2-D inversion

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 $m \le n$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.

 

two-logs
Figure 14 Sonic logs at the source and receiver well respectively. The thin line represents the original log. The thick line represents the corresponding log averaged in 60 layers of equal thickness (21.583 ft)

 

logs-vs-veloc
Figure 15 Average velocity log compared with V_iso (left), V_x (center) and V_z (right). V_z is closer and better correlated with the velocity log.

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 S_z 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 $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$$ 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 V_x is remarkably similar to V_iso, like in the 1-D inversion. The main difference between these two images is that in V_x (Figure ) the events tend to be more horizontally smeared that in V_iso (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 V_iso and V_x. This is because V_z 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 V_x is close to convergency but V_z is far from that point. This in turn introduces artificial anisotropy.

Because V_x and V_z 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

 

s2d-iso-hast
Figure 16 Isotropic 2-D inversion. The image has been divided into 131 x 26 square cells (10 ft² each).

velocity anisotropy (the ratio V_x / V_z 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 V_z > V_x. 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).

 

sx2d-hast
Figure 17 Anisotropic 2-D inversion. Each image has been divided into 131 x 26 square cells (10 ft² each). The spatial resolution of V_z is poor when compared with the spatial resolution of V_x.The ratio V_x / V_z has been separated in four areas that show percentages of anisotropy: white (ratio > 1.25), light gray (1.06 < ratio $\leq$ 1.25), dark gray (0.90 < ratio $\leq$ 1.06) and black (ratio $\leq$ 0.9). The dark gray areas can be considered isotropic.