  In the context of filtering chaotic dynamical systems it is well-known that
partial observations, if sufficiently informative, can be used to control the
inherent uncertainty due to chaos. The purpose of this paper is to investigate,
both theoretically and numerically, conditions on the observations of chaotic
systems under which they can be accurately filtered. In particular, we
highlight the advantage of adaptive observation operators over fixed ones. The
Lorenz '96 model is used to exemplify our findings.
  We consider discrete-time and continuous-time observations in our theoretical
developments. We prove that, for fixed observation operator, the 3DVAR filter
can recover the system state within a neighbourhood determined by the size of
the observational noise. It is required that a sufficiently large proportion of
the state vector is observed, and an explicit form for such sufficient fixed
observation operator is given. Numerical experiments, where the data is
incorporated by use of the 3DVAR and extended Kalman filters, suggest that less
informative fixed operators than given by our theory can still lead to accurate
signal reconstruction. Adaptive observation operators are then studied
numerically; we show that, for carefully chosen adaptive observation operators,
the proportion of the state vector that needs to be observed is drastically
smaller than with a fixed observation operator. Indeed, we show that the number
of state coordinates that need to be observed may even be significantly smaller
than the total number of positive Lyapunov exponents of the underlying system.
