Prestack images of overturned reflections

One of the main advantages of reverse-time migration methods over downward-continuation migration methods is their capability of imaging overturned events, even in presence of lateral velocity variations. However, this potential has not been exploited yet for prestack migration for several reasons. The computational cost is an important obstacle that is slowly being removed by progress in computer technology. In this section I will address two more fundamental problems. First, discrimination between image contributions from reflections generated above an interface from the image contributions from reflections generated below the same interface. These two reflections have usually opposite polarities because they see the same impedance contrast from opposite directions. If their contributions to the image are simply stacked together, they would tend to attenuate each other. Second, the updating of the migration velocity from overturned reflections. Solving the first problem is crucial to the solution of the second one, as graphically illustrated in Figure 5. This figure shows the raypaths for both events. It is evident that the overturned event passes through an area of the velocity field different from the area traversed by the reflection from above. The information on the required velocity corrections, provided by the image obtained using a given velocity function, can thus be inconsistent for the two reflections, even showing errors with opposite signs.

 

imag-rays 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 $\tau$ 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  

$$I\left(z,x,h, \tau \right) = \sum_t S\left(t + \frac{\tau}{... ...2} \right) R\left(t - \frac{\tau}{2},z,x - \frac{h}{2} \right).$$ (4)

Now the image is function of an additional variable $\tau$, that represents the correlation lag in time. Figures 9-10 provide an intuitive understanding of the outcome of the correlation for $\tau=-dt/2$ (Figure 9) and $\tau=dt/2$ (Figure 10). In both cases the two wavefields interfere and they contribute to the image. For negative $\tau$ the image is slightly above the correct location of the reflector, and for positive $\tau$ the image is slightly below it. Therefore, the image of the reflector slowly moves downward (more precisely along the normal to the reflector) as $\tau$ increases. The crucial point is that for reflections generated from below, this movement is in the opposite direction (i.e., upward). This difference in propagation direction allows an easy discrimination of the two reflections by filtering the image according to the propagation direction as $\tau$ progresses. I have not implemented such a filtering yet, but it should be relatively straightforward.

 

image-wave-st-m1 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.

 

image-wave-st-0 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.

 

image-wave-st-p1 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.

 

image-wave-lag-m1 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.