ECO-RAP, Part 1: A new adaptive error covariance localization tool for 4-dimensional ensemble data assimilation

Optimal state estimation requires accurate specifications of the observation error covariance and the forecast error covariance. In the last decade, there has been growing interest in the idea of using a K-member ensemble of forecasts designed to sample the distribution of truth given previous observations to estimate the forecast error covariance. Data assimilation schemes that use such estimates are generally called Ensemble Kalman filters. For very-high dimensional complex systems like the atmosphere and ocean, current computational resources typically limit ensemble sizes to K<100. For such small ensembles, the sample correlations inevitably contain spurious ensemble correlations. Prescribed, non-adaptive moderation or localization functions are widely used in ensemble data assimilation (DA) to reduce the amplitude of spurious ensemble correlations. These functions are poorly suited to four-dimensional (4D) DA problems because true error correlation functions move with the flow while non-adaptive localization functions do not.

A new 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). It is based on the discovery that error propagation information and error correlation width information retained by powers of raw ensemble correlations can be used to propagate and adaptively adjust the width of user-specified correlation functions. The manner in which powers of raw ensemble correlations can achieve this feat will be demonstrated using mathematical analysis and a few important examples. A common test of 4D DA systems is to see if they can accurately estimate a 4D state from a series of observations from a single location. It is shown that 4D DA using raw ensemble Covariances Adaptively Localized with ECO-rap (CALECO) passes this test. To highlight the ability of ECO-RAP localization to adapt to changes in the width of the true error correlation length scale, a three-dimensional (3D) error system was considered in which such changes occurred and it was shown that ECO-RAP localization was superior to non-adaptive localization. When no such variations in error correlation length scale were present, ECO-RAP and non-adaptive localization delivered DA performance of the same quality.