005,24:2162~2165
[35]Gao J et al.Irregular seismic data reconstructionbased on POCS method.Applied Geophysics,2010,7(3):229~238
[36]Candès E,Demanet L.Curvelets and Fourier integraloperators.C R Acad Sci Paris,2003,336(5):395~398
[37]Douma H and Hoop M.Leading-order seismic i ma-ging using curvelets.Geophysics,2007,72(6):S231~S248
[38]Herrmann F J et al.Curvelet-based seismic data pro-cessing:A multiscale and nonlinear approach.Geo-physics,2008,73(1):A1~A5
[39]Candès EJ and Donoho D L.Newtight frames of cur-velets and opti mal representations of objects withpiecewise-c2 singularities.Comm on Pure and ApplMath,2004,57(2):219~266
[40]Hennenfent G and Herrmann F J.Si mply denoise:Wavefield reconstruction via jittered undersampling.Geophysics,2008,73(3):V19~V28
[2]Verschuur D J et al.Adaptive surface-related multi-ple eli mination.Geophysics,1992,57(9):1166~1177
[3]Claerbout J F.Basic Earth I maging∥Stanford Explo-ration Project.http:∥sepwww.stanford.edu/sep/prof/bei1005.pdf,1995
[4]Symes W W.Reverse ti me migration with opti malcheckpointing∥Technical Report 06-18,Depart mentof Computational and Applied Mathematics.Rice U-niversity,Houston,Texas,USA,2006
[5]Liu G C,Fomel S and Chen X.Stacking angle-domaincommon-i mage gathers for normalization of illumina-tion.Geophysical Prospecting,2010(in press)
[6]Naghizadeh M.Parametric Reconstruction of Multi-di mensional Seismic Records[D].University of Al-berta,2009
[7]Sacchi M D and Ulrych T J.Esti mation of the dis-crete Fourier transform,a linear inversion approach.Geophysics,1996,61(4):1128~1136
[8]Duijndam AJ W,Schonewille M Aand Hindriks C OH.Reconstruction of band-li mited signals,irregularlysampled along one spatial direction.Geophysics,1999,64(2):524~538
[9]Xu S et al.Antileakage Fourier transformfor seismicdata regularization.Geophysics,2005,70(4):V87~V95
[10]Zwartjes P and Gisolf A.Fourier reconstruction ofmarine-streamer datainfour spatial coordinates.Geo-physics,2006,71(6):V171~V186
[11]Herrmann F J and Hennenfent G.Non-parametricseismic data recovery with curvelet frames.GeophysJ Int,2008,148(1):233~248
[12]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 Shuang-quan,Wang Shang-xu and Ji Min.Seis-mic data interpolation based on signal hi-fi.OGP,2005,40(5):515~517
[13]Ronen J.Wave-equationtraceinterpolation.Geophys-ics,1987,52(7):973~984
[14]Stolt R H.Seismic data mapping and reconstruction.Geophysics,2002,67(3):890~908
[15]Fomel S.Seismic reflection data interpolation withdifferential offset and shot continuation.Geophysics,2003,68(2):733~744
[16]Malcol m A E,Hoop M V de and LeRousseau J H.The applicability of dip moveout/azi muth moveout inthe presence of caustics.Geophysics,2005,70(1):S1~S17
[17]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):518~522,560Wang Wei-hong,Gao Hong-wei and Liu Hong.Seis-mic trace reconstruction by trace equalization parabol-ic Radon transform.OGP,40(5):518~522,560
[18]Claerbout J F.I mage esti mation by example:geo-physical soundings i mage construction∥Stanford Ex-ploration Project.http:∥sepwww.stanford.edu/sep,2008
[19]崔兴福,刘东奇,张关泉.小波变换实现地震道内插.石油地球物理勘探,2003,38(增刊):93~97
[20]Cohen L.Ti me-Frequency Analysis.Prentice Hall,Inc,1995
[21]Candès E J et al.Fast discrete curvelet transforms.Multiscale Modeling and Si mulation,2005,5(3):861~899
[22]Fomel S and Liu Y.Seislet transform and seisletframe.Geophysics,2010,75(3):V25~V38
[23]Naghizadeh Mand Sacchi D.Beyond alias hierarchicalscale curvelet interpolation of regularly and irregular-ly sampled seismic data.Geophysics,2010,75(6):WB189~WB202
[24]Fomel S and Claerbout J.Multidi mensional recursivefilter preconditioning in geophysical esti mation prob-lems.Geophysics,2003,68(2):577~588
[25]Fomel S.Applications of plane-wave destruction fil-ters.Geophysics,2002,67(6):1946~1960
[26]Crawley S,Claerbout J and Clapp R.Interpolationwith smoothly nonstationary prediction-error filters.SEG Technical Program Expanded Abstracts,1999,18:1154~1157
[27]Liu B and Sacchi M D.Mini mum weighted normin-terpolation of seismic records.Geophysics,2004,69(6):1560~1568
[28]Fomel S.Shaping regularization in geophysical-esti-mation problems.Geophysics,2007,72(2):R29~R36
[29]Hennenfent Get al.Newinsights into one-normsolv-ers fromthe Pareto curve.Geophysics,2008,73(4):A23~A26
[30]Bregman L.The method of successive projection forfinding a common point of convex sets.Soviet MathDokl,1965,6(2):688~692
[31]Candès E J and Romberg J.Practical signal recoveryfrom random projections∥Wavelet Applications inSignal and I mage ProcessingⅪ,Proc SPIE Conf,5914,2005,San Diego
[32]Abma R and Kabir N.3D interpolation of irregulardata with a POCS algorithm.Geophysics,2006,71(6):E91~E97
[33]Donoho D L.De-noising by soft-thresholding.IEEETransaction on Information Theory,1995,41(3):613~627
[34]Hennenfent G and Herrmann F.Sparseness-con-strained data continuation with frames:Applicationsto missing traces and aliased signals in 2/3-D.SEGTechnical Program Expanded Abstracts,2005,24:2162~2165
[35]Gao J et al.Irregular seismic data reconstructionbased on POCS method.Applied Geophysics,2010,7(3):229~238
[36]Candès E,Demanet L.Curvelets and Fourier integraloperators.C R Acad Sci Paris,2003,336(5):395~398
[37]Douma H and Hoop M.Leading-order seismic i ma-ging using curvelets.Geophysics,2007,72(6):S231~S248
[38]Herrmann F J et al.Curvelet-based seismic data pro-cessing:A multiscale and nonlinear approach.Geo-physics,2008,73(1):A1~A5
[39]Candès EJ and Donoho D L.Newtight frames of cur-velets and opti mal representations of objects withpiecewise-c2 singularities.Comm on Pure and ApplMath,2004,57(2):219~266
[40]Hennenfent G and Herrmann F J.Si mply denoise:Wavefield reconstruction via jittered undersampling.Geophysics,2008,73(3):V19~V28
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