** Next:** Non-stationary factorization
** Up:** Smooth lateral variations in
** Previous:** Smooth lateral variations in

If the velocity varies smoothly in space, then we can extend the
stationary theory to cover spatially-variable filtering.
Rather than filtering with the stationary forward and inverse
convolution equations,
| |
(46) |

| (47) |

we can extend the concept of a filter to that of a filter-bank with
one filter for every location in the input/output space
Claerbout (1998a); Margrave (1998).
Now equations () and ()
become
| |
(48) |

| (49) |

Non-stationary convolution and recursive inverse convolution are
indeed true inverse processes, but like stationary polynomial
condition non-stationary inverse filtering has potential stability
problems.
Appendix discusses this in more detail; it also
demonstrates that, in general, the stability of recursive
non-stationary inverse filtering is not guaranteed even if the
individual filters that make up the filter-bank are all minimum
phase.

** Next:** Non-stationary factorization
** Up:** Smooth lateral variations in
** Previous:** Smooth lateral variations in
Stanford Exploration Project

5/27/2001