An Optimal Linear Transformation for Data Assimilation

Linear transformations in state and observation space offer new avenues to improve data-assimilation algorithms. We consider the optimal transformation (for the Kalman filter or best linear unbiased estimator) that diagonalizes the update, in the sense that all covariances become the identity and each coordinate in the observation space depends on a single, unique coordinate in the state space. Although this fact is not new, we illustrate its potential to improve covariance localization in ensemble data assimilation, especially in situations with a range of spatial and temporal scales, and present examples from the Last Millennium Reanalysis.