Error covariance estimation methods based on analysis residuals: theoretical foundation and convergence properties derived from simplified observation networks

Abstract

We examine the theoretical foundation of the method to estimate error covariances based on analysis residuals in observation space, also known as the Desroziers method. Our analysis also includes a method based on a posteriori diagnostics of variational analysis schemes. A mathematical analysis of convergence is carried out with a simplified regular observation network, where we identify stable and unstable fixed‐point solutions, examine their rate of convergence and the conditions for convergence to the truth and compare it with maximum‐likelihood estimation. It is shown that the estimation of variance parameters converges to the truth if the error correlation is specified correctly. The convergence is much faster with the analysis increment method than with the method designed for variational analysis schemes. We also propose a combination of the Desroziers scheme and the maximum‐likelihood estimation method that could be used to estimate the spatial correlation length‐scale of observation errors. The estimation of both full observation and background‐error covariances matrices derived entirely from observation‐based residuals does not change the gain matrix, but the estimation of either one matrices is well‐defined. An analysis of the estimation of the full observation error covariance matrix using a regular observation network reveals that if all eigenvalues of the prescribed background‐error covariance are smaller than their corresponding innovation covariance eigenvalue, then the estimated error covariance converges to the truth but only if the prescribed background error is correctly specified. In the case where some eigenvalues of the background‐error covariance greatly exceed those of the innovation covariance, the convergent observation‐error matrix may become rank‐deficient. Its inverse does not exist.