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Operator aperture

The expression for the kinematics and amplitudes of AMO [equations (6 and 7)] are valid for $\xi_1$ and $\xi_2$ ranging between -1 and 1. However, for finite propagation velocities, the AMO operator has a much narrower aperture, as shown in appendix C. Taking into account this finite aperture is crucial both for accuracy and for efficiency. For a given minimum propagation velocity Vmin, the maximum output time can be evaluated according to the following expressions:

\begin{eqnarray}
\gamma_1=\frac{\frac{\partial t_2}{\partial \xi_1}}{h_{2}t_1\si...
 ...\frac{\partial t_2}{\partial \xi_2}}{h_{1}t_1\sin\Delta \theta}\;,\end{eqnarray} (10)
(11)
 
 \begin{displaymath}
t_2 \leq \frac{2}{V_{min}\sqrt{\left(\gamma_1^2 + \gamma_2^2...
 ..._1\gamma_2\cos\Delta \theta\right)\left(1-{\xi_1}^2\right)}}\;.\end{displaymath} (12)

To avoid truncation artifacts, we recommend the use of a tapering function at the edges of the operator aperture.

 
geoltr
geoltr
Figure 5
Geological in-line section and corresponding velocities of layers. From (Hanson and Witney, 1995).


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previous up next print clean
Next: APPLICATION OF AMO TO Up: INTEGRAL IMPLEMENTATION OF AMO Previous: Operator antialiasing
Stanford Exploration Project
11/11/1997