Non-linear partial derivative and its De Wolf approximation for non-linear seismic inversion

Abstract

We demonstrate that higher order Fréchet derivatives are not negligible and the linear Fréchet derivative may not be appropriate in many cases, especially when forward scattering is involved for large-scale or strong perturbations (heterogeneities). We then introduce and derive the non-linear partial derivative including all the higher order Fréchet derivatives for the acoustic wave equation. We prove that the higher order Fréchet derivatives can be realized by consecutive applications of the scattering operator and a zero-order propagator to the source. The full non-linear partial derivative is directly related to the full scattering series. The formulation of the full non-linear derivative can be used to non-linearly update the model. It also provides a new way for deriving better approximations beyond the linear Fréchet derivative (Born approximation). In the second part of the paper, we derive the De Wolf approximation (DWA; multiple forescattering and single backscattering approximation) for the non-linear partial derivative. We split the linear derivative operator (i.e. the scattering operator) into forward and backward derivatives, and then reorder and renormalize the multiple scattering series before making the approximation of dropping the multiple backscattering terms. This approximation can be useful for both theoretical derivation and numerical calculation. Through both theoretical analyses and numerical simulations, we show that for large-scale perturbations, the errors of the linear Fréchet derivative (Born approximation) are significant and unacceptable. In contrast, the DWA non-linear partial derivative (NLPD), can give fairly accurate waveforms. Application of the NLPD to the least-square inversion leads to a different inversion algorithm than the standard gradient method.