Nonlinear Frechet derivative and its De Wolf approximation

Abstract

We introduce and derive the nonlinear Frechet derivative for the acoustic wave equation. It turns out that the high order Frechet derivatives can be realized by consecutive applications of the scattering operator and a zero-order propagator to the source. We prove that the higher order Frechet derivatives are not negligible and the linear Frechet derivative may not be appropriate in many cases, especially when forward scattering is involved for large scale perturbations. Then we derive the De Wolf approximation (multiple forescattering and single backscattering approximation) for the nonlinear Frechet derivative. We split the linear derivative operator (i.e. the scattering operator) onto forward and backward derivatives, and then reorder and renormalize the nonlinear derivative series before making the approximation by dropping the multiple backscattering terms. Numerical simulations for a Gaussian ball model show significant difference between the linear and nonlinear Frechet derivatives.