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## The stability of non-stationary inverse filtering

A filter is stable if any bounded input produces a bounded output Robinson and Treitel (1980). Therefore, to prove that inverse filtering with a class of filters is stable, we have to demonstrate that all possible members of the class have bounded outputs for all bounded inputs. On the other hand to show that a class of filters is not stable, we just need to find a single example where a bounded input produces an unbounded output.

The stability of stationary recursive inverse filtering depends on the phase of the causal filter: if (and only if) the filter is minimum phase, then its inverse filter is stable. This raises the question: is non-stationary inverse filtering stable if all filters contained in the filter-bank are minimum-phase?

For the case of inverse filtering with a two-point filter (Na=2), equation (9) reduces to x0=y0, and the following formula for k>0:

 xk = yk - a1,(k-1) xk-1. (13)

Recursive substitution then produces an explicit formula for elements of in terms of elements of :
 (14)
For stability analysis, we need to understand how the above series behaves as .If the filters, , are all minimum phase, and there exists a real number, , such that for all i, then
 (15)
The above series will therefore converge, and stability is guaranteed. Furthermore, this proof can easily be extended to gapped two-point minimum-phase non-stationary filters, which correspond to matrices with ones on the main diagonal, and variable coefficients whose magnitude is less than one on a secondary diagonal.

There is a larger class () of stable non-stationary recursive filters that can be obtained by repeatedly multiplying stable bidiagonal matrices. However, given a general non-stationary filter matrix, there is no straightforward way to determine whether it is a member of this stable class. In fact, it is relatively easy to find an example filter-bank consisting of minimum-phase individual filters whose recursive output is unbounded for finite input. Consider the filter-bank, , consisting of minimum-phase filters,
 (16)
Figure 1 shows the impulse response of non-stationary inverse filtering with this filter: clearly an unstable process.

 one Figure 1 Impulsive input (a) and response (b) to non-stationary filtering with filter-bank defined in equation (16).

The instability stems from the fact that as N increases, so does the number of boundaries between different filters. Such rapid non-stationary filter variations, as in the example above, are pathological in the context of seismic applications, where filters are typically quasi-stationary. For these applications instability is rarely observed; however, we must be aware that we are not dealing with an unconditionally stable operator, and instability may rear its ugly head from time-to-time.

Next: Conclusions Up: Theory Previous: Inverse non-stationary convolution and
Stanford Exploration Project
10/25/1999