Analysis of Fraction Skill Score properties for a displaced rainy grid point in a rectangular domain

• Analytical expression for FSS of a displaced rainy grid point is obtained.
• FSS value is dependent on the direction of the displacement.
• The FSS value near a border can be at most 33% larger than in the infinite domain.
• The effect of the nearby corner is stronger than the effect of the nearby border.
• The likelihood of a strong border effect in real datasets increases significantly for large neighborhoods.

The Fraction Skill Score (FSS) is a recently developed and popular metric used for precipitation verification. A compact analytical expression for FSS is derived for a case with a single displaced rainy grid point in a rectangular domain. The existence of an analytical solution is used to determine some properties of FSS, which might also be applicable in other cases since the rain areas of any shape will asymptote towards this solution if the displacement is sufficiently large. The use of the simple square shape of the neighborhood causes the FSS value to be dependent on the direction of the displacements (not only on the displacement size). The effect is limited in scope but can increase or decrease the FSS value by 0.1. Moving a nearby border closer to the rainy points can either increase or decrease the FSS value, depending on the location of the border. The FSS value near a border can be at most 33% larger than the FSS value in the infinite domain, assuming the same neighborhood size and displacement. The effect of the nearby corner is similar to the effect of the nearby border but is stronger. The useful forecast criteria (FSSuseful) is defined as a value of FSS for a precipitation feature with a displacement half the neighborhood size. FSSuseful for a displaced rainy grid point depends on the orientation of the displacement being the largest for displacements that are parallel to the borders and the smallest for a diagonal displacement for which the value can be as low as 0.42. An analysis of a real dataset was also performed, which showed that the border effect is usually small, but in some cases the effect becomes large (an increase of FSS value up to 70% was identified). The likelihood of a strong border effect in real datasets increases significantly if the neighborhood size at FSS = 0.5 is comparable or larger than the domain size.