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![]() | Spitz makes a better assumption for the signal PEF | ![]() |
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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 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:
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 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
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
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signoi90
Figure 1. 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. |
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signoi
Figure 2. 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. |
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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).
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planes90
Figure 3. 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. |
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planes
Figure 4. 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. |
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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.
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![]() | Spitz makes a better assumption for the signal PEF | ![]() |
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