Hessian sparsity and structure

Since the main contributions of the Hessian occurs around the diagonal Chavent and Plessix (1999); Valenciano and Biondi (2004), additional computational savings can be obtain by limiting the computation of equation (9) to ${\bf y}_T$ points close to ${\bf x}_T$. This reduces equation (9) to  

$${\bf H}({\bf x}_T,{\bf x}_T + {\bf a_x})=\sum_{\omega} \sum... ...}_r;\omega) {\bf G}({\bf x}_T + {\bf a_x},{\bf x}_r;\omega),$$ (10)

where ${\bf a_x}=(a_z,a_x,a_y)$ is the ``offset" from the point ${\bf x}_T$. Thus only a few elements of the Hessian matrix are computed (non-stationary filter coefficients).

To be able to perform the multidimensional convolution operation in equation (5), the computed Hessian elements in equation (10) have to be placed on a helix Claerbout (1998). After that, each row of the Hessian matrix is a multidimensional filter applied to the whole model space.

Chavent and Plessix (1999) qualitatively discuss the amount of the spreading away from the diagonal of the Hessian matrix. Future research will address the optimal number of filter coefficients needed to account for the spreading, since this number has a direct impact on the cost of the method.