Models for very large covariance matrices in atmospheric and oceanic sciences

Chris Snyder

With state dimensions exceeding 107, state estimation for large numerical predictions of the atmosphere or ocean requires computational manipulation of covariance matrices that are too large even to be stored on existing computers. I will review some of the methods used to represent and compute with such large covariance matrices. These methods capitalize on prior information related to the dynamics of the atmospheric and oceanic motions. The dynamical information may come from approximate relations derived directly from the governing equations of fluid motion, from heuristic assumptions and from the fluid dynamics embodied in the evolution of solutions from the numerical prediction model. Monte-Carlo techniques, in which the covariance matrix is estimated from a small sample, are a promising and flexible recent approach that avoids some of the assumptions necessary in other covariance models. When the state can be assumed to have a characteristic, finite correlation length, these methods work surprisingly well even with samples much smaller than the state dimension.

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