A multi-scale data assimilation method for the ensemble Kalman filter (EnKF) is proposed for atmospheric models in cases with insufficient observations of fast variables. This method is based on the conservation and invertibility of potential vorticity (PV). The dynamical state variables in the free atmosphere of forecast ensemble members are decomposed into balanced and unbalanced parts by applying PV inversion to the PV anomalies computed from spatially smoothed state variables. The mass variables of the two parts are adjusted to remove additional sampling errors introduced by the decomposition. The forecast error covariances between those parts are ignored in the Kalman gain to suppress spurious error correlations. This approximation makes it possible to apply different covariance localizations to each part. The Kalman gain thus obtained is used to assimilate observations.
The performance of the proposed method is demonstrated with a shallow water model through twin experiments in a perfect model scenario. The results using the same localization radius for the two parts reveal that the proposed EnKF is superior in the accuracy of the analysis to a conventional EnKF unless the ensemble size is sufficiently large. It is found that the adjustment of mass variables is necessary to outperform the conventional EnKF. The benefits of the PV inversion using the Bolin–Charney balance over the quasi-geostrophic inversion are marginal in the experiments.
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