## Models for very large covariance matrices in atmospheric and oceanic sciencesChris SnyderNCAR/MMM/IMAGe AbstractWith state dimensions exceeding 10 ^{7}, 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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