\section{One-way wave-equation in 3-D Riemannian spaces}
Equation~\reqo{weqrc.3d} can be used to describe 
two-way propagation of acoustic waves in a semi-orthogonal 
Riemannian space.
For one-way wavefield extrapolation, we need to modify
the acoustic wave \req{weqrc.3d} by selecting a single 
direction of propagation.
\par
In order to simplify the computations,
we introduce the following notation:
\beqa 
\czz &=& \frac{1  }{\AA^2}                                    \;, \nonumber \\
\cxx &=& \frac{\GG}{\JJ^2}                                    \;, \nonumber \\
\cyy &=& \frac{\EE}{\JJ^2}                                    \;, \nonumber \\
\cxy &=& \frac{\FF}{\JJ^2}                                    \;, \nonumber \\
\cz  &=& \frac{1}{\AA\,\JJ} \eone{\lp\frac{\JJ}{\AA}\rp}{\qz} \;, \nonumber \\
\cx  &=& \frac{1}{\AA\,\JJ} 
\lb \eone{\lp\GG\frac{\AA}{\JJ}\rp}{\qx} -
    \eone{\lp\FF\frac{\AA}{\JJ}\rp}{\qy} \rb \;, \nonumber \\
\cy  &=& \frac{1}{\AA\,\JJ} 
\lb \eone{\lp\EE\frac{\AA}{\JJ}\rp}{\qy} -
    \eone{\lp\FF\frac{\AA}{\JJ}\rp}{\qx} \rb \;.
\label{eqn:coefs.3d}
\eeqa
All quantities in \reqs{coefs.3d}
can be computed by finite-differences for any choice of 
a Riemannian coordinate system
which fulfills the orthogonality condition indicated earlier.
In particular, we can use ray coordinates to compute those 
coefficients. With these notations, the acoustic wave-equation
can be written as:
\beq \label{eqn:weqrc.3d.coefs}
\czz \dtwo{\W}{\qz} +
\cxx \dtwo{\W}{\qx} + 
\cyy \dtwo{\W}{\qy} +
\cz  \done{\W}{\qz} +
\cx  \done{\W}{\qx} + 
\cy  \done{\W}{\qy} +
\cxy \mtwo{\W}{\qx}{\qy} = - \frac{\ww^2}{\vv^2} \W \;.
\eeq
For the particular case of Cartesian coordinates
($\cx=\cy=\cz=0, \cxx=\cyy=\czz=1, \cxy=0$),
the Helmholtz \req{weqrc.3d.coefs} takes the familiar form
\beq 
    \dtwo{\W}{\qz}
 +  \dtwo{\W}{\qx}
 +  \dtwo{\W}{\qy}
 = -\frac{\ww^2}{\vv^2} \W \;.
\eeq
\par
From \req{weqrc.3d.coefs}, we can directly
deduce the modified form
of the dispersion relation for the wave-equation in a
semi-orthogonal {3-D} Riemannian space:
\beq \label{eqn:disp.3d}
- \czz \kz^2
- \cxx \kx^2
- \cyy \ky^2
+i\cz  \kz
+i\cx  \kx
+i\cy  \ky
- \cxy \kx\ky = - \ww^2\ss^2 \;.
\eeq
For one-way wavefield extrapolation, we need to solve
the second order \req{disp.3d} 
%%\beq
%%  \czz \kz^2
%%-i\cz  \kz 
%%+ \lb
%%  \cxx \kx^2 
%%+ \cyy \ky^2
%%-i\cx  \kx   
%%-i\cy  \ky
%%+ \cxy \kx\ky
%%- \ww^2\ss^2 \rb =0 \;,
%%\eeq
for the wavenumber of the extrapolation direction $\kz$,
and select the solution with the appropriate sign to
extrapolate waves in the desired direction:
\beq \label{eqn:oneway.3d}
\kz = i \frac{\cz}{2\czz} \pm
\sqrt{
\frac{\lp\ww\ss\rp^2}{\czz} - \lp\frac{\cz}{2\czz}\rp^2
- \lb \frac{\cxx}{\czz}\kx^2 -i \frac{\cx}{\czz}\kx \rb
- \lb \frac{\cyy}{\czz}\ky^2 -i \frac{\cy}{\czz}\ky \rb
- \frac{\cxy}{\czz} \kx\ky 
}\;.
\eeq
The solution with the positive sign in \req{oneway.3d} corresponds to 
propagation in the positive direction of the extrapolation axis $\qz$.
\par
For the particular case of Cartesian coordinates
($\cx=\cy=\cz=0, \cxx=\cyy=\czz=1, \cxy=0$),
the one-way wavefield extrapolation equation takes the
familiar form
\beq
\kz = \pm\sqrt{\lp\ww\ss\rp^2 -\kx^2 - \ky^2}\;.
\eeq



