A method to infer the bed elevation from glaciers surface measurements (eleva-tion, velocity) and sparse in-situ thickness values is developed and assessed. This inversion method relies on: a statistical model (Deep Neural Network) based on the in-situ thickness measurements, the dedicated RU-SIA flow model (RU for Reduced Uncertainty) natively integrating the surface measurements (altimetry, InSAR) and advanced Variational Data Assimilation processes. The RU-SIA model takes into account basal slipperiness and non uniform vertical profiles (including thermal gradients) via an unique dimensionless parameter. The inversion method is robust; it may be applied to very poorly covered and uncovered areas during airborne campaigns as soon as flows are moderately sheared. Numerical inversions are performed for some large East Antarctica Ice Sheet areas presenting surface velocities ranging from ∼ 5 to 80 m/y. Estimations are provided in uncovered areas during airborne campaigns hence presenting up to now highly uncertain bed elevation values. The estimations are valid for wave lengths greater than ∼ 10 ¯ h due to the considered shallow flow assumption, with a resolution at ∼ ¯ h (¯ h a characteristic thickness value). Detailed analysis and comparisons with the bed topography BedMap2 are presented.