In this Appendix, we describe the main characteristics of our implementation of the Kirchhoff AMO operator. This implementation is based on analysis presented in previous reports Biondi and Chemingui (1994); Chemingui and Biondi (1995); Fomel and Biondi (1995a,b). The AMO integration surface has the shape of a saddle. The exact shape of the saddle depends on the azimuth rotation and offset continuation that are applied to the input data. When the azimuth rotation is small, the saddle has a strong curvature that makes its straightforward integral implementation inaccurate. We address this problem by performing the spatial integration in a transformed coordinate system. In this new coordinate system, the AMO surface is well behaved, and its shape is invariant with respect to the amount of azimuth rotation and offset continuation.
The appropriate coordinate transformation is described by the following chain of transformations
In this new coordinate system, the kinematics of AMO are described by the following simple relationship between the input time t1 and the output time t2.
and the amplitudes (based on Zhang-Black amplitudes for DMO) are described by the following equation
Notice that this expression for the amplitudes does already take into account the Jacobian of the transformation described in (equation (1)).
The result of the AMO integral needs also to be half-differentiated twice, once with a causal and once with an anti-causal differentiator.