Covariance Modeling in Applications of Data Assimilation to High-dimensional Earth and Geospace Systems

Abstract

Data assimilation and inverse methods play a key role in integrating remote-sensing and in-situ Earth and Geospace observations into models of the Earth and Geospace system and subsystems, enabling weather prediction and climate projection of high societal relevance. The problem presents considerable methodological and computational challenges because of the high-dimensionality and non-linearity of the underling dynamics of the Earth and Geospace system and the large volumes of heterogeneous observations from various observing platforms, including Low-Earth-Orbit and geosynchronous-Earth-orbit satellites as well as ground-based and airborne platforms. To make the problem tractable, most data assimilation methods resort to scalable methods derived from well-established Gaussian process-based formulation of the sequential Bayesian estimation problem. One such approach is to use approximate Kalman filters that rely on low-rank approximations of the covariance matrices. In ensemble data assimilation, the covariance is expressed in the subspace spanned by the model ensemble that evolves dynamically. Though the low-rank ensemble-based approximation allows the covariance to track non-linear dynamics with a moderate computational cost, the impact of the observations that lie outside of the subspace represented by the ensemble cannot be effectively incorporated into the assimilation analysis. One heuristic remedy widely used in numerical weather prediction is to adopt a hybrid method that uses the full-rank stationary covariance and the ensemble-based covariance. As a scalable approach to modeling the full-rank nonstationary covariance, the use of a multiresolution-based covariance is discussed.