Estimating river discharge from water level observations using a differentiable hydraulic model with an embedded LSTM model to learn time-varying channel roughness

Identifying the relationship between discharge and water level is challenging in lowland streams because seasonal growth of submerged aquatic vegetation causes complex, time-varying channel roughness. We propose a novel differentiable modelling framework integrating (1) an LSTM model that predicts the dynamics of the channel roughness, with (2) a hydraulic river model providing physics-based routing. The integrated framework is trained to learn time-varying channel roughness based on historical observations of meteorological variables, river water levels, and discharge. Once trained, we demonstrate that the framework can be applied to independent periods for deriving discharge estimates using backpropagation with only observed water levels and meteorological variables as inputs, thereby acting as a time-varying rating curve. Synthetic experiments confirm accurate recovery of (a) time-varying roughness and (b) discharges imposed during hydraulic model simulations. Applied to a real Danish river, the framework achieves an NSE for water levels of 0.76, compared to 0.19 for a constant-roughness benchmark. Incorporating temporal roughness dynamics reduces the Mean Absolute Relative Error (MARE) in discharge estimation from 21.4% to 10.5%. This work takes a significant step towards setting up hydraulic models directly from primary data, bypassing the explicit construction of complex, time-varying rating curves.