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