Statistical methods for interpreting Monte Carlo ensemble forecasts

For complex dynamical systems such as the atmosphere, improved estimates of future behaviour can be obtained by making ensembles of forecasts starting from a set of Monte Carlo perturbed initial conditions. Ensemble forecasting, however, generates an overwhelming amount of data that is difficult to analyse in detail. Fortunately, the main features of interest are often summarised by certain statistics estimated from the sample of forecasts. By considering an ensemble of forecasts as a realisation of a linear mapping from phase space to sample space, it is possible to 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 distances between the different ensemble members. Multivariate skewness and kurtosis can also be estimated from ensembles of forecasts and provide useful information on the reliability of the sample mean and covariance estimated from the ensemble. They can also give useful information on the non-linearity of the evolution in phase space. Entropy can also be defined for an ensemble of forecasts and shows a regular increase due to the smooth and rapid loss of initial information in the first 3 days of a meteorological forecast. These new tools for summarising ensemble forecasts are illustrated using a single ensemble of 51 weather forecasts made at the European Centre for Medium-Range Weather Forecasts for the period 20-30 December 1997.