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 ($\bold{S}$) from the experimental data ($\bold m$). Generally, this leads to a system of non-linear equations  

$$\bold{P}(\sigma) \bold{m} = (\bold{I} - \bold{S}(\sigma)) \bold{m} = \bold{0}\;,$$ (24)

which need to be solved with respect to $\sigma$. One way of solving system (24) is to apply the general Newton's method, which leads to the iteration  

$$\sigma_{k} = \sigma_{k-1} + \frac{\bold{P}(\sigma_{k-1}) \bold{m}}{\bold{S}'(\sigma_{k-1}) \bold{m}}\;,$$ (25)

where the derivative $\bold{S}'(\sigma)$ 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 (25) 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.