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Full waveform inversion (FWI) is a nonlinear PDE constrained optimizationproblem, which seeks to estimate constitutive parameters of a medium such asphase velocity, density, and anisotropy, by fitting waveforms. Attenuation isan additional parameter that needs to be taken into account in viscous media toexploit the full potential of FWI. Attenuation is more easily implemented inthe frequency domain by using complex-valued velocities in the time-harmonicwave equation. These complex velocities are frequency-dependent to guaranteecausality and account for dispersion. Since estimating a complexfrequency-dependent velocity at each grid point in space is not realistic, theoptimization is generally performed in the real domain by processing the phasevelocity (or slowness) at a reference frequency and attenuation (or qualityfactor) as separate real parameters. This real parametrization requires an apriori empirical relation (such as the nonlinear Kolsky-Futterman (KF) orstandard linear solid (SLS) attenuation models) between the complex velocityand the two real quantities, which is prone to generate modeling errors if itdoes not represent accurately the attenuation behavior of the subsurface.Moreover, it leads to a multivariate inverse problem, which is twice largerthan the actual size of the medium and ill-posed due to the cross-talk betweenthe two classes of real parameters. To alleviate these issues, we present amono-variate algorithm that solves directly the optimization problem in thecomplex domain by processing in sequence narrow bands of frequencies under theassumption of band-wise frequency dependence of the sought complex velocities.

Complex-valued Imaging with Total Variation Regularization: An Application to Full-Waveform Inversion in Visco-acoustic Media