Stochastic enlargement of ensembles under a second-moment constraint

Craig H. Bishop

Naval Research Laboratory, Monterey, CA

Xuguang Wang

The Pennsylvania State University, University Park, PA


One approach to correcting known deficiencies in the probability estimates provided by ensemble forecasts is to stochastically enlarge the ensemble and change its covariance by adding stochastic perturbations to each of the ensemble members. Essentially, this is a stochastic or particle representation of a method of probability density smoothing known to statisticians as kernel density estimation. Other researchers had suggested that the stochastic perturbations should be drawn from an archive of best member errors. This approach was tested using random number generators and was shown to give unsatisfactory results depending on the number of raw ensemble members and the degree of under dispersion of the raw ensemble. To improve upon this approach a constrained stochastic enlargement method was proposed and tested. In the constrained method, the distribution from which the stochastic perturbations are drawn is designed to ensure that the members of the stochastically enlarged ensemble are indistinguishable from the verification under a second moment measurement on a seasonally averaged basis. Tests with random number generators showed that the constrained ensemble enlargement procedure reliably predicts forecast error variance regardless of the size and under-dispersion of the raw ensemble. The constrained stochastic enlargement procedure was also applied to a 16 member ensemble of T42 NCAR CCM3 forecasts. In the test categories of rank histogram and skill scores, the constrained stochastic enlargement method gave better scores than the raw ensemble and "best member" stochastic enlargement method. To show the applicability of the constrained enlargement method to non-linear multi-variate user cost functions, it was also applied to the problem of forecasting and trading cumulative cooling degree days (CDDs) in Boston.