Almost sure error bounds for data assimilation in dissipative systems with unbounded observation noise

Data assimilation is uniquely challenging in weather forecasting due to the high dimensionality of the employed models and the nonlinearity of the governing equations. Although current operational schemes are used successfully, our understanding of their long-term error behaviour is still incomplete. In this work, we study the error of some simple data assimilation schemes in the presence of unbounded (e.g. Gaussian) noise on a wide class of dissipative dynamical systems with certain properties, including the Lorenz models and the 2D incompressible Navier-Stokes equations. We exploit the properties of the dynamics to derive analytic bounds on the long-term error for individual realisations of the noise in time. These bounds are proportional to the variance of the noise. Furthermore, we find that the error exhibits a form of stationary behaviour, and in particular an accumulation of error does not occur. This improves on previous results in which either the noise was bounded or the error was considered in expectation only