When the problem is linear we should obtain the model that ``best'' fit the data in only one iteration. When the problem is non-linear one approach is to solve it as a sequence of linearized steps. We usually call these steps external iterations, to differentiate them from the internal iterations needed to solve each linear problem when using iterative techniques such as conjugate gradients. Ideally, if the problem has n unknowns, each external iteration should consists of m CG-steps (m internal iterations), where is the number of different singular values. When dealing with field data, however, we might not be able to do this because of the presence of the noise. Noise can affect the solution of each linearized problem in the following ways: (a) It might be amplified in the model by the smallest singular values recovered when m iterations are performed, (b) It might affect considerably the accuracy of the search directions and consequently, the position of the minimum associated with the solution. Therefore, we have to deal carefully with the noise.
Under the straight-ray assumption, only one external iteration was needed in the 1-D inversion to find the model shown in Figure . By selecting the layer thickness appropriately, we were able to perform the CG-iterations required to reach convergency without being much affected by the noise: thicker layers damped the solution whereas thiner layers introduced instabilities. In 2-D, however, the situation is different. In this case we found that the results were more sensitive to noise in the data than 1-D results. This is not surprising because now we are trying to estimate horizontal variations in Sz which, as explained before, are related to the smallest singular values of the problem (that amplify the noise).
Because of the sensitiveness to the noise of the 2-D inversion, it is necessary to avoid ``many'' CG-iterations at each linearized step. After several tests combining in different ways external and internal iterations with mean-average smoothing of the slowness model, we adopted a conservative approach to minimize the error (12). The approach consisted of the following steps: (1) Compute traveltimes in the given model, calculate the matrix and find the residuals. (2) Approximate the solution of the linear problem (11) by applying few (typically one or two) CG-iterations. (3) Smooth the updated slowness model. (4) Repeat all the previous steps until there is no reduction in the sum (12). When this happens, either quit or increase the number of CG-iterations by one and check if further reductions in the mismatch are obtained. If the problem is linear, the solution is not obtained in only one iteration because of the presence of the noise.
When the previous procedure was applied to estimate an isotropic model from the data, we obtained the image shown in Figure (error = 0.54 ms). In this case, the unknown model was discretized into 131 x 26 square cells (10 ft2 each). It is interesting to notice that adding more degrees of freedom in structure (more cells) does not improve substantially the parameter error obtained with 28 times less degrees of freedom in the 1-D anisotropic inversion. The model shown in Figure is similar to the one obtained by Harris et al. (1990b).
The result of the anisotropic inversion is shown in Figure (error = 0.45 ms). Notice that Vx is remarkably similar to Viso, like in the 1-D inversion. The main difference between these two images is that in Vx (Figure ) the events tend to be more horizontally smeared that in Viso (Figure ). This was expected from the synthetic example shown in Figure .
The events in the vertical component of the velocity tend to be smeared in the direction of the steepest rays and the spatial resolution in this component is poor when compared with Viso and Vx. This is because Vz is not properly sampled by the recording geometry. In the 1-D case, as we said before, this lack of information is compensated by assuming a layered model, which allows to perform more CG-iterations without having problems with the noise. In 2-D this is not possible and therefore, the results obtained can be in a stage where Vx is close to convergency but Vz is far from that point. This in turn introduces artificial anisotropy.
Because Vx and Vz cannot be estimated at the same resolution (at least using only this type of recording geometry), it is not possible to estimate spatial variations in
velocity anisotropy (the ratio Vx / Vz for example) at the same scale of the variations in velocity. Still, an image that shows variations in velocity anisotropy can be useful if it accounts only for the large scale variations that are well resolved by the inversion. Such an image is shown in Figure . This image is divided into four areas: highly anisotropic, moderately anisotropic, isotropic and anisotropic with Vz > Vx. We can see that most of the model is isotropic whereas the anisotropic areas are associated with high isotropic-velocity zones, possibly shales.
The mismatch estimated by equation (12) (error) decreases roughly 50 from the homogeneous to the 1-D inversion and about 60 from the homogeneous to the 2-D inversion. This means that for this data set, by trying to estimate lateral variations in the medium (small singular values) only a 10 reduction in the mismatch is gained with respect to estimating only vertical variations in the model (largest singular values).