A preconditioning scheme for full waveform inversion

Preconditioning

Preconditioning amounts to a change of variable $\mathbf{m=Sp}$ where $\mathbf{p}$ is a new variable used for the inversion and $\mathbf{S}$ the preconditioning operator. In this paper, we define this operator as non-stationary deconvolution with local dip-filters. Having introduced the new variable $\mathbf{p}$ , the iterative scheme in equation (5) becomes:

$$\mathbf{p}_{n+1}=\mathbf{p}_n - \alpha_n \tilde{\mathbf{H}}_n^{-1}\nabla \tilde{f}(\mathbf{p}_n),$$ (6)

where

$$\nabla \tilde{f}(\textbf{p}_n)= \mathbf{S'}\nabla f(\textbf{m}_n)= \mathbf{S'}\nabla f(\textbf{Sp}_n)$$ (7)

and $\mathbf{S'}$ is the transpose of $\mathbf{S}$ . 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 $\mathbf{S}$ and compute its adjoint and inverse (to accommodate any starting guess $\mathbf{p}_0=\mathbf{S}^{-1}\mathbf{m}_0$ ), preconditioning can be easily introduced in any waveform inversion scheme. Once a solution vector $\mathbf{p}_{sol}$ has been found, the final model is obtained by computing

$$\mathbf{m}_{sol}=\mathbf{Sp}_{sol}.$$ (8)

Now we present our choice of preconditioning operator $\mathbf{S}$ .

A preconditioning scheme for full waveform inversion