Summary In this work, we propose a stochastic nonlinear inversion framework for PP and PS seismic data based on the ensemble smoother with multiple data assimilations (ES-MDA) to estimate elastic reservoir properties with uncertainty quantification. The ES-MDA is an iterative ensemble-based data assimilation method that generates an ensemble of solutions of the inverse problem. In our approach, it is applied to a seismic inversion problem in which the full Zoeppritz equations, without linearization, are used to improve the inversion accuracy. The ensemble of updated reservoir realizations obtained by assimilating seismic data allows evaluating the associated model uncertainty. To avoid the model uncertainty be underestimated in the ensemble-based approach, we propose to apply the ES-MDA in a lower-dimensional data space obtained by the re-parameterization of PP and PS seismic data using the singular value decomposition (SVD).