** Next:** Synthetic examples
** Up:** Implicit extrapolation theory
** Previous:** Cross-spectral factorization

Kolmogorov cross-spectral factorization, therefore, provides a tool
to factor the helical 1-D filter of length 2*N*_{x} + 1 into
minimum-phase causal (and maximum-phase anti-causal) filters of
length *N*_{x} +1.
Deconvolution with minimum-phase filters is unconditionally stable.
However, inverse-filtering with the entire filters would be an
expensive operation.
Fortunately, filter coefficients drop away rapidly from either end.
In practice, small-valued coefficients can be safely discarded,
without violating the minimum-phase requirement; so for a
given grid-size, the cost of the matrix inversion scales linearly with
the size of the grid.
The unitary form of equation () can be maintained
by factoring the right-hand-side matrix,
in equation (), with Kolmogorov before
applying it to .

| |
(44) |

| (45) |

Chapter and Appendix extend the
concept of recursive inverse filtering to handle non-stationarity.
There are pitfalls associated with this process, however;
consequently, in this Chapter I limit the examples to the constant
velocity case.

** Next:** Synthetic examples
** Up:** Implicit extrapolation theory
** Previous:** Cross-spectral factorization
Stanford Exploration Project

5/27/2001