Hybrid Neural Network - Variational Data Assimilation algorithm to infer river discharges from SWOT-like data

Estimating discharges Q(x, t) from altimetric measurements only, for ungauged rivers (in particular, those with unknown bathymetry b(x)), is an ill-posed inverse problem. We develop here an algorithm to estimate Q(x, t) without prior flow information other than global open datasets. Additionally, the ill-posedness feature of this inverse problem is re-investigated. Inversions based on a Variational Data Assimilation (VDA) approach enable accurate estimation of spatio-temporal variations of the discharge, but with a bias scaling the overall estimate. This key issue, which was already highlighted in our previous studies, is partly solved by considering additional hydrological information (the drainage area, $$A\ (km^2)$$ ) combined with a Machine Learning (ML) technique. Purely data-driven estimations obtained from an Artificial Neural Network (ANN) provide a reasonably good estimation at a large scale ( $$\approx 10^3$$ m). This first estimation is then employed to define the first guess of an iterative VDA algorithm. The latter relies on the Saint-Venant flow model and aims to compute the complete unknowns (discharge Q(x, t), bathymetry b(x), friction coefficient K(x, t)) at a fine scale (approximately $$10^2$$ m). The resulting complete inversion algorithm is called the H2iVDI algorithm for "Hybrid Hierarchical Variational Discharge Inference". Numerical experiments have been analyzed for 29 heterogeneous worldwide river portions. The obtained estimations present an overall bias (less than 30% for rivers with similar characteristics than those used for calibration) smaller than previous results, with accurate spatio-temporal variations of the flow. After a learning period of the observed rivers (e.g. one year), the algorithm provides two complementary estimators: a dynamic flow model enabling estimations at a fine scale and spatio-temporal extrapolations, and a low complexity estimator (based on a dedicated algebraic low Froude flow model). This last estimator provides reasonably accurate estimations (less than 30% for considered rivers) at a large scale from newly acquired WS measurements in real-time, therefore making it a potentially operational algorithm.