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General properties of the noise and signal filters

Following the preceding definitions, we can define the noise and signal filters more precisely. But first, remind that
\begin{displaymath}
\left( \begin{array}
{c} 
 \hat{{\bf m_n}} \\  \hat{{\bf m_s...
 ..._n}L_s})^{-1}{\bf L_s'\overline{R_n}}\end{array}\right){\bf d},\end{displaymath}   
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}
(28)
${\bf \overline{R_s}}$ and ${\bf \overline{R_n}}$ are signal and noise filtering operators respectively. If we denote
   \begin{eqnarray}
{\bf \overline{R_s}}&=&{\bf I}-{\bf R_s},
\nonumber \\  
\overline{{\bf R_n}}&=&{\bf I}-{\bf R_n}
.\end{eqnarray}
(29)
with ${\bf R_s}={\bf L_s}({\bf L_s'L_s})^{-1}{\bf L_s'}$ and ${\bf R_n}={\bf L_n}({\bf L_n'L_n})^{-1}{\bf L_n'}$ the signal and noise resolution operators, we deduce that ${\bf R_s}$ and ${\bf \overline{R_s}}$, ${\bf R_s}$ and ${\bf \overline{R_s}}$ are complementary operators (definition 2).

It can be shown that ${\bf \overline{R_s}}$, ${\bf \overline{R_n}}$,${\bf R_s}$ and ${\bf R_n}$ are projectors. For ${\bf R_s}$ and ${\bf \overline{R_s}}$, 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}
(30)
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}
(31)
Thus, ${\bf R_s}$ and ${\bf \overline{R_s}}$ are projectors as defined in definition 1. The same proofs work for ${\bf R_n}$ and ${\bf \overline{R_n}}$.

We can prove that ${\bf \overline{R_s}}$ and ${\bf R_s}$, ${\bf \overline{R_n}}$ and ${\bf R_n}$ are mutually orthogonal. For ${\bf R_s}$ and ${\bf \overline{R_s}}$, 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}
(32)
Hence, ${\bf \overline{R_s}}$ and ${\bf R_s}$, ${\bf \overline{R_n}}$ and ${\bf R_n}$ are complementary, mutually orthogonal projectors.


next up previous print clean
Next: Geometric interpretation Up: geometric interpretation of the Previous: Definitions
Stanford Exploration Project
4/29/2001