Explicit vs. implicit Hessian matrix computation

Equation 17 expresses the angle-domain Hessian as a chain of the offset-to-angle operator and the subsurface offset Hessian matrix. This implies that to implement the angle-domain wave-equation inversion using a conjugate gradient algorithm there is no need to explicitly compute the angle Hessian matrix. But the possible drawback is that, for each iteration, the offset-to-angle transformation needs to be performed.

A different strategy might be to explicitly compute the angle-domain Hessian matrix. This can be done by a simple manipulation the terms in equation 17

$$\begin{eqnarray} {\bf H}({\bf x,\Theta};{\bf x',\Theta'}) &=& {\bf T'} (\Theta... ... T'}(\Theta',{\bf h'}){\bf H}'({\bf x,h};{\bf x',h'})\right)'. \end{eqnarray}$$ (18)

Due to the symmetry of the Hessian matrix equation 18 turns into:

$$\begin{eqnarray} {\bf H}({\bf x,\Theta};{\bf x',\Theta'})&=& {\bf T'} (\Theta,{\... ...bf T'}(\Theta',{\bf h'}){\bf H}({\bf x,h};{\bf x',h'})\right)'. \end{eqnarray}$$ (19)

In practice, equation 19 takes the subsurface offset Hessian matrix and applies an offset-to-angle transformation, then transposes the resulting matrix and reapplies the same offset-to-angle transformation.

This explicit angle Hessian matrix computation could be an expensive operation, but it has the advantage of only needing to be performed once. In contrast to the application of a chain of the offset-to-angle operator and the subsurface offset Hessian matrix (implicit approach) which needs to be performed at each conjugate gradient iteration. Each approach has its advantages and disadvantages, thus the specific application will dictate which path to follow.