Generation of pseudo-primaries

Pseudo-primaries can be generated by computing Shan and Guitton (2004)  

$$W(x_p,x_m,\omega)=\sum_{x_s} M(x_s, x_m,\omega)\bar{P}(x_s,x_p,\omega),$$ (1)

where W is the pseudo-primary data, $\omega$ is frequency, x_s is the shot location, x_p is the surface location, $\bar{P}(x_s,x_p,\omega)$ is the complex conjugate of the original trace at (x_s,x_p) and M are the multiple reflections recorded at x_m. In this equation, the cross-correlation of the first-order multiples in M with the primaries and first-order multiples in P produces primaries and zero-lag components, respectively. Cross-correlation of the second-order multiples in M with the primaries, first-order, and second-order multiple reflections in P produces first-order multiples, primaries, and zero-lag components, respectively. With higher orders of multiples this trend continues.

Pseudo-primaries generated in this fashion contain subsurface information that would not be recorded with a non-zero minimum offset. One example of this is a first-order multiple that reflects at the free surface within the recording array, resulting in near offsets being recorded when that wave returns to the surface. An example of this is shown in Figure , where (a) is a single Sigsbee2B shot (including the negative offsets predicted by reciprocity) but with offsets less than 2000 feet removed, and (b) is the corresponding pseudo-primaries for the same area, which is generated in part with (a).

 

shot
Figure 1 (a) Original shot record and (b) pseudo-primaries for the same area from the Sigsbee2B dataset.

We can see in Figure where the first and second-order multiples in P map to in the zero-lag at the top of the image. We can also see a lot of near-offset information present in the pseudo-primaries that is not present in the recorded primaries. However, simply replacing the missing near offsets of the primaries with the corresponding pseudo-primaries would not yield a satisfactory result due to the crosstalk and noise in the pseudo-primary shot.