In 2-D isotropic models, the discretization of the model and prediction of the data using natural pixels produces a matrix that is well conditioned. Data and model parameters can be well resolved. Unfortunately, I don't know a formulation similar to the natural pixels one valid also for anisotropic models and therefore, for estimating slowness anisotropy in 2-D, I still have to deal with the inconveniences inherent to the square pixels discretization.

From the previous results we have confirmed that
it is a more difficult problem to find horizontal variations in *S*_{z}
(smaller singular values)
than it is to find vertical variations in
*S*_{x} (larger singular values).

If the components of the model in the null space are eliminated, the problem in one dimension is better conditioned than the problem in two dimensions for both the isotropic and anisotropic cases. This result is not surprising but it has received little attention. For cross-well geometries, most tomographic traveltime inversions try to estimate 2-D variations even in cases when the medium is known to be horizontally layered. The extra degrees of freedom allowed in the inversion have to be penalized appropriately in the objective function. When the velocity logs are available and the medium is horizontally layered and isotropic, 1-D inversions are not interesting and the intrinsic advantages of the parametrization (fewer unknowns and better conditioning) are not useful. However, if the medium is anisotropic and horizontally layered, a 1-D instead of a 2-D parametrization can be the difference between recovering or not (accurately) both slowness components.

The results in data space tell us also something new: when using iterative techniques to solve the problem, the data are not resolved ``evenly'' as iterations proceed but, instead, some iterations resolve part of the data not resolved by other iterations. This has to be taken into account when interpreting traveltime residuals plotted in this fashion because their spatial distribution may depend on when the iterations are stopped.

12/18/1997