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.

11/17/1997