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.

Evaluation of Analysis by Cross-Validation, Part II: Diagnostic and Optimization of Analysis Error Covariance

“…In this paper, we go further by using principles of error covariance estimation to obtain a near optimal Kalman gain. In addition we impose the innovation covariance consistency [15] and show that all diagnostics of analysis error variance nearly agree with one another. These include the Hollingsworth and Lönnberg [13], the Desroziers et al [14] and new diagnostics that we will introduce.…”

“…In our next step, iter 1, we first re-estimate the correlation length by applying a maximum likelihood method, as in Ménard [15], using the iter 0 error variances that are consistent with the optimal ratioγ obtained in iter 0 and the innovation variance consistency. Then, with this new correlation length, we estimate a new optimal ratioγ (iter 1), which turn out to be very close to the value obtained in iter 0.…”

“…should be equal to the analysis error covariance in observation space but, again, only if the gain is optimal and the innovation covariance consistency is respected [15]. The impasse of the estimation of the true analysis error seems to be tied with using active observations, i.e., using the same observations as those used to create the analysis.…”

“…where K is the gain matrix built from the prescribed error covariances, H is the observation operator for the active observation set, d is the active innovation vector, [14,15,18].…”

“…This implies that the gain matrix using the prescribed error covariances, K in Equation (1), must be identical to the gain using the true error covariances, i.e., K = BH T (HBH T + R) −1 [14,15]. It is important to mention that necessary and sufficient conditions to obtain the true error covariances HBH T and R in observation space, are: 1-the Kalman gain condition, H K = HK true and 2-the innovation covariance consistency,…”