Romain Brossier Université de Grenoble I (Joseph Fourier) Uncertainty Quantification is a major issue for most geophysical tomography problems. In the frame of local optimization, it is known that the inverse Hessian is associated to the posterior covariance matrix and provides useful information on such uncertainties. However, computing and accessing such operator is out of reach for most large scale tomography problems, as the Full Waveform Inversion. In order to go forward uncertainty quantification in FWI, a review of different methods proposed in the literature to probe the (inverse) Hessian and what the Data Assimilation community has developed for decades will be first performed. Then, a combination of Ensemble Kalman Filter and FWI will be presented. This approach will allow accessing uncertainty through the set ensemble members, which defines a low-rank representation of the posterior covariance matrix.