IMPLEMENTATION OF INTEGRAL AMO

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  

$$\left[ \begin{array} {c} z_1 \ z_2 \ \end{array} \... ...\left[ \begin{array} {c} x_m \ y_m \end{array} \right]\;,$$ (1)

where x_m and y_m are the original midpoint coordinates, z₁, and z₂ are the transformed coordinates, and $\alpha_1$ and $\alpha_2$ are respectively the azimuth of the input trace and the azimuth of the output trace. The right matrix represents a space invariant rotational squeezing of the coordinate, while the left matrix is a simple rescaling of the axes by a factor dependent on the azimuth rotation $\alpha=\alpha_2-\alpha_1$.When the azimuth rotation is zero, the transformation described in equation (1) becomes singular. In this case the AMO operator degenerates into the 2-D offset continuation operator Biondi and Chemingui (1994); Fomel (1995). In practice, a simple pragmatic method to avoid the singularity is to set a lower limit for the product $\left\vert{\bf h_1}\right\vert\left\vert{\bf h_2}\right\vert\sin\left(\alpha \right)$.Since the 3-D AMO operator converges smoothly to the 2-D offset continuation operator Fomel and Biondi (1995a), the error introduced by this approximation is negligible. 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.  

$$t_2=t_1\,\sqrt{\frac{1-{z_2}^2}{1-{z_1}^2}}\;,$$ (2)

and the amplitudes (based on Zhang-Black amplitudes for DMO) are described by the following equation  

$$A=t_2\frac{\left(1+{z_2}^2\right)}{\left(1-{z_1}^2\right)\left(1-{z_2}^2\right)}\;.$$ (3)

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.