  Bayesian state estimation of a dynamical system utilising a stream of noisy
measurements is important in many geophysical and engineering applications. We
establish rigorous results on (time-asymptotic) accuracy and stability of these
algorithms with general covariance and observation operators. The accuracy and
stability results for EnKF and EnSRKF for dissipative PDEs are, to the best of
our knowledge, completely new in this general setting. It turns out that a
hitherto unexploited cancellation property involving the ensemble covariance
and observation operators and the concept of covariance localization in
conjunction with covariance inflation play a pivotal role in the accuracy and
stability for EnKF and EnSRKF. Our approach also elucidates the links, via
determining functionals, between the approximate-Bayesian and control-theoretic
approaches to data assimilation. We consider the `model' dynamics governed by
the two-dimensional incompressible Navier-Stokes equations and observations
given by noisy measurements of averaged volume elements or spectral/modal
observations of the velocity field. In this setup, several continuous-time data
assimilation techniques, namely the so-called 3DVar, EnKF and EnSRKF reduce to
a stochastically forced Navier-Stokes equations. For the first time, we derive
conditions for accuracy and stability of EnKF and EnSRKF. The derived bounds
are given for the limit supremum of the expected value of the $L^2$ norm and of
the $\mathbb{H}^1$ Sobolev norm of the difference between the approximating
solution and the actual solution as the time tends to infinity. Moreover, our
analysis reveals an interplay between the resolution of the observations
associated with the observation operator underlying the data assimilation
algorithms and covariance inflation and localization which are employed in
practice for improved filter performance.
