Different applications of the parabolic transforms

Basically, the parabolic transforms maps the NMO-corrected data to the t₀-p domain. The main seismic events will transform to different regions of the mapping. Thus, by putting to 0 the values of U(t₀,p) inside (or outside) these regions, we should be able to separate the events according to their nature.

I mentioned already the multiples elimination method, proposed by Hampson (1986). The primaries should correspond to small values of p, while the multiples (especially water-bottom multiples and peglegs) should map to large values of p. However, when the primaries are too deep, their peglegs tend to have also small p-parameters, since p=1/(t₀V_r²), and the time t₀ is large: it is more difficult to separate these multiples from the primaries at the same t₀.

Then, for land data, the ground-roll, having a very low velocity, should map to regions of very large values of p. Consequently, canceling the U(t₀,p) in this region should suppress the ground-roll.

Finally, as we will see on real marine data, the direct arrivals and their low-frequency trend will also map to large values of p, because they have a linear moveout. However, they actually cover a vast region of the t₀-p domain, and it is more difficult to suppress them without affecting the other events.

However, the role of the mute is essential. After NMO correction, the seismic events at far offsets are stretched, especially for small 0-offset times. Then, they will map to a vast region of p, since they can be described by several parabolas. If a mute is not sufficient, it is also possible to give low weights to these far offsets, since the Toeplitz structure makes weighting easy to use. Furthermore, the seismic events at far offsets don't fit the parabolic modeling, which justifies in this context the under-weighting in this region.

Actually, I will use for my examples two marine shot-gathers, known to contain no dipping events, nor important diffractions. Consequently, the NMO correction can be reliable. The first one contains very strong long-period water-bottom multiples, and I will show on it the efficiency of the multiples elimination. The second one will be used for interpolation purposes. Some traces are missing : the mapping to the t₀-p domain will be performed with irregular sampling, but the mapping back to the time-offset will use a regular sampling, and thus interpolate the missing traces. On this example, I will also use offset-dependent weights, to stress the importance of traces near the missing traces.