It is possible to construct many other operators which try to maximize the frequency content of the difference section.

For example another approach may be to find filters, *A _{1}* and

(8) | ||

(9) | ||

(10) |

(11) |

Since we are estimating an intermediate model simultaneously with the pair of filters, a good initial model speeds convergence. The initial models chosen were and .

With two sets of regression, a good choice of is important. A choice of that is too large results in and does not do a good job of match-filtering. Alternatively, a choice of that is too small does a good job of match-filtering, but a lot of the resolution is lost and the result approaches that of the simple inverse filtering above. Unfortunately an intermediate choice of is the worst of both worlds, as it does a poor job of match-filtering and the bandwidth of the signals are reduced. The results of the inversion with are shown in Figures 8. By playing around with different values of and different filter lengths, I was not able to get a successful result.

Figure 8

Although this approach looks like it may be successful on paper, I believe its failure stems from the size of the model space. Although time domain match-filtering uses a similar least squares formulation, it has only about a dozen elements in the model space, as opposed to a model the size of the data space which has to be estimated in this intermediate model approach.

11/11/1997