The inverse of the cross product operator *LL*^{T} acts as a filter
for the data space.
Each element *A*_{ij} of (*LL*^{T})^{-1} measures the correlation between a data
element *d*_{i} and another data element *d*_{j}. The computation of each
element *A*_{ij} requires the evaluation of an inner product in the
model space. Since the model space is regularly sampled,
the inner products for several imaging operators can be computed analytically
which leads to a fast and affordable evaluation of the elements of *LL*^{T}.
This is the case for the solution we
present in this paper.

Let's consider an irregularly sampled input of *n* seismic traces
and let *L*_{m,di} be
the operator that maps trace *d*_{i} into the model space *m*.
The operator that performs the inverse mapping is therefore *L*^{T}_{m,di}.
In matrix notation, we write the cross-product matrix *LL*^{T} as

(6) |

(7) |

where *A*_{(hi,hj)} is AMO from input offset *h*_{i} to output offset *h*_{j}
and, *I* is the identity operator (mapping from *h*_{i} to *h*_{i}). Comforming
to the definition of AMO Biondi et al. (1996), *A*_{(hi,hj)}
is the adjoint of *A*_{(hj,hi)}; therefore, the filter is Hermitian
with diagonal elements being the identity and off-diagonal elements being
AMO transforms.

This is the fundamental definition of **A** that will allow
a fast and efficient numerical approximation of its inverse,
and thus of the whole prestack imaging inverse problem.
Being a narrow operator, the cost
of applying AMO to prestack data is almost negligible compared
to other imaging operators such as prestack migration.
When the azimuth rotation
or the offset continuation is small, as is the case for geometry
regularization problems, the size of the operator
is very small, but also has a strong curvature.
Biondi et al 1996 discussed the design of an efficient
implementation of the AMO operator that avoids aliasing and simultaneously
takes advantage of the opportunity for saving computation by properly limiting
the spatial extent of the numerical integration.

11/11/1997