Decomposition of the Brier score for weighted forecast-verification pairs

The Brier score is widely used in meteorology for quantifying probability forecast quality. The score can be decomposed into terms representing different aspects of forecast quality, but this implicitly requires each forecast-verification pair to be allocated equal weight. In this note an expression is derived for the decomposed Brier score that accounts for weighted forecast-verification pairs. A comparison of the unweighted and weighted cases using seasonal forecasts from the ENSEMBLES project shows that when weights are assigned proportional to the area represented by each grid point (weighting by cosine of latitude), the weighted forecasts give improved Brier and reliability scores comparedwith the unweighted case. This result is consistent with what is expected, given that tropical predictability is generally better than extratropical predictability.