Localization in the ensemble Kalman Filter

Data assimilation in meteorology seeks to provide a current analysis of the state of the atmosphere to use as initial conditions in a weather forecast. This is achieved by using an estimate of a previous state of the system and merging that with observations of the true state of the system. Ensemble Kalman filtering is one method of data assimilation. Ensemble Kalman filters operate by using an ensemble, or statistical sample, of the state of a system. A known prior state of a system is forecast to an observation time, then observation is assimilated. Observations are assimilated according to a ratio of the errors in the prior state and the observations. An analysis estimate of the system and an analysis estimate of the of the errors associated with the analysis state are also produced. This project looks at some problems within ensemble Kalman filtering and how they may be overcome. Undersampling is a key issue, this is where the size of the ensemble is so small so as to not be statistically representative of the state of a system. Undersampling can lead to inbreeding, filter divergence and the development of long range spurious correlations. It is possible to implement counter measures. Firstly covariance inflation is used to combat inbreeding and the subsequent filter divergence. Covariance localization is primarily used to remove long range spurious correlations but also has the benefit increasing the effective ensemble size. Specifically this project uses an implementation of the ensemble Transform Kalman filter (ETKF) a deterministic ensemble, with a simple model, to demonstrate the behaviour of the filter when undersampling is present. Covariance inflation was implemented, and was found to increase the accuracy of the analysis state. A new method of covariance localization by Schur product for the ETKF was introduced and implemented. This method was not consistent with the equations of the ETKF. The analysis estimate was detrimentally affected by this technique. By using covariance inflation in conjunction with this localization the performance may be improved. In its current state this implementation does not function as desired.