
\title{A modified anisotropic system}
\lefthead{Muir}
\righthead{Anisotropic system}
\footer{SEP--67}
\author{Francis Muir}

\INT

Some years back, Muir \& Dellinger (1985), in an attempt to simplify
the processing of multi-component seismic data, introduced a system of
equations that unified ray and wave theory in an approximate manner.
This allowed, for example, time/offset relations to be converted to
dispersion relations which could then form the basis for anisotropic
imaging schemes.

This paper removes one drawback from that scheme---the use of an
anelliptic factor that had no counterpart in normal experience---and
broadens it sufficiently that it could find application in the three
geometries now in common use, surface-to-surface, borehole-to-surface,
and borehole-to-borehole.

\mhead{WAVE EQUATIONS}

The key equation is the dispersion relation, the equation that links
the temporal and spatial frequencies of an eigenfield in the Fourier
domain. The four variable parameters all have units of velocities squared.
$W_{x}$ and $W_{z}$ are the actual horizontal and vertical square
velocities that apply equally well to waves and rays, and $W_{xnmo}$
and $W_{znmo}$ are the apparent square velocities as determined from move-out
measurements. For example, $W_{xnmo}$ is the square NMO velocity that
we would use every day in surface-to-surface data processing. If the
$x$ subscript seems unusual for a paraxial measurement about the
vertical, remember that in an elliptic world it is the horizontal
velocity that move-out measures.

\begin{equation}
\omega^{2} = \frac {W_{x}^3  k_{x}^6 + W_{x}^2  (2  W_{z} +
W_{znmo})  k_{x}^4  k_{z}^2 + (2 W_{x} + W_{xnmo}) W_{z}^2 k_{x}^2 k_{z}^4
+ W_{z}^6 k_{z}^6}{(W_{x} k_{x}^2 + W_{z} k_{z}^2)^2}
\end{equation}

With minor modifications that are discussed in Muir \& Dellinger (1985),
there are precisely equivalent plane wave and partial differential
equations.

\mhead{RAY EQUATIONS}

For the ray equations we need to define four new parameters, all with
units of inverse velocity squared. These parameters are defined:

\begin{itemize}
\item $M_{x} = W_{x}^{-1}$
\item $M_{z} = W_{z}^{-1}$ 
\item $M_{xnmo} = W_{xnmo}^{-1}$
\item $M_{znmo} = W_{znmo}^{-1}$
\end{itemize}

If we further define:

\begin{itemize}
\item $\phi$ is the angle from horizontal
\item $c$ is $\cos (\phi)$
\item $s$ is $\sin (\phi)$
\end{itemize}

then

\begin{equation}
M(\phi) = \frac {M_{x}^3 c^6 + M_x{x}^2 (2M_{z} + M_{znmo}) c^4 s^2
+ (2M_{x} + M_{xnmo}) c^2 s^4 + M_{z}^3 s^6}{(M_{x} c^2 + M_{z} s^2)^2}
\end{equation}

Again, in a manner shown in the referenced paper, we can develop move-out
equations (about the $x$ or $z$ axes) from this ray equation.

\mhead{CONCLUSION}

Are these wave and ray equations precisely equivalent? No. Equation (1) can 
be made to approximate dispersion relations that have a horizontal symmetry
plane. Approximate it both in the neighbourhood of the $z$ axis for
conventional seismic and VSP data, and in the neighbourhood of the $x$ axis
for cross-hole data. Likewise Equation (2) will then be a good paraxial
approximation in the neighbourhood of both the $z$ and the $x$ axes.

\REF

\reference{Muir,F. and Dellinger, J., 1985, A Practical Anisotropic
System: SEP--{\bf 44}, 55--58.}
