Figure 4 shows how the hyperbolas build up from the individual plane waves. In the left panel there is only one plane wave present, instead of 200 as in Figure 3. The trace at zero spatial offset is the autocorrelation function, and it has two spike as expected: one at zero lag and one corresponding to the reflected event. Other traces, corresponding to cross-correlations, have the two same spikes, but these are offset in time due to the different receiver positions and the apparent velocity of the incident wave. In the right panel of Figure 4 there are 5 plane waves incident, and a hyperbola is beginning to form. Only one part of each plane wave adds coherently with the others, the rest of the energy is smeared out.
The corollary of this is that, in order to see a slope in the pseudo-reflection seismograms, we need incident plane waves with the relevant slowness. This will become an issue when looking at real data where the direction of incoming waves will not be spatially white.
Figure 4 Pseudo-shot gather over model with single horizontal layer. Left panel has one incoming plane wave and right panel has five.