Justification for estimating observation‐error covariances with the Desroziers diagnostic

Abstract

The convergence properties of the iterative Desroziers method, which is commonly used to estimate observation error statistics and correlations based on analysis and background departures, is examined. In the derivation of the Desroziers diagnostic, it is assumed that the error statistics are correctly specified in the analysis. While this is not true in practice, it is possible to apply the diagnostic iteratively to obtain increasingly better estimates of the error statistics and correlations. It is proved here that these iterations converge to the truth only when the background error statistics are correct in the analysis. Bounds on the eigenvalues of the assumed background error covariance matrix determine when this matrix is over‐estimated. If the background covariance is not exact, but is not over‐estimated, it is shown that the iterations converge to a symmetric positive definite matrix that depends on the true and assumed error covariances, and diverge to a singular matrix otherwise. The use of the Desroziers method when background and observation errors are correlated with each other is also addressed, as well as the application of this method to unassimilated observations, with examples.