Spitz makes a better assumption for the signal PEF

INTRODUCTION

Knowledge of signal spectrum and noise spectrum allows us to find filters for optimally separating data $\mathbf d$ into two components, signal $\mathbf s$ and noise $\mathbf n$ (Claerbout, 1999). Actually, it is the inverses of these spectra which are required. In Claerbout's textbook example (Claerbout, 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 $\mathbf d$. A PEF based on data $\mathbf d$ might be expected to be named the data PEF $D$, but Claerbout estimates two different PEFS from $\mathbf d$ 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 $D\approx SN$ idea of Spitz. Here we first provide a straightforward derivation of the Spitz insight and then we show some experimental results.

Spitz makes a better assumption for the signal PEF