Statistical methods for interpreting Monte Carlo ensemble forecasts

For complex dynamical systems such as the atmosphere, improved estimates of future behaviourcan be obtained by making ensembles of forecasts starting from a set of Monte Carlo perturbedinitial conditions. Ensemble forecasting, however, generates an overwhelming amount of datathat is difficult to analyse in detail. Fortunately, the main features of interest are often summarisedby certain statistics estimated from the sample of forecasts. By considering an ensemble offorecasts as a realisation of a linear mapping from phase space to sample space, it is possibleto construct two types of sample covariance matrix. The ensemble covariance can be visualised by constructing multidimensional scaling maps, which show at a glance the relative distancesbetween the different ensemble members. Multivariate skewness and kurtosis can also be estimatedfrom ensembles of forecasts and provide useful information on the reliability of the samplemean and covariance estimated from the ensemble. They can also give useful information onthe non-linearity of the evolution in phase space. Entropy can also be defined for an ensembleof forecasts and shows a regular increase due to the smooth and rapid loss of initial informationin the first 3 days of a meteorological forecast. These new tools for summarising ensembleforecasts are illustrated using a single ensemble of 51 weather forecasts made at the EuropeanCentre for Medium-Range Weather Forecasts for the period 20–30 December 1997.