A Hybrid Matrix Inversion Method for 3‐D Implicit Prestack Depth Migration

Abstract

AbstractIn 3‐D implicit Finite‐Difference depth migration methods, we face a problem to inverse a blocked tri‐diagonal matrix. Generally, the inversion of the matrix may consume a large amount of computational resources, which seriously hinders its application. Using a helical boundary condition, this matrix turns to have a Toeplitz structure and can be inversed through LU decomposition methods, such as the spectral decomposition method and the method of direct elimination decomposition. Here, a new LU decomposition known as hybrid decomposition method, which combines the advantages of the spectral decomposition method and the direct elimination method, is proposed to solve the problem of matrix inversion. The hybrid decomposition method bases on the decomposition results given by the method of spectral decomposition and employs the recursive formula used in the direct elimination method to give a new decomposition result. A numerical comparison has been made among the method of spectral decomposition, the method of direct elimination decomposition and the hybrid decomposition method. The maximum error given by the hybrid decomposition method is about ten times less than that of the method of spectral decomposition and the hybrid decomposition is more efficient than the direct elimination method of decomposition. Therefore, the hybrid method is a compromise of the method of spectral decomposition and the direct elimination method of decomposition. It may help the 3‐D implicit Finite‐Difference method go through its way to real data processing.