# Richard Menard, Environment and Climate Change Canada

## Theoretical foundation of error covariance estimation methods based on analysis residuals, with examples derived from realistic surface observation network

#### 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. An analysis of convergence is carried out with a simplified regular observation network but other more realistic observation network is also considered. 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