Full waveform inversion (FWI) is a nonlinear waveform matching procedure which can provide high-resolution subsurface models. However, viscous effects must be taken into account in attenuating media to exploit the full potential of FWI. In the frequency domain, attenuation is implemented in the time-harmonic wave equation with complex-valued velocities. During the inverse problem, the real and imaginary parts of the velocity are generally processed as two independentreal-valued parameters. In this study, we process instead the velocityas a complex-valued parameter using derivative of real functions of complex variables. Moreover, we implement visco-acoustic frequency FWI with search space extension in the framework of the wavefield reconstruction inversion (WRI) method. We implement WRI with the alternating-direction method of multiplier (ADMM), which makes the parameter-estimation subproblem linear thanks to the bilinearity of the wave equation and provides a suitable framework to cascade nonsmooth regularizations and bound constraints in the objective function. In this study, we review ADMM-based WRI for complex-valued parameters and show preliminary results of joint velocity and attenuation reconstruction when inversion is performed without and with total variation (TV) regularization. We show the key role of TV regularization to decrease the ill-posedness of the velocity-attenuation reconstruction.


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