Moveout, velocity, and stacking

Crossing traveltime curves

Since velocity increases with depth, at wide enough offset a deep enough path will arrive sooner than a shallow path. In other words, traveltime curves for shallow events must cut across the curves of deeper events. Where traveltime curves cross, NMO is no longer a one-to-one transformation. To see what happens to the stacking process I prepared Figures 4.3- using a typical marine recording geometry (although for clarity I used larger $(\Delta t,\Delta x)$) and we will use a typical Texas gulf coast average velocity, $v(z)=1.5+\alpha z$ where $\alpha=.5$.

First we repeat the calculation of Figure 4.2 with constant velocity $\alpha=0$ and more reflectors. We see in Figure 4.3 that the stack reconstructs the model except for two details: (1) the amplitude diminishes with time, and (2) the early waveforms have become rounded.

nmoalfa0 Figure 3. Synthetic CMP gather for constant velocity earth and reconstruction.

Then we repeat the calculation with the Gulf coast typical velocity gradient $\alpha=1/2$. The polarity reversal on the first arrival of the wide offset trace in Figure  is evidence that in practice traveltime curves do cross. (As was plainly evident in Figures , and crossing traveltime curves are even more significant elsewhere in the world.) Comparing Figure 4.3 to Figure 4.4 we see that an effect of the velocity gradient is to degrade the stack's reconstruction of the model. Velocity gradient has ruined the waveform on the shallowest event, at about 400ms. If the plot were made on a finer mesh with higher frequencies, we could expect ruined waveforms a little deeper too.

nmoalfa1 Figure 4. Synthetic CMP gather for velocity linearly increasing with depth (typical of Gulf of Mexico) and reconstruction.

Moveout, velocity, and stacking