A unified approach to ray bending, ray perturbation and paraxial ray theories

Abstract

SUMMARY In the past, perturbation theory has been formulated for the case that either a slowness model was perturbed and the effect of this perturbation on rays was determined, or for the case where the slowness was fixed and where estimates of the ray position were deformed towards the true ray. In this paper both problems are combined in a single perturbation theory. The theory also accommodates arbitrary perturbations to the endpoints of rays and leads to a simple linear differential equation for the ray perturbation. Expressions are derived for the second-order perturbation of the traveltime. This quantity describes the effect of the ray perturbation on the traveltime and of the bias in the traveltime due to the fact that the reference curve need not be a true ray. The second-order traveltime perturbation can be evaluated efficiently by a single integration along the reference curve. In contrast to formalisms using ray-centred coordinates, endpoints perturbations in an arbitrary direction are allowed. This is of importance in tomographic inversions which incorporate earthquake relocations. The cross-term between the slowness perturbations and the source relocations is derived explicitly. The fact that the reference curve does not need to be a true ray in the reference medium allows for an iterative application of ray perturbation theory. The use of the second-order traveltime perturbation allows one to correct for the bias in the traveltime due to the fact that the reference curve is not a ray. A proof is given that the equation for the ray perturbation is consistent with earlier results derived in ray-centred coordinates and the relation with the ray bending theory of Julian & Gubbins (1977) is established. For a fixed-slowness model and for fixed-ray endpoints the two theories are equivalent except at isolated points, this is illustrated with an analogy from classical mechanics. This difference, which results in superior numerical properties for the new algorithm, is illustrated by several numerical examples.