The result with synthetic data is striking (Figure 8): all the artifacts have disappeared, leaving a clean model space. The data are almost entirely recovered. Figure 9 displays the diagonal elements of the matrix at each frequency. We can see that from the lowest to the highest frequencies, the diagonal elements focus at four different locations corresponding to the four parabolic curvatures present in the input data. The cut-off at 70 Hz which corresponds to the highest frequency component present in the data, is used to calculate the model space.
Figure 9 Diagonal elements of the weighting matrix at each frequency. The four stripes correspond to the location of the four curvatures in the radon domain. The cutoff at 70 Hz corresponds to the highest frequency present in the data.
With real data, however, the results suggest strategies to better focus the radon domain. Figure 10 shows the inversion of one CMP gather in the parabolic radon domain when no attempt were made to focus the model space components, that is, no weight in equation (23). The residual is displayed in the right panel of Figure 12. Although the inversion produces a satisfactory fitting of the input data, some aliasing artifacts appear in the radon domain. Figure 11 displays the result of the inversion using the steering-weighting matrices. It shows that fewer artifacts appear in the radon domain. A comparison of the residual with and without weight , shown in Figure 12, demonstrates that the data fitting is satisfactory for both cases. It turns out that the crucial parameter is . I don't have any guideline for choosing it but trial and error. The efficiency of the steering-weighting matrices method is based on the number of parabolic events present in the data. The best results are achieved when few events have to be focused in a large radon domain. However, since for real data cases this requirement may be difficult to meet, I anticipate no or few improvements if we use this method. One solution may be simply to apply it to different patches as suggested by Herrmann et al. (2000).