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Data assimilation is uniquely challenging in weather forecasting due to thehigh dimensionality of the employed models and the nonlinearity of thegoverning equations. Although current operational schemes are usedsuccessfully, our understanding of their long-term error behaviour is stillincomplete. In this work, we study the error of some simple data assimilationschemes in the presence of unbounded (e.g. Gaussian) noise on a wide class ofdissipative dynamical systems with certain properties, including the Lorenzmodels and the 2D incompressible Navier-Stokes equations. We exploit theproperties of the dynamics to derive analytic bounds on the long-term error forindividual realisations of the noise in time. These bounds are proportional tothe amplitude of the noise. Furthermore, we find that the error exhibits a formof stationary behaviour, and in particular an accumulation of error does notoccur. This improves on previous results in which either the noise was boundedor the error was considered in expectation only.

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