\lefthead{Guerra}
\righthead{Weighted offset-to-angle}
\email{guerrac@stanford.edu}
\footer{SEP--134}
\title{Angle-domain parameters computed via weighted slant-stack}
\author{Claudio Guerra}

\vspace{.5cm}

\section{Introduction} 
In the past decade, oil exploration has been 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 (PsDM), in particular wave-equation migration. A variety of wave-equation migration (WEM) 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
Considering that defining the velocity model for PsDM demands some iterations of PsDM -- evaluation of the velocity errors -- update of the velocity model, the use of WEM to perform this task may be impractical. By the time that the computer power was not sufficient to routinely use WEM for PsDM projects, several techniques based on the linear nature of wave propagation were proposed to obtain combined shots in such a way that migration could provide a reasonable image at low cost. 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. Phase-encoded shot records \longcite{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}.
\par
. However, in this The concept of controlled illumination by using areal shots was first introduced by \longcite{Rietveld01}

\section{Areal Shots by Controlled Illumination Synthesis Operator}

The synthesis of areal shots using the concept of controlled illumination is a generalization of the plane wave synthesis from \longcite{Schultz}. 

\begin{equation}
D(\omega,x_s,x_r,z\small{=}0) = \sum_{\bold{x}} G(\omega,x_s,\bold{x}) R(\bold{x}) G(\omega,\bold{x},x_r) S(\omega),
\label{eq:eq1}
\end{equation}

\begin{equation}
Q(\omega,x_s,z\small{=}0) = \sum_{\bold{x}} G^{\dagger}(\omega,x_s,\bold{x}) W(\bold{x}),
\label{eq:eq2}
\end{equation}

\begin{equation}
\tilde{D}(\omega,x_r,z\small{=}0) = \sum_{x_s} D(\omega,x_s,x_r,z=0) Q(\omega,x_s,z=0),
\label{eq:eq3}
\end{equation}


\section{Conclusions}
I showed how to estimate angle-domain parameters from the subsurface-offset domain using what I call weighted offset-to-angle, particularly subsurface-offset Hessian diagonals. The proposed approach provides useful information, which can be confirmed by the amplitude compensation results. The transformation of off-diagonal terms of the subsurface-offset Hessian indicates that the results are not strongly dependent on the amplitude distribution in the ADCIGs. However, it is still not clear how to use these transformed off-diagonal terms in inversion schemes.
\section{Aknowledgements}
I would like to thank Alejandro Valenciano for providing me the subsurface-offset domain Hessian and common-image gathers and for the explanations and fruitful discussions.
 
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