Next: Discussion of the filtering
Up: Approximating the noise covariance
Previous: Filtering the coherent noise
The real data example I discuss here inspired the preceding synthetic
gather. Figure 6a shows the CMP gather,
6b its amplitude spectrum, 6c the noise model
obtained by simple lowpass filtering of
the CMP gather plus random noise, and 6d the amplitude
spectrum of the model. Three types of noise appear in the data:
 1.
 A lowvelocity, lowfrequency event that corresponds to
the main target of the noise attenuation process.
 2.
 Amplitude anomalies that are both local
(offset 0.9 km and time 2.7 s) and trace related (offset 1.9
km).
 3.
 A time shift near offset 2 km.
The next two sections present the results of the noise filtering methods when the
real data set is used, demonstrating that the coherent noise
can be filtered with a single 1D PEF.
 Filtering with a noise model
In this particular case, because I was assuming stationary signal and noise,
I used one PEF for the entire dataset. The coherent noise is not
completely linear and the noncontinuity of the
coherent noise as displayed in Figure 6c
indicates that a onedimensional PEF is preferable.
I choose a 30coefficient filter (a=30,1) to make sure
that the PEF absorbs enough spectral information from the residual.
The result of the inversion is displayed in Figure
7. The residual (Figure
7c) is not perfectly white, but the
coherent noise has been filtered out. Because the noise
model does not incorporate them, the remaining artifacts
are the amplitude anomalies.
The remaining very weak dipping event in the residual (Figure
7c) needs to be understood.
To investigate this issue, I show in Figure 8a the spectrum
of the input data; in 8b, the residual
before the first iteration; and in 8c,
the residual after the inversion ends.
We can see that the estimated PEF has eliminated almost
all the coherent noise spectrum with a remaining tiny peak at 10 Hertz.
As the iterations go on (between Figures 8b and
c), the peak does not change in amplitude and becomes
relatively more important in the spectrum of the residual,
which explains why some coherent information remains in the residual.
The bimodal shape of the spectrum in Figure 8c
shows one mode around 10 Hertz for the coherent noise,
and one mode around 30 Hertz for the amplitude
anomalies. Because I did not reestimate the PEF iteratively, I will not
get any whiter residual. I think that a nonstationary PEF would
treat the noise more efficiently Guitton et al. (2001).
The limited accuracy of the filter I used for the entire residual
caused the imperfect attenuation.
datawz
Figure 6 (a) A real CMP gather. (b)
The amplitude spectrum of this gather. (c) A noise model obtained by lowpassing
the data and adding random zeromean Gaussian noise. (d) The amplitude spectrum
of the noise model.
csyntwz08
Figure 7 Filtering the coherent
noise in real data with a noise model. (a) An estimated model
space. (b) Reconstructed data using this model space. (c) The weighted residual
() after inversion. (d) The difference between
the input data in Figure 6a and the reconstructed
data in 7b.
spwz08
Figure 8 Filtering the coherent
noise in real data with a noise model. (a) The amplitude spectrum of the
input data in Figure 6a. (b) The amplitude spectrum of
the residual before the first iteration.
(c) The amplitude spectrum of the residual after inversion.
 Filtering without a noise model
As an alternative to using a noise model, I estimate the PEF
iteratively. Once again, in the first stage of this method, I iterate with
no PEF in the fitting goal and then estimate the PEF after a
certain number of iterations from the residual.
I iterated 35 times before estimating the first filter.
The number of iterations turns out to be an important element
in the final separation.
Figures 9a and b shows the
spectrum of the residual during the first stage for 10 and 35
iterations. After 10 iterations, I have one peak at 10 Hertz, and a
second peak around 30 Hertz. If I estimate a PEF after 10 iterations,
it will absorb the two remaining events. The separation tends to be
very sensitive to the second peak. Because it overlaps with the
signal spectrum, the noise spectrum at 30 Hertz should be avoided
in the PEF. I then decided to estimate a PEF after 35 iterations
(Figure 9b) since the second peak was
considerably attenuated.
Figure 9c shows the inverse spectrum of the PEF estimated from
the residual with the spectrum in Figure 9b.
The main features of the spectrum are correctly absorbed in the filter.
In the second stage of the inversion scheme,
I reestimated the PEF only twice during the remaining 38 iterations.
There were 73 iterations in total, 35 iterations for the first stage and 38 for the
second. Figure 10 displays the final results.
In Figure 10c, the residual appears fairly
white with little coherent information left. However, there remains
a very weak background event with the same slope
as the coherent noise I wish to attenuate.
To investigate this problem further, I compute the spectrum of the residual
after the inversion ends.
Figure 11a shows the spectrum of the input data;
Figure 11b, the spectrum of the residual; and
Figure 11c, the same spectrum at a smaller scale.
The residual looks white with a slightly higher peak at 10 Hertz,
which explains the residual in Figure 10c.
Nonetheless, the final result is, I think,
very satisfactory and compares favorably with the result displayed in
Figure 7 when a noise model is known.
To demonstrate the usefulness of a PEF in the fitting goal
(equation (1)) further, I computed a comparison, shown in Figure
12, of the normalized objective functions for the
iterative scheme with and without a PEF.
The PEF clearly improves the optimization.
sp3wz08
Figure 9 The amplitude spectrum of the
residual in the first stage after (a) 10 iterations and (b) 35
iterations. (c) The inverse spectrum of the PEF estimated from the
residual with the spectrum in 9b; as expected, 9c
resembles 9b.
cwz08rec
Figure 10 Filtering the coherent
noise in real data without a noise model. (a) An estimated model
space. (b) The reconstructed data using the model space. (c) The weighted residual
() after inversion. (d) The difference between
the input data in Figure 6a and the reconstructed
data in 10b.
sp2wz08
Figure 11 Filtering the coherent
noise in real data without a noise model. (a) The amplitude spectrum of the
input data in Figure 6a.
(b) The amplitude spectrum of the residual after inversion. (c) The
amplitude spectrum of the residual after inversion using a smaller
scale than in 11b.
conv
Figure 12 Convergence of the iterative
schemes with or without a PEF for seven iterations. The PEF (continuous and
dotted lines) allows the best convergence.

 
Next: Discussion of the filtering
Up: Approximating the noise covariance
Previous: Filtering the coherent noise
Stanford Exploration Project
4/29/2001