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

TL;DR

This paper investigates how partial and adaptive observations can be used to accurately filter chaotic systems like the Lorenz '96 model, demonstrating that adaptive strategies require fewer observations for effective control.

Contribution

It provides theoretical conditions and numerical evidence showing adaptive observation operators outperform fixed ones in filtering chaotic systems with fewer observed variables.

Findings

3DVAR filter can recover system state within observational noise bounds

Adaptive observation operators significantly reduce the number of observed variables needed

Less informative fixed operators can still achieve accurate signal reconstruction

Abstract

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.