Gamma selection

After performing the residual Stolt migration and converting to the angle domain, I am left with a volume of dimension $v(z,\alpha,x,\gamma)$, where $\alpha$ is aperture angle. From this volume we need to pick the best $\gamma$ as a function of x and z. I also have the problem that even with the redefinition of the residual Stolt migration problem in Equation (7), events still have some movement at different $\gamma$'s. For now I will ignore the movement problem on the theory that as long as we tend towards the correct solution, the best focusing $\gamma$ will tend towards 1 and the amount of mispositioning at the best focusing $\gamma$ will decrease.

For now I took a rather simple approach. I calculated the semblance for flat events at the different $\gamma$ values. I then picked the best $\gamma$ ratio at each location. I used this field as my data $\bf d$. I used the maximum semblance at each location as a weighting operator $\bf W$ to give more preference to strong events. I used a 2-D gradient operator for my regularization operator $\bf A$ and solved the inversion problem defined by the fitting goals,

$$\begin{eqnarray} \bf 0&\approx&\bf W ( \bf d- \bf m) \\ \nonumber \bf 0&\approx&\epsilon \bf A\bf m,\end{eqnarray}$$ (8)

where $\epsilon$ is the amount of relative smoothing and $\bf m$ is the resulting model. A better method, and a topic for future work, would be to calculate the semblance for a range of moveouts and do a non-linear search for a smooth $\gamma$ function.

To test whether the method works I scanned over $\gamma$ values from .95 to 1.05 on the migration result shown in the center panel of Figure 2. The left panel of Figure 5 shows the selected ratio, $\bf m$, in fitting goals (8) and the right panel shows a histogram of the picked values. Note how we have generally picked the correct $\gamma$ value ($\gamma=.97$).

 

pick
Figure 5 The left plot is the selected $\gamma$ value using fitting goals (8). The right panel is a histogram of the picked values. Note the peak at approximately .97, the inverse of the velocity scaling.