Using the Kalman-Bucy filter in an ensemble framework

Two recent formulations for the analysis step in ensemble Kalman filtering (EnKF) adapt ideas from the continuous Kalman-Bucy filter into an ensemble setting. In the first formulation (Bergemann et al, 2009) –which we label BGR09-, the ensemble of perturbations is updated through an ODE formulation in pseudo-time, while the mean and covariance are updated in the standard EnKF way. In the second formulation (Bergeman and Reich, 2010) –which we label BR10-, the full ensemble is updated in the analysis step as the solution of a single ODE in pseudo-time. We use the Lorenz 3-variable model (1963) to assess the performance of these two methods and we compare them to an ensemble transform Kalman filter, the LETKF of Hunt et al, 2007. Two cases are considered in our experiment: frequent observations (every 8 time steps), where the perturbations are linear, and infrequent observations (every 25 time steps), where perturbations grow nonlinearly. In both, the stability and convergence of the formulations are analyzed under different covariance inflation parameters as well as different number of steps in the numerical solution of the pseudo-time ODEs. For the frequent observations case, both formulations are stable and achieve results comparable to the LETKF using 5 steps in pseudo-time and with very little difference between BGR09 and BR10. For the infrequent observations case, however, at least 50 steps in pseudo-time are needed for BGR09 to achieve a performance comparable to the LETKF, while for BR10 not even 60 steps are enough to achieve this. Moreover, in this case there is a clear difference in the performance between BGR09 and BR10, in which the latter is more unstable than the former. While in the frequent observations case the new continuous formulations can be considered successful, in the case of infrequent observations they fail to provide a good performance without heavy computational expenses.