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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 ensembledoi:10.5194/npg-8-331-2001 fatcat:ofsvjk5qabcdlev6inaogkhioe