Tangential cone condition for the full waveform forward operator in the viscoelastic regime: the non-local case

We discuss mapping properties of the parameter-to-state map of full waveform inversion and generalize the results of [M. Eller and A. Rieder, Inverse Problems 37 (2021) 085011] from the acoustic to the viscoelastic wave equation. In particular we establish injectivity of the Fréchet derivative of the parameter-to-state map for a semi-discrete seismic inverse problem in the viscoelastic regime. Here, the finite dimensional parameter space is restricted to functions having global support in the propagation medium (the non-local case) and that are locally linearly independent. As a consequence we deduce local conditional wellposedness of this nonlinear inverse problem. Furthermore, we show that the tangential cone condition holds, which is an essential prerequisite in the convergence analysis of a variety of inversion algorithms for nonlinear illposed problems.