A Review of Innovation-Based Methods to Jointly Estimate Model and Observation Error Covariance Matrices in Ensemble Data Assimilation

<jats:title>Abstract</jats:title><jats:p>Data assimilation combines forecasts from a numerical model with observations. Most of the current data assimilation algorithms consider the model and observation error terms as additive Gaussian noise, specified by their covariance matrices <jats:inline-formula/> and <jats:inline-formula/>, respectively. These error covariances, and specifically their respective amplitudes, determine the weights given to the background (i.e., the model forecasts) and to the observations in the solution of data assimilation algorithms (i.e., the analysis). Consequently, <jats:inline-formula/> and <jats:inline-formula/> matrices significantly impact the accuracy of the analysis. This review aims to present and to discuss, with a unified framework, different methods to jointly estimate the <jats:inline-formula/> and <jats:inline-formula/> matrices using ensemble-based data assimilation techniques. Most of the methods developed to date use the innovations, defined as differences between the observations and the projection of the forecasts onto the observation space. These methods are based on two main statistical criteria: 1) the method of moments, in which the theoretical and empirical moments of the innovations are assumed to be equal, and 2) methods that use the likelihood of the observations, themselves contained in the innovations. The reviewed methods assume that innovations are Gaussian random variables, although extension to other distributions is possible for likelihood-based methods. The methods also show some differences in terms of levels of complexity and applicability to high-dimensional systems. The conclusion of the review discusses the key challenges to further develop estimation methods for <jats:inline-formula/> and <jats:inline-formula/>. These challenges include taking into account time-varying error covariances, using limited observational coverage, estimating additional deterministic error terms, or accounting for correlated noise.</jats:p>