equations~(\ref{eqn:snells1}) and (\ref{eqn:snells2}) to the form\begin{eqnarray}\label{eqn:connection1}\tau_n \, {{\partial \tau_n} \over {\partial y}} & = & \tau \, {{\partial\tau} \over {\partial y}} \,=\, 4h\,{{\sin{\alpha} \cos{\alpha}\cot{\gamma}} \over {v^2}}\;; \\ \tau_n \, {{\partial \tau_n} \over {\partial h}} & = &\tau \, {{\partial\tau} \over {\partial h}} - {{4h} \over {v^2}} \,=\,-\,4h\,{{\sin^2{\alpha}} \over {v^2}}\;.   \label{eqn:connection2} \end{eqnarray}Without loss of generality, we can assume $\alpha$ to be positive.Consider a plane tangent to a true reflector at the reflectionpoint(Figure \ref{fig:ocobol}).The traveltime of a wave, reflected from the plane, has theknown explicit expression\begin{equation}\tau\,=\,{2 \over v}\,\sqrt{L^2+h^2\,\cos^2{\alpha}}\,\,\,,   \label{eqn:CDP} \end{equation}where $L$ is the length of the normal ray from the midpoint. Asfollows from combining (\ref{eqn:CDP}) and (\ref{eqn:length}),\begin{equation}{\cos{\alpha} \cot{\gamma}} \,=\, {L \over h}   \,\,\,.\label{eqn:ratio} \end{equation}We can now combine equations~(\ref{eqn:ratio}),(\ref{eqn:connection1}), and (\ref{eqn:connection2}) to transforminequality (\ref{eqn:condition}) to the form\begin{equation} h \ll {L \over {\sin{\alpha}}} \,=\, z\, \cot{\alpha}\,\,,   \label{eqn:hm} \end{equation}where $z$ is the depth of the plane reflector under the midpoint.  Forexample, for a dip of 45 degrees, equation~(\ref{eqn:bolondi}) will besatisfied only for offsets that are much smaller than the depth of thereflector.\sideplot{ocobol}{height=2.5in}{Reflection rays andtangent to the reflector in a constant velocity medium (a scheme).}  \subsection{Offset continuation geometry: time rays}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%To study the laws of traveltime curve transformation in the OCprocess, it is convenient to apply the method of characteristics\cite[]{kurant} to the eikonal-type equation~(\ref{eqn:eikonal}). Thecharacteristics of equation~(\ref{eqn:eikonal}) [{\em  bi}-characteristics with respect to equation (\ref{eqn:OCequation})]are the trajectories of the high-frequency energy propagation in theimaginary OC process. Following the formal analogy with seismic rays,I call those trajectories {\em time rays}, where the word {\em time}refers to the fact that the trajectories describe the traveltimetransformation \cite[]{me}.  According to the theory of first-orderpartial differential equations, time rays are determined by a set ofordinary differential equations (characteristic equations) derivedfrom equation (\ref{eqn:eikonal}) :\begin{eqnarray}{{{dy} \over {dt_n}}   =  - {{2 h Y} \over {t_n H}}}\;,\; {{{dY} \over {dt_n}}  =  {Y \over t_n}}\;, \nonumber \\{{{dh} \over {dt_n}}  =  {-{1 \over H}+{{2 h} \over t_n}}}\;,\;{{{dH} \over {dt_n}} = {{Y^2} \over {t_n H}}}\;, \label{eqn:rays} \end{eqnarray}where $Y$ corresponds to $\partial \tau_n \over \partial y$ along aray and $H$ corresponds to $\partial \tau_n \over \partial h$. In thisnotation, equation~(\ref{eqn:eikonal}) takes the form\begin{equation}h\, (Y^2-H^2) = -\, t_n H \label{eqn:rayeikonal} \end{equation}and serves as an additional constraint for the definition of timerays.  System~(\ref{eqn:rays}) can be solved by standard mathematicalmethods \cite[]{ode}. Its general solution takes the parametric form,where the time variable $t_n$ is the parameter changing along a timeray:\begin{eqnarray}y(t_n)  =  C_1-C_2\,t_n^2 \; & ; & \;h(t_n)=t_n \sqrt{C_2^2 t_n^2 + C_3}\;;\\ Y(t_n)  =  {{C_2\,t_n}\over C_3}\; & ; & \;H(t_n)={h \over {C_3\,t_n}}\label{eqn:ray} \end{eqnarray}and $C_1$, $C_2$, and $C_3$ are independent coefficients, constantalong each time ray. To find the values of these coefficients, we canpose an initial-value problem for the system of differentialequations~(\ref{eqn:rays}).  The traveltime curve $\tau_n(y;h)$ for agiven common offset $h$ and the first partial derivative $\partial\tau_n \over \partial h$ along the same constant offset sectionprovide natural initial conditions. A particular case of thoseconditions is the zero-offset traveltime curve. If the first partialderivative of traveltime with respect to offset is continuous, itvanishes at zero offset according to the reciprocity principle(traveltime must be an even function of the offset):\begin{math}t_0\left(y_0\right)=\tau_n(y;0), \left. {\partial \tau_n \over \partial h} \right|_{h=0}=0\,.  \end{math}Applying the initial-value conditions to the generalsolution~(\ref{eqn:ray}) generates the following expressions for theray invariants:\begin{eqnarray}C_1 & = & y+h\,{Y \over H}=y_0-{t_0\left(y_0\right) \overt_0'\left(y_0\right)}\;;\;C_2={{h\,Y} \over {\tau_n^2\,H}}=-{1 \over t_0\left(y_0\right)\,t_0'\left(y_0\right)}\;;\;\nonumber \\C_3 & = & {h \over {\tau_n\,H}}=-{1 \over \left(t_0'\left(y_0\right)\right)^2}\;,\label{eqn:abc} \end{eqnarray}where $t_0'\left(y_0\right)$ denotes the derivative$\frac{d\,t_0}{d\,y_0}$.  Finally, substitutingequations~(\ref{eqn:abc})into~(\ref{eqn:ray}), we obtain an explicit parametric form of the raytrajectories:\begin{eqnarray}\label{eqn:yhray1}y_1\left(t_1\right)  & = & \displaystyle{y+{{h\,Y} \over{t_n^2\,H}}\,\left(t_n^2-t_1^2\right)= y_0+{{t_1^2-t_0^2\left(y_0\right)} \over{t_0\left(y_0\right)\,t_0'\left(y_0\right)}}}\;;\\h_1^2\left(t_1\right) & = & \displaystyle{{{h\,t_1^2} \over {t_n^3\,H}}\,\left(t_n^2+t_1^2\,{{h\,Y^2} \over {t_n\,H}}\right)=t_1^2\,{{t_1^2-t_0^2\left(y_0\right)} \over{\left(t_0\left(y_0\right)\,t_0'\left(y_0\right)\right)^2}}}\;.\label{eqn:yhray2}\end{eqnarray}Here $y_1$, $h_1$, and $t_1$ are the coordinates of the continuedseismic section. Equations~(\ref{eqn:yhray1}) indicatesthat the time ray projections to a common-offset section have aparabolic form. Time rays do not exist for $t_0'\left(y_0\right)=0$ (alocally horizontal reflector) because in this case post-NMO offsetcontinuation transform is not required.The actual parameter thatdetermines a particular time ray is the reflection point location.This important conclusion follows from the known parametric equations\begin{eqnarray}\label{eqn:gur1}t_0(x) & = & \displaystyle{t_v \sec{\alpha}=t_v(x)\,\sqrt{1+u^2\left(t_v'(x)\right)^2}}\;, \\y_0(x) & = & \displaystyle{x+u t_v\tan{\alpha} =x+u^2\,t_v(x)t_v'(x)}\;,\label{eqn:gur2}\end{eqnarray}where $x$ is the reflection point, $u$ is half of the wave velocity ($u=v/2$), $t_v$ is the vertical time (reflector depth divided by $u$), and$\alpha$ is the local reflector dip. Taking into account that the derivative of the zero-offsettraveltime curve is\begin{equation}{{dt_0}\over{dy_0}}={{t_0'(x)}\over{y_0'(x)}}={{\sin{\alpha}}\over u}={{t_v'(x)} \over \sqrt{1+u^2\left(t_v'(x)\right)^2}}\label{eqn:gurtx}\end{equation}and substituting equations~(\ref{eqn:gur1}) and~(\ref{eqn:gur2})into~(\ref{eqn:yhray1}) and~(\ref{eqn:yhray2}), we get\begin{eqnarray}\label{eqn:rayrp1}y_1\left(t_1\right) & = &\displaystyle{x+{{t_1^2-t_v^2\left(x\right)} \over{t_v\left(x\right)\,t_v'\left(x\right)}}}\;;\\u^2 t^2\left(t_1\right) & = & \displaystyle{t_1^2\,{{t_1^2-t_v^2\left(x\right)} \over{\left(t_v\left(x\right)\,t_v'\left(x\right)\right)^2}}}\;,\label{eqn:rayrp2}\end{eqnarray}where $t^2\left(t_1\right)=t_1^2+h_1^2\left(t_1\right)/u^2$.To visualize the concept of time rays, let us consider some simpleanalytic examples of its application to geometric analysis of theoffset-continuation process.\subsubsection{Example 1: plane reflector}\inputdir{Math}The simplest and most important example is the case of a plane dippingreflector. Putting the origin of the $y$ axis at the intersection ofthe reflector plane with the surface, we can express the reflectiontraveltime after NMO in the form\begin{equation}\tau_n(y,h)=p\,\sqrt{y^2-h^2}\;,\label{eqn:planett}\end{equation}where $p=2\,{ \sin{\alpha} \over v}$, and $\alpha$ is the dip angle. The zero-offset traveltime in this case is a straight line:\begin{equation}t_0\left(y_0\right)=p\,y_0\;.\label{eqn:planezott}\end{equation}According to equations~(\ref{eqn:yhray1}-\ref{eqn:yhray2}), the timerays in this case are defined by\begin{equation}y_1\left(t_1\right)={t_1^2 \over {p^2\,y_0}}\;;\;h_1^2\left(t_1\right)=t_1^2\,{{t_1^2-p^2\,y_0^2} \over{p^4\,y_0^2}}\;;\;y_0={{y^2-h^2} \over y}\;.\label{eqn:planerays}\end{equation}The geometry of the OC transformation is shown in Figure\ref{fig:ocopln}.\plot{ocopln}{width=6in,height=3in}{Transformation of  the reflection traveltime curves in the OC process: the case of a  plane dipping reflector. Left: Time coordinate before the NMO  correction. Right: Time coordinate after NMO. The solid lines  indicate traveltime curves at different common-offset sections; the  dashed lines indicate time rays.}\subsubsection{Example 2: point diffractor}  The second example is the case of a point diffractor (the left side  of Figure \ref{fig:ococrv}).  Without loss of generality, the origin  of the midpoint axis can be put above the diffraction point. In this  case the zero-offset reflection traveltime curve has the well-known  hyperbolic form\begin{equation}t_0\left(y_0\right)={\sqrt{z^2+y_0^2} \over u}\;,\label{eqn:pointzott}\end{equation}where $z$ is the depth of the diffractor and $u=v/2$ is half of thewave velocity. Time rays are defined according toequations~(\ref{eqn:yhray1}-\ref{eqn:yhray2}), as follows:\begin{equation}y_1\left(t_1\right)={{u^2\,t_1^2-z^2} \over y_0}\;;\;u^2\,t^2\left(t_1\right)=u^2\,t_1^2+h_1^2\left(t_1\right)=u^2\,t_1^2\,{{u^2\,t_1^2-z^2} \over y_0^2}\;.\label{eqn:pointrays}\end{equation}\plot{ococrv}{width=6in,height=3in}{Transformation of  the reflection traveltime curves in the OC process. Left: the case  of a diffraction point. Right: the case of an elliptic reflector.  Solid lines indicate traveltime curves at different common-offset  sections, dashed lines indicate time rays.}\subsubsection{Example 3: elliptic reflector}The third example (the right side of Figure \ref{fig:ococrv}) is thecurious case of a focusing elliptic reflector. Let $y$ be the centerof the ellipse and $h$ be half the distance between the foci of theellipse. If both foci are on the surface, the zero-offsettraveltime curve is defined by the so-called ``DMO smile''\cite[]{GPR29-03-03740406}:\begin{equation}t_0\left(y_0\right)={t_n \over h}\,\sqrt{h^2-\left(y-y_0\right)^2}\;,\label{eqn:smilezott}\end{equation} where $t_n=2\,z/v$, and $z$ is the small semi-axis of the ellipse.The time-ray equations are\begin{equation}y_1\left(t_1\right)=y+{h^2\over {y-y_0}}\,{{t_1^2-t_n^2} \over t_n^2}\;;\;h_1^2\left(t_1\right)=h^2\,{t_1^2 \over t_n^2}\,\left(1+{h^2\over \left(y-y_0\right)^2}\,{{t_1^2-t_n^2} \over t_n^2}\right)\;.\label{eqn:smilerays}\end{equation}When $y_1$ coincides with $y$, and $h_1$ coincides with $h$, thesource and the receiver are in the foci of the elliptic reflector, andthe traveltime curve degenerates to a point $t_1=t_n$. This remarkablefact is the actual basis of the geometric theory of dip moveout\cite[]{GPR29-03-03740406}.\subsection{Proof of amplitude equivalence} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Let us now consider the connection between the laws of traveltimetransformation and the laws of the corresponding amplitudetransformation.  The change of the wave amplitudes in the OC processis described by the first-order partial differential transportequation~(\ref{eqn:transport}). We can find the general solution ofthis equation by applying the method of characteristics. The solutiontakes the explicit integral form\begin{equation}A_n\left(t_n\right)=A_0\left(t_0\right)\,\exp{\left(\int_{t_o}^{t_n}\left[h\,\left({\partial^2 \tau_n \over \partial y^2}-{\partial^2 \tau_n \over \partial h^2}\right)\,\left(\tau_n\,{\partial \tau_n \over \partial h} \right)^{-1}\right]\,d\tau_n\right)}\;.\label{eqn:ampint}\end{equation}The integral in equation~(\ref{eqn:ampint}) is defined on a curvedtime ray, and $A_n(t_n)$ stands for the amplitude transported alongthis ray. In the case of a plane dipping reflector, the ray amplitudecan be immediately evaluated by substituting the explicit traveltimeand time ray equations from the preceding sectioninto~(\ref{eqn:ampint}). The amplitude expression in this case takesthe simple form\begin{equation}A_n\left(t_n\right)=A_0\left(t_0\right)\,\exp{\left(-\int_{t_o}^{t_n}\frac{d\tau_n}{\tau_n}\right)} = A_0\left(t_0\right)\,{t_0 \over t_n}\;.\label{eqn:ampplane}\end{equation}In order to consider the more general case of a curvilinear reflector,