With the development of the helical coordinate system Claerbout (1998b), recursive inverse filtering is now practical in multi-dimensional space. Non-stationary, or adaptive Widrow and Stearns (1985), recursive filtering is now becoming an important tool for preconditioning a range of geophysical estimation (inversion) problems Clapp et al. (1997); Crawley (1999); Fomel et al. (1997), and enabling 3-D depth migration with a new breed of wavefield extrapolation algorithms discussed in Chapters and .
With these applications in mind, it is important to understand fully the properties of non-stationary filtering and inverse-filtering. Of particular concern is the stability of the non-stationary operators.
I formulate causal non-stationary convolution and combination and their adjoints in such a way that it is apparent that the corresponding non-stationary recursive filters are true inverse processes. Stationary recursive inverse-filtering is stable if, and only if, the filter is minimum-phase. I show that recursive inverse-filtering with a filter-bank consisting of minimum-phase two-point filters is also unconditionally stable. However, I demonstrate that, for a more general set of minimum-phase filters, stability of non-stationary recursive inverse filtering is not guaranteed, even though stationary filtering with each individual filter is stable.