实现稀疏反褶积的预条件双共轭梯度法
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摘要
地震勘探稀疏反褶积计算一般要导出一个Toeplitz矩阵的线性系统,通常可以用矩阵求逆、Levison递推及共轭梯度等方法直接求解。当Toeplitz矩阵的条件数很大时,数值稳定性差,甚至无法求解。使用共轭梯度法,在矩阵的对角元素上加入规则化因子,可以改善这种情况,但不能彻底解决数值稳定性和精度问题。若求解最小二乘问题的原始问题,结果会好些。线性系统形式的细微改变,将导致不同的数值计算特性。在规则化策略基础上,可巧妙地构造稀疏反褶积的问题原型,引入预条件,采用双共轭梯度法求解,从而实现稀疏反褶积,获得较好结果。数值算例表明,预条件双共轭梯度法比直接稀疏反褶积方法收敛快、精度高。
A Toeplitz matrix is derived in seismic deconvolution implementation , which can be solved by matrix inversion, Levision recursion and conjugate grad ient methods. The condition number of the Toeplitz matrix may, however, be too l arge to solve in a satisfied numerical stability and accuracy. Although the solu tion may be improved when the regularization parameter is introduced, it can not give the complete solution. It is almost better to solve the original problems instead of the Toeplitiz matrix linear system. In this paper, the authors presen t a subtle construction of the problem prototype based upon regularization stra tegy by introducing preconditional matrix and solve the new linear system by du al conjugate gradient (DCG) algorithm with fast convergence and high accuracy. M oreover, the sparse deconvolution is implemented by PDCG with satisfied performa nce.
A Toeplitz matrix is derived in seismic deconvolution implementation , which can be solved by matrix inversion, Levision recursion and conjugate grad ient methods. The condition number of the Toeplitz matrix may, however, be too l arge to solve in a satisfied numerical stability and accuracy. Although the solu tion may be improved when the regularization parameter is introduced, it can not give the complete solution. It is almost better to solve the original problems instead of the Toeplitiz matrix linear system. In this paper, the authors presen t a subtle construction of the problem prototype based upon regularization stra tegy by introducing preconditional matrix and solve the new linear system by du al conjugate gradient (DCG) algorithm with fast convergence and high accuracy. M oreover, the sparse deconvolution is implemented by PDCG with satisfied performa nce.
引文
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[4] PressWH.NumericalRecipes[M].NewYork:CambridgeUniversityPress,1992.
[2] RobinsonEA,TreitelS.Geophysicalsignalanalysis[M].prentice-Hall,Inc.1980.
[3] SacchiMD,UlrychTJ.ImprovingresolutionofRadonoperatorsusingamodelre-weightedleastsquaresprocedure[J].JournalofSeis-micExploration,1995,4:315.
[4] PressWH.NumericalRecipes[M].NewYork:CambridgeUniversityPress,1992.
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