Estimating and including observation-error correlations in data assimilation

Usually in data assimilation with geophysical systems, the observation-error covariance matrix R is assumed to be diagonal for simplicity and computational efficiency, although there are studies indicating that several types of satellite observations contain significantly correlated errors. This study brings to light the impact of the off-diagonal terms of R in data assimilation. The adaptive estimation method of Li etal., which allows online estimation of the observation-error variance using innovation statistics, is extended to include off-diagonal terms of R. The extended method performs well with the 40-variable Lorenz model in estimating non-diagonal observation-error covariances. Interestingly, the analysis accuracy is improved when the observation errors are correlated, but only if the observation-error correlations are explicitly considered in data assimilation. Further theoretical considerations relate the impact of observing systems (characterized by both R and an observation operator H) on analysis accuracy. This analysis points out the importance of distinguishing between observation-error correlations (i.e. non-diagonal R) and correlated observations (i.e. non-orthogonal H). In general, observations with a non-diagonal R carry more information, whereas observations with a non-orthogonal H carry less information, but it turns out that the combination of R and H is essential: more information is available from positively (negatively) correlated observations with negatively (positively) correlated errors, resulting in a more accurate analysis.