Commit bc245fbe authored by thomas.forbriger's avatar thomas.forbriger Committed by thomas.forbriger
Browse files

finished

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: 4990
SVN UUID:     67feda4a-a26e-11df-9d6e-31afc202ad0c
parent 1599f889
......@@ -17,6 +17,7 @@
\usepackage{anysize}
\usepackage{amsmath}
\usepackage{graphicx}
\selectlanguage{english}
%----------------------------------------------------------------------
\newcommand{\libstfinv}{\texttt{libstfinv}}
\newcommand{\Fourier}[1]{\ensuremath{\tilde{#1}}}
......@@ -55,9 +56,9 @@ $\So_l=l\,\Delta\So$ for the time series recorded at offset $\Sr_k$ to the
source.
The corresponding Fourier coefficient for the synthetic waveform is
$\Fs\Silk$.
The synthetic waveform is simulated
The synthetic waveform is numerically simulated for
some model of subsurface structure and a time history of the source
used to excite the synthetic wavefield given by
used to excite the synthetic wavefield which is given by
$N$ Fourier coefficients $\Fg_l$.
$\Fd\Silk$, $\Fs\Silk$, and $\Fg_l$ are complex numbers.
......@@ -68,16 +69,7 @@ This engine returns filter coefficients
\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 $\Fq_l\,\Fs\Silk$ to the recorded data.
This is equivalent to minimizing the least-squares objective function
\begin{equation}
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}
with respect to the real and imaginary parts of all $\Fq_l$.
are Fourier coefficients of an optimized time history of the source.
Here $\Fs\Silk\Scc$ is the complex conjugate of $\Fs\Silk$
and
\begin{equation}
......@@ -85,6 +77,16 @@ and
\end{equation}
is the average power of the Fourier coefficients $\Fs\Silk$ scaled by
$\Sf_k$.
The optimized time history of the source provides an improved fit of the
synthetics $\Fq_l\,\Fs\Silk$ to the recorded data $\Fd\Silk$ in a
least-squares sense.
This is equivalent to minimizing the least-squares objective function
\begin{equation}
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}
with respect to the real and imaginary parts of all $\Fq_l$.
While eq.~\eqref{eq:least:squares:error} makes the least-squares approach
obvious, eq.~\eqref{eq:least:squares:solution} demonstrates that the solution
is a water-level deconvolution, essentially.
......@@ -93,7 +95,7 @@ By means of the
scaling coefficients $\Sf_k$ we can make sure that all receivers
$k$ contribute to an equal average amount to
eq.~\eqref{eq:least:squares:error}.
They could be chosen
The coefficients could be chosen
\begin{equation}
\Sf_k=\sqrt{\frac{N}{\SslN\left|\Fs\Silk\right|^2}}
\end{equation}
......@@ -110,13 +112,14 @@ using $\kappa$ to adjust a compensation for a power law attenuation with
offset $\Sr$.
Alternatively, $\Sf_k$ can be made large for small $\Sr_k$, such that the
source correction filter primarily is obtained from near-offset records.
This might by preferable in early stages of the inversion, when the model of
the subsurface structure can hardly produce waves at correct propagation
velocity.
This is achieved by choosing $\kappa$ small or even negative and
might by preferable in early stages of the inversion when the model of
the subsurface structure is hardly able to produce waves at correct
propagation velocity.
With $\Se=0$ the $\Fq_l$ in eq.~\eqref{eq:least:squares:error} optimize the
data fit.
At frequencies where $\Fs\Silk$ become small with respect ot the noise in
data fit only.
At frequencies where $\Fs\Silk$ become small with respect to the noise in
$\Fd\Silk$, the solution in eq.~\eqref{eq:least:squares:solution} develops a
singularity (for $\Fs\Silk\rightarrow0$).
A finite $\Se$ is used to keep the least-squares solution regular
......@@ -128,9 +131,9 @@ average power
\begin{equation}
\frac{1}{M}\Ssk\Sf_k^2\left|\Fs\Silk\right|^2
\end{equation}
of the scaled synthetics $\Sf_k\Fd\Silk$ at frequency $\So_l$ becomes smaller
of the scaled synthetics $\Sf_k\Fs\Silk$ at frequency $\So_l$ becomes smaller
than a fraction $\Se^2$ of the overall average power $\SmE$ of the scaled
synthetics $\Sf_k\Fd\Silk$.
synthetics $\Sf_k\Fs\Silk$.
\end{document}
% ----- END OF libstfinv.tex -----
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