AZIMUTH-MOVEOUT TRANSFORMATION

Azimuth moveout is a crucial component of the proposed common-azimuth imaging procedure; it is the process necessary to transform narrow-azimuth data collected with multiple streamers into effective common-azimuth data. Although the AMO transformation to common azimuth is approximate, AMO removes much of the dip limitations that are intrinsic in standard methods to transform prestack data.

Azimuth moveout transforms 3-D prestack data with a given offset and azimuth to equivalent data with different offset and azimuth. The AMO operator is derived by collapsing in one single step the cascade of an imaging operator and a forward modeling operator Biondi et al. (1996a). In principle, any 3-D prestack imaging operator can be used for defining AMO. AMO has been derived both as a cascade of DMO and ``inverse'' DMO and as the cascade of full 3-D prestack constant velocity migration and its inverse. AMO is applied after NMO, and thus velocity heterogeneities are taken into account, at least at first order, by the NMO step.

AMO is not a single-trace to single-trace transformation, but it is a partial-migration operator that moves events across midpoints according to their dip. Its impulse response is 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 offset vector of the desired output data ${\bf h}_{2}=h_{2} (\cos \theta_{2},\sin \theta_{2})$, where the unit vectors $\bf x$ and $\bf y$ point respectively in the in-line direction and the cross-line direction. The time shift to be applied to the data is a function of the difference vector ${\bf \Delta m}=\Delta m(\cos \Delta \varphi,\sin \Delta \varphi)$ between the midpoint of the input trace and the midpoint of the output trace. The analytical expression of the AMO saddle is,  

$${t}_{2}\left({{\bf \Delta m}},{{\bf h}_{1}},{{\bf h}_{2}},{t... ...\theta_1-\theta_2)-\Delta m^2\sin^2(\theta_1-\Delta \varphi)}}.$$ (1)

The traveltimes t₁ and t₂ are respectively the traveltime of the input data after NMO has been applied, and the traveltime of the results before inverse NMO has been applied.

The surface represented by equation (1) is a skewed saddle; its shape and spatial extent are controlled by the values of the absolute offsets h₁ and h₂, and by the azimuth rotation $\Delta \theta=\theta_{1}-\theta_{2}$. Consistent with intuition, the spatial extent of the operator is maximum when $\Delta \theta=90^{\circ}$and it vanishes when offsets and azimuth rotation tend to zero. Furthermore, it can be easily verified that t₂= t₁ for the zero-dip components of the data; that is, the kinematics of zero-dip data after NMO do not depend on azimuth and offset.

The expression for the AMO saddle is velocity independent, but the lateral aperture of the operator is velocity dependent. An upper bound on the spatial extent of the AMO operator is defined by the region where the expression in equation (1) is valid. This region is delimited by the parallelogram with main diagonal $\left({\bf h}_{1}+ {\bf h}_{2}\right)$ and minor diagonal $\left({\bf h}_{1}- {\bf h}_{2}\right)$,as shown in Figure 1. The effective AMO aperture is often much narrower than the parallelogram and is, for given ${\bf h}_{1}$ and ${\bf h}_{2}$, a function of the minimum velocity V_min and the input traveltime. Because of its limited aperture, AMO is relatively inexpensive to apply.

 

amo-apert Figure 1 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 ${\bf h}_{2}$.

 

amo-max Figure 2 AMO impulse response when $t_1 = 1~{\rm s}$,$h_{1}= 2~{\rm km}$, $h_{2}= 1.8~{\rm km}$,$\theta_{1}= 0^{\circ}$,$\theta_{2}= 30^{\circ}$and no limitations are imposed on the operator aperture.

 

amo-eff Figure 3 AMO impulse response when $t_1 = 1~{\rm s}$,$h_{1}= 2~{\rm km}$, $h_{2}= 1.8~{\rm km}$,$\theta_{1}= 0^{\circ}$,$\theta_{2}= 30^{\circ}$and the aperture is limited assuming $V_{min}= 2~{\rm km/s}$.

Figure 2 shows the surface of the AMO impulse response when t₁=1 s, h₁=2 km, h₂=1.8 km, $\theta_{1}= 0^{\circ}$, and $\theta_{2}= 30^{\circ}$,and no limitations are imposed on the operator aperture. In contrast, Figure 3 shows the surface of the AMO impulse response with the same parameters as in Figure 2, but with the aperture limited by assuming a minimum velocity $V_{min}= 2~{\rm km/s}$.