Introduction

In the past, 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 Fomel and Claerbout (1997); Rickett et al. (1998); Rickett and Claerbout (1998).

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.