Inverse problems for abstract evolution equations II: higher order differentiability for viscoelasticity

Abstract. In this follow-up of [Inverse Problems 32 (2016) 085001] we generalize our previous abstract results so that they can be applied to the viscoelastic wave equation which serves as a forward model for full waveform inversion (FWI) in seismic imaging including dispersion and attenuation. FWI is the nonlinear inverse problem of identifying parameter functions of the viscoelastic wave equation from measurements of the reflected wave field. Here we rigorously derive rather explicit analytic expressions for the Fréchet derivative and its adjoint (adjoint state method) of the underlying parameter-to-solution map. These quantities enter crucially Newton-like gradient decent solvers for FWI. Moreover, we provide the second Fréchet derivative and a related adjoint as ingredients to second degree solvers.