To estimate spatial variations in the elastic constants, the first step is to describe the model as a superposition of homogeneous blocks that incorporate our previous knowledge about the structure. Then, the elliptical group slownesses of the different wave types are estimated tomographically, as explained by Michelena et al. (1992). Finally, local transformations from these elliptical group velocities to elastic constants are performed at each homogeneous block by using the procedure for homogeneous media explained in the preceding section.
Estimation of elastic constants from narrow aperture traveltimes and tomographic estimation of elliptical group velocities have opposite requirements in terms of data aperture. On the one hand, the estimation of elastic constants requires traveltimes from rays that travel as close as possible to one axis [this is how equations (22) and (23) were derived.] One the other hand, the tomographic estimation of elliptical velocities requires wide ray angles to improve the conditioning of the problem, the accuracy of the NMO velocities and the spatial resolution of the result. Fortunately, as the examples will show, there is a common range of ray angles in which accurate results can be obtained with both procedures. However, the accuracy will depend in general on the complexity of the heterogeneity. Problems can be anticipated if the velocity varies rapidly.