Regularization by AMO

The main drawback of the method described above is that smoothing over offset/azimuth cubes by the inverse of the simple roughener operator expressed in equation (13) may result in loss of resolution when geological dips are present. It is well know that dipping events are not flattened by NMO with the same velocity as flat events. However, the method is easily generalized by substitution of the identity matrix in the lower diagonal of ${\bf D}_{{\bf h}}$with an appropriate operator that correctly transforms a common offset-azimuth cube into an equivalent cube with a different offset and azimuth. This can be accomplished by AMO Biondi et al. (1998). Since the cubes to be transformed are uniformly sampled we can use a Fourier-domain formulation of AMO that is both efficient and straightforward to implement Vlad and Biondi (2001). The roughener operator that includes AMO is then expressed as

$$\begin{eqnarray} \widetilde{\bf D}_{{\bf h}}= \frac{1}{1-\rho_D} \left[ { \matri... ...&\ddots&-\rho_D{\bf T}_{{\bf h}_{n-1,n}} &{\bf I} \cr } } \right],\end{eqnarray}$$ (16)

where ${\bf T}_{{\bf h}_{i,i+1}}$is the AMO operator that transforms the offset-azimuth cube i into the offset-azimuth cube i+1. The $\widetilde{\bf D}_{{\bf h}}^{'}\widetilde{\bf D}_{{\bf h}}$ operator can also be easily inverted by recursion and thus the least-squares problem obtained by substituting $\widetilde{\bf D}_{{\bf h}}$ for ${\bf D}_{{\bf h}}$ in equations (12) can also be easily preconditioned and normalized using the same techniques described in equations (14-18).