Delta-Matrix Factorization for Fast Propagation through Solid Layers in a Fluid-Solid Medium

Abstract

Propagator techniques for computation of wave propagation in layered fluid-solid media are well established in underwater acoustics and seismology. Delta matrices for solid layers provide a convenient way of computing the boundary values at a fluid-solid interface, with loss-of-precision control. A certain vector is propagated through a sequence of multiplications with delta matrices. We show that computations of this kind can be performed more efficiently if each delta matrix is decomposed as a product of sparse matrices which are applied in sequence. We propose two kinds of delta-matrix factorizations. In connection with dispersion computations, our first factorization turns out to give a method that is very much related to the powerful "fast form" of Knopoff's method. Our algorithm is slightly more efficient, however. Furthermore, the derivation gives insight into the structure of the delta-matrix propagator and other applications, such as propagation in connection with wavenumber integration for point-source studies, become immediate. Our second delta-matrix factorization gives a dispersion-function method that is significantly faster than the "fast form" of Knopoff's method. The most conspicuous gains in comparison to previous methods, however, appear for applications to wavenumber-integration techniques such as the reflectivity method. Very few arithmetic operations are needed. This factorization also provides a good basis for analyzing the numerical performance of delta-matrix propagation. It becomes quite obvious how to perform scaling so as to avoid overflow for arbitrarily thick homogeneous layers without layer splitting. In connection with multi-frequency computations distinct advantages are obtained.