PowerPoint Presentation - Methods for dealing with spurious covariances arising from small samples in ensemble data assimilation

Covariance localization increases rank of Pb

•If the ensemble has k members, then Pb describes nonzero uncertainty only in a k-dimensional subspace .

•Analysis only adjusted in this subspace.

•If the system is high-dimensionally unstable (if it has more than k positive Lyapunov exponents) then forecast errors will grow in directions not accounted for by the ensemble, and these errors will not be corrected by the analysis.

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	Sampling error manifests itself directly in the form of spurious long-range covariances. Alternaltely, one can think of the sampling error as a manifestation of rank-deficiency in the ensemble (if k < the number of degrees of freedom in the dynamical model). Can’t correct the missing directions, so errors grow and ensemble variance shrinks, leading to filter divergence.