Filter accuracy for the Lorenz 96 model: Fixed versus adaptive observation operators

• Published in: Physica. D Vol. 325; no. C; pp. 1 - 13
• Main Authors: Law, K.J.H., Sanz-Alonso, D., Shukla, A., Stuart, A.M.
• Format: Journal Article
• Language: English
• Published: United States Elsevier B.V 15.06.2016; Elsevier
• Subjects: 3DVAR; Adaptive observations; Chaos theory; Control systems; Dynamical systems; Extended Kalman filter; Filter accuracy; Filtering; Lorenz ’96; Lyapunov exponents; Mathematical models; MATHEMATICS AND COMPUTING; Operators; State vectors; Lorenz ’96; Adaptive observations; 3DVAR; Filter accuracy; Extended Kalman filter
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2015.12.008

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. •It is proven that 3DVAR filter error is the size of the observational noise.•Numerical experiments with 3DVAR and extended Kalman filters show better performance.•Adaptive observation operators numerically allow even fewer observations.