Anti-aliasing in the multiple prediction

The multiple prediction proposal discussed in the preceding section suggests that we can estimate 3-D multiples without trace interpolation. However, as Figure 6 shows, the other problem--aliasing noise--has to be dealt with carefully if there is no missing-streamer interpolation. Like any other Kirchhoff-style operation, anti-aliasing is an important issue in multiple prediction. This issue deserves even more attention in three dimensions, since the cross-line sampling is more sparse than the in-line sampling.

The 3-D estimation of a multiple trace is achieved by stacking a 3-D MCG. As discussed in the preceding section, we can safely stack the 3-D MCG into a 2-D PSMCG along the in-line direction. In the cross-line direction, we must sample the PSMCG more densely to avoid the aliasing noise. Therefore, I propose interpolating the PSMCG directly and then stack it into a multiple trace.

We can interpolate the aliased data in either the F-X Spitz (1991) or the T-X domain Claerbout (1992). I have chosen the time-space domain multi-scale prediction-error filter (MSPEF) theory discussed in Section 8.4 of Claerbout (1992) to interpolate the PSMCG. The basic idea of the theory is that large objects often resemble small objects. Supposing that we have input data with alternate missing traces, we can estimate a PEF with the following shape:  

$$\begin{array} {ccccccccc} a &\cdot &b &\cdot &c &\cdot &d &... ...&\cdot &\cdot &\cdot &1 &\cdot &\cdot &\cdot &\cdot \end{array}$$ (1)

Then we can make the filter smaller by throwing away the zeros (represented by dots) in filter (1) to get  

$$\begin{array} {ccccc} a &b &c &d &e \\ \cdot &\cdot &1 &\cdot &\cdot \end{array}$$ (2)

which has the same dip characteristics as filter (1).

Figure 7 shows two PSMCGs containing crossing events, before and after interpolating the alternative missing traces and the corresponding stacking results. The aliasing noise has decreased significantly after trace interpolation.

 

mcg-interp
Figure 7 Top: a densely sampled ($\Delta_{\rm streamer}$=25m) PSMCG and its stacking result. Bottom: a sparsely sampled ($\Delta_{\rm streamer}$=50m) PSMCG and its stacking result. The aliasing noise has been greatly decreased after the trace interval is halved from 50m to 25m.