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

Full-waveform inversion (FWI) is a nonlinear PDE constrained optimization problem which seeks to estimate the constitutive parameters of a medium by fitting waveforms. Among these parameters, attenuation needs to be taken into account in viscous media to exploit the full potential of FWI. Attenuation is 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 medium. Moreover, it leads to a multivariate inverse problem, which is ill-posed due to the cross-talk between the two classes of real parameters. To alleviate these issues, we solve directly the optimization problem in the complex domain by processing narrow bands of frequencies in sequence under the assumption of bandwise frequency dependence of the complex velocities. Moreover, we use a relaxation method to extend the FWI search space by processing the wave equation as a weak constraint with the alternating direction method of multipliers (ADMM) to mitigate the risk of spurious local minima. To mitigate the ill-posedness of the inversion, three total variation (TV) regularization schemes based upon ADMM and proximity algorithms are presented. In the first, regularization is applied directly on the complex velocities. In the other two, separate TV regularizations are tailored to different attributes of the complex velocities (real and imaginary parts, magnitude and phase). The real phase velocity and attenuation factor are then reconstructed a posteriori at each spatial position from the estimated complex velocity using arbitrary empirical relation. The numerical results first show that the regularization of the amplitude and phase provides the most reliable results. Moreover, they show that the band-by-band design of the inversion limits the sensitivity of the recovered phase velocity and attenuation factor to the attenuation model used for their a posteriori extraction.