Gradient calculation

In SEP-60 I discussed an optimization method that backprojects perturbations in prestack migrated events onto the velocity model. In general, the perturbations are not readily available; picking them on the constant surface location gathers is cumbersome and not very reliable for the data shown here. However, it may be feasible to pick events from the migrated constant-offset sections after they have been locally stacked to reduce noise. The sections resemble the geology, at least for the inner offsets, as can be seen in Figure  and .

A method that avoids picking altogether uses semblance derivatives, and was described in my previous report. Semblance in the stacked image is calculated along stacking trajectories in the prestack migrated data. Instead of picking perturbations in the stacking trajectory, semblance derivatives along the trajectory drive the optimization method. I briefly review the calculations here; for more details I refer the reader to the SEP-60.

The derivative of semblance J with respect to z_m for a certain event is:

$$\displaystyle{{\partial J} \over {\partial z_m}}(\S, h) \ =\... ...S_{+\Delta z}(h))\ -\ J(\S_{-\Delta z}(h))} \over {2\Delta z}},$$ (1)

where h is offset, and (x_m,z_m) the position after migration. $\S$ is the current stacking trajectory, and $\S_{+\Delta z}$ and $\S_{-\Delta z}$ are perturbations thereof. $\S_{+\Delta z}(h)$ is the upward perturbed trajectory; it is the same as $\S$, except for a perturbation by $+\Delta z$ in the z_m direction at offset h:

$$\S_{+\Delta z}(h)\ =\ \left\{ \begin{array} {ll} \ (x_m(h),... ...offset}\ h,\\ \ \S\ & {\rm elsewhere.} \end{array} \right.$$ (2)

Likewise, $\S_{-\Delta z}(h)$ is perturbed by $-\Delta z$ at offset h.

The gradient calculation is not very robust if the trajectory is just shifted at one offset, as described in the above equation. Therefore, the path is perturbed at several offsets, with the perturbations decreasing away from the offset under consideration. This is schematically illustrated in Figure .