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).

11/17/1997