Flow estimation from observed water levels using differentiable modeling for low-lying rivers affected by vegetation and backwater

Simulations of river flows and water levels are crucial for flood predictions and water resources management. Water levels are easy to observe using sensors, while the mapping between water levels and flows in rivers is usually derived from rating curves. However, rating curves frequently do not include geometry, backwater effects, and/or seasonal variations, which can limit their applicability – especially in stream systems that are affected by seasonal vegetation and backwater effects. To address this, we propose a differentiable model that merges a neural network with a physically based, steady-state implementation of the Saint-Venant equations. 

In the setup, the neural network is trained to predict seasonal variations caused by vegetation growth in Manning’s roughness based on inputs of meteorological forcing and time, while the physical model is responsible for converting flow estimates into water levels along the river channel. The framework efficiently estimates model parameters by tracking gradients through both the physical model and the neural network via backpropagation. This allows us to calibrate parameters for both the runoff and the Manning’s roughness from measured water levels, thus overcoming rating curve limitations while accounting for backwater, river geometry, and seasonal variations in roughness. 

We tested the model on a 20 km stretch of the Vejle River, Denmark, which is both heavily vegetated and affected by backwater from the coast. The model was trained across five water level sensors using two years of data (2020-2022). When evaluated against 10 years of observed flow measurements (2007-2017), the model demonstrated a Mean Absolute Relative Error (MARE) of 10% compared to manually gauged discharge observations. This is comparable to the estimated uncertainty of 10% in the discharge measurements.  

The framework enables a calibration of dynamic Manning roughness within a few hours, and therefore offers a scalable solution for estimating river flows from water levels when cross-section information is available. Potential applications span across many disciplines in water resource management.