The procedure for estimating elastic constants from P-, SV- and SH-wave traveltimes can be summarized as tomographic estimation of elliptical velocities and transformation of the elliptical velocities into elastic constants. These two steps have opposite requirements in terms of data aperture. On the one hand, the mapping from elliptical velocities to elastic constants requires velocities estimated from rays that travel as closely as possible to one axis of symmetry (Michelena, 1992c). 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. Therefore, the aperture of the traveltimes used for the inversion should satisfy the following two conditions simultaneously: it shouldn't be too large because otherwise the elliptical approximation may not be adequate, and it shouldn't be too small because otherwise the tomographic estimation of elliptical velocities fails, even if the medium is actually isotropic.
Large ray angles are important for the estimation of moderate and large dips in the medium. Since the procedure doesn't allow the use of large ray angles in the inversion of P- and SV-wave traveltimes, I assume that the dips in the medium are small. If the dips are not small, they can be estimated first from SH-wave, wide-aperture traveltimes (that are truly elliptical), and the result can be used to constrain the boundaries in the inversion of P- and SV-wave data.
The axes of symmetry of the different homogeneous blocks that describe the model are assumed to be vertical or near vertical.[They can also be horizontal or near horizontal. The algorithm works equally well in either case because the axes of symmetry of the ellipses are not constrained to be either the major or the minor axis, as explained in Michelena (1992a).] Therefore, when starting the iterations in the anisotropic traveltime tomography by assuming vertical axes of symmetry, the actual inclinations can be found while the estimation of the elliptical velocities remains accurate, regardless of the wave type. If the axes of symmetry are neither vertical nor close to vertical, we need to find their inclination first by fitting SH-wave traveltimes with heterogeneous elliptically anisotropic models, as explained in Michelena (1992a). Once the inclination of the axes of symmetry of the different blocks is known, the elliptical group velocities of P- and SV-waves at each block are estimated using only rays that travel near the axes of symmetry. This process assumes that the axes of symmetry of the different blocks are in the same plane of the survey, as explained also in Michelena (1992a).
In summary, in the absence of SH-wave traveltimes, the medium is assumed to be horizontally layered with vertical axes of symmetry. Small departures from this initial assumption can also be estimated. Larger variations from this initial guess require elliptical SH-wave traveltimes that allow the use of large data apertures.
When the inclination of the axes of symmetry varies across the medium, the estimated elastic constants are referred to different coordinate frames, one for each different axis of symmetry. For purposes of interpretation, having the elastic constants referred to different frames is not a problem as long as we also use the inclination of the axes of symmetry. However, for further computations (finite difference modeling, for example) it might be necessary to transform the elastic constants to a common frame. This transformation can be done by using Bond's matrices (Auld, 1990).