基于jitter采样和曲波变换的三维地震数据重建
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摘要
传统的地震勘探数据采样必须遵循奈奎斯特采样定理,而野外数据采样可能由于地震道缺失或者勘探成本限制,不一定满足采样定理要求,因此存在数据重建问题.本文基于压缩感知理论,利用随机欠采样方法将传统规则欠采样所带来的互相干假频转化成较低幅度的不相干噪声,从而将数据重建问题转为更简单的去噪问题.在数据重建过程中引入凸集投影算法(POCS),提出采用e-(x~(1/2))(0≤x≤1)衰减规律的阈值参数,构建基于曲波变换三维地震数据重建技术.同时针对随机采样的不足,引入jitter采样方式,在保持随机采样优点的同时控制采样间隔.数值试验表明,基于曲波变换的重建效果优于傅里叶变换,jitter欠采样的重建效果优于随机欠采样,最后将该技术应用于实际地震勘探资料,获得较好的应用效果.
Traditional seismic data sampling must follow the Nyquist sampling theorem,while the field data acquisition can′t meet the sampling theorem due to missing traces or exploration cost limit,so there exits data reconstruction problem.In this paper,based on the theory of compressed sensing,we render coherent aliases of regular under-sampling into harmless incoherent random noise using the random under-sampling,effectively turning the reconstruction problem into a much simpler de-noising problem.We introduce the Projections Onto Convex Sets(POCS) algorithm during the process of reconstruction,choosing the square root exponentially decreased threshold,constructing a curvelet-based recovery strategy of 3D seismic data.At the same time,aiming at the deficiency of simple random under-sampling,we introduce the jittered under-sampling,it shares the benefits of random sampling and controls the maximum gap size.Experiments show that reconstruction effect based on curvelet is better than FFT transform and jittered under-sampling is better than random under-sampling.At last,we apply this technology into practical seismic data and obtain a good application.
Traditional seismic data sampling must follow the Nyquist sampling theorem,while the field data acquisition can′t meet the sampling theorem due to missing traces or exploration cost limit,so there exits data reconstruction problem.In this paper,based on the theory of compressed sensing,we render coherent aliases of regular under-sampling into harmless incoherent random noise using the random under-sampling,effectively turning the reconstruction problem into a much simpler de-noising problem.We introduce the Projections Onto Convex Sets(POCS) algorithm during the process of reconstruction,choosing the square root exponentially decreased threshold,constructing a curvelet-based recovery strategy of 3D seismic data.At the same time,aiming at the deficiency of simple random under-sampling,we introduce the jittered under-sampling,it shares the benefits of random sampling and controls the maximum gap size.Experiments show that reconstruction effect based on curvelet is better than FFT transform and jittered under-sampling is better than random under-sampling.At last,we apply this technology into practical seismic data and obtain a good application.
引文
[1]Herrmann F J.Randomized sampling and sparsity:Gettingmore information from fewer samples.Geophysics,2010,75(6):WB173-WB187.
[2]Trad D O.Five-dimensional interpolation:Recovering fromacquisition constraints.Geophysics,2009,74(6):V123-V132.
[3]马坚伟,唐刚,汤文.基本曲波变换和压缩感知的不完备地震数据恢复.长沙:中国地球物理年会,2010.Ma J W,Tang G,Tang W.Recovery of incomplete seismicdata based on curvelet transform and compressed sensing(inChinese).Changsha:Chinese Geophysical Year Meeting,2010.
[4]孟小红,郭良辉,张致付等.基于非均匀快速傅里叶变换的最小二乘反演地震数据重建.地球物理学报,2008,51(1):235-241.Meng X H,Guo L H,Zhang Z F,et al.Reconstruction ofseismic data with least squares inversion based on nonuniformfast Fourier transform.Chinese J.Geophys.(in Chinese),2008,51(1):235-241.
[5]Donoho D L.Compressed sensing.IEEE Transactions onInformation Theory,2006,52(4):1289-1306.
[6]Herrmann F J,Hennenfent G.Non-parametric seismic datarecovery with curvelet frames.Geophysical JournalInternational,2008,173(1):233-248.
[7]Herrmann F J,Wang D,Hennenfent G,et al.Curvelet-based seismic data processing:A multiscale and nonlinearapproach.Geophysics,2008,73(1):A1-A5.
[8]Neelamani R,Baumstein A,Gillard D,et al.Coherent andrandom noise attenuation using the curvelet transform.TheLeading Edge,2008,27(2):240-248.
[9]Wason H,Herrmann F J,Tim T Y.Sparsity-promotingrecovery from simultaneous data:a compressive sensingapproach.81thAnnual International Meeting of Society ofExploration Geophysicists,2011:6-10.
[10]唐刚,杨慧珠.基于泊松碟采样的地震数据压缩重建.地球物理学报,2010,53(9):2181-2188.Tang G,Yang H Z.Seismic data compression andreconstruction based on Poisson disk sampling.Chinese J.Geophys.(in Chinese),2010,53(9):2181-2188.
[11]Leneman O.Random sampling of random processes:Impulseresponse.Information and Control,1966,9(2):347-363.
[12]Hennenfent G,Herrmann F J.Simply denoise:wavefieldreconstruction via jittered undersampling.Geophysics,2008,73(3):V19-V28.
[13]唐刚.基于压缩感知和稀疏表示的地震数据重建与去噪[博士论文].北京:清华大学,2010.Tang G.Seismic data reconstruction and denoising based oncompressive sensing and sparse representation[Ph.D.thesis](in Chinese).Beijing:Tsinghua University,2010.
[14]Trad D,Ulryeh T,Sacchi M.Latest views of the sparseRadon transform.Geophysics,2003,68(1):386-399.
[15]Wang J F,Ng M,Perz M.Seismic data interpolation bygreedy local Radon transform.Geophysics,2010,75(6):225-234.
[16]Jin S.5Dseismic data regularization by a damped least-normFourier inversion.Geophysics,2010,75(6):103-111.
[17]刘国昌,陈小宏,郭志峰等.基于curvelet变换的缺失地震数据插值方法.石油地球物理勘探,2011,46(2):237-245.Liu G C,Chen X H,Guo Z F,et al.Missing seismic datarebuilding by interpolation based on curvelet transform.OilGeophysical Prospecting(in Chinese),2011,46(2):237-245.
[18]Xu S,Zhang Y,Pham D,et al.Antileakage Fouriertransform for seismic data regularization.Geophysics,2005,70(4):87-95.
[19]Abma R,Kabir N.3Dinterpolation of irregular data with aPOCS algorithm.Geophysics,2006,71(5):E91-E97.
[20]Gao J J,Chen X H,Li J Y,et al.Irregular seismic datareconstruction based on exponential threshold model of POCSmethod.Applied Geophysics,2010,7(3):229-238.
[21]Berg E D,Friedlander M P.Probing the Pareto frontier forbasis pursuit solutions.SIAM J.Sci.Comput.,2008,31(2):890-912.
[22]Bregman L.The method of successive projection for finding acommon point of convex sets.Soviet Math,1965,6(3):688-692.
[23]Ozkan M K,Tekalp A M,Sezan M I.POCS basedrestoration of space-varying blurred images.IEEETransactions on Image Processing,1994,3(4):450-454.
[24]Candès E,Demanet L,Donoho D,et al.Fast discretecurvelet transforms.SIAM Multiscale Modeling andSimulation,2006,5(3):861-899.
[25]Ma J,Plonka G.The curvelet transform.IEEE SignalProcessing Magazine,2010,27(2):118-133.
[2]Trad D O.Five-dimensional interpolation:Recovering fromacquisition constraints.Geophysics,2009,74(6):V123-V132.
[3]马坚伟,唐刚,汤文.基本曲波变换和压缩感知的不完备地震数据恢复.长沙:中国地球物理年会,2010.Ma J W,Tang G,Tang W.Recovery of incomplete seismicdata based on curvelet transform and compressed sensing(inChinese).Changsha:Chinese Geophysical Year Meeting,2010.
[4]孟小红,郭良辉,张致付等.基于非均匀快速傅里叶变换的最小二乘反演地震数据重建.地球物理学报,2008,51(1):235-241.Meng X H,Guo L H,Zhang Z F,et al.Reconstruction ofseismic data with least squares inversion based on nonuniformfast Fourier transform.Chinese J.Geophys.(in Chinese),2008,51(1):235-241.
[5]Donoho D L.Compressed sensing.IEEE Transactions onInformation Theory,2006,52(4):1289-1306.
[6]Herrmann F J,Hennenfent G.Non-parametric seismic datarecovery with curvelet frames.Geophysical JournalInternational,2008,173(1):233-248.
[7]Herrmann F J,Wang D,Hennenfent G,et al.Curvelet-based seismic data processing:A multiscale and nonlinearapproach.Geophysics,2008,73(1):A1-A5.
[8]Neelamani R,Baumstein A,Gillard D,et al.Coherent andrandom noise attenuation using the curvelet transform.TheLeading Edge,2008,27(2):240-248.
[9]Wason H,Herrmann F J,Tim T Y.Sparsity-promotingrecovery from simultaneous data:a compressive sensingapproach.81thAnnual International Meeting of Society ofExploration Geophysicists,2011:6-10.
[10]唐刚,杨慧珠.基于泊松碟采样的地震数据压缩重建.地球物理学报,2010,53(9):2181-2188.Tang G,Yang H Z.Seismic data compression andreconstruction based on Poisson disk sampling.Chinese J.Geophys.(in Chinese),2010,53(9):2181-2188.
[11]Leneman O.Random sampling of random processes:Impulseresponse.Information and Control,1966,9(2):347-363.
[12]Hennenfent G,Herrmann F J.Simply denoise:wavefieldreconstruction via jittered undersampling.Geophysics,2008,73(3):V19-V28.
[13]唐刚.基于压缩感知和稀疏表示的地震数据重建与去噪[博士论文].北京:清华大学,2010.Tang G.Seismic data reconstruction and denoising based oncompressive sensing and sparse representation[Ph.D.thesis](in Chinese).Beijing:Tsinghua University,2010.
[14]Trad D,Ulryeh T,Sacchi M.Latest views of the sparseRadon transform.Geophysics,2003,68(1):386-399.
[15]Wang J F,Ng M,Perz M.Seismic data interpolation bygreedy local Radon transform.Geophysics,2010,75(6):225-234.
[16]Jin S.5Dseismic data regularization by a damped least-normFourier inversion.Geophysics,2010,75(6):103-111.
[17]刘国昌,陈小宏,郭志峰等.基于curvelet变换的缺失地震数据插值方法.石油地球物理勘探,2011,46(2):237-245.Liu G C,Chen X H,Guo Z F,et al.Missing seismic datarebuilding by interpolation based on curvelet transform.OilGeophysical Prospecting(in Chinese),2011,46(2):237-245.
[18]Xu S,Zhang Y,Pham D,et al.Antileakage Fouriertransform for seismic data regularization.Geophysics,2005,70(4):87-95.
[19]Abma R,Kabir N.3Dinterpolation of irregular data with aPOCS algorithm.Geophysics,2006,71(5):E91-E97.
[20]Gao J J,Chen X H,Li J Y,et al.Irregular seismic datareconstruction based on exponential threshold model of POCSmethod.Applied Geophysics,2010,7(3):229-238.
[21]Berg E D,Friedlander M P.Probing the Pareto frontier forbasis pursuit solutions.SIAM J.Sci.Comput.,2008,31(2):890-912.
[22]Bregman L.The method of successive projection for finding acommon point of convex sets.Soviet Math,1965,6(3):688-692.
[23]Ozkan M K,Tekalp A M,Sezan M I.POCS basedrestoration of space-varying blurred images.IEEETransactions on Image Processing,1994,3(4):450-454.
[24]Candès E,Demanet L,Donoho D,et al.Fast discretecurvelet transforms.SIAM Multiscale Modeling andSimulation,2006,5(3):861-899.
[25]Ma J,Plonka G.The curvelet transform.IEEE SignalProcessing Magazine,2010,27(2):118-133.
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