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To design an operator to remove artifacts from a velocity stack,
I am going to use the property of a two-column
PEF to destroy a plane wave Claerbout (1999). Since artifacts from
the far offset have a variable dip; this filter will need to be non-stationary.
To design the operator we need to model the artifacts. The model of the
artifacts can be created by applying the operator to the
first and the last trace of the data (Figure 2). We should be able to
describe such a model with a three-column-wide filter.
However, it is easier to divide the model of the artifacts into
a ``horizontal'' and a ``vertical'' parts.
We then create a pair of two-column PEF and train one on the ``horizontal''
and the other on the ``vertical'' part of the noise model. These two filters may then be
applied to a velocity stack one after another to destroy the artifacts. It is very
important to notice that it is not necessary to use
traces from real data to model the ``noise.''
Figure 2 illustrates how these separated parts of the
``noise'' model will look.
**noise
**

Figure 2 Illustration of different parts
of a noise model

Using this approach, we could create the noise
model and PEF only once and then re-use them when needed.
This would greatly reduce the cost of the procedure and allow for the design of a stable filter.
Designing a stable non-stationary PEF is the most difficult task in
any application that uses a non-stationary PEF. However, in this case I have
the advantage of being able to compute a non-stationaty PEF before
the processing, so I can use various methods of smoothing
the filter coefficients.
I can also change the ``noise'' model, which might
be important if I try to use this filter in any least-squares
inversion schemes later.

** Next:** Application of the filter
** Up:** Karpushin: Removing velocity stack
** Previous:** Introduction
Stanford Exploration Project

6/7/2002