AMO aperture

Although the expression for the kinematics of AMO is velocity independent, the aperture of the operator depends strongly on velocity. The domain of influence of the AMO operator is defined by the region of validity of expression amo_surf.eq, which becomes singular when either of the following conditions is fulfilled:

$$\begin{eqnarray} \frac{\left\vert {\bf \Delta m}\times {\bf h}_{1}\right\vert}{\... ...{1}\times {\bf h}_{2}\right\vert} & = & 1. \EQNLABEL{parallel.eq}\end{eqnarray}$$ (2)

Geometrically, this means that the support of the AMO operator at the recording surface is limited to the region within the parallelogram with s main diagonal $\left({\bf h}_{1}+ {\bf h}_{2}\right)$ and minor diagonal $\left({\bf h}_{1}- {\bf h}_{2}\right)$. The shaded area in Figure amoapert shows a sample parallelogram that represents the maximum possible spatial extent of the AMO operator. More stringent bounds for the AMO aperture were presented by Biondi et. al. 1998. For given ${\bf h}_{1}$ and ${\bf h}_{2}$, these bounds are a function of the input traveltime and the minimum velocity V_min. The parallelogram in Figure amoapert therefore represents the extreme case where either the velocity or the input traveltime is equal to zero. Figure impulse-small shows the effective AMO impulse response for a limited aperture corresponding to a realistic minimum velocity of 2 ${\rm km/s}$.Figure impulse-small is significantly narrower than the whole impulse response shown in Figure impulse-big. This velocity-dependent aperture limitation is important for an efficient use of AMO and makes the cost of applying AMO to the data negligible compared to the cost of full 3-D prestack migration.

 

amoapert
Figure 2 The maximum spatial support of the AMO operator (shaded parallelogram) in the midpoint plane ($\Delta m_x,\Delta m_y$), as a function of the input offset ${\bf h}_{1}$,and the output offset ${\bf h}_{2}$.

 

impulse-small
Figure 3 The effective AMO impulse response when $V_{min}= 2~{\rm km/s}$, and $t_1 = 1~{\rm s}$,$h_{1}= 2~{\rm km}$, $h_{2}= 1.8~{\rm km}$,$\theta_{1}= 0^{\circ}$,$\theta_{2}= 30^{\circ}$,as in Figure 1.

The effective aperture becomes tiny when the azimuth rotation $\Delta \theta$ is small. At the limit, the expression in equation amo_surf.eq is singular when the azimuth rotation vanishes and the AMO surface reduces to a 2-D line. This operator, corresponding to the case of offset continuation Bolondi et al. (1984), has been derived independently by Biondi and Chemingui 1994, Stovas and Fomel 1996, and (in a different form) Bagaini et al. 1994. Its expression is given by the following quadric equation,

$$\begin{eqnarray} \lefteqn{{t}_{2}(\Delta m,h_{1},h_{2},{t}_{1})=} \nonumber \\ &... ...m^2]}}} {\sqrt{2}h_{2}} & h_{2}\leq h_{1}. \EQNLABEL{same_azim.eq}\end{eqnarray}$$ (3)

Taking into account the effective aperture of the AMO operator, it can be shown Fomel and Biondi (1995a) that the 3-D operator monotonously shrinks to a line, and the limit of the kinematics of the 3-D operator [equation amo_surf.eq] approaches the 2-D operator [equation same_azim.eq].