Ensemble Kalman Filter Configurations and Their Performance with the Logistic Map

This paper examines ensemble Kalman filter (EnKF) performance for a number of different EnKF configurations. The study is performed in a perfect-model context using the logistic map as forecast model. The focus is on EnKF performance when the ensemble is small. In accordance with theory, it is found that those configurations that maintain an appropriate ensemble spread are indeed those with the smallest ensemble mean error in a data assimilation cycle. Thus, the deficient ensemble spread produced by the single-ensemble EnKF results in increased ensemble mean error for this configuration. This problem with the conceptually simplest EnKF motivates an examination of a variety of other configurations. These include the configuration with a pair of ensembles and several configurations with overlapping ensembles, such as the four-subensemble configuration (used operationally at the Canadian Meteorological Centre) and the configuration in which observations are assimilated into each member using a gain computed from all of the other members. Also examined is a configuration that uses the jackknife estimator to obtain an estimate of the gain and an estimate of its uncertainty. Using these estimates, a different perturbed gain is then produced for each ensemble member. In general, it is found that these latter configurations outperform both the single-ensemble EnKF and the configuration with a pair of ensembles. In addition to these "stochastic'' filters, the performance of a "deterministic'' filter (which does not use perturbed observations) is also examined.