A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters

The use of perturbed observations in the traditional ensemble Kalman filter (EnKF) results in a suboptimal filter behaviour, particularly for small ensembles. In this work, we propose a simple modification to the traditional EnKF that results in matching the analysed error covariance given by Kalman filter in cases when the correction is small; without perturbed observations. The proposed filter is based on the recognition that in the case of small corrections to the forecast the traditional EnKF without perturbed observations reduces the forecast error covariance by an amount that is nearly twice as large as that is needed to match Kalman filter. The analysis scheme works as follows: update the ensemble mean and the ensemble anomalies separately; update the mean using the standard analysis equation; update the anomalies with the same equation but half the Kalman gain. The proposed filter is shown to be a linear approximation to the ensemble square root filter (ESRF). Because of its deterministic character and its similarity to the traditional EnKF we call it the ‘deterministic EnKF’, or the DEnKF. A number of numerical experiments to compare the performance of the DEnKF with both the EnKF and an ESRF using three small models are conducted. We show that the DEnKF performs almost as well as the ESRF and is a significant improvement over the EnKF. Therefore, the DEnKF combines the numerical effectiveness, simplicity and versatility of the EnKF with the performance of the ESRFs. Importantly, the DEnKF readily permits the use of the traditional Schur product-based localization schemes.