Jessie Levillain (CNES / INSA-Mathematics Institute of Toulouse, France); Kevin Larnier (Hydro-Matters / INSA-Mathematics Institute of Toulouse, France)

Knowledge of water depth h(x,t) [m] in rivers, more precisely the cross-sectional area A(x,t) [m^2], or equivalently the bathymetry b(x) [m] once a geometrical wetted cross-sections model is fixed, is indispensable data for setting up river dynamics models. However, river bathymetry is unknown for a large number of rivers worldwide: it is difficult to measure on a large scale and cannot be done remotely. Knowledge of river bathymetry is important in itself, but also for estimating the river discharges Q(x,t) [m^3/s] of ungauged rivers, as shown, for example, in [1]. Estimating river bathymetry from water surface measurements alone is an ill-posed problem. Given a single measurement b_ref, effective bathymetry profiles b(x) of river portions can be estimated from SWOT-like measurements, as seen in [2], by using the so-called Low-Froude flow equation, highlighted in [3]. However, the use of this equation can be problematic with sparse and noisy data, especially for long portions. This is due to a drift in the estimation that is intrinsic to such an Ordinary Differential Equation.
Given a geometrical wetted-cross section model compatible with SWOT measurements, under flow assumptions (including the low-Froude assumption (1-Fr^2) \approx 1, one can derive a PDE relating the water surface elevation H(x) to the bathymetry profile b(x) only, [4]. This provides an "altimetry-to-bathymetry model" valid for a great majority of worldwide rivers at approximately 1 km reach scale. This model enables robust estimations against noise and without drifting phenomena in the estimation. Moreover, one can derive a formulation adapted to networks, enabling the estimation of bathymetry profiles of a complete river network given a single measurement in the network, [5]. The method has been tested on the synthetic Pepsi rivers and synthetic river networks. Moreover, within its domain of applicability, this method can improve the estimation of the river discharge Q(x,t) presented in [6].

References
[1] Kévin Larnier, Jérôme Monnier, Pierre-André Garambois, and Jonas Verley. River discharge and bathymetry estimation from swot altimetry measurements. Inverse problems in science and engineering, 29(6):759–789, 2021.
[2] Pierre-André Garambois and Jérôme Monnier. Inference of effective river properties from remotely sensed observations of water surface. Advances in Water Resources, 79:103–120, 2015.
[3] Alelign Gessese, Kabila Mawanzi Wa, and Mathieu Sellier. Bathymetry reconstruction based on the zero-inertia shallow water approximation. Theoretical and Computational Fluid Dynamics, 27:721–732, 2013.
[4] Thibault Malou and Jérôme Monnier. Estimation of bathymetry profiles from altimetry data. Mathematics Institute of Toulouse, Internal Research report, 2021.
[5] Jessie Levillain and Jérôme Monnier. Estimation of bathymetry profiles for river networks from swot-like data. Submitted, 2025.
[6] Kévin Larnier, Jérôme Monnier, and Olivier Roustant. Hybrid bayesian-neural network-variational estimations of river characteristics from swot-like data. Submitted, 2025.