Spitz makes a better assumption for the signal PEF |

We assume that the data vector is composed of the signal
and noise components and :

where is a scalar scaling coefficient, reflecting the presumed signal-to-noise ration (Claerbout, 1999).

The formal solution of system (6-7)
has the form of a *projection filter*:

In 1-D or - setting, one can accomplish the division in formulas (8) and (9) directly by spectral factorization and inverse recursive filtering (Soubaras, 1995,1994). A similar approach can be applied in the case of - or - filtering with the help of the helix transform (Claerbout, 1998; Ozdemir et al., 1999) or by solving system (6-7) directly with an iterative method (Abma, 1995).

Claerbout's approach, implemented in the examples of *GEE*
(Claerbout, 1999), is to estimate the signal and noise PEFs and 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 completely embeds the shape of
, our estimate of serves as a predictor of both signal and
noise. We might as well consider it as , the prediction-error
filter for the data.

Spitz (1999) argues that the data PEF can be regarded as the convolution of the signal and noise PEFs and .

This assertion suggests the following algorithm:

- Estimate and .
- Estimate by deconvolving (polynomial division) by .
- Solve the least-square system (6-7).

The modified algorithm is The formal least-squares solution of system (10-11) is

Comparing (12) with (8), we can see that both the numerator and the denominator in the two expressions differ by the same multiplier . This multiplication should not effect the result of projection filtering.

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 - prediction-error filters
were estimated from the same data by applying different filter
templates. The template for is

a a a a a a 1 a a a a a a a a a a awhere the

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).

signoi90
Signal and noise separation with the
original GEE method. The input signal is on the left. Next is that
signal with random noise added. Next are the estimated signal and
the estimated noise.
Figure 1. |
---|

signoi
Signal and noise separation
with the modified Spitz method. The input signal is on the left.
Next is that signal with random noise added. Next are the estimated signal
and the estimated noise.
Figure 2. |
---|

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 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).

planes90
Plane wave separation with the
GEE method. The input signal is on the left. Next is that signal
with noise added. Next are the estimated signal and the estimated
noise.
Figure 3. |
---|

planes
Plane wave separation with the
modified Spitz method. The input signal is on the left. Next is
that signal with noise added. Next are the estimated signal and the
estimated noise.
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.

Spitz makes a better assumption for the signal PEF |

2013-03-03