Imaging of multiples

A similar machinery can be effectively used to image multiples at their correct location in the subsurface. I keep the same imaging principle as developed by Claerbout. The differences stem from the choice of up- and down-going wavefields I extrapolate.

Figure illustrates the basic idea behind the migration of the multiples. In Figure (a), a wavefield generated at S is recorded at the receiver R_n. The reflector location r_a is imaged by extrapolating both the primary wavefield recorded at R_n and the source wavefield at S simultaneously in the subsurface and by crosscorrelating them at each depth step.

 

primult
Figure 2 Illustration of the basic idea of reflector mapping with (a) primaries and (b) multiples. In (a) the primary images the reflector location r_a and the source is impulsive. In (b) the multiple helps to image the reflector location r_b and the source is a primary recorded at R₁.

Similarly in Figure (b), a multiple recorded at R_n can be used to image the reflector location r_b if we use the primary wavefield recorded at the receiver R₁ as a source function. Hence a multiple reflection recorded at any receiver location can be used to image the subsurface if a primary is utilized as a source.

Finding the exact location of R₁ for each multiple and each receiver position R_n would require an earth-model and many tedious computation steps. Fortunately, the imaging condition tells us that the image is formed if and only if the up- and down- going wavefields correlate. Therefore for a given receiver position R_n, we can use as a source function every single primary recorded on the seismic array since only relevant receiver positions R₁ would produce constructive crosscorrelations.

Now if we expand this idea to every receiver position R_n on the seismic array, we can produce an image of the earth by simply taking every recorded primary as the source function and every recorded multiple as the up-going wavefield. The impulsive source becomes an areal-shot record. In theory, any order of multiples can be properly imaged if their corresponding source path exists in the down-going wavefield. Hence first order multiples need primaries as sources, second order multiples need first order multiples, and so on.

Note that the source function needs to be time-reversed before the extrapolation. This is done in the (2#2,x) domain by computing the complex conjugate of the source wavefield.

A similar approach has been presented by () using the so-called ``WRW'' model. Notice that so far, this approach works for surface-related multiples only but could be easily extended to internal-multiple migration.