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AMO aperture

Cascading DMO and inverse DMO allowed us to evaluate the summation path of AMO and the corresponding weighting function. However, this procedure is not sufficient for evaluating the third major component of the integral operator, that is, its aperture (range of integration). To solve this problem, Fomel and Biondi 1995c applied an alternative approach by defining AMO as the cascade of 3D common-offset common-azimuth migration and 3D modeling at a different azimuth and offset. I birefly discuss their derivations in this appendix as part of a complete definition of integral AMO.

The impulse response of the common-offset common-azimuth migration is a symmetric ellipsoid with the center in the input midpoint and axis of symmetry along the input-offset direction. Such an ellipsoid is described by the general formula
z({\bf m})=\sqrt{R^2- \Delta m^2 + \gamma\,
 ... m}\cdot{\bf h}_{1}\right)^2}{h_{1}^2}}\;,
\EQNLABEL{ellips.eq}\end{displaymath} (90)
where z stands for the depth coordinate, ${\bf m}$ is the surface coordinate, ${\bf \Delta m}= {\bf m}- {\bf m}_1$, R is the small semi-axis of the ellipsoid, and $\gamma$ is a nondimensional parameter describing the stretching of the ellipse $(\gamma < 1)$. Deregowski and Rocca 1981 derived the following connections between the geometric properties of the reflector and the coordinates of the corresponding impulse in the data:
R={{v\,t_1}\over 2}\;;\;
\gamma={{{4\,{h_{1}}^2}\over v^2} \over 
t_1^2+ {{4\,{h_{1}}^2}\over v^2}}\;,
\EQNLABEL{rbeta.eq}\end{displaymath} (91)
where v is the propagation velocity.

The impulse response of the AMO operator corresponds kinematically to the reflections from the ellipsoid defined by equation ellips.eq to a different azimuth and different offset. To constrain the AMO aperture, Fomel and Biondi 1995c based their derivations on answering the following question: For a given elliptic reflector defined by the input midpoint, offset, and time coordinates, what points on the surface can form a source-receiver pair valid for a reflection? If a point in the output midpoint-offset space cannot be related to a reflection pattern, it should be excluded from the AMO aperture.

Fermat's principle provides a general method of solving the kinematic reflection problem Goldin (1986). The formal expression for the two-point reflection traveltime is given by
t_2={\sqrt{({\bf s}_2- {\bf m})^2+z^2({\bf m})} \over v}+ 
 ...2- {\bf m})^2+z^2({\bf m})} \over v}\;,
\EQNLABEL{reflecttt.eq}\end{displaymath} (92)
where ${\bf m}$ is the vertical projection of the reflection point to the surface, ${\bf s}_2$ is the source location, and ${\bf r}_2$ is the receiver location for the output trace. According to Fermat's principle, the reflection raypath between two fixed points must correspond to the extremum value of the traveltime. Hence, in the vicinity of a reflected ray,
{\partial t_2 \over \partial {\bf m}}=0\;.
\EQNLABEL{fermat.eq}\end{displaymath} (93)
Solving equation fermat.eq for ${\bf m}$ allows us to find the reflection raypath for a given source-receiver pair on the surface.

To find the solution of fermat.eq, it is convenient to decompose the reflection-point projection ${\bf m}$ into three components: ${\bf m}= {\bf m}_1+ {\bf m}_{\parallel}+ {\bf m}_{\perp}$, where ${\bf m}_{\parallel}$ is parallel to the input offset vector ${\bf h}_{1}$, and ${\bf m}_{\perp}$ is perpendicular to ${\bf h}_{1}$. The plane, drawn through the reflection point and the central line of ellipsoid ellips.eq, must contain the zero-offset (normally reflected) ray because of the cylindrical symmetry of the reflector. The fact that the zero-offset ray is normal to the reflector gives us the following connection between the zero-offset midpoint ${\bf m}_0$ and the ${\bf m}_{\parallel}$ component of the reflection point ${\bf m}$:
{\bf m}_0= 
({\bf m}_1+ {\bf m}_{\parallel}) + z \left({\bf ...
{\bf m}_1+ \gamma\,{\bf m}_{\parallel}\,.
\EQNLABEL{xi2x0.eq}\end{displaymath} (94)
Equation xi2x0.eq evaluates ${\bf m}_{\parallel}$ in terms of ${\bf m}_0$, as follows:
{\bf m}_{\parallel}={{{\bf \Delta m}_{10}}\over{\gamma}}\;.
\EQNLABEL{x02xi.eq}\end{displaymath} (95)
where the length of the vector ${\bf \Delta m}_{10} = {\bf m}_0- {\bf m}_1$ can be determined using the following geometric connection between any given input and output midpoints ${\bf m}_1$ and ${\bf m}_2$, and azimuths $\theta_{1}$ and $\theta_{2}$

\left\vert {\Delta m\over \ {\sin \Delta \theta}} \right\ver...
 ... (\theta_1-\Delta \varphi)}} \right\vert\;.
\EQNLABEL{sines.eq}\end{displaymath} (96)

To find the third component of the reflection point projection ${\bf m}_{\perp}$, we substitute expression x02xi.eq into reflecttt.eq. Choosing a convenient parameterization ${\bf s}_2= {\bf m}_0+ {\bf h}_{2}^s$, ${\bf r}_2= {\bf m}_0+ {\bf h}_{2}^r$, where ${\bf h}_{2}^r-{\bf h}_{2}^s=2\,{\bf h}_{2}$, and ${\bf h}_{2}^r+{\bf h}_{2}^s = 2\,{\bf \Delta m}_{02} = 2\,\left({\bf m}_2-{\bf m}_0\right)$, we can rewrite the two-point traveltime function from reflecttt.eq in the form

t_2 = {\sqrt{R^2 - \gamma\,(1- \gamma) m_{\parallel}^2 +
 ..._{\perp}+ (1 - \gamma)\,{\bf m}_{\parallel}\right)}
\over v}\;+\end{displaymath}

{\sqrt{R^2 - \gamma\,(1- \gamma) m_{\parallel}^2 +
 ... m}_{\perp}+ (1 - \gamma)\,{\bf m}_{\parallel}\right)}
\over v}\end{displaymath} (97)
Fermat's principle fermat.eq leads to a simple linear equation for the length of ${\bf m}_{\perp}$, which has the explicit solution
m_{\perp}=(\gamma -1)\,m_{\parallel}\,\cot{\left(\theta_2 - ...
\sin{\left(\theta_2 - \theta_1\right)}}}\;, 
\EQNLABEL{xiy.eq}\end{displaymath} (98)
where $m_{\parallel}$ is defined by x02xi.eq, and $\Delta m_{02}$satisfies the relationship sines.eq.

Because the reflection point is contained inside the ellipsoid, its projection obeys the evident inequality
z^2 ({\bf m}) = R^2 - m_{\perp}^2 - (1 - \gamma)\,m_{\parallel}^2 \geq 0\;. 
\EQNLABEL{xiyleq.eq}\end{displaymath} (99)
It is inequality xiyleq.eq that defines the aperture of the AMO operator. After transformation transform and algebraic simplifications, it takes the form of inequality t2leq.eq, which is convenient for an efficient implementation of AMO.

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