Random rays and seismic amplitude anomalies

Abstract

We give an explanation of the phenomenon, sometimes observed in exploration seismology, of anomalously large amplitudes which seem inconsistent with the traveltime curves when the data are interpreted as resulting from reflections from smooth interfaces of piece‐wise homogeneous media. Monte Carlo simulations illustrate how this phenomenon can occur when the homogeneous media have small, smooth, random velocity fluctuations which vary on a length scale which is large compared with a wavelength but small compared with the propagation distance. Synthetic gathers of reflections from a single plane‐stratified layer with and without the random lateral inhomogeneities produce an amplitude anomaly which is related to the random occurrence of a caustic; limit theorems for stochastic differential equations provide a theory. Theoretical curves, giving the probability of first occurrence of this phenomenon along a ray as a function of propagation distance (for plane waves and for point and line sources in two and three dimensions) are qualitatively similar: they have an initial flat portion where amplitude anomalies are very unlikely, rise to a peak at the distance most likely for first occurrence, and decay exponentially to zero, thus predicting that the phenomenon will occur at some finite distance with probability one.