Impacts of localisation in the EnKF and EnOI: experiments with a small model

The performance of an inexpensive, ensemble based optimal interpolation (EnOI) scheme that uses a stationary ensemble of model anomalies to approximate forecast error covariances, is compared with that of an ensemble Kalman filter (EnKF). The model to which the methods are applied is a pair of "perfect", one-dimensional, linear advection equations for two related variables. While EnOI is sub-optimal, it can give results that are comparable to those of the EnKF. The computational cost of EnOI is typically about N times less than that of EnKF, where N is the ensemble size. We suggest that EnOI may provide a practical and cost-effective alternative to the EnKF for some applications where computational cost is a limiting factor. We demonstrate that when the ensemble size is smaller than the dimension of the model's sub-space, both the EnKF and EnOI may require localisation around each observation to eliminate effects of sampling error and to increase the effective number of independent ensemble members used to construct an analysis. However, localisation can degrade an analysis if the length-scales of the localising function are too short. We demonstrate that, as the length-scale of the localising function is decreased, localisation can significantly compromise the model's dynamical balances. We also find that localisation artificially amplifies high frequencies for applications of the EnKF. Based on our experiments, for applications where localisation is necessary, the length-scales of the localisation should be larger than the decorrelation length scales of the variables being updated.