On the Linearization of Operators Related to the Full Waveform Inversion in Seismology

In this work we analyze the parameter-to-solution map of the acoustic wave equation with respect to its parameters wave speed and mass density. This map is a mathematical model for the seismic inverse problem where one wants to recover the parameters from measurements of the acoustic potential. We show its complete continuity and Fréchet differentiability. To this end we provide necessary existence, stability and regularity results. Moreover, we discuss various implications of our findings on the inverse problem and comment on the Born series.