Abstract
In order to represent forecast uncertainty in numerical weather prediction, ensemble prediction systems generate several different forecasts of the same weather variable by perturbing initial conditions and model parameters. The resulting ensemble of forecasts is interpreted as a sample of a predictive distribution. It offers valuable probabilistic information, but often turns out to be uncalibrated, i.e. it suffers from biases and typically underestimates the prediction uncertainty. Methods for statistical post-processing have therefore been proposed to re-calibrate the ensemble and turn it into a full predictive probability distribution. Weather variables like temperature depend on factors that are quite variable in space which suggests that post-processing should be done at each site individually. If a predictive distribution is desired at locations where no measurements for calibration are available, post-processing parameters from neighboring stations must be interpolated. We propose an extension of the non-homogeneous Gaussian regression (NGR) approach for temperature post-processing that uses an intrinsically stationary Gaussian random field model for spatial interpolation. This model is able to capture large scale fluctuations of temperature, while additional covariates are integrated into the random field model to account for altitude-related and other local effects. We further discuss the modeling of spatial correlations of forecast errors for temperature. This becomes important whenever probabilistic forecasts at different sites are considered simultaneously, or when the interest is in composite quantities like averages, minima, or maxima of temperature over some domain. Again, we use Gaussian random field models which are now fitted to forecast errors in a training data set, and permit the simulation of re-calibrated temperature fields with corrected spatial correlations. Both of these spatial aspects are illustrated with temperature forecasts by the COSMO-DE-EPS, an ensemble prediction system operated by the German Meteorological Service, and observational data over Germany.