Introduction

The separation of a desired signal from unwanted noise can take many forms, but is a central problem throughout exploration geophysics. Various methods have been developed to address the issue, and we focus on those applicable to cases where the signal and the noise can be predicted with prediction error filters (PEF's) Abma (1995); Nemeth (1996); Soubaras (1994). In particular, we consider the problem of multiples, and focus on their suppression in the angle domain.

We use the projection filtering method Abma (1995); Soubaras (1994) for this paper. Our goal is to efficiently compare the effects of various model choices, so the small loss of mathematical rigor pointed out by Guitton (2003b) is made up for by the relative computational stability. The projection filtering technique is described in detail by Guitton (2003a). We review briefly: We consider data (${\bf d}$) to be the sum of unknown signal (${\bf s}$) and noise (${\bf n}$):  

$${{\bf d} = {\bf s} +{\bf n}}$$ (1)

and we estimate non-stationary PEF's ${\bf S}$ and ${\bf N}$ to be used as whitening operators for the unknown signal and noise, respectively. The signal-noise separation is based on the following fitting goals:  

$${\bf 0} \approx {\bf N}({\bf s}-{\bf d})$$ (2)

and

$${\bf 0} \approx \epsilon{\bf S}{\bf s}.$$ (3)

A noise model is not explicitly estimated in the inversion. It is considered to be the difference between the data and the estimated signal (${\bf n}={\bf d} - {\bf s}$).

The successful application of this method to a particular problem requires careful choices of a number of different parameters and inputs. These include the damping/weighting factor $\epsilon$ and the dimensions of the PEF's. A particularly important decision is the choice of models to be used for the estimation of the PEF's ${\bf S}$ and ${\bf N}$. Sava and Guitton (2003) use muting in the parabolic radon transforms (PRT) domain to discriminate between multiples and primaries in the angle domain, and we use their results as a starting point for a PEF-based approach here. Rather than simply subtracting the multiple model output by the PRT approach from the original data, we use the primary and multiple models to estimate PEF's which are then used in the signal/noise separation technique outlined above. A series of synthetic and real data examples illustrate the effectiveness of the technique, and enable us to test various options for PEF estimation models.