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 W11 and W44 or to check whether the assumption of transverse isotropy is valid or not, in particular when using cross-well measurements where vertical velocities are not well sampled.
The combined application of this technique and tomographic methods to estimate spatial variations of elastic constants will be presented elsewhere.