A Bayesian framework for Stage–Fall–Discharge Laws estimation from SWOT altimetry and slopes

River discharge is traditionally estimated using stage–discharge rating curves, yet their calibration relies on sparse and costly in situ measurements and remains highly uncertain, particularly under extreme flow conditions. The Surface Water and Ocean Topography (SWOT) mission provides unprecedented observations of river surface elevation, slope, and width; however, inferring discharge from these variables alone is fundamentally ill-posed and susceptible to structural biases. Overcoming these limitations requires additional physical constraints at the river-network scale, motivating the use of hydrological–hydraulic closure and data assimilation to derive robust, observation-conditioned rating curves from SWOT data. In this study, we propose a multi-scale Markov Chain Monte Carlo (MCMC) framework to infer three stage–discharge rating curve formulations by jointly exploiting SWOT observations, hydrological model outputs, and in situ discharge measurements. Our results demonstrate the feasibility of using SWOT data to infer reliable  parametric stage-discharge relationships including a conceptual river hydraulic geometry, while revealing spatially coherent patterns in model discharge quality consistent with previous studies. The analysis also highlights current limitations in SWOT data processing quality and establishes a foundation for deriving prior hydraulic knowledge to support future end-to-end hydrology–hydraulic learning frameworks.