The relation between the mean areal rainfall and the fractional area where it rains above a given threshold

Abstract

A mathematical framework based on the decomposition of the rain rate above a given threshold as the product of two random functions will be developed. These functions are the nonzero rain rate and a binary function set to one if the intensity is higher than a threshold τ and zero if this is not the case. Then two theoretical values of the correlation between the mean areal rainfall and the percentage of surface where rainfall is higher than τ will be put forward. The first one assumes independence between the mean of the rainfall above a given threshold on the area where it rains more than this threshold and the percentage of surface covered by this rainfall. The second takes into account a linear relationship between these two variables. The first expression depends only on the coefficients of variation of the latter variables and shows that their variabilities account for most of the correlation between the mean areal rainfall and the percentage of surface with rain rates exceeding τ. -from Authors