This study proposes an inverse method to infer the bed topography beneath ice flows from the surface observations (e.g. altimetry elevations and InSAR velocities) and sparse depth measurements (e.g. from airborne campaigns). The flow model is valid for highly to mildly-sheared regimes (hence mildly-rapid) and takes into account varying vertical thermal profiles; it is depth-integrated (long-wave assumption). The inverse problem is particularly challenging since the assimilated surface signatures integrate the bottom features (bed elevation and friction-slip amount) and the internal deformation due to non constant rate factor vertical profile. The first key step of this multi-physics flow inversion is a re-derivation of the anisothermal xSIA model (lubrication type model for generalized Newtonian fluids) leading to a Reduced Uncertainty (RU) version presenting a single uncertain multi-physic parameter γ; that is the so-called RU-SIA equation. The next key steps are advanced Variational Data Assimilation (VDA) formulations combined with a stochastic extension of γ based on the trend observed in the in-situ measurements (e.g. along the flight tracks). The resulting method provides the first physical-based depth (ice thickness) inversions in mildly-sheared mildly-slippery shallow flows. Numerical results are presented in a poorly monitored inland Antarctica area. The uncertainty of the estimated bedrock elevation is noticeably reduced compared to the current estimations uncertainties. The robustness of the inversion process is demonstrated through numerous numerical experiments and empirical sensitivity analyses. The new RU-SIA model may provide a-posteriori estimations of the thermal basal boundary layer too.