Real Data Examples

125 CMP's from a 2-D prestack data set that was acquired in the Gulf of Mexico were used. Since the Gulf of Mexico usually exhibits flat reflectors, this is suitable for the Dix equation. The region is also faulted which implies discontinuous velocities.

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.

 

vel-vrms
Figure 1 Top: Auto-picked RMS velocity of one CMP from a 2-D prestack dataset. Bottom: Raw RMS velocity map for all 125 CMPs

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.

 

stack1
Figure 2 Top: Stacked data using the raw RMS velocity. Middle: Stack data with lines showing the faults. Bottom: RMS velocity with same lines showing the faults.

The top of Figure shows the interval velocity resulting from solving equations (2) with $\ell_2$ regularization. The bottom of Figure shows the interval velocity when equation (5) is solved with $\ell_2$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.

 

L2vint
Figure 3 Top: Interval velocity computed using conjugate gradients with $\ell_2$ regularization. Bottom: Interval velocity computed using convex optimization with $\ell_2$ regularization.

If we now look at the solutions to equations (2) and (5) solved with a $\ell_1$ 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.

 

L1vint
Figure 4 Top: Interval velocity computed using conjugate gradients with $\ell_1$ regularization. Bottom: Interval velocity computed using convex optimization with $\ell_1$ regularization.

In the $\ell_1$ 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 $\bf cvx$, which is blockier than with conjugate gradients. The difference in ``blockiness'' is due to the different $\epsilon$'s and how they are applied in each case.

 

L1-interp
Figure 5 Same as Figure except with lines marking fault locations.

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:

$$v(z)=v_0+\alpha z,$$ (7)

then we can get a general estimate of the interval velocity. This is shown in Figure . Velocities 20 percent above and below this model are used as the maximum and minimum constraints in equation (6). Figures and show the bounded constrained $\ell_2$ and $\ell_1$ solutions, respectively. As we can see, the low velocities occurring at late times have been attenuated.

 

bounds
Figure 6 Model from which the upper and lower velocity constraints are formed. Upper is $20\%$ greater than this everywhere and the lower is $20\%$ less.

 

L2vint-cvx-bounded
Figure 7 Solution by convex optimization with $\ell_2$ regularization and bound constraints.

 

L1vint-cvx-bounded
Figure 8 Solution by convex optimization with $\ell_1$ regularization and bound constraints.