Knowledge of signal spectrum and noise spectrum allows us to find filters for optimally separating data into two components, signal and noise Claerbout (1999). Actually, it is the inverses of these spectra which are required. In Claerbout's textbook example 1999 he estimates these inverse spectra by estimating prediction-error filters (PEFs) from the data. He estimates both a signal PEF and a noise PEF from the same data .A PEF based on data might be expected to be named the data PEF D, but Claerbout estimates two different PEFS from and calls them the signal PEF S and the noise PEF N. They differ by being estimated with different number of adjustable coefficients, one matching a signal model (two plane waves) having three positions on the space axis, the other matching a noise model having one position on the space axis.
Meanwhile, using a different approach, Spitz (1999) concludes that the signal, noise, and data inverse spectra should be related by D=SN. The conclusion we reach in this paper is that Claerbout's estimate of S is more appropriately an estimate of the data PEF D. To find the most appropriate S and N we should use both the ``variable templates'' idea of Claerbout and the idea of Spitz. Here we first provide a straightforward derivation of the Spitz insight and then we show some experimental results.