paper.tex

国外免费地震资料处理软件包

\begin{equation}
  \label{eqn:pick}
  \left\{\begin{array}{rcl}
      \mathbf{W}\,\mathbf{x} & \approx & \mathbf{W}\,\mathbf{p} \\
      \epsilon \mathbf{D} \mathbf{x} & \approx & \mathbf{0}
    \end{array}\right.\;.
\end{equation}
\new{In the more standard notation, the solution $\mathbf{x}$ minimizes the
least-squares objective function}
\begin{equation}
  \label{eqn:ofpick}
(\mathbf{x}-\mathbf{p})^{T}\, \mathbf{W}^2\,(\mathbf{x}-\mathbf{p}) +
\epsilon^2\,\mathbf{x}^T\,\mathbf{D}^T\,\mathbf{D}\,\mathbf{x}
\end{equation}
Here $\mathbf{p}$ is the vector of blind maximum-semblance picks (possibly in a
predefined fairway), $\mathbf{x}$ is the estimated velocity picks, $\mathbf{W}$ is
the weighting operator with the weight corresponding to the semblance values
at $\mathbf{p}$, $\epsilon$ is the scalar regularization parameter, $\mathbf{D}$
is a roughening operator, and $\mathbf{D}^T$ is the adjoint operator.  The first
least-squares fitting goal in (\ref{eqn:pick}) states that the estimated
velocity picks should match the measured picks where the semblance is high
enough\footnote{Of course, this goal might be dangerous, if the original picks
  $\mathbf{p}$ include regular noise (such as multiple reflections) with high
  semblance value \cite[]{Toldi.sepphd.43}. For simplicity, and to preserve the
  linearity of the problem, I assume that this is not the case.}.  The second
fitting goal tries to find the smoothest velocity function possible.  The
least-squares solution of problem (\ref{eqn:pick}) takes the form
\begin{equation}
  \label{eqn:LS}
  \mathbf{x} = 
  \left(\mathbf{W}^2 + 
    \epsilon^2\,\mathbf{D}^T\,\mathbf{D}\right)^{-1}\,\mathbf{W}^2\,\mathbf{p}\;.
\end{equation}

\inputdir{beivc}

In the case of picking a one-dimensional velocity function from a single
semblance panel, one can simplify the algorithm by choosing $\mathbf{D}$ to be a
convolution with the derivative filter $(1,-1)$. It is easy to see that in
this case the inverted matrix in formula (\ref{eqn:LS}) has a tridiagonal
structure and therefore can be easily inverted with a linear-time algorithm.
The regularization parameter $\epsilon$ controls the amount of smoothing of
the estimated velocity function.  Figure \ref{fig:velpick} shows an example
velocity spectrum and two automatic picks for different values of $\epsilon$.

\plot{velpick}{width=6in}{Semblance panel (left) and automatic
  velocity picks for different values of the regularization parameter.
%  Center: $\epsilon=0.01$, right: $\epsilon=0.1$. 
Higher values of
  $\epsilon$ lead to smoother velocities.}

\par
In the case of picking two- or three-dimensional velocity functions,
one could generalize problem (\ref{eqn:pick}) by defining $\mathbf{D}$
as a 2-D or 3-D roughening operator. I chose to use a more simplistic
approach, which retains the one-dimensional structure of the algorithm. 
I transform system (\ref{eqn:pick}) to the form
\begin{equation}
  \label{eqn:pick2}
  \left\{\begin{array}{rcl}
      \mathbf{W}\,\mathbf{x} & \approx & \mathbf{W}\,\mathbf{p} \\
      \epsilon \mathbf{D} \mathbf{x} & \approx & \mathbf{0} \\
      \lambda \mathbf{x} & \approx & \lambda \mathbf{x_0}
    \end{array}\right.\;,
\end{equation}
where $\mathbf{x}$ is still one-dimensional, and $\mathbf{x}_0$ is the
estimate from the previous midpoint location. The scalar parameter
$\lambda$ controls the amount of lateral continuity in the estimated
velocity function. The least-squares solution to system
(\ref{eqn:pick2}) takes the form
\begin{equation}
  \label{eqn:LS2}
  \mathbf{x} = 
  \left(\mathbf{W}^2 + \epsilon^2\,\mathbf{D}^T\,\mathbf{D}
    + \lambda^2\,\mathbf{I}\right)^{-1}\,
  \left(\mathbf{W}^2\,\mathbf{p} + \lambda^2 \mathbf{x_0}\right)\;,
\end{equation}
where $\mathbf{I}$ denotes the identity matrix. Formula (\ref{eqn:LS2})
also reduces to an efficient tridiagonal matrix inversion.

\new{After the velocity has been picked, an optimally focused image is constructed
by slicing in the time-midpoint-velocity cube. I used simple linear
interpolation for slicing between the velocity grid values. A more accurate
interpolation technique can be easily adopted.}

\section{Examples}

I demonstrate the performance of the method using a simple 2-D synthetic
test and a field data example from the North Sea.

\subsection{Synthetic Test}
\inputdir{sigvc}

The synthetic test uses constant-velocity prestack modeling and migration to
check the validity of the method when all the theoretical requirements are
satisfied. The data were generated from the synthetic reflectivity model
(Figure~\ref{fig:mod}) and included 60~offsets ranging from 0 to 0.5~km.
The exact velocity in the model is~1.5~km/s, and the initial velocity for
starting the continuation process was chosen at 2~km/s.

%\plot{sig-mva2}{width=6in}{Semblance panels for migration
%  velocity analysis at the common image point~0.5~km. Left: after
%  velocity continuation.  Right: after conventional (NMO) velocity
%  analysis. In this structurally simple region, the difference between
%  two methods is small. The correct velocity is 1.5~km/s.}

Figure~\ref{fig:sig-mva1} compares the semblance panels for migration
velocity analysis using velocity continuation and using the
conventional (NMO) analysis. In the top part of the image, both panels
show maximum picks at the correct velocity (1.5~km/s).  The advantage
of velocity continuation is immediately obvious in the deeper part of
the image, where the events are noticeably better focused.

\plot{sig-mva1}{width=6in}{Semblance panels for migration
  velocity analysis at the common image point~1~km. Left: after
  velocity continuation.  Right: after conventional (NMO) velocity
  analysis. In a structurally complex region, velocity continuation
  clearly provides better focusing. The correct velocity is 1.5~km/s.}

The final result of velocity continuation (after picking maximum semblance and
slicing in the velocity cube) is shown in the bottom left plot of
Figure~\ref{fig:sig-all}. For comparison, Figure~\ref{fig:sig-all} also shows
the result of migration with the correct velocity (the top left plot), initial
velocity (the top right plot), and the result of velocity slicing after the
simple NMO correction, corresponding to the conventional MVA (the bottom right
plot). \new{The same velocity picking and slicing program was used in both cases.}
The comparison clearly shows that, in this simple example, velocity
continuation is able to accurately reproduce the correct image without using
any prior information about the migration velocity and without any need for
repeating the prestack migration procedure. \new{Velocity continuation correctly
images events with conflicting dips by properly taking into account both
vertical and lateral shifts in the image position.}

\plot{sig-all}{width=6in,height=6in}{Velocity continuation tested 
  on the synthetic example. Top left: prestack migration with the
  correct velocity of 1.5~km/s. Top right: prestack migration
  with the velocity of 2~km/s. Bottom left: the result of velocity
  continuation. Bottom right: the result from picking migration
  image after only conventional NMO correction.}

\subsection{\new{Field} Data Example}
\inputdir{elfvc}

Figure \ref{fig:elf-migr} compares the result of a constant-velocity prestack
migration with the velocity of 2 km/s, applied to a dataset from the North Sea
(courtesy of Elf Aquitaine) and the result of velocity continuation to the
same velocity from a migration with a smaller velocity of 1.4 km/s (Figure
\ref{fig:elf-migr}a). The two images (Figures \ref{fig:elf-migr}b and
\ref{fig:elf-migr}c) look remarkably similar, in full accordance with the
theory.

\plot{elf-migr}{width=6in}{Constant-offset section of the North Sea
  dataset after migration with the velocity of 1.4 km/s (a), migration
  with the velocity of 2 km/s (b), migration with the velocity of 1.4
  km/s and velocity continuation to 2 km/s (c).}

Figure~\ref{fig:elf-npk} shows a result of two-dimensional velocity
picking after velocity continuation. I used values of $\epsilon=0.1$
and $\lambda=0.1$. The first parameter controls the vertical smoothing
of velocities, while the second parameter controls the amount of
lateral continuity.

%\ref{fig:beivpk} shows a result of two-dimensional velocity picking
%after residual NMO. Figure \ref{fig:beifpk} shows an analogous result

%The difference between residual NMO velocities and
%velocities picked after velocity continuation is small, but clearly
%visible.

%\plot{beivpk}{width=6in,height=3.5in}{Automatic picks of 2-D RMS velocity
%  after residual NMO. The contour spacing is 0.1 km/s, starting from 1.5 km/s.}

\plot{elf-npk}{angle=90,totalheight=8.5in,width=6in}{Automatically
  picked migration velocity after velocity continuation.}

Figure \ref{fig:elf-fmg} shows the final result of velocity
continuation: an image, obtained by slicing through the velocity cube
with the picked imaging velocities. The edges of the salt body in the middle
of the section have been sharply focused by the velocity
continuation process. To transform the already well focused image into the
depth domain, one may proceed in a way similar to
\emph{hybrid migration}: demigration to zero-offset, followed by
post-stack depth migration \cite[]{GEO62-02-05680576}. This step
would require constructing an interval velocity model from the picked imaging
velocities.

\plot{elf-fmg}{angle=90,totalheight=8.5in,width=6in}{Final result of
  velocity continuation: seismic image, obtained by slicing through
  the velocity cube.}

\begin{comment}
Without repeating the details of the procedure, Figures~\ref{fig:pck}
and \ref{fig:img} show picked imaging velocities and the velocity
continuation image for the Blake Outer Ridge data, shown in many other
papers in this report.

%\plot{pck}{width=6in,height=3.5in}{Blake Outer Ridge data. Automatic
%  picks of 2-D imaging velocity after velocity continuation. The contour
%  spacing is 0.01 km/s, starting from 1.5 km/s.}

%\plot{img}{angle=90,totalheight=8.5in,width=6in}{Blake Outer Ridge
%  data. Final result of velocity continuation: seismic image, obtained
%  by slicing through the velocity cube.}
\end{comment}

\section{Conclusions}

Velocity continuation is a powerful method for time migration velocity
analysis.  The strength of this method follows from its ability to take into
account both vertical and lateral movement of the reflection events in seismic
images with the changes of migration velocity.

Efficient practical algorithms for velocity continuation can be constructed
using either finite-difference or spectral methods. When applied in the
post-stack (zero-offset) setting, velocity continuation can be used as a
computationally attractive method of time migration. Both finite-difference
and spectral approaches possess remarkable invertability properties:
continuation to a lower velocity reverses continuation to a higher velocity.
\new{For the finite-difference algorithm, this property is confirmed by synthetic
tests. For the spectral algorithm, it follows from the fact that velocity
continuation reduces to a simple phase-shift unitary operator.}

\new{Including velocity continuation in the practice of migration velocity analysis
can improve the focusing power of time migration and reduce the production
time by avoiding the need for iterative velocity refinement. No prior velocity
model is required for this type of velocity analysis. This conclusion is
confirmed by synthetic and field data examples.}

\section{Acknowledgments}

I thank Jon Claerbout, Biondo Biondi, and Bill Symes for useful and
stimulating discussions, the sponsors of the Stanford Exploration Project for
their financial support, and Elf Aquitaine for providing the data used in this
work. \new{I am also grateful to Paul Fowler, Samuel Gray, Hugh Geiger, and one
anonymous reviewer for thorough and helpful reviews that improved the quality
of the paper.}

\newpage
\bibliographystyle{seg}
\bibliography{SEP2,SEG,paper,spec,velcon}

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