paper.tex

国外免费地震资料处理软件包

Figure~\ref{fig:errgrp2}. These are the Zhang-Uren approximation
\cite[]{SEG-2001-01060109} and the Alkhalifah-Tsvankin approximation, which
follows directly from the normal moveout equation suggested by
\cite{GEO60-05-15501566}:
\begin{equation}
  \label{eq:talts}
  t^2(x) \approx t_0^2 + \frac{x^2}{V_n^2} - 
  \frac{2\,\eta\,x^4}{V_n^2\,\left[t_0^2\,V_n^2 + (1+ 2\,\eta)\,x^2\right]}
\;,
\end{equation}
where $t(x)$ is the moveout curve, $t_0$ is the vertical traveltime, and $V_n
= \sqrt{a}/(1 + 2\,\eta)$ is the NMO velocity. In a homogeneous medium,
equation~(\ref{eq:talts}) corresponds to the group velocity approximation
\begin{equation}
  \label{eq:valts}
  \frac{1}{V_P^2(\Theta)} \approx \frac{\cos^2{\Theta}}{V_z^2} + 
  \frac{\sin^2{\Theta}}
  {V_n^2} - \frac{2\,\eta\,\sin^4{\Theta}}
  {V_n^2\,\left[\cos^2{\Theta}\,V_n^2/V_z^2 + 
  (1+ 2\,\eta)\,\sin^2{\Theta}\right]}
\;,
\end{equation}
where $V_z = \sqrt{c}$. In the notation of this paper, the Alkhalifah-Tsvankin
equation~(\ref{eq:valts}) takes the form
\begin{equation}
  \label{eq:valts2}
  \frac{1}{V_P^2(\Theta)} \approx E(\theta) + 
  \frac{(Q-1)\,A\,C\,
    \sin^2{\Theta}\,\cos^2{\Theta}}{E(\Theta) + (Q^2-1)\,A\,\sin^2{\Theta}}
\end{equation}
and differs from approximation~(\ref{eq:muir2}) by the correction term in the
denominator. Approximation~(\ref{eq:shiftM2}) is noticeably more accurate
\new{for} this example than any of the other approximations considered here.

\new{Another accurate group velocity approximation was suggested by}
\cite{GEO65-04-13161325}. \new{However, the analytical expression is
complicated and inconvenient for practical use. The accuracy of Alkhalifah's
approximation for the Greenhorn shale example is depicted in}
Figure~\ref{fig:errgrp4}.

\plot{errgrp}{height=3in}{Relative error of different group velocity
  approximations for the Greenhorn shale anisotropy. Short dash: Thomsen's
  weak anisotropy approximation. Long dash: Muir's approximation. Solid line:
  suggested approximation.}

\plot{errgrp2}{height=3in}{Relative error of different group velocity
  approximations for the Greenhorn shale anisotropy. Short dash:
  Alkhalifah-Tsvankin approximation. Long dash: Zhang-Uren 
  approximation. Solid line: suggested approximation.}

\plot{errgrp4}{height=3in}{Relative error of different group velocity
  approximations for the Greenhorn shale anisotropy. Dashed line:
  Alkhalifah approximation. Solid line: suggested approximation.}

It is similarly possible to convert a group velocity approximation into the
corresponding moveout equation. In a homogeneous anisotropic medium, the
reflection traveltime $t$ as a function of offset $x$ is
\begin{equation}
  \label{eq:travel}
  t(x) = \frac{2\,\sqrt{(x/2)^2+z^2}}
  {V_P\left(\arctan\left(\frac{x}{2\,z}\right)\right)}\;,
\end{equation}
where $z = t_0\,V_P(0)/2$ is the depth of the reflector.  The moveout equation
corresponding to approximation~(\ref{eq:shiftM2}) is
\begin{eqnarray}
  \nonumber
  t^2(x) & \approx & \frac{1+2\,Q}{2\,(1+Q)}\,H(x) + 
\frac{1}{2\,(1+Q)}\,
\sqrt{H^2(x) + 4\,(Q^2-1)\,\frac{t_0^2\,x^2}{Q\,V_n^2}} \\
& = & \frac{3+4\,\eta}{4\,(1+\eta)}\,H(x) + 
\frac{1}{4\,(1+\eta)}\,
\sqrt{H^2(x) + 16\,\eta\,(1+\eta)\,\frac{t_0^2\,x^2}{(1+2\,\eta)\,V_n^2}}
\;,
  \label{eq:moveout}
\end{eqnarray}
where
$H(x)$ represents the hyperbolic part:
\begin{equation}
  \label{eq:hyper}
  H(x) = t_0^2 + \frac{x^2}{Q\,V_n^2} = 
  t_0^2 + \frac{x^2}{(1+2\,\eta)\,V_n^2}\;.
\end{equation}
For small offsets, the Taylor series expansion of equation~(\ref{eq:moveout})
is
\begin{eqnarray}
\nonumber
t^2(x) & \approx & t_0^2 + \frac{x^2}{V_n^2} - 
(Q-1)\,\frac{x^4}{t_0^2\,V_n^4} +
(Q-1)\,(2\,Q^2-1)\,\frac{x^6}{Q\,t_0^4\,V_n^6} + O(x^8) \\
& = & t_0^2 + \frac{x^2}{V_n^2} - 2\,\eta\,\frac{x^4}{t_0^2\,V_n^4} +
2\,\eta\,(1+8\,\eta+8\eta^2)\,\frac{x^6}{(1+2\,\eta)\,t_0^4\,V_n^6} + 
O(x^8)\;.
\label{eq:moveout2}  
\end{eqnarray}

Figure~\ref{fig:timepp} compares the accuracy of different moveout
approximations assuming reflection from the bottom of a homogeneous
anisotropic layer of 1~km thickness with the elastic parameters of Greenhorn
shale.  Approximation~(\ref{eq:moveout}) appears extremely accurate for
half-offsets up to 1~km and does not develop errors greater than 5~ms even
at much larger offsets.
  
\plot{timepp}{height=3in}{Traveltime moveout error of different group velocity
  approximations for Greenhorn shale anisotropy. The reflector depth is 1~km.
  Short dash: Alkhalifah-Tsvankin approximation. Long dash: Zhang-Uren
  approximation. Solid line: suggested approximation.}

It remains to be seen if the suggested approximation proves to be useful for
describing normal moveout in layered media. The next section discusses its
application for traveltime computation in heterogenous velocity models.

\section{Application: Finite-difference traveltime computation}

\new{As} an essential part of seismic imaging with the Kirchhoff method,
traveltime computation has received a lot of attention in the geophysical
literature.  Finite-difference eikonal solvers
\cite[]{GEO55-05-05210526,GEO56-06-08120821,podvin.gji.91} provide an efficient
and convenient way of computing first arrival traveltimes on regular grids.
Although they have a limited capacity for imaging complex structures
\cite[]{GEO58-04-05640575}, eikonal solvers can be extended in several different
ways to accommodate multiple arrivals
\cite[]{GEO62-02-05770588,SEG-1998-1945,GEO64-01-02300239}. A particularly
attractive approach to finite-difference traveltime computation is the fast
marching method, developed by \cite{paper} in the general context of level
set methods for propagating interfaces \cite[]{osher,book}.
\cite{GEO64-02-05160523} adopt the fast marching method for computing
seismic isotropic traveltimes. Alternative implementations are discussed by
\cite{SEG-1998-1949}, \cite{GPR49-02-01650178}, and \cite{kim}.
The fast marching method possesses a remarkable numerical stability, which
results from a cleverly chosen order of finite-difference evaluation.  The
order selection scheme resembles expanding wavefronts of
\cite{GEO57-03-04780487} and wavefront tracking of
\cite{GEO59-04-06320643}.

While the anisotropic eikonal equation~(\ref{eq:eikonal}) operates with phase
velocities, the kernel of the fast marching eikonal solver can be interpreted
in terms of local ray tracing in a constant-velocity background
\cite[]{Fomel.sep.95.sergey3} and is more conveniently formulated with the help
of the group velocity. \cite{alex} present a thorough extention of the
fast marching method to anisotropic wavefront propagation in the form of
ordered upwind methods. In this paper, I adopt a simplified approach.
Anisotropic traveltimes are computed in relation to an isotropic background.
At each step of the isotropic fast marching method, the local propagation
direction is identified, and the anisotropic traveltimes are computed by local
ray tracing with the group velocity corresponding to the same direction.  This
is analogous to the tomographic linearization approach in ray tracing, where
anisotropic traveltimes are computed along ray trajectories, traced in the
isotropic background \cite[]{pratt}. \cite{tariq} and \cite{schneider}
present different approaches for linearizing the anisotropic eikonal equation.

Many alternative forms of finite-difference traveltime computation in
anisotropic media are presented in the literature
\cite[]{GEO58-09-13491358,SEG-1997-1786,SEG-1999-18751878,ANI00-00-03330338,SEG-2001-12251228,qin,linbin}.
Although the method of this paper has limited accuracy because of the
linearization assumption, it is simple and efficient in practice and serves as
an illustration for the advantages of the explicit group velocity
approximation~(\ref{eq:shiftM2}).  For a more accurate and robust extension of
the fast marching method for anisotropic traveltime calculation, I recommend
the ordered upwind methods of \cite{alex,alex2}.

\inputdir{ell}

Figure~\ref{fig:const} shows finite-difference wavefronts for an isotropic and
an anisotropic homogeneous media, compared with the exact solutions. The
anisotropic media has the parameters of the Greenhorn shale. The
finite-difference error decreases with finer sampling.

\plot{const}{height=2.5in}{Finite-difference wavefronts in an isotropic (left)
  and an anisotropic (right) homogeneous media. The anisotropic media has the
  parameters of the Greenhorn shale. The finite-difference sampling is 100~m.
  The contour sampling is 0.1~s. Dashed curves indicate the exact solution.
  The finite-difference error will be reduced at finer sampling.}

\inputdir{emarm}

Figure~\ref{fig:marm} shows the first arrival wavefronts (traveltime contours)
computed in the ani\-so\-tro\-pic Marmousi model created by
\cite{Alkhalifah.sep.95.tariq3} in comparison with wavefronts for the
isotropic Marmousi model \cite[]{TLE13-09-09270936,TME00-00-00010194}. The model
parameters are shown in Figure~\ref{fig:marmmod}. The observed significant
difference in the wavefront position suggests a difference in the positioning
of seismic images when anisotropy is not properly taken into account.

\plot{marm}{width=5.5in}{Finite-difference wavefronts in the isotropic
  (top) and anisotropic (bottom) Marmousi models. A significant shift
  in the wavefront position suggest possible positioning error when
  seismic imaging does not take anisotropy into account.}

\plot{marmmod}{width=5.5in}{Alkhalifah's anisotropic Marmousi model.
  Top: vertical velocity. Bottom: anelliptic $\eta$ parameter. The
  vertical velocity is taken equal to the NMO velocity $V_n$.}

\section{Conclusions}

I have developed a general approach for approximating both phase and group
velocities in a VTI medium. Suggested approximations use three elastic
parameters as opposed to the four parameters in the exact phase velocity
expression. The phase velocity approximation coincides with the acoustic
approximation of \cite{GEO63-02-06230631,GEO65-04-12391250} but is derived
differently. The group velocity approximation has an analogous form and
similar superior approximation properties. It is important to stress that the
two approximations do not correspond exactly to each other. The exact group
velocity corresponding to the acoustic approximation is different from the
approximation derived in this paper and can be too complicated for practical
use \cite[]{EAE-1999-1049}. The suggested phase and group approximations match
each other in the sense that they have analogous approximation accuracy in the
dual domains.

The group velocity approximation is useful for approximating normal moveout
and diffraction traveltimes in applications to non-hyperbolic velocity
analysis and prestack time migration. It is also useful for traveltime
computations that require ray tracing in locally homogeneous cells. I have
shown examples of such computations utilizing an anisotropic extension of the
fast marching finite-difference eikonal solver.

%I have found no simple way to extend the results of this paper for
%approximating velocities of $qSV$ waves. Such an extension awaits further
%research.

\section{Acknowledgments}

The paper was improved by helpful suggestions from Tariq Alkhalifah and Paul
Fowler.

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