  We study a discrete-in-time data-assimilation algorithm based on nudging
through a time-delayed feedback control in which the observational measurements
have been contaminated by a Gaussian noise process. In the context of the
two-dimensional incompressible Navier-Stokes equations we prove the expected
value of the square-error between the approximating solution and the reference
solution over time is proportional to the variance of the noise up to a
logarithmic correction. The qualitative behavior and physical relevance of our
analysis is further illustrated by numerical simulation.
