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

Covariance localization increases rank of Pb

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

ĄAnalysis only adjusted in this subspace.

ĄIf 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.


	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). CanŐt correct the missing directions, so errors grow and ensemble variance shrinks, leading to filter divergence.