% . . Appendix for radcig
\append{ADCIG coordinate transform} 
%
This appendix addresses how to express operators $\frac{\partial }{\partial x_3}$ and $\frac{\partial}{\partial h_{x_1}}$ in generalized coordinate systems to derive equation~\ref{eqn:final3body}.  I first assume that generalized coordinate systems are related to the Cartesian variables through a bijection (i.e., one-to-one mapping) 
\begin{equation} \label{eqn:mapping}
x_1 = f(\xi_1,\xi_3) \quad {\rm and} \quad x_3 = g(\xi_1,\xi_3)
\end{equation}
with a non-vanishing Jacobian of coordinate transformation, $J_{\boldsymbol{\xi}}$.  The bijection between a generalized and Cartesian coordinate system allows us to rewrite the left-hand-sides of equations \ref{eqn:dtdh} as  \cite[]{Widder}
\begin{equation} \label{eqn:dtdh2}
\frac{\partial t}{\partial {x_1}} = \frac{1}{J_{\boldsymbol{\xi}}}\frac{\partial (t,x_3) }{ \partial
 ({\xi_1},{\xi_3}) }
\quad {\rm and} \quad 
\frac{\partial t}{\partial x_3} = \frac{1}{J_{\boldsymbol{\xi}}}\frac{\partial (x_1,t) }{ \partial
 (\xi_1,\xi_3) }.
\end{equation} 
Expanding the Jacobian notation leads to  
\begin{eqnarray}\label{eqn:PD}
%\left. 
\left[ \begin{array}{c}
\dtdex \dxzdez - \dtdez\dxzdex \\
\dtdez \dxxdex - \dtdex\dxxdez
\end{array} \right] 
%\right|_{\xi_1,t} 
= 
2\, J_{\boldsymbol{\xi}} \,s\, \rcos\, \gamma
\left[ \begin{array}{c}
\rsin \, \alpha \\
\rcos \, \alpha 
\end{array} \right].
\end{eqnarray}
The right-hand-sides of equations~\ref{eqn:PD} are analogous to those derived by \cite{SavaADCIG}.  Cross-multiplying the expressions by factors $\frac{\partial
  x_1}{\partial \xi_3}$ and $\frac{\partial x_3}{\partial \xi_3}$ 
\begin{eqnarray}\label{eqn:PD2}
%\left. 
\left[ \begin{array}{c}
\frac{\partial
  x_1}{\partial \xi_3}
\left(\dtdex \dxzdez - \dtdez\dxzdex\right) \\
\frac{\partial x_3}{\partial \xi_3}
\left(\dtdez \dxxdex - \dtdex\dxxdez\right)
\end{array} \right] 
%\right|_{\xi_1,t} 
= 
2\, J_{\boldsymbol{\xi}} \,s\, \rcos\, \gamma
\left[ \begin{array}{c}
\frac{\partial
  x_1}{\partial \xi_3}\rsin \,\alpha \\
\frac{\partial x_3}{\partial \xi_3}\rcos \,\alpha 
\end{array} \right]
\end{eqnarray}
and adding the two expressions results in 
\begin{equation} \label{eqn:PD3}
\dtdez \left( \dxzdez \dxxdex - \dxxdez \dxzdex \right) = 
2\, J_{\boldsymbol{\xi}} \,s\, \rcos\, \gamma \left( \dxxdez \rsin  \,\alpha + \dxzdez \rcos \,
\alpha \right).
\end{equation}
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\par
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A similar argument can be used to construct the equations for the subsurface-offset axis.  The bijection between the generalized coordinate and Cartesian subsurface-offset axes allows for the left-hand-side of equations~\ref{eqn:dtdh} to be rewritten as 
\begin{equation}
\frac{\partial t}{\partial h_{x_1} } = \frac{1}{J_{\boldsymbol{h}}}\frac{\partial
 (t,h_{x_3}) }{ \partial 
 (h_{\xi_1},h_{\xi_3}) }
\quad {\rm and} \quad 
\frac{\partial t}{\partial h_{x_3}} = \frac{1}{J_{\boldsymbol{h}}}\frac{\partial
 (h_{x_1},t) }{ \partial 
 (h_{\xi_1},h_{\xi_3}) },
\end{equation}
where $J_{\boldsymbol{h}}$ is the subsurface-offset Jacobian of transformation. Expanding the Jacobian notation leads to  
\begin{eqnarray}\label{eqn:PDH}
%\left. 
\left[ \begin{array}{c}
\dtdhex \dhxzdhez - \dtdhez\dhxzdhex \\
\dtdhez \dhxxdhex - \dtdhex\dhxxdhez
\end{array} \right] 
%\right|_{\xi_1,t}
= 
2\, J_{\boldsymbol{h}} \,s\, \rsin\, \gamma
\left[ \begin{array}{c}
\rcos\, \alpha \\
\rsin \, \alpha 
\end{array} \right].
\end{eqnarray}
The right-hand-side of equations~\ref{eqn:PDH} are again analogous to those given by \cite{SavaADCIG}.  Cross-multiplying the expressions by factors $\frac{\partial h_{x_1}}{\partial h_{\xi_1}}$ and $\frac{\partial h_{x_3}}{\partial h_{\xi_1}}$  
\begin{eqnarray}\label{eqn:PDH2}
%\left. 
\left[ \begin{array}{c}
\frac{\partial h_{x_1}}{\partial h_{\xi_1}}
\left(\dtdhex \dhxzdhez - \dtdhez\dhxzdhex\right) \\
\frac{\partial h_{x_3}}{\partial h_{\xi_1}}
\left(\dtdhez \dhxxdhex - \dtdhex\dhxxdhez\right)
\end{array} \right] 
%\right|_{\xi_1,t} 
= 
2\, J_{\boldsymbol{h}} \,s\, \rsin\, \gamma
\left[ \begin{array}{c}
\frac{\partial h_{x_1}}{\partial h_{\xi_1}}
\rcos \, \alpha \\
\frac{\partial h_{x_3}}{\partial h_{\xi_1}}
\rsin \, \alpha 
\end{array} \right],
\end{eqnarray}
and subtracting the two expressions above yields
\begin{equation} \label{eqn:PDH3}
\dtdhex \left( \dhxxdhex \dhxzdhez - \dhxxdhez \dhxzdhex \right) = 
2\, J_{\boldsymbol{h}} \,s\, \rsin\, \gamma \left( \dhxxdhex \rcos \alpha - \dhxzdhex \rsin
\alpha \right).
\end{equation}
An expression for ADCIGs can be obtained by dividing equation~\ref{eqn:PDH3} by equation~\ref{eqn:PD3}
\begin{equation} \label{eqn:final1}
\frac{\dtdhex}{\dtdez} \frac
{\left( \dhxxdhex \dhxzdhez - \dhxxdhez  \dhxzdhex \right)}
{\left( \dxzdez   \dxxdex   - \dxxdez    \dxzdex   \right)}
 =  \rtan
\, \gamma \, \frac{J_{\boldsymbol{h}}}{J_{\boldsymbol{\xi}}} \, \frac
{\left( \dhxxdhex \rcos \,\alpha - \dhxzdhex \rsin \,\alpha \right)}
{\left( \dxzdez   \rcos \,\alpha + \dxxdez   \rsin \,\alpha \right)}.
\end{equation}
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\par
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One question arising from the geometric factors in equation~\ref{eqn:final1} is what do the terms $\dhxxdhex$, $\dhxzdhex$, $\dhxxdhez$ and $\dhxzdhez$ represent?  I assume that the subsurface offset axes are generated by uniform wavefield shifting such that the following equations are valid:
\begin{eqnarray} \label{eqn:something}
\left[\begin{array}{c}
h_{x_1}  \\ 
h_{x_3}  \\
h_{\xi_1}\\
h_{\xi_1}
\end{array} 
\right] = \left[ \begin{array}{c}
x_1   \\ 
x_3   \\
\xi_1 \\
\xi_3
\end{array}\right] 
\quad {\rm such} \,\,\, {\rm that} \quad
\left[\begin{array}{c}
\dhxxdhex \\
\dhxzdhex \\
\dhxxdhez \\
\dhxzdhez
\end{array} 
\right] = \left[ \begin{array}{c}
\dxxdex \\
\dxzdex \\
\dxxdez \\
\dxzdez
\end{array}\right].
\end{eqnarray} 
If the subsurface offset axes were generated by anything other than uniform shifting (e.g. $h_{x_1} = x_1^2$), then the assumptions behind equations~\ref{eqn:something} would not be honored. 
%
\par
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Using these identities in equation~\ref{eqn:PD3} reduces equation~\ref{eqn:final1} to 
\begin{equation} \label{eqn:final3}
- \left. \dezdhex\right|_{\xi_1,t} = \left. \dtdhex \right/\dtdez  =  \rtan\, \gamma \, \frac
{\left(\dxxdex \rcos\,\alpha - \dxzdex \rsin \,\alpha \right)} 
{\left( \dxzdez \rcos\,\alpha + \dxxdez \rsin \,\alpha \right)},
\end{equation}
where the two Jacobian transformations are equivalent (i.e. $J_{\boldsymbol{\xi}}=J_{\boldsymbol{h}}$). 
This completes the derivation of equation~\ref{eqn:final3body}.
