Figure 2 Initial slowness function.
Figure 3 shows every twenty-fifth CMP gathers after NMO correction. Note that these gathers are not perfectly flat and that the noise level is quite high, especially in the deepest part of the section. In addition, there are both missing and bad traces at different offsets. We expect that the estimation of stepouts is robust enough to the noise level present in the gathers to give reasonable dips.
Local stepouts and time shifts are estimated from the CMP gathers. Figure 4 displays the estimated time shifts for the five selected CMP gathers. It is interesting to notice that the time shifts increase with offset. The time shifts are also relatively smooth in the direction thanks to the dip regularization in equation (3). The smoothing in both offset and midpoint directions during the stepouts estimation allows us to have time shifts where traces were originally missing (e.g., gathers four and five in Figure 3). The fact that the estimated time shifts change with midpoint for a fixed time and offset prove that lateral velocity variations exist. These time shifts can be checked by applying a moveout correction to the input gathers in Figure 3 according to the shift values in Figure 4. Figure 5 shows the same gathers after moveout correction. These gathers are now flat and demonstrate that the estimated time shifts after integration of the local stepouts are correct.
The velocity perturbations are then estimated from the time shifts with the tomography. Figure 6 shows estimated slowness perturbations and Figure 7 displays the updated slowness field. We used 40 iterations and set in equation (15) to obtain this result. Lateral velocity variations are visible throughout. In Figure 8, four fault locations interpreted from the seismic are superimposed (Figure 15). These faults locations seem to be aligned with velocity variations in Figure 7. In particular, it is pleasing to see the change of velocities across the different faults.
To check whether the method converged, modeled time shifts are estimated from the slowness perturbations in Figure 6 by applying the forward operator in equation (10). The re-modeled time shifts are shown in Figure 9. Comparing Figures 4 and 9, it appears that the re-modeled time shifts are smoother. Yet, applying these time shifts to the NMO corrected data in Figure 3 yield flat gathers (Figure 10). The difference between Figures 4 and 9 is that the re-modeled time shifts are constrained by the physics of the tomographic inversion, thus giving well-behaved amplitude variations. In Figure 4, however, the time shifts take any value according to estimated dips. The forward operator of the tomographic inversion can be interpreted as a velocity-consistent, time shifts estimator.
Finally, the flattened gathers in Figure 10 are stacked (Figure 14). We compare this result with the stacked section of the data with a 1D slowness model shown in Figure 11. This slowness function flattens the CMP gathers well for every midpoint position (Figure 12). We show the stacked section of the input data with this 1D slowness function in Figure 13. The reflectors are stronger and better defined in Figure 14 wherever the signal level is strong. In the lower part of the section, however, some continuous events in Figure 13 are attenuated in Figure 14. This effect is due to the difficulty to estimate meaningful stepouts when the noise level is too high. Again, four interpreted faults are shown on the stacked section in Figure 15.
Figure 11 A 1D stacking slowness function.