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Modélisations des écoulements fluviaux adaptées aux observations spatiales et assimilations de données altimétriques

Abstract : This PhD work focuses on river modelling adapted to spatial altimetry, which allows the measurement of the height of water in rivers. In order to estimate the discharge using these data, the mathematical models need to be consistent with the spatio-temporal scale of the observations (hundreds of metres and tens of days) and an estimate of some quantities not measured by these altimetry satellites, notably the bottom elevation and a physical parametrization (friction coefficient).The difficulty in estimating the discharge from altimetric data comes in particular from the slope of the free surface, which is also not measured at a fine enough scale.A new methodology to determine local and algebraic discharge estimation laws (Stage-Fall-Discharge laws, SFD) from altimetry data from several satellites (e.g. Jason-3, Sentinel-3A and Sentinel-3B) is proposed. The method is based on a hydrodynamic model calibrated by assimilation of altimetry data. These SFD laws are determined to reproduce the discharge estimated by the hydrodynamic model from noisy altimetry data and physically consistent simulated hydraulic quantities.These laws are successfully obtained on the complex hydrographic network of the Rio Negro-Rio Branco.The method should be applicable for operational estimation of discharge.Modelling adapted to spatial observations therefore requires choosing models that are coherent with the available data and with the observed spatio-temporal scales. As a result, the diffusive wave equation has the advantage of having the water height as state variable, which is directly measured in contrast to the discharge.In this work, a double spatio-temporal scale is introduced to take into account the scale of physics (small scale) and the scale of observations (large scale). The width variations are negligible on the scale of physics, which is not the case on the scale of observations. A diffusive wave equation adapted to the scale of satellite observations is derived. This new diffusive wave equation takes into account width variations through two additional terms compared to the classical equation.A numerical study shows that the observation-scale equation estimates the slope of the free surface and thus the discharge with better accuracy than the classical equation. One of the additional terms in the observation-scale equation is also highlighted by quantifying the importance of the terms of a dictionary based on sparse regression.In order to obtain an estimate of the bottom elevation and the friction coefficient (non-observed by the altimetry satellites), altimetric data are assimilated into the hydrodynamic models by minimising the gap between the modelled height and the measured height. The quality of this data assimilation depends in particular on the estimation of the covariance of the background error, i.e. the error between the background value and the true value of the parameter, that preconditiones the Hessian of the cost function. However, this covariance is usually defined in an empirical manner.Thus, this work proposes a method for estimating the covariance of the background error and the correlation length from the equations governing the physics (here the diffusive wave equations) using the Green kernels.These new operators and the physically consistent correlation length coupled with a decreasing exponential kernel give better results than the empirical operators.
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Thibault Malou. Modélisations des écoulements fluviaux adaptées aux observations spatiales et assimilations de données altimétriques. Mathématiques générales [math.GM]. INSA de Toulouse, 2022. Français. ⟨NNT : 2022ISAT0001⟩. ⟨tel-03630148⟩

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