LU Decomposition with Spectral Factorization in Seismic Imaging

Abstract

AbstractThe inversion of Helmholtz matrix H, which is the finite‐difference representation of the Helmholtz operator, plays a critical role in numerical methods of seismic modeling and 3D depth migration. If a helical boundary condition is introduced, the Helmholtz matrix H has a Topelitz structure and it can be efficiently inversed through the spectral LU decomposition method. This study analyzes the efficiency and characters of the spectral LU decomposition method, the errors it brings in the decomposition of matrix H with different velocity models, and the errors distribution and effects on seismic imaging. Our results indicate that, for constant velocity model, all columns of matrix H have same nonzero values and the errors brought by the spectral LU decomposition method will not affect the computation of wave fields if an absorb boundary condition is used at the same time. For a non‐constant velocity model, each column in matrix H has different nonzero values, the errors increase with variability degree of the velocity model, which are fatal to wave fields calculation. Therefore, once a spectral LU decomposition method is used to calculate the wave field in seismic modeling and imaging, the errors caused by this method should be considered to ensure correct wave fields.