  Chaos is ubiquitous in physical systems. The associated sensitivity to
initial conditions is a significant obstacle in forecasting the weather and
other geophysical fluid flows. Data assimilation is the process whereby the
uncertainty in initial conditions is reduced by the astute combination of model
predictions and real-time data. This chapter reviews recent findings from
investigations on the impact of chaos on data assimilation methods: for the
Kalman filter and smoother in linear systems, analytic results are derived; for
their ensemble-based versions and nonlinear dynamics, numerical results provide
insights. The focus is on characterising the asymptotic statistics of the
Bayesian posterior in terms of the dynamical instabilities, differentiating
between deterministic and stochastic dynamics. We also present two novel
results. Firstly, we study the functioning of the ensemble Kalman filter in the
context of a chaotic, coupled, atmosphere-ocean model with a quasi-degenerate
spectrum of Lyapunov exponents, showing the importance of having sufficient
ensemble members to track all of the near-null modes. Secondly, for the fully
non-Gaussian method of the particle filter, numerical experiments are conducted
to test whether the curse of dimensionality can be mitigated by discarding
observations in the directions of little dynamical growth of uncertainty. The
results refute this option, most likely because the particles already embody
this information on the chaotic system. The results also suggest that it is the
rank of the unstable-neutral subspace of the dynamics, and not that of the
observation operator, that determines the required number of particles. We
finally discuss how knowledge of the random attractor can play a role in the
development of future data assimilation schemes for chaotic multiscale systems
with large scale separation.
