Commit 5e168ff8 authored by thomas.forbriger's avatar thomas.forbriger Committed by thomas.forbriger
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revised second half

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: 4988
SVN UUID:     67feda4a-a26e-11df-9d6e-31afc202ad0c
parent dc909d95
......@@ -70,7 +70,7 @@ This engine returns filter coefficients
such that $\FQl=\Fq_l\,\Fg_l$
are Fourier coefficients of
an optimized source time history providing an improved fit of the
synthetics $\Fq_l\,\Ss\Silk$ to the recorded data.
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
......@@ -89,7 +89,8 @@ 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.
The scaling coefficients $\Sf_k$ can be used to make sure that all receivers
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
......@@ -107,17 +108,29 @@ In the actual implementation we prefer
\end{equation}
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.
With $\Se=0$ the $\Fq_l$ in eq.~\eqref{eq:least:squares:error} miminize the
With $\Se=0$ the $\Fq_l$ in eq.~\eqref{eq:least:squares:error} optimize the
data fit.
At frequencies where $\Fs\Silk\rightarrow0$, the solution in
eq.~\eqref{eq:least:squares:solution} develops a singularity.
A finite $\Se$ is used to stabilize the least-squares solution by introducing
a penalty to eq.~\eqref{eq:least:squares:error} and a water-level to the
denominator of eq.~\eqref{eq:least:squares:solution}.
The $\Fq_l$ are damped (kept artificially small) as the scaled energy of the
synthetics $\Fd\Silk$ at frequency $\So_l$ is smaller than a fraction $\Se^2$
of the average scaled energy of the synthetics $\Fd\Silk$.
At frequencies where $\Fs\Silk$ become small with respect ot 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
by introducing a stabilizing penalty to the objective function in
eq.~\eqref{eq:least:squares:error} and
a water-level to the denominator of eq.~\eqref{eq:least:squares:solution}.
The $\Fq_l$ are damped (kept artificially small) as the
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
than a fraction $\Se^2$ of the overall average power $\SmE$ of the scaled
synthetics $\Sf_k\Fd\Silk$.
\end{document}
% ----- END OF libstfinv.tex -----
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