Design and implementation of data assimilation methods based on Cholesky decomposition

Abstract

In Data Assimilation, analyses of a system are obtained by combining a previous numerical model of the system and observations or measurements from it. These numerical models are typically expressed as a set of ordinary differential equations and/or a set of partial differential equations wherein all knowledge about dynamics and physics of, for instance, the ocean and or the atmosphere are encapsulated. We treat numerical forecasts and observations as random variables and therefore, error dynamics can be estimated by using Bayes’ rule. For the estimation of hyper-parameters in error distributions, an ensemble of model realizations is employed. In practice, model resolutions are several order of magnitudes larger than ensemble sizes, and consequently, sampling errors impact the quality of analysis corrections and besides, models can be highly non-linear and well-common Gaussian assumptions on prior errors can be broken. To overcome these situations, we replace prior errors by a mixture of Gaussians and even more, precision covariance matrices intra-clusters are estimated by means of the modified Cholesky decomposition. Four different methods are proposed, namely the Posterior EnKF with its deterministic and stochastic variations, a Non-Gaussian method and a MCMC filter, which used the Bickel-Levina estimator; these methods are based on a modified Cholesky decomposition and tested with the Lorenz 96 model. Their implementations are shown to provide equivalent solutions compared to another EnKF methods like the LETKF and the EnSRF.