基于SL_0的高分辨率Radon变换及数据重建
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摘要
L_0范数是度量数据稀疏度的最优方法,但它具有的非凸性造成求解困难。本文将光滑L_0范数(Smoothed L_0 Norm,SL_0)稀疏约束引入Radon变换中,降低了其求解难度,并进一步提高了Radon变换的分辨率。即通过构建平滑的连续函数逼近L_0范数,以此作为抛物线Radon变换的目标函数,采用最速下降法和梯度投影原理获得最优解。理论模型和实际地震数据重建试验结果表明,该方法进一步提高了Radon变换分辨率,较好地恢复了缺失地震数据的连续性和AVO特性。
The L_0 norm is the optimal way to measure the sparsity of data,however it is difficult to solve the L_0 due to its non convexity.This paper introduce smoothed L_0 norm(SL_0)into the Radon transform to overcome the difficulty of solving and further to improve the resolution of Radon transform.We first use smoothed continuous functions as objective functions of the parabolic Radon transform to approximate the L_0 norm,and then we use the steepest descent method and gradient projection principle to approach the optimal solution.Experiments on both theoretical model and field data show that the proposed method not only improves the resolution of Radon transform,but also restores the continuity of seismic data and AVO characteristics.
The L_0 norm is the optimal way to measure the sparsity of data,however it is difficult to solve the L_0 due to its non convexity.This paper introduce smoothed L_0 norm(SL_0)into the Radon transform to overcome the difficulty of solving and further to improve the resolution of Radon transform.We first use smoothed continuous functions as objective functions of the parabolic Radon transform to approximate the L_0 norm,and then we use the steepest descent method and gradient projection principle to approach the optimal solution.Experiments on both theoretical model and field data show that the proposed method not only improves the resolution of Radon transform,but also restores the continuity of seismic data and AVO characteristics.
引文
[1]王雄文,王华忠.基于压缩感知的高分辨率平面波分解方法研究.地球物理学报,2014,57(9):2946-2960.Wang Xiongwen,Wang Huazhong.A research of high-resolution plane-wave decomposition based on compressed sensing.Chinese Journal of Geophysics,2014,57(9):2946-2960.
[2]徐彦凯,曹思远,何元.双曲Radon-ASVD方法压制叠前地震数据随机噪声.石油地球物理勘探,2017,52(3):451-457.Xu Yankai,Cao Siyuan,He Yuan.Hyperbolic RadonASVD method for suppressing random noise of prestack seismic data.OGP,2017,52(3):451-457.
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[4]范景文,李振春,宋翔宇等.各向异性高分辨率Radon变换压制多次波.石油地球物理勘探,2016,51(4):665-669.Fan Jingwen,Li Zhenchun,Song Xiangyu et al.Suppression of multiple waves by anisotropic high resolution Radon transform.OGP,2016,51(4):665-669.
[5]谢俊法,孙成禹,韩文功.迭代抛物Radon变换法分离一次波与多次波.石油地球物理勘探,2014,49(1):76-81.Xie Junfa,Sun Chengyu,Han Wengong.Separation of primary and multiple waves by iterative parabolic Radon transform.OGP,2014,49(1):76-81.
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[7]安鹏.基于高分辨率Radon变换的波场分离方法研究[学位论文].山东东营:中国石油大学(华东),2009.An Peng.Wave Field Separation Technology Research Based on High-resolution Radon Transform[D].China University of Petroleum(East China),Dongying,Shandong,2009.
[8]Sacchi M,Ulrych T.High resolution velocity gathers and offset space reconstruction.Geophysics,1995,60(4):1169-1177.
[9]Sacchi M,Porsani M.Fast high resolution parabolic Radon transform.SEG Technical Program Expanded Abstracts,1999,18:1477-1480.
[10]Trad D,Ulrych T,Sacchi M.Latest view of the sparse Radon transform.Geophysics,2003,68(1):386-399.
[11]Figueiredo M A T,Nowak R D,Wright S J.Gradient projection for sparse reconstruction:application to compressed sensing and other inverse problems.IEEEJournal of Selected Topics in Signal Processing,2007,1(4):586-597.
[12]Chen S S,Donoho D L,Saunders M A.Atomic decomposition by basis pursuit.SIAM Journal of Scientific Computing,1998,20(1):33-61.
[13]Mohimani H,Babaie-Zadeh M and Jutten C.Fast sparse representation using smoothed L0 norm.IEEE Transactions on Signal Processing,2007,46(6):1-12.
[14]Mohimani H,Babaie-Zadeh M,Jutten C.A fast approach for overcomplete sparse decomposition based on smoothed L0 norm.IEEE Transactions on Signal Processing,2009,57(1):289-301.
[15]Hyder M M and Mahata K.An improved smoothed L0 approximation algorithm for sparse representation.IEEE Transactions on Signal Processing,2010,58(4):2194-2205.
[16]ZayyaniH,Babaie-ZadehM.Thresholdedsmoothed-L0dictionary learning for sparse representations.IEEEInternational Conference on Acoustics,Speech and Signal Processing,2009,1825-1828.
[17]Ghalehjegh S H,Babaie-Zadeh M,Jutten C.Fast blocksparse decomposition based on SL0.9th International Conference on Latent Variable Analysis and Signal Separation,2010,426-433.
[18]林婉娟,赵瑞珍,李浩.用于压缩感知信号重建的NSL0算法.新型工业化,2011,1(7):78-84.Lin Wanjuan,Zhao Ruizhen,Li Hao.The NSL0algorithm for compressive sensing signal reconstruction.Industrialization,2011,1(7):78-84.
[19]祁锐,李宏伟,张玉洁.基于改进光滑L0范数的块稀疏信号重构算法.计算机工程,2015,41(11):294-298.Qi Rui,Li Hongwei,Zhang Yujie.Block sparse signal reconstruction algorithm based on improved smoothed L0 norm.Computer Engineering,2015,41(11):294-298.
[20]杨良龙,赵生妹,郑宝玉等.基于SL0压缩感知信号重建的改进算法.信号处理,2012,28(6):834-841.Yang Lianglong,Zhao Shengmei,Zheng Baoyu et al.The improved reconstruction algorithm for compressive sensing on SL0.Signal Processing,2012,28(6):834-841.
[21]曹静杰,王彦飞,杨长春.地震数据压缩重构的正则化与零范数稀疏最优化方法.地球物理学报,2012,55(2):596-607.Cao Jingjie,Wang Yanfei,Yang Changchun.Seismic data restoration based on compressive sensing using the regularization and zero-norm sparse optimization.Chinese Journal of Geophysics,2012,55(2):596-607.
[22]张良,韩立国,刘争光等.基于压缩感知和Contourlet变换的地震数据重建方法.石油物探,2017,56(6):804-811.Zhang Liang,Han Liguo,Liu Zhengguang et al.Seismic data reconstruction based on compressed sending and Contourlet transform.GPP,2017,56(6):804-811.
[2]徐彦凯,曹思远,何元.双曲Radon-ASVD方法压制叠前地震数据随机噪声.石油地球物理勘探,2017,52(3):451-457.Xu Yankai,Cao Siyuan,He Yuan.Hyperbolic RadonASVD method for suppressing random noise of prestack seismic data.OGP,2017,52(3):451-457.
[3]薛亚茹,唐欢欢,陈小宏.高阶高分辨率Radon变换地震数据重建方法.石油地球物理勘探,2014,49(1):95-100,131.Xue Yaru,Tang Huanhuan,Chen Xiaohong.Reconstruction method of seismic data using high-order high resolution Radon transform.OGP,2014,49(1):95-100,131.
[4]范景文,李振春,宋翔宇等.各向异性高分辨率Radon变换压制多次波.石油地球物理勘探,2016,51(4):665-669.Fan Jingwen,Li Zhenchun,Song Xiangyu et al.Suppression of multiple waves by anisotropic high resolution Radon transform.OGP,2016,51(4):665-669.
[5]谢俊法,孙成禹,韩文功.迭代抛物Radon变换法分离一次波与多次波.石油地球物理勘探,2014,49(1):76-81.Xie Junfa,Sun Chengyu,Han Wengong.Separation of primary and multiple waves by iterative parabolic Radon transform.OGP,2014,49(1):76-81.
[6]刘喜武,刘洪,李幼铭.高分辨率Radon变换方法及其在地震信号处理中的应用.地球物理学进展,2004,19(1):8-15.Liu Xiwu,Liu Hong,Li Youming.High resolution radon transform and its application in seismic signal processing.Progress in Geophysics,2004,19(1):8-15.
[7]安鹏.基于高分辨率Radon变换的波场分离方法研究[学位论文].山东东营:中国石油大学(华东),2009.An Peng.Wave Field Separation Technology Research Based on High-resolution Radon Transform[D].China University of Petroleum(East China),Dongying,Shandong,2009.
[8]Sacchi M,Ulrych T.High resolution velocity gathers and offset space reconstruction.Geophysics,1995,60(4):1169-1177.
[9]Sacchi M,Porsani M.Fast high resolution parabolic Radon transform.SEG Technical Program Expanded Abstracts,1999,18:1477-1480.
[10]Trad D,Ulrych T,Sacchi M.Latest view of the sparse Radon transform.Geophysics,2003,68(1):386-399.
[11]Figueiredo M A T,Nowak R D,Wright S J.Gradient projection for sparse reconstruction:application to compressed sensing and other inverse problems.IEEEJournal of Selected Topics in Signal Processing,2007,1(4):586-597.
[12]Chen S S,Donoho D L,Saunders M A.Atomic decomposition by basis pursuit.SIAM Journal of Scientific Computing,1998,20(1):33-61.
[13]Mohimani H,Babaie-Zadeh M and Jutten C.Fast sparse representation using smoothed L0 norm.IEEE Transactions on Signal Processing,2007,46(6):1-12.
[14]Mohimani H,Babaie-Zadeh M,Jutten C.A fast approach for overcomplete sparse decomposition based on smoothed L0 norm.IEEE Transactions on Signal Processing,2009,57(1):289-301.
[15]Hyder M M and Mahata K.An improved smoothed L0 approximation algorithm for sparse representation.IEEE Transactions on Signal Processing,2010,58(4):2194-2205.
[16]ZayyaniH,Babaie-ZadehM.Thresholdedsmoothed-L0dictionary learning for sparse representations.IEEEInternational Conference on Acoustics,Speech and Signal Processing,2009,1825-1828.
[17]Ghalehjegh S H,Babaie-Zadeh M,Jutten C.Fast blocksparse decomposition based on SL0.9th International Conference on Latent Variable Analysis and Signal Separation,2010,426-433.
[18]林婉娟,赵瑞珍,李浩.用于压缩感知信号重建的NSL0算法.新型工业化,2011,1(7):78-84.Lin Wanjuan,Zhao Ruizhen,Li Hao.The NSL0algorithm for compressive sensing signal reconstruction.Industrialization,2011,1(7):78-84.
[19]祁锐,李宏伟,张玉洁.基于改进光滑L0范数的块稀疏信号重构算法.计算机工程,2015,41(11):294-298.Qi Rui,Li Hongwei,Zhang Yujie.Block sparse signal reconstruction algorithm based on improved smoothed L0 norm.Computer Engineering,2015,41(11):294-298.
[20]杨良龙,赵生妹,郑宝玉等.基于SL0压缩感知信号重建的改进算法.信号处理,2012,28(6):834-841.Yang Lianglong,Zhao Shengmei,Zheng Baoyu et al.The improved reconstruction algorithm for compressive sensing on SL0.Signal Processing,2012,28(6):834-841.
[21]曹静杰,王彦飞,杨长春.地震数据压缩重构的正则化与零范数稀疏最优化方法.地球物理学报,2012,55(2):596-607.Cao Jingjie,Wang Yanfei,Yang Changchun.Seismic data restoration based on compressive sensing using the regularization and zero-norm sparse optimization.Chinese Journal of Geophysics,2012,55(2):596-607.
[22]张良,韩立国,刘争光等.基于压缩感知和Contourlet变换的地震数据重建方法.石油物探,2017,56(6):804-811.Zhang Liang,Han Liguo,Liu Zhengguang et al.Seismic data reconstruction based on compressed sending and Contourlet transform.GPP,2017,56(6):804-811.