The figure 1 also displays the transform of this shot-gather. The strong events at are due to the presence of strong direct arrivals, and head-waves. The water-bottom peg-legs are still very easy to recognize. Notice also that the event at 1.5 seconds is stronger than the previous events. Actually, it seems that the primaries are stronger with time; this property justifies the use of adaptive algorithms.
To see the effects of this process, I apply three algorithms, with the same length for the filter (n=30). The first one consists of a block method: I solve the minimization problem on time windows of 800 msec (200 samples), with a classical Levinson algorithm; the windows overlap by half their length. The second is the LSL algorithm, with no tapering (). It is interesting to observe what happens before the event at 1.5 sec, because the LSL residuals before this event are supposed not to be influenced by it. The third algorithm is the general Burg's adaptive filtering, with , and no windowing. Each of these algorithms is applied on each trace separately.
I display the results on Figure 2, with which we can compare the outputs of the three processes with the input in the domain on Figure 1.
The Burg-type algorithm seems more robust that the LSL algorithm. The different multiples are efficiently suppressed. Effectively, the Burg-type algorithm takes into account the events after 1.5 sec: they provide a lot of information on the multiplication process, because they are stronger than the previous events. On the other hand, up to this time (1.5 sec), the LSL algorithm does not take them into account, so that the process is less stable. Notice also a better lateral continuity on the residuals of the Burg-type algorithm. This suggests a better numerical stability of this algorithm, less sensitive to incoherent noise.
A doubt could still persist about the undetermined event at 1.2 sec on the residuals of the LSL algorithm. However, I transformed back the residuals in the domain to the time-offset domain, as shown on Figure 3. After applying a velocity analysis on the LSL residuals, it appeared that this event was a pegleg multiple of another primary at 0.7sec. Was it an artifact of the process, or a real unpredictable multiple? I don't know indeed, but it confirms the instability of this algorithm.