\shortnote
\lefthead{Guerra}
\righthead{Residual off2ang}
\email{guerrac@stanford.edu}
\footer{SEP--131}
\title{Residual offset-to-angle .....}
\author{Claudio Guerra}

\vspace{.5cm}

\section{Introduction} 
Migration velocity analysis strongly relies on the analysis of the curvature of the reflectors in angle-domain common-image gathers (ADCIG). Unfortunately, because of uneven sampling of the wavefields, after transformation from the subsurface-offset domain image gathers (ODCIG) \cite{Prucha.sep.103.marie1,Sava.sep.103.paul2}, ADCIGs can show curved events which can mislead migration velocity interpretation.
\par
Motivated by the work of \cite{tang1}, here I propose a method to attenuate undesirable kinematic artifacts in angle-domain common-image gathers (ADCIG) which have their origin in badly sampled receiver or source wavefields. The most direct application of the method is to pre-process ADCIGs prior to migration velocity analysis, in such a way that the remaining curved events are only related to incorrect migration velocities. This method along with illumination compensation **cite{guerra}** are cheap ways to produce reasonable ADCIGs if compared to least-squares inversion \cite{Prucha.sep.103.marie1,Valenciano.sep.124.alejandro1}. 
\par
Numerical results show the bla,bla,bla....

\section{Badly sampled receiver wavefield}

In a locally-constant velocity media, the source-wavefield is represented by the semi-circle
\begin{equation}
(x - x_s)^2 + z^2 = z_m^2 + (x_m - x_s)^2 ,
\label{eq:sw1}
\end{equation}
where $x_s$ is the horizontal shot coordinate and $(x_m,z_m)$ are the coordinates of the image point. Similarly, the receiver-wavefield is the semi-circle given by
\begin{equation}
(x - x_r)^2 + z^2 = z_m^2 + (x_r - x_m)^2 ,
\label{eq:rw1}
\end{equation}
where $x_r$ is the horizontal receiver coordinate. To derive the analytical formula of the image in a ODCIG, $I(z,x,h_x)$, we must apply the multi-offset imaging condition
\begin{equation}
I (z,x,h_x) = \sum_{\omega} D^{\ast}(z,x - h_x,\omega )U(z,x + h_x,\omega ),
\label{eq:ic}
\end{equation}
where $D$ is the downward is the source-wavefield, $U$ is the receiver-wavefield, $\omega$ is frequency, $h_x$ is the subsurface-offset. This means to evaluate equations \ref{eq:sw1} and \ref{eq:rw1} at different subsurface-offsets. So, equations \ref{eq:sw1} and \ref{eq:rw1} reads:
\begin{equation}
(x - h_x - x_s)^2 + z^2 = z_m^2 + (x_m - x_s)^2
\label{eq:sw2}
\end{equation}
and
\begin{equation}
(x + h_x - x_r)^2 + z^2 = z_m^2 + (x_r - x_m)^2 .
\label{eq:rw2}
\end{equation}
By summing these equations and taking into account that the image is formed when $x = x_m$, we get the following relation between depth and subsurface-offset at the image coordinate $x_m$:
\begin{equation}
z = \sqrt{z_m^2 - h_x^2 + h_x(x_r - x_s)} .
\label{eq:ocig1}
\end{equation}
The computation of the ADCIG, $Q$, from the ODCIG, $P$,can be viewed as a process in which we compute (a) the derivative of $z$ with respect to $h_x$ evaluated at $h_x^{\star}$, where the slanted-path operator is tangent to $z$, obtaining the tangent of the reflection angle, $\gamma$, and (b) the depth, $z_0$, at which the slanted-path intersects the depth axis; and place at $Q(\gamma,z_0)$ the amplitude collected at $P(h_x,z)$. The derivative $\frac{dz}{dh_x}$ of the equation \ref{eq:rw2} is
\begin{equation}
\frac{dz}{dh_x} = \tan (\gamma) = \frac{x_r - x_s - 2h_x}{2z} .
\label{eq:dzdh}
\end{equation}
The depth, $z_0$, in the angle gather is given by 
\begin{equation}
z_0 = z - h \tan (\gamma) .
\label{eq:slant}
\end{equation}
Substituting the value for $h_x$ from the equation \ref{eq:dzdh} in equation \ref{eq:ocig1} we get, after some manipulation:
\begin{equation}
z = \sqrt{z_m^2 + \frac{(x_r - x_s)^2}{4}} \cos(\gamma) .
\label{eq:ocig2}
\end{equation}
The slope at $h_x = 0$ is the tangent of the reflection angle, $\gamma_0$, at the image point, $I (z_m,x_m)$, correspondent to the considered source-receiver pair. From this we get:
\begin{equation}
\tan (\gamma_0) = \frac{x_r - x_s}{2z_m} .
\label{eq:gam0}
\end{equation}
From equation \ref{eq:gam0} and \ref{eq:ocig2}, $z$ assumes the form
\begin{equation}
z = z_m\sqrt{1 + \tan^2 (\gamma_0)} \cos(\gamma) = z_m (\sec (\gamma_0) \cos (\gamma)) .
\label{eq:ocig3}
\end{equation}
According to equations \ref{eq:dzdh}, \ref{eq:gam0} and \ref{eq:ocig3}, $h_x$ can be recast as
\begin{equation}
h_x = z_m(\tan (\gamma_0) - \sec (\gamma_0) \sin (\gamma)) .
\label{eq:hx1}
\end{equation}
Finally, substituting equations \ref{eq:ocig3} and \ref{eq:hx1} in the equation \ref{eq:slant} we get
\begin{equation}
z_0 = z_m (\sec (\gamma_0) \sec (\gamma) - \tan (\gamma_0) \tan (\gamma)).
\label{eq:slant2}
\end{equation}
Equation \ref{eq:slant2} if applied to the ADCIG resulting from the transformation to angle of the ODCIG given by equation \ref{eq:ocig1} will focus the smeared energy at the point $(\gamma_0,z_m)$ correspondent to the correct position of the ADCIG from a well-sampled receiver wavefield.

Figure \ref{fig:odcig_bad}, Figure \ref{fig:adcig_bad}, Figure \ref{fig:adcig2_bad}.
\plot{odcig_bad}{width=4.0in,height=6.0in}{ODCIG badly sampled.}
\plot{adcig_bad}{width=4.0in,height=6.0in}{ADCIG resulting from the OFF2ANG of the ODCIG from Figure \ref{fig:odcig_bad}.}
\plot{adcig2_bad}{width=4.0in,height=6.0in}{ADCIG resulting from the residual correction of the ADCIG from Figure \ref{fig:adcig_bad}}.

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