DISCRETE TIME SOLUTION OF PLANE P‐SV WAVES IN A PLANE LAYERED MEDIUM

Abstract

For plane waves at normal incidence to a layered elastic medium, both the forward and inverse discrete time problems have been previously solved. In this paper the forward problem of calculating the waves in a medium of plane, homogeneous, isotropic layers is extended to P and SV body waves at nonnormal incidence, where the horizontal phase velocity of each wave is greater than the shear and compressional velocities of each layer. Vertical traveltimes for P and SV waves through each layer are rounded off to unequal integer multiples of a small time increment Δτ. This gives a 4×4 layer matrix analogous to the 2×2 layer matrix for normal incidence obtained by previous authors. Reflection and transmission responses recorded at the free surface of a layered half space are derived as matrix series in integer powers of the Fourier transform variable [Formula: see text]. These responses are generated recursively by polynomial division and include all multiply reflected P and SV waves with mode conversions. It is shown that the reflection response matrix generated by a source at the free surface equals the product of a constant matrix and the positive time part of the autocorrelation matrix of the transmission response matrix due to a deep source. This is an extension to nonnormal incidence of a theorem proved by Claerbout for acoustic waves at normal incidence.