We constructed a synthetic interval slowness perturbation model (Figure
toldi-steer, left panel) where the perturbations from zero follow a
sinusoidal path, and the anomalies go from positive to negative as you go
from left to right.
We used Toldi's forward operator to compute
stacking velocities at various depth levels (Figure toldi-stack,
left panel), in this case we simulating collecting stacking velocity at 10
evenly spaced depths (compared to 160 depth locations in our interval
slowness model), assuming a cable length of 2 *km*.

Figure 10

Figure 11

We applied a fairly traditional inversion methodology to estimate our interval velocity perturbations:

Where , is the Toldi operator; , is a Laplacian smoother; , is our interval slowness perturbation model; and , is our data, stacking slowness perturbations.

The center panel of Figure toldi-steer shows the inversion result. We tried a variety of values, selecting one that created a rough model, but did a fairly good job recovering the correct interval velocity perturbations. Next, we attempted to recover the interval slowness perturbations, starting from the same stacking slowness perturbations, using the steering filter methodology. We constructed our steering filters to follow the sinusoidal pattern of the model and changed our fitting goal to:

where is our steering filter matrix. As the right panel of Figure toldi-steer shows we did a substantially better job following ``geology'', with the added benefit of better vertically constraining the interval slowness perturbations.

10/9/1997