The accuracy of the O'Doherty-Anstey approximation for wave propagation in highly disordered stratified media

A pulse propagates normally through a stratified medium. It is known from numerical experiments and heuristic arguments that under certain circumstances, including those arising in seismic exploration, the shape and precise travel time of the forward propagating pulse as it travels over large distances depends to a good approximation only upon the autocorrelation function of the medium properties. The expression for this dependence is surprisingly simple and is known as the O'Doherty-Anstey approximation after the authors who first derived it in 1971. We give a theoretical estimate of the error committed in making this approximation. The analysis proceeds by first obtaining a closed equation governing the evolution of the propagation mode of interest. There is an interesting interplay of frequency- and time-domain considerations. Thus the error is frequency dependent, but depends also on the length of time for which the signal is to be computed. In contrast to earlier related work we give a rigorous error estimate, which is derived in a deterministic context, without the use of small or large parameters. Some previous results involving limits as small parameters tend to zero are recovered by application of our general estimate.