Commit dc909d95 authored by thomas.forbriger's avatar thomas.forbriger Committed by thomas.forbriger
Browse files

revised first paragraphs

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: 4987
SVN UUID:     67feda4a-a26e-11df-9d6e-31afc202ad0c
parent 2c037796
......@@ -21,14 +21,18 @@
\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{\Fs}{\ensuremath{\Fourier{s}}}
%\newcommand{\Sg}{\ensuremath{g}}
%\newcommand{\Fq}{\ensuremath{\Fourier{q}}}
\newcommand{\Fs}{\ensuremath{\Fourier{g}}}
\newcommand{\Sg}{\ensuremath{s}}
\newcommand{\Fq}{\ensuremath{\Fourier{c}}}
\newcommand{\Fg}{\ensuremath{\Fourier{\Sg}}}
\newcommand{\FQl}{\ensuremath{\Fourier{\Sg}_{l}^{\text{opt}}}}
\newcommand{\So}{\ensuremath{\omega}}
\newcommand{\Sf}{\ensuremath{f}}
\newcommand{\Sr}{\ensuremath{r}}
\newcommand{\Ssk}{\ensuremath{\sum\limits_{k}}}
\newcommand{\Ssk}{\ensuremath{\sum\limits_{k=1}^{M}}}
\newcommand{\SslN}{\ensuremath{\sum\limits_{l=0}^{N-1}}}
\newcommand{\Silk}{\ensuremath{_{lk}}}
\newcommand{\Scc}{\ensuremath{^{\ast}}}
......@@ -46,43 +50,44 @@ 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
Let $\Fd_{lk}$ be the complex Fourier expansion coefficient at frequency
$\So_l=l\,\Delta\So$ for the time series recorded at offset $\Sr_k$ to 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.
The corresponding Fourier coefficient for the synthetic waveform is
$\Fs\Silk$.
The synthetic waveform is simulated
some model of subsurface structure and a time history of the source
used to excite the synthetic wavefield given by
$N$ Fourier coefficients $\Fg_l$.
$\Fd\Silk$, $\Fs\Silk$, and $\Fg_l$ are complex numbers.
This engine seeks filter coefficients
This engine returns filter coefficients
\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}{M\,\SmE\,\Se^2+
\Ssk\Sf^2_k\,\left|\Fs\Silk\right|^2}
\label{eq:least:squares:solution}
\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
synthetics $\Fq_l\,\Ss\Silk$ to the recorded data.
This is equivalent to minimizing the least-squares objective function
\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
F(\Fq_l;\Se)=\SslN\Ssk\Sf^2_k\,\left|\Fd\Silk-\Fq_l\Fs\Silk\right|^2
+M\,\SmE\,\Se^2\SslN\left|\Fq_l\right|^2
\label{eq:least:squares:error}
\end{equation}
is minimized
with respect to the real and imaginary parts of all $\Fq_l$.
$\Fs\Silk\Scc$ is the complex conjugate of $\Fs\Silk$
Here $\Fs\Silk\Scc$ is the complex conjugate of $\Fs\Silk$
and
\begin{equation}
\SmE=\frac{1}{N}\SslN\Ssk\Sf^2_k\,\left|\Fs\Silk\right|^2
\SmE=\frac{1}{M\,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
is the average power of the Fourier coefficients $\Fs\Silk$ scaled by
$\Sf_k$.
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.
obvious, eq.~\eqref{eq:least:squares:solution} demonstrates that the solution
is a water-level deconvolution, essentially.
The scaling coefficients $\Sf_k$ can be used to make sure that all receivers
$k$ contribute to an equal average amount to
......@@ -91,14 +96,14 @@ They could be chosen
\begin{equation}
\Sf_k=\sqrt{\frac{N}{\SslN\left|\Fs\Silk\right|^2}}
\end{equation}
such that the average energy
such that the average power
\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
\Sf_k=\left(\frac{\Sr_k}{1\,\text{m}}\right)^\kappa
\end{equation}
using $\kappa$ to adjust a compensation for a power law attenuation with
offset $\Sr$.
......
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