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Following the preceding definitions, we can define the noise and
signal filters more precisely. But first, recall that
with
| ![\begin{eqnarray}
{\bf \overline{R_s}}&=&{\bf I}-{\bf L_s}({\bf L_s'L_s})^{-1}{\b...
...line{{\bf R_n}}&=&{\bf I}-{\bf L_n}({\bf L_n'L_n})^{-1}{\bf L_n'}.\end{eqnarray}](img300.gif) |
|
| (109) |
and
are signal and
noise filtering operators respectively. If we define
| ![\begin{eqnarray}
{\bf \overline{R_s}}&=&{\bf I}-{\bf R_s},
\nonumber \\ \overline{{\bf R_n}}&=&{\bf I}-{\bf R_n},\end{eqnarray}](img301.gif) |
|
| (110) |
with
and
the signal and
noise resolution operators, we deduce that
and
,
and
are
complementary operators (definition 2).
It can be shown that
,
,
and
are projectors. Indeed, for
and
, we have
| ![\begin{eqnarray}
{\bf R_sR_s}&=&{\bf L_s}({\bf L_s'L_s})^{-1}{\bf L_s'}{\bf
L_s...
... L_s'L_s})^{-1}{\bf L_s'}, \nonumber \\ {\bf R_sR_s}&=&{\bf R_s},\end{eqnarray}](img306.gif) |
|
| |
| (111) |
and
| ![\begin{eqnarray}
{\bf \overline{R_s}}{\bf \overline{R_s}}&=&({\bf I}-{\bf
R_s})...
... {\bf \overline{R_s}}{\bf \overline{R_s}}&=&{\bf \overline{R_s}}.\end{eqnarray}](img307.gif) |
|
| |
| (112) |
Thus,
and
are projectors as defined
in definition 1. The same proofs work for
and
.
We can prove also that
and
,
and
are mutually orthogonal. For
and
, we have
| ![\begin{eqnarray}
{\bf \overline{R_s}}{\bf R_s}&=&({\bf I}-{\bf R_s}){\bf R_s},
...
...\bf R_s}), \nonumber
\\ {\bf \overline{R_s}}{\bf R_s}&=&{\bf 0}.\end{eqnarray}](img308.gif) |
|
| |
| (113) |
Hence,
and
,
and
are complementary, mutually orthogonal projectors.
Next: Geometric interpretation
Up: Geometric interpretation of the
Previous: Definitions
Stanford Exploration Project
5/5/2005