Introduction

Depending on the degree of anisotropy of the medium and the seismic wavelengths used, the tomograms obtained from cross-well traveltime data assuming that the medium is isotropic may suffer from severe distortions. These distortions are analogous to the well known mispositions occurred in surface seismic when, in an anisotropic environment, the stacking velocity is used to control depth conversion. Eliminating these distortions is one reason to allow the model to be anisotropic in tomographic traveltime inversion. If we do that, we are not only solving an imaging problem. It is well known that anisotropy can be a useful tool for studying lithology and degree of stratification in sedimentary rocks and therefore, taking into account velocity anisotropy in tomographic traveltime inversion also helps to gain extra and useful information about the reservoir.

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.