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Figure 5 Ray paths corresponding to the reflection generated above the reflector and the one generated below the reflector. The rays corresponding to the source wavefield are red (dark in B&W), lines represent the wavefronts and the rays corresponding to the receiver wavefield are green. (light in B&W), |
The reflection from above and the reflection from below can be discriminated by a simple generalization of the imaging principle expressed in equation (2), that includes a time lag in the crosscorrelation. To understand this generalization it is useful to review the process of image formation in reverse time migration. Figures 6-8 sketches this process at three different values in the propagation time t. For simplicity, the sketches represent the process for the familiar reflection from above, but similar considerations would hold also for the reflection from below. The reddish (dark in B&W) lines represent the wavefronts for the source wavefield. The greenish (light in B&W) lines represent the wavefronts for the receiver wavefield. At time t-dt (Figure 6) the two wavefronts do not intersect, and thus they do not contribute to the crosscorrelation. At time t (Figure 7) the two wavefronts begin to interfere, and thus they begin to contribute to the image. The contribution starts in the middle of the reflector at time t, and then it moves to the sides as the time progresses to t+dt (Figure 8). The process described above correlates the wavefields at the same time (t-dt, t, t+dt). However, the wavefields can also be correlated at a non-zero lag over the time axis. In mathematical terms, we can generalize equation (2) as
(4) |
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Figure 6 Wavefronts for the source wavefield (red), and the receiver wavefield (green), for three time steps (t-dt, t, t+dt). The wavefronts at t-dt are highlighted in darker color. The two highlighted wavefronts do not intersect, and thus their contribution to the image is null. |
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Figure 7 Wavefronts as in Figure 6. The two highlighted wavefronts (time t) intersect in the middle of the reflector, and they contribute to the image at the intersecting point. |
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Figure 8 Wavefronts as in Figure 6. The two highlighted wavefronts (time t+dt) intersect at the edges of the reflector, and they contribute to the image at the intersecting points. |
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Figure 9 Wavefronts as in Figure 6. The two highlighted wavefronts (source wavefront at time t-dt and the receiver wavefront at time t+dt) intersect above the reflector, and they contribute to the image at the intersecting point. |
image-wave-lag-p1
Figure 10 Wavefronts as in Figure 6. The two highlighted wavefronts (source wavefront at time t+dt and the receiver wavefront at time t-dt) intersect below the reflector, and they contribute to the image at the intersecting point. |
I have confirmed this intuitive understanding by applying the generalized imaging condition in equation (4) to both the synthetic data set described above, and a synthetic data set with overturned events. I describe the results of the test on the latter in the next section.