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

Controlling Unpredictability with Observations in the Partially Observed Lorenz '96 Model