Analysis of Fraction Skill Score properties for a displaced rainband in a rectangular domain

Abstract

A compact analytical expression of the Fraction Skill Score (FSS) is derived for a case with a single displaced rainband in a rectangular domain. The rainband is oriented parallel to the border and displaced perpendicularly to its orientation. An analytical solution is used to determine some of the properties of the FSS which might also be applicable in other cases. The solution is independent of the length of the rainband (it is valid also for a displaced rainy grid point). The position of the borders perpendicular to the rainband orientation does not influence the FSS value. The position of the other two borders does however influence the FSS value in a complex way; moving a border closer to the rainbands can either increase or decrease the FSS value depending on the location of the borders. FSS is shown to be a monotonically increasing function of the neighbourhood size (regardless of the position of the borders). If the FSS value for a displacement that is half the neighbourhood size is used to define a 'useful' FSS value then the usefulness criterion is somewhat different than presented in the original FSS paper (there is no dependence on the frequency of the observations/forecasts). 'Useful' FSS values are always >1/2 but depend on the position of the borders and the size of the displacement.