\section{Introduction}
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Angle-domain common-image gathers (ADCIGs) are used increasingly in seismic imaging to examine migration velocity model accuracy \cite[]{3DSI}.  The key idea is that migrating with the correct velocity model leads to flat angle gathers that shift neither vertically nor horizontally as a function of reflection opening angle. Migrating with an incorrect velocity, though, leads to inconsistent angle-domain reflectivity and generates residual curvature in the ADCIG volume.  ADCIGs are thus an effective velocity analysis tool and have been incorporated in wave-equation-based inversion schemes to update velocity profiles \cite[]{SavaWEMVA1, SavaWEMVA2}.    
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Wave-equation imaging techniques generate ADCIGs in straightforward manners for both shot-profile \cite[]{deBruin,Rickett,SavaADCIG} and shot-geophone \cite[]{MosherADCIG,PruchaADCIG} migration approaches.  In shot-profile migration, one first generates a subsurface-offset axis at each depth step by correlating the source and receiver wavefields at a number of subsurface shifts.  The second step involves computing an offset-to-angle domain transformation using, for example, post-imaging Radon-based operators \cite[]{SavaADCIG}. 
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Conventional ADCIG theory usually assumes horizontal wavefield shifts, largely because wavefield extrapolation and  imaging are most commonly performed in Cartesian coordinates.  However, a number of studies have noted that these ADCIG results degrade for steeply dipping structures, such as salt flanks \cite[]{BiondiShan2002,BiondiSymes}.  Although this is partially due to problems associated with inaccurate large-angle extrapolation, ADCIGs calculated using horizontal wavefield shifts become decreasingly sensitive for increasingly steep structural dips.  \cite{BiondiSymes} demonstrate that this problem can be addressed by generating ADCIGs with vertical offset-domain common-image gathers (VODCIGs); however, this approach is less desirable computationally because it requires storing the larger wavefield volumes required to calculate the VODCIGs in memory.
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The introduction of shot-profile migration in more general coordinate systems [e.g. tilted Cartesian \cite[]{SEG2004sgj} and elliptical meshes \cite[]{ShraggeGEORWE}] presents an opportunity to circumvent problems associated with generating ADCIGs for steeply dipping structure.  In particular, migration domains can be oriented such that geologic structures with steep dips in Cartesian meshes have relatively gentle dips in generalized coordinate systems, thus improving the robustness of the ADCIG calculation.  Developing an ADCIG theory capable of handling more arbitrary coordinate meshes, though, requires proper treatment of the effects of non-Cartesian geometries.  For example, wavefield extrapolation in non-Cartesian coordinate systems induces local wavenumber stretches, rotations and/or shearing 
(see Chapter 2).
%\cite[]{ShraggeGEONO}.  
Similarly, non-uniform wavefield sampling can lead to anisotropic angle-domain stretching.  These effects can be corrected using Jacobian change-of-variable transformations.   
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The goal of this chapter 
%paper 
is to extend ADCIG theory to non-Cartesian geometries.  I demonstrate that ADCIG theory, as developed in a differential sense \cite[]{SavaADCIG}, remains valid for arbitrary geometries provided that the corresponding derivative operators are properly specified.  Non-Cartesian coordinates do, however, introduce space-domain geometric factors that can render Fourier-based offset-to-angle methods unsuitable. However, I show that ADCIGs can be calculated directly in the Fourier domain for all coordinate systems satisfying the Cauchy-Riemann differentiability criteria \cite[]{Arfken}.  Moreover, ADCIGs can be calculated in all situations using the slant-stack approaches discussed in \cite{PruchaADCIG}.
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I begin by reviewing how to generate subsurface offsets and ADCIGs in Cartesian coordinates.  I then provide an extension to generalized coordinate systems based on Jacobian change-of-variable arguments.  I examine two canonical coordinate systems, tilted Cartesian and elliptical meshes, where the reflection angle can be explicitly calculated using Fourier-based methods, and a third, polar coordinates, where it cannot.  I test the generalized ADCIG theory analytically and numerically using a set of elliptical reflectors, and demonstrate how computing angle gathers in elliptical coordinates can lead to improvements relative to Cartesian coordinates, especially for steeply dipping structure.
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\section{ADCIG theory}
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The ADCIG theory presented in this section draws from that presented in \cite{SavaADCIG}.  In the ensuing development, $\boldsymbol{x}=[x_1,x_3]$ denotes the Cartesian variables and  $\boldsymbol{\xi}=[\xi_1,\xi_3]$ represents a generalized Riemannian coordinate system.   I also use a convention where the extrapolation axis is oriented in the $x_3$ ($\xi_3$) direction for Cartesian (Riemannian) coordinates.  Coordinates $x_2=\xi_2=0$ are assumed throughout.
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\subsection{Generating subsurface offsets axes}
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Shot-profile migration in Cartesian coordinates consists of completing a recursive two-step procedure.  The first step involves propagating the source and receiver wavefields, $S$ and $R$, from depth level $x_3-\Delta x_3$ to  $x_3$ using an extrapolation operator $E_{x_3}[\cdot]$
\begin{eqnarray}
E_{x_3}[S(x_3-\Delta x_3,x_1|\omega)] &=& S(x_3,x_1|\omega), \nonumber \\
E^*_{x_3}[R(x_3-\Delta x_3,x_1|\omega)] & = & R(x_3,x_1|\omega),
\end{eqnarray}
where $*$ denotes the conjugate operator, $\omega$ is angular frequency, and $\Delta x_3$ is the depth step. A subsurface image, $I$, is subsequently computed at each extrapolation step by evaluating an imaging condition
\begin{equation} \label{eqn:subsurface1}
I(x_3,x_1,h_{x_1}) = \sum_\omega S^*(x_3,x_1+h_{x_1}|\omega) R(x_3,x_1-h_{x_1}|\omega),
\end{equation}
where the subsurface-offset axis, $h_{x_1}$, is generated by correlating the source and receiver wavefields at various relative shifts in the $x_1$ direction.  Finally, the ADCIG volume is computed using an offset-to-angle transformation operator, \new{ ${\bf T}_{\boldsymbol{x}}$ }
\begin{equation} \label{eqn:subsurface2}
I(x_3,x_1,\gamma) = {\bf T}_{\boldsymbol{x}} I(x_3,x_1,h_{x_1}),
\end{equation} 
where $\gamma$ is the reflection opening angle shown in Figure~\ref{fig:Figure1}. 
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\plot{Figure1}{width=\textwidth}{Cartoon illustrating the geometry of the ADCIG calculation.  Parameter $\gamma$ is the reflection opening angle, $\alpha$ is geologic dip.  a) Cartesian geometry using coordinates $x_1, x_3$ and $h_{x_1}$.  b) Generalized geometry using coordinates $\xi_1, \xi_3$ and $h_{\xi_1}$.  Adapted from \cite{SavaADCIG}. {\bf NR} }  
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Imaging in generalized coordinate systems follows the same two-step procedure.   However, because of the different migration geometry in the $\boldsymbol{\xi}$-coordinate system, new extrapolation operators, $E_{\xi_3}[\cdot]$, must be used to propagate wavefields.  I specify these operators using Riemannian wavefield extrapolation (RWE).  I do not discuss RWE herein, and refer readers interested in additional information to 
Chapter~2. 
%\cite{Sava.geo.rwepub} and \cite{ShraggeGEONO}.  
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The first generalized coordinate imaging step is performing wavefield extrapolation
\begin{eqnarray}
E_{\xi_3}[S(\xi_3-\Delta \xi_3,\xi_1|\omega)] &=& S(\xi_3,\xi_1|\omega), \nonumber \\
E^*_{\xi_3}[R(\xi_3-\Delta \xi_3,\xi_1|\omega)] &=& R(\xi_3,\xi_1|\omega),
\end{eqnarray}
where $\Delta \xi_3$ is the extrapolation step increment.  Generalized coordinate images are then constructed by evaluating an imaging condition
\begin{equation} \label{eqn:subsurface}
I(\xi_3,\xi_1,h_{\xi_1}) = \sum_\omega S^*(\xi_3,\xi_1 + h_{\xi_1}|\omega) R(\xi_3,\xi_1 - h_{\xi_1}|\omega),
\end{equation}
where $h_{\xi_1}$ is the $\boldsymbol{\xi}$-coordinate equivalent of Cartesian subsurface offset axis $h_{x_1}$.   The generalized coordinate ADCIG volume is generated by applying an offset-to-angle transformation \new{ ${\bf T}_{\boldsymbol{\xi}}$ }
\begin{equation} \label{eqn:subsurface5}
I(\xi_3,\xi_1,\gamma) = {\bf T}_{\boldsymbol{\xi}} I(\xi_3,\xi_1,h_{\xi_1}).
\end{equation}
Conventional ADCIG volumes can be recovered by sinc interpolating each $I(\xi_3,\xi_1,\gamma)$ image  computed via equation~\ref{eqn:subsurface5} to the final Cartesian coordinate volume.  
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Figure~\ref{fig:Figure2} illustrates this process  using the elliptical coordinate system.  Panel~\ref{fig:Figure2}a shows the BP synthetic velocity model \cite[]{bpdata} with an elliptical mesh overlain.  Note that the salt flanks to the right-side of the model are nearly vertical in Cartesian coordinates.   Panel~\ref{fig:Figure2}b shows the velocity model in panel \ref{fig:Figure2}a interpolated to the elliptical coordinate system.   Importantly, the aforementioned salt flanks in the elliptical coordinate system are nearly horizontal, which should lead to ADCIG calculations more robust than in Cartesian coordinates.   However, proving this assertion requires understanding the differences, if any, between the Cartesian and generalized coordinate offset-to-angle operators, ${\bf T}_{\boldsymbol{x}}$ and ${\bf T}_{\boldsymbol{\xi}}$, in equations~\ref{eqn:subsurface2}~and~\ref{eqn:subsurface5}, respectively.
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\plot{Figure2}{width=\textwidth}{Prestack migration test in elliptical  coordinates.  a) Benchmark synthetic velocity model with an overlying elliptical coordinate system. b) Effective slowness model in the transformed elliptical coordinate system in a).  c) Benchmark synthetic velocity model with a different overlying elliptical coordinate system.  d) Effective elliptical coordinate slowness model for the coordinate system in c). {\bf NR}} 
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\subsection{Cartesian coordinate ADCIGs}
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For constant velocity media in conventional Cartesian geometry, a straightforward link exists between differential changes in the travel time, $t$, of rays connecting the source-reflector and reflector-receiver paths to changes in the subsurface offset, $h_{x_1}$, and depth, $x_3$, coordinates.  Figure~\ref{fig:Figure1}a shows the geometry of these variables.    
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Mathematically, these relationships are
\begin{eqnarray}\label{eqn:dtdh}
\left[ \begin{array}{c}
\dtdhxx \\ \dtdxz
\end{array}  \right]
 = 2 \,s \, \rcos \, \alpha
\left[ \begin{array}{c}
\rsin \, \gamma \\ \rcos \, \gamma
\end{array}  \right],
\end{eqnarray}
where $s$ is slowness, $\alpha$ is reflector dip, and $\gamma$ is the reflection opening angle.  The right-hand-side of equations~\ref{eqn:dtdh} are derived by \cite{SavaADCIG}.  Equations~\ref{eqn:dtdh} can be rewritten as 
\begin{equation} \label{eqn:Angle}
- \left. \dxzdhxx \right|_{x_1,t}  = \left. \dtdhxx \right/ \dtdxz = \rtan \, \gamma, 
\end{equation} 
where the negative sign derives from use of the implicit functions theory \cite[]{ImplicitFunction}. \cite{BiondiSymes} note that Cartesian ADCIGs become pathogenically degenerate in situations where $\dtdxz \rightarrow 0$ (i.e. for steeply dipping structures where $\alpha\rightarrow 90^\circ$ in Figure~\ref{fig:Figure1}).  Vertically oriented structures are, generally, not well imaged in Cartesian coordinates because of limited large-angle propagation in downward extrapolation.  
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Finally, because equation~\ref{eqn:dtdh} has no explicit geometric-dependence, Fourier-based methods can calculate the reflection opening angle directly in the wavenumber domain
\begin{equation} \label{eqn:Fangle} 
\rtan \gamma = - \frac{k_{h_{x_1}}}{k_{x_3}},
\end{equation}
where $k_{h_{x_1}}$ and $k_{x_3}$ are the wavenumbers in the $h_{x_1}$ and ${x_3}$ directions, respectively. 
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\subsection{Generalized coordinate ADCIGs} 
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Figure~\ref{fig:Figure1}b illustrates a scenario similar to that in Figure~\ref{fig:Figure1}a, but for a more general coordinate system.  The reflection opening angle,  $\gamma$, and the reflector dip, $\alpha$, obviously remain unchanged in the subsurface; however, the orientations of the $h_{\xi_1}$ and $\xi_3$ axes used to estimate $\gamma$ now differ. The key question is, which quantities in the ADCIG calculation are affected by this change of variables?  
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Answering this question requires properly formulating the derivative operators, $\frac{\partial}{\partial x_3}$ and $\frac{\partial}{\partial h_{x_1}}$, in equations~\ref{eqn:dtdh} in the generalized coordinate system variables $\boldsymbol{\xi} = [\xi_1,\xi_3]$ and $\boldsymbol{h_{\xi}} = [h_{\xi_1},h_{\xi_3}]$.  
Appendix~E 
%Appendix~A 
shows how these derivatives can be specified using Jacobian change-of-variable arguments.  Assuming that the subsurface-offset axes are formed by uniform wavefield shifts, 
%Appendix~A
Appendix~E 
derives the following expression for generalized coordinate ADCIGs:
\begin{equation} \label{eqn:final3body}
- \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}
Note that if the $\boldsymbol{\xi}$-coordinate system satisfies the Cauchy-Riemann differentiability criteria \cite[]{ComplexAnalysis}  
\begin{equation} \label{eqn:CRcriteriabody}
\dxxdex = \dxzdez \quad \hbox{and} \quad \dxzdex = - \dxxdez,
\end{equation}
equation~\ref{eqn:final3body} then reduces to  
\begin{equation} \label{eqn:final2} 
-\left. \dezdhex\right|_{\xi_1,t} =  \rtan \, \gamma.
\end{equation}
This is the generalized coordinate equivalent of the Cartesian expression in equation~\ref{eqn:dtdh}. A physical meaning of the criteria in equations~\ref{eqn:CRcriteriabody} is that the coordinate system must behave isotropically (i.e. dilatationally and rotationally) in the neighborhood of every grid point.  Three canonical examples, two of which satisfy equations~\ref{eqn:CRcriteriabody}, are discussed in the following section.   
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Similar to Cartesian coordinates, elliptical coordinate ADCIGs become insensitive where structural dips cause $\dtdez \rightarrow 0$.  However, this insensitivity can be minimized when using generalized coordinate systems, because structural dips appear at different angles in different translated meshes.  Figures~\ref{fig:Figure2}c-d illustrate this by showing an elliptical coordinate shifts for a shot-location different than that presented in panels~\ref{fig:Figure2}a-b.  Note the changes in structural dip in the right-hand-side of the elliptical coordinate panels. Thus, while ADCIGs calculated on one elliptical grid may be insensitive to certain structure locally, mesh translation ensures that ADCIGs are sensitive globally.  Imaging steep dips in elliptical coordinates, though, is limited by the accuracy of wide-angle one-way wavefield extrapolation.
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Finally, one may calculate reflection opening angles in the wavenumber domain for coordinate systems satisfying equations~\ref{eqn:CRcriteriabody} 
\begin{equation} \label{eqn:STEP3}
\rtan \, \gamma = - \frac{k_{h_{\xi_1}}}{k_{\xi_3}},
\end{equation}
where $k_{h_{\xi_1}}$ and $k_{\xi_3}$ are the wavenumbers in the $h_{\xi_1}$
and ${\xi_3}$ directions, respectively.  While some non-orthogonal coordinate systems might satisfy equations~\ref{eqn:CRcriteriabody}, most practical applications will have orthogonal $k_{h_{\xi_1}}$ and $k_{\xi_3}$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Canonical Examples} 
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This section presents three canonical examples that illustrate the generalized ADCIG theory: tilted Cartesian, polar, and elliptical  coordinate systems.  Figure~\ref{fig:Figure3} presents schematic examples of these three coordinate systems.
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\plot{Figure3}{width=\textwidth}{Canonical coordinate system examples. a) Tilted Cartesian coordinates. b) Polar coordinates. c) elliptical coordinates. {\bf NR} }  
%
\subsection{Tilted Cartesian coordinates} 
%
Tilted Cartesian coordinates are a useful generalized migration coordinate system (see Figure~\ref{fig:Figure3}a). \cite{SEG2004sgj} use this mesh in a plane-wave migration scheme where the coordinate system is oriented toward the plane-wave take-off angle to improve large-angle propagation accuracy.  A tilted Cartesian mesh is defined by  
\begin{eqnarray}
\left[ \begin{array}{c}
x_1\\
x_3 
\end{array}  \right] = 
\left[ \begin{array}{cc}
\rcos \, \theta &-\rsin \,\theta \\
\rsin \, \theta & \rcos \,\theta
\end{array}\right]
\left[ \begin{array}{c}
\xi_1 \\
\xi_3
\end{array} \right],
\end{eqnarray}
where $\theta$ is the tilt angle.  The partial derivative transform matrix is  
\begin{eqnarray}
\left[ \begin{array}{cc}
\dxxdex & \dxxdez \\
\dxzdex & \dxzdez
\end{array} \right] = 
\left[ \begin{array}{cc}
\rcos \, \theta &-\rsin \, \theta \\
\rsin \, \theta & \rcos \, \theta
\end{array}\right],
\end{eqnarray}
which leads to the following ADCIG equation:
\begin{equation}
- \left. \dezdhex\right|_{\xi_1,t}   =  \rtan \, \gamma \, \frac{\left( \rcos \,\theta\, \rcos \, \alpha  +\rsin \,\theta\, \rsin \, \alpha \right) }  {\left(   \rcos \,\theta\, \rcos \, \alpha  +\rsin \,\theta\, \rsin \, \alpha \right) } =  \rtan \, \gamma.
\end{equation}
Thus, calculating ADCIGs in tilted Cartesian coordinates directly recovers the correct reflection opening angle.   Note that setting $\theta=0^\circ$ recovers the Cartesian expression in equation~\ref{eqn:Angle}. 
%
\subsection{Polar coordinates} 
%
The polar coordinate system (see Figure~\ref{fig:Figure3}b), where the extrapolation direction is oriented along the radial direction, is appropriate for generating 2D Green's function estimates.  The polar coordinate system is defined by 
\begin{eqnarray}
\left[ \begin{array}{c}
x_1\\
x_3 
\end{array}  \right] = 
\left[ \begin{array}{c}
a\, \xi_3 \,\rcos \,\xi_1 \\
a\,\xi_3\, \rsin \,\xi_1 \\
\end{array} \right].
\end{eqnarray} 
The partial derivative transformation matrix is 
\begin{eqnarray} \label{eqn:testy}
\left[ \begin{array}{cc}
\dxxdex & \dxxdez \\
\dxzdex & \dxzdez
\end{array} \right] = 
\left[ \begin{array}{cc}
-a \,\xi_3\, \rsin\, \xi_1  & a \, \rcos\, \xi_1 \\
 a \,\xi_3 \,\rcos \,\xi_1  & a \, \rsin\, \xi_1
\end{array}\right],
\end{eqnarray}
which leads to the following ADCIG equation:
\begin{equation}
- \left. \dezdhex\right|_{\xi_1,t}   = \xi_3 \, \rtan \, \gamma \, \frac{ \left(  -\rsin\,\xi_1\,\rcos\,\alpha - \rcos\,\xi_1\,\rsin\,\alpha  \right) } {\left(  \rsin\,\xi_1\rcos\,\alpha+\rcos\,\xi_1 \,\rsin,\alpha  \right) }  = - \xi_3 \,\rtan \, \gamma.
\end{equation}
Thus, one cannot calculate ADCIGs directly with Fourier-based methods in polar coordinates because of the spatial geometric dependence on $\xi_3$.  However, polar-coordinate ADCIGs can be calculated using slant-stack processing in the trace domain \cite[]{PruchaADCIG}, because the geometric factor $\xi_3$ is no more than a local weight applied to the velocity model used to calculate the angle gathers.
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\subsection{Elliptical coordinates} 
%
Elliptical coordinates (see Figure~\ref{fig:Figure3}c) are a useful coordinate system for performing 2D shot-profile migration 
(see Chapter 2).
%\cite[]{ShraggeGEONO}. 
An elliptical mesh is defined by
\begin{eqnarray}
\left[ \begin{array}{c}
x_1\\
x_3 
\end{array}  \right] = 
\left[ \begin{array}{c}
a\, {\rm cosh}\, \xi_3 \, \rcos \,\xi_1\\
a\, {\rm sinh}\, \xi_3 \, \rsin \,\xi_1
\end{array} \right].
\end{eqnarray}
The partial derivative transformation matrix is
\begin{eqnarray}
\left[ \begin{array}{cc}
\dxxdex & \dxxdez \\
\dxzdex & \dxzdez
\end{array} \right] = 
a \left[ \begin{array}{cc}
 {\rm cosh} \,\xi_3 \, \rsin  \,\xi_1 & {\rm sinh} \,\xi_3 \, \rcos \,
 \xi_1\\  
-{\rm sinh} \,\xi_3 \, \rcos \,\xi_1 & {\rm cosh} \,\xi_3 \, \rsin
 \,\xi_1 
\end{array}\right],
\end{eqnarray}
which leads to the following ADCIG equation:
\begin{equation} \label{eqn:ELLIP}
- \left. \dezdhex\right|_{\xi_1,t}   =  \rtan
\, \gamma \, \frac{
\left( 
 {\rm cosh} \,\xi_3 \, \rsin \,\xi_1 \, \rcos \,\alpha 
- {\rm sinh} \,\xi_3 \, \rcos \,\xi_1 \, \rsin \,\alpha
\right) } {\left( 
  {\rm cosh} \,\xi_3 \, \rsin \,\xi_1 \, \rcos \,\alpha 
- {\rm sinh} \,\xi_3 \, \rcos \,\xi_1 \, \rsin \,\alpha 
 \right) } = \rtan \, \gamma.
\end{equation}

Thus, calculating ADCIGs in elliptical coordinates with Fourier-based methods will directly recover the true reflection opening angle.  

\section{Numerical Examples} 
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This section presents numerical tests of the generalized theory by comparing the ADCIG volumes for elliptical and Cartesian coordinate systems.  I generated the results using a shot-profile migration algorithm altered to account for elliptical geometry by replacing the Cartesian extrapolator, $E_{x_3}[\cdot]$,  with an elliptical coordinate operator, $E_{\xi_3}[\cdot]$.  In both coordinate systems, the implemented extrapolation operators were accurate to roughly $\pm 80^\circ$ \cite[]{GEO50-10-16341637} with respect to the extrapolation axis.  The reader is referred to 
Chapter 3
%\cite{ShraggeGEORWE} 
for further implementation details on shot-profile migration in elliptical coordinates.
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I calculated ADCIG image volumes for each shot-profile by following a three-step procedure: 1) generate image volume $I(\xi_1,\xi_3,h_{\xi_1})$ by computing the image for 64 subsurface shifts in $h_{\xi_1}$ at each point in every extrapolation step; 2) calculate ADCIG volume $I(\xi_1,\xi_3,\gamma)$ using the procedure described in \cite{SavaADCIG}; and 3) output the image $I(x_1,x_3,\gamma)$ by sinc-based interpolation of single-shot ADCIGs to the global volume.   Steps 1-3 were repeated for all shot-profile sections contributing to the final image.
%
\subsection{Test 1: Elliptical Isochrons}
%
The first test demonstrates the impulse response of the ADCIG imaging operator. I do this by imaging the elliptical isochron responses of two offset point sources in a constant velocity medium (see Figure~\ref{fig:Figure4}).  The source wavefield impulse is at time $t=0$~s and at $x=-1.12$~km, while the receiver wavefield impulses are at $t=2.0$, 2.5, 3.0, and 3.5~s and $x=1.12$~km for a total source-receiver offset of $2h=2.24$~km.  Correlating these two wavefield volumes leads to four elliptical isochrons specified by
\begin{eqnarray}
x_1 (t,h) & = & \frac{vt}{4} \left( \rcos \, \theta_1 + \rcos \, \theta_2 \right) \\
x_3 (t,h) & = & \frac{vt}{4} \left(1 - \frac{h^2}{v^2t^2} \right) 
\left( \rsin \, \theta_1 + \rsin \, \theta_2 \right),
\end{eqnarray}
where, given the source take-off angle $\theta_1$, the receiver take-off angle $\theta_2$ is obtained by
\new{
\begin{equation}
\theta_2 = {\rm cos}^{-1} \left[ \frac{ {\rm cos}\,\theta_1 (\phi^2+1)-2\phi} {\phi^2- 2 \phi\,{\rm cos}\,\theta_1+1} \right]. 
%\theta_2 = {\rm cos}^{-1} \left( \frac{vt}{2h}  \left[ \frac{1}{\frac{2}{1-\frac{4h^2}{v^2t^2}} - %\frac{1}{1-\frac{2h \rcos\,\theta_1}{vt}} } -1 \right]  \right). 
\end{equation}
where $\phi=2h/vt$. }  The reflection opening angle, given by $\gamma = 90 - \frac{(\theta_1 +\theta_2)}{2}$, is shown color-coded on the scatterplot in Figure~\ref{fig:Figure4}. 
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\plot{Figure4}{width=\textwidth}{Theoretical results for an elliptical isochron for four travel times in a constant velocity medium.  The elliptical surfaces are color-coded according to reflection opening angle. {\bf NR}}
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%
Figure~\ref{fig:Figure5} shows the ADCIG volumes calculated in both (a single) elliptical and Cartesian coordinate system.  To generate this image, I first calculated an elliptical coordinate (EC) $I(\xi_1,\xi_3,h_{\xi_1})$ volume by correlating the source and receiver wavefields at 64 subsurface shifts in $h_{\xi_1}$ at each point in every extrapolation step.  I then input this volume to a Fourier-based offset-to-angle transformation operator to generate the EC ADCIG volume, $I(\xi_1,\xi_3,\gamma)$, which I interpolated to Cartesian coordinates to generate the desired image volume, $I(x_1,x_3,\gamma)$.  For the ADCIG transformation, I choose to limit the maximum opening angle to $\gamma =\pm 45^\circ$.  
%These 64 image shifts lead to ADCIGs with an angular range between $-45^\circ < \gamma < 45^\circ$ %with a sampling increment of $\Delta \gamma = 1.875^\circ$.}
%
\par
%
Panel \ref{fig:Figure5}a presents the elliptical coordinate image extracted at $\gamma=-24^\circ$ from the ADCIG volume $I(x_1,x_3,\gamma)$.  The ADCIG volumes consist of elliptically shaped reflectors (panels~\ref{fig:Figure5}a) that, ideally, are localized in the angle domain (panels~\ref{fig:Figure5}b-c and f-h).  The analytical ADCIG locations are represented by black dots.  The breadth of the analytic ADCIG response derives from the coarse binning in the angular domain.  The analytically and numerically generated results are well matched.  Figure~\ref{fig:Figure5}a also shows three vertical lines indicating the locations from left to right of the three ADCIGs in panels \ref{fig:Figure5}b-d.  Again, the analytic and numerical ADCIGs are well matched, though less so at shallower depths due to the increased spreading about  the image focal point.
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\par
%
The Cartesian coordinate (CC) ADCIGs are presented in panels~\ref{fig:Figure5}e-h.  Panel \ref{fig:Figure5}e shows the Cartesian image again extracted at $\gamma=-24^\circ$ in the angle domain.  Panels \ref{fig:Figure5}f-h present three ADCIGs at the same locations as in panels \ref{fig:Figure5}b-d.  The Cartesian image volumes are well-matched to the elliptical coordinate examples, and good agreement between the theoretical results and the wavefield volume is observed in both images.  Energy is focused in the neighborhood of the correct locations. The angle gathers are not always centered relative to the true location, though, which is more noticeable at shallower depths where the Cartesian and elliptical ADCIG volumes both overestimate the reflection opening angle.  
%
\plot{Figure5}{width=\textwidth,height=4in}{Elliptical reflector comparison tests between analytically (black bullets) and numerically generated ADCIG volumes.  Panels a-d are computed in elliptical coordinates (EC), while panels e-h are in Cartesian coordinates (CC).  a) EC image extracted at the -24$^\circ$ reflection angle.  b) EC angle gather at -1.5~km. c) EC gather at 0.75~km. d) EC gather at 0.0~km.  e) CC image extracted at the -24$^\circ$ reflection angle.  f) CC angle gather at -1.5~km. g) CC gather at 0.75~km. h) CC gather at 0.0~km. {\bf ER}} 
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\par
%
Figure~\ref{fig:Figure6} presents the results of a test similar to that shown in Figure~\ref{fig:Figure5}, but with the velocity model rescaled by \new{0.95}.  Again, the black dots show the location of the true image point (assuming a true velocity model).  Note that the image points in each ADCIG remain well-focused, but shift nearer to the surface and to wider angles.  Thus,  imaging with an overly slow velocity model will generate, as expected, reflectors that exhibit upward curvature at wider angles.

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\plot{Figure6}{width=\textwidth,height=4in}{Elliptical reflector comparison tests between analytically (black bullets) and numerically generated ADCIG volumes using a velocity scaled by factor \new{0.95}.  Panels a-d are computed in elliptical coordinates (EC), while panels e-h are in Cartesian coordinates (CC).  a) EC image extracted at the -24$^\circ$ reflection angle.  b) EC angle gather at -1.5~km. c) EC gather at 0.75~km. d) EC gather at 0.0~km.  e) CC image extracted at the -24$^\circ$ reflection angle.  f) CC angle gather at -1.5~km. g) CC gather at 0.75~km. h) CC gather at 0.0~km. {\bf ER}} 
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%A second observation from Figure~\ref{fig:Figure5} is that while elliptical coordinate ADCIGs focus at the correct reflection opening angle, they tend to have greater residual curvature away from the correct image position than in the Cartesian coordinate images.  Figure~\ref{fig:AngleExplanation} illustrates why this occurs using sketches of near- and far-offset shifts for Cartesian, tilted Cartesian, and elliptical coordinate systems.  Each panel has two sets of  perpendicular axes, the intersection of each indicating the points being cross-correlated.  The red lines show the length of the $\xi_1$ coordinate axis, while the blue lines give the Euclidean distance between the two correlated points.   
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%\sideplot{AngleExplanation}{width=\textwidth}{Sketches illustrating wavefield shift differences between elliptical and (tilted) Cartesian coordinates for near- and far-offset shifts.  The intersection points of the black and gray perpendiculars indicate the points being correlated.  Red lines show the length along the coordinate system axes, while blue lines relate the Euclidean distance between the two points.  Red and blue lines are equal for Cartesian and tilted Cartesian coordinate systems for both near- and far-offset shifts, but are unequal at far offsets for the elliptical coordinates, which generates residual curvature.}  
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%In the Cartesian and tilted Cartesian panels, the red and blue line lengths are the same for both near- and far-offsets.  However, the blue and red lines in the elliptical panel are not the same length and become increasingly disparate at greater offsets.  The distance along the $\xi_1$ axis is thus greater than the Euclidean distance, leading to a greater apparent  velocity and downward residual ADCIG curvature.  However, as the next tests demonstrate, this residual curvature is largely stacked out in the final volumes and does not contaminate the ADCIG results.
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\subsection{Test 2: BP velocity model}
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The second test compares elliptical and Cartesian coordinate ADCIG volumes computed for the BP synthetic velocity model.  Images computed in elliptical coordinates used only one coordinate system per shot.  For the one-sided data set, I used a (surface) migration aperture of 12~km and located the source and farthest offset receiver (at 8~km) points 2~km in from the edges of the computational mesh.  (Note that the migration aperture effectively expands during wavefield extrapolation because the coordinate mesh expands outward.)  I found this initial migration geometry to produce the best results for the BP synthetic model through iterative testing.  Generally, the optimal elliptical coordinate migration geometry is controlled by the velocity model.
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Figure~\ref{fig:Figure7} shows slices all clipped at the 99$^{th}$ percentile from the corresponding elliptical and Cartesian ADCIG image volumes.
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\plot{Figure7}{width=\textwidth,height=3.5in}{Vertical elliptical and Cartesian ADCIGs slices using the correct migration velocity model.  a) elliptical coordinate image with three vertical lines showing the locations of ADCIG gathers from left to right in panels b-d.  e) Cartesian coordinate image with three vertical lines showing the locations of ADCIG gathers from left to right in panels f-h.  {\bf CR}}
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Panel~\ref{fig:Figure7}a shows an elliptical coordinate image with three vertical lines indicating the angle-gather locations from left to right in Figures~\ref{fig:Figure7}b-d. The three panels show flat reflectors with predominantly one-sided reflectivity.  This result is expected because the input migration data were not in a split-spread geometry.   (This statement holds for all subsequent images calculated using this data set.) The only significant exceptions occur in panel~\ref{fig:Figure7}b within salt where energy is exhibited for both positive and negative reflection angles.  I attribute this to the reversal of source and receiver wavefield orientations within the salt.  Note also that the curved events in the ADCIG volumes are the images of the multiple reflections off of the water bottom and top salt.
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The image in panel~\ref{fig:Figure7}d has a wide reflection zone between 3.75-4.25~km in depth, which occurs because the shown angle gather is a vertical slice through the nearly vertical salt flank.  This creates the appears of low-frequency noise, which is the appropriate response for a near-vertical reflector.  Panel~\ref{fig:Figure7}e shows the Cartesian image for the same location as panel~\ref{fig:Figure7}a, while panels~\ref{fig:Figure7}f-h are extracted from the same locations as panels~\ref{fig:Figure7}b-d.  The Cartesian angle gathers look similar to those in elliptical coordinates, except for the salt flanks to the right-hand-side of panel~\ref{fig:Figure7}h. 
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A final observation from Figure~\ref{fig:Figure7} is that ADCIGs calculated via subsurface correlations will generate artifacts at locations near salt-sediment interfaces - whether in an elliptical or a Cartesian coordinate system.  This geologic setting leads to situations where a wavefield sample inside a salt body is correlated with another sample located in the sediment with a significantly different velocity.  This significant velocity difference violates one of the theoretical ADCIG assumptions, namely that the velocity remains constant across the correlation window.  Hence, one must be careful not to interpret ADCIG artifacts as signal useful for migration velocity analysis. 
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Figure~\ref{fig:Figure8} shows horizontal slices through the image volume that better resolve the vertical salt flank.  Panel~\ref{fig:Figure8}a presents the elliptical coordinate image, with three horizontal lines showing the ADCIG slice locations from top to bottom.  The right-hand sides of panels~\ref{fig:Figure8}b-d display the well-focused vertical salt-flank reflector.  This demonstrates the robustness of the ADCIG calculation in elliptical coordinate systems.
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\plot{Figure8}{width=\textwidth,height=7in}{Horizontal elliptical and Cartesian ADCIGs slices using the correct migration velocity model.  a) elliptical coordinate image with three horizontal lines showing the locations of horizontal ADCIG gathers from top to bottom in panels b-d.  e) Cartesian coordinate image with three horizontal lines showing the locations of horizontal ADCIG gathers  from top to bottom in panels f-h. {\bf CR}}
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Panel~\ref{fig:Figure8}e shows the Cartesian coordinate image with three horizontal lines showing the locations of the ADCIG slices.  The right-hand salt-flank reflector in panel~\ref{fig:Figure8}f is similarly well-resolved, largely because the structural dip is relatively low.  However, the salt-flank images in panels~\ref{fig:Figure8}g-h are somewhat blurred out.  I attribute this to the combined effects of inaccurate large-angle extrapolation and insensitivity of the ADCIG calculation to steep structural dip.
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An additional test examines how the ADCIG volumes change when introducing an incorrect migration velocity profile.  Figure~\ref{fig:Figure9} presents ADCIG volumes similar to those shown in Figure~\ref{fig:Figure7} after using a migration velocity profile rescaled by \new{0.95}.  
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\plot{Figure9}{width=\textwidth,height=3.5in}{Vertical elliptical and Cartesian ADCIGs slices using an incorrect  migration velocity model.  a) elliptical coordinate image with three vertical lines showing the locations of vertical ADCIG gathers  from left to right  in panels b-d.  e) Cartesian coordinate image with three vertical lines showing the locations of vertical ADCIG gathers  from left to right  in panels f-h. {\bf CR}}
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Both images are poorly focused and have residual curvature indicating an incorrect migration velocity.  Because the reflectors are near vertical, though, the sensitivity of horizontal gathers is weak.  This low sensitivity is greatly improved when examining the horizontal slices in Figure~\ref{fig:Figure10} taken at the same locations as in Figure~\ref{fig:Figure8}.
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\plot{Figure10}{width=\textwidth,height=7in}{Horizontal elliptical and Cartesian ADCIGs slices using an incorrect migration velocity model.  a) elliptical coordinate image with three horizontal lines showing the locations of horizontal ADCIG gathers  from top to bottom in panels b-d.  e) Cartesian coordinate image with three horizontal lines showing the locations of horizontal ADCIG gathers  from top to bottom in panels f-h. {\bf CR}}
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The elliptical angle gathers in panels~\ref{fig:Figure10}b-d, and especially to the right-hand side in panel~\ref{fig:Figure10}d, show much greater residual curvature.  This indicates that the elliptical coordinate horizontal ADCIGs have greater sensitivity to velocity error for near-vertical structures  than Cartesian coordinate horizontal ADCIGs.  The imaging enhancements afforded by elliptical coordinates should improve any migration velocity analysis approach that uses residual curvature in steeply dipping reflectors to compute velocity model updates.
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\subsection{Discussion}
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Extending the above theory of generalized coordinate ADCIGs to 3D coordinate systems should be fairly straightforward, though more difficult to implement numerically.  \cite{Sergey3DADCIG} presents a theory for 3D Cartesian coordinates that specifies the differential travel-time expressions required to express the reflection opening angle, $\gamma$, in 3D Cartesian ADCIGs [see equation~16 in \cite{Sergey3DADCIG}].  Applying Jacobian change-of-variable transformations to these equations should yield a 3D expression for reflection angle. Similar to 3D Cartesian coordinates, though, this quantity will depend on geologic dips and needs to be computed by one of the two algorithms suggested in \cite{Sergey3DADCIG}.
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Given that a 3D expression can be formulated, there are a number of coordinate systems well-suited to imaging steep geologic dips where 3D ADCIG volumes could be a good diagnostic tool for velocity analysis.  \cite{ShraggeGEORWE} discusses how a judicious choice of 3D coordinate system depends greatly on the acquisition geometry and the desired migration geometry.  For example, the migration geometries employed in shot-profile migration of wide-azimuth data sets are well-matched with 3D ellipsoidal meshes that enable high-angle and turning-wave propagation in all directions.  Evaluating ADCIG image focussing in such a 3D coordinate geometry would then provide information on velocity model accuracy for steeply dipping reflectors - such as salt flanks.  These somewhat speculative extensions, though, are beyond the scope of this 
chapter
%paper 
and remain an active area of research.
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\section{Conclusions}
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I extend the Cartesian ADCIG theory to 2D generalized coordinate systems.  The generalized ADCIG expressions related the reflection opening angle to differential travel-time operators and spatially varying weights derived from the non-Cartesian geometry.  I show that these geometric expressions cancel out for coordinate systems satisfying the Cauchy-Riemann differentiability criteria,  which include tilted Cartesian and elliptical meshes.  The procedure for calculating ADCIGs in elliptical coordinates is very similar to that in Cartesian coordinates.  I validate the approach by comparing analytically and numerically generated ADCIG volumes, and with tests on the BP synthetic data set.  ADCIGs calculations are more robust where computed in elliptical coordinates than in Cartesian coordinate.  I assert that this result is due to improved large-angle propagation and enhanced sensitivity to steep structural dips afforded by the coordinate transforms.  Finally, the imaging advantages afforded by elliptical coordinates should improve the procedure of any migration velocity analysis approach that uses residual ADCIG curvature on steeply dipping reflectors to compute velocity model updates.
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\section{Acknowledgments}
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I would like to thank Biondo Biondi, Alejandro Valenciano, Antoine Guitton and Brad Artman for helpful discussions, and BP for releasing the synthetic data used in the examples.  I thank Tamas Nemeth, Samuel Gray, and an anonymous reviewer for helpful comments on the early stages of this manuscript. 

