The cross-product filter

The inverse of L^TL (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^TL (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 $\bf A$ in terms of its AMO elements as

$${\bf A}= \left[ \begin{array} {ccccc} I & A_{(h_1,h_2)} & A... ... A_{(h_n,h_3)} &...... & I \end{array} \right] \EQNLABEL{equ7}$$ (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 $\bf A$ 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.