emerging from the point $\{t_n,h,y\}=\{t_1,h_1,y_1\}$. Thecommon-offset slices of the characteristic conoid are shown in theleft plot of Figure \ref{fig:cont}.\plot{cont}{width=6in}{Constant-offset sections of the characteristic conoid - ``offsetcontinuation fronts'' (left), and branches of the conoid used in theintegral OC operator (right). The upper part of the plots(small times) corresponds to continuation to smaller offsets; the lower part (large times) corresponds to larger offsets.}As a second-order differential equation of the hyperbolic type,equation (\ref{eqn:OCequation}) describes two different processes. Thefirst process is ``forward'' continuation from smaller to largeroffsets, the second one is ``reverse'' continuation in the oppositedirection.  These two processes are clearly separated in thehigh-frequency asymptotics of operator~(\ref{eqn:integral}). To obtainthe asymptotic representation, it is sufficient to note that ${1  \over \sqrt{\pi}}\, {H(t) \over \sqrt{t}}$ is the impulse responseof the causal half-order integration operator and that $H(t^2-a^2)\over \sqrt{t^2-a^2}$ is asymptotically equivalent to $H(t-a) \over{\sqrt{2a}\,\sqrt{t-a}}$ $(t, a >0)$.  Thus, the asymptotical form ofthe integral offset-continuation operator becomes\begin{eqnarray}P^{(\pm)}(t_n,h,y) & = &{\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm)}_0(\xi;h_1,h,t_n)\,P^{(0)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\nonumber \\ & \pm  & {\bf I}^{1/2}_{\pm\,t_n}\,\int w^{(\pm)}_1(\xi;h_1,h,t_n)\,P^{(1)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;.\label{eqn:asintegral} \end{eqnarray}Here the signs ``$+$'' and ``$-$'' correspond to the type ofcontinuation (the sign of ${h-h_1}$), ${\bf D}^{1/2}_{\pm\,t_n}$ and${\bf I}^{1/2}_{\pm\,t_n}$ stand for the operators of causal andanticausal half-order differentiation and integration applied withrespect to the time variable $t_n$, the summation paths$\theta^{(\pm)}(\xi;h_1,h,t_n)$ correspond to the two non-negativesections of the characteristic conoid~(\ref{eqn:conoid}) (Figure\ref{fig:cont}):\begin{equation}t_1=\theta^{(\pm)}(\xi;h_1,h,t_n)={t_n \over h}\,\sqrt{{U \pm V} \over 2 }\;,\label{eqn:summation}\end{equation}where $U=h^2+h_1^2-\xi^2$, and $V=\sqrt{U^2-4\,h^2\,h_1^2}$; $\xi$ isthe midpoint separation (the integration parameter), and $w^{(\pm)}_0$and $w^{(\pm)}_1$ are the following weighting functions:\begin{eqnarray}w^{(\pm)}_0 & = & {1 \over \sqrt{2\,\pi}}\,{\theta^{(\pm)}(\xi;h_1,h,t_n) \over \sqrt{t_n\,V}}\;,\label{eqn:w0} \\w^{(\pm)}_1 & = & {1 \over \sqrt{2\,\pi}}\,{{\sqrt{t_n}\, h_1} \over {\sqrt{V}\,\theta^{(\pm)}(\xi;h_1,h,t_n)}}\;. \label{eqn:w1} \end{eqnarray}  Expression~(\ref{eqn:summation}) for the summation path of the OCoperator was obtained previously by \cite{stovas} and\cite{SEG-1994-1541}. A somewhat different form of it is proposed by\cite{GEO61-06-18461858}. I describe the kinematic interpretationof formula~(\ref{eqn:summation}) in Appendix~B.In the high-frequency asymptotics, it is possible to replace the twoterms in equation~(\ref{eqn:asintegral}) with a single term\cite[]{GEO68-03-10321042}. The single-term expression is\begin{equation}P^{(\pm)}(t_n,h,y)  = {\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm)}(\xi;h_1,h,t_n)\,P^{(0)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;,\end{equation}where\begin{eqnarray}w^{(+)} & = & \sqrt{\theta^{(+)}(\xi;h_1,h,t_n) \over {2\,\pi}}\;{{h^2-h_1^2-\xi^2} \over {V^{3/2}}}\;, \label{eqn:wOC12} \\w^{(-)} & = & {{\theta^{(-)}(\xi;h_1,h,t_n)} \over \sqrt{2\,\pi t_n}}\;{{h_1^2-h^2 +\xi^2} \over {V^{3/2}}}\;. \label{eqn:wOC21} \end{eqnarray}A more general approach to true-amplitude asymptotic offsetcontinuation is developed by \cite{tygel}.The limit of expression~(\ref{eqn:summation}) for the output offset $h$approaching zero can be evaluated by L'Hospitale's rule. As one wouldexpect, it coincides with the well-known expression for the summationpath of the integral DMO operator\cite[]{GPR29-03-03740406}\begin{equation}t_1=\theta^{(-)}(\xi;h_1,0,t_n)=\lim_{h \rightarrow 0} {{t_n \over h}\,\sqrt{{U - V} \over 2 }}={{t_n\,h_1} \over \sqrt{h_1^2-\xi^2}}\;.\label{eqn:DMOsummation}\end{equation}I discuss the connection between offset continuation and DMO in thenext section.\section{Offset continuation and DMO}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Dip moveout represents a particular case of offset continuation forthe output offset equal to zero. In this section, I consider the DMOcase separately in order to compare the solutions of equation(\ref{eqn:OCequation}) with the Fourier-domain DMO operators, whichhave been the standard for DMO processing since Hale's outstandingwork \cite[]{Hale.sepphd.36,GEO49-06-07410757}.Equation~(\ref{eqn:integral}) transforms to the time-wavenumber domainwith the help of integral tables:\begin{equation}\widetilde{P}(t_n,h,k)=H(t_n)\,\left(\widetilde{P}_0(t_n,h,k) +t_n\,\widetilde{P}_1(t_n,h,k)\right)\;,\label{eqn:pp0p1} \end{equation}where\begin{eqnarray}\widetilde{P}_0  & = & {\partial \over {\partial t_n}}\,\int_{\left(h_1/h\right)\,t_n}^{t_n}\widetilde{P}^{(0)}_1\left(\left|t_1\right|,k\right)\,J_0\left(k\,\sqrt{\left({h^2 \over t_n^2}-{h_1^2 \over t_1^2}\right)\,\left(t_n^2-t_1^2\right)}\right)\,dt_1\;,\label{eqn:p0} \\\widetilde{P}_1  & = & \int_{\left(h_1/h\right)\,t_n}^{t_n}h_1\,\widetilde{P}^{(1)}_1\left(\left|t_1\right|,k\right)\,J_0\left(k\,\sqrt{\left({h^2 \over t_n^2}-{h_1^2 \over t_1^2}\right)\,\left(t_n^2-t_1^2\right)}\right)\,{dt_1 \over t_1^2}\;,\label{eqn:p1} \end{eqnarray}\begin{eqnarray}\widetilde{P}^{(j)}_1(t_1,k) & = & \int\,P\,^{(j)}_1(t_1,y_1)\exp (-iky_1)\,dy_1\;(j=0,1)\;,\label{eqn:tildej} \\\widetilde{P}(t_n,h,k)  & = & \int\,P(t_n,h,y)\exp (-iky)\,dy\;(j=0,1)\;.\label{eqn:tilde}\end{eqnarray}Setting the output offset to zero, we obtain the following DMO-likeintegral operators in the $t$--$k$ domain:\begin{equation}\widetilde{P}(t_0,0,k)=H(t_0)\,\left(\widetilde{P}_0(t_0,k) +t_0\,\widetilde{P}_1(t_0,k)\right)\;,\label{eqn:dp0p1} \end{equation}where\begin{equation}\widetilde{P}_0(t_0,k)  = - {\partial \over {\partial t_0}}\,\int_{t_0}^{\infty}\widetilde{P}^{(0)}_1\left(\left|t_1\right|,k\right)\,J_0\left({{k\,h_1}\over t_1}\,\sqrt{t_1^2-t_0^2}\right)\,dt_1\;,\label{eqn:d0} \end{equation}\begin{equation}\widetilde{P}_1(t_0,k)  = - \int_{t_0}^{\infty}h_1\,\widetilde{P}^{(1)}_1\left(\left|t_1\right|,k\right)\,J_0\left({{k\,h_1}\over t_1}\,\sqrt{t_1^2-t_0^2}\right)\,{dt_1 \over t_1^2}\;,\label{eqn:d1} \end{equation}the wavenumber $k$ corresponds to the midpoint axis $y$, and $J_0$ isthe zeroth-order Bessel function.  The Fourier transformof~(\ref{eqn:d0}) and (\ref{eqn:d1}) with respect to the time variable$t_0$ reduces to known integrals \cite[]{table} and creates explicitDMO-type operators in the frequency-wavenumber domain, as follows:\begin{equation}\widetilde{\widetilde{P}}_0(\omega_0,k)  = i\,\int_{-\infty}^{\infty}\widetilde{P}^{(0)}_1\left(\left|t_1\right|,k\right)\,{\sin{\left(\omega_0\,|t_1|\,A\right)} \over A}\,dt_1\;,\label{eqn:dw0} \end{equation}\begin{equation}\widetilde{\widetilde{P}}_1(\omega_0,k)  = i\, \int_{-\infty}^{\infty}h_1\,\widetilde{P}^{(1)}_1\left(\left|t_1\right|,k\right)\,{\sin{\left(\omega_0\,|t_1|\,A\right)} \over A}{dt_1 \over t_1^2}\;,\label{eqn:dw1} \end{equation}where \begin{equation}A=\sqrt{1+{(k\,h_1)^2 \over (\omega_0\,t_1)^2}}\;,\label{eqn:a} \end{equation}\begin{equation}\widetilde{\widetilde{P}}_j(\omega_0,k)=\int\,\widetilde{P}_j(t_0,k)\,\exp (i\omega_0 t_0)\,dt_0\;.\label{eqn:FFT} \end{equation}It is interesting to note that the first term of the continuation tozero offset~(\ref{eqn:dw0}) coincides exactly with the imaginary partof Hale's DMO operator \cite[]{GEO49-06-07410757}. However, unlike Hale's,operator~(\ref{eqn:dp0p1}) is causal, which means that its impulseresponse does not continue to negative times. The non-causality ofHale's DMO and related issues are discussed in more detail by\cite{stovas}.%I include a brief summary of this discussion in Appendix~\ref{chapter:hale}.Though Hale's DMO is known to provide correct reconstruction of thegeometry of zero-offset reflections, it does not account properly forthe amplitude changes \cite[]{GEO58-01-00470066}. The preceding section of thispaper shows that the additional contribution to the amplitudeis contained in the second term of the OC operator(\ref{eqn:integral}), which transforms to the second term in the DMOoperator~(\ref{eqn:dp0p1}). Note that this term vanishes at the input offsetequal to zero, which represents the case of the inverse DMO operator.Considering the inverse DMO operator as the continuation from zerooffset to a non-zero offset, we can obtain its representation in the$t$-$k$ domain from equations~(\ref{eqn:pp0p1}-\ref{eqn:p1}) as\begin{equation}\widetilde{P}(t_n,h,k)  = H(t_n) {\partial \over {\partial t_n}}\,\int_{0}^{t_n}\widetilde{P}_0\left(\left|t_0\right|,k\right)\,J_0\left({{k\,h}\over t_n}\,\sqrt{t_n^2-t_0^2}\right)\,dt_0\;,\label{eqn:i0} \end{equation}Fourier transforming equation~(\ref{eqn:i0}) with respect to the timevariable $t_0$ according to equation~(\ref{eqn:FFT}), we get theFourier-domain version of the ``amplitude-preserving'' inverse DMO:\begin{equation}\widetilde{P}(t_n,h,k) = {H(t_n) \over {2\,\pi}}\,{\partial \over \partial t_n}\,\int_{-\infty}^{\infty}\widetilde{\widetilde{P}}_0(\omega_0,k)\, {{\sin{\left(\omega_0\,|t_n|\,A\right)} \over {\omega_0\,A}}}\,d\omega_0\;,\label{eqn:IDMO} \end{equation}\begin{equation}A=\sqrt{1+{(k\,h)^2 \over (\omega_0\,t_n)^2}}\;.\label{eqn:a1} \end{equation}Comparing operator~(\ref{eqn:IDMO}) with Ronen's version of inverse DMO\cite[]{GEO52-07-09730984}, one can see that if Hale's DMO is denotedby ${\bf D}_{t_0}\,{\bf H}$, then Ronen's inverse DMO is ${\bfH^{T}\,D}_{-t_0}$, while the amplitude-preserving inverse~(\ref{eqn:IDMO})is ${\bf D}_{t_n}\,{\bf H^T}$. Here ${\bf D}_t$ is the derivativeoperator $\left( \partial \over \partial t\right)$, and ${\bf H^T}$stands for the adjoint operator defined by the dot-product test\begin{equation}{\bf (Hm,d)=(m,H^{T}d)},\label{eqn:dot}\end{equation}where the parentheses denote the dot product:\begin{equation}{\bf (m_1,m_2)}=\int\!\int\,m_1(t_n,y)\,m_2(t_n,y)\,dt_n\,dy\;.\nonumber\end{equation}  In high-frequency asymptotics, the difference between the amplitudesof the two inverses is simply the Jacobian term ${d\,t_0 \over  d\,t_n}$, asymptotically equal to ${t_0 \over t_n}$. This differencecorresponds exactly to the difference between Black's definition ofamplitude preservation \cite[]{GEO58-01-00470066} and the definition used in BornDMO \cite[]{born,GEO56-02-01820189}, as discussed above. While operator(\ref{eqn:IDMO}) preserves amplitudes in the Born DMO sense, Ronen'sinverse satisfies Black's amplitude preservation criteria. This meansRonen's operator implies that the ``geometric spreading'' correction(multiplication by time) has been performed on the data prior to DMO.To construct a one-term DMO operator, thus avoiding the estimation ofthe offset derivative in~(\ref{eqn:w1}), let us consider the problemof inverting the inverse DMO operator~(\ref{eqn:IDMO}). One of thepossible approaches to this problem is the least-squares iterativeinversion, as proposed by \cite{GEO52-07-09730984}. Thisrequires constructing the adjoint operator, which is Hale's DMO (orits analog) in the case of Ronen's method. The iterative least-squaresapproach can account for irregularities in the data geometry\cite[]{IZO,SEG-1994-1545} and boundary effects, but it is computationallyexpensive because of the multiple application of the operators. Analternative approach is the asymptotic inversion, which can be viewedas a special case of preconditioning the adjoint operator\cite[]{SEG-1988-S17.5,SEG-1996-0032}. The goal of the asymptotic inverse is toreconstruct the geometry and the amplitudes of the reflection eventsin the high-frequency asymptotic limit.According to Beylkin's theory of asymptotic inversion, also known asthe {\em generalized Radon transform} \cite[]{beylkin}, two operators of theform\begin{equation}D(\omega)=\int X(t,\omega)\,M(t)\,\exp\left[i\omega \phi (t,\omega)\right]\,dt\label{eqn:b0}\end{equation}and