Toeplitz矩阵循环延拓后的特征值数值分析
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摘要
本文采用数值分析的方法探讨Toeplitz矩阵延拓成ω循环矩阵时特征值的逼近程度.对于对称共轭型Toeplitz矩阵,采用ω=±i时对应的循环矩阵特征值的逼近程度较好;对于其它Toeplitz矩阵,采用共轭转置将其转化为对称共轭型矩阵后,才有利于特征值的逼近.可将本文方法广泛应用于地球物理中的数值计算(如位场计算、信号处理中的反褶积、地震资料的偏移处理等).
This paper discusses how ω circulant matrices approach Toeplitz matrices and the changes in the corresponding eigenvalues.For symmetric conjugate Toeplitz matrices,choosing ω=±i circulant matrices can approach perfectly;while for any other Toeplitz mactrices,applying conjugate operators to them to become symmetric conjugate ones and then,a good approach of eigenvalues can often be expected.The ideas in the paper can be widely used geophysical computation such as potential field、deconvolution and migration of seismic data.
This paper discusses how ω circulant matrices approach Toeplitz matrices and the changes in the corresponding eigenvalues.For symmetric conjugate Toeplitz matrices,choosing ω=±i circulant matrices can approach perfectly;while for any other Toeplitz mactrices,applying conjugate operators to them to become symmetric conjugate ones and then,a good approach of eigenvalues can often be expected.The ideas in the paper can be widely used geophysical computation such as potential field、deconvolution and migration of seismic data.
引文
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[14]杨文采.地球物理反演的理论与方法[M].北京:地质出版社,1997.Yang W C.Theory and Methods of Geophysical Inversion(inChinese)[M].Beijing:Geology Press,1997.
[15]Clearbout J.Multidimensional recursive filters via a helix[J].Geophysics,1998,63(5):1532-1541.
[16]陈生昌,张博.基于波场垂向和水平方向外推的高陡构造偏移[J].地球物理学报,2012,55(4):1300-1306.Chen S C,Zhang B.Steeply-dipping structures migrationbased on the wavefield vertical-and horizontal-extrapolation[J].Chinese J.Geophys.(in Chinese),2012,55(4):1300-1306.
[17]刘定进,杨瑞娟,罗申玥,等.稳定的保幅高阶广义屏地震偏移成像方法研究[J].地球物理学报,2012,55(7):2402-2411.Liu D J,Yang R J,Luo S Y,et al.The method ofpreserved-amplitude seismic migration imaging with stablegeneralized high order screen[J].Chinese J.Geophys.(inChinese),2012,55(7):2402-2411.
[18]杨仁虎,常旭,刘伊克.叠前逆时偏移影响因素分析[J].地球物理学报,2010,53(8):1902-1913.Yang R H,Chang X,Liu Y K.The influence factorsanalyses of imaging precision in pre-stack reverse timemigration[J].Chinese J.Geophys.(in Chinese),2010,53(8):1902-1913.
[2]Sacchi M D,Wang J F,Kuehl H.Regularized migration/inversion:New generation of seismic imaging algorithm.CSEG recorder,2006Special Edition.54-59.(期刊类型)
[3]Wang Y F,Yang C C.Accelerating migration deconvolutionusing a nonmonotone gradient method[J].Geophysics,2010,75(4):131-137.
[4]王彦飞,杨长春,段秋梁.地震偏移反演成像的迭代正则化方法研究[J].地球物理学报,2009,52(6):1615-1624.Wang Y F,Yang C C,Duan Q L.On iterative regularizationmethods for migration deconvolution and inversion in seismicimaging[J].Chinese J.Geophys.(in Chinese),2009,52(6):1615-1624.
[5]李振华,王彦飞,杨长春.正则化偏移成像的全局优化快速算法[J].地球物理学报,2011,54(3):828-834.Li Z H,Wang Y F,Yang C C.A fast global optimizationalgorithm for regularized migration imaging[J].Chinese J.Geophys.(in Chinese),2011,54(3):828-834.
[6]Tikhonov A N,Arsenin V Y.Solutions of Ill Posed Problems[M].New York:John Wiley and Sons.1977.
[7]肖庭延,于慎根,王彦飞.反问题的数值解法[计算方法丛书][M].北京:科学出版社,2003.Xiao T Y,Yu S G,Wang Y F.Numerical Methods forInverse Problem[Series of the Cmputational Methods](inChinese)[M].Beijing:Science Press,2003.
[8]梅金顺,刘洪.ω循环型边界条件[J].地球物理学报,2003,46(6):835-841.Mei J S,Liu H.ω-circulant boundary condition[J].ChineseJ.Geophys.(in Chinese),2003,46(6):835-841.
[9]梅金顺,刘洪.预条件方程组及其应用[J].地球物理学报,2004,47(4):718-722.Mei J S,Liu H.Preconditioned equation sets and theirapplications[J].Chinese J.Geophys.(in Chinese),2004,47(4):718-722.
[10]程乾生.信号数字处理的数学原理[M].北京:石油工业出版社,1979.Cheng Q S.Mathematical Principles of Data Processing forSeismic Signal(in Chinese)[M].Beijing:Petroleum IndustryPress,1979.
[11]蒋增荣,曾泳泓,余品能.快速算法[M].长沙:国防科技大学出版社,1993.Jiang Z R,Zeng Y H,Yu P N.Fast Algorithm(in Chinese)[M].Changsha:National Defence Scientific UniversityPress,1993.
[12]梅金顺,刘洪.Toeplitz方程组的近似计算[J].地球物理学进展,2003,18(1):128-133.Mei J S,Liu H.Approximate computation of Toeplitzsystems of equations[J].Progress in Geophysics(inChinese),2003,18(1):128-133.
[13]徐仲,张凯院,陆全.Toeplitz矩阵类的快速算法[M].西安:西北工业大学出版社,1999.Xu Z,Zhang K Y,Lu Q.Quick Algorithm of the ToeplitzMatrices(in Chinese)[M].Xi’an:Northwest IndustryUniversity Press,1999.
[14]杨文采.地球物理反演的理论与方法[M].北京:地质出版社,1997.Yang W C.Theory and Methods of Geophysical Inversion(inChinese)[M].Beijing:Geology Press,1997.
[15]Clearbout J.Multidimensional recursive filters via a helix[J].Geophysics,1998,63(5):1532-1541.
[16]陈生昌,张博.基于波场垂向和水平方向外推的高陡构造偏移[J].地球物理学报,2012,55(4):1300-1306.Chen S C,Zhang B.Steeply-dipping structures migrationbased on the wavefield vertical-and horizontal-extrapolation[J].Chinese J.Geophys.(in Chinese),2012,55(4):1300-1306.
[17]刘定进,杨瑞娟,罗申玥,等.稳定的保幅高阶广义屏地震偏移成像方法研究[J].地球物理学报,2012,55(7):2402-2411.Liu D J,Yang R J,Luo S Y,et al.The method ofpreserved-amplitude seismic migration imaging with stablegeneralized high order screen[J].Chinese J.Geophys.(inChinese),2012,55(7):2402-2411.
[18]杨仁虎,常旭,刘伊克.叠前逆时偏移影响因素分析[J].地球物理学报,2010,53(8):1902-1913.Yang R H,Chang X,Liu Y K.The influence factorsanalyses of imaging precision in pre-stack reverse timemigration[J].Chinese J.Geophys.(in Chinese),2010,53(8):1902-1913.
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