Curvelet阈值迭代法在地震数据去噪和插值中的应用研究
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摘要
.基于Curvelet变换的阈值迭代法在地震数据随机噪声衰减和缺失道插值中的应用是本文研究的重点。
本文在前人的工作基础上,通过对Curvelet变换及其稀疏性和阈值迭代法的分析、研究和总结,将Curvelet变换与阈值迭代法结合起来,并在此基础上,把地震数据随机噪声衰减和地震缺失道插值问题描述为一个l1模最优化问题。其求解方法即为阈值迭代法。其中在去噪问题中,通过与传统的中值滤波法、FX反褶积法和近些年发展起来的小波阈值法去噪相对比,理论模型和实际资料的计算结果表明,Curvelet阈值迭代法去噪不仅得到了更高的信噪比,而且对有效信号的损失更少;另外,对Curvelet阈值迭代法去噪结果进行方向控制,进一步提高了信噪比。在缺失道插值问题中,对均匀缺失和随机缺失的地震道数据实现了较为精确的插值,取得了较高的信噪比,验证了随机缺失道插值效果好于均匀缺失道插值效果。
This paper combines the curvelet transform that developed in recently years with the thresholding iterative method that solves optimizational inversion problem. The primary study is the application of curvelet-based thresholding iterative method in seismic data random noise attenuation and missing trace interpolation. Especially, we discuss detailedly seismic data random noise attenuation, and we compare it with traditional median filter, FX deconvolution and wavelet threshold methods, and finally, we know that curvelet thresholding iterative method not only increases sufficiently signal noise ratio, but also damages the least effective signal. Then, we make direction controlling after curvelet thresholding iterative denoising, this further increases signal noise ratio. In the aspect of seismic data missing trace interpolation, we interpolate respectively for uniform missing trace modeling data and nuniform missing trace modeling data, and proved the feasibility of this method.
Fast Discrete Curvelet Transform is a basic tool in this paper’s research. Curvelet transform was proposed firstly by Candès and Donoho in 1999. Its initial application fields contain computer image denoising, image fusion, SAR denoising, deconvolution and so on. The digial curvelet transform used in these initial applications was generally based on tranditional structure, that is, it was achieved by pre-processing image firstly and then ridgelet transform. This algotiehm is more redundant. In the past several years, making curvelet transform become easier use and understanding was its primary research content. Candès et al proposed new curvelet transform frame system in 2002, and it was called second generation curvelet transform. This new theory frame system provides the possibility for increasing the speed of curvelet transform. From then, they proposed two fast discrete realization methods based on second curvelet transform in 2006. These methods become easier and more rapid and decrease the redundancy of traditional algorithm. It is a solid basis for curvelet-based seismic data processing. Our paper makes use of this transform and it makes practical operation easy to realize.
Thresholding iterative method is a basic theory in our paper’s research. It is used to solve sparsity constraint optimization inversion problems. Daubechies et al have derived strictly equation, and described detailedly its continuity, convergence and stability. This method assumpts that signal is sparse. We verify that curvelet transform has much better sparsity than Fourier and wavelet transform on seismic data. So we know that curvelet-based thresholding interative method has better solving precision. Moreover, we describe seismic data denoising and interpolation problems as a one-norm optimization problem, and then solve it with curvelet-based thresholding iterative method.
Seismic data random noise attenuation is the main study content. We compare curvelet thresholding iterative method with conventional median filter, FX deconvolution and wavelet threshold methods in the modeling experiment, and finally, come to the conclusions: median filter could not obtain satisfied denoising result, and it damages much signal. This phenomenon will become worse, especially when noisy data have low signal noise ratio; FX deconvolution have good denoising result for seismic data random noise, however, it damages seriously high frequency signal, and when signal noise ratio of data is low, it losses more effective signal; Wavelet threshold method can obtain well denoising result that could be close to FX deconvolution method, but the denoised seismic data become fuzzy because of defect of wavelet transform itself, this decreases the resolution in seismic data. Also, this method damages much signal. Curvelet-based thresholding iterative method is very effective for seismic data random noise attenuation. It not only attenuates a lot noise, but also damages less signal. So it is a effective method to denoising. In addition, direction controlling which based on the result of curvelet thresholding iterative denoising increases signal noise ratio of data, this is a highlight in the paper. Finally, we realize curvelet thresholding iterative denoising in field data, and obtain fine result.
We realize seismic data missing trace interpolation based on curvelet thresholding iterative in the last part of the paper. We compare the interpolation results of uniform missing trace with nonuniform missing trace, and vetify that interpolation result is much better in the condition of nonuniform missing trace. This is because that the sparsity of nonuniform missing trace is better than uniform missing condition in Fourier domain. In addition, interpolation result is not perfect for 2D data when the percentage of missing trace is big, so we prospect a method of curvelet thresholding iterative in focus domain. Focus transform will increase furtherly the sparsity of data, and this will obtain much better interpolation result in theory.
本文在前人的工作基础上,通过对Curvelet变换及其稀疏性和阈值迭代法的分析、研究和总结,将Curvelet变换与阈值迭代法结合起来,并在此基础上,把地震数据随机噪声衰减和地震缺失道插值问题描述为一个l1模最优化问题。其求解方法即为阈值迭代法。其中在去噪问题中,通过与传统的中值滤波法、FX反褶积法和近些年发展起来的小波阈值法去噪相对比,理论模型和实际资料的计算结果表明,Curvelet阈值迭代法去噪不仅得到了更高的信噪比,而且对有效信号的损失更少;另外,对Curvelet阈值迭代法去噪结果进行方向控制,进一步提高了信噪比。在缺失道插值问题中,对均匀缺失和随机缺失的地震道数据实现了较为精确的插值,取得了较高的信噪比,验证了随机缺失道插值效果好于均匀缺失道插值效果。
This paper combines the curvelet transform that developed in recently years with the thresholding iterative method that solves optimizational inversion problem. The primary study is the application of curvelet-based thresholding iterative method in seismic data random noise attenuation and missing trace interpolation. Especially, we discuss detailedly seismic data random noise attenuation, and we compare it with traditional median filter, FX deconvolution and wavelet threshold methods, and finally, we know that curvelet thresholding iterative method not only increases sufficiently signal noise ratio, but also damages the least effective signal. Then, we make direction controlling after curvelet thresholding iterative denoising, this further increases signal noise ratio. In the aspect of seismic data missing trace interpolation, we interpolate respectively for uniform missing trace modeling data and nuniform missing trace modeling data, and proved the feasibility of this method.
Fast Discrete Curvelet Transform is a basic tool in this paper’s research. Curvelet transform was proposed firstly by Candès and Donoho in 1999. Its initial application fields contain computer image denoising, image fusion, SAR denoising, deconvolution and so on. The digial curvelet transform used in these initial applications was generally based on tranditional structure, that is, it was achieved by pre-processing image firstly and then ridgelet transform. This algotiehm is more redundant. In the past several years, making curvelet transform become easier use and understanding was its primary research content. Candès et al proposed new curvelet transform frame system in 2002, and it was called second generation curvelet transform. This new theory frame system provides the possibility for increasing the speed of curvelet transform. From then, they proposed two fast discrete realization methods based on second curvelet transform in 2006. These methods become easier and more rapid and decrease the redundancy of traditional algorithm. It is a solid basis for curvelet-based seismic data processing. Our paper makes use of this transform and it makes practical operation easy to realize.
Thresholding iterative method is a basic theory in our paper’s research. It is used to solve sparsity constraint optimization inversion problems. Daubechies et al have derived strictly equation, and described detailedly its continuity, convergence and stability. This method assumpts that signal is sparse. We verify that curvelet transform has much better sparsity than Fourier and wavelet transform on seismic data. So we know that curvelet-based thresholding interative method has better solving precision. Moreover, we describe seismic data denoising and interpolation problems as a one-norm optimization problem, and then solve it with curvelet-based thresholding iterative method.
Seismic data random noise attenuation is the main study content. We compare curvelet thresholding iterative method with conventional median filter, FX deconvolution and wavelet threshold methods in the modeling experiment, and finally, come to the conclusions: median filter could not obtain satisfied denoising result, and it damages much signal. This phenomenon will become worse, especially when noisy data have low signal noise ratio; FX deconvolution have good denoising result for seismic data random noise, however, it damages seriously high frequency signal, and when signal noise ratio of data is low, it losses more effective signal; Wavelet threshold method can obtain well denoising result that could be close to FX deconvolution method, but the denoised seismic data become fuzzy because of defect of wavelet transform itself, this decreases the resolution in seismic data. Also, this method damages much signal. Curvelet-based thresholding iterative method is very effective for seismic data random noise attenuation. It not only attenuates a lot noise, but also damages less signal. So it is a effective method to denoising. In addition, direction controlling which based on the result of curvelet thresholding iterative denoising increases signal noise ratio of data, this is a highlight in the paper. Finally, we realize curvelet thresholding iterative denoising in field data, and obtain fine result.
We realize seismic data missing trace interpolation based on curvelet thresholding iterative in the last part of the paper. We compare the interpolation results of uniform missing trace with nonuniform missing trace, and vetify that interpolation result is much better in the condition of nonuniform missing trace. This is because that the sparsity of nonuniform missing trace is better than uniform missing condition in Fourier domain. In addition, interpolation result is not perfect for 2D data when the percentage of missing trace is big, so we prospect a method of curvelet thresholding iterative in focus domain. Focus transform will increase furtherly the sparsity of data, and this will obtain much better interpolation result in theory.
引文
[1]张军华,吕宁,田连玉,等.地震资料去噪方法技术综合评价[J].地球物理学进展,2006,21(2):546-553.
[2]熊翥.我国物探技术的进步及展望[J].石油地球物理勘探,2003,38(3):334-337.
[3] Ramesh N, Domimique G, Mohamed T, et al. Coherent and random noise attenu- ation using the curvelet transform [J].The Leading Edge,2008,27(2):240-248.
[4]黄薇.Curvelet变换及其在图像处理中的应用[D].西安:西安理工大学,2007.
[5]赵心.基于Curvelet变换的图像去噪方法研究与应用[D].青岛:山东科技大学,2007.
[6]吴芳平.Curvelet变换及其在图像处理中的应用研究[D].广州:暨南大学,2008.
[7] Herrmann F J, G Hennenfent. Non-parametric seismic data recovery with curve- let frames [J].Geophysical Journal International,2008,173(1):233-248.
[8] Candès E, Donoho D. Curvelets: A surprisingly effective nonadaptive represent- tation of objects with edges [M].TN: Vanderbilt University Press,1999.
[9] Candès E, Donoho D. New tight frames of curvelets and optimal representations of objects with C2
[10]Candès E, Demanet L. Curvelets and Fourier integral operators [R].To appear in Comptes-Rendus del’Academie des Sciences,Paris,Serie I,2002. singularities [R].Technical Report,Stanford,2002.
[11]Candès E, F Guo. New multiscale transforms, minimum total variation synthesis: Applications to edge-preserving image reconstruction [J].Signal Processing,2002, 82:1519–1543.
[12]Candès E, Demanet L, Donoho D, et al. Fast discrete curvelet transforms [J].SI- AM Multiscale Modeling and Simulation,2006,5:861-899.
[13]Herrmann F J, Verschuur E. Curvelet-domain multiple elimination with sparsen- ess constraints [C].Society of Exploration Geophysicists,Expanded Abstracts, p1333-1336,2004.
[14]G Hennenfent, Herrmann F J. Three-term amplitude-versus-offset(AVO) inverse- on revisited by curvelet and wavelet transforms [C].Society of Exploration Geo- physicists,Expanded Abstracts,p211-214,2004.
[15]Huub D, Maarten V. Wave-character preserving pre-stack map migration using curvelets [C].Society of Exploration Geophysicists,Expanded Abstracts, p961-964,2004.
[16]Herrmann F J, Deli Wang, G Hennenfent, et al. Seismic data processing with cu- rvelets: A multiscale and nonlinear approach [C].Society of Exploration Geophy- sicists,Expanded Abstracts,p2220-2224,2007.
[17]Herrmann F J, Deli Wang, G Hennenfent. Multiple prediction from incomplete data with the focused curvelet transform [C].Society of Exploration Geophysi- cists,Expanded Abstracts,p2505-2509,2007.
[18]何劲,李宏伟,张帆.基于Curvelet变换的图像消噪[J].现代电子技术,2008(2):140-144.
[19]刘国高,周建东,符进祥.Curvelet变换及其在图像去噪中的应用研究[J].电子测量技术,2008(7):7-11.
[20]卢成武,宋国乡.带曲波域约束的全变差正则化抑噪方法[J].2008,36(4):646-649.
[21]吴芳平,狄红卫.基于Curvelet变换的软硬阈值折衷图像去噪[J].光学技术,2007,33(5):688-690.
[22]邓承志,曹汉强,汪胜前.Curvelet变换域自适应收缩图像去噪[J].应用科学学报,2008,26(1):22-27.
[23]张素芳,徐义贤,雷栋.基于Curvelet变换的多次波去除技术[J].石油地球物理勘探,2006,41(3):262-265.
[24]包乾宗,高静怀,陈文超.Curvelet域垂直地震剖面波场分离[J].西安交通大学学报,2007,41(6):650-654.
[25]彭才,常智,朱仕军.基于曲波变换的地震数据去噪方法[J].石油物探,2008,47(5):461-464.
[26]张恒磊,张云翠,宋双,等.基于Curvelet域的叠前地震资料去噪方法[J].石油地球物理勘探,2008,43(5):508-513.
[27]Deli Wang, Rayan Saab, ?zgür Yilmaz, et al. Bayesian wavefield separation by transform-domain sparsity promotion [J].Geophysics,2008,73(5):A33-A38.
[28]刘冰.基于Curvelet变换的地震数据去噪方法研究[D].长春:吉林大学,2008.
[29]Bednar J B. Applications of median filtering to deconvolution, pulse estimation, and statistical editing of seismic data [J].Geophysics,1983,48(12):1598-1610.
[30]李承楚.关于中值滤波的理论基础[J].石油地球物理勘探,1986,21(4):372-379.
[31]Duncan G, Beresford G. Some analyses of 2-D median f-k filters [J]. Geophysics, 1995,60(4):1157-1168.
[32]Canlaes L L. Random noise reduction [A].In:54th
[33]Yu S, Cai X, Su Y. Seismic signal enhancement by polynomial fitting [J]. Appli- SEG Mtg[C],1984.525-572.ed Geophysics,1989,1:57-65.
[34]Brian Russell, Dan Hampson, Joong Chun. Noise elimination and the radon tran- sform, Part1 [J].The Leading Edge,1990,9(11):18-23.
[35]Brian Russell, Dan Hampson, Joong Chun. Noise elimination and the radon tran- sform, Part2 [J].The Leading Edge,1990,9(11):31-37.
[36]Alan R Michell, Panos G Kelamis. Efficient tau-p hyperbolic velocity filtering [J].Geophysics,1990,55(5):619-625.
[37]Jones I F, Levy S. Signal-to-noise ratio enhancement in multichannel seismic da- ta via the Karhunen-Love transform [J].Geophysical Prospecting,1987,35:12-32.
[38]Al-Yahya KM. Application of the partial Karhunen-Love transform to suppress random noise in seismic seismic sections [J].Geophysical Prospecting,1991,39(1): 77-93.
[39]Rongfeng Zhang, Ulrych T J. Phasical wavelet frame denoising [J].Geophysics, 2003,68(1):225-231.
[40]高静怀,毛剑,满蔚仕,等.叠前地震资料噪声衰减的小波域方法研究[J].地球物理学报,2006,49(4):1155-1163.
[41]夏洪瑞,朱勇,周开明.小波变换及其在去噪中的应用[J].石油地球物理勘探,1994,29(3):274-285.
[42]刘法启,张关泉.小波变换与F-K算法在滤波中的应用[J].石油地球物理勘探,1996,31(6):782-792.
[43]宗涛,孟鸿鹰,贾玉兰,等.小波包域前后向预测去噪[J].地球物理学报,1998,41(增刊):337-346.
[44]王振国,汪恩华.小波包相关阈值去噪[J].石油物探,2002,41(4):400-405.
[45]Abma R, Claerbout J. Lateral prediction for noise attenuation by t-x and f-x tech- niques [J].Geophysics,1995,60(6):1887-1896.
[46]Bekara M, Van der Baan M. Local singular value decomposition for signal enha- ncement of seismic data [J].Geophysics,2007,72(2):V59-V65.
[47]张山.f_x域内提高地震资料信噪比[J].石油地球物理勘探,1992, 27(5):648-653.
[48]杨晓春,李小凡,张美根.地震波反演方法研究的某些进展及其数学基础[J].地球物理学进展,2001,16(4):96-109.
[49]Daubechies I, M Defrise, C De Mol. An iterative thresholding algorithm for line- ar inverse problems with a sparsity constraint [J].Communications on Pure and Applied Mathematics,2004,57:1413-1457.
[50]张军.求解不适定问题的快速Landweber迭代法[J].数学杂志,2005,25(3):333-335.
[51]Andress Kirsch. An Introduction to the Mathematical Theory of Inverse Proble- ms [M]. Springer Verlag New York, Inc. 1996.
[52]Hennenfent G, Ewout van den B, Michael P F, et al. New Insights into One-norm Solvers from the Pareto Curve [J].Geophysics,2008,73(4):A23-A26.
[53]Hennenfent G, Herrmann F J. Simply denoise: Wavefield reconstruction via jitte- red undersampling [J].Geophysics,2008,73(3):V19-V28.
[54]Elad M, J L Starck, P Querre, et al. Simulataneous Cartoon and Texture Image Inpainting using Morphological Component Analysis (MCA) [C]. Appl. Comput. Harmon. Anal,2005,19:340-358.
[55]Jakubowicz H. Wavefield reconstruction [C].64th Ann. Internat. Mtg., Soc. Expl.Geophys.Expanded Abstracts,1994,1557-1560.
[56]Larner K, Gibson B, Rothman D. Trace interpolation and the design of seismic surveys [J].Geophysics,1981,46(4):407-409.
[57]Fomel S. Theory of differential offset continuation [J].Geophysics,2003, 68(2): 718-732.
[58]Fomel S, J G Berryman, R G Clapp, et al. Iterative resolution estimation in least- squares Kirchoff migration [J].Geophys. Pros.,50,577-588.
[59]Schonewille M A, Duijndam A J W. Decomosition of signal and noise for sparse- ly sampled data using a mixed-Fourier/Radon transform [C]. 66th
[60]Spitz S. Seismic trace interpolation in the f-x domain [J].Geophysics,1991,56(6): 785-794. Ann. Internat. Mtg., Soc, Expl. Geophys. Expanded Abstract,1996.1438-1441.
[61]Soubaras R. Spatial interpolation of aliased seismic data [C]. 67th
[62]Duijndam A J W, Schonewille M, Hindriks K. Reconstruction of seismic signals, irregularly sampled along on spatial coordinate [J].Geophysics,1999, 64(2):524- 538. Ann. Intermat. Mtg., Soc. Exp. Geophys. Expanded Abstracts,1997,1167-1190.
[63]刘喜武,刘洪,刘彬.反假频非均匀地震数据重建方法研究[J].地球物理学报,2004,47(2):299-305.
[64]何樵登.地震勘探原理和方法[M].北京:地质出版社,1986.
[65]伊尔马兹.地震资料分析—地震资料处理、反演和解释[M].北京:石油工业出版社,2006.
[2]熊翥.我国物探技术的进步及展望[J].石油地球物理勘探,2003,38(3):334-337.
[3] Ramesh N, Domimique G, Mohamed T, et al. Coherent and random noise attenu- ation using the curvelet transform [J].The Leading Edge,2008,27(2):240-248.
[4]黄薇.Curvelet变换及其在图像处理中的应用[D].西安:西安理工大学,2007.
[5]赵心.基于Curvelet变换的图像去噪方法研究与应用[D].青岛:山东科技大学,2007.
[6]吴芳平.Curvelet变换及其在图像处理中的应用研究[D].广州:暨南大学,2008.
[7] Herrmann F J, G Hennenfent. Non-parametric seismic data recovery with curve- let frames [J].Geophysical Journal International,2008,173(1):233-248.
[8] Candès E, Donoho D. Curvelets: A surprisingly effective nonadaptive represent- tation of objects with edges [M].TN: Vanderbilt University Press,1999.
[9] Candès E, Donoho D. New tight frames of curvelets and optimal representations of objects with C2
[10]Candès E, Demanet L. Curvelets and Fourier integral operators [R].To appear in Comptes-Rendus del’Academie des Sciences,Paris,Serie I,2002. singularities [R].Technical Report,Stanford,2002.
[11]Candès E, F Guo. New multiscale transforms, minimum total variation synthesis: Applications to edge-preserving image reconstruction [J].Signal Processing,2002, 82:1519–1543.
[12]Candès E, Demanet L, Donoho D, et al. Fast discrete curvelet transforms [J].SI- AM Multiscale Modeling and Simulation,2006,5:861-899.
[13]Herrmann F J, Verschuur E. Curvelet-domain multiple elimination with sparsen- ess constraints [C].Society of Exploration Geophysicists,Expanded Abstracts, p1333-1336,2004.
[14]G Hennenfent, Herrmann F J. Three-term amplitude-versus-offset(AVO) inverse- on revisited by curvelet and wavelet transforms [C].Society of Exploration Geo- physicists,Expanded Abstracts,p211-214,2004.
[15]Huub D, Maarten V. Wave-character preserving pre-stack map migration using curvelets [C].Society of Exploration Geophysicists,Expanded Abstracts, p961-964,2004.
[16]Herrmann F J, Deli Wang, G Hennenfent, et al. Seismic data processing with cu- rvelets: A multiscale and nonlinear approach [C].Society of Exploration Geophy- sicists,Expanded Abstracts,p2220-2224,2007.
[17]Herrmann F J, Deli Wang, G Hennenfent. Multiple prediction from incomplete data with the focused curvelet transform [C].Society of Exploration Geophysi- cists,Expanded Abstracts,p2505-2509,2007.
[18]何劲,李宏伟,张帆.基于Curvelet变换的图像消噪[J].现代电子技术,2008(2):140-144.
[19]刘国高,周建东,符进祥.Curvelet变换及其在图像去噪中的应用研究[J].电子测量技术,2008(7):7-11.
[20]卢成武,宋国乡.带曲波域约束的全变差正则化抑噪方法[J].2008,36(4):646-649.
[21]吴芳平,狄红卫.基于Curvelet变换的软硬阈值折衷图像去噪[J].光学技术,2007,33(5):688-690.
[22]邓承志,曹汉强,汪胜前.Curvelet变换域自适应收缩图像去噪[J].应用科学学报,2008,26(1):22-27.
[23]张素芳,徐义贤,雷栋.基于Curvelet变换的多次波去除技术[J].石油地球物理勘探,2006,41(3):262-265.
[24]包乾宗,高静怀,陈文超.Curvelet域垂直地震剖面波场分离[J].西安交通大学学报,2007,41(6):650-654.
[25]彭才,常智,朱仕军.基于曲波变换的地震数据去噪方法[J].石油物探,2008,47(5):461-464.
[26]张恒磊,张云翠,宋双,等.基于Curvelet域的叠前地震资料去噪方法[J].石油地球物理勘探,2008,43(5):508-513.
[27]Deli Wang, Rayan Saab, ?zgür Yilmaz, et al. Bayesian wavefield separation by transform-domain sparsity promotion [J].Geophysics,2008,73(5):A33-A38.
[28]刘冰.基于Curvelet变换的地震数据去噪方法研究[D].长春:吉林大学,2008.
[29]Bednar J B. Applications of median filtering to deconvolution, pulse estimation, and statistical editing of seismic data [J].Geophysics,1983,48(12):1598-1610.
[30]李承楚.关于中值滤波的理论基础[J].石油地球物理勘探,1986,21(4):372-379.
[31]Duncan G, Beresford G. Some analyses of 2-D median f-k filters [J]. Geophysics, 1995,60(4):1157-1168.
[32]Canlaes L L. Random noise reduction [A].In:54th
[33]Yu S, Cai X, Su Y. Seismic signal enhancement by polynomial fitting [J]. Appli- SEG Mtg[C],1984.525-572.ed Geophysics,1989,1:57-65.
[34]Brian Russell, Dan Hampson, Joong Chun. Noise elimination and the radon tran- sform, Part1 [J].The Leading Edge,1990,9(11):18-23.
[35]Brian Russell, Dan Hampson, Joong Chun. Noise elimination and the radon tran- sform, Part2 [J].The Leading Edge,1990,9(11):31-37.
[36]Alan R Michell, Panos G Kelamis. Efficient tau-p hyperbolic velocity filtering [J].Geophysics,1990,55(5):619-625.
[37]Jones I F, Levy S. Signal-to-noise ratio enhancement in multichannel seismic da- ta via the Karhunen-Love transform [J].Geophysical Prospecting,1987,35:12-32.
[38]Al-Yahya KM. Application of the partial Karhunen-Love transform to suppress random noise in seismic seismic sections [J].Geophysical Prospecting,1991,39(1): 77-93.
[39]Rongfeng Zhang, Ulrych T J. Phasical wavelet frame denoising [J].Geophysics, 2003,68(1):225-231.
[40]高静怀,毛剑,满蔚仕,等.叠前地震资料噪声衰减的小波域方法研究[J].地球物理学报,2006,49(4):1155-1163.
[41]夏洪瑞,朱勇,周开明.小波变换及其在去噪中的应用[J].石油地球物理勘探,1994,29(3):274-285.
[42]刘法启,张关泉.小波变换与F-K算法在滤波中的应用[J].石油地球物理勘探,1996,31(6):782-792.
[43]宗涛,孟鸿鹰,贾玉兰,等.小波包域前后向预测去噪[J].地球物理学报,1998,41(增刊):337-346.
[44]王振国,汪恩华.小波包相关阈值去噪[J].石油物探,2002,41(4):400-405.
[45]Abma R, Claerbout J. Lateral prediction for noise attenuation by t-x and f-x tech- niques [J].Geophysics,1995,60(6):1887-1896.
[46]Bekara M, Van der Baan M. Local singular value decomposition for signal enha- ncement of seismic data [J].Geophysics,2007,72(2):V59-V65.
[47]张山.f_x域内提高地震资料信噪比[J].石油地球物理勘探,1992, 27(5):648-653.
[48]杨晓春,李小凡,张美根.地震波反演方法研究的某些进展及其数学基础[J].地球物理学进展,2001,16(4):96-109.
[49]Daubechies I, M Defrise, C De Mol. An iterative thresholding algorithm for line- ar inverse problems with a sparsity constraint [J].Communications on Pure and Applied Mathematics,2004,57:1413-1457.
[50]张军.求解不适定问题的快速Landweber迭代法[J].数学杂志,2005,25(3):333-335.
[51]Andress Kirsch. An Introduction to the Mathematical Theory of Inverse Proble- ms [M]. Springer Verlag New York, Inc. 1996.
[52]Hennenfent G, Ewout van den B, Michael P F, et al. New Insights into One-norm Solvers from the Pareto Curve [J].Geophysics,2008,73(4):A23-A26.
[53]Hennenfent G, Herrmann F J. Simply denoise: Wavefield reconstruction via jitte- red undersampling [J].Geophysics,2008,73(3):V19-V28.
[54]Elad M, J L Starck, P Querre, et al. Simulataneous Cartoon and Texture Image Inpainting using Morphological Component Analysis (MCA) [C]. Appl. Comput. Harmon. Anal,2005,19:340-358.
[55]Jakubowicz H. Wavefield reconstruction [C].64th Ann. Internat. Mtg., Soc. Expl.Geophys.Expanded Abstracts,1994,1557-1560.
[56]Larner K, Gibson B, Rothman D. Trace interpolation and the design of seismic surveys [J].Geophysics,1981,46(4):407-409.
[57]Fomel S. Theory of differential offset continuation [J].Geophysics,2003, 68(2): 718-732.
[58]Fomel S, J G Berryman, R G Clapp, et al. Iterative resolution estimation in least- squares Kirchoff migration [J].Geophys. Pros.,50,577-588.
[59]Schonewille M A, Duijndam A J W. Decomosition of signal and noise for sparse- ly sampled data using a mixed-Fourier/Radon transform [C]. 66th
[60]Spitz S. Seismic trace interpolation in the f-x domain [J].Geophysics,1991,56(6): 785-794. Ann. Internat. Mtg., Soc, Expl. Geophys. Expanded Abstract,1996.1438-1441.
[61]Soubaras R. Spatial interpolation of aliased seismic data [C]. 67th
[62]Duijndam A J W, Schonewille M, Hindriks K. Reconstruction of seismic signals, irregularly sampled along on spatial coordinate [J].Geophysics,1999, 64(2):524- 538. Ann. Intermat. Mtg., Soc. Exp. Geophys. Expanded Abstracts,1997,1167-1190.
[63]刘喜武,刘洪,刘彬.反假频非均匀地震数据重建方法研究[J].地球物理学报,2004,47(2):299-305.
[64]何樵登.地震勘探原理和方法[M].北京:地质出版社,1986.
[65]伊尔马兹.地震资料分析—地震资料处理、反演和解释[M].北京:石油工业出版社,2006.