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Instead of removing the noise by filtering (i.e., equation
()), the forward operator can be improved
to model both noise and signal components. This technique treats
the coherent noise as components of the data. Therefore, a coherent
noise modeling operator and signal modeling operator
are introduced to give Nemeth (1996)
in equation ().
The model space then becomes the vector
, where is the model
space for the noise and is the model space for the
signal. Introducing an adequate noise modeling
operator , the data residual will become IID
and can be approximated more safely with a diagonal
operator with constant variance. In terms of fitting goal and omitting
the regularization term, we have now, assuming where is the identity matrix:
| |
(21) |

The cost function becomes
| |
(22) |

and the estimated inverse for (see Appendix
for details)
| |
(23) |

with and . Both rows in equations () are the
solutions of a weighted least-squares problem Menke (1989) for the following
fitting goals:
| |
(24) |

where and are the residuals for the
noise and signal components. Equation () is true
because and are
(1) projection operators, and (2) signal and noise filtering
operators, respectively (see Appendix for
details). It is important to realize that in practice, the projection
operators in equation () are not directly estimated and
and are rather computed iteratively from the
fitting goal in equation ().
What is interesting in equation (), however, is that
the first fitting goal is very similar to equation (),
the difference stemming from the choice of weighting (or filtering)
operator. The modeling approach can be then interpreted as a
weighting of the data residual with a projection filter that
annihilates coherent noise. Therefore, the noise covariance matrix
is approximated as follows:

| |
(25) |

To summarize, the filtering approach attempts approximating
the noise covariance operator with prediction-error filters,
thus using the property that is the
power spectrum of the noise. In contrary, the modeling approach
tackles the noise problem at its source by trying to model both
the noise and signal simultaneously. Nonetheless, this technique
can still be seen as a way of approximating with
projection filters (i.e., equation ()).
Numerous authors
Abma and Claerbout (1995); Ozdemir et al. (1999); Soubaras (1994); Soubaras (1995); Abma (1995)
have proved that projection filters were more desirable for
signal/noise separation than simple prediction-error filters. The main
reason is that the spectrum of projection filters is in the range of zero
to one. Therefore, the modeling approach should be used
as much as possible for coherent seismic noise attenuation. As an
illustration, Chapter demonstrates on
an interpolation problem of noisy data the benefits of the modeling
approach compared to the filtering one.

In the next section, practical considerations are addressed for both
noise filtering and noise modeling approaches. In particular,
strategies for choosing the filters and operators are detailed.
In addition, a pattern-based approach for signal/noise separation
is briefly introduced.

** Next:** Coherent noise filtering and
** Up:** Proposed solutions to attenuate
** Previous:** A filtering method
Stanford Exploration Project

5/5/2005