Passive seismic imaging applied to synthetic data

SYNTHETIC DATA

Cole (1995) tested this conjecture on both synthetic and real data. However his field data was very noisy, and he did not draw any solid conclusions.

With this as a starting point, however, I continued modeling a single reflection, using a program based on the flow

loop over each plane wave {
calculate a random slowness, $p$
calculate the time delay due to a reflection, $\Delta t$
loop over each frequency, $\omega$ {
calculate a random amplitude
loop over each spatial location, x {
multiply each frequency by a factor $(1 + r e^{i \omega \Delta t}) \; e^{i \omega p x}$
}
}
}

Having produced synthetics, it was then possible to go ahead and cross-correlate traces to try and create pseudo-reflection seismograms. Figure 3 is a pseudo-shot gather generated by cross-correlating one trace with every other. The center panel shows how the clarity of the signal was improved by applying a $\sqrt{-i\omega}$ filter. The black line which has been overlain in the left panel corresponds to the expected hyperbola which would be observed in a real shot gather, offset by 0.05 s so it does not obscure the data. Therefore the kinematics in this case appear to be consistent with the conjecture.

first
Figure 3. Pseudo-shot gather over model with single horizontal layer and 200 incoming plane waves. The left panel is raw cross-correlations, the center panel has a half differentiation filter applied and the right panel is labeled with the correct kinematics shifted by 0.05 s.

Passive seismic imaging applied to synthetic data