paper.tex

国外免费地震资料处理软件包

separation. Changing the $\epsilon$ parameter in
equation~(\ref{eqn:s2}) could clean up the signal estimate, but it
would also bring some of the signal into the subtracted noise. A
better strategy is to separate the events by using both the difference
in frequency and the difference in slope. For that purpose, I adopted
the following algorithm:
\begin{enumerate}
\item Use a frequency-based separation (or, alternatively, a simple
  low-pass filtering) to obtain an initial estimate of the ground-roll
  noise.
\item Select a window around the initial noise. The further
  separation will happen only in that window.
\item Estimate the noise dip from the initial noise estimate.
\item Estimate the signal dip in the selected data window as the
  complimentary dip component to the already known noise dip.
\item Use the signal and noise dips together with the signal and noise
  frequencies to perform the final separation. This is
  achieved by cascading single-dip plane-wave destruction filters with
  local 1-D three-coefficient PEFs aimed at destroying a particular
  frequency.
\end{enumerate}
The separation result is shown in Figure~\ref{fig:dune-sn}. The
separation goal has been fully achieved: the estimated ground-roll noise is
free of the signal components, and the estimated signal is free of the
noise.


\plot{dune-dat}{width=6in,height=3.5in}{Ground-roll-contaminated data
  from Saudi Arabian sand dunes. A reciever cable out of a 3-D shot gather.
  }

\plot{dune-exp}{width=6in,height=7in}{Signal and noise separation
  based on frequency. Top: estimated signal. Bottom: estimated noise.}

\plot{dune-sn}{width=6in,height=7in}{Signal and noise separation based
  on both apparent dip and frequency in the considered receiver cable. 
  Top: estimated signal. Bottom: estimated
  noise.}
\par

\begin{comment}
The left plot in Figure~\ref{fig:ant-dat} shows another test example:
a shot gather from \cite{yilmaz}, which is contaminated by nearly
linear low-velocity noise. In this case, a simple dip-based separation
was sufficient for achieving a good result. The algorithm proceeds as
follows:
\begin{enumerate}
\item Bandpass the original data with an appropriate low-pass filter
  to obtain an initial noise estimate (the right plot in
  Figure~\ref{fig:ant-dat}.) 
\item Estimate the local noise dip from the initial noise model.
\item Estimate the signal dip from the input data as the complimentary
  dip component to the already known noise dip.
\item Estimate the noise by an iterative optimization of
  system~(\ref{eqn:s1}-\ref{eqn:s2}) and subtract it from the data to
  get the signal estimate.
\end{enumerate}
Figure~\ref{fig:ant-sn} shows the separation result. The signal and
noise components are accurately separated.

%\plot{ant-dat}{angle=90,totalheight=8.7in,width=5.075in}{Left: Input
%  noise-contaminated shot gather. Right: Result of low-pass
%  filtering. The filtered data is is used as a noise model to estimate
%  the noise dip.}

%\plot{ant-sn}{angle=90,totalheight=8.7in,width=5.075in}{Signal and
%  noise separation based on dip. Left: estimated signal. Right:
%  estimated noise.}
\par
\end{comment}

The examples in this subsection show that when the signal and noise
components have distinctly different local slopes, we can
successfully separate them with plane-wave destruction filters.

\section{Conclusions}

Plane-wave destruction filters with an improved finite-difference
design can be a valuable tool in processing multidimensional seismic
data. On several examples, I showed their good performance in such
problems as fault detection, missing data interpolation, and noise
attenuation. Although only 2-D examples were demonstrated, it
is straightforward to extend the method to 3-D applications by
considering two orthogonal plane-wave slopes.
\par
The similarities
and differences between plane-wave destructors and $T$-$X$
prediction-error filters can be summarized as follows:
\par
\noindent Similarities:
\begin{itemize}
\item Both types of filters operate in the original time-and-space
  domain of recorded data.
\item Both filters aim to predict local plane-wave events in the data.
\item In most problems, one filter type can be replaced by the other,
  and certain techniques, such as Claerbout's trace interpolation
  method, are common for both approaches.
\end{itemize}
Differences:
\begin{itemize}
\item The design of plane-wave destructors is purely deterministic and
  follows the plane-wave differential equation. The design of $T$-$X$
  PEF has statistical roots in the framework of the maximum-entropy
  spectral analysis \cite[]{Burg.sepphd.6}. In principle, $T$-$X$ PEF
  can characterize more complex signals than local plane waves.
\item In the case of PEF, we estimate filter coefficients. In the
  case of plane-wave destructors, the estimated quantity is the local
  plane-wave slope.  Several important distinctions follow from that
  difference:
\begin{itemize}
\item The filter-estimation problem is linear. The slope estimation
  problem, in the case of the improved filter design, is non-linear,
  but allows for an iterative linearization. In general, non-linearity
  is an undesirable feature because of local minima and the dependence
  on initial conditions. However, we can sometimes use it creatively.
  For example, it helped to avoid aliased dips in the trace
  interpolation example.
\item Non-stationarity is handled gracefully in the local slope
  estimation. No local windows are required to produce a smoothly
  varying estimate of the local slope. This is a much more difficult
  issue for PEFs because of the largely under-determined problem.
\item Local slope has a clearly interpretable physical meaning, which
  allows for easy quality control of the results. The coefficients
  of $T$-$X$ PEFs are much more difficult to interpret.
\end{itemize}
\item The efficiency of the two approaches is difficult to compare.
  Plane-wave destructors are generally more efficient to apply because
  of the small number of filter coefficients. However, they
  may require more computation at the estimation stage because of the
  non-linearity problem.
\end{itemize}

\section{Acknowledgments}
This work was partially accomplished at
the Stanford Exploration Project (SEP).

I would like to thank Jon Claerbout, Robert Clapp, Matthias Schwab, and
other SEP members for developing and maintaining the reproducible
research technology, which helped this research. 

Suggestions from two anonymous reviewers helped to improve the paper.

\bibliographystyle{segnat}
\bibliography{SEP2,SEG,wave}

\newpage

\append{Determining filter coefficients by Taylor expansion}

This appendix details the derivation of equations~(\ref{eqn:b3})
and~(\ref{eqn:b5}). The main idea to match the frequency responses 
of the approximate plane-wave filters to the response of the exact 
phase-shift operator at low frequencies.

The Taylor series expansion of the phase-shift operator $e^{i \omega
  \sigma}$ around the zero frequency $\omega=0$ takes the form
\begin{equation}
  e^{i \omega \sigma} \approx 1 + i\, \,\sigma\,\omega - 
  \frac{\sigma^2\,\omega^2}{2} - i\, \frac{\sigma^3\,\omega^3}{6} +
  \mbox{O}\left(\omega^4\right)
  \label{eq:eseries}
\end{equation}
The Taylor expansion of the six-point implicit finite-difference operator 
takes the form
\begin{eqnarray}
  \nonumber
\frac{B_3(Z_t)}{B_3(1/Z_t)} & = &
\frac
{b_{-1}\,Z_t^{-1} + b_0 + b_1\,Z_t}
{b_1\,Z_t + b_0 + b_{-1}\,Z_t^{-1}} = 
\frac
{b_{-1}\,e^{-i \omega} + b_0 + b_1\,e^{i \omega}}
{b_1\,e^{-i \omega} + b_0 + b_{-1}\,e^{i \omega}} \\
\nonumber
 & \approx &
1 - \frac{2\,i \,\left( b_{-1} - b_1 \right) \,\omega}{b_0 + b_{-1} + b_1} - 
  \frac{2\,\left( b_{-1} - b_1 \right)^2\,\omega^2}
  {\left( b_0 + b_{-1} + b_1 \right)^2} \\
& &  + 
  \frac{i\,
     \left( b_{-1} - b_1 \right) \,
     \left[ b_0^2 - b_0\,\left( b_{-1} + b_1 \right) 
       4\,\left( b_{-1}^2 - 4\,b_{-1}\,b_1 + 
          {b_1}^2 \right)  \right] \,\omega^3}
    {3\,\left(b_0 + b_{-1} + b_1\right)^3} + \ldots
\label{eq:bseries}
\end{eqnarray}
Matching the corresponding terms of expansions~(\ref{eq:eseries}) 
and~(\ref{eq:bseries}), we arrive at the system of nonlinear equations
\begin{eqnarray}
  \sigma & = & \frac{2\,\left( b_1 - b_{-1} \right) }
  {b_0 + b_{-1} +b_1} 
  \label{eq:sys1} \\
  \sigma^2 & = & \frac{4\,\left( b_1 - b_{-1} \right)^2}
  {\left(b_0 + b_{-1} +b_1 \right)^2} \label{eq:sys2}
  \\
  \sigma^3 & = & \frac{2\,\left( b_1 - b_{-1} \right) \,
    \left[ b_0^2 - b_0\,\left( b_{-1} +b_1 \right)  + 
         4\,\left( b_{-1}^2 - 4\,b_{-1}\,b_1 + 
            b_1^2 \right)  \right] }
      {\left(b_0 + b_{-1} + b_1 \right)^3}
      \label{eq:sys3}
\end{eqnarray}
System~(\ref{eq:sys1}-\ref{eq:sys3}) does not uniquely constrain the
filter coefficients $b_{-1}$, $b_0$, and $b_1$ because
equation~(\ref{eq:sys2}) simply follows from~(\ref{eq:sys1}) and
because all the coefficients can be multiplied simultaneously by an
arbitrary constant without affecting the ratios in
equation~(\ref{eq:bseries}). I chose an additional constraint in the 
form
\begin{equation}
  \label{eq:b0}
  B_3(1) = b_{-1} + b_0 + b_1 = 1\;,
\end{equation}
which ensures that the filter $B_3(Z_t)$ does not alter the zero
frequency component. System~(\ref{eq:sys1}-\ref{eq:sys3}) with the
additional constraint~(\ref{eq:b0}) resolves uniquely to
the coefficients of filter~(\ref{eqn:b3}) in the main text:
\begin{eqnarray}
   b_{-1} & = & \frac{(1-\sigma)(2-\sigma)}{12}\;;   
   \label{eq:sol1}
   \\
  b_0 & = & \frac{(2+\sigma)(2-\sigma)}{6}\;; 
  \label{eq:sol2}
  \\
  b_1 & = & \frac{(1+\sigma)(2+\sigma)}{12}\;.
  \label{eq:sol3}
\end{eqnarray}

The $B_5$ filter of equation~(\ref{eqn:b5}) is constructed in a
completely analogous way, using longer Taylor expansions to constrain
the additional coefficients. Generalization to longer filters is
straightforward.

The technique of this appendix aims at matching the filter responses
at low frequencies. One might construct different filter families by
employing other criteria for filter design (least squares fit, 
equiripple, etc.)

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