Solution steering with space-variant filters

Space variable filters

Steering filters are effective in spreading information along a given direction, but are limited to a single dip. If it is inappropriate to apply a single smoothing direction to the entire model there are two general courses of action:

Patching

(Schwab and Claerbout, 1995; Claerbout, 1992b) Redefine our problem into a series of problems, each on a small subset of the data where the stationarity assumption is valid, then recombine the data. This approach leads to problems in determining subsets where the stationarity condition is satisfied and how to effectively remove patching boundaries from the final output.

Space varying filters

Filters that vary with location but are spatially smooth. In many ways this is the a more appealing approach. In the past, space varying filters have not been used because they impose significant memory issues (a filter at every location) and must be spatially smooth. By choosing steering filters for our regularization operator and using helix enabled polynomial division, these weaknesses are significantly diminished. We can construct and store relatively small filters which are much easier to smooth (smoothing the preferential dip direction is sufficient). In addition the polynomial division produced inverse filters will have an even higher level of smoothness because each filter spreads information over large, overlapping regions at each iteration.

Solution steering with space-variant filters