A new spatial-scale decomposition of the Brier score: Application to the verification of lightning probability forecasts

A new scale decomposition of the Brier score for the verification of probabilistic forecasts defined on a spatial domain is introduced. The technique is illustrated on the Canadian Meteorological Centre (CMC) lightning probability forecasts. Probability forecasts of lightning occurrence in 3-h time windows and 24-km spatial resolution are verified against lightning observations from the North American Lightning Detection Network (NALDN) on a domain encompassing Canada and the northern United States. Verification is performed for lightning occurrences exceeding two different thresholds, to assess the forecast performance both for modest and intense lightning activity. Observation and forecast fields are decomposed into the sum of components on different spatial scales by performing a discrete 2D Haar wavelet decomposition. Wavelets, rather than Fourier transforms, were chosen because they are locally defined, and therefore more suitable for representing discontinuous spatial fields characterized by the presence of a few sparse nonzero values, such as lightning. Verification at different spatial scales is performed by evaluating Brier score and Brier skill score for each spatial-scale component. Reliability and resolution are also evaluated on different scales. Moreover, the bias on different scales is assessed, along with the ability of the forecasts to reproduce the observed-scale structure.