The approach taken for obtaining the RMS velocities is the same as that in Valenciano et al. (2003), but will briefly be recreated here. First, velocity analysis was preformed on each CMP. Then an auto-picker was used to pick the maximum stacking power corresponding to the best RMS velocity at each CMP position. An example velocity analysis with picks for a single CMP is shown on the top of Figure . The velocity values from all the CMPs can be combined to form a complete RMS velocity model space. This is shown on the bottom of Figure . Please note that the velocities picked in Figure are in slowness rather than velocity, while the complete RMS model space is in velocity. This is because the conjugate gradient method used a slowness model, but the convex optimization failed with slowness and velocity had to be used. This limitation may be because the values were all close to forcing the solution down or due to the narrow range of slowness values. Thus the images computed with conjugate gradients were computed in slowness and inverted to velocity, while the convex optimization images were computed directly with velocity values.

Figure 1

Figure shows the stacked section which displays the faulting mentioned above. The middle panel has the faults highlighted and the bottom panels shows the same faults on the raw RMS velocity. It is interesting to note that the raw RMS velocity shows the faults clearly.

Figure 2

The top of Figure shows the interval velocity resulting from solving equations (2) with regularization. The bottom of Figure shows the interval velocity when equation (5) is solved with regularization. Note that all interval velocities are clipped at their respective maximums. Both images in Figure are very similar showing that convex optimization is at least equivalent to conjugate gradients in terms of quality of solution.

Figure 3

If we now look at the solutions to equations (2) and (5) solved with a regularization, shown in Figure , we can see that, as expected, a much blockier solution is found. As in the previous figure, the top panel of Figure , created by conjugate gradients, is very similar to the bottom image, solved with convex optimization.

Figure 4

In the regularization image, the faults do show up faintly. If we overlay the same lines shown in Figure onto Figure , this becomes more obvious as shown in Figure . The faults may be slightly more obvious in the problem solved with , which is blockier than with conjugate gradients. The difference in ``blockiness'' is due to the different 's and how they are applied in each case.

Figure 5

There appears to be some low velocity anomalies near the bottom of the
interval velocity solution. This is predominately seen in the blocky
models, but there are uncharacteristically low velocities at late times
in all the models. As we can see from the stack in Figure
, there is no evidence to support such velocities. To correct this we can constrain the solution
further by adding bounds when solving the convex optimization
problem. If we assume that the interval velocity *v*(*z*) increases
linearly with depth:

(7) |

Figure 6

Figure 7

Figure 8

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