Multi-Parameter FWI - An Illustration of the Hessian Operator Role for Mitigating Trade-off between Parameter Classes
Résumé
Full Waveform Inversion is a powerful tool for quantitative estimation of subsurface parameters (P-wave, S-wave velocities, density, attenuation, anisotropy parameters). This methods has been applied successfully to 2D acoustic and elastic reconstructions, as well as to 3D acoustic reconstructions. Most of the applications of Full Waveform Inversion have been devoted to mono-parameter reconstructions of wave velocities. Multi-parameter Full Waveform Inversion aims at reconstructing simultaneously different class of subsurface parameters. This is a very challenging task: the similarity of the sensitivity of the data to different classes of parameters is the source of trade-off (or cross-talk) which renders the Full Waveform Inversion problem even more undetermined than in the mono-parameter context. This can related to the similarity of the diffraction patterns of different classes parameters for a given propagation regime. In order to overcome this difficulty, the role of the Hessian operator should be crucial. The off-diagonal blocks of this operator accounts for the trade-off between parameters. Incorporating the inverse Hessian operator within the Full Waveform Inversion scheme may help to alleviate this difficulty. On this basis, we provide in this study a very simple example for which we can compute exactly the Hessian operator we use to illustrate these issues.