  We adapt a previously introduced continuous in time data assimilation
(downscaling) algorithm for the 2D Navier-Stokes equations to the more
realistic case when the measurements are obtained discretely in time and may be
contaminated by systematic errors. Our algorithm is designed to work with a
general class of observables, such as low Fourier modes and local spatial
averages over finite volume elements. Under suitable conditions on the
relaxation (nudging) parameter, the spatial mesh resolution and the time step
between successive measurements, we obtain an asymptotic in time estimate of
the difference between the approximating solution and the unknown reference
solution corresponding to the measurements, in an appropriate norm, which shows
exponential convergence up to a term which depends on the size of the errors. A
stationary statistical analysis of our discrete data assimilation algorithm is
also provided.
