The model space needed to perform seismic traveltime tomography consists of two separate models: one for the velocities and one for the heterogeneities. The selection of each of these models should be made depending on any prior knowledge we may have about the true medium. Even if we don't know anything about the medium, the selection should be guided by the type of information we want the extract from the data.

The most common choice for the model of the velocities is also the simplest one: isotropic. By contrast, the model of heterogeneities is often the most complex: a fine 2-D grid (McMechan, 1983). Although the selection of the model for the velocities is usually guided by a simple form of prior knowledge (i.e, ``isotropic velocities usually work''), the selection for the model of heterogeneities implicitly assumes that we don't know anything about the spatial variations expected in the medium. When available, this information should be introduced in the model as proposed by Van Trier (1988), Harlan (1989) and Michelena (1990), among others.

From the inversion point of view,
velocity and heterogeneities are closely related.
When the model for the velocities is isotropic but the real medium
is not, the image of heterogeneities obtained
is distorted. On the contrary,
if we want to estimate velocity anisotropy
in the medium but the heterogeneities are
not properly described, we might get wrong estimates for the velocities.
These problems can be solved by describing properly both velocities and
heterogeneities and then estimating both *simultaneously*, as I
propose in this paper.
However, this might be difficult to do for a general anisotropic and
heterogeneous medium.

Velocity anisotropy, heterogeneity, or both at the same time, can affect the traveltimes nonlinearly depending on how strong they are and how complex the model that describes them is. When one of these nonlinearities is neglected, the computations related to the other one can be simplified. For example, assuming weak anisotropy, Pratt and Chapman (1992) and Chapman and Pratt (1992) proposed to do the ray tracing in isotropic media and to use those rays to invert in the anisotropic model. Michelena et al. (1992) simplify the problem even more and present examples where the velocity contrasts are small enough such that the rays traced on the elliptically anisotropic models are straight and the only nonlinearity that remains is in the estimation of the anisotropy. Another way to simplify the simultaneous estimation of velocities and heterogeneities is by assuming simple models for both. This is the approach I follow in this paper.

The simplest anisotropic velocity model we can assume is elliptical. Elliptical anisotropy is a useful model when the aperture of the experiment is such that the slowness surface is not properly sampled and therefore, it is difficult to estimate more than two velocities: one based on the arrivals along a certain axis (direct velocities) and other based on the curvature around that axis (NMO velocities).

Elliptical anisotropy is not a good
approximation for velocities at all angles
in a transversely
isotropic media. However, it is a
good *paraxial* approximation that can be used in
two steps
(once around the horizontal and once around the vertical)
to approximate globally
the slowness surface and impulse response of any
wave type in a transversely isotropic medium
(Muir (1990) and Dellinger and Muir (1991)). Because the approximation
uses two perpendicular ellipses, it is called the
double elliptic approximation.
What is needed to use this approximation is an inversion
procedure that fits traveltimes with ellipses once
a model for heterogeneities
has been chosen.

The simplest model to describe the heterogeneities of the medium is 1-D. If dips are present, we can allow the interfaces of the 1-D model to have variable slope. However, if the spatial variations in the medium are more complex, we may need to use models that specifically account for them, as proposed by Van Trier (1988) and Harlan (1989). It may seem that with a fine 2-D parametrization we should have, in one hand, enough degrees of freedom to explain all kinds of ``unexpected'' variations in the medium and in the other hand, we should be able to explain the data better. This is only partially true. Michelena at al. (1992) show that for certain media ( 1-D, weakly anisotropic) field data can be fitted equally well either with 1-D anisotropic models or with 2-D isotropic models, the latter having 2 orders of magnitude more of degrees of freedom. In the same paper, Michelena et al. also show that the problem of estimating velocity anisotropy in layered media is more stable and the results are more accurate when the discretization is 1-D than when the discretization is 2-D.

The model for velocities and heterogeneities I will use in this paper is a compromise between the simplicity of 1-D isotropic and the complexity of 2-D anisotropic models (Michelena at al., 1992). This compromise addresses three issues: how general the medium can be, how stable the inversion procedure is, and how difficult the ray tracing is. Such an ``intermediate'' model consists of consists of homogeneous elliptically anisotropic blocks separated by straight interfaces of variable dip and intersect. Interfaces may change position during the inversion process but are never allowed to cross in the area of interest. Byun (1982) proposes a procedure to trace rays in this type of model. In Michelena (1992), I review Byun's procedure and show the details of the actual implementation when the axes of symmetry are tilted.

Certain types of azimuthally anisotropic media can be also approximated with this model of velocity and heterogeneity, in particular those formed by dipping transversely isotropic layers. This is done by considering also the inclination of the axis of symmetry as a variable. The estimation of this variable may help the interpreter to characterize the amount of structural deformation that have undergone the different formations in the reservoir and may also help in the identification of structural unconformities.

A theoretical discussion of the technique is followed by examples with cross-well synthetic data and field data from BP's Devine test site.

11/18/1997