paper.tex

国外免费地震资料处理软件包

\begin{comment}
\subsection{Gap interpolation}

\inputdir{hole}

Irregular gaps occur in the recorded data for many different reasons,
and prediction-error filters are known as a powerful method for
interpolating them. Interpolating irregularly spaced data also reduces
to gap interpolation after binning. 
\par
Figure~\ref{fig:hole0} shows a simple synthetic example of gap
interpolation from \cite{gee}.  The input data has a large
elliptic gap cut out from a two plane-wave model. I estimate both dip
components from the input data by using the method of
equations~(\ref{eqn:lin2}-\ref{eqn:reg2}). The initial values for the
two local dips were 1 and 0, and the estimated values are close to the
true dips of 2 and -1 (two middle plots in Figure~\ref{fig:hole0}.)
Although the estimation program did not make any assumption about dip
being constant, it correctly estimated nearly constant values with the
help of regularization equations~(\ref{eqn:reg1}-\ref{eqn:reg2}). The
rightmost plot in Figure~\ref{fig:hole0} shows the result of gap
interpolation with a two-plane local plane-wave destructor. The result
is nearly perfect and compares favorably with the analogous result of
the $T$-$X$ PEF technique \cite[]{gee}.

\plot{hole0}{width=6in,height=3.5in}{Synthetic gap interpolation
  example. From left to right: original data, input data, first
  estimated dip, second estimated dip, interpolation output.}
\par
Figure~\ref{fig:seab} is another benchmark gap interpolation example
from \cite{gee}. The data are ocean depth measurements from one
day SeaBeam acquisition. The data after normalized binning are shown
in the left plot of Figure~\ref{fig:seab}. From the known part of the
data, we can partially see a certain elongated and faulted structure
on the ocean floor. Estimating a smoothed dominant dip in the data and
interpolating with the plane-wave destruction filters produces the
image in the right plot of Figure~\ref{fig:seab}. The V-shaped
acquisition pattern is somewhat visible in the interpolation result,
which might indicate the presence of a fault. Otherwise, the result
is both visually pleasing and fully agreeable with the data.
\cite{Clapp.sep.105.bob3} shows on the same data example how to
obtain multiple statistically equivalent realizations of the
interpolated data.

\inputdir{seab}

\plot{seab}{width=6in,height=3.5in}{Depth of the ocean from SeaBeam
  measurements.  Left plot: after binning. Right plot: after binning
  and gap interpolation.}
\par
A 3-D interpolation example is shown in Figure~\ref{fig:passfill}. The
input data resulted from a passive seismic experiment
\cite[]{Cole.sepphd.86} and originally contained many gaps because of
instrument failure. I interpolated the 3-D gaps with a pair of two
orthogonal plane-wave destructors in the manner proposed by
\cite{Schwab.sep.84.271} for $T$-$X$ prediction filters. The
interpolation result shows a visually pleasing continuation of locally
plane events through the gaps. It compares favorably with an analogous
result of a stationary $T$-$X$ PEF.

\inputdir{blast}

\plot{passfill}{width=6in,height=3.5in}{3-D gap interpolation in
  passive seismic data. The left 12 panels are slices of the input
  data.  The right 12 panels are the corresponding slices in the
  interpolation output.}
\par
We can conclude that plane-wave destructors provide an effective
method of gap filling and missing data interpolation.
\end{comment}

\subsection{Trace interpolation beyond aliasing}

\inputdir{alias}

\cite{GEO56-06-07850794} popularized the application of
prediction-error filters to regular trace interpolation and showed how
the spatial aliasing restriction can be overcome by scaling the
lower frequencies of $F$-$X$ PEFs. An analogous technique for $T$-$X$
filters was developed by \cite{Claerbout.blackwell.92,gee} and
was applied for 3-D interpolation with non-stationary PEFs by
\cite{Crawley.sepphd.104}. The $T$-$X$ technique implies
stretching the filter in all directions so that its dip spectrum is
preserved while the coefficients are estimated at alternating
traces. After the filter is estimated, it is scaled back and used for
interpolating missing traces between the known ones.  A very
similar method works for finite-difference plane wave destructors,
only we need to take special care of the aliased dips at the dip
estimation stage.

A simple synthetic example of interpolation beyond aliasing is shown
in Figure~\ref{fig:aliasp0}. The input data are clearly aliased and
non-stationary. To take the aliasing into account, I estimate the two
dips present in the data with the slope estimation technique of
equations~(\ref{eqn:lin2}) and~(\ref{eqn:reg1}-\ref{eqn:reg2}). The
first dip corresponds to the true slope, while the second dip
corresponds to the aliased dip component. In this example, the true
dip is non-negative everywhere and is easily distinguished from the
aliased one. In the more general case, an additional interpretation
may be required to determine which of the dip components is
contaminated by aliasing.  Throwing away the aliased dip and
interpolating intermediate traces with the true dip produces the
accurate interpolation result shown in the right plot of
Figure~\ref{fig:aliasp0}. Three additional traces were inserted between
each of the neighboring input traces.

\plot{aliasp0}{width=5.5in,height=2.75in}{Synthetic example of
  interpolation beyond aliasing with plane-wave destruction filters.
  Left: input aliased data, right: interpolation output.Three
  additional traces were inserted between each of the neighboring
  input traces.}

\inputdir{sean}

\par
Figure~\ref{fig:sean2} shows a marine 2-D shot gather from a deep
water Gulf of Mexico survey before and after subsampling in the offset
direction. The data are similar to those used by
\cite{Crawley.sepphd.104}. The shot gather has long-period
multiples and complicated diffraction events caused by a salt body.
The amplitudes of the hyperbolic events are not as uniformly
distributed as in the synthetic case of Figure~\ref{fig:aliasp0}.
Subsampling by a factor of two (the right plot in
Figure~\ref{fig:sean2}) causes clearly visible aliasing in the
steeply dipping events.  The goal of the experiment is to interpolate
the missing traces in the subsampled data and to compare the result
with the original gather shown in the left plot of
Figure~\ref{fig:sean2}.

\plot{sean2}{angle=90,totalheight=8.69in,width=5.056in}{2-D marine 
  shot gather. 
  Left: original.  Right: subsampled by a factor of two in the offset
  direction.}
\par
A straightforward application of the dip estimation
equations~(\ref{eqn:lin2}-\ref{eqn:reg2}) applied to aliased data can
easily lead to erroneous aliased dip estimation because the aliased dip
may get picked instead of the true dip. In order to avoid
this problem, I chose a slightly more complex strategy. The algorithm for trace
interpolation of aliased data consists of the following steps:
\begin{enumerate}
\item Applying Claerbout's $T$-$X$ methodology, stretch a two-dip
  plane-wave destruction filter and estimate the dips from decimated
  data. 
\item The second estimated dip will be degraded by aliasing. Ignore
  this initial second-dip estimate.
\item Estimate the second dip component again by fixing the first dip
  component and using it as the initial estimate of the second
  component. This trick prevents the nonlinear estimation algorithm
  from picking the wrong (aliased) dip in the data.
\item Downscale the estimated two-dip filter and use it for
  interpolating missing traces.
\end{enumerate}
The two estimated dip components are shown in
Figure~\ref{fig:sean2-dip}. The first component contains only positive
dips. The second component coincides with the first one in the areas
where only a single dip is present in the data. In other areas, it
picks the complementary dip, which has a negative value for
back-dipping hyperbolic diffractions.

\plot{sean2-dip}{angle=90,totalheight=8.69in,width=5.056in}{Two components
  of the
  estimated dip field for the decimated 2-D marine shot gather.}
\par
Figure~\ref{fig:sean2-int} shows the interpolation result and the
difference between the interpolated traces and the original traces,
plotted at the same clip value. The method succeeded in the sense that
it is impossible to distinguish interpolated traces from the
interpolation result alone. However, it is not ideal, because some of
the original energy is missing in the output. A close-up comparison
between the original and the interpolated traces in
Figure~\ref{fig:sean2-close} shows that imperfection in more detail.
Some of the steepest events in the middle of the section are poorly
interpolated, and in some of the other places, the second dip
component is continued instead of the first one.

\plot{sean2-int}{angle=90,totalheight=8.69in,width=5.056in}{Left: 
  2-D marine shot
  gather after trace interpolation. Right: Difference between the
  interpolated and the original gather. The error is zero at the location of
  original traces and fairly random at the location of inserted traces.}
\plot{sean2-close}{angle=90,totalheight=8.69in,width=5.056in}{Close-up 
  comparison of
  the interpolated (right) and the original data (left).}
\par
One could improve the interpolation result considerably  by including
another dimension. To achieve a better result, we can use a pair of
plane-wave destructors, one predicting local plane waves in the offset
direction and the other predicting local plane waves in the shot
direction.

\subsection{Signal and noise separation}

\inputdir{signoi}

Signal and noise separation and noise attenuation are yet another
important application of plane-wave prediction filters. A random noise
attention has been successfully addressed by
\cite{SEG-1984-S10.1}, \cite{SEG-1986-POS2.10},
\cite{GEO60-06-18871896}, \cite{SEG-1995-0711}, and others. A
more challenging problem of coherent noise attenuation has only recently
joined the circle of the prediction technique applications
\cite[]{TLE18-01-00550058,morganSEG,SEG-2001-13051308}.
\par
The problem has a very clear interpretation in terms of the local dip
components. If two components, $\mathbf{s}_1$ and $\mathbf{s}_2$ are
estimated from the data, and we can interpret the first component as
signal, and the second component as noise, then the signal and noise
separation problem reduces to solving the least-squares system
\begin{eqnarray}
  \label{eqn:sn1}
  \mathbf{C}(\mathbf{s}_1) \mathbf{d}_1 & \approx & 0 \;, \\
  \label{eqn:sn2}
  \epsilon \mathbf{C}(\mathbf{s}_2) \mathbf{d}_2 & \approx & 0 \;
\end{eqnarray}
for the unknown signal and noise components $\mathbf{d}_1$ and
$\mathbf{d}_2$ of the input data $\mathbf{d}$:
\begin{equation}
  \label{eqn:dsn}
  \mathbf{d}_1 + \mathbf{d}_2 = \mathbf{d}.
\end{equation}
The scalar parameter $\epsilon$ in equation~(\ref{eqn:sn2}) reflects
the signal to noise ratio. We can combine equations~(\ref{eqn:sn1}-\ref{eqn:sn2})
and~(\ref{eqn:dsn}) in the explicit system for the
noise component $\mathbf{d}_2$:
\begin{eqnarray}
  \label{eqn:s1}
  \mathbf{C}(\mathbf{s}_1) \mathbf{d}_2 & \approx & 
  \mathbf{C}(\mathbf{s}_1) \mathbf{d}\;, \\
  \label{eqn:s2}
  \epsilon \mathbf{C}(\mathbf{s}_2) \mathbf{d}_2 & \approx & 0 \;.
\end{eqnarray}
\par
Figure~\ref{fig:sn2} shows a simple example of the described approach.
I estimated two dip components from the input synthetic data and
separated the corresponding events by solving the least-squares
system~(\ref{eqn:s1}-\ref{eqn:s2}). The separation result is visually
perfect.

\plot{sn2}{width=4.8in,height=2.8in}{Simple example of dip-based single
  and noise separation. From left to right: ideal signal, input data,
  estimated signal, estimated noise.}
\par

\inputdir{dune}

Figure~\ref{fig:dune-dat} presents a significantly more complicated
case: a receiver line from of a 3-D land shot gather from Saudi
Arabia, contaminated with three-dimensional ground-roll, which appears 
hyperbolic in the cross-section.
The same dataset has been used previously by \cite{morganSEG}.
The ground-roll noise and the reflection events have a significantly
different frequency content, which might suggest separating
them on the base of frequency alone. The result of frequency-based
separation, shown in Figure~\ref{fig:dune-exp} is, however, not ideal:
part of the noise remains in the estimated signal after the