Preconditioning amounts to a change of variable
is a new variable used for the inversion and
the preconditioning operator.
In this paper, we define this operator as non-stationary deconvolution with local dip-filters. Having introduced the new variable
, the iterative scheme in equation (5) becomes:
is the transpose of
. Therefore, with preconditioning, we obtain a new gradient direction. In our scheme, we will opt for
a preconditioning operator that steers the gradient toward a geologically constrained solution. Note that in Equation (6) the approximate
Hessian in equation (5) is blind to this change of variable as it is built from gradient and solution step vectors only.
Assuming that we can estimate
and compute its adjoint and inverse (to accommodate any starting guess
preconditioning can be easily introduced in any waveform inversion scheme. Once a solution vector
has been found, the final model is obtained
Now we present our choice of preconditioning operator
A preconditioning scheme for full waveform inversion