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Obtaining accurate high-resolution representations of model outputs isessential to describe the system dynamics. In general, however, only spatially-and temporally-coarse observations of the system states are available. Theseobservations can also be corrupted by noise. Downscaling is a process/scheme inwhich one uses coarse scale observations to reconstruct the high-resolutionsolution of the system states. Continuous Data Assimilation (CDA) is a recentlyintroduced downscaling algorithm that constructs an increasingly accuraterepresentation of the system states by continuously nudging the large scalesusing the coarse observations. We introduce a Discrete Data Assimilation (DDA)algorithm as a downscaling algorithm based on CDA with discrete-in-timenudging. We then investigate the performance of the CDA and DDA algorithms fordownscaling noisy observations of the Rayleigh-B\'enard convection system inthe chaotic regime. In this computational study, a set of noisy observationswas generated by perturbing a reference solution with Gaussian noise beforedownscaling them. The downscaled fields are then assessed using various error-and ensemble-based skill scores. The CDA solution was shown to converge towardsthe reference solution faster than that of DDA but at the cost of a higherasymptotic error. The numerical results also suggest a quadratic relationshipbetween the $\ell_2$ error and the noise level for both CDA and DDA. Cubic andquadratic dependences of the DDA and CDA expected errors on the spatialresolution of the observations were obtained, respectively.

Continuous and Discrete Data Assimilation with Noisy Observations for the Rayleigh-Benard Convection: A Computational Study