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.
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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.