  We analyze the performance of a data-assimilation algorithm based on a linear
feedback control when used with observational data that contains measurement
errors. Our model problem consists of dynamics governed by the two-dimension
incompressible Navier-Stokes equations, observational measurements given by
finite volume elements or nodal points of the velocity field and measurement
errors which are represented by stochastic noise. Under these assumptions, the
data-assimilation algorithm consists of a system of stochastically forced
Navier-Stokes equations. The main result of this paper provides explicit
conditions on the observation density (resolution) which guarantee explicit
asymptotic bounds, as the time tends to infinity, on the error between the
approximate solution and the actual solutions which is corresponding to these
measurements, in terms of the variance of the noise in the measurements.
Specifically, such bounds are given for the the limit supremum, as the time
tends to infinity, of the expected value of the $L^2$-norm and of the $H^1$
Sobolev norm of the difference between the approximating solution and the
actual solution. Moreover, results on the average time error in mean are
stated.
