Commit 2e7524f5 authored by thomas.forbriger's avatar thomas.forbriger Committed by thomas.forbriger
Browse files

proceeding

This is a legacy commit from before 2015-03-01.
It may be incomplete as well as inconsistent.
See COPYING.legacy and README.history for details.


SVN Path:     http://gpitrsvn.gpi.uni-karlsruhe.de/repos/TFSoftware/trunk
SVN Revision: 4982
SVN UUID:     67feda4a-a26e-11df-9d6e-31afc202ad0c
parent 3b30662f
...@@ -59,6 +59,7 @@ This engine seeks filter coefficients ...@@ -59,6 +59,7 @@ This engine seeks filter coefficients
\begin{equation} \begin{equation}
\Fq_l=\frac{\Ssk\Sf^2_k\,\Fs\Silk\Scc\,\Fd\Silk}{\SmE\,\Se^2+ \Fq_l=\frac{\Ssk\Sf^2_k\,\Fs\Silk\Scc\,\Fd\Silk}{\SmE\,\Se^2+
\Ssk\Sf^2_k\,\left|\Fs\Silk\right|^2} \Ssk\Sf^2_k\,\left|\Fs\Silk\right|^2}
\label{eq:least:squares:solution}
\end{equation} \end{equation}
such that $\FQl=\Fq_l\,\Fg_l$ such that $\FQl=\Fq_l\,\Fg_l$
are Fourier coefficients of are Fourier coefficients of
...@@ -68,7 +69,7 @@ the least-squares error ...@@ -68,7 +69,7 @@ the least-squares error
\begin{equation} \begin{equation}
R^2=\SslN\Ssk\Sf^2_k\,\left|\Fd\Silk-\Fq_l\Fs\Silk\right|^2 R^2=\SslN\Ssk\Sf^2_k\,\left|\Fd\Silk-\Fq_l\Fs\Silk\right|^2
+\SmE\,\Se^2\SslN\left|\Fq_l\right|^2 +\SmE\,\Se^2\SslN\left|\Fq_l\right|^2
\label{eq:least:squares} \label{eq:least:squares:error}
\end{equation} \end{equation}
is minimized is minimized
with respect to the real and imaginary parts of all $\Fq_l$. with respect to the real and imaginary parts of all $\Fq_l$.
...@@ -79,8 +80,13 @@ and ...@@ -79,8 +80,13 @@ and
\end{equation} \end{equation}
is the average energy of the Fourier coefficients $\Fs\Silk$ scaled by is the average energy of the Fourier coefficients $\Fs\Silk$ scaled by
$\Sf_k$. $\Sf_k$.
While eq.~\eqref{eq:least:squares:error} makes the least-squares approach
obvious, eq.~\eqref{eq:least:squares:solution} shows that the solution
esentially is a water-level deconvolution.
The scaling coefficients $\Sf_k$ can be used to make sure that all receivers The scaling coefficients $\Sf_k$ can be used to make sure that all receivers
$k$ contribute to an equal average amount to eq.~\eqref{eq:least:squares}. $k$ contribute to an equal average amount to
eq.~\eqref{eq:least:squares:error}.
They could be chosen They could be chosen
\begin{equation} \begin{equation}
\Sf_k=\sqrt{\frac{N}{\SslN\left|\Fs\Silk\right|^2}} \Sf_k=\sqrt{\frac{N}{\SslN\left|\Fs\Silk\right|^2}}
...@@ -96,6 +102,14 @@ In the actual implementation we prefer ...@@ -96,6 +102,14 @@ In the actual implementation we prefer
\end{equation} \end{equation}
using $\kappa$ to adjust a compensation for a power law attenuation with using $\kappa$ to adjust a compensation for a power law attenuation with
offset $\Sr$. offset $\Sr$.
With $\Se=0$ the $\Fq_l$ in eq.~\eqref{eq:least:squares:error} miminize the
data fit.
At frequencies where $\Fs\Silk\rightarrow0$, the solution in
eq.~\eqref{eq:least:squares:solution} develops a singularity.
A finite $\Se$ is used to stabilize the least-squares solution by introducing
a penalty to eq.~\eqref{eq:least:squares:error} and a water-level to the
denominator of eq.~\eqref{eq:least:squares:solution}.
\end{document} \end{document}
% ----- END OF libstfinv.tex ----- % ----- END OF libstfinv.tex -----
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