Ray perturbation theory is concerned with the change in ray paths and travel times due to changes in the slowness model or the end-point conditions of rays. Several different formulations of ray perturbation theory have been developed. Even for the same physical problem different perturbation equations have been derived. The reasons for this is that ray perturbation theory contains a fundamental ambiguity. One can move a point along a curve without changing the shape of the curve. This means that the mapping from a reference curve to a perturbed curve is not uniquely defined, because one may associate a point on the reference curve with different points on the perturbed curve. The mapping that is used is usually defined implicitly by the choice of the coordinate system or the independent parameter. In this paper, a formalism is developed where one can specify explicitly the mapping from the reference curve to the perturbed curve by choosing a stretch factor that relates increments in arc length along the reference curve and the perturbed curve. This is incorporated in a theory that is accurate to first order in the ray position and to second order in the travel time. The second order travel time perturbation describes the effect of changes in the position of the ray on the travel time. For first arrivals, rays are paths of minimum travel time. The travel time along a ray estimate is therefore necessarily longer than the travel time along the true ray.

The resulting travel time bias is described to leading order by the second order perturbation of the travel time. This quantity may be of great importance in nonlinear travel time tomography. Several existing perturbation equations are shown to correspond to the general equation derived in this paper for specific choices of the stretch factor. Depending on the stretch factor, analytical solutions for the ray perturbation can be found for different models of the reference slowness. For two-point ray tracing problems, the arc length (measured in terms of the independent parameter) may change with the ray perturbation. It is shown both numerically and analytically that for two-point ray tracing problems, one must introduce a tuning parameter in the perturbation problem. Leaving out such a tuning parameter, as is done in current Hamiltonian ray perturbation theory, may lead to erroneous solutions. In the formulation of this paper, paraxial ray perturbations, slowness perturbations, and pure ray bending are treated in a uniform fashion. This may be very useful in nonlinear tomographic inversions which include earthquake relocation. ¿ American Geophysical Union 1993