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Observation of transient shear-wave propagation in soft tissue is of great interest for the study of tissue viscoelastic properties. Previously, we introduced a technique, supersonic shear imaging (SSI), able to generate transient shear waves using the acoustic radiation force and to image their propagation in soft tissues in real time. Inversion methods were used to recover elasticity from the shear-wave propagation. We now present a precise and robust inversion algorithm taking into account not only elastic, but also viscous, properties of soft tissues. Based on a Voigt model, this algorithm is designed to provided quantitative and local estimation of soft tissue elasticity and viscosity. The influence of viscosity on transient shear waves is modeled and analyzed using a 3D analytical formulation of the mechanical Green's function in a viscoelastic medium. The spatial and temporal shape of experimental shear waves induced in soft tissues using SSI can only be accurately modeled by taking into account tissue shear viscosity. The respective influences of viscosity, elasticity or diffraction on the shear wave shape are carefully studied and discriminated. Taking advantage of the previous modeling, a local inverse problem permitting the recovery of shear elasticity and viscosity is presented and validated using the Green's function based simulation tool. The role of viscosity on the accuracy of the elasticity estimation is studied. The influence of out of plane shear propagation on the inversion algorithm is discussed. In media exhibiting shear viscoelasticity heterogeneities, finite differences simulations are used to study the spatial resolution of the algorithm and its sensitivity to signal-to-noise ratio. Experiments on calibrated tissue-mimicking phantoms having different viscoelastic properties are presented, validating the simulation results.