Commit a98de2f2 authored by thomas.forbriger's avatar thomas.forbriger Committed by thomas.forbriger
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started with text for Fourier domain least squares

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: 4980
SVN UUID:     67feda4a-a26e-11df-9d6e-31afc202ad0c
parent ab1a7c53
This is a legacy version of the repository. It may be incomplete as well as
inconsistent. See README.history for details. For the old stock of the
repository copyright and licence conditions apply as specified for versions
commited after 2015-03-01. Use recent versions as a base for new development.
The legacy version is only stored to keep a record of history.
% this is <libstfinv.tex>
% ----------------------------------------------------------------------------
% $Id: $
%
% Copyright (c) 2013 by Thomas Forbriger (BFO Schiltach)
%
% libstfinv
%
% REVISIONS and CHANGES
% 13/02/2013 V1.0 Thomas Forbriger
%
% ============================================================================
%
\documentclass{article}
\usepackage{ngerman}
\usepackage{pslatex}
\usepackage{anysize}
\usepackage{amsmath}
\usepackage{graphicx}
%----------------------------------------------------------------------
\newcommand{\libstfinv}{\texttt{libstfinv}}
\newcommand{\Fourier}[1]{\ensuremath{\tilde{#1}}}
\newcommand{\Fd}{\ensuremath{\Fourier{d}}}
\newcommand{\Fs}{\ensuremath{\Fourier{s}}}
\newcommand{\Fg}{\ensuremath{\Fourier{g}}}
\newcommand{\Fq}{\ensuremath{\Fourier{q}}}
\newcommand{\FQl}{\ensuremath{\Fourier{g}_{l}^{\text{opt}}}}
\newcommand{\So}{\ensuremath{\omega}}
\newcommand{\Sf}{\ensuremath{f}}
\newcommand{\Sr}{\ensuremath{r}}
\newcommand{\Ssk}{\ensuremath{\sum\limits_{k}}}
\newcommand{\SslN}{\ensuremath{\sum\limits_{l=0}^{N-1}}}
\newcommand{\Silk}{\ensuremath{_{lk}}}
\newcommand{\Scc}{\ensuremath{^{\ast}}}
\newcommand{\SmE}{\ensuremath{\overline{E}}}
\newcommand{\Se}{\ensuremath{\epsilon}}
%----------------------------------------------------------------------
\begin{document}
\title{\libstfinv}
\author{Thomas Forbriger}
\date{(BFO Schiltach)}
\maketitle
\section{Internals of \libstfinv\ engines}
\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
source.
The corresponding Fourier coefficient for the synthatic waveform calculated
for some subsurface model is $\Fs\Silk$.
The source time history used to excite the synthetic wavefield is given by
the $N$ Fourier coefficients $\Fg_l$.
$\Fd$, $\Fs$, and $\Fg$ are complex numbers.
This engine seeks filter coefficients
\begin{equation}
\Fq_l=\frac{\Ssk\Sf^2_k\,\Fs\Silk\Scc\,\Fd\Silk}{\SmE\,\Se^2+
\Ssk\Sf^2_k\,\left|\Fs\Silk\right|^2}
\end{equation}
such that $\FQl=\Fq_l\,\Fg_l$
are Fourier coefficients of
an optimized source time history providing an improved fit of the
synthetics to the recorded data, i.e.
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
\end{equation}
is minimized
with respect to the real and imaginary parts of all $\Fq_l$.
$\Fs\Silk\Scc$ is the complex conjugate of $\Fs\Silk$
\begin{equation}
\SmE=\frac{1}{N}\SslN\Ssk\Sf^2_k\,\left|\Fs\Silk\right|^2
\end{equation}
\end{document}
% ----- END OF libstfinv.tex -----
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