AMO impulse response

The impulse response of AMO is described by a saddle in the output midpoint domain. The shape of the saddle depends on the offset vector of the input data ${\bf h}_{1}=h_{1}\cos\theta_{1}{\bf x}+h_{1}\sin\theta_{1}{\bf y}=h_{1}(\cos\theta_{1},\sin\theta_{1})$ and on the output offset vector ${\bf h}_{2}=h_{2}(\cos \theta_{2},\sin \theta_{2})$,where the unit vectors $\bf x$ and $\bf y$ point respectively in the in-line direction and the cross-line direction. The analytical expression of the AMO saddle, as given in Appendix A, is,

$${t}_{2}\left({{\bf \Delta m}},{{\bf h}_{1}},{{\bf h}_{2}},{t... ...ta m^2\sin^2(\theta_1-\Delta \varphi)}}. \EQNLABEL{amo_surf.eq}$$ (1)

The traveltimes t₁ and t₂ are the traveltime of the input data after normal moveout correction and the traveltime of the results before inverse NMO. The midpoint vector ${\bf \Delta m}=\Delta m(\cos \Delta \varphi,\sin \Delta \varphi)$ is the difference between the input and output midpoint location vectors.

The surface represented by equation amo_surf.eq is a skewed saddle; its shape is controlled by the values of the absolute offsets h₁ and h₂, and by the azimuth rotation $\Delta \theta=\theta_{1}-\theta_{2}$. The spatial extent of the operator has a maximum for rotation of 90 degrees and vanishes when offsets and azimuth rotation tend to zero. Figure impulse-big shows the surface of the AMO impulse response when t₁ is equal to 1 s, h₁ is equal to 2 km, h₂ is equal to 1.8 km, $\theta_{1}$ is equal to zero degrees, and $\theta_{2}$ is equal to 30 degrees.

 

impulse-big
Figure 1 The full AMO impulse response $\left(V_{min}\simeq 0 \right)$ when $t_1 = 1~{\rm s}$,$h_{1}= 2~{\rm km}$, $h_{2}= 1.8~{\rm km}$,$\theta_{1}= 0^{\circ}$,$\theta_{2}= 30^{\circ}$.