paper.tex

国外免费地震资料处理软件包

\title{Regularizing seismic inverse
problems by model re-parameterization using plane-wave construction}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\lefthead{Fomel \& Guitton}
\righthead{Plane-wave construction}

\author{Sergey Fomel\footnotemark[1] and
  Antoine Guitton\footnotemark[2]}

\address{
\footnotemark[1]Bureau of Economic Geology, \\
John A. and Katherine G. Jackson School of Geosciences \\
The University of Texas at Austin \\
University Station, Box X \\
Austin, TX 78713-8972 \\
\footnotemark[2]3DGeo Development Inc. \\
4633 Old Ironsides Drive, Suite 401 \\
Santa Clara, CA 95054}

\maketitle

\begin{abstract}
  We define plane-wave construction (PWC), an operator for generating
  data aligned along predefined locally variable slopes, as the
  inverse of plane-wave destruction, an operator used for measuring
  the slopes.  PWC can be applied for efficient \old{preconditioning}
  \new{regularization} of \old{geophysical} \new{seismic} estimation
  problems. Using simple examples, we demonstrate the applicability of
  PWC for enhancing the coherency of seismic images, improving
  velocity estimation methods, and separating primaries and multiples
  with a pattern-based approach.
\end{abstract}

\section{Introduction}

\cite{GEO67-06-19461960} describes applications of plane-wave
destruction filters, a concept originally introduced by \cite{pvi}.
High-order accurate destruction filters are used for estimating local
slopes of seismic events and can be applied for such problems as fault
detection, data regularization, and noise attenuation.  Estimating
local slopes can also replace traditional velocity analysis and enable
velocity-independent time-domain seismic imaging \cite[]{pmig}.

In this paper, we introduce \emph{plane-wave construction} (PWC), a
formal inverse of the plane-wave destruction operator. We show that
PWC can be used as an efficient \old{preconditioner} \new{regularizer}
for speeding up iterative optimization that involves models with local
plane-wave structure.  In that sense, one can view PWC as a
higher-order generalization of \emph{steering filters}
\cite[]{SEG-1998-1851,clapp}. We illustrate its applications with
field data examples.

The immediate effect of PWC is smoothing the data along dominant event
slopes. In application to coherency enhancement in seismic images,
iterative least-squares inversion with PWC \old{preconditioning}
\new{parameterization} extracts portions of the image aligned with the
dominant slopes. Using PWC in combination with iterative reweighting
allows us to preserve fault geometry in the coherency enhancing
process. When applied to the classic velocity estimation problem, PWC
enforces consistency between the velocity structure and reflector
geometry. In the multiple elimination problem, PWC
\old{preconditioning} \new{parameterization} enables separating
primary and multiple events on the basis of differences in their local
slopes.

\new{PWC requires seismic traces to be sequentially
ordered. Therefore, it may not be immediately applicable to 3-D
problems.  Transformation from plane-wave construction to plane-wave
shaping \cite[]{shape} addresses this problem \cite[]{us}.}

\begin{comment}
A more powerful approach to regularization involves shaping operators
that project the estimated model into the space of acceptable models
\cite[]{shape}. In the last section of this paper, we show how to
transform plane-wave construction into plane-wave shaping by following
an analogy with triangle filtering.
\end{comment}

\section{Plane-wave construction defined}

Let us represent a seismic section $\mathbf{s}$ as a collection of
traces: $\mathbf{s} = \left[\mathbf{s}_1 \; \mathbf{s}_2 \; \ldots \;
\mathbf{s}_N\right]^T$. A plane-wave destruction operator
\cite[]{GEO67-06-19461960} effectively predicts each trace from its
neighbor and subtracts the prediction from the original trace. In the
linear operator notation, we can write the plane-wave destruction
operation as
\begin{equation}
  \label{eq:pwd}
  \mathbf{r} = \mathbf{D\,s}\;,
\end{equation}
where $\mathbf{r}$ is the destruction residual, and $\mathbf{D}$ is the
destruction operator defined as
\begin{equation}
  \label{eq:d}
  \mathbf{D} = \mathbf{I}_N - \mathbf{P} =
  \left[\begin{array}{ccccc}
      \mathbf{I} & 0 & 0 & \cdots & 0 \\
      - \mathbf{P}_{1,2} & \mathbf{I} & 0 & \cdots & 0 \\
      0 & - \mathbf{P}_{2,3} & \mathbf{I} & \cdots & 0 \\
      \cdots & \cdots & \cdots & \cdots & \cdots \\
      0 & 0 & \cdots & - \mathbf{P}_{N-1,N} & \mathbf{I} \\
    \end{array}\right]\;,
\end{equation}
where $\mathbf{I}$ stands for the identity operator, $\mathbf{P}$ is
the prediction operator defined by \cite{GEO67-06-19461960}, and
$\mathbf{P}_{i,j}$ describes prediction of trace $j$ from trace
$i$. \new{Prediction of a trace consists of shifting the original
trace along the dominant event slope. The dominant slope is estimated
by minimizing the prediction error (the output of $\mathbf{D}$) by
regularized least-squares optimization. Regularization constraints the
estimated slopes to vary smoothly inside the data space.}

The plane-wave construction (PWC) operator $\mathbf{C}$ is simply the inverse of
$\mathbf{D}$:
\begin{eqnarray}
  \nonumber 
  \mathbf{C} & = & 
  \left[\begin{array}{ccccc}
      \mathbf{I} & 0 & 0 & \cdots & 0 \\
      - \mathbf{P}_{1,2} & \mathbf{I} & 0 & \cdots & 0 \\
      0 & - \mathbf{P}_{2,3} & \mathbf{I} & \cdots & 0 \\
      \cdots & \cdots & \cdots & \cdots & \cdots \\
      0 & 0 & \cdots & - \mathbf{P}_{N-1,N} & \mathbf{I} \\
    \end{array}\right]^{-1} \\
  \nonumber
  &  = & 
  \left[\begin{array}{ccccc}
      \mathbf{I} & 0 & 0 & \cdots & 0 \\
      \mathbf{P}_{1,2} & \mathbf{I} & 0 & \cdots & 0 \\
      \mathbf{P}_{1,2}\,\mathbf{P}_{2,3} & \mathbf{P}_{2,3} & \mathbf{I} & \cdots & 0 \\
      \cdots & \cdots & \cdots & \cdots & \cdots \\
      \mathbf{P}_{1,2} \cdots \mathbf{P}_{N-1,N} & 
      \mathbf{P}_{2,3} \cdots \mathbf{P}_{N-1,N} & \cdots & 
      \mathbf{P}_{N-1,N} & \mathbf{I} \\
    \end{array}\right] \\
  &  = & 
  \left[\begin{array}{ccccc}
      \mathbf{I} & 0 & 0 & \cdots & 0 \\
      \mathbf{P}_{1,2} & \mathbf{I} & 0 & \cdots & 0 \\
      \mathbf{P}_{1,3} & \mathbf{P}_{2,3} & \mathbf{I} & \cdots & 0 \\
      \cdots & \cdots & \cdots & \cdots & \cdots \\
      \mathbf{P}_{1,N} & 
      \mathbf{P}_{2,N} & \cdots & 
      \mathbf{P}_{N-1,N} & \mathbf{I} \\
    \end{array}\right]
  \;,
  \label{eq:c}
\end{eqnarray}

For efficiency, it is convenient to apply PWC as a recursive triangular inversion. The output
of
\begin{equation}
  \label{eq:pwc}
  \mathbf{c} = \mathbf{C\,s} = \left[\mathbf{c}_1 \; \mathbf{c}_2 \; \ldots \;
    \mathbf{s}_N\right]^T
\end{equation}
is computed recursively as follows:
\begin{equation}
  \label{eq:rec}
  \mathbf{c}_1 = \mathbf{s}_1\;,\quad
  \mathbf{c}_k = \mathbf{s}_k + \mathbf{P}_{k-1,k}\,\mathbf{c}_{k-1}\;\quad
  k=2,3,\ldots,N
\end{equation}
$\mathbf{C}$ is a smoothing operator along local
plane-waves. \new{Re-parameterization by $\mathbf{C}$ provides
effective regularization and} can help \old{accelerating}
\new{accelerate the} convergence of iterative optimization in inverse
problems \cite[]{harlan,GEO68-02-05770588}.  When applied for model
\old{preconditioning} \new{re-parameterization} in least-squares
inversion of the forward modeling operator
\begin{equation}
  \label{eq:for}
  \mathbf{d} = \mathbf{L\,m} = \mathbf{L\,C\,p}\;,
\end{equation}
plane-wave construction leads to the formal inversion
\begin{equation}
  \label{eq:inv}
  \widehat{\mathbf{m}} = \mathbf{C}\,\widehat{\mathbf{p}} =
\mathbf{C\,C}^T\,\left(\mathbf{L\,C\,C}^T\,\mathbf{L}^T + \epsilon^2\,\mathbf{I}_N\right)^{-1}\,\mathbf{d}\;.
\end{equation}
Here $\mathbf{m}$ is the model, $\mathbf{p}$ is the \old{preconditioned} \new{re-parameterized}
model, $\mathbf{d}$ is the observed data, $\epsilon$ is the
regularization parameter, and $\widehat{\mathbf{m}}$ is the
regularized model estimate. The $\mathbf{C}\,\mathbf{C}^T$ operator
plays the role of the model covariance operator. % \cite[]{tarantola}.
In large-scale
problems, the inversion in equation~(\ref{eq:inv}) is computed efficiently by
an iterative conjugate-gradient algorithm.  A different approach to recursive
filter preconditioning, based on the helix transformation
\cite[]{GEO63-05-15321541}, was suggested by \cite{GEO68-02-05770588}.

The total cost of plane-wave construction is simply proportional to the data
size and to the cost of an elementary prediction $\mathbf{P}_{k-1,k}$, which
operates at the speed of tridiagonal inversion for a matrix of the
trace-length size \cite[]{GEO67-06-19461960}. This operation is comfortably
efficient in practical applications.

\section{Applications}

\inputdir{beic}

In this section, we describe three practical applications of
plane-wave construction in solving inverse problems with
  model \old{preconditioning} \new{re-parameterization}.

\subsection{Coherency enhancement}

\plot{bei-slp}{width=0.9\columnwidth}{Left: seismic image after prestack time
  migration. Right: local dips estimated with plane-wave destruction.}
\plot{bei-vg}{width=0.9\columnwidth}{Left: seismic image after coherency
  enhancing. Right: difference between the coherency-enhanced image and the