REGULARIZATION

The steering filter methodology has the most potential as a regularization operator in large inversion problems. For our final example we use inverse steering-filters in conjunction with another operator, in this case a tomography operator, to improve the inversion result. For our tomography operator we chose Toldi's interval to stacking velocity operatorToldi (1985). Generally, Toldi1985 related perturbations in interval slowness to perturbations in stacking slowness in simple slowness models.

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.

 

toldi-steer
Figure 10 Left, slowness perturbation model; center, inversion result using Laplacian smoother; right, inversion result using steering filters.

 

toldi-stack
Figure 11 Left, input stacking slowness; right, calculated stacking slowness of steering filter inversion model.

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

$$\begin{eqnarray} {\bold T} {\bold m} &\approx& {\bold d} \ {\bold \epsilon} {\bold A} {\bold m} &\approx& 0 .\end{eqnarray}$$ (20) (21)

Where ${\bold T}$, is the Toldi operator; $\bold A$, is a Laplacian smoother; $\bold m$, is our interval slowness perturbation model; and $\bold d$, is our data, stacking slowness perturbations.

The center panel of Figure toldi-steer shows the inversion result. We tried a variety of ${\bold \epsilon}$ 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:

$$\begin{eqnarray} {\bold A^{-1} x} &=& {\bold m} \ {\bold T} {\bold A}^{-1} {\bold x} &\approx& {\bold d} \end{eqnarray}$$ (22) (23)

where $\bold A$ 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.