TRAVEL-TIME TOMOGRAPHY IN ANISOTROPIC MEDIA .1. THEORY

In principle, crosshole, traveltime tomography is ideal for directly detecting and measuring seismic anisotropy. The traveltimes of multiple rays with wide angular coverage will be sensitive both to inhomogeneities and to anisotropy. In practice, the traveltimes will depend only on a limited number of the anisotropic velocity parameters, and the data may not be adequate even to determine these parameters uniquely. In addition, trade-offs may exist between anisotropy and inhomogeneities. In this paper, we use the linear perturbation theory for traveltimes in general, weakly anisotropic media to discuss the dependence of traveltimes in 2-D crosshole tomographic experiments on the anisotropic parameters. In a companion paper, we apply the results to synthetic and real data examples. We show that when measurements are restricted to a 2-D plane, the qP and qS traveltimes depend on subsets of the complete set of 21 anisotropic velocity parameters. Formulae are developed for the differential coefficients of the traveltimes with respect to these parameters in piecewise homogeneous and in linearly interpolated models. It is shown how in a generally oriented model element, the local parameters are related to the same parameters in the global model. The parameters that can be determined from 2-D tomographic data do not in general determine the full nature of the anisotropy. Rather, these parameters serve only to describe the intersection of the slowness sheet with the 2-D plane. Since many models may fit this description, additional information on symmetry properties and orientations is required. For example, if a priori information suggests that the anisotropy is transversely isotropic (TI), then we can determine some of the TI parameters and some information on the orientation of the axis of symmetry. Formulae are given relating the general parameters to those of a TI system with general orientation of the symmetry axis. The general formulae for qS traveltimes are intrinsically more complicated than those for qP. In the qS case, the traveltime perturbation depends on the polarization, which in turn depends on the perturbation. This makes the general problem non-linear even for small perturbations. However, the mean qS traveltime and the traveltime dependence on various subsets of parameters are linear. Although linear perturbation theory is invalid for qS rays, degenerate perturbation theory is valid for the calculation of the traveltimes and could be used in a non-linear inversion scheme.