Ensemble covariances adaptively localized with ECO-RAP. Part 1: tests on simple error models

In atmospheric data assimilation (DA), observations over a 6-12-h time window are used to estimate the state. Non-adaptive moderation or localization functions are widely used in ensemble DA to reduce the amplitude of spurious ensemble correlations. These functions are inappropriate (1) if true error correlation functions move a comparable distance to the localization length scale over the time window and/or (2) if the widths of true error correlation functions are highly flow dependent. A method for generating localization functions that move with the true error correlation functions and that also adapt to the width of the true error correlation function is given. The method uses ensemble correlations raised to a power (ECO-RAP). A gallery of periodic one-dimensional error models is used to show how the method uses error propagation information and error correlation width information retained by powers of raw ensemble correlations to propagate and adaptively adjust the width of the localization function. It is found that ECO-RAP localization outperforms non-adaptive localization when the true errors are propagating or the error correlation length scale is varying and is as good as non-adaptive localization when such variations in error covariance structure are absent.