Commit 3b30662f authored by thomas.forbriger's avatar thomas.forbriger Committed by thomas.forbriger
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scaling factors

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: 4981
SVN UUID:     67feda4a-a26e-11df-9d6e-31afc202ad0c
parent a98de2f2
......@@ -41,6 +41,10 @@
\date{(BFO Schiltach)}
\maketitle
\section{Internals of \libstfinv\ engines}
While the \texttt{doxygen} generated documentation gives the relation in a
implementation specific quantities, this document uses the end-user
perspective.
\subsection{Fourier domain least squares}
Let $\Fd_{lk}$ be the complex Fourier expansion coefficient for frequency
$\So_l=l\,\Delta\So$ of the time series recorded at offset $\Sr_k$ from the
......@@ -64,13 +68,34 @@ the least-squares error
\begin{equation}
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
\label{eq:least:squares}
\end{equation}
is minimized
with respect to the real and imaginary parts of all $\Fq_l$.
$\Fs\Silk\Scc$ is the complex conjugate of $\Fs\Silk$
and
\begin{equation}
\SmE=\frac{1}{N}\SslN\Ssk\Sf^2_k\,\left|\Fs\Silk\right|^2
\end{equation}
is the average energy of the Fourier coefficients $\Fs\Silk$ scaled by
$\Sf_k$.
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}.
They could be chosen
\begin{equation}
\Sf_k=\sqrt{\frac{N}{\SslN\left|\Fs\Silk\right|^2}}
\end{equation}
such that the average energy
\begin{equation}
\frac{1}{N}\SslN\Sf^2_k\,\left|\Fs\Silk\right|^2=1
\end{equation}
for each $k$.
In the actual implementation we prefer
\begin{equation}
\Sf_k=\frac{\Sr_k}{1\,\text{m}}^\kappa
\end{equation}
using $\kappa$ to adjust a compensation for a power law attenuation with
offset $\Sr$.
\end{document}
% ----- END OF libstfinv.tex -----
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