we need to take into account the connection between the traveltimederivatives in~(\ref{eqn:ampint}) and the geometry of the reflector.As follows directly from the trigonometry of the incident andreflected rays triangle (Figure \ref{fig:ocoray}),\begin{eqnarray}h & = & {r-s \over 2}=D\,{{\cos{\alpha}\,\sin{\gamma}\,\cos{\gamma}} \over{\cos^2{\alpha}-\sin^2{\gamma}}}\;,\label{eqn:geoh}\\y & = & {r+s \over 2}=x+D\,{{\cos^2{\alpha}\,\sin{\alpha}} \over{\cos^2{\alpha}-\sin^2{\gamma}}}\;,\label{eqn:geoy}\\y_0 & = & x+D\,\sin{\alpha}\;,\label{eqn:geoy0}\end{eqnarray}where $D$ is the length of the normal ray. Let $\tau_0=2\,D/v$ be thezero-offset reflection traveltime. Combiningequations~(\ref{eqn:geoh}) and~(\ref{eqn:geoy0}) with(\ref{eqn:length}), we can get the following relationship:\begin{equation}a={\tau_n\over\tau_0}={{\cos{\alpha}\,\cos{\gamma}}\over\left(\cos^2{\alpha}-\sin^2{\gamma}\right)^{1/2}}=\left(1+{{\sin^2{\alpha}\,\sin^2{\gamma}}\over{\cos^2{\alpha}-\sin^2{\gamma}}}\right)^{1/2}={h\,\over\sqrt{h^2-\left(y-y_0\right)^2}}\;,\label{eqn:A}\end{equation}which describes the ``DMO smile''~(\ref{eqn:smilezott}) found by\cite{GPR29-03-03740406} in geometric terms.Equation~(\ref{eqn:A}) allows for a convenient change of variablesin equation~(\ref{eqn:ampint}). Let the reflection angle $\gamma$ be aparameter monotonically increasing along a time ray. In this case,each time ray is uniquely determined by the position of the reflectionpoint, which in turn is defined by the values of $D$ and $\alpha$.According to this change of variables, we candifferentiate~(\ref{eqn:A}) along a time ray to get\begin{equation}{{d\tau_n}\over\tau_n}=-{{\sin^2{\alpha}}\over{2\,\cos^2{\gamma}\,\left(\cos^2{\gamma}-\sin^2{\alpha}\right)}}\,d\left(\cos^2{\gamma}\right)\;.\label{eqn:dt2tg}\end{equation}Note also that the quantity $h\,\left(\tau_n\,{\partial \tau_n \over    \partial h} \right)^{-1}$ in equation~(\ref{eqn:ampint}) coincidesexactly with the time ray invariant $C_3$ found inequation~(\ref{eqn:abc}). Therefore its value is constant along eachtime ray and equals\begin{equation}h\,\left(\tau_n\,{\partial \tau_n \over \partial h}\right)^{-1}=-{v^2 \over 4\, \sin^2{\alpha}}\;.\label{eqn:c3}\end{equation}Finally, as shown in Appendix~\ref{chapter:deriv},\begin{equation}\tau_n\,\left({\partial^2 \tau_n \over \partial y^2}-{\partial^2 \tau_n \over \partial h^2}\right)=4\,{\cos^2{\gamma}\over v^2}\,\left({\sin^2{\alpha}+DK}\over{\cos^2{\gamma}+DK}\right)\;,\label{eqn:curve}\end{equation}where $K$ is the reflector curvature at the reflection point.Substituting (\ref{eqn:dt2tg}),~(\ref{eqn:c3}), and~(\ref{eqn:curve})into~(\ref{eqn:ampint}) transforms the integral to the form\begin{eqnarray*} \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 = \end{eqnarray*}\begin{equation} = -{1 \over 2}\,\int_{\cos^2{\gamma_0}}^{\cos^2{\gamma}}\left({1 \over {\cos^2{\gamma'}-\sin^2{\alpha}}}-{1 \over{\cos^2{\gamma'}+DK}}\right)\,d\left(\cos^2{\gamma'}\right) \label{eqn:int2g}\end{equation}which we can evaluate analytically. The final equation for theamplitude transformation is\begin{eqnarray}A_n & = & A_0\,{\sqrt{\cos^2{\gamma}-\sin^2{\alpha}}\over\sqrt{\cos^2{\gamma_0}-\sin^2{\alpha}}}\,\left({\cos^2{\gamma_0}+DK}\over{\cos^2{\gamma}+DK}\right)^{1/2}=\nonumber \\& = & A_0\,{{\tau_0\,\cos{\gamma}}\over{\tau_n\,\cos{\gamma_0}}}\,\left({\cos^2{\gamma_0}+DK}\over{\cos^2{\gamma}+DK}\right)^{1/2}\;. \label{eqn:ampcurve}\end{eqnarray}In case of a plane reflector, the curvature $K$ is zero, andequation~(\ref{eqn:ampcurve}) coincides with~(\ref{eqn:ampplane}).In the general case can be rewritten as\begin{equation}A_n={{c\,\cos{\gamma}}\over{\tau_n\,\sqrt{\cos^2{\gamma}+DK}}}\;,\label{eqn:ampray}\end{equation}where $c$ is constant along each time ray (it may vary with the reflection pointlocation on the reflector but not with the offset). We should compare equation(\ref{eqn:ampray}) with the known expression for the reflection wave amplitude of the leadingray series term in 2.5-D media \cite[]{cwp}:\begin{equation}A={{C_R(\gamma) \Psi}\over G}\;,\label{eqn:amptrue}\end{equation}where $C_R$ stands for the angle-dependent reflection coefficient, $G$ is thegeometric spreading\begin{equation}G=v \tau {\sqrt{\cos^2{\gamma}+DK}\over \cos{\gamma}}\;,\label{eqn:GS}\end{equation}and $\Psi$ includes other possible factors (such as the sourcedirectivity) that we can either correct or neglect in the preliminaryprocessing.  It is evident that the curvature dependence of theamplitude transformation (\ref{eqn:ampray}) coincides completely withthe true geometric spreading factor (\ref{eqn:GS}) and that the angledependence of the reflection coefficient is not accounted for the offsetcontinuation process. If the wavelet shape of the reflected wave onseismic sections [$R_n$ in equation~(\ref{eqn:raymethod})] isdescribed by the delta function, then, as follows from the knownproperties of this function,\begin{equation}A\,\delta\left(t-\tau(y,h)\right)=\left|{{dt_n} \over {dt}}\right|\,A\,\delta\left(t_n-\tau_n(y,h)\right) ={t \over t_n}\,A\,\delta\left(t_n-\tau_n(y,h)\right)\;, \label{eqn:deltafun} \end{equation}which leads to the equality\begin{equation}A_n=A\,{t \over t_n}\;.\label{eqn:a2an} \end{equation}Combining equation~(\ref{eqn:a2an}) with equations~(\ref{eqn:amptrue})and~(\ref{eqn:ampray}) allows us to evaluate the amplitude aftercontinuation from some initial offset $h_0$ to another offset $h_1$,as follows:\begin{equation}A_1={{C_R(\gamma_0) \Psi_0}\over G_1}\;.\label{eqn:ampfin}\end{equation}According to equation~(\ref{eqn:ampfin}), the OC process described byequation (\ref{eqn:OCequation}) is amplitude-preserving in the sensethat corresponds to the definition of Born DMO\cite[]{born,GEO56-02-01820189}. This means that the geometric spreadingfactor from the initial amplitudes is transformed to the truegeometric spreading on the continued section, while the reflectioncoefficient stays the same. This remarkable dynamic property allowsAVO (amplitude versus offset) analysis to be performed by a dynamiccomparison between true constant-offset sections and the sectionstransformed by OC from different offsets.  With a simple trick, theoffset coordinate is transferred to the reflection angles for the AVOanalysis.  As follows from~(\ref{eqn:A}) and~(\ref{eqn:length}),\begin{equation}{\tau_n^2 \over \tau\,\tau_0}=\cos{\gamma}\;.\label{eqn:AVO}\end{equation}If we include the ${t_n^2 \over t\,t_0}$ factor in the DMO operator(continuation to zero offset) and divide the result by the DMO sectionobtained without this factor, the resultant amplitude of the reflectedevents will be directly proportional to $\cos{\gamma}$, where thereflection angle $\gamma$ corresponds to the initial offset. Ofcourse, this conclusion is rigorously valid for constant-velocity2.5-D media only.\cite{GEO58-01-00470066} suggest a definition of true-amplitude DMO differentfrom that of Born DMO. The difference consists of two importantcomponents:\begin{enumerate}\item {\em True-amplitude DMO addresses preserving the peak amplitude    of the image wavelet instead of preserving its spectral density.}  In the terms of this paper, the peak amplitude corresponds to the  pre-NMO amplitude $A$ from formula~(\ref{eqn:amptrue}) instead of  corresponding to the spectral density amplitude $A_n$.  A simple  correction factor $t \over t_n$ would help us take the difference  between the two amplitudes into account. Multiplication by $t \over  t_n$ can be easily done at the NMO stage.\item {\em Seismic sections are multiplied by time to correct for the    geometric spreading factor prior to DMO (or, in our case, offset    continuation) processing.} \end{enumerate} As followsfrom~(\ref{eqn:GS}), multiplication by $t$ is a valid geometricspreading correction for plane reflectors only.  It is theamplitude-preserving offset continuation based on the OCequation~(\ref{eqn:OCequation}) that is able to correct for thecurvature-dependent factor in the amplitude. To take into account thesecond aspect of Black's definition, we can consider the modifiedfield $\hat{P}$ such that\begin{equation}\hat{P}\left(y,h,t_n\right)=t\,P\left(y,h,t_n\right)\;.\label{eqn:p2pt}\end{equation}Substituting~(\ref{eqn:p2pt}) into the OC equation~(\ref{eqn:OCequation}) transforms thelatter to the form\begin{equation}h \, \left( {\partial^2 \hat{P} \over \partial y^2} - {\partial^2 \hat{P} \over\partial h^2} \right) \, = \, t_n \, {\partial^2 \hat{P} \over {\partial t_n \,\partial h}}\, -{\partial \hat{P} \over \partial h}\; . \label{eqn:BSZequation} \end{equation}Equations~(\ref{eqn:BSZequation}) and~(\ref{eqn:OCequation}) differonly with respect to the first-order damping term $\partial \hat{P}\over \partial h$.  This term affects the amplitude behavior but notthe traveltimes, since the eikonal-type equation~(\ref{eqn:eikonal})depends on the second-order terms only.  Offset continuation operatorsbased on~(\ref{eqn:BSZequation}) conform to Black's definition oftrue-amplitude processing.\cite{menorm} describe an alternative approach toconfirming the kinematic and amplitude validity of the offsetcontinuation equation. Applying equation~(\ref{eqn:OCequation})directly on the Kirchhoff model of prestack seismic data shows thatthe equation is satisfied to the same asymptotic order of accuracyas the Kirchhoff modeling approximation \cite[]{haddon,norm}.\section{Integral offset continuation operator}%%%%%%%%%%%%%%%%%%%%%%Equation~(\ref{eqn:OCequation}) describes a continuous process ofreflected wavefield continuation in the time-offset-midpoint domain.In order to find an integral-type operator that performs the one-stepoffset continuation, I consider the following initial-valueproblem for equation~(\ref{eqn:OCequation}):{\em Given a post-NMO constant-offset section at half-offset $h_1$}\begin{equation}\left.P(t_n,h,y)\right|_{h=h_1}=P^{(0)}_1(t_n,y) \label{eqn:bound0} \end{equation}{\em and its first-order derivative with respect to offset}\begin{equation}\left.\partial P(t_n,h,y)\over \partial h\right|_{h=h_1}=P^{(1)}_1(t_n,y)\;, \label{eqn:bound1} \end{equation}{\em find the corresponding section $P^{(0)}(t_n,y)$ at offset $h$.}Equation~(\ref{eqn:OCequation}) belongs to the hyperbolic type, withthe offset coordinate $h$ being a ``time-like'' variable and themidpoint coordinate $y$ and the time $t_n$ being ``space-like''variables.  The last condition~(\ref{eqn:bound1}) is required for theinitial value problem to be well-posed \cite[]{kurant}. From a physicalpoint of view, its role is to separate the two different wave-likeprocesses embedded in equation~(\ref{eqn:OCequation}), which areanalogous to inward and outward wave propagation. We will associatethe first process with continuation to a larger offset and the secondone with continuation to a smaller offset.  Though the offsetderivatives of data are not measured in practice, they can beestimated from the data at neighboring offsets by a finite-differenceapproximation. Selecting a propagation branch explicitly, for exampleby considering the high-frequency asymptotics of the continuationoperators, can allow us to eliminate the need forcondition~(\ref{eqn:bound1}). In this section, I discuss the exactintegral solution of the OC equation and analyze its asymptotics.The integral solution of problem~(\ref{eqn:bound0}-\ref{eqn:bound1})for equation~(\ref{eqn:OCequation}) is obtained in with the help ofthe classic methods of mathematical physics\cite[]{me,Fomel.sepphd.107}. It takes the explicit form\begin{eqnarray}P(t_n,h,y) & = &\int\!\!\int P^{(0)}_1(t_1,y_1)\,G_0(t_1,h_1,y_1;t_n,h,y)\,dt_1\,dy_1\nonumber \\ + & & \int\!\!\int P^{(1)}_1(t_1,y_1)\,G_1(t_1,h_1,y_1;t_n,h,y)\,dt_1\,dy_1\;,\label{eqn:integral} \end{eqnarray}where the Green's functions $G_0$ and $G_1$ are expressed as\begin{eqnarray}G_0(t_1,h_1,y_1;t_n,h,y) & = & \mbox{sign}(h-h_1)\,{H(t_n) \over \pi}\,{\partial \over \partial t_n}\,\left\{H(\Theta) \over \sqrt{\Theta}\right\}\;,\label{eqn:green0} \\G_1(t_1,h_1,y_1;t_n,h,y) & = & \mbox{sign}(h-h_1)\,{H(t_n) \over \pi}\,h\,{t_n \over t_1^2}\,\left\{H(\Theta) \over \sqrt{\Theta}\right\}\;,\label{eqn:green1} \end{eqnarray}and the parameter $\Theta$ is\begin{equation}\Theta(t_1,h_1,y_1;t_n,h,y)  = \left(h_1^2/t_1^2-h^2/t_n^2\right)\,\left(t_1^2-t_n^2\right)-\left(y_1-y\right)^2\;.\label{eqn:gamma}\end{equation}$H$ stands for the Heaviside step-function.  From equations~(\ref{eqn:green0}) and~(\ref{eqn:green1}) one can seethat the impulse response of the offset continuation operator isdiscontinuous in the time-offset-midpoint space on a surface definedby the equality\begin{equation}\Theta(t_1,h_1,y_1;t_n,h,y)  = 0\;,\label{eqn:conoid}\end{equation}which describes the ``wavefronts'' of the offset continuation process.In terms of the theory of characteristics \cite[]{kurant}, the surface$\Theta=0$ corresponds to the characteristic conoid formed by thebi-characteristics of equation~(\ref{eqn:OCequation}) -- time rays