  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 amplitude 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.
