An inversion algorithm to estimate the discharge of rivers observed by the forthcoming SWOT mission (wide swath altimetry) is elaborated and assessed into details. The algorithm relies on an advanced variational data assimilation formulation applied to the Saint-Venant equations (1D shallow-water) and permanent low Froude flow relations (direct and inverse algebraic models). This hierarchical based modeling approach makes possible to estimate the three key unknowns, that are the time-dependent inflow discharge Qin (t), the space varying bathymetry b(x) and a varying roughness coefficient K, from altimetry measurements only. The flow model is build up from effective cross sections defined from the altimetry measurements. The numerical results are analyzed on three river portions (⇠ 100 km long) presenting highly frequent flow variations compared to the observation frequency. Two scenarios of observation are considered: frequent satellite overpasses corresponding to the SWOT Cal-Val orbit (1 day period) and SWOT like data corresponding to 21 days period with 1 to 4 passes at mid-latitudes. The numerical experiments demonstrate that the estimations of the discharge Qin (t) are accurate, the bathymetry profile b(x) too. Various prior configurations of Q and b are considered in view of worldwide applications. Past the learning period (i.e. after the assimilation of a complete measurements set, typically after one year), the low complexity algebraic models provide accurate discharge estimations in real-time from newly acquired measurements.