Pré-Publication, Document De Travail Année : 2020

Theoretical Issues in the Application of Bayesian Data Assimilation in Context of Meteorology. Part 2: Stability and convergence

Résumé

The elicitation of the background error covariance matrix (becm) is a major difficulty of Bayesian assimilation used in atmospheric and oceanic sciences based on Kalman filter in the form of 3Dvar or 4Dvar. In a preliminary companion paper, the definition of a becm and link to earlier information were seen poorly practicable. Here, more decisive arguments are obtained by reviewing mathematical studies, many in optimal control, a field dual to filtering. When the system is discretized, and in the simplest steady and linear case, controlla-bility and detectability conditions are generally fulfilled implying convergence in theory, not necessarily in practice as the limit may be very large. The non steady or nonlinear case may be stabilized by covariance inflation. However, mathematical works about inflation disprove the common belief that the converged result would be the becm of the system. Considering the complete infinite dimensional system is even more disappointing. The becm is replaced by a trace class operator solution, in the steady linear case, to an infinite-dimensional Riccati equation. This equation has never been studied corresponding to the specific features of geophysical assimilation of data. These correspond to limit cases where the existence of solutions is no longer guaranteed. There is no ground, accordingly, for presuming filter convergence in terms of the true infinite-dimensional system. Counterexamples are proposed. This conclusion jeopardizes the physical meaning of any finite-dimensional becm that could be computed based on discretization.

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hal-02434502 , version 1 (10-01-2020)

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Jean-Pierre Issartel, X Busch, Maithili Sharan. Theoretical Issues in the Application of Bayesian Data Assimilation in Context of Meteorology. Part 2: Stability and convergence. 2020. ⟨hal-02434502⟩
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