From surface seismic measurements, whether reflection or refraction, it is
possible to obtain the horizontal component of the slowness. However,
for estimating anisotropy, additional
subsurface information (layer thicknesses or
vertical slownesses) is required (Levin, 1978). For this reason, in recent
studies where anisotropy has been quantified, either a different geometry like
VSP has been used (Byun and Corrigan, 1990; White et al., 1983) or the surface
seismic information has been combined with well-logs (Banik, 1984). However,
based on the observation that velocity anisotropy does not affect *P*-wave
moveout considerably, Winterstein (1986) estimated the required layer
thicknesses using velocities obtained form *P*-wave velocity analysis and
then, from *SH*-wave velocity analysis, he was able to estimate velocity
anisotropy. Dellinger (1989) concludes that because of the ill-conditioning
of the problem, it is not possible to estimate with high accuracy a 2-D vector
velocity field from VSP-like geometries.

From cross-well measurements, fewer attempts have been made to estimate velocity
anisotropy. Karrenbach (1989) proposes a practical scheme for
estimating velocity and *Q* in homogeneous transversely isotropic media,
fitting elliptical curves to the dispersion relations. Cunha-Filho (1990)
fits elliptical curves in layered models to cross-well traveltimes.
Chapman and Pratt (1990) and Pratt and Chapman (1990) estimate
velocity anisotropy in a general 2-D medium assuming, in contrast
with the works of Karrenbach and Cunha-Filho, weak anisotropy. This
allows the ray tracing to be performed in isotropic media.

The main difference between tomographic velocity estimation from surface measurements and cross-well measurements is that the former requires to know a priori the depths of the reflectors whereas the later does not. The only positions needed to estimate velocities from cross-well traveltimes are the source-receiver locations. When sources and receivers are located at depth, the cross-well configuration eliminates one non-linearity of the tomographic inversion of reflection traveltimes: unknown positions of reflectors. By eliminating this non-linearity, the estimation of other non-linear effects in the traveltimes, such as velocity anisotropy, should be simpler when cross-well traveltimes are used.

In this paper we present a tomographic technique to estimate velocity
anisotropy if the medium is transversely isotropic
with vertical symmetry axis.
The technique presented here generalizes the
idea of tomographic inversion in isotropic media (McMechan, 1983)
where
the model is discretized into orthogonal regions and the
Jacobian is related to the intersection of the
rays with all those regions. Instead of
using a circular relationship between time and distance, we assume
that an elliptical relation.
Both
components of the slowness are estimated *simultaneously*,
without using any additional information.

When fitting ellipses to the traveltimes we obtain:
*S*_{x} and *S*_{znmo} (for cross-well geometries) and
*S*_{xnmo} and *S*_{z} (for VSP-like geometries). In the previous notation,
``*nmo*'' highlights the slownesses poorly sampled by the corresponding
recording geometry. These slownesses do not necessarily correspond
to the
true slownesses of the medium.
However, they can be used to estimate the real slowness
surface using an approximate expression derived by Muir (1990).

We study the effects of the limited view of the measurements (from cross-well geometries) in the estimation of both slowness components. We conclude that, our technique is stable when used to invert 1-D (layered) models if the range of ray angles is ``wide enough''. In 2-D models, the estimation of lateral variations in the vertical component of the slowness is particularly difficult from cross-well geometries alone. Consequently, spatial variations in velocity anisotropy cannot be estimated at the same scale of variations in velocity.

The theory presented in the first part of the paper is illustrated with synthetic examples and applications to field data from a cross-well geometry.

12/18/1997