Three synthetic experiments were then carried out on each Earth model:
The modeling for the cross-correlation experiments was done independently for each incoming wave; these results were then cross-correlated and then added together (Figure 5). This approach can be justified by considering the following Claerbout (1979):
If B1 = the impulse response at 1, B2 = the impulse response at 2, and N = white noise.Since white noise is defined so that the expectation of ,the cross-correlation of the two white responses equals the cross-correlation of the impulse responses.
Therefore B1 N = the white response at 1, and B2 N = the white response at 2.
Figure 5 Flow used to generate common shot gathers by cross-correlating finite-difference noise seismograms.
The advantage of modeling each incident wave separately is that the seismograms are free from random noise: the results are the same as would be expected from cross-correlating infinite time series. The effects of the finite length time series are therefore removed from the experiment, allowing the issues of signal to noise to be separated from issues of resolution.