Solution steering with space-variant filters

FUTURE WORK AND CONCLUSIONS

We show that by using helicon enabled inverse operators built from small steering filters we can quickly obtain esthetically pleasing models. Tests on smooth models, with a single dip at each location proved successful. The methodology does not adequately handle models with multiple dips at each location and presupposes some knowledge of the desired final model. A different approach would be to estimate the steering filters ($\mathbf S$) from the experimental data ($\mathbf m$). Generally, this leads to a system of non-linear equations

$$\mathbf{P}(\sigma) \mathbf{m} = (\mathbf{I} - \mathbf{S}(\sigma)) \mathbf{m} = \mathbf{0}\;,$$ (20)

which need to be solved with respect to $\sigma$. One way of solving system (20) is to apply the general Newton's method, which leads to the iteration

$$\sigma_{k} = \sigma_{k-1} + \frac{\mathbf{P}(\sigma_{k-1}) \mathbf{m}}{\mathbf{S}'(\sigma_{k-1}) \mathbf{m}}\;,$$ (21)

where the derivative $\mathbf{S}'(\sigma)$ can be computed analytically. It is interesting to note that if we start with $\sigma=0$ and apply the first-order filter (8), then the first iteration of scheme (21) will be exactly equivalent to the slope-estimation method of Claerbout (1992a), used by Bednar (1997) for calculating coherency attributes. Finally, the steering filter regularization methodology needs to be tried in conjunction with a variety of operators and applied to real data problems.

Solution steering with space-variant filters