地震时频分析的加权l_1范数稀疏正则化及交替方向乘子算法
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摘要
地震时频分析在地震信号处理中具有重要意义.本文研究一种基于反演的稀疏算法来对反射地震记录进行时频分析.首先使用窗口逆Fourier变换来形成正演问题,然后建立一个加权l_1范数约束的最小化模型,用于求解未知模型参数向量(Fourier频率域系数).为了实现最小化问题,本文提出应用加权交替方向乘子法(ADMM)进行求解.数值试验部分针对短时Fourier变换(STFT)、连续小波变换(CWT)和本文提出的算法进行了对比结果分析.从比较结果可以看出,本文提出的优化模型和相关算法可以得到比STFT和CWT更高分辨率的地震数据的频谱分解.
Seismic time-frequency analysis is of great importance in seismic signal processing. We study a sparse inversion-based algorithm for computing the time-frequency analysis of reflection seismograms. We first formulate the forward problem using the windowed inverse Fourier transform, and then we establish a weighted l_1-norm constrained minimization model for solving the unknown model parameter vector(the Fourier frequency coefficients). To realize the minimization problem, an alternating directions method of multipliers(ADMM) is applied. Numerical experiments based on the well-known short time Fourier transform(STFT), the continuous wavelet transform(CWT) and the proposed algorithm are analyzed. It indicates from the comparison results that the proposed model and the related algorithm can produce a spectral decomposition of the seismic data with high resolution than that of the STFT and CWT.
Seismic time-frequency analysis is of great importance in seismic signal processing. We study a sparse inversion-based algorithm for computing the time-frequency analysis of reflection seismograms. We first formulate the forward problem using the windowed inverse Fourier transform, and then we establish a weighted l_1-norm constrained minimization model for solving the unknown model parameter vector(the Fourier frequency coefficients). To realize the minimization problem, an alternating directions method of multipliers(ADMM) is applied. Numerical experiments based on the well-known short time Fourier transform(STFT), the continuous wavelet transform(CWT) and the proposed algorithm are analyzed. It indicates from the comparison results that the proposed model and the related algorithm can produce a spectral decomposition of the seismic data with high resolution than that of the STFT and CWT.
引文
1 Peyton L,Bottjer R,Partyka G.Interpretation of incised valleys using new 3-D seismic techniques:A case history using spectral decomposition and coherency.Leading Edge,1998,17:1294-1298
2 Partyka G,Gridley J,Lopez J.Interpretational applications of spectral decomposition in reservoir characterization.Leading Edge,1999,18:353-360
3 Chopra S,Castagna J P,Portniaguine O.Seismic resolution and thin-bed reflectivity inversion.Canad Soc Explorat Geophys Recorder,2006,31:19-25
4 Chopra S,Castagna J P,Portniaguine O.Thin-bed reflectivity inversion.In:Proceedings of the 75th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.New Orleans:Society of Exploration Geophysicists(SEG),2006,2057-2061
5 Partyka G A.Spectral decomposition.SEG distinguished lecture.Http://www.seg.org/education/lectures-courses/distinguished-lecturers/spring2005/partykaabstract,2005
6 Portniaguine O,Castagna J P.Inverse spectral decomposition.In:Proceedings of the 74th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.Denver:Society of Exploration Geophysicists(SEG),2004,1786-1789
7 Portniaguine O,Castagna J P.Spectral inversion.In:Proceedings of the 75th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.Houston:Society of Exploration Geophysicists(SEG),2005,1638-1641
8 Puryear C I,Castagna J P.An algorithm for calculation of bed thickness and reflection coefficients from amplitude spectrum.In:Proceedings of the 76th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.New Orleans:Society of Exploration Geophysicists(SEG),2006,1767-1770
9 Marfurt K J,Kirlin R L.Narrow-band spectral analysis and thin-bed tuning.Geophysics,2001,66:1274-1283
10 Castagna J P,Sun S,Siegfried R W.Instantaneous spectral analysis:Detection of low-frequency shadows associated with hydrocarbons.Leading Edge,2003,22:120-127
11 Sinha S,Routh P S,Anno P D,et al.Spectral decomposition of seismic data with continuous-wavelet transform.Geophysics,2005,70:P19-P25
12 Puryear C I,Castagna J P.Layer-thickness determination and stratigraphic interpretation using spectral inversion:Theory and application.Geophysics,2008,73:R37-R48
13 Puryear C I,Portniaguine O N,Cobos C M,et al.Constrained least-squares spectral analysis:Application to seismic data.Geophysics,2012,77:V143-V167
14 Herrmann P,Mojesky T,Magesan M,et al.De-aliased,high-resolution Radon transforms.In:Proceedings of the 70th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.Calgary:Society of Exploration Geophysicists(SEG),2000,1953-1956
15 Mitsuhata Y,Uchida T,Murakami Y,et al.The Fourier transform of controlled-source time-domain electromagnetic data by smooth spectrum inversion.Geophys J Int,2001,144:123-135
16 Oldenburg D W.Calculation of Fourier transforms by the Backus-Gilbert method.Geophys J Int,1976,44:413-431
17 Sacchi M D,Ulrych T J.Improving resolution of Radon operators using a model re-weighted least-squares procedure.J Seismic Expl,1995,4:315-328
18 Sacchi M D,Ulrych T J.Estimation of the discrete Fourier transform,a linear inversion approach.Geophysics,1996,61:1128-1136
19 Sacchi M D,Ulrych T J,Walker C J.Interpolation and extrapolation using a high-resolution discrete Fourier transform.IEEE Trans Signal Process,1998,46:31-38
20 Thorson J R,Claerbout J F.Velocity-stack and slant-stack stochastic inversion.Geophysics,1985,50:2727-2741
21 Trad D,Ulrych T,Sacchi M.Latest views of the sparse Radon transform.Geophysics,2003,68:386-399
22 Ulrych T J,Sacchi M D,Graul J M.Signal and noise separation:Art and science.Geophysics,1999,64:1648-1656
23 Tarantola A.Book reviews.Geophys J Int,1988,94:167-168
24 Wang Y,Yang C,Li X.Regularizing kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval using smoothness constraint.J Geophys Res,2008,113:D13101
25 Wang Y F.Computational Methods for Inverse Problems and Their Applications.Beijing:Higher Education Press,2007
26 Tibshirani R.Regression shrinkage and selection via the LASSO.J R Stat Soc Ser B Stat Methodol,1996,58:267-288
27 Kim S J,Koh K,Lustig M,et al.An interior-point method for large-scale-regularized least squares.IEEE J Sel Top Signal Process,2007,1:606-617
28 Dai Y H,Fletcher R.Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming.Numer Math,2005,100:21-47
29 van den Berg E,Friedlander M P.Probing the Pareto frontier for basis pursuit solutions.SIAM J Sci Comput,2009,31:890-912
30 Figueiredo M A T,Nowak R D,Wright S J.Gradient projection for sparse reconstruction:Application to compressed sensing and other inverse problems.IEEE J Sel Top Signal Process,2007,1:586-597
31 Wang Y,Ma S.Projected Barzilai-Borwein method for large-scale nonnegative image restoration.Inverse Probl Sci Eng,2007,15:559-583
32 Wang Y,Fan S,Feng X.Retrieval of the aerosol particle size distribution function by incorporating a priori information.J Aerosol Sci,2007,38:885-901
33 Wang Y,Yang C,Cao J.On Tikhonov regularization and compressive sensing for seismic signal processing.Math Model Methods Appl Sci,2012,22:1150008
34 Chen S S,Donoho D L,Saunders M A.Atomic decomposition by basis pursuit.SIAM Rev,2001,43:129-159
35 Tropp J A,Gilbert A C.Signal recovery from random measurements via orthogonal matching pursuit.IEEE Trans Inform Theory,2007,53:4655-4666
36 Wen Z,Yin W,Goldfarb D,et al.A fast algorithm for sparse reconstruction based on shrinkage,subspace optimization,and continuation.SIAM J Sci Comput,2010,32:1832-1857
37 Yuan Y X.A review on subspace methods for nonlinear optimization.In:Proceedings of International Congress of Mathematicians.Soeul:Kyung Moon SA Co.Ltd.,2014,807-827
38 Wang Y,Cao J,Yang C.Recovery of seismic wavefields based on compressive sensing by an l_1-norm constrained trust region method and the piecewise random subsampling.Geophys J Int,2011,187:199-213
39 He B S,Xu M H,Yuan X M.Solving large-scale least squares covariance matrix problems by alternating direction methods.SIAM J Matrix Anal Appl,2011,32:136-152
40 He B S,Yuan X M.On nonergodic convergence rate of Douglas-Rachford alternating direction method of multipliers.Numer Math,2015,130:567-577
41 Wang Y F.Sparse optimization methods for seismic wavefields recovery.Tr Inst Mat Mekh UrO RAN,2012,18:42-55
42 Wen Z,Yang C,Liu X,et al.Alternating direction methods for classical and ptychographic phase retrieval.Inverse Problems,2012,28:28-11
43 Yang J,Yuan X.Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization.Math Comp,2013,82:301-329
44 Dai Y H,Han D,Yuan X,et al.A sequential updating scheme of the Lagrange multiplier for separable convex programming.Math Comp,2017,86:315-343
2 Partyka G,Gridley J,Lopez J.Interpretational applications of spectral decomposition in reservoir characterization.Leading Edge,1999,18:353-360
3 Chopra S,Castagna J P,Portniaguine O.Seismic resolution and thin-bed reflectivity inversion.Canad Soc Explorat Geophys Recorder,2006,31:19-25
4 Chopra S,Castagna J P,Portniaguine O.Thin-bed reflectivity inversion.In:Proceedings of the 75th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.New Orleans:Society of Exploration Geophysicists(SEG),2006,2057-2061
5 Partyka G A.Spectral decomposition.SEG distinguished lecture.Http://www.seg.org/education/lectures-courses/distinguished-lecturers/spring2005/partykaabstract,2005
6 Portniaguine O,Castagna J P.Inverse spectral decomposition.In:Proceedings of the 74th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.Denver:Society of Exploration Geophysicists(SEG),2004,1786-1789
7 Portniaguine O,Castagna J P.Spectral inversion.In:Proceedings of the 75th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.Houston:Society of Exploration Geophysicists(SEG),2005,1638-1641
8 Puryear C I,Castagna J P.An algorithm for calculation of bed thickness and reflection coefficients from amplitude spectrum.In:Proceedings of the 76th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.New Orleans:Society of Exploration Geophysicists(SEG),2006,1767-1770
9 Marfurt K J,Kirlin R L.Narrow-band spectral analysis and thin-bed tuning.Geophysics,2001,66:1274-1283
10 Castagna J P,Sun S,Siegfried R W.Instantaneous spectral analysis:Detection of low-frequency shadows associated with hydrocarbons.Leading Edge,2003,22:120-127
11 Sinha S,Routh P S,Anno P D,et al.Spectral decomposition of seismic data with continuous-wavelet transform.Geophysics,2005,70:P19-P25
12 Puryear C I,Castagna J P.Layer-thickness determination and stratigraphic interpretation using spectral inversion:Theory and application.Geophysics,2008,73:R37-R48
13 Puryear C I,Portniaguine O N,Cobos C M,et al.Constrained least-squares spectral analysis:Application to seismic data.Geophysics,2012,77:V143-V167
14 Herrmann P,Mojesky T,Magesan M,et al.De-aliased,high-resolution Radon transforms.In:Proceedings of the 70th Annual International Meeting of the Society of Exploration Geophysicists,Expanded Abstracts.Calgary:Society of Exploration Geophysicists(SEG),2000,1953-1956
15 Mitsuhata Y,Uchida T,Murakami Y,et al.The Fourier transform of controlled-source time-domain electromagnetic data by smooth spectrum inversion.Geophys J Int,2001,144:123-135
16 Oldenburg D W.Calculation of Fourier transforms by the Backus-Gilbert method.Geophys J Int,1976,44:413-431
17 Sacchi M D,Ulrych T J.Improving resolution of Radon operators using a model re-weighted least-squares procedure.J Seismic Expl,1995,4:315-328
18 Sacchi M D,Ulrych T J.Estimation of the discrete Fourier transform,a linear inversion approach.Geophysics,1996,61:1128-1136
19 Sacchi M D,Ulrych T J,Walker C J.Interpolation and extrapolation using a high-resolution discrete Fourier transform.IEEE Trans Signal Process,1998,46:31-38
20 Thorson J R,Claerbout J F.Velocity-stack and slant-stack stochastic inversion.Geophysics,1985,50:2727-2741
21 Trad D,Ulrych T,Sacchi M.Latest views of the sparse Radon transform.Geophysics,2003,68:386-399
22 Ulrych T J,Sacchi M D,Graul J M.Signal and noise separation:Art and science.Geophysics,1999,64:1648-1656
23 Tarantola A.Book reviews.Geophys J Int,1988,94:167-168
24 Wang Y,Yang C,Li X.Regularizing kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval using smoothness constraint.J Geophys Res,2008,113:D13101
25 Wang Y F.Computational Methods for Inverse Problems and Their Applications.Beijing:Higher Education Press,2007
26 Tibshirani R.Regression shrinkage and selection via the LASSO.J R Stat Soc Ser B Stat Methodol,1996,58:267-288
27 Kim S J,Koh K,Lustig M,et al.An interior-point method for large-scale-regularized least squares.IEEE J Sel Top Signal Process,2007,1:606-617
28 Dai Y H,Fletcher R.Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming.Numer Math,2005,100:21-47
29 van den Berg E,Friedlander M P.Probing the Pareto frontier for basis pursuit solutions.SIAM J Sci Comput,2009,31:890-912
30 Figueiredo M A T,Nowak R D,Wright S J.Gradient projection for sparse reconstruction:Application to compressed sensing and other inverse problems.IEEE J Sel Top Signal Process,2007,1:586-597
31 Wang Y,Ma S.Projected Barzilai-Borwein method for large-scale nonnegative image restoration.Inverse Probl Sci Eng,2007,15:559-583
32 Wang Y,Fan S,Feng X.Retrieval of the aerosol particle size distribution function by incorporating a priori information.J Aerosol Sci,2007,38:885-901
33 Wang Y,Yang C,Cao J.On Tikhonov regularization and compressive sensing for seismic signal processing.Math Model Methods Appl Sci,2012,22:1150008
34 Chen S S,Donoho D L,Saunders M A.Atomic decomposition by basis pursuit.SIAM Rev,2001,43:129-159
35 Tropp J A,Gilbert A C.Signal recovery from random measurements via orthogonal matching pursuit.IEEE Trans Inform Theory,2007,53:4655-4666
36 Wen Z,Yin W,Goldfarb D,et al.A fast algorithm for sparse reconstruction based on shrinkage,subspace optimization,and continuation.SIAM J Sci Comput,2010,32:1832-1857
37 Yuan Y X.A review on subspace methods for nonlinear optimization.In:Proceedings of International Congress of Mathematicians.Soeul:Kyung Moon SA Co.Ltd.,2014,807-827
38 Wang Y,Cao J,Yang C.Recovery of seismic wavefields based on compressive sensing by an l_1-norm constrained trust region method and the piecewise random subsampling.Geophys J Int,2011,187:199-213
39 He B S,Xu M H,Yuan X M.Solving large-scale least squares covariance matrix problems by alternating direction methods.SIAM J Matrix Anal Appl,2011,32:136-152
40 He B S,Yuan X M.On nonergodic convergence rate of Douglas-Rachford alternating direction method of multipliers.Numer Math,2015,130:567-577
41 Wang Y F.Sparse optimization methods for seismic wavefields recovery.Tr Inst Mat Mekh UrO RAN,2012,18:42-55
42 Wen Z,Yang C,Liu X,et al.Alternating direction methods for classical and ptychographic phase retrieval.Inverse Problems,2012,28:28-11
43 Yang J,Yuan X.Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization.Math Comp,2013,82:301-329
44 Dai Y H,Han D,Yuan X,et al.A sequential updating scheme of the Lagrange multiplier for separable convex programming.Math Comp,2017,86:315-343