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From this dataset, the goal is to estimate in a least-squares sense
a velocity panel with the hyperbolic radon
transform of Chapter (
). This velocity
panel can be later used for velocity analysis or multiple attenuation
Alvarez and Larner (2004); Foster and Mosher (1992); Thorson and Claerbout (1985).
A conjugate-gradient (CG) solver is used for the iterations.
Figure
a illustrates the effects of the coherent
noise on the model space after inversion of the data in Figure
a. Some energy corresponding to both
the signal and the noise is present in Figure
a,
but overall, it is quite difficult to recognize events corresponding
to the signal. The remodeled data in Figure
b
(i.e.,
) indicate that the inversion is fitting
both the noise and signal, which creates artifacts in
Figure
a. In addition, note that the residual (i.e.,
) in Figure
c is not IID: a lot
of coherent energy remains. Therefore, the filtering and modeling
techniques are needed to obtain (1) IID residuals, (2) a better
velocity panel, and (3) noise-free remodeled data.
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In this example, the noise is assumed to be known. For the filtering
approach, a PEF is used for
in
equation (
). For the modeling approach,
and
,
being the same PEF that for the
filtering approach.
First, as an illustration on how a simple
prediction-filter differs from a projection filter, Figure
a displays the spectrum of the PEF
only and
Figure
b the spectrum of the projection
filter
with
.The PEF is estimated from the known noise, and we notice that its
spectrum in Figure
a is the smallest at the noise
location. The amplitude varies a lot with local minima and a
maximum amplitude far from one, however. The spectrum
of the projection filter in Figure
b is better behaved:
it equals zero at the noise location and is flat almost everywhere.
The maximum energy is not exactly one because a damping term was added
for the division.
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Figure displays the result of the inversion with the
filtering approach. The estimated model in Figure
a
is now noise-free and the hyperbolas are well focused in the velocity
space. The residual in Figure
d has no coherent
energy left and is IID. The remodeled data in Figure
b are also basically noise free, and so is the
unweighted residual (
) in Figure
c.
Therefore, the filtering technique worked very well and delivered the
expected results.
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Figure shows the result of the modeling
approach. Remember that the inverse of the
PEF estimated
from the true noise is used for the noise operator
and that the hyperbolic radon transform is used to model
the signal. The balancing parameter
in equation
(
) is chosen by trial and error.
The estimated model after inversion is shown in Figure
a. Similar to what we observed with the filtering
approach, the model is noise free and easy to interpret. The remodeled
data in Figure
b show a little bit of coherent noise.
The problem stems from the fact that the two modeling operators
overlap: the inverse PEF can model some tails of hyperbolas and
the radon transform can fit some of the coherent noise
(because the velocity of the noise is within the range of the slowness
scan). Although not crucial here, a regularization term would improve
this result Nemeth (1996). The estimated noise of Figure
c proves that the separation worked very well. A
little bit of one hyperbola has been absorbed, however. The residual
in Figure
d is IID which confirms that the goals
of this technique were reached.
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