Sometimes a very small amount of signal will be left in the
residuals even after extensive adjustment of the parameters. While this
may signify that the optimal combination of parameters has not been
attained yet, in practice the result is better than either *f-x*
or *t-x*
decon. Given the way the method works (averaging the results of
*f-x* and *t-x* decons), in order for signal to be overlooked by
ERNA, it has to be overlooked by *both* types of decon. The left
panels of Figure show the final residuals after
applying several ERNA iterations to each of the three datasets
presented in Figures , , and
. Hints of signal can be distinguished, especially
in the prestack seismic residual. However, it must be kept in mind
that the amplitude in the panels has been gained for visualization.
The Root Mean Square (RMS) values of the residuals throughout the iterations (normalized
to the value of the input data) are shown in the right
side panels in the same Figure . The first iteration
removes most of the noise; the rest only extract leftover signal from
the residuals. After a small number of iterations ERNA converges to
its practical limit.

Experience with data strongly affected by coherent noise shows
that the method does not react well to crossing dips, especially when
one set of events is less coherent than the other. Applying the method
to such data results in a patchwork of small regions in which a single
dip dominates, much like the results of a dip field estimator that
attempts to find the most energetic local dip. Another instance when
ERNA may not function optimally is the case of very simple
synthetics, which contain extended surfaces of constant (especially
zero) values between events. This is because *f-x* and *t-x* "decons" may
have trouble with such areas. "Spiking" very clean real data with
noise (Figure ) can be a good alternative to using a
simple synthetic.

`Fx2d` is cheaper to apply than `Txdec`, the cost of which increases with
the cube of the filter size. `Txdec` can be applied on a 3-D
cube, doing true 3-D spatial prediction, while with `Fx2d` one has
to resort to a surrogate involving multiple passes. The cost of 3-D *
t-x* decon can be prohibitive nonetheless, so a 2-D decon may have to be
used. Simply looping 2-D noise attenuation operators along the third
dimension of a cube will eliminate data that is coherent along the
third dimension, but is not coherent in the filtering plane. This is
easily visible in the residual. To simulate the effects of true 3-D
noise attenuation one has to take the residual (not the result) of the
first 2-D filtering along the first dimension, and to filter it along
another dimension, obtaining a second result and a second
residual. This second residual is then filtered along the third
dimension to obtain a third filtering result and a third residual. The
final result is the sum of the three filtering results.

A potential objection to the method is that the noise passed by the
spatial prediction filtering programs consists only of coding
artifacts due to the different size of the windows. The response is
that when varying the parameters, including window sizes, inside the
same method, the artifacts remain similar enough as not to destruct
their coherence when superimposed. However, the patterns passed by any
of the *f-x* runs are different from those passed by the *
t-x*, and they interfere to destroy their coherence. The two spatial prediction
filtering methods employed are supposed in theory to produce similar
results, by working in two different domains. However, we see that in
practice they produce different artefacts. A more quantitative
analysis of the properties of the noise passed by *f-x* and *t-x* decon may be warranted.

My experience showed that ERNA can benefit velocity analysis in the case of 2D prestack seismic data when automatic velocity analysis is affordable for each midpoint. Denoising common-offset planes greatly improved the ability of automatic velocity picking programs to output results that are consistent across midpoint.

*F-x* decon is commonly implemented in seismic processing packages,
and it also exists in SEPlib, as does *t-x* decon. There is therefore
no need to write any code in order to implement the ERNA technique. Everything can be accomplished in a Makefile, a
shell script or any other form of batch file.

7/8/2003