A preconditioning scheme for full waveform inversion |

where

and 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 by computing

(8) |

Now we present our choice of preconditioning operator .

A preconditioning scheme for full waveform inversion |

2010-11-26