  Data assimilation methodologies are designed to incorporate noisy
observations of a physical system into an underlying model in order to infer
the properties of the state of the system. Filters refer to a class of data
assimilation algorithms designed to update the estimation of the state as data
is acquired sequentially. For linear problems subject to Gaussian noise
filtering can be performed exactly using the Kalman filter. For nonlinear
systems it can be approximated in a systematic way by particle filters. However
in high dimensions these particle filtering methods can break down. Hence, for
the large nonlinear systems arising in applications such as oceanography and
weather forecasting, various ad hoc filters are used, based on Gaussian
approximations. In this work, we study the accuracy and stability of these ad
hoc filters in the context of the 2D incompressible Navier-Stokes equation. The
ideas readily generalize to a range of dissipative partial differential
equations (PDEs). By working in this infinite dimensional setting we provide an
analysis which is useful for the understanding of high dimensional filtering,
and is robust to mesh-refinement. We describe theoretical results showing that,
in the small observational noise limit, the filters can be tuned to perform
accurately in tracking the signal itself (filter accuracy), provided the system
is observed in a sufficiently large low dimensional space; roughly speaking
this space should be large enough to contain the unstable modes of the
linearized dynamics. The tuning corresponds to what is known as variance
inflation in the applied literature. Numerical results are given which
illustrate the theory. The positive results herein concerning filter stability
complement recent numerical studies which demonstrate that the ad hoc filters
can perform poorly in reproducing statistical variation about the true signal.
