I have shown how
to estimate the elastic constants of homogeneous TI media from
*P*-, *SV*-, and *SH*-wave traveltimes
near a single axis of symmetry (either from VSP or cross-well geometries).
For heterogeneous TI media, the procedure can be easily generalized
by using tomographic techniques (Michelena et al. 1992;
Michelena, 1992a).

The technique uses the parameters obtained by fitting
traveltimes near one axis with elliptical models.
For *SH*-wave traveltimes, the estimation of the corresponding
TI elastic constants is trivial because *SH*-wave phase
velocities are also elliptical in TI media.
For *P*- and *SV*-wave traveltimes,
four parameters are needed to estimate
the phase velocities at all angles from measurements near one axis. These
parameters are the direct and *NMO* phase velocities for
*P*- and *SV*-waves.
The transformation from elliptical parameters
to elastic constants is simple for both VSP and cross-well
geometries.

The accuracy of the estimation of the elastic
constants from the elliptical parameters depends on the accuracy
of the estimation of the *NMO* velocities.
For small ray angles the approximation is excellent when using
either VSP or cross-well geometries because the elliptical approximations
are accurate in both cases. In practice, however, using only small
ray angles may hinder an accurate estimation of *NMO* velocities.
For intermediate ray angles, the estimation of
vertical *P*-wave velocities
from cross-well geometries
is more accurate than the estimation of horizontal *P*-wave
velocities from VSP.
This is because the elliptical approximation for *P*-wave impulse
response
fits wider angles around the horizontal than around the vertical
where the group velocity is smaller.
For large ray angles the approximation doesn't work because
elliptical fits are not as accurate.

When compressional and shear velocity logs are also available,
they can be used either to add redundancy in the estimation of *W _{11}* and

The combined application of this technique and tomographic methods to estimate spatial variations of elastic constants will be presented elsewhere.

11/17/1997