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

TL;DR

This study compares continuous and discrete data assimilation algorithms for reconstructing high-resolution states of Rayleigh-Benard convection from noisy, coarse observations, highlighting their convergence behaviors and error dependencies.

Contribution

It introduces a Discrete Data Assimilation algorithm based on CDA and evaluates both methods' performance with noisy data in a chaotic convection system.

Findings

CDA converges faster than DDA but has higher asymptotic error.

Error scales quadratically with noise level for both methods.

Error depends on spatial resolution, with DDA showing quadratic and CDA cubic relationships.

Abstract

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.