\shortnote
\lefthead{Guerra}
\righthead{Rotation of coordinates}
\email{guerrac@stanford.edu}
\footer{ }
\title{Rotation of coordinates for Prestack exploding-reflectors modeling}
\author{Claudio Guerra}

\vspace{.5cm}

\section{Rotation of coordinates} 
Here, I derive the coordinate transformation necessary to set correct initial conditions for the source and receiver wave-field modeling from subsurface-offset domain common-image gathers poorly focused because of velocity inaccuracies. I start from Figure~\ref{fig:cig01} where $\bf{I_x}$$(z_{\xi}^i,x_{\xi}^i)$ is the image point considered in the computation of the horizontal subsurface-offset gather (HODCIG). $\bf{S_x}$$ = (z_{\xi}^s,x_{\xi}-h_x)$ and $\bf{R_x}$$ = (z_{\xi}^r,x_{\xi}^i+h_x)$ are the vector positions of the source and receiver propagated into the earth by wave-field extrapolation according to a velocity model, in this case slower than the actual velocity. Notice that the source- and receiver-depth coordinates are different from the depth coordinate of the image point $\bf{I_x}$ to accomodate velocity errors and illumination artifacts. $I_0$$(z_0,x_0)$ is the image point involved in the computation of geological offset gather and $\bf{S_0}$$ = (z_0^s,x_0-h_0)$ and $\bf{R_0}$$ = (z_0^r,x_0+h_0)$ are the correspondent source and receiver vector positions, respectively. The source- and receiver-depth coordinates are different from the image point 
\par
Consider a coordinate system centered at $(z_{\xi}^i,x_{\xi}^i)$. So, in this new coordinate system, $\bf{S_x}$$ = S_x(0,-h_x)$ and $\bf{R_x} = R_x(0,+h_x)$. After rotation according to the apparent dip, $-\alpha$, $\bf{S_x}$ and $\bf{R_x}$ originate the vectors $\bf{S_x^*}=(z_{\xi}^{*s},x_{\xi}^{*s})$ and $\bf{R_x^*}=(z_{\xi}^{*r},x_{\xi}^{*r})$. The stretch of the rotated offset axes generates the vectors $\bf{S_0^*}=(z_0^{*s},x_0^{*s})$ and $\bf{R_0^*}=(z_0^{*r},x_0^{*r})$. By shifting the coordinate system and vectors to a new position, $I_0(z_0,x_0)$, the new coordinates of the vectors are 
\plot{cig01}{width=6.0in}{Geometry of the problem.}

\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}



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