Data assimilation using noisy time-averaged measurements

Highlights: • We couple chaotic dynamical systems with noisy time-averaged partial observations. • Analytical bounds on the error of the corresponding synchronizing signals are proved. • In the zero-noise case, small enough time-average windows give exact synchronization. • In the presence of noise, convergence is to within a factor of the noise's variance. • Numerics show synchronization occurs under conditions milder than required by theory. We study the synchronization of chaotic systems when the coupling between them contains both time averages and stochastic noise. Our model dynamics – inspired by the partial differential equations which govern the atmosphere – are given by the Lorenz equations which are a system of three ordinary differential equations in the variables $X$, $Y$ and $Z$. Our theoretical results show that coupling two copies of the Lorenz equations using a feedback control which consists of time averages of the $X$ variable leads to exact synchronization provided the time-averaging window is known and sufficiently small. In the presence of noise the convergence is to within a factor of the variance of the noise. We also consider the case when the time-averaging window is not known and show that it is possible to tune the feedback control to recover the size of the time-averaging window. Further numerical computations show that synchronization is more accurate and occurs under much less stringent conditions than our theory requires.