## 4 Numerical Results
Since this pipe is parallel to the direction of the Bscan, it appears as a straight line with regard to the part of the pipe contained in the upper layer. However, when the pipe enters the second layer on the left-hand side, it appears to progressively dip, coherently with the progressive decrease in the average velocity of the waves along their path from the transmitting dipole up to the pipe and back to the receiving dipole. Finally,  
Figure 4: Sixth processed Bscan on the long side. Focusing achieved after the propagation velocity of the upper layer. Common time–depth conversion is applied. the interface between the two buried layers is clearly visible as a slanted line. In Figure 5, the same Bscan of Figure 4 is shown, but in this case, focusing was achieved according to the propagation velocity allowing for better focusing of the pipe in the lower layer (P\({}_{2}\)). This velocity, found by trials, is 12 cm/ns. In Figure 5, we can see that good focusing of pipe P\({}_{2}\) is achieved, but obviously at the cost of defocusing pipe P\({}_{3}\). In Figure 5, the same time-depth conversion as in Figure 4 is applied. The apparent bending of pipe P\({}_{1}\) also remains, as can be seen. It is also to be said that the problem is more complex than what Figure 5 represents. For example, if we had more pipes in the second layer, the propagation velocity allowing the best focusing of one of them would not necessarily guarantee the best focusing of the other ones. We are exploiting a heuristic model, and so in more complex cases, we should choose a compromised solution optimizing the focusing of the presumably most important targets while minimizing the defocusing of the other ones as much as possible. In particular, targets displaced in the second layer but very close to the interface between the two layers would be indeed better focused making use of the propagation velocity of the upper layer. In fact, in this case, the propagation of the waves occurs mostly in the upper layer. In Figure 6, we propose the application of a combined time-depth conversion to the result of Figure 4. As can be seen, the combined time-depth conversion allows us to restore the correct depth of the pipes because now pipes P\({}_{2}\) and P\({}_{3}\) appear to be at about the same depth. Moreover, the apparent bending of pipe P\({}_{1}\) is much more mitigated. However, pipe P\({}_{2}\) appears strongly distorted because we have not applied joined migration. In particular, if a slicing procedure were applied based on all the Bscans processed in this way, we would achieve severe distortion of the image of pipe P\({}_{2}\). Figure 5: Sixth processed Bscan on the long side. Focusing achieved after the propagation velocity allowing the best focusing of the pipe in the lower layer. Common time–depth conversion is applied. Figure 6: Sixth processed Bscan on the long side. Combined time–depth conversion is applied to the image in Figure 4.  
In Figure 7, we propose the result of joined migration without combined time-depth conversion. As can be seen, this time pipes P\({}_{2}\) and P\({}_{3}\) are both well-focused. However, P\({}_{2}\) is imaged deeper than its actual position, and P\({}_{1}\) is again bent on the left-hand side. The conclusion is that only the application of both joined migration (or more generally, an algorithm devoted to the focusing of the target in both layers in the time domain) joined to combined time-depth conversion can provide good imaging in the case at hand. In Figure 8, both joined migration and combined time-depth conversion are applied. As said, the propagation velocity exploited for the focusing of pipe P\({}_{2}\) is 12 cm/ns, whereas that exploited for the combined time-depth conversion is 8 cm/ns. As can be seen from Figure 8, now both pipes P\({}_{2}\) and P\({}_{3}\) are well focused and appear to be displaced at the same depth level as they are. Moreover, the bending effect of pipe P\({}_{1}\) is meaningfully mitigated.