The 3DVAR filter is prototypical of methods used to combine observed data
with a dynamical system, online, in order to improve estimation of the state of
the system. Such methods are used for high dimensional data assimilation
problems, such as those arising in weather forecasting. To gain understanding
of filters in applications such as these, it is hence of interest to study
their behaviour when applied to infinite dimensional dynamical systems. This
motivates study of the problem of accuracy and stability of 3DVAR filters for
the Navier-Stokes equation.

We work in the limit of high frequency observations and derive continuous
time filters. This leads to a stochastic partial differential equation (SPDE)
for state estimation, in the form of a damped-driven Navier-Stokes equation,
with mean-reversion to the signal, and spatially-correlated time-white noise.
Both forward and pullback accuracy and stability results are proved for this
SPDE, showing in particular that when enough low Fourier modes are observed,
and when the model uncertainty is larger than the data uncertainty in these
modes (variance inflation), then the filter can lock on to a small
neighbourhood of the true signal, recovering from order one initial error, if
the error in the observations modes is small. Numerical examples are given to
illustrate the theory.