paper.tex

国外免费地震资料处理软件包

  original image.}

The simplest kind of regularized inversion involves $\mathbf{L}$ as
the identity operator $\mathbf{I}_N$. The corresponding application of
PWC \old{preconditioning} \new{parameterization} results in an
effective deconvolution that enhances structural consistency and
continuity of seismic images.

A coherency enhancement operator is
\begin{equation}
  \label{eq:vg}
  \mathbf{h = H\,s} = \mathbf{C\,W\,C}^T\,\left(\mathbf{C\,W\,C}^T + \epsilon^2\,\mathbf{I}_N\right)^{-1}\,\mathbf{s}\;,
\end{equation}
where a diagonal weighting operator $\mathbf{W}$ is added to prevent
plane-wave smoothing across structural discontinuities. We define
$\mathbf{W}$ as a diagonalized magnitude of $\mathbf{p}$ from an
unweighted inversion in equation~(\ref{eq:inv}). When repeated
iteratively, this method corresponds to iteratively reweighted
least-squares with a simulated $L_1$ norm for vector $\mathbf{p}$
\cite[]{GEO68-01-03860399}.

We apply coherency enhancement to a time-migrated seismic image from a
historic Gulf of Mexico dataset \cite[]{bei}, shown in
Figure~\ref{fig:bei-slp}.  Estimating local event slopes (right plot
in Figure~\ref{fig:bei-slp}) defines the PWC operator. The output of
coherency enhancement using equation~(\ref{eq:vg}) and the
corresponding noise component removed from the data are shown in
Figure~\ref{fig:bei-vg}. Coherency enhancement highlights locally
continuous reflectors while preserving the geometry of faults. A
similar effect was described by \cite{TLE21-03-02380243}, who called
it ``Van Gogh'' filtering.

\subsection{Velocity estimation}

\plot*{bei-pdxw}{width=0.88\textwidth}{Seismic image from
Figure~\ref{fig:bei-vg} overlaid on top of the interval velocity model
estimated by PWC-parameterized Dix inversion.}
\plot{bei-pdx}{width=0.9\columnwidth}{Left: migration velocity used for
  prestack time migration. Right: migration velocity predicted by regularized
  Dix inversion.}


%Plane-wave construction can be used as a model preconditioning
%operator \cite[]{GEO59-05-08180829,GEO68-02-05770588} for regularizing
%inverse problems in the cases, where estimated models have certain
%structure. In these applictions, plane-wave construction acts as an
%accurate steering filter in terminology of \cite{clapp}.

To demonstrate an application of PWC to the seismic velocity
estimation problem, we chose a simple Dix inversion formulation
\cite[]{GEO20-01-00680086} applied for interval velocity estimation
using the Gulf of Mexico dataset from Figure~\ref{fig:bei-slp}. When
Dix inversion is formulated as a regularized estimation problem, the
forward operator $\mathbf{L}$ \new{in equations~(\ref{eq:for})
and~(\ref{eq:inv})} turns into simple integration
\cite[]{Clapp.sep.97.bob1,alejandro}. \old{Preconditioning}
\new{Parameterization} by plane-wave construction using the dip field
estimated from the image forces the estimated velocity to follow the
geological structure.

Figure~\ref{fig:bei-pdxw} shows the resultant interval velocity model.
Figure~\ref{fig:bei-pdx} shows a comparison between the input and
predicted RMS (root-mean-square) migration velocity.  We can see that
both goals of model estimation are achieved: the estimated model
explains the observed data while following a geological structure
consistent with the seismic image.

\subsection{Multiple suppression}

Separation of primary and multiple reflections is one of the most
important tasks in seismic data processing. A distinguishable
characteristic of surface-related multiple events is different slopes
because of different apparent velocities. The advantage of using
slope-based prediction is that, for estimating dominant slopes of
multiple events, we can utilize models of the multiples that may have
incorrect amplitudes and wavelets as long as they correctly predict
event geometry \cite[]{antoine}. An example is shown in
Figure~\ref{fig:cmp}, which contains a multiple-infested CMP gather
from the Mobil AVO dataset \cite[]{CSI00-00-00010213}, its prediction
with the SRME (surface-related multiple elimination) method
\cite[]{GEO57-09-11661177} and two dominant slopes estimated from the
data and corresponding to primary and multiple reflections. Even
though SRME does not provide correct amplitudes and wavelets in
prediction of the multiple events, it can guide the slope estimation
method toward extracting the dominant slopes of multiple events.

\inputdir{cmp}

\plot{cmp}{width=\columnwidth}{a: CMP gather from the
Mobil AVO dataset, b: multiple model from SRME prediction, c: estimated
dominant slope of the primary reflection events, d: estimated dominant
slope of the multiple reflection events.}

The model of the data is now \cite[]{GEO65-02-05740583,FBR20-03-01610167}
\begin{equation}
\label{eq:splusn}
  \mathbf{d} = \mathbf{C}_p\,\mathbf{p}+\mathbf{C}_n\,\mathbf{n} = 
  \left[\begin{array}{cc} \mathbf{C}_p & \mathbf{C}_n \end{array}\right]\,
  \left[\begin{array}{c} \mathbf{p} \\ \mathbf{n} \end{array}\right]\;,
\end{equation}
where $\mathbf{C}_p$ is plane-wave construction along primary slopes
and \old{$\mathbf{C}_p$} \new{$\mathbf{C}_n$} is plane-wave
construction along multiple slopes. In accordance with
equation~(\ref{eq:inv}), the least-squares estimates of the multiples
is
\begin{equation}
\label{eq:mult}
  \widehat{\mathbf{m}} = \mathbf{C}_n\,\widehat{\mathbf{n}} =
  \mathbf{C}_n\,\mathbf{C}_n^T\,
  \left(\mathbf{C}_p\,\mathbf{C}_p^T + \mathbf{C}_n\,\mathbf{C}_n^T + \epsilon^2\,\mathbf{I}_N\right)^{-1}\,\mathbf{d}\;.
\end{equation}

Figure~\ref{fig:super} shows the estimated primary and multiple events
and comparison of velocity semblance scans before and after multiple
suppression. A large portion of the multiple energy is successfully
removed from the data.

\plot{super}{width=\columnwidth}{a: estimated primary
reflections (data with multiples removed), b: estimated multiple
reflections, c: velocity scan of the original gather, d: velocity scan of
the gather after multiple suppression.}

\begin{comment}

\section{From plane-wave construction to shaping}

Shaping \cite[]{shape} is a powerful approach to
regularization. Shaping acts as a smoothing operator that projects the
estimated model into the space of acceptable models.

To transform plane-wave construction to shaping, we can use the
prediction operator~$\mathbf{P}$ from equation~(\ref{eq:d}) and define
a box smoother of length $k$ as
\begin{equation}
  \label{eq:bk}
  \mathbf{B}_k = \frac{1}{k}\,\left(\mathbf{I}_N + \mathbf{P} + 
  \mathbf{P}^2 + \cdots +  \mathbf{P}^k\right)\;.
\end{equation}
Implementing equation~(\ref{eq:bk}) directly requires many computational
operations. Noting that
\begin{equation}
  \label{eq:rec}
  \left(\mathbf{I}_N - \mathbf{P}\right)\,\mathbf{B}_k =
    \frac{1}{k}\,\left(\mathbf{I}_N - \mathbf{P}^{k+1}\right)\;,
\end{equation}
we can rewrite equation~(\ref{eq:bk}) in the compact form
\begin{equation}
  \label{eq:bcomp}
  \mathbf{B}_k = 
  \frac{1}{k}\,\left(\mathbf{I}_N - \mathbf{P}\right)^{-1}\,
  \left(\mathbf{I}_N - \mathbf{P}^{k+1}\right) = \frac{1}{k}\,\mathbf{C}\,\left(\mathbf{I}_N - \mathbf{P}^{k+1}\right)\;,
\end{equation}
which can be implemented economically.  Finally, combining two
generalized box smoothers creates a symmetric generalized triangle smoothing operator suitable for shaping regularization
\begin{equation}
  \label{eq:tk}
  \mathbf{T}_k = \mathbf{B}_k^T\,\mathbf{B}_k\;.
\end{equation}
A triangle shaper uses local predictions from both the left and the
right neighbors of a record and averages with triangle weights.

The least-squares inversion regularized by shaping takes the form
\cite[]{shape}
\begin{equation}
  \label{eq:shape}
  \widehat{\mathbf{m}} = 
  \mathbf{B}_k\,\left[\epsilon^2\,\mathbf{I}_N + 
    \mathbf{B}_k^T\,\left(\mathbf{L}^T\,\mathbf{L} - 
      \epsilon^2\,\mathbf{I}_N\right)\,
    \mathbf{B}_k\right]^{-1}\,
  \mathbf{B}_k^T\,\mathbf{L}^T\,\,\mathbf{d}\;.
\end{equation}

Figure~\ref{fig:bei-smo} shows several impulse responses for shaping with
$k=20$ for the seismic image used in this study.

%\plot{bei-smo}{width=\columnwidth}{Impulse responses of PWC shaping
%  for different points inside the image space.}

\end{comment}

\section{Conclusions}

We have introduced plane-wave construction (PWC), an operator that
generates models aligned with predefined locally variable dips. PWC is
defined as the inverse of plane-wave destruction, an operator used for
measuring local dips.  It is applicable as a model
\old{preconditioner} \new{regularizer} in \old{geophysical}
\new{seismic} estimation problems such as coherency enhancement,
\old{seismic} velocity estimation, and multiple suppression. As an
efficient operator for characterizing data elongated over dominant
smoothly variable slopes, PWC can be incorporated for regularization
of any inversion problems that operate with locally planar data. We
anticipate more applications of the proposed method.

\old{An important limitation of the PWC method is that it is only applicable to
2-D problems, where seismic traces can be simply ordered. A possible
extension to 3-D problems is in transformation from plane-wave
construction to plane-wave shaping and in
application of shaping regularization}.

\section{Acknowledgments}

The first author thanks Norsk Hydro for partially supporting this
research. 

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