avz.tex

国外免费地震资料处理软件包

$S_2$, which range from $1$ to $2$, the expression (\ref{eqn:tsvmal}) is 
positive. This means that the anisotropic approximation
(\ref{eqn:TIapprox}) overestimates traveltimes in the isotropic
heterogeneous model even more than does the shifted hyperbola 
(\ref{eqn:malovichko}) shown in Figure~\ref{fig:nmofrz}b. 
Below, we examine which of the two approximations is more suitable when
the model includes both vertical heterogeneity and anisotropy.

\subsection{Vertically heterogeneous VTI model} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In a model that includes vertical heterogeneity and anisotropy, both
factors influence bending of the rays. The weak anisotropy
approximation, however, allows us to neglect the effect of anisotropy on ray
trajectories and consider its influence on traveltimes only. This
assumption is analogous to the linearization, conventionally done for
tomographic inversion. Its application to weak anisotropy has been
discussed by \cite{GEO61-06-18831894}. According to the
linearization assumption, we can retain isotropic equation
(\ref{eqn:hp}) describing the ray trajectories and rewrite equation
(\ref{eqn:tp}) in the form
\begin{equation}
t(p) = 2\,\int_{0}^{z}\,{{dz} \over {V_g(z,\psi(z))\,\cos{\psi(z)}}}\;,
\label{eqn:TItp} 
\end{equation}
where $V_g$ is the anisotropic group velocity, which varies both with the
depth $z$ and with the ray angle $\psi$ and has the expression
(\ref{eqn:vg}). Differentiation of the parametric traveltime equations
(\ref{eqn:TItp}) and (\ref{eqn:hp}) and linearization with respect to Thomsen's
anisotropic parameters shows that the general form of equations
(\ref{eqn:a0})--(\ref{eqn:a3}) remains valid if we replace the definitions 
of the root-mean-square velocity $V_{rms}$ and the parameters $M_k$ 
by 
\begin{eqnarray}
V_{rms}^2 & = & {1 \over t_z}\,\int_{0}^{t_z}\,V_z^{2}(t)\,
\left[1 + 2\,\delta(t)\right]\,dt\;,
\label{eqn:TIvrms} \\
M_k & = & {1 \over t_z}\,\int_{0}^{t_z}\,V_z^{2k}(t)\,
\left[1 + 2\,\delta(t)\right]^{2k}\,\left[1 + 8\,\eta(t)\right]\,dt \qquad 
(k = 2,3, \ldots)\;.
\label{eqn:TImk} 
%\\
%S_k & = & {M_k \over {V_{rms}^{2k}}}\qquad (k = 2, 3, \ldots)\;.
%\label{eqn:TIsk}
\end{eqnarray}
%??? Am I right removing this eq? It is the same as eq.(29). --
In homogeneous media, expressions (\ref{eqn:TIvrms}) 
and (\ref{eqn:TImk}) transform series (\ref{eqn:taylor}) with
coefficients (\ref{eqn:a0})--(\ref{eqn:a3}) into the form equivalent to
series (\ref{eqn:TItaylor}). Two important conclusions follow from 
%the mathematical form of 
equations (\ref{eqn:TIvrms}) and (\ref{eqn:TImk}). 
First, if the mean value of the anisotropic coefficient $\delta$ 
is less than zero, the presence of anisotropy can reduce the difference
between the effective root-mean-square velocity and the effective vertical 
velocity $\widehat{V}_z=z/t_z$. In this case, the influence of anisotropy
and heterogeneity partially cancel each other, and the moveout curve
may behave at small offsets as if the medium were homogeneous and
isotropic. This behavior has been noticed by
\cite{GEO58-10-14541467}. On the other hand, if the anellipticity
coefficient $\eta$ is positive and different from zero, it can
significantly increase the values of the heterogeneity parameters
$S_k$ defined by equations~(\ref{eqn:sk}). Then, the nonhyperbolicity 
of reflection moveouts at
large offsets is stronger than that in isotropic media.
\par
To exemplify the general theory, let us consider a simple analytic
model with constant anisotropic parameters and the vertical velocity
linearly increasing with depth according to the equation
\begin{equation}
V_z(z) = V_z(0)\,(1 + \beta \,z) = V_z(0)\,e^{\kappa(z)}\;,
\label{eqn:vzlin} 
\end{equation}
where $\kappa$ is the logarithm of the velocity change. In this case,
the analytic expression for the RMS velocity $V_{rms}$ is found
from equation (\ref{eqn:TIvrms}) to be
\begin{equation}
V_{rms}^2 = V_z^2(0)\,(1 + 2\,\delta)\,{{e^{2\kappa}-1}\over {2\,\kappa}}\;,
\label{eqn:vrmslin} 
\end{equation}
while the mean vertical velocity is
\begin{equation}
\widehat{V}_z = {z \over t_z} = 
V_z(0)\,{{e^{\kappa}-1}\over {\kappa}}\;,
\label{eqn:hatvzlin} 
\end{equation}
where $\kappa=\kappa(z)$ is evaluated at the reflector depth.
Comparing equations (\ref{eqn:vrmslin}) and (\ref{eqn:hatvzlin}), we can see 
that the squared RMS velocity $V_{rms}^2$ equals to the squared mean velocity
$\widehat{V}_z^2$ if
\begin{equation}
1 + 2\,\delta = {{2\,\left(e^\kappa - 1\right)} \over
{\kappa\,\left(e^\kappa + 1\right)}}\;.
\label{eqn:deltalin} 
\end{equation}
For small $\kappa$, the estimate of $\delta$ from equation
(\ref{eqn:deltalin}) is
\begin{equation}
\delta \approx - {\kappa^2 \over 24}\;.
\label{eqn:appdeltalin} 
\end{equation}
For example, if the vertical velocity near the reflector is twice
that at the surface (i.e., $\kappa = \ln 2 \approx 0.69$), 
having the anisotropic
parameter $\delta$ as small as~$-0.02$ is sufficient to cancel out the
influence of heterogeneity on the normal-moveout velocity.  The values of
parameters $S_2$ and $S_3$, found from equations~(\ref{eqn:sk}), (\ref{eqn:TIvrms}) and~(\ref{eqn:TImk}), are
\begin{eqnarray}
S_2 & = & (1 + 8\,\eta)\,\kappa\,{{e^{2\kappa}+1} \over {e^{2\kappa} - 1}}\;,
\label{eqn:lins2} \\
S_3 & = & {4 \over 3}\,
(1 + 8\,\eta)\,\kappa^2\,{{e^{4\kappa} + e^{2\kappa}+1} \over 
{\left(e^{2\kappa} - 1\right)^2}}\;.
\label{eqn:lins3}
\end{eqnarray} 
Substituting equations (\ref{eqn:lins2}) and (\ref{eqn:lins3}) into 
the estimates (\ref{eqn:malerror}) and (\ref{eqn:tsverror}) and linearizing 
them both in $\eta$ and in $\kappa$, we
find that the error of anisotropic traveltime approximation
(\ref{eqn:TIapprox}) in the linear velocity model is 
\begin{equation}
{{\Delta t^2(l)} \over t^2(0)} = 
-{{\kappa^2\,(1 - 8\,\eta)} \over 12}\,
\left({l \over {t_0\,V_n}}\right)^6\;,
\label{eqn:lintsverr}
\end{equation}
while the error of the shifted-hyperbola approximation
(\ref{eqn:malovichko}) is
\begin{equation}
{{\Delta t^2(l)} \over t^2(0)} = 
\left({{\kappa^2\,(1 - 8\,\eta)} \over 24} - \eta\right)\,
\left({l \over {t_0\,V_n}}\right)^6\;.
\label{eqn:linmalerr}
\end{equation}
%??? Please put the correct sign into one of the previous two eqs. Otherwise,
%eq.(53) does not seem correct -- that is why I thought it is incorrect. --
Comparing 
equations (\ref{eqn:lintsverr}) and (\ref{eqn:linmalerr}), we 
conclude that if the medium is elliptically anisotropic $(\eta=0)$, 
the shifted hyperbola can be twice as accurate as the anisotropic equation 
(assuming the optimal choice of parameters). The accuracy of the latter,
however, increases when the anellipticity coefficient $\eta$ grows and
becomes higher than that of the shifted hyperbola if $\eta$ satisfies
the approximate inequality 
\begin{equation}
\eta \geq {\kappa^2 \over {8\,(1 + \kappa^2)}}\;.
\label{eqn:lineta}
\end{equation}
%??? -- I got from equations (\ref{eqn:lintsverr}) and (\ref{eqn:linmalerr})
%\[
%  \eta \geq {\kappa^2 \over {4\,(3 + 2\,\kappa^2)}}\;.
%\]
%Who is right?
For instance, if $\kappa = \ln 2$, inequality~(\ref{eqn:lineta}) yields
$\eta \geq 0.03$, a quite small value.

%\begin{comment}
%\subsection{Stolt Stretch}
%%%%%%%%%%%%%%%%%%%%%
%Stolt stretch
%\cite{GEO43.01.00230048,Levin.sep.42.373,Claerbout.blackwell.85} is
%a method of extending constant-velocity frequency-domain migration to
%the case of a vertically variable velocity. The method consists of
%stretching the time axis according to the equation
%\begin{equation}
%\tau(t_z)=
%\left({{2 \over V_0^2}\,\int_0^{t_z}\,t\,V_{rms}^2(t)\,dt}\right)^{1/2}\;,
%\label{eqn:ss} 
%\end{equation}
%double Fourier transform, and migration according to the dispersion
%relation
%\begin{equation}
%\omega_m(k,\omega_0)=
%\left(1-{1\over W}\right)\,\omega_0+
%{{\mbox{sign}(\omega_0)}\over W}\,
%\sqrt{\omega_0^2 - W\,V_0^2\,k_x^2}\;,
%\label{eqn:sdispersion} 
%\end{equation}
%where $V_0$ is a constant frame velocity, $\omega_0$ and $\omega_m$
%are the frequencies before and after the migration, corresponding to
%the stretched time coordinate, $k_x$ is the wavenumber, and $W$ is a
%constant parameter ($W=1$ in the constant velocity case). Fomel
%\shortcite{Fomel.sep.84.61} has shown that the optimal choice of the
%Stolt stretch parameter $W$ for a particular traveltime depth $t_z$ is
%given by the expression
%\begin{equation}
%W=1-{{V_0^2\,\tau^2\left(t_z\right)} \over{V_{rms}^2\left(t_z\right) t_z^2}}\,
%\left({{V^2\left(t_z\right)} \over {V_{rms}^2\left(t_z\right)}}
%-S_2\left(t_z\right)
%\right)\;\;.
%\label{eqn:main} 
%\end{equation}
%This expression remains valid in the case of a vertically
%heterogeneous VTI medium if the values of $V_{rms}$ and $S_2$ are
%computed according to equations (\ref{eqn:TIvrms}) and
%(\ref{eqn:TIsk}). The method of cascaded migrations
%\cite{GEO52.05.06180643} can improve the performance of Stolt
%migration in the case of variable velocity \cite{GEO53.07.08810893}.
%However, this method affects only the isotropic part of the model and
%cannot change the contribution of the anisotropic parameters.
%Therefore, in the anisotropic case, it is important to incorporate
%anisotropic parameters into the Stolt stretch correction.
%\end{comment}