\lefthead{Guerra and Biondi}
\righthead{  }
\footer{SEP--137}
\title{Wave equation migration velocity analysis with time-shift imaging condition}
\author{Claudio Guerra and Biondo Biondi}

\maketitle

%\begin{abstract}
%Analyze the asymptotic behavior of the rotation integrals
%\end{abstract}

\section{Time-shift imaging condition}
Based on the theory of reflector mapping \cite[]{claerbout71}, the image is formed at points where source and receiver wavefields coincide both in time of wavefield propagation and spatially. In the context of migration by downward-continuation, this translates into point-wise multiplying the receiver wavefield, $R$, with the complex conjugate of the source wavefield, $S^{\star}$, at every image location ,$\bold{x}$, and summing over all frequencies, $\omega$, which represents to extract the image, $I$,at zero time. For a single shot we have
\begin{equation}
I(\bold{x}) = \sum_{\omega} S^{\star}(\bold{x},\omega)R(\bold{x},\omega).
\label{eq:e1}
\end{equation}
This imaging condition, however, does not yield an image suited for estimating velocity errors. To overcome the limitation of the above imaging condition, different generalizations have been proposed \cite[]{rickett,biosymes,sava:2052}. They consist of spatially shifting the wavefields with respect to each other prior to the point-wise multiplication. The spatial shifts are interpreted as subsurface offsets, $\bold{h}$, which can be computed in any direction. For the case of complete acquisition aperture and correct velocity, negligible amplitudes are computed for subsurface offsets different from zero. Departures from the zero offset are  indicative of illumination problems or velocity errors. In general, the image is computed by
\begin{equation}
I(\bold{x},\bold{h}) = \sum_{\omega} S^{\star}(\bold{x}-\bold{h},\omega)R(\bold{x}+\bold{h},\omega).
\label{eq:e2}
\end{equation}
\cite {sava:S209} introduced the space- and time-shift imaging condition, from which the time-shift imaging condition is a special case. It is computed by applying a phase-shift to the wavefields prior to summation over frequencies
\begin{equation}
I(\bold{x},\tau) = \sum_{\omega} S^{\star}(\bold{x},\omega)R(\bold{x},\omega)e^{i2\omega\tau}.
\label{eq:e3}
\end{equation}
Time-shift image gathers present linear events with slope equal to the local velocity at the reflector. If migration is performed with a faster velocity, energy concentrates at negative time-shifts and shallower than the correct depth, indicating that source and receiver rays cross above the imaging point (where $\tau = 0$). If migration is performed with a slower velocity, energy concentrates at positive time-shifts and deeper than the correct depth, indicating that rays cross below the imaging point. For the case of correct velocity, energy concentrates at  $\tau = 0$.

\section{Using time-shift imaging condition for velocity update}
\cite {sava:S209} show that time-shift image gathers can be transformed into angle-domain common image gathers by slant-stacking. In 2D, the transformation can be carried out by a Radon transform
\begin{equation}
I(\bold{x},\gamma) = \int I(\bold{x},\tau) d\tau \arrowvert_{\zeta(\gamma,\alpha,\tau) },
\label{eq:e4}
\end{equation}
where $\gamma$ is apparent the reflection angle, $\alpha$ is the apparent geological dip, and
\begin{equation}
\zeta(\gamma,\alpha,\tau) = z - v\tau \sec \alpha \sec \gamma}
\label{eq:e5}
\end{equation}
is the stacking path.
\par
This transformation yields images suited for velocity update. Using angle gathers, in wave-equation migration velocity analysis 
\cite[]{sava05}, a residual moveout parameter is picked and used to compute the image perturbation. \cite{shen:VE49} use angle gathers, computed from subsurface offset gathers, to obtain the perturbed image by applying the differential semblance operator ($\bf{DSO}$). They show that the results of velocity inversion using $\bf{DSO}$ in the angle domain ($\bf{DSO_{\gamma}}$) to compute the image perturbation are of lower quality than that obtained using DSO in the offset domain ($\bf{DSO_{\h}}$). They claim that the different performances are due to the fact that the Hessian is better conditioned in the case of the offset domain $\bf{DSO}$.
\par
To apply a DSO-like strategy when using time-shift image gathers, transformation to the angle domain requires information about the geological dip. If this information is not available or not reliable, the slant-stack transformation should include the apparent geological dip dimension, turning the computation of the perturbed image more expensive. So, it is desirable to derive an offset-domain DSO-like operator to be used in the time-shift gathers.
\par
The equivalence between the time-shift $\bf{DSO}$ ($\bf{DSO_{\tau}}$) with $\bf{DSO_{\gamma}}$, if exists, can be expressed by
\begin{equation}
\bf{DSO_{\tau}} I(\bold{x},\tau) = \bf{R_{\gamma}}^{-1}  \bf{DSO_{\gamma}} \bf{R_{\gamma}} I(\bold{x},\tau),
\label{eq:e6}
\end{equation}
where $\bf{R_{\gamma}}$ is the time-shift to angle transformation (equation \ref{eq:eq4}). The perturbed image, after application of forward and adjoint operators, is
\begin{equation}
\bf{DSO_{\tau}}^{\star} \bf{DSO_{\tau}} I(\bold{x},\tau) = \bf{R_{\gamma}}^{\star} \bf{DSO_{\gamma}}^{\star} \bf{DSO_{\gamma}} \bf{R_{\gamma}} I(\bold{x},\tau).
\label{eq:e7}
\end{equation}

 This additional dimension in the slant-stack process makes it more expensive.  should be performed is more expensive than for the subsurface-offset gathers because  However, there is the cost of performing 

\section {Numerical examples}

\section{Acknowledgements}

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