Generation of pseudo-primaries

Pseudo-primaries can be generated by computing a slightly modified version of cross-correlation of primaries and a multiple model Shan and Guitton (2004)  

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

where W is the pseudo-primary data, $\omega$ is frequency, x_s is the shot location, x_p and x_m are receiver locations, $\bar{D}(x_s,x_p,\omega)$ is the complex conjugate of the original trace at (x_s,x_p) and $D(x_s,x_p,\omega)$ is that same data at x_m. In this equation, the result of the cross-correlation of primaries, first-order multiples and second-order multiples in D are outlined in the table below, with the first and second columns corresponding to the inputs to the cross-correlation and the third column corresponding to the output.

Input 1 Input 2 Output

first-order multiples first-order multiples zero-lag

second-order multiples second-order multples zero-lag

first-order multiples primaries pseudo-primaries

second-order multiples first-order multiples pseudo-primaries

second-order multiples primaries pseudo-first-order multiples

With higher order multiples the trend in this table continues. This produces similar results to cross-correlating primaries with a multiple model Shan and Guitton (2004), as the additional correlation of the primaries on one term is already taking place with the identical primaries on the other term of the autocorrelation.

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. This is shown in Figures and , where Figure is a cube of the input Sigsbee2B shots (including the negative offsets predicted by reciprocity) but with offsets less than 2000 feet removed, and Figure is the corresponding cube of pseudo-primaries for the same area. Put briefly, the source coverage of the pseudo-primary data is much greater than that of the input data because all receivers in the original data become sources for the pseudo-primaries.

 

input
Figure 1 Input dataset missing the nearest 2000' of offsets on either side.

 

pseudo
Figure 2 Pseudo-primaries created by autocorrelation and summation of Figure . Note that the near offsets have been filled in.

Figure contains 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-primaries.

 

pseudoex
Figure 3 A single virtual shot from figure that has been expanded over the input shot axis.

The crosstalk in the pseudoprimaries is largely a function of the number of shots that are summed over in the input data. Figure shows the shot on the right panel of Figure , but without the summing over shots in equation 1 where instead of summing over shots each shot is plotted along the front face of the cube. It shows how the stacking procedure greatly increases the signal-to-noise ratio.