Discussion

Sometimes data is stationary. Then Fourier analysis gives a more efficient approach. The ideas above are more relevant to cases where stationarity is less valid. Likewise, when the required filter is very long, comparable to a trace length, Fourier analysis would be more appropriate.

On the other hand, with spatial filtering applications a local PEF is more appropriate. In my book GEE, I explain how to build a time-variable PEF. It seems an alternative could be based on a training data set that varies locally. I see this as perhaps theoretically superior. In the GEE example, the filter itself is stated to vary smoothly. Now I would be proposing that the training data set be varying smoothly. In general, PEFs tend to ``look bad'' because their frequency content is inverse to that of the signal. This could mean that smoothing a PEF is not nearly such a good idea as using a training data set.

Imagine we seek a new PEF upon the arrival of each new trace, or perhaps even upon the arrival of each new data point. Naturally, the Wilson-Burg spectral factorization method might be helpful. Generally however, I am not sure how to proceed.