Adaptive Tuning of Numerical Weather Prediction Models: Simultaneous Estimation of Weighting, Smoothing, and Physical Parameters

Abstract

In Wahba et al. it was shown how the randomized trace method could be used to adaptively tune numerical weather prediction models via generalized cross validation (GCV) and related methods. In this paper a ?toy? four-dimensional data assimilation model is developed (actually one space and one time variable), consisting of an equivalent barotropic vorticity equation on a latitude circle, and used to demonstrate how this technique may be used to simultaneously tune weighting, smoothing, and physical parameters. Analyses both with the model as a strong constraint (corresponding to the usual 4D-Var approach) and as a weak constraint (corresponding theoretically to a fixed-interval Kalman smoother) are carried out. The conclusions are limited to the particular toy problem considered, but it can be seen how more elaborate experiments could be carried out, as well as how the method might be applied in practice. The authors have considered five adjustable parameters, two related to a distributed coefficient in the equivalent barotropic vorticity equation (?physical? parameters), one governing the relative weight given to observations versus forecast, one governing the relative weight given to observations versus goodness of fit to the model (in the weak constraint case), and one governing the damping of high-frequency oscillations in the analysis at the final time point (?smoothing? parameter). The weighting parameters and the smoothing parameter can, if desired, be interpreted as ratios of parameters in prior covariances. Analyses are made with a low-resolution model of the dynamics of the equivalent barotropic vorticity equation given noisy forecast (initial conditions) and noisy wind observations, and compared with nature evolved from exact initial conditions using a high-resolution forward integration. The authors found that these five (carefully chosen) parameters are simultaneously tunable on line, that is, simultaneously with the analysis, and 1) that the analysis is equally and strongly sensitive to both the choice of the observed versus forecast weighting parameter and the choice of the smoothing parameter; 2) that the analysis with the model as a weak constraint, based on the tuned estimate of the parameter governing how close the analysis satisfies the model, is somewhat better than the analysis using the model as a strong constraint, although estimation of this tuning parameter varies much more than the other parameters with replications of the experiment; and 3) good estimates of the physical parameters are obtained; however, these estimates are closer to those that make the model integrated forward with perfect initial conditions best fit nature, and these are not exactly the ?true? parameters.