paper.tex

国外免费地震资料处理软件包

\plot{arg}{width=6in}{Phase of the implicit offset-continuation
  operators in comparison with the exact solution.  The offset
  increment is assumed to be equal to the midpoint spacing.  The left
  plot corresponds to $\Omega=1$, the right plot to $\Omega=10$.}

\inputdir{ocimp}

\plot{off-imp}{width=6in,height=3.5in}{Inverse DMO impulse responses
  computed by the Fourier method (left) and by finite-difference
  offset continuation (right). The offset is 1~km.}

\section{Application}
I start numerical testing of the proposed regularization first on a
constant-velocity synthetic, where all the assumptions behind the
offset continuation equation are valid.  
%
%Encouraged by the results, I
%proceed to tests on the Marmousi synthetic dataset, which is
%associated with a highly inhomogeneous velocity model.

\subsection{Constant-velocity synthetic}

\inputdir{cup}

\sideplot{cup}{width=3in}{Reflector model for the constant-velocity test}

A sinusoidal reflector shown in Figure~\ref{fig:cup} creates
complicated reflection data, shown in Figures~\ref{fig:data}
and~\ref{fig:tslice}. To set up a test for regularization by offset
continuation, I removed 90\% of randomly selected shot gathers from
the input data.  The syncline parts of the reflector lead to
traveltime triplications at large offsets. A mixture of different dips
from the triplications would make it extremely difficult to
interpolate the data in individual common-offset gathers, such as
those shown in Figure~\ref{fig:data}.  The plots of time slices
after NMO (Figure~\ref{fig:tslice}) clearly show that the data are
also non-stationary in the offset direction.  Therefore, a simple
offset interpolation scheme is also doomed.

\plot{data}{width=5.5in,height=7.33in}{Prestack common-offset gathers for
  the constant-velocity test. Left: ideal data (after NMO). Right:
  input data (90\% of shot gathers removed).  Top, center, and
  bottom plots correspond to different offsets.}

\plot{tslice}{width=5.5in,height=7.33in}{Time slices of the prestack
  data for the constant-velocity test. Left: ideal data (after NMO).
  Right: input data (90\% of random gathers removed).  Top, center,
  and bottom plots correspond to time slices at 0.3, 0.4, and 0.5 s.}

\par
Figure~\ref{fig:fslice} shows the reconstruction process on individual
frequency slices. Despite the complex and non-stationary character of
the reflection events in the frequency domain, the offset continuation
equation is able to accurately reconstruct them from the decimated
data.

\plot{fslice}{width=5.5in,height=7.33in}{Interpolation in frequency
  slices.  Left: input data (90\% of the shot gathers removed). Right:
  interpolation output. Top, bottom, and middle plots correspond to
  different frequencies. The real parts of the complex-valued data are
  shown.}

\par
Figure~\ref{fig:all} shows the result of interpolation after the data
are transformed back to the time domain. The offset continuation
result (right plots in Figure~\ref{fig:all}) reconstructs the ideal
data (left plots in Figure~\ref{fig:data}) very accurately even in
the complex triplication zones, while the result of simple offset
interpolation (left plots in Figure~\ref{fig:all}) fails as expected.
The simple interpolation scheme applied the offset derivative 
$\frac{\partial}{\partial h}$
in place of the offset continuation equation and thus did not take
into account the movement of the events across different midpoints.

\plot{all}{width=5.5in,height=7.33in}{Interpolation in common-offset
  gathers.  Left: output of simple offset interpolation.  Right:
  output of offset continuation interpolation. Compare with
  Figure~\ref{fig:data}. Top, center, and bottom plots correspond
  to different common-offset gathers.}

\par
The constant-velocity test results allow us to conclude that, when all
the assumptions of the offset continuation theory are met, it provides
a powerful method of data regularization. 

Being encouraged by the synthetic results, I proceed to a
three-dimensional real data test.

\begin{comment}

\subsection{3-D data regularization with the offset continuation equation}

3-D differential offset continuation amounts to applying two
differential filters, operating on the in-line and cross-line
projections of the offset and midpoint coordinates. The corresponding
system of differential equations has the form
\begin{equation} 
\displaystyle
  \left\{\begin{array}{rcl} 
      \displaystyle
      h_1 \, \left( {\partial^2 \tilde{P} \over \partial y_1^2} - 
        {\partial^2 \tilde{P} \over \partial h_1^2} \right) - 
      i\,\Omega \, {\partial \tilde{P} \over   {\partial h_1}} & = & 0\;; \\
      \displaystyle
      h_2 \, \left( {\partial^2 \tilde{P} \over \partial y_2^2} - 
        {\partial^2 \tilde{P} \over \partial h_2^2} \right) - 
      i\,\Omega \, {\partial \tilde{P} \over   {\partial h_2}} & = & 0\;,
    \end{array}\right.
  \label{eqn:OC-3D} 
\end{equation}
where $y_1$ and $y_2$ correspond to the in-line and cross-line
midpoint coordinates, and $h_1$ and $h_2$ correspond to the in-line
and cross-line offsets.  The projection approach is justified in the
theory of azimuth moveout \cite[]{amo,GEO63-02-05740588}.
 
The result of a 3-D data regularization test is shown in
Figure~\ref{fig:off4}. The input data is a subset of a 3-D marine
dataset from the North Sea, complicated by salt dome reflections and
diffractions. The same dataset was used previously for testing azimuth
moveout \cite[]{GEO63-02-05740588}. I used neighboring offsets in the
in-line and cross-line directions and the differential 3-D offset
continuation to reconstruct the empty traces in a selected midpoint
cube. Although the reconstruction is not entirely accurate, it
successfully fulfills the following goals:
\begin{itemize}
\item The input traces are well hidden in the interpolation result. It
  is impossible to distinguish between input and interpolated traces.
\item The main structural features are restored without using any
  assumptions about structural continuity in the midpoint domain. Only
  the physical offset continuity is used.
\end{itemize}

%\plot{off4}{width=5.5in,height=7.33in}{3-D data regularization test.
%  Top: input data, the result of binning in a 50 by 50 meters offset
%  window. Bottom: regularization output. Data from neighboring offset
%  bins in the in-line and cross-line directions were used to
%  reconstruct missing traces.}

\end{comment}

%In the next section, I
%deal with the much more complicated case of Marmousi.

%%\subsection{Marmousi synthetic}

%%The famous Marmousi synthetic was modeled over a very complicated
%%velocity and reflector structure \cite[]{TLE13-09-09270936}. The dataset
%%has been used in numerous studies of various seismic processing and
%%imaging techniques. Figure~\ref{fig:marm} shows the near and far
%%common-offset gathers from the Marmousi dataset. The structure of the
%%reflection events is extremely complex and contains multiple
%%triplications and diffractions.

%%\plot{marm}{width=6in}{Common-offset gathers of the Marmousi dataset.
%%  Left: near offset. Right: far offset.}

%%\par
%%To test the proposed interpolation method, I set the goal of
%%interpolating the missing near offsets in the Marmousi dataset.
%%Additionally, I attempted to interpolate intermediate shot gathers so
%%that all common-midpoint gathers receive the same offset fold. In the
%%original dataset, both receiver and shot spacing are equal to 25
%%meters, which creates a checkerboard pattern in the offset-midpoint
%%plane. This acquisition pattern is typical for 2-D seismic surveys.
%%\par
%%Interpolation of near offsets can reduce imaging artifacts in
%%different migration methods. \cite{Ji.sepphd.90} used near-offset
%%interpolation for accurate wavefront-synthesis migration of the
%%Marmousi dataset. He developed an interpolation technique based on
%%the hyperbolic Radon transform inversion. Ji's method produces fairly
%%good results, but is significantly more expensive that the offset
%%continuation approach explored in this paper.
%%\par
%%Figure~\ref{fig:mslice} shows the input and interpolated Marmousi data
%%in the log-stretch frequency domain. We can see that the data in the
%%frequency slices also have a very complicated structure. Nevertheless,
%%the offset continuation method is able to reconstruct the missing
%%portions of the data in a visually pleasing way. The data are not
%%extrapolated off the sides of the common-offset gathers. This behavior
%%is physically reasonable, because such an extrapolation would involve
%%assumptions about unilluminated portions of the subsurface.

%%%\plot{mslice}{width=6in,height=8in}{Interpolation of the Marmousi
%%%  dataset in frequency slices.  Left: input data. Right: interpolation
%%%  output.  Top, center, and bottom plots correspond to different
%%%  frequencies.  Real parts of the complex-valued data are shown.}

%%\par
%%Figure~\ref{fig:mshot} shows one of the shot gathers obtained after
%%transforming the data back into time domain and resorting them into
%%shot gathers. The positive offset part of the shot gather was
%%reconstructed from a common receiver gather by using reciprocity.
%%Comparing the top and bottom plots, we can see that many different
%%events in the original shot gather are nicely extended into near
%%offsets by the interpolation procedure.

%%%\plot{mshot}{width=6in,height=8in}{Interpolation of near offsets in a
%%%  Marmousi shot gather. The shot position is 4500 m.  Top: input data.
%%%  Bottom: interpolation output.}

%%\par
%%In addition to interpolating near offsets, I have reconstructed the
%%intermediate shot gathers in order to equalize the CMP fold.
%%Figure~\ref{fig:mishot} shows an example of an artificial shot gather
%%created by such a reconstruction. An sample CMP gather before and
%%after interpolation is shown in Figure~\ref{fig:mcmp}. Examining the
%%bottom part of the section, we can see that that the interpolation
%%process tends to put more continuity in the near offsets than could be
%%expected from the data. In other places, the interpolation succeeds in
%%producing a visually pleasant result.

%%%\plot{mishot}{width=6in}{This shot gather at 4525 m is a result of
%%%  data interpolation.}

%%%\plot{mcmp}{width=6in,height=8in}{Interpolation of near and
%%%  intermediate offsets in a Marmousi CMP gather. The midpoint position
%%%  is 4500 m. Top: input data.  Bottom: interpolation output.}

%\section{Discussion}

% \par
%In the range of possible interpolation methods \cite[]{EAE-1998-2051},
%the offset continuation approach clearly stands on the more efficient
%side. The efficiency is achieved both by the small size of the
%finite-difference filter and by the method's ability to decompose and
%parallelize the method across different frequencies. Part of the
%efficiency gain could probably be sacrificed for achieving more
%accurate results.  Here are some interesting ideas one could try:
%\begin{itemize}
%\item Instead of fixing the offset continuation filter in a
%  data-independent way, one could estimate some of its coefficients
%  from the data. In particular, the second term in
%  equation~(\ref{eqn:OC}) can be varied to better account for the
%  effects of variable velocity and amplitude variation with offset.
%  Theoretical extensions of offset continuation to the variable
%  velocity case were studied by \cite{SEG-1997-1901} and
%  \cite{SEG-1999-19331936}.

%\item Formulating the problem in the pre-NMO domain would allow us to
%  consider several velocities by convolving several continuation