预条件共轭梯度法在地震数据重建方法中的应用
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摘要
基于最小平方的Fourier地震数据重建方法最终转化为求解一个线性方程组,其系数矩阵是Toeplitz矩阵,可以用共轭梯度法求解该线性方程组.共轭梯度法的迭代次数受系数矩阵病态程度的影响,地震数据的非规则采样程度越高,所形成的系数矩阵病态程度越高,就越难收敛和得到合理的计算结果.本文研究了基于Toeplitz矩阵的不同预条件的构造方法,以及对共轭梯度法收敛性的影响.通过预条件的使用,加快了共轭梯度法的迭代速度,改进了共轭梯度算法的收敛性,提高了计算的效率.数值算例和实际地震数据重建试验证明了预条件共轭梯度法对计算效率有很大的提高.
Seismic data reconstruction based on the least-squares Fourier method is ultimately transformed into solving a linear equation.The coefficient matrix is Toeplitz matrix.The conjugate gradient method can be used to solve the linear equations.Pathological extent of matrix affects iterations of the conjugate gradient method.The more irregular sampled seismic data is,the more pathological the matrix is,then it is more difficult to get convergence and reasonable results.We study different construction methods of preconditions based on Toeplitz matrix and the effects of convergence of the conjugate gradient method.Through the use of preconditions,we can speed up the iterative speed of the conjugate gradient method,improve the convergence of conjugate gradient method and the efficiency of computation.Numerical examples and real seismic data reconstruction experiment show that the preconditioned conjugate gradient method has greatly improved efficiency of calculation.
Seismic data reconstruction based on the least-squares Fourier method is ultimately transformed into solving a linear equation.The coefficient matrix is Toeplitz matrix.The conjugate gradient method can be used to solve the linear equations.Pathological extent of matrix affects iterations of the conjugate gradient method.The more irregular sampled seismic data is,the more pathological the matrix is,then it is more difficult to get convergence and reasonable results.We study different construction methods of preconditions based on Toeplitz matrix and the effects of convergence of the conjugate gradient method.Through the use of preconditions,we can speed up the iterative speed of the conjugate gradient method,improve the convergence of conjugate gradient method and the efficiency of computation.Numerical examples and real seismic data reconstruction experiment show that the preconditioned conjugate gradient method has greatly improved efficiency of calculation.
引文
[1]Schonewille M A.Fourier reconstruction of irregularlysampled seismic data[Ph.D.thesis].Delft:Delft Universityof Technology,2000.
[2]刘喜武,刘洪,刘彬.反假频非均匀地震数据重建方法研究.地球物理学报,2004,47(2):299-305.Liu X W,Liu H,Liu B.A study on algorithm forreconstruction of de-alias uneven seismic data.Chinese J.Geophys.(in Chinese),2004,47(2):299-305.
[3]孟小红,刘国峰,周建军.大间距地震数据重建方法研究.地球物理学进展,2006,21(3):687-691.Meng X H,Liu G F,Zhou J J.The study of reconstructionof large gap seismic data.Progress in Geophys.(in Chinese),2006,21(3):687-691.
[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]Zwartjes P,Gisolf A.Fourier reconstruction with sparseinversion.Geophysical Prospecting,2007,55(2):199-221.
[6]Zwartjes P M,Sacchi M D.Fourier reconstruction ofnonuniformly sampled,aliased seismic data.Geophysics,2007,72(1):V21-V32.
[7]Duijndam A J W,Schonewille M A,Hindriks C O H.Reconstruction of band-limited signals,irregularly sampledalong one spatial direction.Geophysics,1999,64(2):524-538.
[8]熊登.叠前地震数据规则化、重建及噪音压制[博士论文].北京:中国科学院研究生院,2008.Xiong D.Prestack seismic data regularization,reconstructionand noise suppression[Ph.D.thesis](in Chinese).Beijing:Graduate University,Chinese Academy of Sciences,2008.
[9]高建军,陈小宏,李景叶等.基于非均匀Fourier变换的地震数据重建方法研究.地球物理学进展,2009,24(5):1741-1747.Gao J J,Chen X H,Li J Y,et al.Study on reconstruction ofseismic data based on nonuniform Fourier transform.Progress in Geophys.(in Chinese),2009,24(5):1741-1747.
[10]李冰,刘洪,李幼铭.三维地震数据离散光滑插值的共轭梯度法.地球物理学报,2002,45(5):691-699.Li B,Liu H,Li Y M.3-D seismic data discrete smoothinterpolation using conjugate gradient.Chinese J.Geophys.(in Chinese),2002,45(5):691-699.
[11]梅金顺,刘洪.ω循环型边界条件.地球物理学报,2003,46(6):835-841.Mei J S,Liu H.ω-circulant boundary condition.Chinese J.Geophys.(in Chinese),2003,46(6):835-841.
[12]梅金顺,刘洪.Toeplitz方程组的近似计算.地球物理学进展,2003,18(1):128-133.Mei J S,Liu H.Approximate computation of Toeplitzsystems of equations.Progress in Geophys.(in Chinese),2003,18(1):128-133.
[13]梅金顺,刘洪.预条件方程组及其应用.地球物理学报,2004,47(4):718-722.Mei J S,Liu H.Preconditioned equation sets and theirapplications.Chinese J.Geophys.(in Chinese),2004,47(4):718-722.
[14]Chan R H,Ng N K.Conjugate gradient methods for Toeplitzsystems.SIAM Review,1996,38(3):427-482.
[15]Strang G.Introduction to Applied Mathematics.Cambridge:Wellesley-Cambridge Press,1986.
[16]Chan T F.An optimal circulant Preconditioner for Toeplitzsystems.SIAM J.Sci.Stat.Comput.,1988,9(4):766-771.
[17]Chan R H.Circulant preconditioners for Hermitian Toeplitzsystems.SIAM J.Matrix Anal.Appl.,1989,10(4):542-550.
[18]Chan R H,Yip A M,Ng M K.The best circulantpreconditioners for hermitian Toeplitz systems.SIAM J.Numer.Anal.,2001,38(3):876-896.
[19]Chan R H,Tso T M,Sun H W.Circulant preconditionersfrom B-splines.Proceedings to the SPIE Symposium onAdvanced Signal Processing:Algorithms,Architectures,andImplementations VII,1997,3162:338-347.
[20]Chan R H,Yeung M C.Circulant preconditionersconstructed from kernels.SIAM J.Numer.Anal.,1992,29(4):1093-1103.
[2]刘喜武,刘洪,刘彬.反假频非均匀地震数据重建方法研究.地球物理学报,2004,47(2):299-305.Liu X W,Liu H,Liu B.A study on algorithm forreconstruction of de-alias uneven seismic data.Chinese J.Geophys.(in Chinese),2004,47(2):299-305.
[3]孟小红,刘国峰,周建军.大间距地震数据重建方法研究.地球物理学进展,2006,21(3):687-691.Meng X H,Liu G F,Zhou J J.The study of reconstructionof large gap seismic data.Progress in Geophys.(in Chinese),2006,21(3):687-691.
[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]Zwartjes P,Gisolf A.Fourier reconstruction with sparseinversion.Geophysical Prospecting,2007,55(2):199-221.
[6]Zwartjes P M,Sacchi M D.Fourier reconstruction ofnonuniformly sampled,aliased seismic data.Geophysics,2007,72(1):V21-V32.
[7]Duijndam A J W,Schonewille M A,Hindriks C O H.Reconstruction of band-limited signals,irregularly sampledalong one spatial direction.Geophysics,1999,64(2):524-538.
[8]熊登.叠前地震数据规则化、重建及噪音压制[博士论文].北京:中国科学院研究生院,2008.Xiong D.Prestack seismic data regularization,reconstructionand noise suppression[Ph.D.thesis](in Chinese).Beijing:Graduate University,Chinese Academy of Sciences,2008.
[9]高建军,陈小宏,李景叶等.基于非均匀Fourier变换的地震数据重建方法研究.地球物理学进展,2009,24(5):1741-1747.Gao J J,Chen X H,Li J Y,et al.Study on reconstruction ofseismic data based on nonuniform Fourier transform.Progress in Geophys.(in Chinese),2009,24(5):1741-1747.
[10]李冰,刘洪,李幼铭.三维地震数据离散光滑插值的共轭梯度法.地球物理学报,2002,45(5):691-699.Li B,Liu H,Li Y M.3-D seismic data discrete smoothinterpolation using conjugate gradient.Chinese J.Geophys.(in Chinese),2002,45(5):691-699.
[11]梅金顺,刘洪.ω循环型边界条件.地球物理学报,2003,46(6):835-841.Mei J S,Liu H.ω-circulant boundary condition.Chinese J.Geophys.(in Chinese),2003,46(6):835-841.
[12]梅金顺,刘洪.Toeplitz方程组的近似计算.地球物理学进展,2003,18(1):128-133.Mei J S,Liu H.Approximate computation of Toeplitzsystems of equations.Progress in Geophys.(in Chinese),2003,18(1):128-133.
[13]梅金顺,刘洪.预条件方程组及其应用.地球物理学报,2004,47(4):718-722.Mei J S,Liu H.Preconditioned equation sets and theirapplications.Chinese J.Geophys.(in Chinese),2004,47(4):718-722.
[14]Chan R H,Ng N K.Conjugate gradient methods for Toeplitzsystems.SIAM Review,1996,38(3):427-482.
[15]Strang G.Introduction to Applied Mathematics.Cambridge:Wellesley-Cambridge Press,1986.
[16]Chan T F.An optimal circulant Preconditioner for Toeplitzsystems.SIAM J.Sci.Stat.Comput.,1988,9(4):766-771.
[17]Chan R H.Circulant preconditioners for Hermitian Toeplitzsystems.SIAM J.Matrix Anal.Appl.,1989,10(4):542-550.
[18]Chan R H,Yip A M,Ng M K.The best circulantpreconditioners for hermitian Toeplitz systems.SIAM J.Numer.Anal.,2001,38(3):876-896.
[19]Chan R H,Tso T M,Sun H W.Circulant preconditionersfrom B-splines.Proceedings to the SPIE Symposium onAdvanced Signal Processing:Algorithms,Architectures,andImplementations VII,1997,3162:338-347.
[20]Chan R H,Yeung M C.Circulant preconditionersconstructed from kernels.SIAM J.Numer.Anal.,1992,29(4):1093-1103.
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