Inversion

As a first experiment, we have constructed a synthetic model similar to the sections in Figures 7 and 8. As we said, our goal is to convert the differences in focusing between the two images, or the perturbation in the image, into a better slowness model, that is, to find the perturbation in the slowness.

Figure 9 represents the background slowness model (S). We used this model to generate the synthetic data at the surface (S), and then to compute the background wavefield (U) and the background image (R).

The top panel of Figure 10 shows the perturbation in slowness ($\Delta S$). We used this model to generate the scattered wavefield ($\Delta W$), the perturbation wavefield ($\Delta U$), and the perturbation in image ($\Delta R$).

We start the inversion by assuming zero perturbation in slowness. The middle panel of Figure 10 represents the perturbation in slowness obtained at the first iteration. At this stage, we have obtained only a small perturbation in slowness, which is not totally concentrated at the right location. An important part of the energy of the section is spread, for example in the region around the midpoint 2.2-2.4km and around the depth 1.3-1.4km. This artifact is the result of the still imperfect definition of the slowness anomaly, possibly caused by the proximity of the edge.

By the 20^th iteration, shown in the bottom panel of Figure 10, the perturbation in slowness is much better shaped, and the artifact at depth is much weaker. Also, the absolute magnitude of the anomaly is getting very close to the correct value: s_max=0.088 s/km for the original, and s_max=0.084 s/km for the inversion at the 20^th iteration.

 

backslo Figure 9 The background slowness (S).

 

inversion Figure 10 Top: the perturbation in slowness ($\Delta S$). Middle: the perturbation recovered at the first iteration. Bottom: the perturbation recovered at the 20th iteration.