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Up: Proposed solutions to attenuate
Previous: A filtering method
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:
| ![\begin{displaymath}
\begin{array}
{rclcl}
{\bf 0} &\approx& {\bf r_d} &=& {\bf L_s m_s+L_n m_n - d}.
\end{array}\end{displaymath}](img76.gif) |
(21) |
The cost function becomes
| ![\begin{displaymath}
f({\bf m_s},{\bf m_n})=\Vert{\bf L_sm_s}+{\bf L_nm_n}-{\bf d})\Vert^2,\end{displaymath}](img77.gif) |
(22) |
and the estimated inverse for
(see Appendix
for details)
| ![\begin{displaymath}
\left( \begin{array}
{c}
\hat{{\bf m_s}} \\ \hat{{\bf m_n...
..._s}L_n})^{-1}{\bf L_n'\overline{R_s}}\end{array}\right){\bf d},\end{displaymath}](img78.gif) |
(23) |
with
and
. Both rows in equations (
) are the
solutions of a weighted least-squares problem Menke (1989) for the following
fitting goals:
| ![\begin{displaymath}
\begin{array}
{ccccl}
{\bf 0} &\approx &{\bf r_{ds}}& =& {\...
...dn}}& =& {\bf \overline{R_s}}({\bf L_n m_n -
d}),
\end{array}\end{displaymath}](img81.gif) |
(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:
| ![\begin{displaymath}
\begin{array}
{rclcl}
\overline{{\bf R_n}}'\overline{{\bf R...
... =& \overline{{\bf
R_n}} &\approx& \bf{C_d^{-1}}.
\end{array}\end{displaymath}](img86.gif) |
(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