paper.tex

国外免费地震资料处理软件包

%  filters. This could be an appropriate approach for interpolating
%  both primary and multiple reflections.
  
%\item Missing data interpolation problems can be greatly accelerated
%  by preconditioning \cite[]{Fomel.sep.94.sergey1,Fomel.sep.95.sergey1}.
%  Finding an appropriate preconditioning for the offset continuation
%  method is an open problem.  The non-stationary nature of the
%  continuation filter make this problem particularly challenging.

%\end{itemize}

Analogously to integral azimuth moveout operator
\cite[]{GEO63-02-05740588}, differential offset continuation can be
applied in 3-D for regularizing seismic data prior to prestack imaging.

  
  In the next section, I return to the 2-D case to consider an
  important problem of shot gather interpolation.

  \section{Shot continuation}
  
  Missing or under-sampled shot records are a common example of data
  irregularity \cite[]{Crawley.sepphd.104}. The offset continuation
  approach can be easily modified to work in the shot record domain.
  With the change of variables $s = y - h$, where $s$ is the shot
  location, the frequency-domain equation~(\ref{eqn:OC}) transforms to
  the equation
  \begin{equation}
    h \, \left( 2\,{\partial^2 \tilde{P} \over \partial s \partial h} - 
      {\partial^2 \tilde{P} \over \partial h^2} \right) - 
    i\,\Omega \, \left({\partial \tilde{P} \over   {\partial h}} -
        {\partial \tilde{P} \over   {\partial s}}\right) = 0 \;.
    \label{eqn:SC} 
  \end{equation}
  Unlike equation~(\ref{eqn:OC}), which is second-order in the
  propagation variable $h$, equation~(\ref{eqn:SC}) contains only
  first-order derivatives in $s$. We can formally write its solution
  for the initial conditions at $s=s_1$ in the form of a phase-shift
  operator:
    \begin{equation}     
      \widehat{\widehat{P}}(s_2) = \widehat{\widehat{P}}(s_1)\,
      \exp{\left[i\,k_h\,\left(s_2-s_1\right)\,
          \frac{k_h\,h-\Omega}{2\,k_h\,h-\Omega}\right]}\;,
    \label{eqn:SCshift} 
  \end{equation}
  where the wavenumber~$k_h$ corresponds to the half-offset~$h$.
  Equation~(\ref{eqn:SCshift}) is in the mixed offset-wavenumber
  domain and, therefore, not directly applicable in practice. However,
  we can use it as an intermediate step in designing a
  finite-difference shot continuation filter. Analogously to the cases
  of plane-wave destruction and offset continuation, shot continuation
  leads us to the rational filter
  \begin{equation}
    \label{eqn:SCpass}
    \hat{P}_{s+1}(Z_h) = 
    \hat{P}_{s} (Z_h) \frac{S(Z_h)}{\bar{S}(1/Z_h)}\;,
  \end{equation}
  The filter is non-stationary, because the coefficients of $S(Z_h)$
  depend on the half-offset $h$. We can find them by the Taylor
  expansion of the phase-shift equation~(\ref{eqn:SCshift}) around
  zero wavenumber $k_h$. For the case of the half-offset sampling
  equal to the shot sampling, the simplest three-point filter is
  constructed with three terms of the Taylor expansion. It takes the
  form
  \begin{equation}
    \label{eqn:SCfilt}
    S(Z_h) = - \left(\frac{1}{12} + i\,\frac{h}{2\,\Omega}\right)\,Z_h^{-1} + 
    \left(\frac{2}{3} - i\,\frac{\Omega^2 + 12\,h^2}{12\,\Omega\,h}\right) +
    \left(\frac{5}{12} + i\,\frac{\Omega^2 + 18\,h^2}{12\,\Omega\,h}\right)\,
    Z_h\;.
  \end{equation}
  
  Let us consider the problem of doubling the shot density.  If we use
  two neighboring shot records to find the missing record between
  them, the problem reduces to the least-squares system
  \begin{equation}
    \label{eqn:SCls}
    \left[\begin{array}{c}
        \mathbf{S} \\
        \mathbf{\bar{S}}
      \end{array}\right]\,\mathbf{p}_s \approx
    \left[\begin{array}{c}
        \mathbf{\bar{S}}\,\mathbf{p}_{s-1} \\
        \mathbf{S}\,\mathbf{p}_{s+1}
      \end{array}\right]\;,
  \end{equation}
  where $\mathbf{S}$ denotes convolution with the numerator of
  equation~(\ref{eqn:SCpass}), $\mathbf{\bar{S}}$ denotes convolution
  with the corresponding denominator, $\mathbf{p}_{s-1}$ and
  $\mathbf{p}_{s+1}$ represent the known shot gathers, and $\mathbf{p}_s$
  represents the gather that we want to estimate. The least-squares
  solution of system~(\ref{eqn:SCls}) takes the form
  \begin{equation}
    \label{eqn:SCsol}
    \mathbf{p}_s = \left(
      \mathbf{S}^T\,\mathbf{S} +
      \mathbf{\bar{S}}^T\,\mathbf{\bar{S}}
    \right)^{-1}\,
    \left(\mathbf{S}^T\,\mathbf{\bar{S}}\,\mathbf{p}_{s-1} +
      \mathbf{\bar{S}}^T\,\mathbf{S}\,\mathbf{p}_{s+1}\right)\;.
  \end{equation}
  If we choose the three-point filter~(\ref{eqn:SCfilt}) to construct
  the operators~$\mathbf{S}$ and $\mathbf{\bar{S}}$, then the inverted
  matrix in equation~(\ref{eqn:SCsol}) will have five non-zero
  diagonals. It can be efficiently inverted with a direct banded
  matrix solver using the $LDL^T$ decomposition \cite[]{golub}. Since
  the matrix does not depend on the shot location, we can perform the
  decomposition once for every frequency so that only a triangular
  matrix inversion will be needed for interpolating each new shot.
  This leads to an extremely efficient algorithm for interpolating
  intermediate shot records.

  Sometimes, two neighboring shot gathers do not fully constrain the
  intermediate shot. In order to add an additional constraint, I
  include a regularization term in equation~(\ref{eqn:SCsol}), as
  follows:
  \begin{equation}
    \label{eqn:SCreg}
    \mathbf{p}_s = \left(
      \mathbf{S}^T\,\mathbf{S} +
      \mathbf{\bar{S}}^T\,\mathbf{\bar{S}} + 
        \epsilon^2\,\mathbf{A}^T\mathbf{A}
    \right)^{-1}\,
    \left(\mathbf{S}^T\,\mathbf{\bar{S}}\,\mathbf{p}_{s-1} +
      \mathbf{\bar{S}}^T\,\mathbf{S}\,\mathbf{p}_{s+1}\right)\;,
  \end{equation}
  where $\mathbf{A}$ represents convolution with a three-point
  prediction-error filter (PEF), and $\epsilon$ is a scaling
  coefficient. The appropriate PEF can be estimated from
  $\mathbf{p}_{s-1}$ and $\mathbf{p}_{s+1}$ using Burg's algorithm
  \cite[]{GEO37-02-03750376,Burg.sepphd.6,Claerbout.fgdp.76}. A
  three-point filter does not break the five-diagonal structure
  of the inverted matrix.  The PEF regularization attempts to preserve
  offset dip spectrum in the under-constrained parts of the estimated
  shot gather.
  
  Figure~\ref{fig:shotin} shows the result of a shot interpolation
  experiment using the constant-velocity synthetic from
  Figure~\ref{fig:data}. In this experiment, I removed one of the
  shot gathers from the original NMO-corrected data and interpolated
  it back using equation~(\ref{eqn:SCreg}). Subtracting the true shot
  gather from the reconstructed one shows a very insignificant error,
  which is further reduced by using the PEF regularization (right
  plots in Figure~\ref{fig:shotin}).  The two neighboring shot gathers
  used in this experiment are shown in the top plots of
  Figure~\ref{fig:shot3}.  For comparison, the bottom plots in
  Figure~\ref{fig:shot3} show the simple average of the two shot
  gathers and its corresponding prediction error. As expected, the
  error is significantly larger than the error of shot
  continuation. An interpolation scheme based on local dips in the
  shot direction would probably achieve a better result, but it is
  significantly more expensive than the shot continuation scheme
  introduced above.
  
  \plot{shot3}{width=6in,height=7in}{Top: Two synthetic shot gathers
    used for the shot interpolation experiment. An NMO correction has
    been applied. Bottom: simple average of the two shot gathers
    (left) and its prediction error (right).}
  \plot{shotin}{width=6in,height=3.5in}{Synthetic shot interpolation
    results. Left: interpolated shot gathers. Right: prediction errors
    (the differences between interpolated and true shot gathers),
    plotted on the same scale.}
 
\inputdir{elfshot}
 
  A similar experiment with real data from a North Sea marine dataset
  is reported in Figure~\ref{fig:elfshotin}. I removed and
  reconstructed a shot gather from the two neighboring gathers shown
  in Figure~\ref{fig:elfshot3}. The lower parts of the gathers are
  complicated by salt dome reflections and diffractions with
  conflicting dips. The simple average of the two input shot gathers
  (bottom plots in Figure~\ref{fig:elfshotin}) works reasonably well
  for nearly flat reflection events but fails to predict the position
  of the back-scattered diffractions events. The shot continuation
  method works well for both types of events (top plots in
  Figure~\ref{fig:elfshotin}). There is some small and random residual
  error, possibly caused by local amplitude variations.
   
  \plot{elfshot3}{width=6in,height=3.5in}{Two real marine shot gathers
    used for the shot interpolation experiment. An NMO correction has
    been applied.}  
  
  \plot{elfshotin}{width=6in,height=7in}{Real-data shot interpolation
    results.  Top: interpolated shot gather (left) and its prediction
    error (right). Bottom: simple average of the two input shot
    gathers (left) and its prediction error (right).}
  
  Analogously to the case of offset continuation, it is possible to
  extend the shot continuation method to three dimensions. A simple
  modification of the proposed technique would also allow us to use
  more than two shot gathers in the input or to extrapolate missing
  shot gathers at the end of survey lines.

%  \section{Stack and partial stack as a data regularization problem}

\section{Conclusions}

Differential offset continuation provides a valuable tool for
interpolation and regularization of seismic data. Starting from
analytical frequency-domain solutions of the offset continuation
differential equation, I have designed accurate finite-difference
filters for implementing offset continuation as a local convolutional
operator. A similar technique works for shot continuation across
different shot gathers. Missing data are efficiently interpolated by
an iterative least-squares optimization. The differential filters have
an optimally small size, which assures high efficiency.

Differential offset continuation serves as a bridge between integral
and convolutional approaches to data interpolation. It shares the
theoretical grounds with the integral approach but is applied in a
manner similar to that of prediction-error filters in the
convolutional approach.

Tests with synthetic and real data demonstrate that the proposed
interpolation method can succeed in complex structural situations
where more simplistic methods fail.

\section{Acknowledgments}

The financial support for this work was provided by the sponsors of
the Stanford Exploration Project (SEP).

%The 3-D North Sea dataset was released to SEP by Conoco and its
%partners, BP and Mobil. 

For the shot continuation test, I used a
North Sea dataset, released to SEP by Elf Aquitaine.

I thank Jon Claerbout and Biondo Biondi for helpful discussions about
the practical application of differential offset continuation.

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\bibliography{SEG,SEP2,oc}

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