paper.tex

国外免费地震资料处理软件包

\begin{equation}
{\mbox{amp}_t \over \mbox{amp}_x}=
\left|{\partial x \over \partial t}\right|\,{\Delta t \over \Delta x}=
{\Delta t \over \delta t} \leq 1\;.
\label{eqn:Jacobian}
\end{equation}

According to the proposed modification, Hale's antialiasing principle
is reformulated, as follows:
\begin{quote}
{\em In the steep part of an integral operator, never allow successive
time shifts applied to the input trace to differ by more than one time
sampling interval. In the flat part of the operator, never allow
successive space shifts to differ by more than one space sampling
interval.}
\end{quote}

\inputdir{flt}

Figure \ref{fig:amotra}, borrowed from \cite{Claerbout.bei.95},
illustrates the basic idea of the proposed technique. It clearly shows
the difference between the flat and steep parts of migration
hyperbolas. To observe the reciprocity, rotate the figure by 90
degrees.

\plot{amotra}{height=2in}{Figure borrowed from
  \cite{Claerbout.bei.95} to illustrate the reciprocity
  antialiasing. The flat parts of the hyperbolas require interpolation
  in time. The steep parts of the hyperbolas require interpolation in
  space.}

\inputdir{mod}

To compare the proposed antialiasing method with the temporal
filtering method, I test the antialiased migration program on simple
2-D synthetic tests. Figure \ref{fig:amomod} shows a simple model and
the modeling results from modeling without antialiasing, with temporal
filtering, and with the proposed reciprocity method.  The modeling
results were migrated with the corresponding migration operators to
obtain the image of the model in Figure \ref{fig:amomig}.  Both the
temporal filtering and the proposed method succeed in removing the
major aliasing artifacts. However, the reciprocity method demonstrates
a higher resolution and a better preservation of the frequency
content.

\plot{amomod}{width=4.5in}{Top left is a synthetic model. Top
right is modeling without antialiasing. Bottom left is modeling with
reciprocity antialiasing (the proposed method). Bottom right is modeling
with antialiasing by temporal filtering.}

\plot{amomig}{width=4.5in}{Top left plot is the synthetic
model. The other plots are migrations of the corresponding data shown
in the previous figure . Top right is a migration without
antialiasing. Bottom left is a migration with reciprocity antialiasing
(the proposed method). Bottom right is a migration with triangle
filter antialiasing.}

\inputdir{sig}

These properties are examined more closely in the next synthetic
example. Figure \ref{fig:amosmo} shows a more sophisticated synthetic
model that contains a fault, an unconformity and layered structures
\cite[]{Claerbout.bei.95}. For better displaying, I extract the central
part of the model and compare it with the migration results of
different methods in Figure \ref{fig:amosmi}. Comparing the plots
shows that the reciprocity method successfully removes the aliasing
artifacts (round-off errors) of the aliased (nearest neighbor
interpolation) migration.  At the same time, it is less harmful to the
high-frequency components of the data than triangle filtering. This
conclusion finds an additional support in Figure \ref{fig:amospe} that
displays the average spectrum of the image traces for different
methods. Both of the antialiasing methods remove the high-frequency
artifacts of the nearest neighbor modeling and migration. The
reciprocity method performs it in a gentler way, preserving the
high-frequency components of the model.

\plot{amosmo}{height=2.5in}{Synthetic model used to test
the antialiased migration program.}

\plot{amosmi}{width=6in}{Top left plot is a
zoomed portion of the synthetic model. The other plots are migrated
images. Top right is a migration without antialiasing. Bottom left is
a migration with reciprocity antialiasing (the proposed method). Bottom
right is a migration with triangle filter antialiasing.}

\plot{amospe}{height=2.5in}{Top is the spectrum of the
model. The other plots are the spectra of the migrated images. The
second plot corresponds to the modeling/migration without account for
antialiasing. The third plot is modeling/migration with the
reciprocity antialiasing. The bottom plot is modeling/migration with
triangle antialiasing.}  

The algorithm sequence of the antialiased migration is illustrated in
Figures \ref{fig:amormo} and \ref{fig:amormi}. The two plots in Figure
\ref{fig:amormo} show the steep-dip and flat-dip modeling respectively. The
superposition of these two terms is the resultant antialiased data
shown in the left plot of Figure \ref{fig:amormm}. The right plot of
Figure \ref{fig:amormm} shows the migrated image obtained by adding the
flat-dip (left of Figure \ref{fig:amormi}) and steep-dip (right of Figure
\ref{fig:amormi}) migrations.

\plot{amormo}{width=6in,height=2.25in}{Antialiased modeling. Left
corresponds to the flat-dip term. Right corresponds to the steep-dip
term.}
\plot{amormi}{width=6in,height=2.25in}{Antialiased
migration. Left corresponds to the flat-dip term. Right corresponds to
the steep-dip term.}
\plot{amormm}{width=6in,height=2.25in}{Antialiased
modeling and migration. Left is the superposition of the flat-dip and
steep-dip modeling. Right is superposition of the flat-dip and
steep-dip migration.}

\begin{comment}
The efficiency of the antialiased migration is compared in the CPU
time chart in Figure \ref{fig:amochp}.  The test data set included 500
by 250 data points with $\Delta t=0.004$ sec, and $\Delta x = 25$ m.
The figure shows that the performance of the reciprocity antialiasing
increases with increase of the migration velocity. This behavior can
be explained by the fact that high-velocity migration hyperbolas
require a smaller number of expensive computations in the steep
(aliased) parts. It allows us to expect a high performance of the
method in application to the curvilinear operators with limited
aperture (dip moveout, azimuth moveout, shot continuation).

%\plot{amochp}{height=1.5in}{CPU time of migration programs
%on HP 9000-735 versus the constant migration velocity used in the
%experiment.}
\end{comment}

\subsection{3-D antialiasing}
\inputdir{imp}

The proposed method of antialiasing is easily generalized to the case
of a three-dimensional integral operator. In this case, we need to
consider three different parameterizations: $t(x,y)$, $x(t,y)$, and
$y(t,x)$ and switch from one of them to another according to the rule:
\begin{itemize}
\item if $\Delta t \geq {{\Delta x} \, {|\partial t / \partial x|}}$
and      $\Delta t \geq {{\Delta y} \, {|\partial t / \partial y|}}$,
use $t(x,y)$,
\item if $\Delta x \geq {{\Delta t} \, {|\partial x / \partial t|}}$
and      $\Delta x \geq {{\Delta y} \, {|\partial x / \partial y|}}$,
use $x(t,y)$,
\item if $\Delta y \geq {{\Delta t} \, {|\partial y / \partial t|}}$
and      $\Delta y \geq {{\Delta x} \, {|\partial y / \partial x|}}$,
use $y(t,x)$.
\end{itemize}

Following \cite{GEO66-02-06540666}, I illustrate 3-D
antialiasing by applying prestack time migration on a single input
trace. The results are shown in Figures~\ref{fig:imp-noa},
\ref{fig:imp-aal} and \ref{fig:imp-all}. The result without any
antialiasing protection (Figure~\ref{fig:imp-noa}) contains clearly
visible aliasing artifacts caused by the steeply dipping parts of the
operator. Antialiasing by temporal filtering
(Figure~\ref{fig:imp-aal}) removes the artifacts but also attenuates
the steeply dipping events. Antialiasing by the proposed reciprocal
parameterization (Figure~\ref{fig:imp-all}) removes the aliasing
artifacts while preserving the steeply dipping events and the image
resolution.

\plot{imp-noa}{width=6in}{Prestack 3-D time migration of a single
  input trace. Top: time slice at 1 s. Bottom: vertical slice. No
  antialiasing protection has been applied. As a result, aliasing
  artifacts are clearly visible in the image.}

\plot{imp-aal}{width=6in}{Prestack 3-D time migration of a single
  input trace. Top: time slice at 1 s. Bottom: vertical slice.
  Antialiasing by temporal filtering has been applied. Aliasing artifacts
  are removed, steeply dipping events are attenuated.}

\plot{imp-all}{width=6in}{Prestack 3-D time migration of a single
  input trace. Top: time slice at 1 s. Bottom: vertical slice.  The
  proposed reciprocal antialiasing has been applied. Aliasing
  artifacts are removed, steeply dipping events are preserved.}

\section{Conclusions}
%%%%
I have introduced a new method of antialiasing integral operators,
modified from Hale's approach to antialiased dip moveout. The method
compares favorably with the popular temporal filtering technique. The
main advantages are:
\begin{enumerate}
\item Accurate handling of variable operator dips.
\item Consequent preservation of the high-frequency part of the data
spectrum, leading to a higher resolution.
\item Easy control of operator amplitudes.
\item Easy generalization to 3-D.
\end{enumerate}
The method possesses a sufficient numerical efficiency in practical
implementations. Its most appropriate usage is for antialiasing
operators with analytically computed summation paths, such as prestack
time migration, dip moveout, azimuth moveout, and shot continuation.

\section{Acknowledgments}

I thank Biondo Biondi for many helpful discussions. The financial
support for this work was provided by the sponsors of the Stanford
Exploration Project.

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