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Data assimilation (DA) aims at optimally merging observational data and modeloutputs to create a coherent statistical and dynamical picture of the systemunder investigation. Indeed, DA aims at minimizing the effect of observationaland model error, and at distilling the correct ingredients of its dynamics. DAis of critical importance for the analysis of systems featuring sensitivedependence on the initial conditions, as chaos wins over any finitely accurateknowledge of the state of the system, even in absence of model error. Clearly,the skill of DA is guided by the properties of dynamical system underinvestigation, as merging optimally observational data and model outputs isharder when strong instabilities are present. In this paper we reverse theusual angle on the problem and show that it is indeed possible to use the skillof DA to infer some basic properties of the tangent space of the system, whichmay be hard to compute in very high-dimensional systems. Here, we focus ourattention on the first Lyapunov exponent and the Kolmogorov-Sinai entropy, andperform numerical experiments on the Vissio-Lucarini 2020 model, a recentlyproposed generalisation of the Lorenz 1996 model that is able to describe in asimple yet meaningful way the interplay between dynamical and thermodynamicalvariables.

Inferring the instability of a dynamical system from the skill of data assimilation exercises