Tuesday, 15 January 2002: 4:44 PM
The value of perfection
Perfect ensembles under perfect models yield accountable PDF forecasts. Nothing else does. But does it matter? For large enough analysis errors and/or small enough ensemble sizes it may prove difficult to distinguish between perfect and imperfect ensembles. Relative entropy yields a measure of value for perfect ensembles under perfect models, slowly decreasing until the for ensemble is indistinguishable from the climatology and the forecast is useless. Yet this information theoretic statistic fails to distinguish 'good' information from 'bad' information, where by 'bad' information we mean a real difference between the evolved ensemble in hand and the corresponding perfect ensemble due, for instance, to model error or an initial ensemble inconsistent with the long term dynamics of the system (so-called ``off the attractor'' points). The minimal spanning tree generalization of rank histograms to higher dimensions can quantify the forecast time when, on average, the analysis is not consistent with a random sample from the evolved ensemble. This information is complementary to that provided by relative entropy. Methods for combining these two measures are discussed; both are illustrated in moderate dimensional chaotic systems (where a perfect model exists), and the minimal spanning tree is applied to the atmosphere as modeled by the operational ECMWF ensemble system.