On Figure 4, I display the input shot-gather. I also represent the NMO-corrected data; the velocity analysis was again performed after a first attempt to remove of the water-bottom multiples in the domain (Darche, 1990).

To perform the interpolation, I will first transform the data to the *t _{0}*-

Figure 4

In fact, I overweight the far offsets, to emphasize the interpolation
process on the last traces. I will use a weighting matrix *W* identical
at all frequencies, but in which the values *w*(*h*) are 1 in the first 60
traces, and 4 in the last traces, in the region where traces are missing.
The corresponding values *U*(*t _{0}*,

Notice on Figure 4 that it is difficult to separate
the primaries and the multiples peaks in the *t _{0}*-

To perform the interpolation, I use the modeling operator with
modified parameters: 119 traces, regularly sampled, so that the energy
in the *t _{0}*-

The interpolation result has several interesting features. First, as expected, the far offsets have been successfully interpolated, since the parabolic transform was overweighted in this region. This is true either for the hyperbolic events (primaries and multiples) or the linear events, as it can be seen on the residuals. However, the near traces have been slightly modified. Especially, they have been roughened, losing some lateral coherency. Once again, the choice of the weighting was reducing the role of these traces, and it is not surprising to see some variations.

Actually, the choice of the weights is quite subjective. Effectively, I said that the parabolic model is not reliable at far offsets, and overweighting the far offsets might seem contradictory. On the other hand, the curvature of a parabola is better determined at far offsets. So, there is a trade-off between these criteria. I noticed on this data set that, if I underweight the last traces, the events are not well restored, and they tend to disappear on the last traces. Moreover, some strong aliased features at far offsets appear with this kind of weighting.

Another point is that it could be preferable to suppress (by dip-filtering)
from the beginning the linear events, which cannot be modeled as parabolas
after NMO correction. If we keep them, they tend to spread their energy
on a vast region of *p*-parameters.

In conclusion, the interpolation can be considered successful, in the sense that the missing traces have been efficiently restored. The choice of the weights matters, according to the purpose of the process: not disturbing the near traces (far offsets underweighted), or restoring precisely the far traces (far offsets overweighted). A trade-off is to choose an uniform weighting.

Figure 5

1/13/1998