  Data assimilation (DA) aims at optimally merging observational data and model
outputs to create a coherent statistical and dynamical picture of the system
under investigation. Indeed, DA aims at minimizing the effect of observational
and model error, and at distilling the correct ingredients of its dynamics. DA
is of critical importance for the analysis of systems featuring sensitive
dependence on the initial conditions, as chaos wins over any finitely accurate
knowledge 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 under
investigation, as merging optimally observational data and model outputs is
harder when strong instabilities are present. In this paper we reverse the
usual angle on the problem and show that it is indeed possible to use the skill
of DA to infer some basic properties of the tangent space of the system, which
may be hard to compute in very high-dimensional systems. Here, we focus our
attention on the first Lyapunov exponent and the Kolmogorov-Sinai entropy, and
perform numerical experiments on the Vissio-Lucarini 2020 model, a recently
proposed generalisation of the Lorenz 1996 model that is able to describe in a
simple yet meaningful way the interplay between dynamical and thermodynamical
variables.
