For the 3x3, 2x2, and 1x1 block systems of weak constraint 4D-Var, adding observations moves negative eigenvalues away from zero while shrinking some positive eigenvalues, and explicit spectral intervals are derived for all three systems.
Inexact constraint preconditioners for linear systems arising in interior point methods
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Spectral estimates for saddle point matrices arising in weak constraint four-dimensional variational data assimilation
For the 3x3, 2x2, and 1x1 block systems of weak constraint 4D-Var, adding observations moves negative eigenvalues away from zero while shrinking some positive eigenvalues, and explicit spectral intervals are derived for all three systems.