Adjoints and low-rank covariance representation

Quantitative measures of the uncertainty of Earth system estimates can be as important as the estimates themselves. Direct calculation of second moments of estimation errors, as described by the covariance matrix, is impractical when the number of degrees of freedom of the system state is large and the sources of uncertainty are not completely known. Theoretical analysis of covariance equations can help guide the formulation of low-rank covariance approximations, such as those used in ensemble and reduced-state approaches for prediction and data assimilation. We use the singular value decomposition and recently developed positive map techniques to analyze a family of covariance equations that includes stochastically forced linear systems. We obtain covariance estimates given imperfect knowledge of the sources of uncertainty and we obtain necessary conditions for low-rank approximations to be appropriate. The results are illustrated in a stochastically forced system with time-invariant linear dynamics.