Introduction

Previously, applications of non-stationary inverse filtering by recursion have been limited to problems in 1-D, such as time-varying deconvolution Claerbout (1998a). Theory presented no way of extending polynomial division to higher dimensions.

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.