paper.tex

国外免费地震资料处理软件包

\begin{equation}
  \label{eq:qpdefhat}
  \hat{q} = \frac{l\,(a-l) + (l+f)^2}
  {c\,(a-l)}\;.
\end{equation}

\cite{Muir.sep.44.55} also suggested approximating the VTI
group velocity with an analogous expression
\begin{equation}
  \label{eq:muir2}
\frac{1}{V^2_{P}(\Theta)} \approx E(\Theta) + 
\frac{(Q-1)\,A\,C\,
\sin^2{\Theta}\,\cos^2{\Theta}}{E(\Theta)}
\end{equation}
where $A=1/a$, $C=1/c$, $Q = 1/q$, $\Theta$ is the group angle, and
$E(\Theta)$ is the elliptical part:
\begin{equation}
\label{eq:muel2}
E(\Theta) = A\,\sin^2{\Theta} + C\,\cos^2{\Theta}\;.
\end{equation}
Equations~\eqref{eq:muph} and~\eqref{eq:muir2} are consistent in the sense
that both of them are exact for elliptic anisotropy ($q=Q=1$) and
accurate to the first order in $(q-1)$ or $(Q-1)$ in the general case of
transversally isotropic media.

To the same approximation order, the connection between the phase and group
directions is
\begin{equation}
\label{eq:t2T}
\tan{\Theta}  = \tan{\theta}\,\frac{a}{c}\, 
\left(1 - (q-1)\,\frac
  {a\,\sin^2{\theta} - c\,\cos^2{\theta}}
  {a\,\sin^2{\theta} + c\,\cos^2{\theta}}\right)\;.
\end{equation}
%or
%\begin{equation}
%  \label{eq:T2t}
%  \sin^2{\theta}  =  
%  \frac{A^2\,\sin^2{\Theta}}{\left(A^2\,\sin^2{\Theta} + 
%      C^2\,\cos^2{\Theta}\right)}\,
%  \left[1 - \frac{2\,(Q-1)\,C^2\,\cos^2{\Theta}\,
%      \left(C\,\cos^2{\Theta} - A\,\sin^2{\Theta}\right)}
%    {\left(A\,\sin^2{\Theta} + C\,\cos^2{\Theta}\right)\,
%      \left(A^2\,\sin^2{\Theta} + C^2\,\cos^2{\Theta}\right)}\right]\;.
%\end{equation}

\section{Shifted hyperbola approximation for the phase velocity}

Despite the beautiful symmetry of Muir's approximations~(\ref{eq:muph})
and~(\ref{eq:muir2}), they are less accurate in practice than some other
approximations, most notably the weak anisotropy approximation of
\cite{GEO51-10-19541966}, which can be written as 
\cite[]{GEO61-02-04670483}
\begin{equation}
  \label{eq:thoms}
  v_P^2(\theta) \approx c\,\left(1 + 2\,\epsilon\,\sin^4{\theta} + 
  2\,\delta\,\sin^2{\theta}\,\cos^2{\theta}\right)\;,
\end{equation}
where 
\begin{equation}
  \label{eq:epsdelta}
  \epsilon = \frac{a-c}{2\,c}\quad\mbox{and}\quad
  \delta = \frac{(l + f)^2 - (c - l)^2}{2\,c\,(c - l)}\;.
\end{equation}

Note that both approximations involve the anellipticity factor ($q-1$ or
$\epsilon-\delta$) in a linear fashion. If the anellipticity effect is
significant, the accuracy \new{of Muir's equations can be improved by
replacing the linear approximation with a nonlinear one}. There are, of
course, infinitely many nonlinear expressions that share the same
linearization. In this study, I focus on the shifted hyperbola approximation,
which follows from the fact that an expression of the form
 \begin{equation}
   \label{eq:lin}
   x + \frac{\alpha}{x}   
 \end{equation}
is the linearization (Taylor series expansion) of the form
\begin{equation}
  \label{eq:nonlin}
  x\,(1-s) + s\,\sqrt{x^2 + \frac{2\,\alpha}{s}}
\end{equation}
for small $\alpha$. Linearization does not depend on the parameter $s$, which
affects only higher-order terms in the Taylor expansion.
Expression~(\ref{eq:nonlin}) is reminiscent of the shifted hyperbola
approximation for normal moveout in vertically heterogeneous media
\cite[]{malov,Sword.sep.51.313,GEO53-02-01430157,GEO59-06-09830999} and the
Stolt stretch correction in the frequency-wavenumber migration
\cite[]{GEO43-01-00230048,mystolt}. It is evident that Muir's
approximation~(\ref{eq:muph}) has exactly the right form~(\ref{eq:lin}) to be
converted to the shifted hyperbola approximation~(\ref{eq:nonlin}). 

Thus, we seek an approximation of the form
\begin{equation}
  \label{eq:shiftm}
  v_P^2(\theta) \approx e(\theta)\,(1-s) + s\,\sqrt{e^2(\theta) + 
    \frac{2\,(q-1)\,a\,c\,
      \sin^2{\theta}\,\cos^2{\theta}}{s}}
\end{equation}
with $e(\theta)$ defined by equation~(\ref{eq:muel}). The plan is to select a
value of the additional parameter $s$ to fit the exact phase velocity
expression~(\ref{eq:qp}) and then to constrain $s$ so that it depends only on
the three parameters already present in the original
approximation~\eqref{eq:muph}.

One can verify that the velocity curvature $d^2 v_P/d \theta^2$ around the
vertical axis $\theta=0$ for approximation~(\ref{eq:shiftm}) depends on the
chosen value of $q$ but does not depend on the value of the shift parameter
$s$. This means that the velocity profile $v_P(\theta)$ becomes sensitive to
$s$ only further away from the vertical direction.  This separation of
influence between the approximation parameters is an important and attractive
property of the shifted hyperbola approximation. I find an appropriate value
for $s$ by fitting additionally the fourth-order derivative $d^4 v_P/d
\theta^4$ at $\theta=0$ to the corresponding derivative of the exact
expression. The fit is achieved when $s$ has the value
\begin{equation}
  \label{eq:sval}
  s = \frac{c-l}{2}\,\frac{(a-l)\,(c-l) - (l+f)^2}
  {a\,(c-l)^2 - c\,(l+f)^2}\;.  
\end{equation}
It is more instructive to express it in the form
\begin{equation}
  \label{eq:sval2}
  s = \frac{1}{2}\,\frac{(a-c)\,(q-1)\,(\hat{q}-1)}
  {a\,\left(1 - \hat{q} - q\,(1-q)\right) - 
    c\,\left((\hat{q}-1)^2+\hat{q}\,(q-\hat{q})\right)}\;,  
\end{equation}
where $q$ and $\hat{q}$ are defined by equations~\eqref{eq:qpdef}
and~\eqref{eq:qpdefhat}.  In this form of the expression, $\hat{q}$ appears as
the extra parameter that we need to eliminate.  This parameter was defined by
fitting the velocity profile curvature around the horizontal axis, which would
correspond to infinitely large offsets in a surface seismic experiment. One
possible way to constrain it is to set $\hat{q}$ equal to $q$, which implies
that the velocity profile has similar behavior near the vertical and the
horizontal axes. Setting $\hat{q} \approx q$ in equation~\eqref{eq:sval2}
yields
\begin{equation}
  \label{eq:sappr}
  s \approx \lim_{\hat{q} \rightarrow q} s = \frac{1}{2}\;.
\end{equation}
Substituting \eqref{eq:sappr} in equation~\eqref{eq:shiftm} produces the final
approximation
\begin{equation}
  \label{eq:shiftm2}
  v_P^2(\theta) \approx \frac{1}{2}\,e(\theta) + 
  \frac{1}{2}\,\sqrt{e^2(\theta) + 
    4\,(q-1)\,a\,c\,\sin^2{\theta}\,\cos^2{\theta}}\;.
\end{equation}

Approximation~\eqref{eq:shiftm2} is exactly equivalent to the \emph{acoustic
  approximation} of \cite{GEO63-02-06230631,GEO65-04-12391250},
derived with a different set of parameters by formally setting the $S$-wave
velocity ($l=v_S^2$) in equation~(\ref{eq:qp}) to zero. A similar
  approximation is analyzed by \cite{ANI00-00-03490361}.
Approximation~\eqref{eq:shiftm2} was proved to possess a remarkable accuracy
even for large phase angles and significant amounts of anisotropy.
Figure~\ref{fig:errphp} compares the accuracy of different approximations using
the parameters of the Greenhorn shale. The acoustic approximation appears
especially accurate for phase angles up to about 25 degrees and does not
exceed the relative error of 0.3\% even for larger angles.

\plot{errphp}{height=3in}{Relative error of different phase velocity
  approximations for the Greenhorn shale anisotropy. Short dash: Thomsen's
  weak anisotropy approximation. Long dash: Muir's approximation. Solid line:
  suggested approximation (similar to Alkhalifah's acoustic approximation.)}

\section{Shifted hyperbola approximation for the group velocity}

Similar strategy is applicable for approximating the group velocity. Applying
the shifted hyperbola approach to ``unlinearize'' Muir's
approximation~(\ref{eq:muir2}), we seek an approximation of the form
\begin{equation}
  \label{eq:shiftM}
  \frac{1}{V_P^2(\Theta)} \approx E(\Theta)\,(1-S) + S\,\sqrt{E^2(\Theta) + 
    \frac{2\,(Q-1)\,A\,C\,
      \sin^2{\Theta}\,\cos^2{\Theta}}{S}}
\end{equation}
An approximation of this form with $S$ set to $1/2$ was proposed earlier by
\cite{SEG-2001-01060109}. Similarly to the case of the phase velocity
approximation, I constrain the value of $S$ by Taylor fitting of the velocity
profiles near the vertical angle.

Although there is no simple explicit expression for the transversally
isotropic group velocity, we can differentiate the parametric representations
of $V_P$ and $\Theta$ in terms of the phase angle $\theta$ that follow from
equation~(\ref{eq:group}). The group velocity is an even function of the angle
$\Theta$ because of the VTI symmetry. Therefore, the odd-order derivatives are
zero at the axis of symmetry ($\Theta=\theta=0$). Fitting the second-order
derivative $d^2 V_P/d\Theta^2$ at $\theta=0$ produces $Q=1/q=1+2\,\eta$,
consistent with Muir's approximation~(\ref{eq:muir2}). Fitting additionally
the fourth-order derivative $d^4 V_P/d\Theta^4$ at $\theta=0$ produces
\begin{equation}
  \label{eq:Sval}
  S = \frac{1}{2}\,\frac{
    \left[(l+f)^2 + l\,(c-l)\right]^2\,\left[(c-l)\,(a-l) - (l+f)^2\right]}{
    a^2\,c\,(c-l)\,(l+f)^2 - \left[l\,(c-l) + (l+f)^2\right]^3}
\end{equation}
or, equivalently,
\begin{equation}
  \label{eq:Sval2}
  S = \frac{1}{2}\,\frac{(C-A)\,(Q-1)\,(\hat{Q}-1)}
  {C\,\left(\hat{Q}\,(Q^2-Q-1) + 1\right) + 
    A\,\left(\hat{Q}-Q^3+Q^2-1\right)}\;,
\end{equation}
where $\hat{Q}=1/\hat{q}$. \new{As in} the
previous section, I approximate the optimal value of $S$ by setting $\hat{Q}$
equal to $Q$, as follows:
\begin{equation}
  \label{eq:Sappr}
  S \approx \lim_{\hat{Q} \rightarrow Q} S = \frac{1}{2\,(1+Q)} = 
  \frac{1}{4\,(1 + \eta)}\;.
\end{equation}
Selected in this way, the value of $S$ depends on the anelliptic parameter $Q$
(or $\eta$) and, for small anellipticity, is close to $1/4$, which is
different from the value of $1/2$ in the approximation of
\cite{SEG-2001-01060109}.

The final group velocity approximation takes the form
\begin{equation}
  \label{eq:shiftM2}
  \frac{1}{V^2_{P}(\Theta)} \approx \frac{1+2\,Q}{2\,(1+Q)}\,E(\Theta) + 
\frac{1}{2\,(1+Q)}\,
\sqrt{E^2(\Theta) + 4\,(Q^2-1)\,A\,C\,\sin^2{\Theta}\,\cos^2{\Theta}}\;.
\end{equation}
In Figure~\ref{fig:errgrp}, the accuracy of approximation~(\ref{eq:shiftM2})
is compared with the accuracy of Muir's approximation~(\ref{eq:muir2}) and the
accuracy of the weak anisotropy approximation \cite[]{GEO51-10-19541966} for the
elastic parameters of the Greenhorn shale. The weak anisotropy approximation,
used in this comparison, is
\begin{equation}
  \label{eq:thoms2}
  V_P^2(\Theta) \approx c\,\left(1 + 2\,\epsilon\,\sin^4{\Theta} + 
    2\,\delta\,\sin^2{\Theta}\,\cos^2{\Theta}\right)\;,
\end{equation}
where $\epsilon$ and $\delta$ are Thomsen's parameters, defined in
equations~(\ref{eq:epsdelta}). A similar form (in a different
parameterization) was introduced by \cite{GEO54-12-15641574}.

Approximation~(\ref{eq:shiftM2}) turns out to be remarkably accurate
\new{for} this example. It appears nearly exact for group angles up to 45
degrees from vertical and does not exceed 0.3\% relative error even at
\new{larger} angles. It is compared with two other approximations in