Search

The extended Kalman filter (EKF) is a popular state estimation method for nonlinear dynamical models. The model error covariance matrix is often seen as a tuning parameter in EKF, which is often simply postulated by the user. In this paper, we study the filter likelihood technique for estimating the parameters of the model error covariance matrix. The approach is based on computing the likelihood of the covariance matrix parameters using the filtering output. We show that (a) the importance of the model error covariance matrix calibration depends on the quality of the observations, and that (b) the estimation approach yields a well-tuned EKF in terms of the accuracy of the state estimates and model predictions. For our numerical experiments, we use the two-layer quasi-geostrophic model that is often used as a benchmark model for numerical weather prediction.

Algorithmic numerical weather prediction (NWP) skill optimization has been tested using the Integrated Forecasting System (IFS) of the European Centre for Medium-Range Weather Forecasts (ECMWF). We report the results of initial experimentation using importance sampling based on model parameter estimation methodology targeted for ensemble prediction systems, called the ensemble prediction and parameter estimation system (EPPES). The same methodology was earlier proven to be a viable concept in low-order ordinary differential equation systems, and in large-scale atmospheric general circulation models (ECHAM5). Here we show that prediction skill optimization is possible even in the context of a system that is (i) of very high dimensionality, and (ii) carefully tuned to very high skill. We concentrate on four closure parameters related to the parameterizations of sub-grid scale physical processes of convection and formation of convective precipitation. We launch standard ensembles of medium-range predictions such that each member uses different values of the four parameters, and make sequential statistical inferences about the parameter values. Our target criterion is the squared forecast error of the 500 hPa geopotential height at day three and day ten. The EPPES methodology is able to converge towards closure parameter values that optimize the target criterion. Therefore, we conclude that estimation and cost function-based tuning of low-dimensional static model parameters is possible despite the very high dimensional state space, as well as the presence of stochastic noise due to initial state and physical tendency perturbations. The remaining question before EPPES can be considered as a generally applicable tool in model development is the correct formulation of the target criterion. The one used here is, in our view, very selective. Considering the multi-faceted question of improving forecast model performance, a more general target criterion should be developed. This is a topic of ongoing research.