Catastrophe theory, caustics and traveltime diagrams in seismology

Abstract

A commonly used technique in the analysis of body wave seismograms is to display the times of arrival of geometric rays as a function of position. When caustics are present, these traveltime diagrams are multi‐valued functions of the position coordinate. Catastrophe theory may be used to classify the caustics and their associated traveltime diagrams. For waves which are excited by a compact source region in an earth model whose properties vary as a function of depth only or depth and range only, both the caustics and the associated traveltime curves consist of certain sections of one of the cuspoid catastrophes. Causality and the observation that transverse sections of the caustics are almost always seen provide additional constraints. The simplest non‐trivial example of a traveltime curve consistent with all of these constraints is the familiar triplication. As an example of how the ideas presented might be applied, a stable and rapidly converging iterative algorithm is presented which allows the complete traveltime curve through a triplication to be accurately reconstructed from a knowledge of Ti(pi, Ri) for a small number of first arrivals in the vicinity of the triplication. Such information, which would be difficult to obtain without the aid of the ideas presented, can easily be incorporated into a variety of inversion algorithms (i.e. Herglotz‐Wiechert or τ–p). The algorithm, including a Herglotz–Wiechert inversion, is demonstrated using synthetic data. Copyright © 1987, Wiley Blackwell. All rights reserved