AMO OPERATOR

We define AMO as an operator that transforms 3-D prestack data with a given offset and azimuth to equivalent data with different offset and azimuth. To derive the AMO operator we collapse in one single step the cascade of an imaging operator and a forward modeling operator. In principle, any 3-D prestack imaging operator can be used for defining AMO. We initially cascaded DMO and ``inverse'' DMO, but, to derive an accurate expression for the spatial aperture of AMO, we had to use full 3-D prestack constant velocity migration and its inverse. As expected, the kinematics of AMO, as defined after NMO, are independent from its derivation.

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's 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, as we derive it in Appendices A and B, 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 has a maximum for rotation of 90 degrees, 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. Figure  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.

 

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

The expression for the kinematics is velocity independent, but the lateral aperture of the operator does depend on velocity. An upper bound on the spatial extent of the AMO operator is defined by the region where the expression in equation (1) is valid. Equation (1) becomes singular when either of the following conditions are fulfilled

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

The geometric interpretation of these conditions is that the support of the AMO operator is limited to the region within the parallelogram with 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  shows an example parallelogram that represents the maximum possible spatial extent of the AMO operator. In appendix C we derive more stringent bounds for the AMO aperture, that are, for given ${\bf h}_{1}$ and ${\bf h}_{2}$, function of the minimum velocity V_min and of the input traveltime. The parallelogram in Figure  is thus the worse case, when either the velocity or the input traveltime is equal to zero. Figure  shows the effective AMO impulse response when the velocity-dependent aperture limitation, corresponding to a realistic minimum velocity of 2 ${\rm km/s}$,is applied to the impulse response shown in Figure . The surface shown in Figure  is significantly narrower than the whole impulse response shown in Figure . This velocity-dependent aperture limitation is important for an efficient use of AMO and it contributes to make the cost of applying AMO to the data negligible compared to the cost of applying a 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 ${\bf h}_{2}$.

 

impsmallcrop
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 (1) 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,

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

The apparent dichotomy between the 3-D and the 2-D solutions is reconciled when the effective aperture of the AMO operator is taken into account, and 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 (1)] approaches the 2-D operator [equation (3)].

While the kinematics of AMO are independent from its derivation, the amplitude term varies according to the derivation. We present, and used for the AMO applications shown in this paper, the AMO amplitude that is related to Zhang-Black DMO. It can be shown that the choice of the Zhang-Black's Jacobian yields an amplitude-preserving AMO operator, at least when applied on regularly sampled common offset-azimuth cubes Chemingui and Biondi (1995). This particular choice of the Jacobian results into the following expression for the amplitude term,

$$\begin{eqnarray} \lefteqn{A\left({{\bf \Delta m}},{{\bf h}_{1}},{{\bf h}_{2}},{t... ..._1-\Delta \varphi)} \over {h_{2}^2\sin^2\Delta \theta}}} \right)}.\end{eqnarray}$$ (4)

Notice that the frequency $\left\vert\omega_2\right\vert$ enters as multiplicative factor in the expression for AMO amplitudes. This term can be applied to the output data in the time domain by cascading a causal half-differentiator with an anti-causal half-differentiator.