  Obtaining accurate high-resolution representations of model outputs is
essential to describe the system dynamics. In general, however, only spatially-
and temporally-coarse observations of the system states are available. These
observations can also be corrupted by noise. Downscaling is a process/scheme in
which one uses coarse scale observations to reconstruct the high-resolution
solution of the system states. Continuous Data Assimilation (CDA) is a recently
introduced downscaling algorithm that constructs an increasingly accurate
representation of the system states by continuously nudging the large scales
using the coarse observations. We introduce a Discrete Data Assimilation (DDA)
algorithm as a downscaling algorithm based on CDA with discrete-in-time
nudging. We then investigate the performance of the CDA and DDA algorithms for
downscaling noisy observations of the Rayleigh-B\'enard convection system in
the chaotic regime. In this computational study, a set of noisy observations
was generated by perturbing a reference solution with Gaussian noise before
downscaling them. The downscaled fields are then assessed using various error-
and ensemble-based skill scores. The CDA solution was shown to converge towards
the reference solution faster than that of DDA but at the cost of a higher
asymptotic error. The numerical results also suggest a quadratic relationship
between the $\ell_2$ error and the noise level for both CDA and DDA. Cubic and
quadratic dependences of the DDA and CDA expected errors on the spatial
resolution of the observations were obtained, respectively.
