The cross-product filter

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

 

$$LL^T = \left[ \begin{array} {cccc} \left[ L_{(m,d_1)}L^T_{... .... &\left[ L_{(m,d_n)}L^T_{(m,d_n)}\right] \end{array} \right]$$ (6)

Each inner product $\left[ L_{(m,d_i)}L^T_{(m,d_j)}\right]$ is therefore a reconstruction of a data trace with input offset h_i to a new trace with offset h_j. We recognize this mapping as our previously-defined AMO transformation. Therefore, we call this cross product filter A, and we write it in terms of its AMO elements as

 

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