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.

4/27/2000