•Study about discharge inversion in the context of remotly sensed observations of rivers.•The concept of an effective bathymetry-friction pair is introduced.•Several inverse formulations are proposed from the zero inertia shallow water equations, with an inversion algorithm.•Inversions of topography and friction pairs, and discharge are performed and analyzed on a set of synthetic and real river flows. The future SWOT mission (Surface Water and Ocean Topography) will provide cartographic measurements of inland water surfaces (elevation, widths and slope) at an unprecedented spatial and temporal resolution. Given synthetic SWOT like data, forward flow models of hierarchical-complexity are revisited and few inverse formulations are derived and assessed for retrieving the river low flow bathymetry, roughness and discharge (A0,K,Q). The concept of an effective low flow bathymetry A0 (the real one being never observed) and roughness K, hence an effective river dynamics description, is introduced. The few inverse models elaborated for inferring (A0,K,Q) are analyzed in two contexts: (1) only remotely sensed observations of the water surface (surface elevation, width and slope) are available; (2) one additional water depth measurement (or estimate) is available. The inverse models elaborated are independent of data acquisition dynamics; they are assessed on 91 synthetic test cases sampling a wide range of steady-state river flows (the Froude number varying between 0.05 and 0.5 for 1km reaches) and in the case of a flood on the Garonne River (France) characterized by large spatio-temporal variabilities. It is demonstrated that the most complete shallow-water like model allowing to separate the roughness and bathymetry terms is the so-called low Froude model. In Case (1), the resulting RMSE on infered discharges are on the order of 15% for first guess errors larger than 50%. An important feature of the present inverse methods is the fairly good accuracy of the discharge Q obtained, while the identified roughness coefficient K includes the measurement errors and the misfit of physics between the real flow and the hypothesis on which the inverse models rely; the later neglecting the unobserved temporal variations of the flow and the inertia effects. A compensation phenomena between the indentifiedvalues of K and the unobserved bathymetry A0 is highlighted, while the present inverse models lead to an effective river dynamics model that is accurate in the range of the discharge variability observed. In Case (2), the effective bathymetry profile for 80km of the Garonne River is retrieved with 1% relative error only. Next, accurate effective topography-friction pairs and also discharge can be inferred. Finally, defining river reaches from the observation grid tends to average the river properties in each reach, hence tends to smooth the hydraulic variability.