An Adaptive Ensemble Kalman Filter

Abstract

To the extent that model error is nonnegligible in numerical models of the atmosphere, it must be accounted for in 4D atmospheric data assimilation systems. In this study, a method of estimating and accounting for model error in the context of an ensemble Kalman filter technique is developed. The method involves parameterizing the model error and using innovations to estimate the model-error parameters. The estimation algorithm is based on a maximum likelihood approach and the study is performed in an idealized environment using a three-level, quasigeostrophic, T21 model and simulated observations and model error. The use of a limited number of ensemble members gives rise to a rank problem in the estimate of the covariance matrix of the innovations. The effect of this problem on the two terms of the log-likelihood function is that the variance term is underestimated, while the ?2 term is overestimated. To permit the use of relatively small ensembles, a number of strategies are developed to deal with these systematic estimation problems. These include the imposition of a block structure on the covariance matrix of the innovations and a Richardson extrapolation of the log-likelihood value to infinite ensemble size. It is shown that with the use of these techniques, estimates of the model-error parameters are quite acceptable in a statistical sense, even though estimates based on any single innovation vector can be poor. It is found that, with temporal smoothing of the model-error parameter estimates, the adaptive ensemble Kalman filter produces fairly good estimates of the parameters and accounts rather well for the model error. In fact, its performance in a data assimilation cycle is almost as good as that of a cycle in which the correct model-error parameters are used to increase the spread in the ensemble.