On the mathematical foundation of full waveform inversion in viscoelastic vertically transverse isotropic media *

We present a mathematical framework for viscoelastic full waveform inversion (FWI) in vertically transverse isotropic media. FWI can be formulated as the nonlinear inverse problem of identifying parameters in the underlying attenuating anisotropic wave equation given partial wave field measurements (seismograms). From a mathematical point of view, one has to solve an operator equation for the full waveform forward operator, which is the corresponding parameter-to-state map. We give a rigorous definition of this operator, show its Fréchet differentiability, and explicitly characterize the adjoint operator of its Fréchet derivative. Thus, we provide the main ingredients to implement Newton-type/gradient-based regularization schemes for FWI. Our approach can be directly applied to other concepts of anisotropy.