An example with simple reflectivity models

This section offers a pictorial description of the theory presented in the preceding section. We use two simple examples to highlight the main features of the method. In both cases, the reflectivity model consists of two flat interfaces, one shallow and one deep. The velocity models are different as follows:

• In the first model, we started with a constant velocity background of 2 km/s, on which we superimposed a positive Gaussian perturbation with a magnitude of 0.25 km/s, as shown in the right panel of Figure 2.
• In the second model, we superimposed a negative Gaussian anomaly with the magnitude of 0.5 km/s on the background velocity (Figure 2), while the perturbation remained the same as in the first model (the right panel of Figure 2). The main purpose of selecting this second velocity model was to demonstrate the robustness of the forward and adjoint operators to triplications in the wavefield.

 

slow
Figure 2 Background slowness (labeled S in Figure 1) - left; Perturbation in slowness (labeled $\Delta S$ in Figure 1) - right.

 

data
Figure 3 The data (labeled D in Figure 1) measured at the surface for each of the background velocity models. The top panel corresponds to the case of the constant background velocity, while the bottom panel corresponds to the case of the Gaussian anomaly in the background velocity.

 

image
Figure 4 The image (labeled R in Figure 1) obtained after migrating by downward continuation the data in Figure 3. The top panel corresponds to the case of the constant background velocity, while the bottom panel corresponds to the case of the Gaussian anomaly in the background velocity.

 

dimage
Figure 5 The perturbation in image (labeled $\Delta R$ in Figure 1), that is, the data migrated with the perturbation in slowness. The top panel corresponds to the case of the constant background velocity (Kjartansson's V), while the bottom panel corresponds to the case of the Gaussian anomaly in the background velocity.

 

rays
Figure 6 The perturbation in slowness (labeled $\Delta S$ in Figure 1) corresponding to part of the perturbation in image (Figure 5). The ``fat rays'' are the result of the back-projection of the perturbation in slowness. The top panel corresponds to the case of the constant background velocity, while the bottom panel corresponds to the case of the Gaussian anomaly in the background velocity.

We started by creating the synthetic data (D) that correspond to each of the individual models (Figure 3). In the second case, the reflection from the deeper interface creates a triplication caused by the Gaussian anomaly in the background velocity. Then, we migrated the synthetic data using the correct velocity models in each case, and obtained the background images (R) shown in Figure 4.

We then repeated the same succession of operations, considering the background velocity models on which we superimposed the perturbation anomaly. We then created the data and migrated it with the perturbation in slowness. What we obtained is the perturbation in image depicted in Figure 5. The shape of the image at a given depth is known in the literature as Kjartansson's V Kjartansson (1979). In the case of the nonconstant background, the triplications of the wavefield created a more complex perturbation in the image, which is especially visible at the level of the deeper reflector. For this case, Kjartansson's V becomes a W (Figure 5).

Finally, we back-projected into the velocity model the perturbations we obtained in the images (Figure 6). To clarify how the back-projection operator works, we have isolated in each panel a single event of the perturbation in image, for a fixed reflection ray parameter. As expected, we have obtained ``fat rays'' showing which regions of the velocity model are influenced by the perturbation in image. The top panel of Figure 6 displays the straight fat rays corresponding to the constant velocity background. The bottom left panel, shows the rays for a similar perturbation in the image as in the first case, while the bottom right panel displays the rays for the perturbation in image in a region where the wavefield has triplicated when propagating through the anomaly in the background.