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\lefthead{Guerra}
\righthead{Areal shot migration}
\email{guerrac@stanford.edu}
\footer{SEP--134}
\title{Angle gathers from areal shot migration with controlled illumination}
\author{Claudio Guerra}

\ABS{In very complex ares, wave-equation migration is mandatory to obtain reasonable images. Because of its high cost and small flexibility relatively to Kirchhoff migration, however, it is rarely used in migration velocity analysis. The accuracy of the velocity model, therefore, can be compromised by the use of an inadequate method to perform migration iterations. Fortunately, the linearity which governs the wave propagation permits to generate areal shots by linearly combining several shots as a way to decrease the cost of migration velocity analysis using wave-equation migration. Here, I show the usefulness of areal shot migration with controlled illumination to generate data suited for migration velocity analysis. ***  }
 
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\section{Introduction} 
Oil exploration is being conducted in increasingly more geologically complex areas. The imaging challenges along with the continuous growth of computer power has augmented the relative importance of depth migration, in particular wave-equation migration. A variety of wave-equation migration algorithms have been developed to balance cost and final image quality. Velocity model definition for depth migration, however, remains strongly dependent on Kirchhoff algorithms because of their flexibility and low cost when compared with wave-equation migration velocity analysis. In areas where the high-frequency approximation is violated and the wavefields become too complex, this dependence might result in inaccurate velocity models that yield poor quality or unreliable images even when a state-of-the-art wave-equation algorithm is used to produce the final image. In these areas it is therefore desirable to use wave-equation migration to generate image gathers in order to evaluate the accuracy of the velocity model.
\par
Every iteration in velocity model definition for depth migration includes: 1) migration; 2) measurement of some property that diagnoses the accuracy of the current velocity model; and, 3) update of the velocity model. When using wave-equation migration, step 1 can be by far the most expensive of all. It is important, therefore, to decrease the cost of applying wave-equation migration for wave-equation velocity analysis to be cost effective.
\par
Based on the linearity of wavefield extrapolators, different strategies have been formulated to reduce the computational cost of wave-equation migration. The synthesis of plane waves was first introduced by \longcite{Schultz}. \longcite{Rietveld01} used pre-defined wavefields of arbitrary shape to synthesize areal shots after propagation back to the surface. \longcite{Rietveld02} proposed a depth migration method using the concept of controlled illumination, which can be considered as a generalization of Schultz and Claerbout's method. They show images with superior quality when compared with Phase-encoded shot records \cite{romero} is a technique that provides the ability to combine shot records with minimum  cross-talk. Recently, the combination of phase-encoding and areal shots has been used to drastically decrease the computational effort \cite{Sun01}. \longcite{liu} provide a general framework to evaluate plane-wave composition in prestack source plane-wave migration and show the equivalence of shot profile and plane-wave migrations.
\par
Here, I show the usefulness of the low cost areal shot migration with controlled illumination to generate angle gather -- ADCIG -- for migration velocity analysis. This will be done by comparing the resulting ADCIGs with ADCIGs from shot profile migration.
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\section{Areal Shot migration by Controlled Illumination}
Areal shot migration is an economical way to perform wave-equation migration. The receiver wavefield at all spatial coordinates $(x,z)$ is computed by downward propagating a linear combination, known as areal shots, of all the shot gathers in the original dataset. Similarly, the same linear combination is considered to generate the areal source to be downward propagated. The synthesis of areal shots using the concept of controlled illumination, first introduced by \longcite{Rietveld01}, is a generalization of the plane wave synthesis from \longcite{Schultz}. Here, all the wavefield extrapolations are performed with the one-way wave equation, but similarly could be used two-way extrapolators \cite{boechat}. Conceptually, in 2D, a wavefield, $P(x,z)$, with an arbitrary shape is pre-defined in depth by
\begin{equation}
P(x,z) = \delta(x - x_i) \delta(z - F(x_i)),
\label{eq:eq1}
\end{equation}
where $F(x_i)$ is a function that defines the arbitrary shape of the wavefield. Here, I make use of the most commonly used shape of planar wavefields. The correspondent wavefield at $z=0$, $\tilde{P}(x,\omega)$, is obtained by propagating $P(x,z)$ back to the surface for all the angular frequency components, $\omega$, of a particular source wavelet.
\par
$P(x,\omega)$ is then used as the areal source wavefield for the areal shot migration and as the synthesis operator to be convolved with all the shot gathers, $R(x_s,x_r,\omega)$, to generate an areal shot, $\tilde{R}(x,\omega)$, according to 
\begin{equation}
\tilde{R}(x,\omega) = \sum_{x_s} \sum_{x_r} P(x=x_s,\omega) R(x_s,x_r,\omega).
\label{eq:eq2}
\end{equation}
\par
The migration of a single areal shot is represented by the downward continuation of $\tilde{P}(x,\omega)$ and $\tilde{R}(x,\omega)$ and application of an imaging condition. In this paper, I use the correlation imaging condition \cite{claerbout}. The resulting image, $I(x,z)$, is given by
\begin{equation}
I(x,z) = \sum_{\omega} \tilde{P}^{\star}(x,z,\omega) \tilde{R}(x,z,\omega),
\label{eq:eq3}
\end{equation}
where $\star$ stands for the complex conjugate.
\par
As \longcite{liu} showed for source plane-wave migration, the areal shot migration with controlled illumination also is affected by the nonlinearity of the imaging condition operator which causes artifacts, known as cross-talk, resulting from the cross-correlation of unrelated source and receivers wavefields. To decrease the cross-talk, therefore, several areal shots are computed by means of applying synthesis operators with different tilt angles. Additionally, this procedure permits to image a wider range of angle dips. Areal shot migration is expected to be less expensive than shot-profile migration because the number of areal shots is much less than the original shot gathers. For 2D data this difference can be of one order of magnitude and in 3D, up to three orders of magnitude.
\par 
The imaging condition expressed in equation \ref{eq:eq3} is not suitable for migration velocity analysis. For source plane-wave migration it is straightforward to generate common-image-point gathers (in ray-parameter) that can be used to evaluate the accuracy of the velocity model. For areal shot migration with controlled illumination, however, it is necessary to compute local-subsurface-offset, $h$, gathers -- ODCIG -- by applying a different imaging condition in which the wavefields are shifted horizontally by $2h$ prior to correlation \cite{rickett}.  Now, the prestack image, $I(x,h,z)$, is expressed as
\begin{equation}
I(x,h,z) = \sum_{\omega} \tilde{P}^{\star}(x-h,z,\omega) \tilde{R}(x+h,z,\omega).
\label{eq:eq4}
\end{equation}
\par
Then, the image $I(x,h,z)$ can be used as input to compute local-angle-domain common-image gathers -- ADCIG, by applying a slant-stack transfomation \cite{sava01}.
\par
In the next section, I show the areal data and results of areal shot migration with controlled illumination on Sigsbee2b dataset \cite{paffenholz}.

\section{Examples}

\section{Conclusions}

\section{Acknowledgments}

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