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\section{VERTICAL HETEROGENEITY}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Vertical heterogeneity is another reason for nonhyperbolic
moveout. We start this section by reviewing well-known results for
isotropic media. Although these results can be interpreted
in terms of an effective anisotropy, we show that it has different properties
than those for the VTI model. We then extend the theory to vertically heterogeneous
VTI media and perform a comparative analysis of various three-parameter 
nonhyperbolic approximations.

\inputdir{Math}

\subsection{Vertically heterogeneous isotropic model} 
%%%%%%%%%%%%%%%%%%%%%%
Nonhyperbolicity of reflection moveout in vertically heterogeneous
isotropic media has been extensively studied using the
Taylor series expansion in the powers of the offset
\cite[]{der,taner,GPR21-04-07830795}. The most important property of
vertically heterogeneous media is that the ray parameter 
\[
p \equiv {{\sin{\psi}(z)} \over {V_z(z)}}
\]
does not change along
any given ray (Snell's law). This fact leads to the explicit parametric
relationships 
\begin{eqnarray}
t(p) & = & 2\,\int_{0}^{z}\,{{dz} \over {V_z(z)\,\cos{\psi(z)}}} =
\int_{0}^{t_z}\,{{dt_z} \over {\sqrt{1 - p^2\,V_z^2(t_z)}}}\;,
\label{eqn:tp} \\ 
l(p) & = & 2\,\int_{0}^{z}\,{{dz}\,\tan{\psi(z)}} =
\int_{0}^{t_z}\,{{p\,V_z^2(t_z)\,dt_z} \over 
{\sqrt{1 - p^2\,V_z^2(t_z)}}}\;,
\label{eqn:hp}
\end{eqnarray}
where
\begin{equation}
t_z = t(0) = 2\,\int_{0}^{z}\,{{dz} \over {V_z(z)}}\;.
\label{eqn:tz}
\end{equation}
Straightforward differentiation of parametric equations (\ref{eqn:tp})
and (\ref{eqn:hp}) yields the first four coefficients of the Taylor
series expansion
\begin{equation}
t^2(l) = a_0 + a_1\,l^2 + a_2\,l^4 + a_3\,l^6 + \ldots
\label{eqn:taylor}
\end{equation}
in the vicinity of the vertical zero-offset ray.  Series
(\ref{eqn:taylor}) contains only even powers of the offset $l$ because
of the reciprocity principle: the pure-mode reflection traveltime is an even
function of the offset.  The Taylor series coefficients for the isotropic case
are defined as follows:
\begin{eqnarray}
a_0 & = & t_z^2\;,
\label{eqn:a0} \\
a_1 & = & {1 \over V_{rms}^2}\;,
\label{eqn:a1} \\
a_2 & = & {{1 - S_2} \over {4\,t_z^2\,V_{rms}^4}}\;,
\label{eqn:a2} \\
a_3 & = & {{2\,S_2^2 -  S_2 - S_3} \over {8\,t_z^4\,V_{rms}^6}}\;,
\label{eqn:a3}
\end{eqnarray}
where 
\begin{eqnarray}
V_{rms}^2 = M_1 \, ,
\label{eqn:a31}
\end{eqnarray}
\begin{eqnarray}
M_k & = & {1 \over t_z}\,\int_{0}^{t_z}\,V_z^{2k}(t)\,dt \qquad
(k = 1, \, 2, \, \ldots)\;,
\label{eqn:mk} \\
S_k & = & {M_k \over {V_{rms}^{2k}}} \qquad (k = 2, \, 3, \, \ldots)\;.
\label{eqn:sk}
\end{eqnarray}
Equation (\ref{eqn:a1}) shows that, at small offsets, the reflection
moveout has a hyperbolic form with the normal-moveout velocity $V_n$
equal to the root-mean-square velocity $V_{rms}$. At large offsets,
however, the hyperbolic approximation is no longer accurate.  Studying
the Taylor series expansion (\ref{eqn:taylor}), \cite{malov}
introduced a three-parameter approximation for the reflection
traveltime in vertically heterogeneous isotropic media.
%\cite{malov,Sword.sep.51.313}. 
His equation has the form of a
shifted hyperbola \cite[]{castle,nmo}:
\begin{equation}
t(l) = \left(1 - {1 \over S}\right)\,t_0 + 
{1 \over S} \sqrt{t_0^2 + S\,{l^2 \over V_n^2}}\;.
\label{eqn:malovichko}
\end{equation}
\par
If we set the zero-offset traveltime $t_0$ equal to the vertical
traveltime $t_z$, the velocity $V_n$ equal to $V_{rms}$, and the {\em
parameter of heterogeneity} $S$ equal to $S_2$, equation
(\ref{eqn:malovichko}) guarantees the correct coefficients $a_0$, $a_1$, and
$a_2$ in the Taylor series (\ref{eqn:taylor}). Note that the parameter $S_2$
is related to the variance $\sigma^2$ of the squared velocity
distribution, as follows:
\begin{equation}
\sigma^2 = M_2 - V_{rms}^4 = V_{rms}^4\,(S_2 -1)\;.
\label{eqn:sigma2}
\end{equation}
According to equation (\ref{eqn:sigma2}), this parameter is always greater
than unity (it equals 1 in homogeneous media). In many 
practical cases, the value of $S_2$ lies between~$1$ and~$2$.  We can
roughly estimate the accuracy of approximation (\ref{eqn:malovichko}) at
large offsets by comparing the fourth term of its Taylor series with
the fourth term of the exact traveltime expansion (\ref{eqn:taylor}).
According to this estimate, the error of Malovichko's approximation is
\begin{equation}
{{\Delta t^2(l)} \over t^2(0)} = {1 \over 8} (S_3-S_2^2)\,
\left({l \over {t_0\,V_n}}\right)^6\;.
\label{eqn:malerror}
\end{equation}
As follows from the definition of the parameters $S_k$ 
[equations~(\ref{eqn:sk})] and the Cauchy-Schwartz inequality, the expression
(\ref{eqn:malerror}) is always nonnegative. 
This means that the shifted-hyperbola approximation
tends to overestimate traveltimes at large offsets. As the offset
approaches infinity, the limit of this approximation is 
\begin{equation}
\lim_{l \rightarrow \infty} t^2(l) = {1 \over S}\,{l^2 \over V_n^2}\;.
\label{eqn:Mhlimit}
\end{equation}
\par
Equation (\ref{eqn:Mhlimit}) indicates that the effective horizontal
velocity for Malovichko's approximation (the slope of the shifted
hyperbola asymptote) differs from the normal-moveout velocity. One 
can interpret this difference as evidence of some \emph{effective}
depth-variant anisotropy. However, the anisotropy implied in
equation (\ref{eqn:malovichko}) differs from the true anisotropy in a 
homogeneous transversely isotropic medium [see equation (\ref{eqn:vg})]. 
To reveal this difference, let us substitute
the effective values $t(l) = {\sqrt{4\,z^2 + l^2} / {V_g(\psi)}}$,
$~t_0 = {2\,z / V_z}$, $~l = 2\,z\,\tan{\psi}$, and $S = {V_x^2 / V_n^2}$ 
into equation (\ref{eqn:malovichko}). After eliminating the
variables $z$ and $l$, the result takes the form
\begin{equation}
{1 \over {V_g(\psi)}} = {1 \over V_z}\,\left\{
\cos{\psi}\left(1 - {V_n^2 \over V_x^2}\right) +
\sqrt{{V_z^2 \over V_x^2}\,\sin^2{\psi} +
{V_n^4 \over V_x^4}\,\cos^2{\psi}}\right\}.
\label{eqn:mal2vg}
\end{equation}
If the anisotropy is induced by vertical heterogeneity, $V_x \ge V_n \ge V_z$.  
Those inequalities follow from the definitions of $V_{rms}$, $t_z$,
$S_2$, and the Cauchy-Schwartz inequality. They reduce to equalities only
when velocity is constant.  Linearizing expression
(\ref{eqn:mal2vg}) with respect to Thomsen's anisotropic parameters $\delta$
and $\epsilon$, we can transform it to the form analogous to
that of equation (\ref{eqn:vgeta}):
\begin{equation}
V_g^2(\psi) = V_z^2\,\left[\,1 + 2\,\delta\,\sin^2{\psi} + 
2\,\eta\,(1 - \cos{\psi})^2\right]\;.
\label{eqn:vgvz}
\end{equation}
Figure \ref{fig:nmofrz} illustrates the difference between the VTI
model and the effective anisotropy implied by the Malovichko approximation. 
The differences are noticeable in both the shapes of the effective wavefronts
(Figure \ref{fig:nmofrz}a) and the moveouts (Figure \ref{fig:nmofrz}b).

\plot{nmofrz}{width=6in,height=1.5in}{Comparison of the
wavefronts (a) and moveouts (b) in the VTI (solid) and vertically
inhomogeneous isotropic media (dashed). The values of the effective 
vertical, horizontal, and NMO velocities are the same
in both media and correspond to Thomsen's parameters $\epsilon = 0.2$ and
$\delta = 0.1$.}
%??? -- Label the plots with a and b. 
%??? -- Remove the numbers from the b-plot.
%??? -- Replace the dash-dotted line with dashed.}   

\par
In deriving equation (\ref{eqn:vgvz}), we have assumed the correspondence
\begin{equation}
S =  {V_x^2 \over V_n^2} = 
{{1 + 2\,\epsilon} \over {1 + 2\,\delta}} \approx 1 + 2\,\eta\;.
\label{eqn:sone}
\end{equation}
We could also have chosen the value of the parameter of heterogeneity $S$ that
matches the coefficient $a_2$ given by equation (\ref{eqn:a2}) with
the corresponding term in the Taylor series (\ref{eqn:TItaylor}).Then, the value of $S$ is \cite[]{GEO62-06-18391854}
\begin{equation}
S = 1 + 8\,\eta\;.
\label{eqn:stwo}
\end{equation}
The difference between equations (\ref{eqn:sone}) and (\ref{eqn:stwo}) is an
additional indicator of the fundamental difference between homogeneous 
VTI and vertically heterogeneous isotropic media. The three-parameter 
anisotropic approximation (\ref{eqn:TIapprox}) can match the
reflection moveout in the isotropic model up to 
the fourth-order term in the Taylor series expansion if the value of
$\eta$ is chosen in accordance with equation (\ref{eqn:stwo}). We can
estimate the error of such an approximation with an equation analogous
to (\ref{eqn:malerror}): 
\begin{equation}
{{\Delta t^2(l)} \over t^2(0)} = {1 \over 8} (S_3-2 + 3\,S_2-2\,S_2^2)\,
\left({l \over {t_0\,V_n}}\right)^6\;.
\label{eqn:tsverror}
\end{equation}
The difference between the error estimates (\ref{eqn:malerror}) and
(\ref{eqn:tsverror}) is
\begin{equation}
{{\Delta t^2(l)} \over t^2(0)} = {1 \over 8} (2 - S_2)\,(S_2-1)\,
\left({l \over {t_0\,V_n}}\right)^6\;.
\label{eqn:tsvmal}
\end{equation}
For usual values of 
%the parameter of heterogeneity