Results

I applied the downward continuation migration and the preconditioned inversion scheme to a synthetic dataset provided to us by SMAART JV, using different velocity models. The correct velocity model can be seen in Figure . The result of migration using this model is Figure . In the CRP-depth panel, note the sudden decrease in amplitude of the reflectors as they pass beneath the salt edge, particularly within the oval. There are also strong artifacts in the shadow zone beneath the salt (inside the oval) which make it difficult to pick out any true events. In the p_h-depth panel, note the ``holes'' in the events at the mid-range of ray parameters (inside the oval). These holes are caused by the poor illumination under the salt edge. The steeply dipping events in the p_h-depth panel are artifacts caused by aliasing along the offset axis in Fourier space.

 

vel.corvel Figure 1 The correct velocity model.

 

mig.corvel
Figure 2 The result of downward continuation migration using the correct velocity model. Note the low amplitudes of events as they pass beneath the salt in the CRP-depth panel and the artifacts obscuring events beneath the salt (indicated by ovals). In the offset ray parameter-depth panel note the holes in the events at the mid-range of offset ray parameters (particularly within the oval).

The result of 3 iterations of conjugate-gradient preconditioned least-squares inversion using the correct velocity model can be seen in Figure . Note that the artifacts have been largely cleaned up. It is now possible to reliably pick out events beneath the salt (see inside the oval). In the CRP-depth panel, the amplitude of the events is maintained farther beneath the salt (particularly within the oval). The holes in the p_h-depth panel (inside the oval) are being filled in.

 

geop.corvel
Figure 3 The result of 3 iterations of preconditioned inversion using the correct velocity model. Note the more consistent amplitudes of events as they pass beneath the salt in the CRP-depth panel and the lack of artifacts obscuring events beneath the salt (inside ovals). In the offset ray parameter-depth panel note the filling in of the holes in the events at the mid-range of offset ray parameters (inside ovals).

To test the sensitivity of the preconditioned inversion, the first incorrect velocity model I tested simply increased the correct velocities by $5\%$. As expected, the migration result using this velocity model (Figure ) shows the events positioned deeper than they should be and moveout along the offset ray parameter axis. The ovals on this figure are placed in the same absolute positions as the the ovals in Figures  and , not relative to the events themselves.

 

mig.hivel
Figure 4 The result of downward continuation migration using a velocity model $5\%$ higher than the correct model. The events are all positioned deeper than they should be and there is moveout along the offset ray parameter axis. The ovals still indicate the loss of amplitudes under the salt edge and the poor imaging beneath the salt in the CRP-depth panel and holes in the events in the p_h panel.

Recall that the preconditioning operator acts horizontally along the offset ray parameter axis. It is this sensitivity that we are interested in observing in the result of 3 iterations of preconditioned inversion using the high velocity model (Figure ). Note that once again the preconditioned inversion has cleaned up many of the artifacts. In the CRP-depth panel, the events extend farther under the salt, in a similar way to the inversion result using the correct velocity (Figure ). The more interesting result is the p_h-depth panel. The inversion is still successfully filling in the holes along the events at the mid-range of offset ray parameters. At large p_h, where the moveout is more pronounced, the preconditioning has made some attempt to change the dips to be more horizontal, but the moveout is still visible. This means that this result is most likely not safe to use for velocity analysis, but this preconditioned inversion technique was never intended as a velocity tool. Overall, this result indicates that this technique can produce a better image than migration alone, even when the velocity model is incorrect by up to $5\%$.

 

geop.hivel
Figure 5 The result of 3 iterations of preconditioned inversion using the $5\%$ too high velocity model. Despite the use of the incorrect velocity model, the image is quite comparable to the result using the correct velocity (Figure ). In the CRP-depth panel, the events extend farther under the salt and events under the salt can be seen (inside the ovals). In the p_h-depth panel, the holes in the events are filled in (inside the oval).

 

vel.smoothvel Figure 6 The smoothed velocity model. Note that the canyon in the top of the salt has disappeared.

A more extreme velocity model I tested was a severely smoothed one (Figure ). This model has been smoothed so much that the canyon in the top of the salt has disappeared. As expected, the migration result from this model isn't very good (Figure ). The depth positioning of events is fairly good away from the salt, but becomes poor near the salt. The salt top and bottom are very poorly imaged. The events in the p_h-depth panel appear to be mostly random. Once again, the ovals indicate the same absolute regions as the ovals in Figures  and .

 

mig.smoothvel
Figure 7 The result of downward continuation migration using a severely smoothed velocity model. The events in the CRP-depth panel are properly imaged away from the salt but are mispositioned near the salt. The offset ray parameter-depth panel is completely uninformative. The ovals indicate the same absolute regions as the ovals in Figures  and .

The result of 3 iterations of preconditioned inversion using this smoothed velocity model can be seen in Figure . Although many of the artifacts have been cleaned up, overall the image is not any better than the migration result. The events in the p_h-depth panel are more horizontal, but they are not more believable than the events in the p_h-depth of the migration result. This is a reassuring result, as it indicates that the regularization was not able to artificially introduce events where the data indicated otherwise.

 

geop.smoothvel
Figure 8 The result of 3 iterations of preconditioned inversion using the smoothed velocity model. The result is cleaner than the migration result, but not more believable. The ovals indicate the same absolute regions as the ovals in Figures  and