We assume that the data vector is composed of the signal and noise components and :
The formal solution of system (6-7) has the form of a projection filter :
Claerbout's approach, implemented in the examples of GEE Claerbout (1999), is to estimate the signal and noise PEFs S and N from the data by specifying different shape templates for these two filters. The filter estimates can be iteratively refined after the initial signal and noise separation. In some examples, such as those shown in this paper, the signal and noise templates are not easily separated. When the signal template behaves as an extension of the noise template so that the shape of S completely embeds the shape of N, our estimate of S serves as a predictor of both signal and noise. We might as well consider it as D, the prediction-error filter for the data.
Spitz (1999) argues that the data PEF D can be regarded as the convolution of the signal and noise PEFs S and N.
This assertion suggests the following algorithm:
Figure 1 shows a simple example of signal and noise separation taken from GEE Claerbout (1999). The signal consists of two crossing plane waves with random amplitudes, and the noise is spatially random. The data and noise T-X prediction-error filters were estimated from the same data by applying different filter templates. The template for D is
a a a a a a 1 a a a a a a a a a a awhere the a symbol represents adjustable coefficients. The data filter shape has three columns, which allows it to predict two plane waves with different slopes. The noise filter N has only one column. Its template is
1 a a aThe noise PEF can estimate the temporal spectrum but would fail to capture the signal predictability in the space direction. Figure 2 shows the result of applying the modified Spitz method according to equations (10-11). Comparing figures 1 and 2, we can see that using a modified system of equations brings a slightly modified result with more noise in the signal but more signal in the noise. It is as if has changed, and indeed this could be the principal effect of neglecting the denominator in equation (3).
To illustrate a significantly different result using the Spitz insight we examine the new situation shown in Figures 3 and 4. The wave with the positive slope is considered to be regular noise; the other wave is signal. The noise PEF N was estimated from the data by restricting the filter shape so that it could predict only positive slopes. The corresponding template is
a 1 aThe data PEF template is
a a a a 1 a a a a a a a aUsing the data PEF as a substitute for the signal PEF produces a poor result, shown in Figure 3. We see a part of the signal sneaking into the noise estimate. Using the modified Spitz method, we obtain a clean separation of the plane waves (Figure 4).
Clapp and Brown (1999, 2000) and Brown et al. (1999) show applications of the least-squares signal-noise separation to multiple and ground-roll elimination.