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Méthodes variationnelles d'ensemble et optimisation variationnellepour les géosciences

Abstract : Data assimilation consists in the estimation of a physical system state. This estimation should optimally combine erroneous observations together with imperfect numerical simulations of this system. In practice, this estimation is the initial state of a dynamical system. It can be used to precisely predict the system evolution. Especially in geophysics where data sets are substantial.A first strategy is based on the maximum a posteriori estimation. Thus the estimation is solution of an optimization problem. This strategy called 4DVar often requires the computation of the model and observation operator adjoints. This operation is time consuming in the forecasting system development. A second strategy analyses the system state sequentially. It is based on “ensemble” techniques. Here, perturbations of the system state allow to estimate its statistics sequentially thanks to the Kalman equations.Both strategies was recently successfully combined in EnVar methods currently used in operating systems. They benefit from both: an efficient treatment of the operators nonlinearity through variational optimization techniques together with statistics and derivatives estimation through ensembles. The IEnKS is an archetype of such EnVar methods. It uses a data assimilation window (DAW) which is time shifted each cycle to combine both strategies. Various DAW parameterizations lead to non equivalent assimilations when the system dynamics are non linear.In particular, long DAWs reduce the frequency of the prior density Gaussian approximation. This results in a performance improvement but only to some extent. After, the cost function variational minimization fails because of its complex shape. A solution called “Quasi static variational assimilation” gradually adds observations to the cost function during multiple minimizations. The thesis second chapter generalizes the QSVA to EnVar methods. Theoretical and numerical aspects of the QSVA applied to the IEnKS are adressed.However, the QSVA relies on the absence of model error. Indeed, the information contained in a remote in time observation may be deteriorated by model error. The thesis third chapter is dedicated to model error introduction in the IEnKS. The IEnKS-Q, a 4D ensemble variational method solving sequentially the smoothing problem with model error, is built in this chapter. Unfortunately, with model error, a state trajectory is not anymore determined by its initial condition. The number of parameters required to describe its statistics increase with the DAW length. When this number is paired with the number of model evaluations, the consequences on the computing time are disastrous. A proposed solution is to dissociate those quantities with anomaly matrices decompositions. In this case, the IEnKS-Q is as expensive as an IEnKS in terms of model evaluations
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Anthony Fillion. Méthodes variationnelles d'ensemble et optimisation variationnellepour les géosciences. Ingénierie de l'environnement. Université Paris-Est, 2019. Français. ⟨NNT : 2019PESC1012⟩. ⟨tel-02497496⟩

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