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 carefully if there is no missing-streamer interpolation. Like any other Kirchhoff-style operation, anti-aliasing is an important issue in the multiple prediction. This issue deserves even more attention in three dimension, 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, the 3-D MCG can be safely stacked into a 2-D PSMCG along the in-line direction. In the cross-line direction, the PSMCG has to be more densely sampled to avoid the aliasing noise. Therefore, we propose to interpolate 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 PEF theory discussed in Section 8.4 of Claerbout (1992) to interpolate the PSMCG. The basic idea embedded in the theory is that large objects often resemble small objects. Supposing that we have an 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 been greatly reduced 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.