Waves in stratified viscoelastic media with microstructure

A pulse propagates obliquely through a perfectly stratified, isotropic, viscoelastic medium, over a large distance compared with the length scale on which the medium varies. If the ratio of these lengths is ϵ², we shall assume the medium differs from a slowly varying medium by 0(ϵ) (in a weak sense). We show by a perturbation technique how an approximation to the field may be rapidly calculated. The method is closely related to that of Burridge and Chang (1989), but extends the region of validity farther into the wave coda by using the sample autocorrelation instead of the theoretical, ensemble averaged, autocorrelation of reflectivity. In the numerical examples illustrating this naive theory we obtain very good agreement with exact computations using a layer-matrix code for broad pulses and using an 'exact' time-domain code for the impulse response in Goupillaud media at normal incidence. The main error is a small but growing error of timing late in the coda. This may be corrected by using throughout the travel times appropriate to the (local) effective medium. A complete analysis of this correction has yet to be made. The approximate code is between 100 and 1000 times faster than the time-stepping finite-difference code, and about 90 times faster than the layer matrix code for broad pulses for which only low frequencies need be computed.