4.2.1.1 Rating curve model specification: The first step consists in defining a probabilistic model that relates observed water surface elevations (WSE) to discharge (Q) through a commonly used power-law relationship: Q = a (WSE-z 0 ) b Eq.2 a is a scaling coefficient controlling the magnitude of the discharge, b is an exponent shaping the Q–WSE relationship, reflecting channel geometry and flow regime characteristics (e.g., roughness, cross-section), z ₀ is the offset or base elevation, generally representing the river bed elevation or the WSE level at which flow begins.
In phase 2 of this study, the rating curve is enhanced by incorporating dynamic water surface slope measurements S , derived from the Surface Water and Ocean Topography (SWOT) mission over few cases as a proof of concept and relevance. The inclusion of slope data is motivated by hydraulic theory, particularly the Manning equation, where discharge depends on both hydraulic radius and slope. The enhanced rating curve equation is formulated as: Q = a S 1/2 (WSE-z 0 ) b Eq.3 where: S is the water surface slope, estimated from SWOT observations, which varies spatially and temporally (trough a climate mean). The other parameters retain their physical meaning as before.
This formulation explicitly integrates river slope variability, allowing for a more physically representative and accurate estimation of discharge. It accounts for changes in hydraulic driving forces and improves the adaptability of the rating curve across different river systems and flow regimes. 4.2.1.2 Prior distribution: The Bayesian framework requires assigning prior probability distributions to each parameter, which can be either informative (based on prior studies or expert knowledge) or uninformative (non-committal). For this study, the following normal prior distributions were adopted, reflecting typical ranges observed in diverse hydrological contexts: a (scaling factor):
- Must be non-negative - Typical mean value: 800, standard deviation: 300, resulting in a plausible range: [0 – 1700] b (exponent): - Must be positive - Commonly ranges between 0 and 3 (Chow et al., 1988) - Mean: 1.5, standard deviation: 0.5, allowing flexibility for most river geometries and hydraulic conditions z ₀ (offset or river bed elevation): - Estimated as the minimum WSE minus 5 meters (to ensure discharge begins near the riverbed) D.3 River Discharge ATBD. Reference: RD_CCI-0006-ATBD - Issue 2.0 – 31/07/2025 Open/Public © 2019 CLS. All rights reserved. Proprietary and Confidential. 14/31