By doing inversion without taking care of the coherent noise, the velocity panel in Figure a is obtained. The model space is contaminated with strong noise coming from the residual in Figure c. When noise is present in the data, least-squares with a simple damping can help to mitigate the noise effects. This was not done in this case, however. The reconstructed data in Figure b is relatively noise free, which indicates that most of the energy in the model space (Figure a) is in the null space of the operator.
For the filtering and modeling techniques to work, a noise model is needed. As suggested in a preceding section, the noise model is estimated from the residual of least-squares inversion assuming that (i.e., Figure c). From this noise model, a PEF is estimated. Figure a displays the result of the inversion after 40 iterations of CG. The few hyperbolas present in the data are clearly mapped in the model space without any artifact. The remodeled data in Figure b show almost no ground-roll remaining. The unweighted residual in Figure c () contains only coherent noise and the residual in Figure d is almost IID: a little bit of ground-roll is still present.
The modeling technique leads to similar results to the filtering technique, as shown in Figure . The main difference comes from the residual in Figure d. With the modeling method, the residual is smaller than with the filtering technique and is IID. As a final comparison between the two techniques, Figure shows how the objective function (normalized) decreases in both cases. The modeling technique has the best convergence properties.
Figure 11 A comparison of convergence between the filtering and modeling approach for the land data example.