适用于三维隐式叠前深度偏移中矩阵求逆的混合算法

A HYBRID METHOD OF MATRIX INVERSION SUITED FOR 3D IMPLICIT PRESTACK DEPTH MIGRATION

在深度偏移方法中 ,把二维隐式方法推广到三维 ,就会面对一个分块对角矩阵求逆问题 .通常 ,这种矩阵的求逆将耗费大量计算时间 ,严重制约了三维隐式方法偏移在实际资料处理中的广泛应用 .在螺旋边界条件下 ,该矩阵H具有Toeplitz结构的正定厄密矩阵 ,其快速求逆可由谱法LU分解或直解法快速实现 .本文结合谱法LU分解和直接解法方法的优点 ,提出了一种混合算法 .文中采用谱分解方法建立起矩阵列元素的谱分解表 ,并采用直解法的递推公式 ,可以快速给出矩阵的分解 .通过与谱法分解和直解法在分解精度和分解速度两方面的比较表明 ,本文方法与谱法相比 ,在非均匀介质中亥姆霍兹算子矩阵分解时的精度提高 10倍 ;在计算速度方面 ,混合方法比简化后的直解法快 .因此 ,该方法的提出 ,在计算精度许可的条件下 ,最大限度地减少三维隐式差分偏移中矩阵求逆占用的时间 ,从而使得该方法能真正用于实际地震资料的处理 .

In depth wavefield migration, in order to extend the 2-D implicit method to 3-D case, a block-like diagonal matrix has to be inverted. General speaking, the matrix inversion will consume a large amount of computation time, which heavily hinder the application of 3-D implicit scheme in waveform migration. Under the helix boundary condition, the Helmhotz matrix has Toeplitz structure with positive definite property. The inversion of matrix can be realized by spectrum LU decomposition and direct solution. This paper proposes a hybrid scheme, which combines the advantages of spectrum LU decomposition and direct solution. Based on the table of matrix element of spectrum decomposition, the Helmhotz matrix can be decomposed by recursion of the direct solution. A numerical comparison has been made with the spectrum LU decomposition and direct solution. The accuracy of Helmhotz matrix is raised 10 times than that of spectrum, and the calculation speed is faster than that of the direct solution. In this way, the method can reduce the time of matrix inversion in 3-dimensional implicit difference migration and can be used to real data processing.