The inverse of *L*^{T}*L*
(*LL*^{T}) filters
the model (data) space to *correct* the
adjoint operator for the interdependencies between model (data) elements.
Each element *A*_{ij} of *L*^{T}*L* (*LL*^{T})
describes the correlation between the model parameter *m*_{i}
(data parameter *d*_{i})
and the model parameter *m*_{j}
(data parameter *d*_{j}).
Chemingui and Biondi 1997 showed that
the cross-product matrix is, in short, an AMO matrix.
We write the cross-product filter in terms
of its AMO elements as

(6) |

where *A*_{(hi,hj)} is the 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.
Since AMO has a narrow operator, the cost
of applying it to prestack data is almost negligible compared
to that of 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. Biondi et al 1996
discussed the design of an efficient
implementation of the AMO operator that
saves computation by properly limiting
the spatial extent of the numerical integration.

10/9/1997