\begin{equation}\widetilde{M}(t)=\int Y(t,\omega)\,D(\omega)\,\exp\left[-i\omega \phi (t,\omega)\right]\,d\omega\label{eqn:b1}\end{equation}constitute a pair of asymptotically inverse operators($\widetilde{M}(t)$ matching $M(t)$ in the high-frequency asymptotics)if\begin{equation}X(t,\omega)\,Y(t,\omega)={Z(t,\omega) \over {2\,\pi}}\;,\label{eqn:belka}\end{equation}where $Z$ is the ``Beylkin determinant''\begin{equation}Z(t,\omega)=\left|\partial \omega \over \partial \hat{\omega}\right|\;\mbox{for}\;\hat{\omega}=\omega\,{\partial \phi(t,\omega) \over \partial t}\;.\label{eqn:det}\end{equation}With respect to the high-frequency asymptotic representation, we canrecast (\ref{eqn:IDMO}) in the equivalent form by moving the timederivative under the integral sign:\begin{equation}\widetilde{P}(t_n,k) \approx {H(t_n) \over {2\,\pi}}\,\mbox{Re}\left[\int_{-\infty}^{\infty}A^{-2}\widetilde{\widetilde{P}}_0(\omega_0,k)\, \exp\left(-i \omega_0\,|t_n|\,A\right)\,d\omega_0\right]\label{eqn:IDMO2} \end{equation}Now the asymptotic inverse of~(\ref{eqn:IDMO2}) is evaluated bymeans of Beylkin's method (\ref{eqn:b0})-(\ref{eqn:b1}), which leadsto an amplitude-preserving one-term DMO operator of the form\begin{equation}\widetilde{\widetilde{P}}_0(\omega_0,k)  = \mbox{Im}\left[\int_{-\infty}^{\infty} B\widetilde{P}^{(0)}_1\left(\left|t_1\right|,k\right)\,\exp\left(i \omega_0\,|t_1|\,A\right)\,dt_1\right]\;,\label{eqn:born} \end{equation}where \begin{equation}B = A^2 {\partial \over \partial \omega_0}\left(\omega_0\,{{\partial (t_n\,A)} \over \partial t_n}\right) = A^{-1}\,(2\,A^2 - 1)\;.\label{eqn:jack} \end{equation}  The amplitude factor~(\ref{eqn:jack}) corresponds exactly to that ofBorn DMO \cite[]{born} in full accordance with the conclusions of theasymptotic analysis of the offset-continuation amplitudes. Ananalogous result can be obtained with the different definition ofamplitude preservation proposed by \cite{GEO58-01-00470066}. Inthe time-and-space domain, the operator asymptotically analogous to(\ref{eqn:born}) is found by applying either the stationary phasetechnique \cite[]{GEO55-05-05950607,GEO58-01-00470066} or Goldin's method ofdiscontinuities \cite[]{goldintomo,Goldin.sep.67.171}, which is thetime-and-space analog of Beylkin's asymptotic inverse theory\cite[]{stovas}. The time-and-space asymptotic DMO operator takes theform\begin{equation}P_0(t_0,y) = {\bf D}^{1/2}_{-t_0}\,\int w_0(\xi;h_1,t_0)\,P^{(0)}_1(\theta^{(-)}(\xi;h_1,0,t_0),y_1-\xi)\,d\xi\;,\label{eqn:TADMO}\end{equation} where the weighting function $w_0$ is defined as \begin{equation}w_0(\xi;h_1,t_0)=\sqrt{t_0 \over {2\,\pi}}\, {{h_1\,(h_1^2+\xi^2)} \over (h_1^2-\xi^2)^2}\;.\label{eqn:TAw}\end{equation}\section{Offset continuation in the log-stretch domain}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%The log-stretch transform, proposed by \cite{GPR30-06-08130828}and further developed by many other researchers, is a usefultool in DMO and OC processing. Applying a log-stretch transform of theform\begin{equation}\sigma = \ln\left|t_n \over t_* \right|\;, \label{eqn:log}\end{equation}where $t_*$ is an arbitrarily chosen time constant, eliminates thetime dependence of the coefficients in equation~(\ref{eqn:OCequation})and therefore makes this equation invariant to time shifts. After thedouble Fourier transform with respect to the midpoint coordinate $y$and to the transformed (log-stretched) time coordinate $\sigma$, thepartial differential equation~(\ref{eqn:OCequation}) takes the form ofan ordinary differential equation,\begin{equation}h\,\left({{d^2 \widehat{\widehat{P}}} \over {dh^2}} + k^2\,\widehat{\widehat{P}}\right) =i\Omega\,{{d \widehat{\widehat{P}}} \over {dh}}\;,\label{eqn:LSequation}\end{equation}where\begin{equation}\widehat{\widehat{P}}(h) = \int\!\int P(t_n=t_*\,\exp(\sigma),h,y)\,\exp(i\Omega\sigma - iky)\,d\sigma\,dy\;. \label{eqn:LSFT}\end{equation}Equation~(\ref{eqn:LSequation}) has the known general solution,expressed in terms of cylinder functions of complex order $\lambda ={{1+i\Omega} \over 2}$ \cite[]{watson}\begin{equation}\widehat{\widehat{P}}(h) = C_1(\lambda)\,(kh)^{\lambda}\,J_{-\lambda}(kh)+C_2(\lambda)\,(kh)^{\lambda}\,J_{\lambda}(kh)\;,\label{eqn:gensol}\end{equation}where $J_{-\lambda}$ and $J_{\lambda}$ are Bessel functions, and $C_1$and $C_2$ stand for some arbitrary functions of $\lambda$ that do notdepend on $k$ and $h$.In the general case of offset continuation, $C_1$ and $C_2$ areconstrained by the two initial conditions~(\ref{eqn:bound0}) and(\ref{eqn:bound1}). In the special case of continuation from zero offset, wecan neglect the second term in~(\ref{eqn:gensol}) as vanishing at the zerooffset. The remaining term defines the following operator of inverseDMO in the ${\Omega,k}$ domain:\begin{equation}\widehat{\widehat{P}}(h) = \widehat{\widehat{P}}(0)\,Z_{\lambda}(kh)\;,\label{eqn:OKDMO}\end{equation}where $Z_{\lambda}$ is the analytic function\begin{eqnarray}\nonumberZ_{\lambda}(x) & = & \Gamma(1-\lambda)\,\left(x \over 2\right)^{\lambda}\,J_{-\lambda}(x)={}_0F_1\left(;1-\lambda;-\frac{x^2}{4}\right) \\& = &\sum_{n=0}^{\infty} {(-1)^n \over n!}\,{\Gamma(1-\lambda) \over \Gamma(n+1-\lambda)}\,\left(x \over 2\right)^{2n}\;,\label{eqn:z}\end{eqnarray}$\Gamma$ is the gamma function and ${}_0F_1$ is the confluenthypergeometric limit function \cite[]{ab}.The DMO operator now can be derived as the inversion of operator(\ref{eqn:OKDMO}), which is a simple multiplication by$1/Z_{\lambda}(kh)$. Therefore, offset continuation becomes amultiplication by $Z_{\lambda}(kh_2)/Z_{\lambda}(kh_1)$ (the cascadeof two operators). This fact demonstrates an important advantage ofmoving to the log-stretch domain: both offset continuation and DMO are simplefilter multiplications in the Fourier domain of the log-stretched timecoordinate.In order to compare operator~(\ref{eqn:OKDMO}) with the known versionsof log-stretch DMO, we need to derive its asymptotic representationfor high frequency $\Omega$. The required asymptotic expressionfollows directly from the definition of function $Z_{\lambda}$ inequation~(\ref{eqn:z}) and the known asymptotic representation for a Besselfunction of high order \cite[]{watson}:\begin{equation}J_{\lambda}(\lambda z) \stackrel{\lambda \rightarrow \infty}{\approx} {{(\lambda z)^{\lambda}\,\exp\left(\lambda\,\sqrt{1-z^2}\right)} \over{e^{\lambda}\,\Gamma(\lambda+1)\,(1-z^2)^{1/4}\,\left\{1+\sqrt{1-z^2}\right\}^{\sqrt{1-z^2}}}}\;.\label{eqn:carlini}\end{equation}Substituting approximation~(\ref{eqn:carlini}) into~(\ref{eqn:z}) andconsidering the high-frequency limit of the resultant expressionyields\begin{equation}Z_{\lambda}(kh) \approx \left\{{1+\sqrt{1-\left(kh \over \lambda\right)^2}} \over2\right\}^{\lambda}\, {{\exp\left(\lambda\,\left[1 - \sqrt{1-\left(kh \over\lambda\right)^2}\right]\right)} \over \left(1-\left(kh \over \lambda\right)^2\right)^{1/4}} \approx{F(\epsilon)\,e^{i\Omega\,\psi(\epsilon)}}\;,\label{eqn:AOKDMO}\end{equation}where $\epsilon$ denotes the ratio ${2\,k\,h} \over\Omega$,\begin{equation}F(\epsilon)=\sqrt{{1+\sqrt{1+\epsilon^2}} \over{2\,\sqrt{1+\epsilon^2}}}\,\exp\left({1-\sqrt{1+\epsilon^2}} \over 2\right)\;,\label{eqn:F}\end{equation}and\begin{equation}\psi(\epsilon)={1 \over 2}\,\left(1 - \sqrt{1+\epsilon^2} +\ln\left({1 + \sqrt{1+\epsilon^2}} \over 2\right)\right)\;.\label{eqn:psi}\end{equation}The asymptotic representation~(\ref{eqn:AOKDMO}) is valid for highfrequency $\Omega$ and $|\epsilon| \leq 1$. Thephase function $\psi$ defined in~(\ref{eqn:psi}) coincides preciselywith the analogous term in Liner's \emph{exact log DMO}\cite[]{GEO55-05-05950607}, which provides the correctgeometric properties of DMO. Similar expressions for the log-stretchphase factor $\psi$ were derived in different ways by\cite{GEO61-03-08150820} and \cite{GEO61-04-11031114}.However, the amplitude term $F(\epsilon)$ differs from the previouslypublished ones because of the difference in the amplitude preservationproperties.  A number of approximate log DMO operators have been proposed in theliterature. As shown by \cite{GEO55-05-05950607}, all of them butexact log DMO distort the geometry of reflection effects at largeoffsets. The distortion is caused by the implied approximations of thetrue phase function $\psi$. Bolondi's OC operator\cite[]{GPR30-06-08130828} implies $\psi(\epsilon) \approx -{\epsilon^2  \over 8}$, Notfors' DMO \cite[]{GEO52-12-17181721} implies$\psi(\epsilon) \approx 1 - \sqrt{1+(\epsilon /2)^2}$, and the ``fullDMO'' \cite[]{SEG-1987-S14.1} has $\psi(\epsilon) \approx {1 \over 2}\ln\left[1-(\epsilon / 2)^2\right]$. All these approximations arevalid for small $\epsilon$ (small offsets or small reflector dips) andhave errors of the order of $\epsilon^4$ (Figure \ref{fig:pha}).  Therange of validity of Bolondi's operator is defined inequation~(\ref{eqn:hm}).    \sideplot{pha}{height=1.5in}{ Phase functions of the log    DMO operators. Solid line: exact log DMO; dashed line: Bolondi's    OC; dashed-dotted line: Bale's full DMO; dotted line: Notfors'    DMO.  }    In practice, seismic data are often irregularly sampled in space but  regularly sampled in time. This makes it attractive to apply offset  continuation and DMO operators in the $\{\Omega,y\}$ domain, where  the frequency $\Omega$ corresponds to the log-stretched time and  $y$ is the midpoint coordinate. Performing the inverse Fourier  transform on the spatial frequency transforms the inverse DMO  operator~(\ref{eqn:OKDMO}) to the $\{\Omega,y\}$ domain, where the  filter multiplication becomes a convolutional operator:\begin{equation}\widehat{P}(\Omega,h,y) ={\widehat{F}(\Omega) \over \sqrt{2\,\pi}}\,\int_{|\xi|<h}{ h \over {h^2-\xi^2}}\,\widehat{P_0}(\Omega,y-\xi)\,\exp\left(-{i\Omega \over 2}\,\ln\left(1-{\xi^2 \over h_1^2}\right)\right) \,d\xi\;.\label{eqn:OXDMO} \end{equation}Here $\widehat{F}(\Omega)$ is a high-pass frequency filter:\begin{equation}\widehat{F}(\Omega)={{\Gamma(1/2-i\Omega/2)}\over {\sqrt{1/2}\,\Gamma(-i\Omega/ 2)}}\;.\label{eqn:hat} \end{equation}At high frequencies $\widehat{F}(\Omega)$ is approximately equal to$(- i \Omega)^{1/2}$, which corresponds to the half-derivativeoperator $\left(\partial \over \partial \sigma \right)^{1/2}$, which,in turn, is equal to the $\left(t_n {\partial \over \partial t_n}\right)^{1/2}$ term of the asymptotic OCoperator~(\ref{eqn:asintegral}). The difference between the exactfilter $\widehat{F}$ and its approximation by the half-orderderivative operator is shown in Figure \ref{fig:flt}. This differenceis a measure of the validity of asymptotic OC operators.\plot{flt}{width=6in}{ Amplitude (left) and phase (right) of  the time filter in the log-stretch domain. The solid line is for the  exact filter; the dashed line for its approximation by the  half-order derivative filter. The horizontal axis corresponds to the  dimensionless log-stretch frequency $\Omega$.}Inverting operator~(\ref{eqn:OXDMO}), we can obtain the DMO operator in the$\{\Omega,y\}$ domain.\section{Discussion}The differential model for offset continuation is based on severalassumptions. It is important to fully realize them in order tounderstand the practical limitations of this model.\begin{itemize}\item The \emph{constant velocity} assumption is essential for  theoretical derivations. In practice, this limitation is not too  critical, because the operators act locally. DMO and  offset continuation algorithms based on the constant-velocity  assumptions are widely used in practice \cite[]{DMO00-00-04960496}.\item The \emph{single-mode} assumption does not include multiple  reflections in the model. If multiple events (with different  apparent velocities) are present in the data, they might require  extending the model. Convolving two (or more) differential offset  continuation operators, corresponding to different velocities, we  can obtain a higher-order differential operator for predicting  multiple events.\item The \emph{continuous AVO} assumption implies that the  reflectivity variation with offset is continuous and can be  neglected in a local neighborhood of a particular offset. While the  offset continuation model correctly predicts the geometric spreading  effects in the reflected wave amplitudes, it does not account for  the variation of the reflection coefficient with offset.\item The \emph{2.5-D} assumption was implicit in the derivation of  the offset continuation equation. According to this assumption, the  reflector does not change in the cross-line direction, and we can  always consider the reflectio