### scaling factors

This is a legacy commit from before 2015-03-01.
It may be incomplete as well as inconsistent.
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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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