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 V_min, 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)

 

$$t_2 \leq \frac{2}{V_{min}\sqrt{\left(\gamma_1^2 + \gamma_2^2... ..._1\gamma_2\cos\Delta \theta\right)\left(1-{\xi_1}^2\right)}}\;.$$ (12)

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

 

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