% wavefield
\def\W{{U}}
% laplacian
\def\LAP{\Delta}
\def\dd {\nabla^2}
\def\bxi{{\bf \xi}}

% partial derivatives
\def\de#1{  \frac{\partial}{\partial #1} }
\def\dtwo#1#2{  \frac{\partial^2 #1}{\partial #2^2}   }
\def\done#1#2{  \frac{\partial   #1}{\partial #2  }   }
\def\eone#1#2{  \frac{\partial   }  {\partial #2  } #1}
\def\mtwo#1#2#3{\frac{\partial^2 #1}{\partial #2\partial #3}   }

\def\mmm#1#2{m^{#1#2}}
\def\ii{i}
\def\jj{j}
\def\z{z}
\def\x{x}
\def\y{y}

\def\qx{\xi_1}
\def\qy{\xi_2}
\def\qz{\xi_3}
\def\qi{\xi_i}
\def\qj{\xi_j}


\def\kz{k_{\qz}}
\def\kx{k_{\qx}}
\def\ky{k_{\qy}}
\def\ki{k_{\xi_i}}
\def\kj{k_{\xi_j}}

\def\xone{x_1}
\def\xtwo{x_2}
\def\xthr{x_3}

\def\DF#1#2{h_{#1#2}}
%%\def\hxx{\DF{\qx}{\qx}}
%%\def\hyy{\DF{\qy}{\qy}}
%%\def\hzz{\DF{\qz}{\qz}}
%%\def\hxy{\DF{\qx}{\qy}}

\def\EE{g_{11}}
\def\FF{g_{12}}
\def\GG{g_{22}}
\def\JJ{(\EE\GG-\FF^2)}
\def\HH{g_{13}}
\def\LL{g_{23}}
\def\PP{\Phi}
\def\AA{g_{33}}

\def\ww{\omega}
\def\ss{s}
\def\vv{v}

\def\xvec{\bf{x}}

\def\dxdex{ \frac{\partial x_k}{\partial \qx} }
\def\dxdey{ \frac{\partial x_k}{\partial \qy} }
\def\dxdez{ \frac{\partial x_k}{\partial \qz} }

% Partials
\def\ddex{ \frac{\partial}{\partial \qx} }
\def\ddey{ \frac{\partial}{\partial \qy} }
\def\ddez{ \frac{\partial}{\partial \qz} }
\def\ddexx{ \frac{\partial^2}{\partial \qx \partial \qx} }
\def\ddeyy{ \frac{\partial^2}{\partial \qy \partial \qy} }
\def\ddezz{ \frac{\partial^2}{\partial \qz \partiql \qz} }
\def\ddexy{ \frac{\partial^2}{\partial \qx \partial \qy} }
\def\ddexz{ \frac{\partial^2}{\partial \qx \partial \qz} }
\def\ddeyz{ \frac{\partial^2}{\partial \qy \partiql \qz} }

\def\gdet{ \left| \mathbf{g} \right| }
\def\gdetf{ \sqrt{\gdet} }


\def\hxxdef{\dxdex \dxdex}
\def\hyydef{\dxdey \dxdey}
\def\hzzdef{\dxdez \dxdez}
\def\hxydef{\dxdex \dxdey}
\def\hxzdef{\dxdex \dxdez}
\def\hyzdef{\dxdey \dxdez}

\def\c{c}
\def\cxx{\c_{\qx\qx}}
\def\cx {\c_{\qx   }}
\def\cyy{\c_{\qy\qy}}
\def\cy {\c_{\qy   }}
\def\czz{\c_{\qz\qz}}
\def\cz {\c_{\qz   }}
\def\cxy{\c_{\qx\qy}}

\def\ko{\; k_o}
\def\ofpo{ \lp \qz,\qx,\qy \rp}
\def\ofzo{ \lp \z ,\x ,\y  \rp}

% ray coordinates
\def\g{\gamma}
\def\t{\tau}
\def\hgg{\DF{\g}{\g}}
\def\htt{\DF{\t}{\t}}

\def\ofto{ \lp \t ,\g \rp}
\def\ctt{\c_{\t\t}}
\def\ct {\c_{\t  }}
\def\cgg{\c_{\g\g}}
\def\cg {\c_{\g  }}

\def\kt{k_\t}
\def\kg{k_\g}
\def\kn{k_{\nu}}
% discretization
\def\Wpo{\W^{\qx+1}_{\qz}}
\def\Woo{\W^{\qx}  _{\qz}}
\def\Wmo{\W^{\qx-1}_{\qz}}

\def\Wp {\W^{\qx+1}_{\qz+1}}
\def\Wo {\W^{\qx}  _{\qz+1}}
\def\Wm {\W^{\qx-1}_{\qz+1}}

% LISTS
\def\bit{\begin{itemize}}
\def\eit{  \end{itemize}}

\def\ben{\begin{enumerate}}
\def\een{  \end{enumerate}}

\def\itt{\item}

% SEPARATORS
\def\lp{\left(}
\def\rp{\right)}

\def\lb{\left[}
\def\rb{\right]}

%\def\ll{\left|}
%\def\rl{\right|}

\def\lc{\left\{}
\def\rc{\right\}}

% EQUATIONS
%\def\begin{equation}{\begin{equation}}
%\def\end{equation}{  \end{equation}} 
\def\req#1{equation~(\ref{eqn:#1})}
\def\reqs#1{equations~(\ref{eqn:#1})}
\def\reqo#1{(\ref{eqn:#1})}

%\def\bea{\begin{array}}
%\def\eea{  \end{array}}
%\def\begin{eqnarray}{\begin{eqnarray}}
%\def\end{eqnarray}{  \end{eqnarray}}

%\def	iny \begin{eqnarray}{\tiny   \begin{eqnarray}}
%\def\end{equation}t{\nonumber \end{eqnarray} \normalsize}

% VECTORS
\def\iv{\vec{i}} % x unit vector
\def\jv{\vec{j}} % y unit vector

\def\abs#1{\left| #1 \right|}	% vector modulus
\def\vdp#1#2{\lp#1\cdot#2\rp}   % dot product

% SIDE-BY-SIDE objects
\def\sidetwo#1#2{     \matrix{ {#1} & {#2}               \cr}}
\def\sidethree#1#2#3{ \matrix{ {#1} & {#2} & {#3}        \cr}}
\def\sidefour#1#2#3#4{\matrix{ {#1} & {#2} & {#3} & {#4} \cr}}

% OVER-UNDER objects
\def\overtwo#1#2{     \matrix{ {#1} \cr {#2}                   \cr}}
\def\overthree#1#2#3{ \matrix{ {#1} \cr {#2} \cr {#3}          \cr}}
\def\overfour#1#2#3#4{\matrix{ {#1} \cr {#2} \cr {#3} \cr {#4} \cr}}


\def\dtdhxx{\frac{\partial t}{\partial h_{x_1}}}
\def\dtdxz {\frac{\partial t}{\partial x_3}}
\def\dtdhx {\frac{\partial t}{\partial h_{x_1}}}
\def\dtdhz {\frac{\partial t}{\partial h_{x_3}}}
\def\dtdhex{\frac{\partial t}{\partial h_{\xi_1}}}
\def\dtdhez{\frac{\partial t}{\partial h_{\xi_3}}}

\def\dtdxx{\frac{\partial t}{\partial x_1}}
\def\dtdxz{\frac{\partial t}{\partial x_3}}
\def\dtdex{\frac{\partial t}{\partial \xi_1}}
\def\dtdez{\frac{\partial t}{\partial \xi_3}}
\def\dzdh{\frac{\partial z}{\partial h}}


% e over x
\def\dexdxz{\frac{\partial \xi_1}{\partial x_3}}
\def\dexdxx{\frac{\partial \xi_1}{\partial x_1}}
\def\dezdxz{\frac{\partial \xi_3}{\partial x_3}}
\def\dezdxx{\frac{\partial \xi_3}{\partial x_1}}

% x over e
\def\dxxdex{\frac{\partial x_1}{\partial \xi_1}}
\def\dxzdex{\frac{\partial x_3}{\partial \xi_1}}
\def\dxxdez{\frac{\partial x_1}{\partial \xi_3}}
\def\dxzdez{\frac{\partial x_3}{\partial \xi_3}}

% hx over he
\def\dhxxdhex{\frac{\partial h_{x_1}}{\partial h_{\xi_1}}}
\def\dhxxdhez{\frac{\partial h_{x_1}}{\partial h_{\xi_3}}}
\def\dhxzdhex{\frac{\partial h_{x_3}}{\partial h_{\xi_1}}}
\def\dhxzdhez{\frac{\partial h_{x_3}}{\partial h_{\xi_3}}}

% he over hx
\def\dhexdhxx{\frac{\partial h_{\xi_1}}{\partial h_{x_1}}}
\def\dhexdhxz{\frac{\partial h_{\xi_1}}{\partial h_{x_3}}}
\def\dhezdhxx{\frac{\partial h_{\xi_3}}{\partial h_{x_1}}}
\def\dhezdhxz{\frac{\partial h_{\xi_3}}{\partial h_{x_3}}}

% hx over e
\def\dhxxdex{\frac{\partial h_{x_1}}{\partial {\xi_1}}}
\def\dhxxdez{\frac{\partial h_{x_1}}{\partial {\xi_3}}}
\def\dhxzdex{\frac{\partial h_{x_3}}{\partial {\xi_1}}}
\def\dhxzdez{\frac{\partial h_{x_3}}{\partial {\xi_3}}}

% he over x
\def\dhexdxx{\frac{\partial h_{\xi_1}}{\partial {x_1}}}
\def\dhexdxz{\frac{\partial h_{\xi_1}}{\partial {x_3}}}
\def\dhezdxx{\frac{\partial h_{\xi_3}}{\partial {x_1}}}
\def\dhezdxz{\frac{\partial h_{\xi_3}}{\partial {x_3}}}

% hx over x
\def\dhxxdxx{\frac{\partial h_{x_1}}{\partial {x_1}}}
\def\dhxxdxz{\frac{\partial h_{x_1}}{\partial {x_3}}}
\def\dhxzdxx{\frac{\partial h_{x_3}}{\partial {x_1}}}
\def\dhxzdxz{\frac{\partial h_{x_3}}{\partial {x_3}}}

% he over e
\def\dhexdex{\frac{\partial h_{\xi_1}}{\partial {\xi_1}}}
\def\dhexdez{\frac{\partial h_{\xi_1}}{\partial {\xi_3}}}
\def\dhezdex{\frac{\partial h_{\xi_3}}{\partial {\xi_1}}}
\def\dhezdez{\frac{\partial h_{\xi_3}}{\partial {\xi_3}}}

% x over he
\def\dxxdhex{\frac{\partial x_1}{\partial h_{\xi_1}}}
\def\dxxdhez{\frac{\partial x_1}{\partial h_{\xi_3}}}
\def\dxzdhex{\frac{\partial x_3}{\partial h_{\xi_1}}}
\def\dxzdhez{\frac{\partial x_3}{\partial h_{\xi_3}}}

% x over hx
\def\dxxdhxx{\frac{\partial x_1}{\partial h_{x_1}}}
\def\dxxdhxz{\frac{\partial x_1}{\partial h_{x_3}}}
\def\dxzdhxx{\frac{\partial x_3}{\partial h_{x_1}}}
\def\dxzdhxz{\frac{\partial x_3}{\partial h_{x_3}}}

% e over he
\def\dexdhex{\frac{\partial \xi_1}{\partial h_{\xi_1}}}
\def\dexdhez{\frac{\partial \xi_1}{\partial h_{\xi_3}}}
\def\dezdhex{\frac{\partial \xi_3}{\partial h_{\xi_1}}}
\def\dezdhez{\frac{\partial \xi_3}{\partial h_{\xi_3}}}

% rm font
\def\rtan{{\rm tan}}
\def\rcos{ {\rm cos} }
\def\rsin{ {\rm sin} }
\def\rsinh{ {\rm sinh} }
\def\rcosh{ {\rm cosh} }
\def\w{\omega}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% . . Appendix A
%
\append{Riemannian geometry overview}
%
Geometry in a generalized 3D Riemannian space is described by a symmetric
metric tensor, $g_{jk}=g_{kj}$, that relates the geometry in a general
non-orthogonal coordinate system, $\boldsymbol{\xi}=[\xi_1,\xi_2,\xi_3]$, to an underlying
Cartesian mesh, $\boldsymbol{x}=[x_1,x_2,x_3]$ \cite[]{Guggenheimer}.  In
matrix form, the metric tensor is written,    
\begin{equation} \label{eqn:metric}
\left[g_{jk}\right] 
= 
\left[\begin{array}{ccc}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33} 
\end{array}\right] 
\end{equation}
where $\EE$, $\FF$, $\GG$, $\HH$, $\LL$, and $\AA$ are functions
linking the two coordinate systems through, 
\begin{eqnarray} \label{eqn:pderels}
\EE=\hxxdef, \;\;\;\; \FF=\hxydef, \;\;\;\; \GG = \hyydef,\nonumber \\
\HH=\hxzdef, \;\;\;\; \LL=\hyzdef, \;\;\;\; \AA = \hzzdef.  
\end{eqnarray} The associated (or inverse) metric
tensor, $g^{jk}$, is given by,   
\begin{equation} \label{eqn:ametric}
  \left[g^{jk}\right] =\frac{1}{\left|\mathbf{g}\right|}
  \left[\begin{array}{ccc}
      \GG\AA -\LL^2 & \HH\LL-\FF\AA & \FF\LL-\HH\GG \\
      \HH\LL-\FF\AA & \EE\AA-\HH^2  & \FF\HH-\EE\LL \\
      \FF\LL-\HH\GG & \FF\HH-\EE\LL & \EE\GG-\FF^2 
    \end{array}\right],
\end{equation}
and has the following metric discriminant, $\left| \mathbf{g}\right|$,
\begin{equation} \label{eqn:det}
  |\mathbf{g}| =g_{11}g_{22}g_{33} - g^2_{12}g_{33} - g^2_{23}g_{11} -
  g^2_{13}g_{22} + 2g_{12}g_{13}g_{23}.
\end{equation}
A weighted metric tensor, $m^{jk}=\gdetf \, g^{jk}$, is also used
throughout the thesis.   
%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% . . Appendix B
%
\append{Generalized phase-shift operators}
%
This appendix specifies a phase-shift approximation to the extrapolation wavenumber used to propagate wavefields described in equation~\ref{eqn:kzfinal}.  The relationship between the extrapolation wavenumber $\kz$, the two other wavenumbers $\kx$ and $\ky$, and the geometry variables is given by
\begin{equation} \label{eqn:kzfinalB}
\kz =
- a_1 \kx
- a_2 \ky 
+ i a_3 
\pm
\lb
  a_4^2 \ww^2 
- a_5^2 \kx^2 
- a_6^2 \ky^2
- a_7 \kx\ky 
+ i a_8 \,\kx 
+ i a_9 \,\ky  
- a_{10}^2  
\rb^{\frac{1}{2}},
\end{equation} 
where the non-stationary coefficients, $a_j$ in equation~\ref{eqn:kzfinalB}, are presented in vector $\mathbf{a}$,   
\begin{eqnarray}
 \label{eqn:ametric2B}
\mathbf{a}
= &
\left[
\frac{m^{13}}{ m^{33}} \;\;\;\;\;
\frac{m^{23}}{ m^{33} }\;\;\;\;\;
\frac{ n^3 } {2 m^{33} } \;\;\;\;\;
\frac{ \gdetf \ss }{ \sqrt{m^{33}} } \;\;\;\;\;\;
\sqrt{ \frac{m^{11}}{ m^{33}} - \lp \frac{m^{13}}{m^{33}}  \rp^2 }
\;\;\;\;\; 
\sqrt{\frac{m^{22}}{ m^{33} } - \lp \frac{ m^{23}}{m^{33}}  \rp^2 }
\;\;\;\;\; 
... \right. \nonumber \\
\, & \left. ... 
\lb \frac{2\,m^{12}}{m^{33}} - \frac{2\, m^{13}\,m^{23}}{ \lp m^{33}
  \rp^2} \rb \;\;\;\;\; 
\lb \frac{n^1}{m^{33}} - \frac{m^{13}\, n^3}{ \lp m^{33}\rp^2}\rb
\;\;\;\;\; 
\lb \frac{n^2}{m^{33}} - \frac{m^{23}\, n^3}{ \lp m^{33}\rp^2}\rb
\;\;\;\;\; 
\frac{ n^3 }{2 m^{33} }
\right]^{\mathbf{T}}.
\end{eqnarray}
The extrapolation wavenumber defined in equations~\ref{eqn:kzfinalB} and \ref{eqn:ametric2B} generally cannot be implemented exactly in the Fourier domain due to a simultaneous spatial dependence (i.e. a function of both $\boldsymbol{\xi}$ and ${\bf k}_{\boldsymbol{\xi}}$). This can be addressed using a multi-coefficient version of the split-step Fourier approximation \cite[]{GEO55-04-04100421} that uses Taylor expansions to separate $\kz$ into two parts: $\kz \approx \kz^{PS} + \kz^{SSF}$. Wavenumbers  $\kz^{PS}$ and $\kz^{SSF}$ represent a pure Fourier ($\omega-{\bf k}_{\boldsymbol{\xi}}$) domain phase-shift and a mixed ($\omega-\boldsymbol{\xi}$) domain split-step correction, respectively.  
%
\par
The phase-shift term is given by,   
\begin{small}
\begin{equation}
\kz^{PS} = 
- b_1 \kx
- b_2 \ky 
+ i b_3  
\pm
\lb
  b_4^2 \ww^2 
- b_5^2 \kx^2 
- b_6^2 \ky^2
- b_7 \kx\ky 
+ i b_8 \,\kx 
+ i b_9 \,\ky  
- b_{10}^2  
\rb^{\frac{1}{2}},
\end{equation}
\end{small}
where $b_j=b_j(\xi_3)$ are reference values of $a_j=a_j(\xi_1,\xi_2,\xi_3)$.  The split-step approximation is developed by performing a Taylor expansion about each coefficient $a_j$ and evaluating the results at stationary reference values $b_j$.  Assuming that the stationary values of $\kx$ and $\ky$ are zero, the split-step correction is  as follows,
\begin{equation} \label{eqn:kzssf}
\kz^{SSF}=
\left. \frac{\partial \kz}{\partial a_3} \right|_0 \lp a_3-b_3 \rp +
\left. \frac{\partial \kz}{\partial a_4} \right|_0 \lp a_4-b_4 \rp + 
\left. \frac{\partial \kz}{\partial a_{10}} \right|_0 \lp a_{10}-b_{10} \rp,
\end{equation}
where ``$0$'' denotes ``with respect to a reference medium''.  The partial differential expressions in equation~\ref{eqn:kzssf} are,    
\begin{equation}
\left. \frac{\partial \kz}{\partial a_3} \right|_0 
= b_3,
\;\;\;\;\;
\left. \frac{\partial \kz}{\partial a_4} \right|_0
= \frac{b_4 \,\ww^2}{\sqrt{b_4^2 \, \ww^2 - b_{10}^2 }},
\;\;\;\;\;
\left. \frac{\partial \kz}{\partial a_{10}} \right|_0
= - \frac{b_{10}}{\sqrt{b_{10}^2 \, \ww^2 - b_{10}^2 }},
\;\;\;\;\;
\end{equation}
resulting in the following split-step Fourier correction wavenumber, 
\begin{equation}
\kz^{SSF} = i\,b_3 \, \lp a_3-b_3 \rp +
\frac{b_4 \, \ww^2 \, \lp a_4-b_4 \rp}{\sqrt{b_4^2 \, \ww^2 - b_{10}^2
}} - \frac{b_{10} \, \lp a_{10}-b_{10} \rp}{\sqrt{b_4\,\ww^2 -
    b_{10}^2 }}. 
\end{equation}
%
\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% . . Appendix C
%
\append{RWE wavenumber approximations}
%
The extrapolation wavenumber developed in equation~\ref{eqn:kzfinal} is appropriate for any non-orthogonal Riemannian geometry.  However, there are a number of situations where symmetry or partial orthogonality are present.  Moreover, kinematic approximations can be made where one ignores all imaginary wavenumber components.  These situations are discussed herein.
%
\par
%
{\bf 3D Semi-orthogonal coordinate systems} - Semi-orthogonal
coordinate systems occur where one coordinate ($\xi_3$) is orthogonal
to the other two coordinates ($\xi_1$ and $\xi_2$)
\cite[]{Sava.geo.rwepub}.  In these cases the $m^{13}$ and $m^{23}$
components of the weighted metric tensor are identically zero, which
leads to the following extrapolation wavenumber, 
\begin{small}
\begin{equation} 
\kz = 
i a_3
\pm   
\lb
  a_4^2 \ww^2 
- a_5^2 \kx^2 
- a_6^2 \ky^2
- a_7 \kx\ky 
+ i a_8 \kx 
+ i a_9 \ky  
- a^2_{10}  
\rb^{\frac{1}{2}},
\end{equation}
\end{small}
where, 
\begin{small}
\begin{equation}
\mathbf{a} = 
\lb 
0 \;\;\;
0 \;\;\;
\frac{ n^3}{2 m^{33}} \;\;\;
\frac{ \gdetf \ss}{ \sqrt{m^{33}} } \;\;\;
\sqrt{\frac{m^{11}}{m^{33}}} \;\;\;
\sqrt{\frac{m^{22}}{m^{33}}} \;\;\;
\frac{ 2 m^{12} }{ m^{33} }  \;\;\;
\frac{n^1}{m^{33}} \;\;\;
\frac{n^2}{m^{33}} \;\;\;
\frac{n^3}{2 m^{33}}
\rb^{\mathbf{T}}.
\end{equation}
\end{small}
which are identical to the coefficients recovered by
\cite{Sava.geo.rwepub}. 
% 
\par
%
{\bf 3D kinematic coordinate systems} - One approximation that reduces the computational and memory costs is to consider any term in equation~\ref{eqn:kzfinal} with an imaginary number to be purely an amplitude factor. (Note that this is not purely correct because the square root of an imaginary number generally has a real component.)  This 'kinematic approximation' leads to extrapolation wavenumber,   
\begin{small}
\begin{equation}
\hat{\kz} =
- a_1 \kx
- a_2 \ky
\pm
\lb
  a_4^2 \ww^2 
- a_5^2 \kx^2 
- a_6^2 \ky^2
- a_7 \kx\ky 
- a_{10}^2  
\rb^{\frac{1}{2}},
\end{equation}
\end{small}
where, 
\begin{eqnarray}
\mathbf{a} = 
\left[
\frac{ m^{13} }{ m^{33} }\;\;\;
\frac{ m^{23} }{ m^{33} } \;\;\;
0\;\;\;
\frac{\gdetf\ss }{ \sqrt{m^{33}} }\;\;\;
\sqrt{\frac{m^{11} }{m^{33} } - \lp \frac{ m^{13} }{ m^{33} } \rp^2
}\;\;\;
...
\right. & \, \nonumber \\
\left.
...
\sqrt{\frac{m^{22}}{ m^{33} } - \lp \frac{ m^{23} }{ m^{33} } \rp^2}\;\;\;
\frac{ 2\,m^{12} }{ m^{33} } -\frac{ 2\, m^{13} m^{23}} {\lp
  m^{33}\rp^2 }\;\;\; 
0\;\;\;
0\;\;\;
\frac{n^3}{2 m^{33}}
\rb^{\mathbf{T}}.
& \, \nonumber
\end{eqnarray}
%\begin{tiny}
%\begin{equation}
%\mathbf{a} = 
%\lb
%\frac{ m^{13} }{ m^{33} }\;\;\;
%\frac{ m^{23} }{ m^{33} } \;\;\;
%0\;\;\;
%\frac{\gdetf\ss }{ \sqrt{m^{33}} }\;\;\;
%\sqrt{\frac{m^{11} }{m^{33} } - \lp \frac{ m^{13} }{ m^{33} } \rp^2 }\;\;\;
%\sqrt{\frac{m^{22}}{ m^{33} } - \lp \frac{ m^{23} }{ m^{33} } \rp^2}\;\;\;
%\frac{ 2\,m^{12} }{ m^{33} } -\frac{ 2\, m^{13} m^{23}} {\lp
%  m^{33}\rp^2 }\;\;\; 
%0\;\;\;
%0\;\;\;
%\frac{n^3}{2 m^{33}}
%\rb^{\mathbf{T}}.
%\end{equation}
%\end{tiny}%
\par
%
{\bf 3D kinematic semi-orthogonal coordinate systems} - Combining the
two above restrictions yields the following extrapolation wavenumber,  
\begin{small}
\begin{equation}
\hat{\kz} =
\pm
\lb
  a_4^2 \ww^2 
- a_5^2 \kx^2 
- a_6^2 \ky^2
- a_7 \kx\ky 
- a_{10}^2  
\rb^{\frac{1}{2}},
\end{equation}
\end{small}
where,
\begin{small}
\begin{equation}
\mathbf{a} = 
\lb
0 \;\;\;
0 \;\;\;
0 \;\;\;
\frac{ \gdetf \ss }{ \sqrt{m^{33}} } \;\;\;
\sqrt{\frac{m^{11}}{ m^{33} } } \;\;\;
\sqrt{\frac{m^{22}}{ m^{33} } }\;\;\;
\frac{ 2\, m^{12} }{ m^{33} } \; \;\;
0 \;\;\; 
0 \;\;\;
\frac{n^3}{2 m^{33}}
\rb^{\mathbf{T}}.
\end{equation}  
\end{small}
% 
\par
%
{\bf 2D non-orthogonal coordinate systems} - Two-dimensional
situations are handled by identifying $\qy=0$.  All derivatives in the
associated metric tensor $m^{jk}$ with respect coordinate $\qy$ are
identically zero, and the resulting 2D non-orthogonal coordinate
system wavenumber is, 
%
\begin{small}
\begin{equation} \label{eqn:2DNO}
\kz =
- a_1 \kx+
i a_3 
\pm
\lb
  a_4^2 \ww^2 
- a_5^2 \kx^2 
+ i a_8 \kx 
- a_{10}^2  
\rb^{\frac{1}{2}},
\end{equation}
\end{small}
where,
\begin{small}
\begin{equation}
\mathbf{a} = 
\lb
\frac{ m^{13} }{ m^{33} }\;\;\;
0 \;\;\;
\frac{ n^3 }{ 2 m^{33} } \;\;\;
\frac{ \gdetf \ss }{ \sqrt{m^{33}} } \;\;\;
\sqrt{ \frac{ m^{11} } {m^{33} } - \lp \frac{ m^{13} }{ m^{33} } \rp^2 } \;\;\;
0 \;\;\;
0 \;\;\;
\frac{n^1}{m^{33}} - \frac{m^{13}n^{3}}{(m^{33})^2} \;\;\;
0 \;\;\;
\frac{n^3}{2 m^{33}}
\rb^{\mathbf{T}}.
\end{equation}
\end{small}
%
\par
%
{\bf 2D non-orthogonal kinematic coordinate systems} -
Two-dimensional kinematic situations are handled through identity 
$\qy=0$.  Again, all derivatives in the associated metric tensor
$m^{jk}$ with respect coordinate $\qy$ are identically zero, and
the 2D non-orthogonal kinematic extrapolation wavenumber is 
\begin{small}
\begin{equation} \label{eqn:testtest}
\hat{\kz} =
- a_1 \kx
\pm
\lb
  a_4^2 \ww^2 
- a_5^2 \kx^2 
- a_{10}^2  
\rb^{\frac{1}{2}},
\end{equation}
\end{small}
where,
\begin{small}
\begin{equation}
\mathbf{a} = 
\lb
\frac{ m^{13} }{ m^{33} } \;\;\;
0 \;\;\;
0 \;\;\;
\frac{ \gdetf \ss }{ \sqrt{m^{33}} } \;\;\;
\sqrt{ \frac{ m^{11} }{ m^{33} } - \lp \frac{ m^{13} }{ m^{33} } \rp^2 } \;\;\;
0 \;\;\; 
0 \;\;\;
0 \;\;\;
0 \;\;\;
\frac{n^3}{2 m^{33}}
\rb^{\mathbf{T}}.
\end{equation}
\end{small}
%
\par
%
{\bf 2D orthogonal coordinate systems} - Two-dimensional situations
are handled with $\qy=\HH=0$. Accordingly, all derivatives in the
associated  metric tensor $m^{jk}$ with respect coordinate $\qy$ are identically
zero, and the 2D orthogonal coordinate system is represented by 
\begin{small}
\begin{equation}
\kz =
i a_3  
\pm
\lb
  a_4^2 \ww^2 
- a_5^2 \kx^2 
+ i a_8 \kx 
- a_{10}^2
\rb^{\frac{1}{2}},
\end{equation}
\end{small}
where,
\begin{small}
\begin{equation}
\mathbf{a} = 
\lb
0 \;\;\;
0 \;\;\;
\frac{ n_3 }{ 2 m^{33} } \;\;\;
\frac{ \gdetf \ss }{ \sqrt{m^{33}} } \;\;\;
\sqrt{\frac{m^{11} }{m^{33} } } \;\;\;
0 \;\;\; 
0 \;\;\;
\frac{n^1}{m^{33}} \;\;\;
0 \;\;\;
\frac{n^3}{2 m^{33}}
\rb^{\mathbf{T}}.
\end{equation}
\end{small}
%
\par
%
{\bf 2D orthogonal kinematic coordinate systems} - The above two
approximations can be combined to yield the following extrapolation
wavenumber for 2D orthogonal kinematic coordinate systems,  
\begin{small}
\begin{equation}
\hat{\kz} =
\pm
\lb
  a_4^2 \ww^2 
- a_5^2 \kx^2 
- a_{10}^2  
\rb^{\frac{1}{2}},
\end{equation}
\end{small}
where,
\begin{small}
\begin{equation}
\mathbf{a} = 
\lb
0 \;\;\;
0 \;\;\;
0 \;\;\;
\frac{ \gdetf \ss }{ \sqrt{m^{33}} } \;\;\;
\sqrt{\frac{ m^{11} }{m^{33} }} \;\;\;
0 \;\;\; 
0 \;\;\;
0 \;\;\;
0 \;\;\;
\frac{n^3}{2 m^{33}}
\rb^{\mathbf{T}}.
\end{equation}
\end{small}

% . . APPENDIX 4
\append{Elliptic coordinate systems}

Exploring constant coordinate surfaces provides additional insight into some characteristics of the elliptic coordinate system.  As illustrated by the following trigonometric identities, curves of constant $\xi_1$ represent hyperbolas   
\begin{equation} \label{eqn:xi1curve}
\frac{x_1^2}{a^2\, {\rm cos}^2\,\xi_1} - \frac{x_3^2}{a^2 \,{\rm sin}^2\,\xi_1} = {\rm cosh}^2 \, \xi_3-{\rm sinh}^2 \, \xi_3 = 1, 
\end{equation}
while curves of constant $\xi_3$ form ellipses
\begin{equation} \label{eqn:xi2curve}
\frac{x_1^2}{a^2 \,{\rm cosh}^2\, \xi_3} + \frac{x_3^2}{a^2\, {\rm sinh}^2\,\xi_3} =  {\rm cos}^2\xi_1+{\rm sin}^2\xi_1 = 1.
\end{equation}
Thus, outward extrapolation in the $\xi_3$ direction would step a wavefield through a family of elliptic surfaces defined by equation~\ref{eqn:xi2curve}.   
%
\par
%
Equation~\ref{eqn:xi1curve} may also be used to derive an expression that defines the local extrapolation axis angle, $\theta(\boldsymbol{\xi})$, relative to vertical reference.  Taking the total derivative of equation~\ref{eqn:xi1curve},
\begin{equation}
\frac{2 \, x_1\, {\rm d}x_1}{a^2\,{\rm cosh}^2\, \xi_3} + \frac{2\,  x_3 \, {\rm d} x_3}{a^2\,{\rm sinh}^2\, \xi_3} =0,
\end{equation}
and further manipulating the result yields the local extrapolation
axis angle $\theta(\boldsymbol{\xi})$ 
\begin{equation}
{\rm tan}\,\theta =\frac{{\rm d}x_1}{{\rm d}x_3} = \frac{x_3\, {\rm
  cos}^2  \, \xi_1}{x_1\, {\rm 
  sin}^2\,\xi_1} = {\rm tanh}\, \xi_3 \,{\rm cot}\,\xi_1. 
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% . . Appendix E
\input{appendix}
