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 and on the offset vector of the desired output data , where the unit vectors and 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 between the midpoint of the input trace and the midpoint of the output trace. The analytical expression of the AMO saddle is,

(1) |

The surface represented by equation (1) is a skewed
saddle; its shape and spatial extent are controlled by the values of
the absolute offsets *h _{1}* and

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 and minor diagonal
,as shown in Figure 1.
The effective AMO aperture is often much narrower
than the parallelogram and is,
for given and , 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
The maximum spatial support of the AMO operator (shaded parallelogram)
in the midpoint plane (),
as a function of the input offset ,and the output offset .Figure 1 |

amo-max
AMO impulse response
when
,,
,,and no limitations are imposed on the operator aperture.
Figure 2 |

amo-eff
AMO impulse response
when
,,
,,and the aperture is limited assuming
.Figure 3 |

Figure 2 shows the surface of the AMO
impulse response when *t _{1}*=1 s,

11/11/1997