对比抛物Radon正变换几种矩阵求解方法
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摘要
抛物Radon变换法(Parabolic Radon Transform)在地震资料处理中有广泛的应用。PRT可对不同频率的地震数据解耦处理,这一特点使得抛物Radon变换的计算效率比双曲Radon变换有数量级上的提高。在频率域求解时,需要对每一个频率成份求解同样大小的线性方程组。求解抛物Radon正变换的计算方法主要有Levinson递推法、共轭梯度法、Cholesky分解法和直接矩阵求逆法。最小平方抛物Radon正变换所形成的矩阵具有Toeplitz结构,可采用Levinson递推法进行计算。高分辨率抛物Radon正变换所形成矩阵的Toeplitz结构被破坏,一般采用共轭梯度法或Cholesky分解法进行求解。这里详细推导了复Toeplitz矩阵的Levinson递推算法,并分别对求解方程的四种方法进行了讨论,最后给出抛物Radon正变换求解的数值算例,并对所给出的四种方程求解方法的计算效率及计算精度进行了对比。
Parabolic Radon transform(PRT) has widely been used in seismic data processing. PRT can deal with seismic data for any frequency that means decoupling in frequency domain and this characteristic is superior in calculation efficiency to hyperbolic Radon transform. The calculation matrix is the same size for any frequency, and the solving approaches to PRT mainly used as follow: Levinson recursion algorithm, conjugate gradient algorithm, Cholesky decomposition approach and direct matrix inverse algorithm. The Levinson recursion algorithm is adopted in traditional least-square PRT because of the solved matrix has Toeplitz structure, and the Toeplitz structure is destroyed in matrix in high resolution PRT, therefore the conjugate gradient algorithm or the Cholesky decomposition is used to solve the linear equations. The detailed calculation steps of Levinson recursion algorithm are derived in this paper for complex Toeplitz matrix, and the four algorithms mentioned are also discussed in this paper. And the paper also gives a simple parabolic Radon forward transform example to demonstrate the efficiency and precision of different algorithms.
Parabolic Radon transform(PRT) has widely been used in seismic data processing. PRT can deal with seismic data for any frequency that means decoupling in frequency domain and this characteristic is superior in calculation efficiency to hyperbolic Radon transform. The calculation matrix is the same size for any frequency, and the solving approaches to PRT mainly used as follow: Levinson recursion algorithm, conjugate gradient algorithm, Cholesky decomposition approach and direct matrix inverse algorithm. The Levinson recursion algorithm is adopted in traditional least-square PRT because of the solved matrix has Toeplitz structure, and the Toeplitz structure is destroyed in matrix in high resolution PRT, therefore the conjugate gradient algorithm or the Cholesky decomposition is used to solve the linear equations. The detailed calculation steps of Levinson recursion algorithm are derived in this paper for complex Toeplitz matrix, and the four algorithms mentioned are also discussed in this paper. And the paper also gives a simple parabolic Radon forward transform example to demonstrate the efficiency and precision of different algorithms.
引文
[1]YILMAZ O.Seismic data processing[M].Society of Exploration Geophysicists,1987.
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[7]KOSTOV C.Toeplitz structure in slant-stack inver-sion:60th Annual Internat.Mtg.,Soc.Expl[C].Geophys.Expanded abstracts,1990.
[8]刘喜武,刘洪.实现稀疏反褶积的预条件双共轭梯度法[J].物探化探计算技术.2003,25(3):215.
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[2]HAMPSON D.Inverse velocity stack for multiple elimi-nation:56thAnn.Internat.Mtg.,Soc.Expl[M].Geophys,Expanded Abstracts,1986.
[3]王维红.叠前地震数据重建和多次波衰减:加权和结合多次波预测的Radon变换方法[D].北京:中国科学院地质与地球物理研究所,2005.
[4]PRESS W H,FLANNERY B P,TEUKOLSKY S A,et al.Numerical recipes[M].Cambridge Univ.Press,1986.
[5]SACCHI M,ULRYCH T.High-resolution velocity gather and offset space reconstruction[J].Geophysics,1995,60:1169.
[6]DARCHE G.Spatial interpolation using a fast parabolic Radon transform,60th Annual Internat.Mtg.,Soc.Expl[C].Geophys.,Expanded abstracts,1990.
[7]KOSTOV C.Toeplitz structure in slant-stack inver-sion:60th Annual Internat.Mtg.,Soc.Expl[C].Geophys.Expanded abstracts,1990.
[8]刘喜武,刘洪.实现稀疏反褶积的预条件双共轭梯度法[J].物探化探计算技术.2003,25(3):215.
[9]SACCHI M D,PORSANI M.Fast high resolution para-bolic Radon transform,69th Annual Internat.Mtg.,Soc.Expl[C].Geophys.,Expanded abstracts,1999:1477.
[10]阮百尧,熊彬.大型对称变带宽方程组的Cholesky分解法[J].物探化探计算技术.2000,22(4):361.
[11]KABIR M M N,MARFURT K J.Toward true ampli-tude multiple removal[J].The Leading Edge,1999,18:66.
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